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604170

The consensus operator for combining beliefs.

The consensus operator provides a method for combining possibly conflicting beliefs within the Dempster-Shafer belief theory, and represents an alternative to the traditional Dempster's rule. This paper describes how the consensus operator can be applied to dogmatic conflicting opinions, i.e., when the degree of conflict is very high. It overcomes shortcomings of Dempster's rule and other operators that have been proposed for combining possibly conflicting beliefs.

Introduction Ever since the publication of Shafer's book A Mathematical Theory of Evidence [1] there has been continuous controversy around the so-called Dempster's rule. The purpose of Dempster's rule is to combine two conflicting beliefs into a single belief that reflects the two conflicting beliefs in a fair and equal way. Dempster's rule has been criticised mainly because highly conflicting beliefs tend to produce counterintuitive results. This has been formulated in the form of examples by Zadeh [2] and Cohen [3] among others. The problem with Dempster's rule is due to its normalisation which redistributes conflicting belief masses to non-conflicting beliefs, and thereby tends to eliminate any conflicting characteristics in the resulting belief mass distribution. An alternative called the non-normalised Dempster's rule proposed by Smets [4] avoids this particular problem by allocating all conflicting belief masses to the empty set. Smets explains this by arguing that 1 The work reported in this paper has been funded in part by the Co-operative Research Centre for Enterprise Distributed Systems Technology (DSTC) through the Australian Federal Government's CRC Programme (Department of Industry, Science & Resources). Preprint of article published in Artificial Intelligence Journal, Vol.141/1-2 (2002), p.157-170 the presence of highly conflicting beliefs indicates that some possible event must have been overlooked (the open world assumption) and therefore is missing in the frame of discernment. The idea is that conflicting belief masses should be allocated to this missing (empty) event. Smets has also proposed to interpret the amount of mass allocated to the empty set as a measure of conflict between separate beliefs [5]. In this paper we describe an alternative rule for combining conflicting belief functions called the consensus operator. The consensus operator forms part of subjective logic which is described in [6]. Our consensus operator is different from Dempster's rule but has the same purpose; namely of combining possibly conflicting beliefs. The definition of the consensus operator in [6] and earlier publications does not cover combination of conflicting dogmatic beliefs, i.e. highly conflicting beliefs. This paper extends the definition of the consensus operator to also cover such cases. A comparison between the consensus operator and the two variants of Dempster's rule is provided in the form of examples. Subjective logic is a framework for artificial reasoning with uncertain beliefs which for example can be applied to legal reasoning [7] and authentication in computer networks [8]. In subjective logic, beliefs must be expressed on binary frames of dis- cernment, and coarsening is necessary if the original frame of discernment is larger than binary. Section 2 describes some basic elements from the Dempster-Shafer theory as well as some new concepts related to coarsening. Section 3 describes the opinion metric which is the binary belief representation used in subjective logic. Section 4 describes the consensus operator which operates on opinions and section 5 provides a comparison between the consensus operator and the two variants of Dempster's rule. A discussion of our results is provided in section 6. Representing Uncertain Beliefs The first step in applying the Dempster-Shafer belief model [1] is to define a set of possible states of a given system, called the frame of discernment denoted by . The powerset of , denoted by 2 , contains all possible unions of the sets in including itself. Elementary sets in a frame of discernment will be called atomic sets because they do not contain subsets. It is assumed that only one atomic set can be true at any one time. If a set is assumed to be true, then all supersets are considered true as well. An observer who believes that one or several sets in the powerset of might be true can assign belief masses to these sets. Belief mass on an atomic set x 2 2 is interpreted as the belief that the set in question is true. Belief mass on a non-atomic set x 2 2 is interpreted as the belief that one of the atomic sets it contains is true, but that the observer is uncertain about which of them is true. The following definition is central in the Dempster-Shafer theory. be a frame of discernment. If with each subset x 2 2 a number m (x) is associated such 2: 3: then m is called a belief mass assignment 2 on , or BMA for short. For each subset x 2 2 , the number m (x) is called the belief mass 3 of x. A belief mass m (x) expresses the belief assigned to the set x and does not express any belief in subsets of x in particular. A BMA is called dogmatic if m (see [5] p.277) because the total amount of belief mass has been committed. In contrast to belief mass, the belief in a set must be interpreted as an observer's total belief that a particular set is true. The next definition from the Dempster-Shafer theory will make it clear that belief in x not only depends on belief mass assigned to x but also on belief mass assigned to subsets of x. be a frame of discernment, and let m be a BMA on . Then the belief function corresponding with m is the function b : defined by: Similarly to belief, an observer's disbelief must be interpreted as the total belief that a set is not true. The following definition is ours. Definition 3 (Disbelief Function) Let be a frame of discernment, and let m be a BMA on . Then the disbelief function corresponding with m is the function defined by: The disbelief in x is equal to the belief in x, and corresponds to the doubt of x in Shafer's book. However, we choose to use the term `disbelief' because we feel that Called basic probability assignment in [1] 3 Called basic probability number in [1] for example the case when it is certain that a set is false can better be described by 'total disbelief' than by `total doubt'. Our next definition expresses uncertainty regarding a given set as the sum of belief masses on supersets or on partly overlapping sets of x. Definition 4 (Uncertainty Function) Let be a frame of discernment, and let m be a BMA on . Then the uncertainty function corresponding with m is the function defined by: y 6 x The sum of the belief, disbelief and uncertainty functions is equal to the sum of the belief masses in a BMA which according to Definition 1 is equal to 1. The following equality is therefore trivial to prove: For the purpose of deriving probability expectation values of sets in 2 , we will show that knowing the relative number of atomic sets is also needed in addition to belief masses. For any particular set x the atomicity of x is the number of atomic sets it contains, denoted by jxj. If is a frame of discernment, the atomicity of is equal to the total number of atomic sets. Similarly, if x; y 2 2 then the overlap between x and y relative to y can be expressed in terms of atomic sets. Our next definition captures this idea of relative atomicity: (Relative Atomicity) Let be a frame of discernment and let x; y 2 2 . Then the relative atomicity of x to y is the function a defined by: jx \ yj It can be observed that (x \ and that (y x) ) 1). In all other cases the relative atomicity will be a value between 0 and 1. The relative atomicity of an atomic set to its frame of discernment, denoted by a(x=), can simply be written as a(x). If nothing else is specified, the relative atomicity of a set then refers to the frame of discernment. A frame of discernment with a corresponding BMA can be used to determine a probability expectation value for any given set. The greater the relative atomicity of a particular set the more the uncertainty function will contribute to the probability expectation value of that set. Definition 6 (Probability Expectation) Let be a frame of discernment with BMA , then the probability expectation function corresponding with m is the function defined by: y Definition 6 is equivalent to the pignistic probability justified by e.g. Smets & Kennes in [9], and corresponds to the principle of insufficient reason: a belief mass assigned to the union of n atomic sets is split equally among these n sets. In order to simplify the representation of uncertain beliefs for particular sets we will define a focused frame of discernment which will always be binary, i.e. it will only contain (focus on) one particular set and its complement. The focused frame of discernment and the corresponding BMA will for the set in focus produce the same belief, disbelief and uncertainty functions as the original frame of discernment and BMA. The definitions of the focused frame of discernment and the focused BMA are given below. Definition 7 (Focused Frame of Discernment) Let be a frame of discernment and let x 2 2 . The frame of discernment denoted by e x containing only x and x, where x is the complement of x in is then called a focused frame of discernment with focus on x. Definition 8 (Focused Belief Mass Assignment) Let be a frame of discernment with BMA m where b(x), d(x) and u(x) are the belief, disbelief and uncertainty functions of x in 2 , and let a(x) be the real relative atomicity of x in . Let e x be the focused frame of discernment with focus on x. The corresponding focused BMA and relative atomicity a e x (x) on e x is defined according to: a e x u(x) for u(x) 6= 0 a e x (2) When the original frame of discernment contains more than 2 atomic sets, the relative atomicity of x in the focused frame of discernment e x is in general different from 1although e x per definition contains exactly two sets. The focused relative atomicity of x in e x is defined so that the probability expectation value of x is equal in and e x , and the expression for a e x (x) can be determined by using Definition 6. A focused relative atomicity represents the weighted average of relative atomicities of x to all other sets in function of their uncertainty belief mass. Working with focused BMAs makes it possible to represent the belief function of any set in 2 using a binary frame of discernment, making the notation very compact. 3 The Opinion Space For purpose of having a simple and intuitive representation of uncertain beliefs we will define a 3-dimensional metric called opinion but which will contain a 4th redundant parameter in order to allow a simple and compact definition of the consensus operator. It is assumed that all beliefs are held by individuals and the notation will therefore include belief ownership. Let for example agent A express his or her beliefs about the truth of set x in some frame of discernment. We will denote A's belief, disbelief, uncertainty and relative atomicity functions as b A x , d A x , u A x and a A x respectively, where the superscript indicates belief ownership and the subscript indicates the belief target. Definition 9 (Opinion Metric) Let be a binary frame of discernment containing sets x and x, and let m be the BMA on held by A where b A x , d A x and u A A's belief, disbelief and uncertainty functions on x in 2 respectively, and let a A x represent the relative atomicity of x in . Then A's opinion about x, denoted by x , is the tuple: The three coordinates (b; d; u) are dependent through Eq.(1) so that one is redun- dant. As such they represent nothing more than the traditional Bel (Belief) and Pl (Plausibility) pair of Shaferian belief theory, where Bel = b and ever, using (Bel, Pl) instead of (b; d; u) would have produced unnecessary complexity in the definition of the consensus operator below. Eq.(1) defines a triangle that can be used to graphically illustrate opinions as shown in Fig.1. UncertaintyProbability axis a E( x Projector Director Fig. 1. Opinion triangle with ! x as example As an example the position of the opinion as a point in the triangle. The horizontal base line between the belief and disbelief corners is called the probability axis. As shown in the figure, the probability expectation value 0:7 and the relative atomicity can be graphically represented as points on the probability axis. The line joining the top corner of the triangle and the relative atomicity point is called the director. The projector is parallel to the director and passes through the opinion point ! x . Its intersection with the probability axis defines the probability expectation value which otherwise can be computed by the formula of Definition 6. Opinions situated on the probability axis are called dogmatic opinions, representing traditional probabilities without uncertainty. The distance between an opinion point and the probability axis can be interpreted as the degree of uncertainty. Opinions situated in the left or right corner, i.e. with either are called absolute opinions, corresponding to TRUE or FALSE states in binary logic. 4 The Consensus Operator The consensus opinion of two possibly conflicting argument opinions is an opinion that reflects both argument opinions in a fair and equal way, i.e. when two observers have beliefs about the truth of x resulting from distinct pieces of evidence about x, the consensus operator produces a consensus belief that combines the two separate beliefs into one. If for example a process can produce two outcomes x and x, and A and B have observed the process over two different time intervals so that they have formed two independent opinions about the likelihood of x to occur, then the consensus opinion is the belief about x to occur which a single agent would have had after having observed the process during both periods. x x ) be opinions respectively held by agents A and B about the same state x, and let x u A x . When u A 0, the relative dogmatism between ! A x is defined by so that x =u A x . Let ! A;B x ) be the opinion such that: for 1: b A;B x u A x )= 2: d A;B x u A x )= 3: u A;B x )= 4: a A;B x u A x (a A x )u A x for 1: b A;B b A x 2: d A;B d A x 3: u A;B 4: a A;B a A x x is called the consensus opinion between ! A x , representing an imaginary agent [A; B]'s opinion about x, as if that agent represented both A and B. By using the symbol '' to designate this operator, we define ! A;B It is easy to prove that the consensus operator is both commutative and associative which means that the order in which opinions are combined has no importance. It can also be shown that the consensus opinion satisfies Eq.(1), i.e. that b A;B x +d A;B independence must be assumed, which for example translates into not allowing an agent's opinion to be counted more than once, and also that that the argument opinions must be based on distinct pieces of evidence. Briefly said, the consensus operator is obtained by mapping beta-probability density functions to the opinion space. It can be shown that posteriori probabilities of binary events can be represented by the beta-pdf (see e.g. [10] p.298). The beta- family of density functions is a continuous family of functions indexed by the two parameters and . The parameters of beta-distributions, which for example can represent the number of observations of events, can be combined by simple addi- tion, and thus a way of combining evidence emerges. We refer the reader to [6] for a detailed description of how the consensus operator can be derived from the combination of beta-distributions. The consensus of two totally uncertain opinions results in a new totally uncertain opinion, although the relative atomicity is not well defined in that case. Two observers would normally agree on the relative atomicity, and in case of two totally uncertain opinions we require that they do so, so that the consensus relative atomicity for example can be defined as a A;B x . In [6] it is incorrectly stated that the consensus operator can not be applied to two dogmatic opinions, i.e. when The definition above rectifies this so that dogmatic opinions can be combined. This result is obtained by computing the limits of (b A;B x ) as u A using the relative dogmatism between A and B defined by x =u A x . This result makes the consensus operator more general than Dempster's rule because the latter excludes the combination of totally conflicting beliefs. In order to understand the meaning of the relative dogmatism , it is useful to consider a process with possible outcomes fx; xg that produces times as many x as x. For example when throwing a fair dice and some mechanism makes sure that A only observes the outcome of 'six' and B only observes the outcome of `one', 'two', `three', 'four' and `five', then A will think the dice only produces 'six' and will think that the dice never produces 'six'. After infinitely many observations A and B will have the conflicting dogmatic opinions ! A 6 ) and respectively. On the average B observes 5 times more events than A so that B remains 5 times more dogmatic than A as u A meaning that the relative dogmatism between A and B is 1=5. By combining their opinions according to the case where inserting the value of , the combined opinion about obtaining a 'six' with the dice can be computed as ! A;B 6 ), which is exactly what one would expect. In the last example, the relative dogmatism was finite and non-zero, but it is also possible to imagine extreme relative dogmatisms, e.g. an infinitesimal (i.e. close to zero). This is related to the concept of epsilon belief functions which has been applied to default reasoning by Benferhat et al. in [11]. Epsilon belief functions are opinions with b; d; u 2 f0; "; 1 "g, i.e. opinions situated close to a corner of the triangle in Fig.1. Without going into details it can be shown that some properties of extreme relative dogmatisms seem suitable for default reasoning. For example, when the relative dogmatism between A and B is infinite ( the consensus opinion is equal to A's argument opinion x ), and when the relative dogmatism is infinitesimal ( the consensus opinion is equal to B's argument opinion (! A;B x ). However, with three agents A, B, and C where the consensus opinion x is non-conclusive as long as the relationship between " 1 and " 2 is unknown. 5 Comparing the Consensus Operator with Dempster's Rule This section describes three examples that compare Dempster's rule, the non-normalised Dempster's rule and the consensus operator. The definition of Dempster's rule and the non-normalised rule is given below. In order to distinguish between the consensus operator and Dempster's rule, the latter will be denoted by 0 . Definition 11 Let be a frame of discernment, and let m A be BMAs on . Then m A is a function m A 1: m A (z) K ; and 2: m A (z) (z) and K 6= 1 in Dempster's rule, and where in the non-normalised version. 5.1 Example 1: Dogmatic Conflicting Beliefs We will start with the well known example that Zadeh [2] used for the purpose of criticising Dempster's rule. Smets [4] used the same example in defence of the non-normalised version of Dempster's rule. Suppose that we have a murder case with three suspects; Peter, Paul and Mary and two highly conflicting testimonies. Table 1 gives the witnesses' belief masses in Zadeh's example and the resulting belief masses after applying Dempster's rule, the non-normalised rule and the consensus operator. Non-normalised Consensus rule Dempster's rule operator Peter Mary Table Comparison of operators in Zadeh's example Because the frame of discernment in Zadeh's example is ternary, a focused binary frame of discernment must be derived in order to apply the consensus operator. The focused opinions are: The above opinions are all dogmatic, and the case where must be invoked. Because of the symmetry between W 1 and W 2 we determine the relative dogmatism between W 1 and W 2 to be 1. The consensus opinion values and their corresponding probability expectation values can then be computed as: Peter Mary The column for the consensus operator in Table 1 is obtained by taking the 'belief' coordinate from the consensus opinions above. Dempster's rule selects the least suspected by both witnesses as the guilty. The non-normalised version acquits all the suspects and indicates that the guilty has to be someone else. This is explained by Smets [4] with the so-called open world interpretation of the frame of discern- ment. In [5] Smets also proposed to interpret m(;) (= 0.9999 in this case) as a measure of the degree of conflict between the argument beliefs. The consensus operator respects conflicting beliefs by giving the average of beliefs to Peter and Mary, whereas the non-conflicting beliefs on Paul is kept unal- tered. This result is consistent with classical estimation theory (see e.g. comments to Smets [4] p.278 by M.R.B.Clarke) which is based on taking the average of probability estimates when all estimates have equal weight. 5.2 Example 2: Conflicting Beliefs with Uncertainty In the following example uncertainty is introduced by allocating some belief to the set Maryg. Table 2 gives the modified BMAs and the results of applying the rules. Non-normalised Consensus rule Dempster's rule operator Peter 0.98 0.00 0.490 0.0098 0.492 Mary Table Comparison of operators after introducing uncertainty in Zadeh's example The frame of discernment in this modified example is again a ternary, and a focused binary frame of discernment must be derived in order to apply the consensus operator. The focused opinions are given below: The consensus opinion values and their corresponding probability expectation values are: Peter Mary The column for the consensus operator in Table 2 is obtained by taking the 'be- lief' coordinate from the consensus opinions above. When uncertainty is intro- duced, Dempster's rule corresponds well with intuitive human judgement. The non-normalised Dempster's rule however still indicates that none of the suspects are guilty and that new suspects must be found, or alternatively that the degree of conflict is still high, despite introducing uncertainty. The consensus operator corresponds well with human judgement and gives almost the same result as Dempster's rule, but not exactly. Note that the values resulting from the consensus operator have been rounded off after the third decimal. The belief masses resulting from Dempster's rule in Table 2 add up to 1. The 'be- lief' parameters of the consensus opinions resulting from the consensus operator do not add up to 1 because they are actually taken from 3 different focused frames of discernment, but the following holds: Peter Mary 5.3 Example 3: Harmonious Beliefs The previous example seemed to indicate that Dempster's rule and the consensus operator give very similar results in the presence of uncertainty. However, this is not always the case as illustrated by the following example. Let two and W 2 have equal beliefs about the truth of x. The agents' BMAs and the results of applying the rules are give in Table 3. Non-normalised Consensus rule Dempster's rule operator Table Comparison of operators i.c.o. equal beliefs The consensus opinion about x and the corresponding probability expectation value are: It is difficult to give an intuitive judgement of these results. It can be observed that Dempster's rule and the non-normalised version produce equal results because the witnesses' BMAs are non-conflicting. The two variants of Dempster's rule amplify the combined belief twice as much as the consensus operator and this difference needs an explanation. The consensus operator produces results that are consistent with statistical analysis (see [6]) and in the absence of other criteria for intuitive or formal judgement, this constitutes a strong argument in favour of the consensus operator. 6 Discussion and Conclusion In addition to the three belief combination rules analysed here, numerous others have been presented in the literature, e.g. the rule proposed by Yager [12] that transfers conflicting belief mass m A (y) to whenever x \ the rule proposed by Dubois & Prade [13] that transfers conflicting belief mass (y) to x[y whenever These rules are commutative, but unfortunately they are not associative, which seems counterintuitive. Assuming that beliefs from different sources should be treated in the same way, why should the result depend on the order in which they are combined? After analysing the rules of Dempster, Smets, Yager, Dubois & Prade as well as simple statistical average, Murphy [14] rejects the rules of Yager and Dubois & Prade for their lack of as- sociativity, and concludes that Dempster's rule performs best for its convergence properties, accompanied by statistical average to warn of possible errors when the degree of conflict is high. Our consensus operator seems to combine both the desirable convergence properties of Dempster's rule when the degree of conflict is low, and the natural average of beliefs when the degree of conflict is high. As mentioned in Lef-evre et al. [15], Dempster's rule and it's non-normalised version require that all belief sources are reliable, whereas Yager's and Dubois & Prade's rules require that at least one of the belief sources is reliable for the result to be meaningful. The consensus operator does not make any assumption about reliability of the belief sources, but does of course not escape the 'garbage in, garbage out' principle. An argument that could be used against our consensus operator, is that it does not give any indication of possible belief conflict. Indeed, by looking at the result only, it does not tell whether the original beliefs were in harmony or in conflict, and it would have been nice if it did. A possible way to incorporate the degree of conflict is to add an extra 'conflict' parameter. This could for example be the belief mass assigned to ; in Smets' rule, which in the opinion notation can be defined as c A;B x d A x where c A;B 1]. The consensus opinion with conflict parameter would then be expressed as ! A;B x ). The conflict parameter would only be relevant for combined belief, and not for original beliefs. A default value could for example indicate original belief, because a default value could be misunderstood as indicating that a belief comes from combined harmonious beliefs, even though it is an original belief. Opinions can be derived by coarsening any frame of discernment and BMA through the focusing process, where focusing on different states produces different opin- ions. In this context it is in general not meaningful to relate belief, disbelief and uncertainty functions from opinions that focus on different states even though the opinions are derived from the same frame of discernment and belief mass assign- ment. The only way to relate such opinions is through the probability expectation value E(! x ) (which can also be written as E(x)), and this leads to interesting re- sults. The proof of the following theorem can be found in [6]. Theorem 1 (Kolmogorov Axioms) Given a frame of discernment with a BMA m , the probability expectation function E with domain 2 satisfies: 1: E(x) 0 for all x 2: 3: If x are pairwise disjoint, then E([ j2 j This shows that probability theory can be built on top of belief theory through the probability expectation value. As such belief functions should not be interpreted as probabilities, instead there is a surjective (onto) mapping from the belief space to the probability space. Belief and possibility functions have been interpreted as upper and lower probability bounds respectively (see e.g. Halpern & Fagin [16] and de Cooman & Ayles [17]). Belief functions can be useful for estimating probability values but not to set bounds, because the probability of a real event can never be determined with absolute certainty, and neither can upper and lower bounds to it. Our view is that probability always is a subjective notion, inasmuch as it is a 1- dimensional belief measure felt by a given person facing a given event. Objective, physical or real probability is a meaningless notion. This view is shared by e.g. de Finetti [18]. In the same way, an opinion as defined here, is a 3-dimensional belief measure felt by a given person facing a given event. It has also been suggested to interpret belief functions as evidence (see e.g. Fagin and Halpern [16]). Belief can result from evidence in the form of observing an event or knowing internal properties of a system, or from more subjective and intangible experience. Statistical evidence can for example be translated into belief functions, as described in [6], and other types of evidence can be intuitively translated into belief functions, but belief and evidence are not the same. We prefer to leave belief functions as a distinct concept in its own right, and in general not try to interpret them as anything else. The opinion metric described here provides a simple and compact notation for beliefs in the Shaferian belief model. We have presented an alternative to Dempster's rule which is consistent with probabilistic and statistical analysis, and which seems more suitable for combining highly conflicting beliefs as well as for combining harmonious beliefs, than Dempster's rule and its non-normalised version. The fact that a binary focused frame of discernment must be derived in order to apply the consensus operator puts no restriction on its applicability. The resulting beliefs for each event can still be compared and can form the basis for decision making. --R A Mathematical Theory of Evidence. Review of Shafer's A Mathematical Theory of Evidence. An expert system framework for non-monotonic reasoning about probabilistic assumptions Belief functions. The transferable belief model for quantified belief representation. A Logic for Uncertain Probabilities. Legal Reasoning with Subjective Logic. An Algebra for Assessing Trust in Certification Chains. The transferable belief model. Statistical Inference. Belief Functions and Default Reasoning. On the Dempster-Shafer framework and new combination rules Representation and combination of uncertainty with belief functions and possibility measures. Combining belief functions when evidence conflicts. A generic framework for resolving the conflict in the combination of belief structures. Two views of belief: Belief as generalised probability and belief as evidence. Supremum preserving upper probabilities. The value of studying subjective evaluations of probability. --TR On the Dempster-Shafer framework and new combination rules Two views of belief The transferable belief model Supremum preserving upper probabilities Combining belief functions when evidence conflicts Belief functions and default reasoning A logic for uncertain probabilities --CTR Audun Jsang , Daniel Bradley , Svein J. Knapskog, Belief-based risk analysis, Proceedings of the second workshop on Australasian information security, Data Mining and Web Intelligence, and Software Internationalisation, p.63-68, January 01, 2004, Dunedin, New Zealand Audun Jsang, Probabilistic logic under uncertainty, Proceedings of the thirteenth Australasian symposium on Theory of computing, p.101-110, January 30-February 02, 2007, Ballarat, Victoria, Australia Audun Jsang , Ross Hayward , Simon Pope, Trust network analysis with subjective logic, Proceedings of the 29th Australasian Computer Science Conference, p.85-94, January 16-19, 2006, Hobart, Australia Weiru Liu, Analyzing the degree of conflict among belief functions, Artificial Intelligence, v.170 n.11, p.909-924, August 2006 Philippe Smets, Analyzing the combination of conflicting belief functions, Information Fusion, v.8 n.4, p.387-412, October, 2007

dempster's rule;belief;conflict;subjective logic;consensus operator

604172

Coherence in finite argument systems.

Argument Systems provide a rich abstraction within which divers concepts of reasoning, acceptability and defeasibility of arguments, etc., may be studied using a unified framework. Two important concepts of the acceptability of an argument p in such systems are credulous acceptance to capture the notion that p can be 'believed'; and sceptical acceptance capturing the idea that if anything is believed, then p must be. One important aspect affecting the computational complexity of these problems concerns whether the admissibility of an argument is defined with respect to 'preferred' or 'stable' semantics. One benefit of so-called 'coherent' argument systems being that the preferred extensions coincide with stable extensions. In this note we consider complexity-theoretic issues regarding deciding if finitely presented argument systems modelled as directed graphs are coherent. Our main result shows that the related decision problem is (p)2 -complete and is obtained solely via the graph-theoretic representation of an argument system, thus independent of the specific logic underpinning the reasoning theory.

Introduction Since they were introduced by Dung [8], Argument Systems have provided a fruitful mechanism for studying reasoning in defeasible contexts. They have proved useful both to theorists who can use them as an abstract framework for the study and comparison of non-monotonic logics, e.g. [2,5,6], and for those who wish to explore more concrete contexts where defeasibility is central. In the study of reasoning in law, for example, they have been used to examine the resolution of conflicting norms, e.g. [12], especially where this is studied through the mechanism of Corresponding author. Email address: ped@csc.liv.ac.uk (Paul E. Dunne). Preprint submitted to Elsevier Science 13 May 2002 a dispute between two parties, e.g. [11]. The basic definition below is derived from that given in [8]. argument system is a Ai, in which X is a set of arguments and A X X is the attack relationship for H. Unless otherwise stated, X is assumed to be finite, and A comprises a set of ordered pairs of distinct arguments. A pair hx; yi 2 A is referred to as 'x attacks (or is an attacker of y' or 'y is attacked by x'. For R, S subsets of arguments in the system H(hX ; Ai), we say that a) s 2 S is attacked by R if there is some r 2 R such that hr; si 2 A. acceptable with respect to S if for every y 2 X that attacks x there is some z 2 S that attacks y. c) S is conflict-free if no argument in S is attacked by any other argument in S. d) A conflict-free set S is admissible if every argument in S is acceptable with respect to S. e) S is a preferred extension if it is a maximal (with respect to ) admissible set. f) S is a stable extension if S is conflict free and every argument y 62 S is attacked by S. g) H is coherent if every preferred extension in H is also a stable extension. An argument x is credulously accepted if there is some preferred extension containing it; x is sceptically accepted if it is a member of every preferred extension. The graph-theoretic representation employed by finite argument systems, naturally suggests a unifying formalism in which to consider various decision problems. To place our main results in a more general context we start from the basis of the decision problems described by Table 1 in is an argument system as in Defn. 1; x an argument in X ; and S a subset of arguments in X . Polynomial-time decision algorithms for problems (1) and (2) are fairly obvious. The results regarding problems (3-7) are discussed below. In this article we are primarily concerned with the result stated in the final line of Table 1: our proof of this yields (8) as an easy Corollary. Before proceeding with this, it is useful to discuss important related work of Dimopoulos and Torres [7], in which various semantic properties of the Logic Programming paradigm are interpreted with respect to a (directed) graph translation of reduced negative logic programs: graph vertices are associated with rules and the concept of 'attack' modelled by the presence of edges hr; si whenever there is a non-empty intersection between the set of literals defining the head of r and the negated set of literals in the body of s, i.e. if z 2 body(s) then :z is in this negated set. Although [7] does not employ the terminology - in terms of credulous accep- tance, admissible sets, etc - from [8] used in the present article it is clear that similar forms are being considered: the structures referred to as 'semi-kernel', 'maximal Problem Decision Question Complexity 3 PREF-EXT(H; S) Is S a preferred extension? CO-NP-complete. stable extension? NP-complete. 5 CA(H; x) Is x in some preferred extension? NP-complete 6 IN-STAB(H; x) Is x in some stable extension? NP-complete 7 ALL-STAB(H; x) Is x in every stable extension? CO-NP-complete. 8 SA(H; x) Is x in every preferred extension? (p) 9 COHERENT(H) Is H coherent? (p) Table Decision Problems in Finite Argument Systems and their Complexity semi-kernel' and `kernel' in [7] corresponding to 'admissible set', 'preferred exten- sion' and `stable extension' respectively. The complexity results for problems (3-6) if not immediate from [7, Thm 5.1, Lemma 5.2, Prop. 5.3] are certainly implied by these. In this context, it is worth drawing attention to some significant points regarding [7, Thm. 5.1] which, translated into the terminology of the present article states: The problem of deciding whether an argument system H(X ; A) has a non-empty preferred extension is NP-complete. First, this implies the complexity classification for PREF-EXT stated, even when the subset S forming part of an instance is the empty set. A second point, also relevant to our proof of (9) concerns the transformation used: present a translation of propositional formulae in 3-CNF (this easily generalises for arbitrary CNF formulae) into a finite argument system H . It is not difficult, however, given to define CNF-formulae H whose satisfiability properties are dependent on the presence of particular structures within H, e.g. stable extensions, admissible subsets containing specific arguments, etc. We thus have a mechanism for transforming a given H into an 'equivalent' system F the point being that F may provide a 'better' basis for graph-theoretic analyses of structures within H. Our final observation, concerns problem (7): although the given complexity classification is neither explicitly stated in nor directly implied by the results of [7], that ALL-STAB is CO-NP-complete can be shown using some minor 're-wiring' of the argument graph G constructed from an instance of 3-SAT. 1 The concept of coherence was formulated by [8, Defn. 31(1), p. 332], to describe those argument systems whose stable and preferred extensions coincide. One significant benefit of coherence as a property has been established in recent work of Vreeswijk and Prakken[13] with respect to proof mechanisms for establishing sceptical acceptance: problem (8) of Table 1. In [13] a sound and complete reasoning method for credulous acceptance - using a dialogue game approach - is presented. This approach, as the authors observe, provides a sound and complete mechanism for sceptical acceptance in precisely those argument systems that are coherent. Thus a major advantage of coherent argument systems is that proofs of sceptical acceptance are (potentially) rather more readily demonstrated in coherent systems via devices such as those of [13]. The complexity of sceptical acceptance is considered (in the context of membership in preferred extensions) for various non-monotonic Logics by [5], where completeness results at the third-level of the polynomial-time hierarchy are demonstrated. Although [5] argue that their complexity results 'dis- credit sceptical reasoning as . "unnecessarily" complex', it might be argued that within finite systems where coherence is 'promised' this view may be unduly pes- simistic. Notwithstanding our main result that testing coherence is extremely hard, there is an efficiently testable property that can be used to guarantee coherence. Some further discussion of this is presented in Section 3. In the next section we present the main technical contribution of this article, that COHERENT is (p) 2 -complete: the complexity class (p) comprising those problems decidable by CO-NP computations given (unit cost) access to an NP oracle. Alterna- tively, (p) 2 can be viewed as the class of languages, L, membership in which is certified by a (deterministic) polynomial-time testable ternary relation R L WXY such that, for some polynomial bound p(jwj) in the number of bits encoding w, Our result in Theorem 2 provides some further indications that decision questions concerning preferred extensions are (under the usual complexity-theoretic assump- tions) likely to be harder than the analogous questions concerning stable exten- sions: line (8) of Table 1 is an easy Corollary of our main theorem. Similar conclusions had earlier been drawn in [5,6], where the complexity of reasoning problems in a variety of non-monotonic Logics is considered under both preferred and stable semantics. This earlier work establishes a close link between the complexity of the reasoning problem and that of the derivability problem for the associated logic. One feature of our proof is that the result is established purely through a graph-theoretic interpretation of argument, similar in spirit, to the approach adopted in [7]: thus, This involves removing all except the edge hAux; Ai for edges hA; xi or hx; Ai: then the differing complexity levels may be interpreted in purely graph-theoretic terms, independently of the Logic that the graph structure is defined from. In Section 3 we discuss some consequences of our main theorem in particular with respect to its implications for designing dialogue game style mechanisms for Sceptical Reasoning. Conclusions are presented in Section 4. 2 Complexity of Deciding Coherence Theorem 2 COHERENT is (p) In order to clarify the proof structure we establish it via a series of technical lem- mata. The bulk of these are concerned with establishing (p) -hardness, i.e with reducing a known (p) -complete problem to COHERENT. We begin with the, comparatively easy, proof that COHERENT(H) is in (p) . . Proof: Given an instance, H(X ; A) of COHERENT, it suffices to observe that, i.e. H is coherent if and only if for each subset S of X : either S is not a preferred extension or S is a stable extension. Since, :PREF-EXT(H; S) is in NP, i.e. (p) 1 and STAB-EXT(H; S) in P, we have COHERENT in (p) 2 as required. 2 The decision problem we use as the basis for our reduction is QSAT 2 . An instance of QSAT 2 is a well-formed propositional formula, (X; Y), defined over disjoint sets of propositional variables, loss of generality we may assume using only the Boolean operations ^, _, and :; and negation is only applied to variables in X [ Y . An instance, (X; Y) of QSAT 2 is accepted if and only if 8 X 9 Y no matter how the variables in X are instantiated ( X ) there is some instantiation Y ) of Y such that h X ; Y i satisfies . That QSAT 2 is (p) -complete was shown in [14]. We start by presenting some technical definitions. The first of these describes a standard presentation of propositional formulae as directed rooted trees that has often been widely used in applications of Boolean formulae, see e.g. [9, Chapter 4] Definition 4 Let (Z) be a well-formed propositional formula (wff) over the vari- AND AND Fig. 1. T (z ables using the operations f^; _; :g with negation applied only to variables of . The tree representation of (denoted T ) is a rooted directed tree with root vertex denoted (T ) and inductively defined by the following rules. a) If single literal z or :z - then T consists of a single vertex labelled w. is formed from the k tree representations hT i i by directing edges from each (T i ) into a new root vertex labelled ^. c) If is formed from the k tree representations hT i i by directing edges from each (T i ) into a new root vertex labelled _. In what follows we use the term node of T to refer to an arbitrary tree vertex, i.e. a leaf or internal vertex. In the tree representation of , each leaf vertex is labelled with some literal w, (several leaves may be labelled with the same literal), and each internal vertex with an operation in f^; _g. We shall subsequently refer to the internal vertices of T as the gates of the tree. Without loss of generality we may assume that the successor of any ^-gate (tree vertex labelled ^) is an _-gate (tree vertex labelled _) and vice- versa. The size of (Z) is the number of gates in its tree representation T . For formulae of size m we denote by hg the gates in T with g m always taken as the root (T ) of the tree. Finally for any edge hh; gi in T we refer to the node h as an input of the gate g. 2 Definition 5 For a formula, (Z), an instantiation of its variables is a mapping, : associating a truth value or unassigned status () with each variable z i . We use i to denote (z i ). An instantiation is total if every variable is assigned a value in ftrue; falseg and partial otherwise. We define a partial ordering We note that since any gate may be assumed to have at most n distinct literals among its inputs, our measure of formula size as 'number of gates' is polynomially equivalent to the more usual measure of size as 'number of literal occurrences', i.e. leaf nodes. over instantiations and - to Z by writing < - if: for each i with and there is at least one i, for which Given (Z) any instantiation induces a mapping from the nodes defining T onto values in ftrue; false; g. Assuming the natural generalisations of ^ and _ to the domain htrue; false; i, 3 we define for h a node in T , its value (h; ) under the instantiation of Z as if h is a leaf node labelled z i or :z i and is a leaf node labelled z i and i 6= is a leaf node labelled :z i and i 6= is an _-gate with inputs hh is an ^-gate with inputs hh where is clear from the context, we write (h) for (h; ). With this concept of the value induced at a node of T via an instantiation , we can define a partition of the literals and gates in T that is used extensively in our later analysis. The value partition Val() of T comprises 3 sets hTrue(); False(); Open()i. T1) The subset True() consists of literals and gates, h, for which T2) The subset False() consists of literals and gates, h, for which T3) The subset Open() consists of literals and gates, h, for which The following properties of this partition can be easily proved: Fact 6 a) b) If < -, then True( For example in Fig. 1 under the partial instantiation with all other variables unassigned, we have: g. At the heart of our proof that QSAT 2 is polynomially reducible to COHERENT is a translation from the tree representation T of a formula (X; Y) to an argument system H (X ; A ). It will be useful to proceed by presenting a preliminary trans- are true or at least one x j is false; _ k are false or at least one is true. z1 z2 z3 -z4 z4 Fig. 2. The Argument System R from the formula of Fig. 1 lation that, although not in the final form that will be used in the reduction, will have a number of properties that will be important in deriving our result. Definition 7 Let (Z) be a propositional formula with tree representation T having size m. The Argument Representation of , is the argument system R defined as follows. R contains the following arguments literal arguments fz ng. X2 For each gate g k of T , an argument :g k (if g k is an _-gate) or an argument k is an ^-gate). If g m , i.e the root of T , happens to be an _-gate, then an additional argument g m is included. We subsequently denote this set of arguments by G . The attack relationship - A - over X contains: ng m is an _-gate in T , A3 If g k is an ^-gate with inputs fh rg. k is an _-gate with inputs fh rg. Fig. 2 shows the result of this translation when it is applied to the tree representation of the formula in Fig. 1. The arguments defining R fall into one of two sets: 2n arguments corresponding to the 2n distinct literals over Z; and m (or m+ 1) 'gate' arguments. The key idea is the following: any instantiation of the propositional variables Z of , induces the partition Val() of literals and gates in T . In the argument system R the attack relationship for gate arguments, reflects the conditions under which the corresponding argument is admissible (with respect to the subset of literal arguments marked out by ). For example, suppose g 1 is an _-gate with literals z 1 , :z 2 , z 3 as its in- puts. In the simulating argument system, g 1 is represented by an argument labelled :g 1 which is attacked by the (arguments labelled with) literals z 1 , :z 2 , and z 3 : the interpretation being that "the assertion 'g 1 is false' is attacked by instantiations in which z 1 or :z 2 or z 3 are true". Similarly were g 1 an ^-gate it would appear in R as an argument labelled g 1 which was attacked by literals :z 1 , z 2 , and :z 3 : the interpretation now being that "the assertion 'g 1 is true' is attacked by instantiations in which z 1 or :z 2 or z 3 are false". With this viewpoint, any instantiation will induce a selection of the literal arguments and a selection of the gate arguments (i.e. those for which no attacking argument has been included). Suppose is an instantiation of Z. The key idea is to map the partition of the tree representation T as Val() onto an analogous partition of the literal and gate arguments in R . Given this partition comprises 3 sets, hIn(); Out(); Poss()i defined by: An argument p is in the subset In() of X if: (p is the argument z i , or (p is the argument :z i , or is in False()) or is in True()) An argument p is in the subset Out() of X if: (p is the argument z i , or (p is the argument :z i , or is in True()) or is in False()) An argument p is in the subset Poss() of X if: With the formulation of the argument system R (X ; A ) from the formula (Z) and the definition of the partition hIn(); Out(); Poss()i via the value partition of T we are now ready to embark on the sequence of technical lemmata which will culminate in the proof of Theorem 2. Our proof strategy is as follows. We proceed by characterising the set of preferred extensions of R showing - in Lemma 8 through Lemma 11 - that these consist of exactly the subsets defined by In( Z is a total instantiation of Z. In Lemma 12 we deduce that these are all stable extensions and thus that R is itself coherent. In the remaining lemmata, we consider the argument systems arising by transforming instances (X; Y) of QSAT 2 . In these, however, we add to the basic system defined by R (which will have 4n literal arguments and m (or gate arguments) an additional set of 3 control arguments one of which attacks all of the Y-literal arguments: we denote this augmented system by H As will be seen in Lemma 15, it follows easily from Lemma 10 that for any h X ; Y i satisfying (X; Y) the subset In( X ; Y ) is a stable extension of both R and H . The crucial property provided by the additional control arguments in H is proved in Lemma 16: if for X there is no Y for which h X ; Y i satisfies (X; Y) then the subset In( X ) (defined from R ) is a preferred but not stable extension of H , denotes the set In( which every y i is unassigned. The reason for introducing the control arguments in moving from R to H is that not a preferred extension of R : although it is admissible, it could be extended by adding, for example, Y-literal arguments. The design of H will be such that unless the gate argument g m can be used in an admissible extension of already maximal in H and not a stable extension since the control arguments are not attacked. Finally, in Lemma 17, it is demonstrated that the only preferred extensions of H are those arising as a result of Lemma 15 and Lemma 16. Theorem 2 will follow easily from Lemma 17, since the argument g m - corresponding to the root node (T ) of the instance (X; Y) - must necessarily belong to any stable extension in H : hence H is coherent if and only if for each instantiation X there is an instantiation Y such that h X ; Y i satisfies (X; Y), i.e. for which in the system R and thence in the corresponding stable extension of H . We employ the following notational conventions: X , Y , (and instantiations of X, Y , (and Z); for an argument p in X , g p (resp. h p ) denotes the corresponding gate (resp. node) in T , hence if g p is an _-gate, then p is the argument labelled :g denotes the set of all preferred (resp. stable) extensions in the argument system M , where M is one of R or H . Z In( Z ) is conflict-free. Proof: Let Z be an instantiation of Z and consider the subset In( Z ) of X in R . Suppose that there are arguments p and q in In( Z ) for which hp; qi 2 A . It cannot be the case that h literal over z i , since exactly one of fz is in True( exactly one of the corresponding literal arguments is in In( Z ). Thus q must be a gate argument. Suppose g q is an _-gate: q 2 In( only if g q 2 False( Z ) and therefore h p , which (since hp; qi 2 A ) must be an input of g q is also in False( Z ). This leads to a contradiction: if h p is a gate then it is an ^-gate, so is a literal u i , then h p 2 False( would mean that :u i 2 True( Z ) and hence u i 62 In( Z ). The remaining possibility is that g q is an ^-gate: q 2 In( If h p is a gate it must be an input of g q and an _-gate: h p 2 True( Finally if the input h p is a literal u i in T then in R the literal :u i attacks q: u Z ). We deduce that In( must be conflict-free. 2 Lemma 9 8 Z In( Z ) is admissible. Proof: From Lemma 8, In( Z ) is conflict-free, so it suffices to show for all arguments Z ) that attack some q 2 In( Z ) there is an argument r 2 In( that attacks p. Let p, q be such that p 62 In( Z ) and hp; qi 2 A . If q is a literal argument, u i say, then p must be the literal argument :u i and choosing provides a counter-attacker to p. Suppose q is a gate argument. One of the inputs to g q must be the node h p . If g q is an _-gate then g q 2 False( Z ) and Z ). If h p is a literal u i then the literal argument attacks is an ^-gate then h p 2 False( Z ) implies there is some input h r to h p with h r 2 False( Z ), so that r = :h r is in In( r is an _-gate or literal) and r attacks p. Similarly, if g q is an ^-gate then g q 2 True( Z ) and Z ). If h p is a literal u i then the attacking argument (on q in R ) is the literal Z ) provides a counter-attack on p. If h p is an _-gate then h p 2 True( Z ) indicates that some input h r of h p is in True( so that r = h r is in In( Z ) and r attacks p. No more cases remain thus In( Z ) is admissible. 2 Z In( Proof: From Lemma 8, 9 and the fact that every argument in X is allocated to either In( Z ) or Out( Z ) by Z , cf. Fact 6(a), it suffices to show that for any argument Z ) there is some q 2 In( Z ) such that p and q conflict. Certainly this is the case for literal arguments, u 2 Out( Z ) since the complementary literal :u is in In( Z ) is a gate argument. If g p is an _-gate then Z ) and hence some input h q of g p must be in Z ). The argument q corresponding to this input node will therefore be in In( Z ). If g p is an ^-gate then p 2 Out( Z ) and some input h q of g p must be in False( Z ). The argument :h q will be in In( Z ) and conflicts with p. 2 Lemma Proof: First observe that all S 2 PE R must contain exactly n literal arguments: exactly one representative from fz for each i. Let us call such a subset of the literal arguments a representative set and suppose that U is any representative set with S U any preferred extension containing U. We will show that there is exactly one possible choice for S U and that this is S (U) is the instantiation of Z by: z Consider the following procedure that takes as input a representative set U and returns a subset S U 2 PE R with U S U . g. We can note three properties of this procedure. Firstly, it always halts: once the literal arguments in the representative set U and their complements have been removed from T U (in Steps 2 and 4), the directed graph-structure remaining is acyclic and thus has at least one argument that is attacked by no others. Thus each iteration of the main loop removes at least one argument from T U which eventually becomes empty. Secondly, the set S U is in PE R : the initial set (U) is admissible and the arguments removed from T U at each iteration are those that have just been added to S U (Step 2) as well as those attacked by such arguments (Step 4); in addition the arguments added to S U at each stage are those that have had counter-attacks to all potential attackers already placed in S U . Finally for any given U the subset returned by this procedure is uniquely defined. In summary, every S 2 PE R is defined through exactly one representative set, U S , and every representative set U develops to a unique S U 2 PE R . Each representative set, U, however, has the form In( hence the unique preferred extension, S U , consistent with U is In( Lemma 12 The argument system R (X ; A ) is coherent. Proof: The procedure of Lemma 11 only excludes an argument, q, from the set S U under construction if q is attacked by some argument p 2 S U . Thus, S U is always a stable extension, and since Lemma 11 accounts for all S 2 PE R , we deduce that R is coherent. 2 Although our preceding three results characterise R as coherent, this, in itself, does not allow R be used directly as the transformation for instances (X; Y) of . The overall aim is to construct an argument system from (X; Y) which is coherent if and only if (X; Y) is a positive instance of QSAT 2 . The problem with R is that, even though (X; Y) may be a positive instance, there could be instan- which fail to satisfy (X; Y) but give rise to a stable extension In order to deal with this difficulty, we need to augment R (giving a system H ) in such a way that the admissible set In( X ) is a preferred (but not stable) extension (in H ) only if no instantiation Y allows h X ; Y i to satisfy (X; Y). Thus, in our augmented system, we will have exactly two mutually exclusive possibilities for each total instantiation X of X: either there is no Y for which true, in which event the set produce a non-stable preferred extension of H ; or there is an appropriate Y , in which case In( X ; Y ) (of which In( X ) is a proper subset, cf. Fact 6(b)) Fig. 3. An Augmented Argument Representation H will yield a stable extension in H . Definition 13 For (X; Y) an instance of QSAT 2 , the Augmented Argument Representation of - denoted H X are the arguments arising in the Argument Representation of (X; Y) - R - as given in Definition 7 and C are 3 new arguments called the control arguments. The attack relationship B contains all of the attacks A in the system R together with new attacks, ng Using the relabelling of variables in our example formula - Figs. 1,2 - as hx 1 the Augmented Argument Representation for the system in Fig. 2 is shown in Fig. 3 Lemma 14 If S 2 PE H then C i 62 S for any of fC g. If S 2 SE H then Proof: Suppose S 2 PE H . If g m 2 S then each of the control arguments is attacked by g m and so cannot be in S. If g m 62 S then C 3 62 S since the only counter-attack to C 2 is the argument C 1 which conflicts with C 3 . By similar reasoning it follows that C 2 62 S and C 1 62 S. For the second part of the lemma, given S 2 SE H , since 6 S, there must be some attacker of these in S. The only choice for this attacker is g m . 2 Lemma Proof: From Lemma 10 and 12, the subset In( X ; Y ) is in SE R . Furthermore, since that the gate argument g m of R is in In( For the augmented system, H , the arguments in In( attacks on Y-literal arguments by the control argument C 1 are attacked in turn by the gate argument g m . In addition, using the arguments of Lemma 10 no arguments in Out( X ; Y ) can be added to the set In( conflict. Thus Lemma is such that no instantiation Y of Y , leads to h Proof: The subset In( X ) of R can be shown to be admissible (in both R and H ) by an argument similar to that of Lemma 9. 4 Suppose for all Y , we have false, and consider any subset S of W in H for which In( X ) S. We show that S 62 PE H . Assume the contrary holds. From Lemma 14 no control argument is in S. If g m 2 S then S must contain a representative set, V Y say, of the Y - literal arguments matching some instantiation Y . From the argument used to prove Lemma is the only preferred extension in R consistent with the literal choices indicated by X and Y , and thus would be the only such possibility for H . Now we obtain a contradiction since g m 62 In( and so cannot be used in H to counter the attack by C 1 on the representative set V Y . Thus we can assume that g m 62 S. From this it follows that no Y-literal argument is in S (as g m is the only attacker of the control argument C 1 which attacks Y - literals). Now consider the gates in T topologically sorted, i.e. assigned a number such that all of the inputs for a gate numbered (g) are from literals or gates h with (h) < (g). Let q be an argument such that g q is the first gate in this topological ordering for which q 2 S=In( X ). We must have would already be excluded from any admissible set having In( X ) as a subset. Consider the set of arguments in W that attack q. At least one attacker, p, must be a node h p in T for which h p 2 Open( X ). Now our proof is completed: S has no available counter-attack to the attack by on q since such could only arise from a Y-literal argument (all of which have been excluded) or from another gate argument r with g r 2 Open( X ), however, contradicts the choice of q. Fig. 4 illustrates the possibilities. We conclude that the subset In( X ) of W is in PE H whenever there is no Y with which true, and since the control arguments are not attacked, 4 A minor addition is required in that since X is a partial instantiation (of hX; Yi) it has to be shown that all arguments p that attack arguments q 2 In( X ) belong to the subset are not in Poss( X ). With the generalisation of ^ and _ to allow unassigned values, it is not difficult to show that if p 2 Poss( X ) then any argument q attacked by p in R cannot belong to In( X ). AND q= -g in Open() q in S() Inputs in False() p=h in Open() p not in S() Inputs in True() r in Open() k(r)<k(g) or Inputs in False() Inputs in True() AND q=g in Open() q in S() p=-h in Open() p not in S() r in Open() or r in Y: r not in S() r in Y: r not in S() (a) (b) Fig. 4. Final cases in the proof of Lemma 16: q 2 Poss( X ) is not admissible Lemma 17 If S 2 SE H then Proof: Consider any S 2 PE H . It is certainly the case that S has as a subset some representative set, V X from the X-literal arguments. Suppose we modify the procedure described in the proof of Lemma 11, to one which takes as input a representative set V of the X-literals and returns a subset S V of the arguments W of H in the following way: g. The set S V is an admissible subset of W that contains only X-literal arguments and a (possibly empty) subset G of the gate arguments G . Furthermore, given V , there is a unique S V returned by this procedure. It follows that for any S 2 PE H , for the representative set V associated with S. This set, V , matches the literal arguments selected by some instantiation (V) of X, and so as in the proof of Lemma 11, we can deduce that S This suffices to complete the proof: we have established that every set S in PE H contains a subset instantiation X : from Lemma 16, In( X ) is not maximal if and only if The proof of our main theorem is now easy to construct. Proof: (of Theorem 2) It has already been shown that COHERENT 2 (p) 2 in Lemma 3. To complete the proof we need only show that (X; Y) is a positive instance of QSAT 2 if and only if H is coherent. First suppose that for all instantiations X there is some instantiation Y for which holds. From Lemma 15 and Lemma 17 it follows that all preferred extensions in H are of the form In( X ; Y ), and these are all stable extensions, hence H is coherent. Similarly, suppose that H is coherent. Let X be any total instantiation of X. Suppose, by way of contradiction, that for all Y , From Lemma 16, In( X ) is a preferred extension in this case, and hence (since H was assumed to be coherent) a stable extension. From Lemma 14 this implies that could only happen if i.e. the value of is determined in this case, independently of the instantiation of Y , contradicting the assumption that ( X ; Y ) was false for every choice of Y . Thus we deduce that (X; Y) is a positive instance of QSAT 2 if and only if H is coherent so completing the proof that COHERENT is (p) An easy Corollary of the reduction in Theorem 2 is Corollary Proof: That SA 2 (p) 2 follows from the fact that x is sceptically accepted in only if: for every subset S of X either S is not a preferred extension or x is in S. To see that SA is (p) 2 -hard, we need only observe that in order for H to be coherent, the gate argument g m must occur in in every preferred extension of H in the reduction of Theorem 2 Thus, H is coherent if and only if g m is sceptically accepted in H . 2 3 Consequences of Theorem 2 and Open Questions A number of authors have recently considered mechanisms for establishing credulous acceptance of an argument p in a finitely presented system H(X ; dialogue games. The protocol for such games assumes two players - the Defender, (D) and Challenger, (C) - and prescribe a move (or locution) repertoire together with the criteria governing the application of moves and concepts of 'winning' or 'losing'. The typical scenario is that following D asserting p the players take alternate turns presenting counter-arguments (consistent with the structure of H) to the argument asserted by their opponent in the previous move. A player loses when no legal move (within the game protocol) is available. An important example of such a game is the TPI-dispute formalism of [13] which provides a sound and complete basis for credulous argumentation. An abstract framework for describing such games was presented in [11], and is used in [3] also to define a game-theoretic approach to Credulous Acceptance. Coherent systems are important with respect to the game formalism of [13]: TPI-disputes define a sound and complete proof theory for both Sceptical and Credulous games on coherent argument systems; the Sceptical Game is not, however, complete in the case of incoherent systems. The sequence of moves describing a completed Credulous Game (for both [3,13]) can be interpreted as certificates of admissibility or inadmissibility for the argument disputed. It may be noted that this view makes apparent a computational difficulty arising in attempting to define similar 'Sceptical Games' applicable to incoherent systems: the shortest certificate that CA(H; x) holds, is the size of the smallest admissible set containing x - it is shown in [10] that there is always a strategy for D that can achieve this; it is also shown in [10] that TPI-disputes won by C, i.e. certificates that :CA(H; x), can require exponentially many (in jX moves. 5 If we consider a sound and complete dialogue game for sceptical reasoning, then the moves of a dispute won by D constitute a certificate of membership in a (p) -complete language: we would expect such certificates 'in general' to have exponential length; similarly, the moves in a dispute won by C constitute a certificate of membership in a (p) complete language and again these are 'likely' to be exponentially long. Thus a further motivation of coherent systems is that sceptical acceptance is 'at worst' CO- NP-complete: short certificates that an argument is not sceptically accepted always exist. The fact that sceptical acceptance is 'easier' to decide for coherent argument sys- tems, raises the question of whether there are efficiently testable properties that can be exploited in establishing coherence. The following is not difficult to prove: Fact 19 If H(X ; A) is not coherent then it contains a (simple) directed cycle of odd length. Thus an absence of odd cycles (a property which can be efficiently decided) ensures that the system is coherent. An open issue concerns coherence in random systems. One consequence of [4] is that random argument systems of n arguments in which each attack occurs (independently) with probability p, almost surely have a stable extension when p is a fixed probability in the range 0 p 1. Whether a similar result can be proven for coherence is open. As a final point, we observe that the interaction between graph-theoretic models of argument systems and propositional formulae may well provide a fruitful source 5 Since these are certificates of membership in a CO-NP-complete language, this is unsur- prising: [10] relates dispute lengths for such instances to the length of validity proofs in the CUT-free Gentzen calculus. of further techniques. We noted earlier that [7] provides a translation from CNF- formulae, into an argument system H ; our constructions above define similar translations for arbitrary propositional formulae. We can equally, however, consider translations in the reverse direction, e.g. given H(X ; it is not difficult to see that the fz:hz;xi2Ag z) is satisfiable if and only H has a stable extension. Similar encodings can be given for many of the decision problems of Table 1. Translating such forms back to argument systems, in effect gives an alternative formulation of the original argument system from which they were generated, and thus these provide mechanisms whereby any system, H can be translated into another system H dec with properties of concern holding of H if and only if related properties hold in H dec . Potentially this may permit both established methodologies from classical propositional logic 6 and graph-theory to be imported as techniques in argumentation. In this article the complexity of deciding whether a finitely presented argument system is coherent has been considered and shown to be (p) -complete, employing techniques based entirely around the directed graph representation of an argument system. An important property of coherent systems is that sound and complete methods for establishing credulous acceptance adapt readily to provide similar methods for deciding sceptical acceptance, hence sceptical acceptance in coherent systems is CO-NP-complete. In contrast, as an easy corollary of our main result it can be shown that sceptical acceptance is (p) 2 -complete in general. Finally we have outlined some directions by which the relationship between argument systems, propositional formulae, and graph-theoretic concepts offers potential for further re-search --R reasoning using classical logic. An abstract Dialectical proof theories for the credulous preferred semantics of argumentation frameworks. Preferred arguments are harder to compute than stable extensions. Finding admissible and preferred arguments can be very hard. Graph theoretical structures in logic programs and default theories. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning The Complexity of Boolean Networks. Two party immediate response disputes: Properties and efficiency. Dialectic semantics for argumentation frameworks. Logical Tools for Modelling Legal Argument. Credulous and sceptical argument games for preferred semantics. Complete sets and the polynomial-time hierarchy --TR The complexity of Boolean networks Kernels in random graphs On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and <italic>n</italic>-person games reasoning using classical logic Graph theoretical structures in logic programs and default theories An abstract, argumentation-theoretic approach to default reasoning Dialectic semantics for argumentation frameworks Preferred Arguments are Harder to Compute than Stable Extension Dialectical Proof Theories for the Credulous Preferred Semantics of Argumentation Frameworks --CTR P. M. Dung , R. A. Kowalski , F. Toni, Dialectic proof procedures for assumption-based, admissible argumentation, Artificial Intelligence, v.170 n.2, p.114-159, February 2006 Pietro Baroni , Massimiliano Giacomin , Giovanni Guida, Self-stabilizing defeat status computation: dealing with conflict management in multi-agent systems, Artificial Intelligence, v.165 n.2, p.187-259, July 2005 Paul E. Dunne , T. J. M. Bench-Capon, Two party immediate response disputes: properties and efficiency, Artificial Intelligence, v.149 n.2, p.221-250, October Trevor J. M. Bench-Capon , Sylvie Doutre , Paul E. Dunne, Audiences in argumentation frameworks, Artificial Intelligence, v.171 n.1, p.42-71, January, 2007

argument Systems;sceptical reasoning;coherence;credulous reasoning;computational complexity

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Typed compilation of recursive datatypes.

Standard ML employs an opaque (or generative) semantics of datatypes, in which every datatype declaration produces a new type that is different from any other type, including other identically defined datatypes. A natural way of accounting for this is to consider datatypes to be abstract. When this interpretation is applied to type-preserving compilation, however, it has the unfortunate consequence that datatype constructors cannot be inlined, substantially increasing the run-time cost of constructor invocation compared to a traditional compiler. In this paper we examine two approaches to eliminating function call overhead from datatype constructors. First, we consider a transparent interpretation of datatypes that does away with generativity, altering the semantics of SML; and second, we propose an interpretation of datatype constructors as coercions, which have no run-time effect or cost and faithfully implement SML semantics.

Introduction The programming language Standard ML (SML) [9] provides a distinctive mechanism for defining recursive types, known as a datatype declaration. For example, the following declaration defines the type of lists of integers: datatype intlist = Nil | Cons of int * intlist This datatype declaration introduces the type intlist and two constructors: Nil represents the empty list, and Cons combines an integer and a list to produce a new list. For instance, the expression Cons (1, Cons (2, Cons (3,Nil))) has type intlist and corresponds to the list [1, 2,3]. Values of this datatype are deconstructed by a case analysis that examines a list and determines whether it was constructed with Nil or with Cons, and in the latter case, extracting the original integer and list. An important aspect of SML datatypes is that they are generative. That is, every datatype declaration defines a type that is distinct from any other type, including those produced by other, possibly identical, datatype declarations. The formal Definition of SML [9] makes this precise by stating that a datatype declaration produces a new type name, but does not associate that name with a defini- tion; in this sense, datatypes are similar to abstract types. Harper and Stone [7] (hereafter, HS) give a type-theoretic interpretation of SML by exhibiting a translation from SML into a simpler, typed internal language. This translation is faithful to the Definition of SML in the sense that, with a few well-known exceptions, it translates an SML program into a well-typed IL program if and only if the SML program is well-formed according to the Definition. Consequently, we consider HS to be a suitable foundation for type-directed compilation of SML. Furthermore, it seems likely that any other suitable type-theoretic interpretation (i.e., one that is faithful to the Definition) will encounter the same issues we explore in our analysis. Harper and Stone capture datatype generativity by translating a datatype declaration as a module containing an abstract type and functions to construct and deconstruct values of that type; thus in the setting of the HS interpretation, datatypes are abstract types. The generativity of datatypes poses some challenges for type-directed compilation of SML. In particular, although the HS interpretation is easy to understand and faithful to the Definition of SML, it is inefficient when implemented na-vely. The problem is that construction and deconstruction of datatype values require calls to functions exported by the module defining the datatype; this is unacceptable given the ubiquity of datatypes in SML code. Conventional compilers, which disregard type information after an initial type-checking phase, may dispense with this cost by inlining those functions; that is, they may replace the function calls with the actual code of the corresponding functions to eliminate the call overhead. A type-directed compiler, however, does not have this option since all optimizations, including inlining, must be type- preserving. Moving the implementation of a datatype constructor across the module boundary violates type abstraction and thus results in ill-typed intermediate code. This will be made more precise in Section 2. In this paper, we will discuss two potential ways of handling this performance problem. We will present these alternatives in the context of the TILT/ML compiler developed at CMU [11, 14]; they are relevant, however, not just to TILT, but to understanding the definition of the language and type-preserving compilation in general. The first approach is to do away with datatype generativity alto- gether, replacing the abstract types in the HS interpretation with concrete ones. We call this approach the transparent interpretation of datatypes. Clearly, a compiler that does this is not an implementation of Standard ML, and we will show that, although the modified language does admit inlining of datatype constructors, it has some unexpected properties. In particular, it is not the case that every well-formed SML program is allowed under the transparent interpretation. In contrast, the second approach, which we have adopted in the most recent version of the TILT compiler, offers an efficient way of implementing datatypes in a typed setting that is consistent with the In particular, since a value of recursive type is typically represented at run time in the same way as its unrolling, we can observe that the mediating functions produced by the HS interpretation all behave like the identity function at run time. We replace these functions with special values that are distinguished from ordinary functions by the introduction of "coercion types". We call this the coercion interpretation of datatypes, and argue that it allows a compilation strategy that generates code with a run-time efficiency comparable to what would be attained if datatype constructors were inlined. The paper is structured as follows: Section 2 gives the details of the HS interpretation of datatypes (which we also refer to as the opaque interpretation of datatypes) and illustrates the problems with inlin- ing. Section 3 discusses the transparent interpretation. Section 4 gives the coercion interpretation and discusses its properties. Section 5 gives a performance comparison of the three interpretations. Section 6 discusses related work and Section 7 concludes. 2 The Opaque Interpretation of Datatypes In this section, we review the parts of Harper and Stone's interpretation of SML that are relevant to our discussion of datatypes. In par- ticular, after defining the notation we use for our internal language, we will give an example of the HS elaboration of datatypes. We will refer to this example throughout the paper. We will also review the way Harper and Stone define the matching of structures against signatures, and discuss the implications this has for datatypes. This will be important in Section 3, where we show some differences between signature matching in SML and signature matching under our transparent interpretation of datatypes. Types s,t ::= - | a | d Recursive Types d Terms e ::= - | x | roll d (e) | unroll d (e) Typing Contexts G ::= e | Figure 1. Syntax of Iso-recursive Types where X is a metavariable, such as a or t where Figure 2. Shorthand Definitions 2.1 Notation Harper and Stone give their interpretation of SML as a translation, called elaboration, from SML into a typed internal language (IL). We will not give a complete formal description of the internal language we use in this paper; instead, we will use ML-like syntax for examples and employ the standard notation for function, sum and product types. For a complete discussion of elaboration, including a thorough treatment of the internal language, we refer the reader to Harper and Stone [7]. Since we are focusing our attention on datatypes, recursive types will be of particular importance. We will therefore give a precise description of the semantics of the form of recursive types we use. The syntax for recursive types is given in Figure 1. Recursive types are separated into their own syntactic subcategory, ranged over by d. This is mostly a matter of notational convenience, as there are many times when we wish to make it clear that a particular type is a recursive one. A recursive type has the form - and each a j is a type variable that may appear free in any or all of t 1 , ., t n . Intuitively, this type is the ith in a system of n mutually recursive types. As such, it is isomorphic to t i with each a j replaced by the jth component of the recursive bun- dle. Formally, it is isomorphic to the following somewhat unwieldy type: (where, as usual, we denote by t[s 1 , ., s n /a 1 , ., a n ] the simultaneous capture-avoiding substitution of s 1 , ., s n for a 1 , ., a n in t). Since we will be writing such types often, we use some notational conventions to make things clearer; these are shown in Figure 2. Using these shorthands, the above type may be written as The judgment forms of the static semantics of our internal language are given in Figure 3, and the rules relevant to recursive types are given in Figure 4. Note that the only rule that can be used to judge two recursive types equal requires that the two types in question are the same (ith) projection from bundles of the same length whose respective components are all equal. In particular, there is no "un- Well-formed context. Well-formed type. Equivalence of types. Well-formed term. Figure 3. Relevant Typing Judgments Figure 4. Typing Rules for Iso-recursive Types rolling" rule stating that d # expand(d); type theories in which this equality holds are said to have equi-recursive types and are significantly more complex [5]. The recursive types in our theory are iso- recursive types that are isomorphic, but not equal, to their expan- sions. The isomorphism is embodied by the roll and unroll operations at the term level; the former turns a value of type expand(d) into one of type d, and the latter is its inverse. 2.2 Elaborating Datatype Declarations The HS interpretation of SML includes a full account of datatypes, including generativity. The main idea is to encode datatypes as recursive sum types but hide this implementation behind an opaque signature. A datatype declaration therefore elaborates to a structure that exports a number of abstract types and functions that construct and deconstruct values of those types. For example, consider the following pair of mutually recursive datatypes, representing expressions and declarations in the abstract syntax of a toy language: datatype | LetExp of dec * exp and dec = ValDec of var * exp | SeqDec of dec * dec The HS elaboration of this declaration is given in Figure 5, using ML-like syntax for readability. To construct a value of one of these datatypes, a program must use the corresponding in function; these functions each take an element of the sum type that is the "un- rolling" of the datatype and produce a value of the datatype. More concretely, we implement the constructors for exp and dec as follows Notice that the types exp and dec are held abstract by the opaque signature ascription. This captures the generativity of datatypes, since the abstraction prevents ExpDec.exp and ExpDec.dec from being judged equal to any other types. However, as we mentioned in Section 1, this abstraction also prevents inlining of the in and structure ExpDec :> sig type exp type dec val exp in : var val exp out : exp -> var val dec in : (var * exp) (dec * dec) -> dec val dec out : dec -> (var * exp) (dec * dec) struct fun exp in fun exp out fun dec in fun dec out Figure 5. Harper-Stone Elaboration of exp-dec Example out functions: for example, if we attempt to inline exp in in the definition of VarExp above, we get but this is ill-typed outside of the ExpDec module because the fact that exp is a recursive type is not visible. Thus performing inlining on well-typed code can lead to ill-typed code, so we say that inlining across abstraction boundaries is not type-preserving and therefore not an acceptable strategy for a typed compiler. The problem is that since we cannot inline in and out functions, our compiler must pay the run-time cost of a function call every time a value of a datatype is constructed or case-analyzed. Since these operations occur very frequently in SML code, this performance penalty is significant. One strategy that can alleviate this somewhat is to hold the implementation of a datatype abstract during elaboration, but to expose its underlying implementation after elaboration to other code defined in the same compilation unit. Calls to the constructors of a locally-defined datatype can then be safely inlined. In the setting of whole-program compilation, this approach can potentially eliminate constructor call overhead for all datatypes except those appearing as arguments to functors. However, in the context of separate compilation, the clients of a datatype generally do not have access to its implementation, but rather only to the specifications of its constructors. As we shall see in Section 3, the specifications of a datatype's constructors do not provide sufficient information to correctly predict how the datatype is actually implemented, so the above compilation strategy will have only limited success in a true separate compilation setting. 2.3 Datatypes and Signature Matching Standard ML makes an important distinction between datatype dec- larations, which appear at the top level or in structures, and datatype specifications, which appear in signatures. As we have seen, the HS interpretation elaborates datatype declarations as opaquely sealed structures; datatype specifications are translated into specifications of structures. For example, the signature datatype intlist = Nil | Cons of int * intlist contains a datatype specification, and elaborates as follows: struct Intlist : sig type intlist val intlist in : intlist -> intlist val intlist out : intlist intlist structure M will match S if M contains a structure Intlist of the appropriate signature. 1 In particular, it is clear that the structure definition produced by the HS interpretation for the datatype intlist defined in Section 1 has this signature, so that datatype declaration matches the specification above. What is necessary in general for a datatype declaration to match a specification under this interpretation? Since datatype declarations are translated as opaquely sealed structures, and datatype specifications are translated as structure specifications, matching a datatype declaration against a spec boils down to matching one signature- the one opaquely sealing the declaration structure-against another signature. Suppose we wish to know whether the signature S matches the signature T; that is, whether a structure with signature S may also be given the signature T. Intuitively, we must make sure that for every specification in T there is a specification in S that is compatible with it. For instance, if T contains a value specification of the form t, then S must also contain a specification val t. For an abstract type specification of the form type t occurring in T, we must check that a specification of t also appears in furthermore, if the specification in S is a transparent one, say imp , then when checking the remainder of the specifications in T we may assume in both signatures that imp . Transparent type specifications in T are similar, but there is the added requirement that if the specification in T is type spec and the specification in S is type imp must be equivalent. Returning to the specific question of datatype matching, a specification of the form datatype (where the t i may be sum types) elaborates to a specification of a structure with the following signature: sig val val t n in : t n -> t n val t only datatypes to match datatype spec- ifications, so the actual HS elaboration must use a name for the datatype that cannot be guessed by a programmer. structure ExpDec :> sig (* . specifications for in and out functions same as before . *) (* . same structure as before . *) Figure 6. The Transparent Elaboration of Exp and Dec In order to match this signature, the structure corresponding to a datatype declaration must define types named t 1 , ., t n and must contain in and out functions of the appropriate type for each. (Note that in any structure produced by elaborating a datatype declaration under this interpretation, the t i 's will be abstract types.) Thus, for example, if m # n then the datatype declaration datatype . and matches the above specification if and only if s since this is necessary and sufficient for the types of the in and out functions to match for the types mentioned in the specification. 3 A Transparent Interpretation of Datatypes A natural approach to enabling the inlining of datatypes in a type-preserving compiler is to do away with the generative semantics of datatypes. In the context of the HS interpretation, this corresponds to replacing the abstract type specification in the signature of a datatype module with a transparent type definition, so we call this modified interpretation the transparent interpretation of datatypes (TID). 3.1 Making Datatypes Transparent The idea of the transparent interpretation is to expose the implementation of datatypes as recursive sum types during elaboration, rather than hiding it. In our expdec example, this corresponds to changing the declaration shown in Figure 5 to that shown in Figure 6 (we continue to use ML-like syntax for readability). Importantly, this change must extend to datatype specifications as well as datatype declarations. Thus, a structure that exports a datatype must export its implementation transparently, using a signature similar to the one in the figure-otherwise a datatype inside a structure would appear to be generative outside that structure, and there would be little point to the new interpretation. As we have mentioned before, altering the interpretation of datatypes to expose their implementation as recursive types really creates a new language, which is neither a subset nor a superset of Standard ML. An example of the most obvious difference can be seen in Figure 7. In the figure, two datatypes are defined by seemingly identical declarations. In SML, because datatypes are generative, the two types List1.t and List2.t are distinct; since the variable l has type List1.t but is passed to List2.Cons, which expects List2.t, the function switch is ill-typed. Under the transparent interpretation, however, the implementations of both datatypes are exported transparently as -a.unit + int * a. Thus under this interpretation, List1.t and List2.t are equal and so switch is a well-typed function. It is clear that many programs like this one fail to type-check in SML but succeed under the transparent interpretation; what is less structure struct datatype Cons of int * t structure struct datatype Cons of int * t fun switch | switch (List1.Cons obvious is that there are some programs for which the opposite is true. We will discuss two main reasons for this. 3.2 Problematic Datatype Matchings Recall that according to the HS interpretation, a datatype matches a datatype specification if the types of the datatype's in and out functions match the types of the in and out functions in the speci- fication. (Note: the types of the out functions match if and only if the types of the in functions match, so we will hereafter refer only to the in functions.) Under the transparent interpretation, how- ever, it is also necessary that the recursive type implementing the datatype match the one given in the specification. This is not a trivial requirement; we will now give two examples of matchings that succeed in SML but fail under the transparent interpretation. 3.2.1 A Simple Example A very simple example of a problematic matching is the following. Under the opaque interpretation, matching the structure struct datatype of u * u | B of int against the signature sig datatype of v | B of int amounts to checking that the type of the in function for u defined in the structure matches that expected by the signature once u * u has been substituted for v in the signature. (No definition is substituted for u, since it is abstract in the structure.) After sub- stitution, the type required by the signature for the in function is which is exactly the type of the function given by the structure, so the matching succeeds. Under the transparent interpretation, however, the structure defines u to be u imp int but the signature specifies u as int. In order for matching to succeed, these two types must be equivalent after we have substituted u imp * u imp for v in the spec- ification. That is, it is required that Observe that the type on the right is none other than -a.expand(u imp ). (Notice also that the bound variable a does not appear free in the body of this -type. Hereafter we will write such types with a wildcard in place of the type variable to indicate that it is not used in the body of the -.) This equivalence does not hold for iso-recursive types, so the matching fails. 3.2.2 A More Complex Example Another example of a datatype matching that is legal in SML but fails under the transparent interpretation can be found by reconsidering our running example of exp and dec. Under the opaque interpretation, a structure containing this pair of datatypes matches the following signature, which hides the fact that exp is a datatype: sig type exp datatype | SeqDec of dec * dec When this datatype specification is elaborated under the transparent interpretation, however, the resulting IL signature looks like: sig type exp where dec spec a. Elaboration of the declarations of exp and dec, on the other hand, produces the structure in Figure 6, which has the signature: sig where we define exp imp dec imp Matching the structure containing the datatypes against the signature can only succeed if dec spec # dec imp (under the assumption that exp # exp imp ). This equivalence does not hold because the two -types have different numbers of components. 3.3 Problematic Signature Constraints The module system of SML provides two ways to express sharing of type information between structures. The first, where type, modifies a signature by "patching in" a definition for a type the signature originally held abstract. The second, sharing type, asserts that two or more type names (possibly in different structures) refer to the same type. Both of these forms of constraints are restricted so that multiple inconsistent definitions are not given to a single type name. In the case of sharing type, for example, it is required that all the names be flexible, that is, they must either be abstract or defined as equal to another type that is abstract. Under the opaque interpretation, datatypes are abstract and therefore flexible, meaning they can be shared; under the transparent inter- pretation, datatypes are concretely defined and hence can never be shared. For example, the following signature is legal in SML: type s datatype type s datatype sharing type We can write an equivalent signature by replacing the sharing type line with where type which is also valid SML. Neither of these signatures elaborates successfully under the transparent interpretation of datatypes, since under that interpretation the datatypes are transparent and therefore ineligible for either sharing or where type. Another example is the following signature: signature type s val datatype sharing type (Again, we can construct an analogous example with where type.) Since the name B.t is flexible under the opaque interpretation but not the transparent, this code is legal SML but must be rejected under the transparent interpretation. 3.4 Relaxing Recursive Type Equivalence We will now describe a way of weakening type equivalence (i.e., making it equate more types) so that the problematic datatype matchings described in Section 3.2 succeed under the transparent interpretation. (This will not help with the problematic sharing constraints of Section 3.3.) The ideas in this section are based upon the equivalence algorithm adopted by Shao [8] for the FLINT/ML compiler. To begin, consider the simple u-v example of Section 3.2.1. Recall that in that example, matching the datatype declaration against the spec required proving the equivalence where the type on the right-hand side is just - . expand(u imp ). By simple variations on this example, it is easy to show that in general, for the transparent interpretation to be as permissive as the opaque, the following recursive type equivalence must hold: We refer to this as the boxed-unroll rule. It says that a recursive type is equal to its unrolling "boxed" by a -. An alternative formulation, equivalent to the first one by transitivity, makes two recursive types equal if their unrollings are equal, i.e.: Intuitively, this rule is needed because datatype matching succeeds under the opaque interpretation whenever the unrolled form of the datatype implementation equals the unrolled form of the datatype spec (because these are both supposed to describe the domain of the in function). Although the boxed-unroll equivalence is necessary for the transparent interpretation of datatypes to admit all matchings admitted by the opaque one, it is not sufficient; to see this, consider the problematic exp-dec matching from Section 3.2.2. The problematic constraint in that example is: dec # spec # dec imp where dec (substituting exp imp for exp in dec imp has no effect, since the variable does not appear free). Expanding the definitions of these types, we get the constraint: -a.var * exp imp + a * a # The boxed-unroll rule is insufficient to prove this equivalence. In order to apply boxed-unroll to prove these two types equivalent, we must be able to prove that their unrollings are equivalent, in other words that var * exp imp var * exp imp dec imp * dec imp But we cannot prove this without first proving dec # spec # dec imp , which is exactly what we set out to prove in the first place! The boxed-unroll rule is therefore unhelpful in this case. The trouble is that proving the premise of the boxed-unroll rule (the equivalence of expand(d 1 ) and expand(d 2 may require proving the conclusion (the equivalence of d 1 and d 2 ). Similar problems have been addressed in the context of general equi-recursive types. In that setting, deciding type equivalence involves assuming the conclusions of equivalence rules when proving their premises [1, 2]. Applying this idea provides a natural solution to the problem discussed in the previous section. We can maintain a "trail" of type- equivalence assumptions; when deciding the equivalence of two recursive types, we add that equivalence to the trail before comparing their unrollings. Formally, the equivalence judgement itself becomes G;A # s # t, where A is a set of assumptions, each of the form t 1 # t 2 . All the equivalence rules in the static semantics must be modified to account for the trail. In all the rules except those for recursive types, the trail is simply passed unchanged from the conclusions to the premises. There are two new rules that handle recursive types: The first rule allows an assumption from the trail to be used; the second rule is an enhanced form of the boxed-unroll rule that adds the conclusion to the assumptions of the premise. It is clear that the trail is just what is necessary in order to resolve the exp-dec anomaly described above; before comparing the unrollings of dec spec and dec imp , we add the assumption dec spec # dec imp to the trail; we then use this assumption to avoid the cyclic dependency we encountered before. In fact, the trailing version of the boxed-unroll rule is sufficient to ensure that the transparent interpretation accepts all datatype matchings accepted by SML. To see why, consider a datatype spec- ification datatype (where the t i may be sum types in which the t i may occur). Suppose that some implementation matches this spec under the opaque interpretation; the implementation of each type t i must be a recursive type d i . Furthermore, the type of the t i in function given in the spec is t i # t i , and the type of its implementation is . Because the matching succeeds under the opaque interpretation, we know that these types are equal after each d i has been substituted for t i ; thus we know that expand(d i each i. When the specification is elaborated under the transparent interpre- tation, however, the resulting signature declares that the implementation of each t i is the appropriate projection from a recursive bundle determined by the spec itself. That is, each t i is transparently specified as - i ( # t).( # t). In order for the implementation to match this transparent specification, it is thus sufficient to prove the following theorem: Theorem 1 If #i # 1.n, G; / 1.n, G; / Proof: See Appendix A. # While we have given a formal argument why the trailing version of the boxed-unroll rule is flexible enough to allow the datatype matchings of SML to typecheck under the transparent interpreta- tion, we have not been precise about how maintaining a trail relates to the rest of type equivalence. In fact, the only work regarding trails we are aware of is the seminal work of Amadio and Cardelli [1] on subtyping equi-recursive types, and its later coinductive axiomatization by Brandt and Henglein [2], both of which are conducted in the context of the simply-typed l-calculus. Our trailing boxed-unroll rule can be viewed as a restriction of the corresponding rule in Amadio and Cardelli's trailing algorithm so that it is only applicable when both types being compared are recursive types. It is not clear, though, how trails affect more complex type systems that contain type constructors of higher kind, such as Gi- rard's F w [6]. In addition to higher kinds, the MIL (Middle Intermediate Language) of TILT employs singleton kinds to model sharing [13], and the proof that MIL typechecking is decidable is rather delicate and involved. While we have implemented the above trailing algorithm in TILT for experimental purposes (see Section 5), the interaction of trails and singletons is not well-understood. As for the remaining conflict between the transparent interpretation and type sharing, one might argue that the solution is to broaden SML's semantics for sharing constraints to permit sharing of rigid components. The problem is that the kind of sharing that would be necessary to make the examples of Section 3.3 typecheck under the transparent interpretation would require some form of type unification. It is difficult to determine where to draw the line between SML's sharing semantics and full higher-order unification, which is undecidable. Moreover, unification would constitute a significant change to SML's semantics, disproportionate to the original problem of efficiently implementing datatypes. 4 A Coercion Interpretation of Datatypes In this section, we will discuss a treatment of datatypes based on coercions. This solution will closely resemble the Harper-Stone interpretation, and thus will not require the boxed-unroll rule or a trail algorithm, but will not incur the run-time cost of a function call at constructor application sites either. 4.1 Representation of Datatype Values The calculus we have discussed in this paper can be given the usual structured operational semantics, in which an expression of the form roll d (v) is itself a value if v is a value. (From here on we will assume that the metavariable v ranges only over values.) In fact, it can be shown without difficulty that any closed value of a datatype d must have the form roll d (v) where v is a closed value of type expand(d). Thus the roll operator plays a similar role to that of the inj 1 operator for sum types, as far as the high-level language semantics is concerned. Although we specify the behavior of programs in our language with a formal operational semantics, it is our intent that programs be compiled into machine code for execution, which forces us to take a slightly different view of data. Rather than working directly with high-level language values, compiled programs manipulate representations of those values. A compiler is free to choose the representation scheme it uses, provided that the basic operations of the language can be faithfully performed on representations. For exam- ple, most compilers construct the value inj 1 (v) by attaching a tag to the value v and storing this new object somewhere. This tagging is necessary in order to implement the case construct. In particular, the representation of any value of type must carry enough information to determine whether it was created with inj 1 or inj 2 and recover a representation of the injected value. What are the requirements for representations of values of recursive It turns out that they are somewhat weaker than for sums. The elimination form for recursive types is unroll, which (unlike case) does not need to extract any information from its argument other than the original rolled value. In fact, the only requirement is that a representation of v can be extracted from any representation of roll d (v). Thus one reasonable representation strategy is to represent roll d (v) exactly the same as v. In the companion technical report [15], we give a more precise argument as to why this is rea- sonable, making use of two key insights. First, it is an invariant of the TILT compiler that the representation of any value fits in a single machine register; anything larger than 32 bits is always stored in the heap. This means that all possible complications having to do with the sizes of recursive values are avoided. Second, we define representations for values, not types; that is, we define the set of machine words that can represent the value v by structural induction on v, rather than defining the set of words that can represent values of type t by induction on t as might be expected. The TILT compiler adopts this strategy of identifying the representations of roll d (v) and v, which has the pleasant consequence that the roll and unroll operations are "no-ops". For instance, the untyped machine code generated by the compiler for the expression roll d (e) need not differ from the code for e alone, since if the latter evaluates to v then the former evaluates to roll d (v), and Types Terms e ::= - | L# a.fold d | L# a.unfold d | v@( # t;e) Figure 8. Syntax of Coercions the representations of these two values are the same. The reverse happens for unroll. This, in turn, has an important consequence for datatypes. Since the in and out functions produced by the HS elaboration of datatypes do nothing but roll or unroll their arguments, the code generated for any in or out function will be the same as that of the identity function. Hence, the only run-time cost incurred by using an in function to construct a datatype value is the overhead of the function call itself. In the remainder of this section we will explain how to eliminate this cost by allowing the types of the in and out functions to reflect the fact that their implementations are trivial. 4.2 The Coercion Interpretation To mark in and out functions as run-time no-ops, we use coer- cions, which are similar to functions, except that they are known to be no-ops and therefore no code needs to be generated for coercion applications. We incorporate coercions into the term level of our language and introduce special coercion types to which they belong. Figure 8 gives the changes to the syntax of our calculus. Note that while we have so far confined our discussion to monomorphic datatypes, the general case of polymorphic datatypes will require polymorphic coercions. The syntax we give here is essentially that used in the TILT compiler; it does not address non-uniform datatypes. We extend the type level of the language with a type for (possibly polymorphic) coercions, a value of this type is a coercion that takes length(# a) type arguments and then can change a value of type t 1 into one of type t 2 (where, of course, variables from # a can appear in either of these types). When # a is empty, we will Similarly, we extend the term level with the (possibly polymorphic) coercion values L# a.fold d and L# a.unfold d ; these take the place of roll and unroll expressions. Coercions are applied to (type and value) arguments in an expression of the form v@( # t;e); here v is the coercion, # t are the type arguments, and e is the value to be coerced. Note that the coercion is syntactically restricted to be a value; this makes the calculus more amenable to a simple code generation strategy, as we will discuss in Section 4.3. The typing rules for coercions are essentially the same as if they were ordinary polymorphic functions, and are shown in Figure 9. With these modifications to the language in place, we can elaborate the datatypes exp and dec using coercions instead of functions to implement the in and out operations. The result of elaborating this pair of datatypes is shown in Figure 10. Note that the interface is exactly the same as the HS interface shown in Section 2 except that the function arrows (->) have been replaced by coercion arrows (#). This interface is implemented by defining exp and dec in the same way as in the HS interpretation, and implementing the in and out coercions as the appropriate fold and unfold values. The elaboration of a constructor application is superficially similar to the opaque interpretation, but a coercion application is generated instead of a function call. For instance, LetExp(d,e) elaborates as exp in@(inj 2 (d,e)). 4.3 Coercion Erasure We are now ready to formally justify our claim that coercions may be implemented by erasure, that is, that it is sound for a compiler to consider coercions only as "retyping operators" and ignore them when generating code. First, we will describe the operational semantics of the coercion constructs we have added to our internal language. Next, we will give a translation from our calculus into an untyped one in which coercion applications disappear. Finally, we will state a theorem guaranteeing that the translation is safe. The operational semantics of our coercion constructs are shown in Figure 11. We extend the class of values with the fold and unfold coercions, as well as the application of a fold coercion to a value. These are the canonical forms of coercion types and recursive types respectively. The two inference rules shown in Figure 11 define the manner in which coercion applications are evaluated. The evaluation of a coercion application is similar to the evaluation of a normal function application where the applicand is already a value. The rule on the left specifies that the argument is reduced until it is a value. If the applicand is a fold, then the application itself is a value. If the applicand is an unfold, then the argument must have a recursive type and therefore (by canonical forms) consist of a fold applied to a value v. The rule on the right defines unfold to be the left inverse of fold, and hence this evaluates to v. As we have already discussed, the data representation strategy of TILT is such that no code needs to be generated to compute foldv from v, nor to compute the result of cancelling a fold with an unfold. Thus it seems intuitive that to generate code for a coercion application v@( # t;e), the compiler can simply generate code for e, with the result that datatype constructors and destructors under the coercion interpretation have the same run-time costs as Harper and Stone's functions would if they were inlined. To make this more precise, we now define an erasure mapping to translate terms of our typed internal language into an untyped language with no coercion application. The untyped nature of the target language (and of machine language) is important: treating v as foldv would destroy the subject reduction property of a typed language. Figure 12 gives the syntax of our untyped target language and the coercion-erasing translation. The target language is intended to be essentially the same as our typed internal language, except that all types and coercion applications have been removed. It contains untyped coercion values fold and unfold, but no coercion application form. The erasure translation turns expressions with type annotations into expressions without them (l-abstraction and coercion values are shown in the figure), and removes coercion applications so that the erasure of v@( # t;e) is just the erasure of e. In particular, for any value v, v and foldv are identified by the translation, which is consistent with our intuition about the compiler. The operational semantics of the target language is analogous to that of the source. The language with coercions has the important type-safety property that if a term is well-typed, its evaluation does not get stuck. An important theorem is that the coercion-erasing translation preserves the safety of well-typed programs: Theorem 2 (Erasure Preserves Safety) If G # e : t, then e - is safe. That is, if e - # f , then f is not stuck. Proof: See the companion technical report [15]. # G,# a # d type G,# a # d type Figure 9. Typing Rules for Coercions structure ExpDec :> sig type exp type dec val exp in : var val exp out : exp # var val dec in : (var * exp) val dec out : dec # (var * exp) struct val exp in = foldexp val exp out = unfoldexp val dec in = fold dec val dec out = unfold dec Figure 10. Elaboration of exp and dec Under the Coercion Interpretation Values v ::= - | L# a.fold t | L# a.unfold t | (L# a.fold t )@(# s;v) Figure 11. Operational Semantics for Coercions fold | unfold (L# a.fold d (L# a.unfold d unfold Figure 12. Target Language Syntax; Type and Coercion Erasure Test life 8.233 2.161 2.380 leroy 5.497 4.069 3.986 boyer 2.031 1.559 1.364 simple 1.506 1.003 0.908 tyan 16.239 8.477 9.512 msort 1.685 0.860 1.012 pia 1.758 1.494 1.417 lexgen 11.052 5.599 5.239 Figure 13. Performance Comparison Note that the value restriction on coercions is crucial to the soundness of this "coercion erasure" interpretation. Since a divergent expression can be given an arbitrary type, including a coercion type, any semantics in which a coercion expression is not evaluated before it is applied fails to be type-safe. Thus if arbitrary expressions of coercion type could appear in application positions, the compiler would have to generate code for them. Since values cannot diverge or have effects, we are free to ignore coercion applications when we generate code. Performance To evaluate the relative performance of the different interpretations of datatypes we have discussed, we performed experiments using three different versions of the TILT compiler: one that implements a na-ve Harper-Stone interpretation in which the construction of a non-locally-defined datatype requires a function call 2 ; one that implements the coercion interpretation of datatypes; and one that implements the transparent interpretation. We compiled ten different benchmarks using each version of the compiler; the running times for the resulting executables (averaged over three trials) are shown in Figure 13. All tests were run on an Ultra-SPARC Enterprise server; the times reported are CPU time in seconds. The measurements clearly indicate that the overhead due to datatype constructor function calls under the na-ve HS interpretation is significant. The optimizations afforded by the coercion and transparent interpretations provide comparable speedups over the opaque interpretation, both on the order of 37% (comparing the total running times). Given that, of the two optimized approaches, only the coercion interpretation is entirely faithful to the semantics of SML, and since the theory of coercion types is a simpler and more orthogonal extension to the HS type theory than the trailing algorithm of Section 3.4, we believe the coercion interpretation is the more robust choice. 6 Related Work Our trail algorithm for weakened recursive type equivalence is based on the one implemented by Shao in the FLINT intermediate language of the Standard ML of New Jersey compiler [12]. The typing rules in Section 3.4 are based on the formal semantics for FLINT given by League and Shao [8], although we are the first to give a formal argument that their trailing algorithm actually works. It is important to note that SML/NJ only implements the transpar- In particular, we implement the strategy described at the end of Section 2.2. ent interpretation internally: the opaque interpretation is employed during elaboration, and datatype specifications are made transparent only afterward. As the examples of Section 3.3 illustrate, there are programs that typecheck according to SML but not under the transparent interpretation even with trailing equivalence, so it is unclear what SML/NJ does (after elaboration) in these cases. As it happens, the final example of Section 3.3, which is valid SML, is rejected by the SML/NJ compiler. Curien and Ghelli [4] and Crary [3] have defined languages that use coercions to replace subsumption rules in languages with subtyp- ing. Crary's calculus of coercions includes roll and unroll for recursive types, but since the focus of his paper is on subtyping he does not explore the potential uses of these coercions in detail. Nev- ertheless, our notion of coercion erasure, and the proof of our safety preservation theorem, are based on Crary's. The implementation of Typed Assembly Language for the x86 architecture (TALx86) [10] allows operands to be annotated with coercions that change their types but not their representations; these coercions include roll and unroll as well as introduction of sums and elimination of universal quantifiers. Our intermediate language differs from these in that we include coercions in the term level of the language rather than treating them specially in the syntax. This simplifies the presentation of the coercion interpretation of datatypes, and it simplified our implementation because it required a smaller incremental change from earlier versions of the TILT compiler. However, including coercions in the term level is a bit unnatural, and our planned extension of TILT with a type-preserving back-end will likely involve a full coercion calculus. 7 Conclusion The generative nature of SML datatypes poses a significant challenge for efficient type-preserving compilation. Generativity can be correctly understood by interpreting datatypes as structures that hold their type components abstract, exporting functions that construct and deconstruct datatype values. Under this interpretation, the inlining of datatype construction and deconstruction operations is not type-preserving and hence cannot be performed by a typed compiler such as TILT. In this paper, we have discussed two approaches to eliminating the function call overhead in a type-preserving way. The first, doing away with generativity by making the type components of datatype structures transparent, results in a new language that is different but neither more nor less permissive than, Standard ML. Some of the lost expressiveness can be regained by relaxing the rules of type equivalence in the intermediate language, at the expense of complicating the type theory. The fact that the transparent interpretation forbids datatypes to appear in sharing type or where type signature constraints is unfortunate; it is possible that a revision of the semantics of these constructs could remove the restriction. The second approach, replacing the construction and deconstruction functions of datatypes with coercions that may be erased during code generation, eliminates the function call overhead without changing the static semantics of the external language. However, the erasure of coercions only makes sense in a setting where a recursive-type value and its unrolling are represented the same at run time. The coercion interpretation of datatypes has been implemented in the TILT compiler. Although we have presented our analysis of SML datatypes in the context of Harper-Stone and the TILT compiler, the idea of "co- ercion types" is one that we think is generally useful. Terms that serve only as retyping operations are pervasive in typed intermediate languages, and are usually described as "coercions" that can be eliminated before running the code. However, applications of these informal coercions cannot in general be erased if there is no way to distinguish coercions from ordinary functions by their types; this is a problem especially in the presence of true separate compilation. Our contribution is to provide a simple mechanism that permits coercive terms to be recognized as such and their applications to be safely eliminated, without requiring significant syntactic and metatheoretic overhead. --R Subtyping recursive types. Coinductive axiomatization of recursive type equality and subtyping. Typed compilation of inclusive subtyping. Coherence of sub- sumption Recursive subtyping revealed. Formal semantics of the FLINT intermediate language. David Mac- Queen A realistic typed assembly language. Implementing the TILT internal language. An overview of the FLINT/ML compiler. Deciding type equivalence in a language with singleton kinds. Typed compilation of recursive datatypes. --TR Subtyping recursive types Coinductive axiomatization of recursive type equality and subtyping Deciding type equivalence in a language with singleton kinds Typed compilation of inclusive subtyping Recursive subtyping revealed --CTR Derek Dreyer, Recursive type generativity, ACM SIGPLAN Notices, v.40 n.9, September 2005 Vijay S. Menon , Neal Glew , Brian R. Murphy , Andrew McCreight , Tatiana Shpeisman , Ali-Reza Adl-Tabatabai , Leaf Petersen, A verifiable SSA program representation for aggressive compiler optimization, ACM SIGPLAN Notices, v.41 n.1, p.397-408, January 2006 Dimitrios Vytiniotis , Geoffrey Washburn , Stephanie Weirich, An open and shut typecase, Proceedings of the 2005 ACM SIGPLAN international workshop on Types in languages design and implementation, p.13-24, January 10-10, 2005, Long Beach, California, USA

recursive types;coercions;typed compilation;standard ML

604189

A typed interface for garbage collection.

An important consideration for certified code systems is the interaction of the untrusted program with the runtime system, most notably the garbage collector. Most certified code systems that treat the garbage collector as part of the trusted computing base dispense with this issue by using a collector whose interface with the program is simple enough that it does not pose any certification challenges. However, this approach rules out the use of many sophisticated high-performance garbage collectors. We present the language LGC, whose type system is capable of expressing the interface of a modern high-performance garbage collector. We use LGC to describe the interface to one such collector, which involves a substantial amount of programming at the type constructor level of the language.

Introduction In a certified code system, executable programs shipped from a producer to a client are accompanied by certificates This material is based on work supported in part by NSF grants CCR- 9984812 and CCR-0121633, and by an NSF fellowship. Any opinions, findings, and conclusions or recommendations in this publication are those of the authors and do not reflect the views of this agency. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. TLDI'03, January 18, 2003, New Orleans, Louisiana, USA. ACM 1-58113-649-8/03/0001. $5.00. that provide evidence of their safety. The validity of a cer- tificate, which can be mechanically verified by the client, implies that the associated program is safe to execute. Examples of certified code frameworks include Typed Assembly Language [6] and Proof-Carrying Code [7, 8]. Most past research on certified code has focused on the safety of the untrusted mobile code itself. However, it is also important to consider the safety implications of the runtime system to which that code is linked. There are two options for dealing with this issue. One choice is to treat the runtime system as part of the untrusted code, and certify its safety. The other choice is to simply assume the runtime system is correct-i.e., treat it as part of the trusted computing base that includes the certificate verifier. Of course, even if the runtime itself is assumed correct, the interaction of the program with the runtime must be certified to conform to the appropriate interface. An important part of the runtime system for many modern languages is the garbage collector. Frameworks in which the runtime system must be certified must use certification technology capable of proving a garbage collector safe. Work on this approach includes that of Wang and Appel [10, 9] and of Monnier et al. [4]. Of the systems that take the second approach, many assume the existence of a trusted conservative garbage collector; the advantage of this is that the application interface of a conservative collector is so simple that it can almost be ignored. There are performance benefits to be gained by using a more precise collector; how- ever, the interface of such a collector is more subtle, and the issue of certifying program conformance to this interface can no longer be ignored. In order to use a better garbage collector for certified code applications, the interface of such a collector must be described and expressed in a type system. The topic of this paper is the specification of the interface for a particular, modern garbage collector, namely that of Cheng and Blelloch [1, 2], implemented in the TILT/ML runtime system. After informally describing the behavior of this collector and its interface to a running program, we present a language whose type system can express this inter- This language, called LGC, is built up from a simple stack based language we call LGC - by extension with the typing constructs necessary to express various elements of the collector's interface. As we present LGC, we describe the interface to Cheng's garbage collector, the precise definition of which involves a substantial amount of programming in the language of type constructors. Finally, we discuss the expressiveness of the LGC language. int G() { . G() (newer frames) F's other locals G's arguments G's other locals F's frame G's frame (older frames) F's arguments return address return address (to F) int F() { Figure 1: Frames on a Stack 1.1 The Garbage Collector's Interface The first part of a garbage collector's job is to find the root set-those registers, globals and stack locations that contain pointers into the heap. This task is the part of garbage collection that requires compiler cooperation, and the part that makes assumptions about the behavior of the program. In this section we describe a simplified form of the root-finding algorithm used in TILT/ML. We will ignore complications such as an optimization for callee-save registers, and assume that all roots are stored on the stack. We can therefore ignore the additional work of finding roots among the registers or global variables. The garbage collector assumes that the stack is laid out as a sequence of frames, each belonging to the particular function that created it. Each frame contains a number of data slots (including function arguments, local variables, and temporaries), as well as a return address. A section of a stack is illustrated in Figure 1. As usual, the stack is shown growing downwards. In the figure, the function F has called the function G; thus, the return address position in G's frame will contain a location somewhere inside the code of F. In fact, the return address found in G's frame uniquely identifies the point in the program from which G was called, and therefore also determines the layout of the frame above its own. The garbage collector uses this property to "parse" the stack. When the program is compiled, the compiler emits type information that is collected by the runtime system into a GC table, which is a mapping from return addresses (iden- tifying function call sites) to information about the stack frame of the function containing the call site. When the collector begins looking for roots, the newest frame on the stack is that of the collector itself; the return address in this frame can be looked up in the GC table to find a description of the next frame, which belongs to the untrusted program. The collector then moves through the stack, performing the following steps for each frame: 1. Using the return address from below the frame being examined, find the GC table entry that describes this frame. 2. Using this GC table entry, determine the following information . The locations of pointers in the current frame. These are roots. . The location of the return address within the current frame. . The size of the current frame. 3. Using this information, find the start of the next frame and look up its GC table entry. These steps are repeated until the base of the stack is reached. Clearly, a correct GC table is essential for the operation of the garbage collector. An incorrect value in the table could lead to a variety of errors, from a single root pointer being ignored to derailment of the entire stack-parsing pro- cess. Put another way, it is crucial that the program structure its use of the stack consistently with the frame descriptions in the GC table. In this paper, we will present a language in which the stack's layout can be precisely controlled, giving us the ability to guarantee that the structure of the stack during collection will be consistent with the collector's expectations. 2 A Language With a GC Interface The main goal of this paper is to describe a type system in which the shape of the program stack can be made to fit the pattern expected by the garbage collector; we must therefore have a language in which stack manipulation is explicit and which is expressive enough to describe the stack in very precise terms. In this section, we begin to describe our language, which we call LGC. We start with a simple core language we call LGC - , which is a simple stack-based language that does not have the sophisticated type constructs to support garbage collection; we will then discuss the refinements necessary to enforce compliance with a GC table. The syntax and typing rules for full LGC are given in Appendix A. 2.1 The Core Language The syntax of LGC - is given in Figure 2. The language is essentially a polymorphic #-calculus with integers, booleans, tuples and sum types, plus a stack that is handled much the same way as in stack-based typed assembly language (STAL) [5]. The details of the language that do not directly relate to garbage collection are not particularly important for our purposes-indeed, there are many possible language designs that would work equally well-and so we will only discuss those aspects briefly here. Here and throughout the paper, we consider expressions that di#er only in the names of bound variables to be identical, and we denote by E[E1 , . , En/X1 , . , the result of the simultaneous capture-avoiding substitution of E1 through En for the variables X1 through Xn in E. Programs An LGC - program consists of a sequence of mutually recursive code block definitions, followed by an ex- pression. Each block has the form #; sp:#).e, indicating that it must be instantiated with some number and kind of type constructor arguments specified by #, and then may be invoked whenever the stack has type #; invoking the block results in the evaluation of e. Notice that no data other than the stack itself is passed into a block; this means that Kinds k ::= T | ST Constructors c, # | int | bool | code(# 0 | #1 - #n | #1 | null | Values v ::= x | n | b | # | v[c] | pack #c, v# as # Expressions e ::= halt v | jump v | if v then e1 else e2 | case v of inj 1 x1 # e1 | - | inj n xn # en | let d in e Declarations d ::= Blocks B ::= #; sp:#).e Programs P ::= letcode Type Contexts #:k Contexts #, x:# Memory Types # Figure 2: Syntax of LGC - all function arguments and results must be passed on the stack. The return address in a function call must also be passed on the stack, leading to a continuation-passing style for programs. Also, because all the code blocks in a program appear at the top level, programs must undergo closure conversion before translation into LGC - . Expressions The body of each block is an expression. The expressions in LGC - include a halt instruction which stops the computation, a jump instruction which takes a code label and transfers control to the corresponding block, an if- then-else construct, case analysis on sums and a form of let-binding that performs one operation, possibly binding the result to a variable, and continues with another expres- sion. The bindings that may occur in a let are a simple value binding, arithmetic operations injection into sum types allocation of tuples projection from tuples unpacking of values of existential type (#, which binds a new constructor variable # in addition to the variable x) and stack operations. The stack operations are reading writing (sp(i) := v), pushing of a value onto the stack (push v) and popping of i values o# the stack (pop i). The syntactic values in our language are variables (x), numerals (n), boolean constants (b), code labels (#), instantiations of polymorphic values with constructors (v[c]), and packages containing a constructor and a value (pack #c, v# as # ), which have existential type. Types and Kinds Our type theory has two kinds, T and ST, which classify constructors. Constructors of kind T are called types, and describe values; constructors of kind ST are called stack types and describe the stack. The constructors themselves include constructor variables (#), the base types int and bool, code label types (code(# 0), n-ary products (#1 - #n ) and sums (#1 types (#:k.# ), the empty stack type null and non-empty stack types of the form #. (In STAL one writes these as nil and #, respectively, but in LGC we prefer to use this ML-like list notation for actual lists of constructors.) The metavariable c will be used to range over all constructors; we will also use the names # and # when we intend that the constructor being named is a type or a stack type, respec- tively. We will extend the kind and constructor levels of the type system later in the paper in order to more precisely describe the shape of the stack. Static Semantics The typing rules for this simple language are generally the expected ones. Due to space considera- tions, we will not present them all here; they are generally similar to those for the full LGC language, whose rules are in Appendix A. We will, however, discuss some of the typing rules before turning to examine how programs in this language interact with a garbage collector. One of the simplest typing rules for expressions in the language is the one for the unconditional jump instruction: This rule states that it is legal to jump to a fully-instantiated code pointer (that is, one that does not expect any more constructor arguments) provided the stack type # expected by the code pointer is the same as the current stack type. To jump to a polymorphic code pointer, one must first instantiate it by applying it the appropriate number and kind of type arguments. The rule for doing this is the following: Another simple typing rule is the one for binding a value to a variable: Read algorithmically, this rule can be understood as follows: To check that the expression let in e is valid, first find the type # of the value v; then check that e is valid under the assumption that x has type # . 2.2 Requirements for Garbage Collection For the goal of certifying interaction with a garbage collec- tor, LGC - is unacceptably simplistic. In fact, the syntax and typing rules we have discussed so far appear to ignore the collector completely. In this section, we begin to identify the specific shortcomings of the language; once we have done this, the remainder of the paper will be devoted to adding the necessary refinements to the language, resulting in the full LGC type system. As we have already explained, if a program in our language is to work properly with a garbage collector, the collector must be able to find the roots whenever it is invoked. In practice, the garbage collector is usually invoked when a program attempts to create a new object in the heap and there is insu#cient space available. The expressions in our language that perform allocation are tuple formation and injection into sum types; this means that the garbage collector may need to be able to find the root set during evaluation of an expression of the form let or let in e. (We will only discuss tuples, since the modifications necessary for sums are exactly analogous.) A na-ve version of the typing rule for tuple allocation would be the following: #, x:#1 - #n ); sp:# e There are two main changes that must be made to this rule. First, we must force all the roots to be on the stack where they can be found by the collector; and second, we must force the stack to have a structure the collector can parse. The first problem stems from the fact that we have variables in our language that are not stack allocated, but we want to assume for the sake of simplicity that the garbage collector scans only the stack when looking for roots. A free occurrence of a variable y in the expression e above could denote a pointer; that pointer could be used in the evaluation of e, but if there is no copy of that pointer on the stack, the garbage collector may not identify it as live. The solution is to force the program to "dump" the contents of all its variables to the stack whenever a collection might occur. To accomplish this, we require the continuation e to be closed except for the result x of the allocation. The rule now looks like this: Note that a more realistic abstract machine would have registers instead of variables; in order to support GC table certification on such a machine we would have to apply the techniques we discuss in the remainder of this paper to the register file as well as the stack. This seems straightforward, but for the sake of simplicity we will limit our discussion in this paper to a collector that can only find roots in the stack. The second modification that must be made to the allocation rule is significantly more di#cult to formulate. In fact, the rest of this paper is devoted to adding a single additional premise to the rule, namely one that stipulates that the stack type # is parsable. That is, we must describe the structure that the stack must have in order to be scanned by the collector, and express that structure in a way that can be enforced by the type system. The type system of LGC - is not up to this task, so before continuing we must endow it with the expressive power to meet our needs. 2.3 Enriching the Constructor Language In order to be able to give a typing constraint in the allocation rule that precisely describes the required structure of the stack, we must enrich the constructor level of our lan- guage. For this purpose, we add a number of constructs from Kinds k ::= - | j | k1 # k2 | k1 - k2 | k1 | -j.k | 1 Constructors c ::= - | #:k.c | c1 c2 | #c1 , c2 # i c | | | # | unit | void | -(c) | +(c) Figure 3: Kinds and Constructors from LX Crary and Weirich's LX type theory [3]. These additions to our language are shown in Figure 3. In addition to function spaces, products and sums over kinds (k1 # k2 , k1 - k2 , k1 +k2 ), LX provides inductive kinds -j.k, where j is a kind variable that may appear in positive positions within k. At the type constructor level, we change the syntax of product and sum types to -(c) and +(c), where in each case c is a constructor of kind -j.1+T-j and represents a list of types. To keep the notation for LGC simple, we allow the syntax from LGC - to serve as shorthand, defined as follows: (The analogous notation is used for sums.) Finally, we have a kind 1 whose sole element is the constructor #, and we add the types unit and void to the language. The type unit has the sole element #, while the type void contains no values. The introduction forms and elimination forms for arrows, sums and products at the constructor level are the usual ones; inductive kinds are introduced with a fold construct and eliminated with primitive recursion constructors of the form pr(j, #:k, #:j # k # , c). If well-formed, this constructor will be a function of kind -j.k # k # [-j.k/j]; c is the body of the function, in which # may appear as the parameter and # is the name of the function itself to be used for recursive calls. For example, if we define the kind senting the natural numbers), then we can define the function iter as follows: The constructor iter is a function taking a function from types to types, a type and a natural number, and returning the result of iterating the function on the given type the specified number of times. Clearly, the pr notation is somewhat unwieldy to read and write, so we will use an ML-like notation for working in the LX constructor language. We will, for many purposes, combine the notions of inductive and sum kind and define datakinds, akin to ML's datatypes. For example, we could write the definition of N above as follows: datakind The function iter would be more readably expressed in ML curried function notatin this way: | iter # (Succ #(iter #) We will often write functions in this style, being careful only to write functions that can be expressed in the primitive recursion notation of LX. To further simplify the presenta- tion, we will also use the familiar ML constructors list and option to stand for the analogous datakinds. 2.4 Approaching Garbage Collection The language LX was originally designed for intensional type analysis. The basic methodology was to define a datakind of analyzable constructors, which we will call TR (for "type representation"), a function interp : TR # T to turn a constructor representation (suitable for analysis) into an actual type (suitable for adorning a variable binding), and a to turn a constructor into the type of a value that represented it at run time. In addition to explaining the somewhat mysterious operation of run-time type analysis in more primitive terms, this had the e#ect of isolating a particular subset of types for analysis: only those types that appeared in the image of the interp mapping could be passed or analyzed at run time. For garbage collection, we want to do something similar: we want to isolate the set of stack types that are structured such that the collector can parse the stack using the algorithm outlined in Section 1.1. We can then add the appropriate stack structure condition to the allocation rule by asserting that the current stack type lies in that set. To do this, the remainder of this paper will define the following LX objects: 1. A datakind SD (for "stack descriptor"), whose elements will be passed around in our programs in place of stack types. 2. A datakind DD (for "data descriptor"), whose elements will be static representations of GC tables. Every program in our language will designate one particular constructor to be its static GC table, or SGCT. This constructor SGCT will have kind DD . 3. A constructor interpS : DD # SD # ST, that will turn a stack descriptor into a stack type provided that it only uses stack frames whose shapes are determined by a particular static GC table. We will be careful to write interpS such that for any constructors s : SD stacks of type interpS SGCT s will always be parsable. Once we have definitions for all of these, ensuring that the stack is parsable for garbage collection is simple: if the current stack type is #, we need to require that there exist some constructor s : SD such that course, we do not want the type-checker to have to guess the appropriate s, so we change the syntax slightly to make it the programmer's responsibility. The next version of our allocation rule (modulo the definitions of these new LX ex- pressions) is: There will be one more development of this rule in Section 3.1. In order for this expression of the interface to the garbage collector in terms of SGCT to guarantee correct programs, we must be sure that the actual data structure used as the GC table agrees with its static representation. While LX is capable of expressing a type for the GC table that guarantees this, we have chosen a simpler approach. Rather than forcing the program to provide its own GC table in both "static" constructor form and "dynamic" value form, we assume that the type-checker in our certified code system transforms the static GC table into a real GC table and provides the latter to the runtime system before the program starts. Thus, we consider the generation of the GC table itself from the static representation to be part of the trusted computing base. The remaining sections of this paper will present the definitions of the kinds SD and DD , and will describe the behavior of interpS and the auxiliary functions needed to define it. The definition of interpS is nontrivial and involves an unusual amount of programming at the type constructor level of the language. The complete code for the special kinds and constructors used in our GC interface can be found in Appendix B. 3 Describing the Stack Since the collector requires the stack to be structured as a sequence of frames, our LX representation of the stack type will be essentially a list of frame descriptors, which we will represent by constructors of another datakind, called FD. A frame descriptor must allow two major operations: (1) since lists of descriptors are passed around in the program instead of stack types, it must be possible to interpret a descriptor to get the partial stack type it represents, and (2) since we are structuring the stack this way so as to ensure agreement with a GC table, it must be possible to check a descriptor against an entry in the table. The individual entries in the static representation of the GC table will be constructors of a kind called FT , for frame template, that we will also define shortly. 3.1 Labels and Singletons As we have mentioned before, a key property of the stack layout required by the garbage collector is that the return address of one frame determines, via the GC table, the expected shape of the next older frame. As a result, in order for our constraint on the stack's type before a collection to guarantee proper functioning of the collector, we must ensure that the value stored in the return address position in each frame corresponds, via the static GC table, to the type of the next frame. To make this happen, we must be able to reason about labels-i.e., pointers to code-at the constructor level of our language. We therefore lift label literals from the value level of the language to the constructor level, and add a new primitive kind, L, to classify them. In addition, we add a construct for forming singleton types from labels. Using this construct, we will be able to force the return address stored in a stack frame to have precisely the value it must in order to correctly predict the shape of the next frame on the stack. The syntax and typing rules for labels and singletons are shown in Figure 4. If c is the label of a code block of type # , #) is the type that contains only instances of c. In order to make use of values of singleton type, we introduce Kinds k ::= - | L Constructors c ::= - | # | Figure 4: Syntax and Typing for Labels the coercion blur, which forgets the identity of a singleton value, yielding a value which is an appropriate operand to a jump instruction. Since values of singleton type are code labels, which are usually polymorphic, we have found it necessary to add a way to apply a label to a constructor argument while maintaining its singleton type; this is accomplished by writing v{c}. The sensitivity of the garbage collector to labels found in the stack raises another issue that must be addressed in the typing rule for allocation. In order for the collector to begin the process of scanning the stack, it must be able to find the GC table entry for its caller's frame (i.e., the newest program frame). It is therefore necessary to associate a label with each allocation site, and require that the first frame descriptor in the stack descriptor correspond to that label. (Since this label is intended to denote the return address of the call to the garbage collector, we must assume that all such labels in the program are distinct.) We also define a function retlab of kind DD # SD # L that extracts the label of the newest frame of the given stack descriptor; making one final change to the syntax of allocation to include a label, the final typing rule is as follows: 3.2 Stack Descriptors The general structure of the kind SD is given in Figure 5, along with an illustration of the interpretation of a constructor of this kind into a stack type by interpS . (The validity checking performed by interpS will be discussed in the next section.) As the kind definitions show, a stack descriptor is either "empty"-in which case it carries the label identifying the return address of the top frame-or it consists of a frame descriptor and a descriptor for the rest of the stack. A frame descriptor consists of a label, which identifies the point in the program that "owns" the frame 1 , the re- That is, the return address of the currently pending function call executed by the function instance that created the frame. kind kind list - T - Slot list datakind Base of L | Cons of FD - SD Figure 5: Structure and Interpretation of Stack Descriptors turn type of the function whose frame it is, and two lists of slots. The kind Slot of slots is not defined here; we address its definition in the next section. A slot describes a single location on the stack; a constructor of kind Slot must support (1) interpretation into a type in the fashion indicated by the arrows in the illustration, and (2) examination to determine what the specification of this slot in the GC table ought to be. The first list of slots in a frame descriptor corresponds to the slots that come before the return address, the second list describes the slots after the return address. As shown in the diagram, interpS builds each frame of the stack type by interpreting the slots into types, and constructs the return address using the function's return type as specified in the frame descriptor, forming a singleton with the label associated with the next frame. The code for interpS is in Appendix B. In keeping with the usual LX methodology, it is our intention that LGC programs pass constructors of kind SD where programs in a GC-ignorant language would pass stack types. For example, the code of a function that takes two integers and returns a boolean (such as a comparison func- might have the type: Unfortunately, this type does not quite capture the relationship between the return address (of type code(-; and the caller's frame (which is hidden "inside" #). A code block with this type will be unable to perform any allocation, because its return address does not have a singleton type. In order to give the return address a singleton type, we must extract the label from the calling frame using the function retlab mentioned in Section 3.1. We then use the following more accurate type in place of the above: A more detailed example of the use of stack descriptors (but with no allocation) is shown in Figure 6. The most interesting part of the function shown in the figure is the recursive call. If we let then the return address of the recursive call to factcode, factreturn{#}, has type specifies a slot of type int, and define code(-;int# (interpS SGCT #))->0) if b then return address off stack pop 2 in ; clear away our frame call blur(ra) ; return else push x in ; push argument push factreturn{#} in ; push return address call int# (interpS SGCT (Cons(factframe,#))) ) . pop 3 in ; clear away our frame push result in call blur(ra) ; return Figure Using Stack and Frame Descriptors if we observe that retlab SGCT (Cons(factframe, factreturn, then the type of the address in the call instruction is: To see that the stack type in this code type matches the current stack at the call site, observe that the first value on the stack is the return address, whose type we have already seen to be equal to the one required for the call. The second value on the stack is the argument to the recursive call, which has type int. Finally, #1 describes the function's own frame and the pre-existing stack. In particular, #r# int#0 , where #r stands for the type of the original return address and is the unknown base portion of the stack. Checking Frame Validity In addition to enforcing the property that the stack is a sequence of frames, the condition must also guarantee that the frames themselves are correctly described by the GC table. To accomplish this, we ensure that the equality can only hold if the frame descriptors in s are consistent with the information about them contained in SGCT , the GC table's constructor-level representation. Since the actual GC table is a mapping from return addresses to frame layout information, it makes sense to structure SGCT as a mapping from labels to frame layouts as well. The basic structure of DD , the kind of SGCT , is given in Figure 7. The static GC table is structured as a list of pairs, each consisting of a label and a constructor of kind FT , which stands for frame template. A frame template is essentially an LX constructor representation of the information in a real GC table entry; it consists of two lists of table kind kind list - TSlot list datakind | ConsDD of L - FT - DD Figure 7: Structure of the Static GC Table slots (constructors of kind TSlot), which correspond to the two lists of slots in a frame descriptor. Checking a frame descriptor for validity therefore consists of looking up the label from the frame descriptor in SGCT and checking each of the slots in the FD against the table slots in the FT . We will give definitions for Slot and TSlot and discuss this consistency checking shortly. First, however, we must make one final addition to LGC. In order to be able to write the all-important lookupDD function that finds the frame template for a given label, our constructor language must be able to compare labels for equality. The syntax and semantics of label equality at the constructor level are given in Figure 8. The constructor definitionally equal to c3 if the labels c1 and c2 are the same, c4 if they are not the same. Note that the reduction rules for ifeq only apply when c1 and c2 are label literals, so the equational theory remains well-behaved. With these constructs in place, lookupDD is easy to write using primitive recursion. 4.1 Monomorphic Programs In this section we will give definitions of Slot and TSlot that allow "monomorphic" programs to be written in LGC. By Constructors Figure 8: Label Equality datakind KnownSlot of TR datakind Trace of 1 | NoTrace of 1 True of 1 | False of 1 Figure 9: Monomorphic Slots and Table Slots "monomorphic" we here mean programs in which all the values a function places in its stack frame have types that are known at compile time. 2 If the type of every value in a function's stack frame has a non-variable type, then it can be determined statically whether each slot in the frame contains a pointer that must be traced. More importantly, the traceability of any slot will be the same for every instance of the function. Consequently, the GC table only needs one bit for each slot, and all that needs to be checked at each allocation site is whether the types of all the slots have the traceabilities specified in the table. The definitions for Slot and TSlot are given in Figure 9 along with the kinds of two constructor functions we will use to check frames. In the case of monomorphic code, a slot is simply a type representation in the usual style of LX; a table slot is simply a flag indicating whether a location is traceable or not. We will not discuss the definition of TR further, as any representation of types that can be coded in LX will do for the purposes of this paper. We do, how- ever, assume the existence of the usual interpretation and representation functions as usual for LX, interp turns a type representation into the type it represents, and R turns a type representation into the type of the value representing that type. The stack interpretation function interpS must make use of interp to translate a slot (which is really a type representation) into a type; we will use R in the next section, when we cover polymorphic programs. The function checkFD checks that a frame descriptor is valid with respect to the static GC table. First, it must look up the frame descriptor's label to get the corresponding frame template if there is one. If there is no frame template for that label, the descriptor is rejected as invalid. The 2 Note that no nontrivial program in a stack-based language can really be totally monomorphic, since every function must be parametric in the stack type so that it can be called at any time. con con Figure 10: Static GC Table for Factorial Example function Slot2TSlot simply decides whether a given type representation is traceable; given a frame template, checkFD applies the Slot2TSlot to each of the slots in the frame descriptor and uses eqTSlot to determine whether the resulting TSlot matches the corresponding one in the frame template. To ensure that the stack can be parsed by the garbage collector, the interpretation function interpS calls checkFD on each of the frame descriptors it sees. This portion of the code of interpS is essentially the following: (Cons #fd , case checkFD SGCT fd of True => . | False => void# null In the case where the frame descriptor is not valid with respect to SGCT , the body of interpS reduces to void# null, which is an unsatisfiable stack type since the type void is uninhabited. If the stack type is interpS SGCT s at some reachable program point, then obviously interpS SGCT s must be inhabited. Therefore, reduction of this definition must not have taken the False branch, so it follows that all the frame descriptors in s must be valid in the sense of checkFD . The static GC table for the factorial example from Figure 6 is shown in Figure 10. Of course, this is a bit unrealistic since we have shown a "program" with only one function call site, so as a result there is only one entry in the GC table. If we were to add anything to the factorial program, such as a main program body that calls the function factcode, the GC table and its static representation would have to be augmented with descriptions of any new call sites we introduced 4.2 Polymorphic Programs It is a little more di#cult to adapt LGC to certifying polymorphic programs, because in such programs a function may have arguments or local variables whose types are di#erent each time the function is called. The TILT garbage collector handles such stack locations by requiring that, in any instance of a polymorphic frame, a value representing the type of each of these slots is available. The slot in the GC table corresponding to a location whose type is statically un- known, rather than directly giving traceability information, tells the collector where the representation can be found. TILT allows some flexibility in where the representations are stored: they can be on the stack, in a heap-allocated record with a pointer to the record on the stack, or in global storage. For our purposes, we will assume a simple, flat ar- rangment in which the type representations for a frame are all stored in that frame. The new definitions of Slot and TSlot to account for polymorphism are shown in Figure 11. We also slightly modify the definition of FD , the kind of frame descriptors. Any frame in a polymorphic program will in general be parametric in some number of "unknown" types; since frame descriptors must be interpretable to give the type of the stack, datakind datakind KnownSlot of TR | VarSlot of N | RepSlot of N kind list - T - Slot list - TR list datakind Trace of 1 | NoTrace of 1 | Var of N | Rep of N list # N # TR Figure 11: Frames and Tables for Polymorphic Programs a frame descriptor represents a single instance of a polymorphic frame. Therefore, the version of FD for polymorphic programs includes a list of type representations that "instan- tiate" the frame descriptor by providing representations of all the unknown types of values in the frame. Each individual slot in a frame descriptor may now take one of three forms: it may be a slot whose type is known at compile time, as before; or it may be a slot whose type is one of the unknown types associated with the frame; or it may be the slot that holds the representation of one of those types. These three possibilities are reflected in the new definition of Slot ; in the case of unknown-type and representation slots, the frame descriptor will carry a natural number indicating which of the type parameters gives the type of, or is represented by, the slot. Similarly, there are now four choices for a slot in the static GC table. A slot may be known to be traceable; it may be known to be un- traceable; it may contain a value of variable type; or it may contain a representation. These four possibilites correspond to the arms of the new TSlot datakind. The interpretation of slots into types is now a bit more complicated as well; for slots of known type the operation is unchanged, but for variable and representation slots interpS must look up the appropriate type representation in the list given by the frame descriptor. Once this representation is obtained, variable slots are turned into types using interp as before, while representation slots are turned into types using the R function described before. We therefore write the function interpsl , which interprets a single slot given the list of type representations from the frame descriptor. fun interpsl trs (KnownSlot | interpsl trs (VarSlot (case nth trs n of SOME tr => interp tr | NONE => void) | interpsl trs (RepSlot (case nth trs n of SOME tr => R tr | NONE => void) Notice that slots specifying invalid indices into the list of representations are given type void, to ensure that the frame described by the invalid descriptor cannot occur at run time. In addition to the possibility of bad indices in variable and representation slots, there is another new way in which a frame descriptor may be invalid: the definition of FD allows a frame to contain a VarSlot for which it does not contain a corresponding RepSlot . Fortunately, the property that the set of indices given in VarSlot 's is contained in the set of indices given in RepSlot 's is easy to check primitive recur- sively. This responsibility falls to the polymorphic version of the function checkFD . An example of a simple polymorphic function in LGC is shown in Figure 12. The code in this figure defines a function which, for any type representation #, takes a value of type interp # and boxes it-that is, allocates and returns a one-field tuple of type -[interp #] containing that value. The stack descriptor provided at the allocation site adds a descriptor for the current frame to the pre-existing stack descriptor #. This new frame descriptor contains two slots corresponding to the two values (other than the return address) that make up the function's stack frame: the first, RepSlot Zero, describes the run-time representation of #, which itself has type R#; the second, VarSlot Zero, describes the argument to the function, which has type 5 Expressiveness In order to experiment with the expressive power of LGC, we have implemented a type-checker for the language, including a "prelude" of constructor and kind definitions giving the meanings of TR, SD , DD, interpS and so on. We have also implemented a translation from a GC-ignorant source language into LGC, demonstrating that LGC is expressive enough to form the basis of a target of a general-purpose compiler. The syntax of the source language is shown in Figure 13. Its design was driven solely by the goal of removing all explicit GC-related constructs while enabling a straightforward translation into LGC. We will briefly mention some of the issues that shaped the design of the source language, since they highlight the unusual properties of a language designed with a garbage collection interface in mind. Implicit Stack Operations Since the garbage collector requires the stack to have a certain structure, it would be very inconvenient to allow the source program unrestricted use of stack manipulation operations. Therefore, we chose to remove the stack almost completely from the syntax of the source language. Source-level functions accept arguments and return results in the usual manner; the translation to LGC takes care of turning parameter-passing into stack ma- nipulation. In addition, since return addresses play such a critical role in scanning the stack, we cannot allow source programs to manipulate those either. As a result, the source language abandons continuation-passing style for a more familiar return instruction (which we merge with the halt instruction since their semantics are similar). Locals Since the source program cannot manipulate the stack, support for storing intermediate results there must be built into the language. A somewhat unfortunate consequence is that all decisions about what will and will not be stored on the stack must have been made before translation into LGC. If the design of LGC were to be applied to a compiler targeting typed assembly language, this would correspond to the fact that register allocation must be completed before generation of GC tables can begin. In order to use stack space for local variables and temporary storage, each code block in a source program begins with lalloc i, which indicates that the block wishes to allocate i mutable local variables on the stack. Special forms of declarations at the expression level provide access to these locals. and pop 3 push cell call blur(ra) Figure 12: Polymorphic Allocation Example Types # | int | bool | ns | code(#e ; #1 , . , #n ) #) | #1 , . , #n) #1 - #n Expressions e ::= return v | if v then e1 else e2 | let d in e Declarations d ::= Blocks B ::= #xe :#e .#1 , . , #n) #.lalloc(i).e Programs P ::= letcode Type Contexts # Contexts #, x:# Memory Types # Figure 13: Syntax of the Source Language Closures Since LGC requires all code blocks to be closed and hoisted to the top level, a translation from a higher-level language in which functions may be nested must perform closure conversion as part of translation into LGC. Since the interface of the garbage collector seems to have little impact on the closure conversion transformation itself, we chose to keep the source-to-LGC translation simple by assuming closure conversion had already been performed. Therefore, the source language also requires all code blocks to be at the top level. However, we do not include existential types in the source language, as providing representations for all the types hidden with existentials would add to the bloat associated with the translation. Since many of these representations turn out to be unnecessary, we find it more economical to introduce existential types as closures at the same time as the translation to LGC. We include in the source language special types of closures (#1 , . , #n) #) and operations for creating them. Every code block expects a special argument, which is the environment of the closure; code pointers are made into functions using the closure operation, which packs a code value together with an environment 6 Conclusion We have presented a language in whose type system the interface to a modern high-performance garbage collector can be expressed. In so doing, we have demonstrated that code certification is indeed compatible with the use of sophisti- cated, accurate garbage collection technology. We have described the interface of one such collector in our language, and implemented a prototype type-preserving translation from a GC-ignorant source language into our target language The alert reader will have noticed the absence of an operational semantics or safety proof in this paper. An operational semantics is completely straightforward, except that the two rules that perform heap allocation must each have an additional side condition requiring that the stack be parsable. A type safety proof is boilerplate, based on the proof for LX by Crary and Weirich [3], except that in the cases of injection and allocation it must be shown that the typing conditions on the stack imply that it is parsable. However, it is not clear how to give a formal definition of parsability that is any simpler than our specification in Appendix B, so such a proof would be unenlightening. The interface of our garbage collector is subtle, and expressing this interface in a type system requires a fair amount of programming at the level of type constructors. Type-checking programs in this language, in turn, involves deciding equivalences of a lot of large constructors that are many reduction steps away from normal form. Our prototype type-checker for LGC decides equivalence using a straightforward, recursive weak-head-normalize and compare algorithm, and while our implementation is not yet serious enough to reach any conclusions about e#ciency, preliminary results indicate the amount of work involved is not unreasonably large. This paper has examined a garbage collector interface based on the one used by the TILT/ML compiler, but considerably simpler. However, we believe that what we have described is su#cient to handle most of the issues that arise in a real collector. For instance, it does not appear di#- cult to account for registers (which TILT treats essentially the same as stack slots) or global variables (whose types are fixed). Our proof-of-concept implementation does not address the possibility of translating higher-order polymorphism into LGC. Higher-order polymorphism arises in the setting of compiling the full ML language, in which abstract parameterized types can occur. TILT is able to use a similar GC table format to the one we have described, even for higher-order polymorphic programs; it performs a program transformation called reification to introduce variable bindings for any types of registers or stack locations that are unknown at compile time. We believe that by performing something similar to reification we can translate programs with higher-order polymorphism into LGC, but this remains a topic for future work. --R Scalable Real-Time Parallel Garbage Collection for Symmetric Multiprocessors A parallel Flexible type anal- ysis Principled scavenging. From System F to typed assembly language. Safe, untrusted agents using proof-carrying code Managing Memory With Types. --TR Proof-carrying code Flexible type analysis From system F to typed assembly language Type-preserving garbage collectors Principled scavenging A parallel, real-time garbage collector Managing memory with types --CTR Feng , Zhong Shao , Alexander Vaynberg , Sen Xiang , Zhaozhong Ni, Modular verification of assembly code with stack-based control abstractions, ACM SIGPLAN Notices, v.41 n.6, June 2006 Andrew McCreight , Zhong Shao , Chunxiao Lin , Long Li, A general framework for certifying garbage collectors and their mutators, ACM SIGPLAN Notices, v.42 n.6, June 2007

typed compilation;garbage collection;certified code;type systems

604198

Context-specific sign-propagation in qualitative probabilistic networks.

Qualitative probabilistic networks are qualitative abstractions of probabilistic networks, summarising probabilistic influences by qualitative signs. As qualitative networks model influences at the level of variables, knowledge about probabilistic influences that hold only for specific values cannot be expressed. The results computed from a qualitative network, as a consequence, can be weaker than strictly necessary and may in fact be rather uninformative. We extend the basic formalism of qualitative probabilistic networks by providing for the inclusion of context-specific information about influences and show that exploiting this information upon reasoning has the ability to forestall unnecessarily weak results.

Introduction Qualitative probabilistic networks are qualitative abstractions of probabilistic networks [Wellman, 1990] , introduced for probabilistic reasoning in a qualitative way. A qualitative probabilistic network encodes statistical variables and the probabilistic relationships between them in a directed acyclic graph. Each node A in this digraph represents a variable. An a probabilistic influence of the variable A on the probability distribution of the variable B; the influence is summarised by a qualitative sign indicating the direction of shift in B's distribution. For probabilistic inference with a qualitative network, an efficient algorithm, based upon the idea of propagating and combining signs, is available [Druzdzel & Henrion, 1993 ] . Qualitative probabilistic networks can play an important role in the construction of probabilistic networks for real-life application domains. While constructing the digraph of a probabilistic network is doable, the assessment of all probabilities required is a much harder task and is only performed when the network's digraph is considered robust. By eliciting signs from domain experts, the obtained qualitative probabilistic network can be used to study and validate the reasoning behaviour of the network prior to probability assessment; the signs can further be used as constraints on the probabilities to be assessed [Druzdzel & Van der Gaag, 1995 ] . To be able to thus exploit a qualitative probabilistic network, it This work was partly funded by the EPSRC under grant should capture as much qualitative information from the application domain as possible. In this paper, we propose an extension to the basic formalism of qualitative networks to enhance its expressive power for this purpose. Probabilistic networks provide, by means of their digraph, for a qualitative representation of the conditional independences that are embedded in a joint probability distribu- tion. The digraph in essence captures independences between nodes, that is, it models independences that hold for all values of the associated variables. The independences that hold only for specific values are not represented in the digraph but are captured instead by the conditional probabilities associated with the nodes in the network. Knowledge of these latter independences allows further decomposition of conditional probabilities and can be exploited to speed up inference. For this purpose, a notion of context-specific independence was introduced for probabilistic networks to explicitly capture independences that hold only for specific values of variables [Boutilier et al., 1996; Zhang & Poole, 1999 ] . A qualitative probabilistic network equally captures independences between variables by means of its digraph. Since its qualitative influences pertain to variables as well, independences that hold only for specific values of the variables involved cannot be represented. In fact, qualitative influences implicitly hide such context-specific independences: if the influence of a variable A on a variable B is positive in one context, that is, for one combination of values for some other variables, and zero in all other contexts - indicating independence - then the influence is captured by a positive sign. Also, positive and negative influences may be hidden: if a variable A has a positive influence on a variable B in some context and a negative influence in another context, then the influence of A on B is modelled as being ambiguous. As context-specific independences basically are qualitative by nature, we feel that they can and should be captured explicitly in a qualitative probabilistic network. For this purpose, we introduce a notion of context-specific sign. We extend the basic formalism of qualitative networks by providing for the inclusion of context-specific information about influences and show that exploiting this information upon inference can prevent unnecessarily weak results. The paper is organised as follows. In Section 2, we provide some preliminaries concerning qualitative probabilistic networks. We present two examples of the type of information that can be hidden in qualitative influences, in Section 3. We present our extended formalism and associated algorithm for exploiting context-specific information in Section 4. In Section 5, we discuss the context-specific information that is hidden in the qualitative abstractions of two real-life probabilistic networks. In Section 6, we briefly show that context-specific information can also be incorporated in qualitative probabilistic networks that include a qualitative notion of strength of influences. The paper ends with some concluding observations in Section 7. Qualitative probabilistic networks A qualitative probabilistic network models statistical variables as nodes in its digraph; from now on, we use the terms variable and node interchangeably. We assume, without loss of generality, that all variables are binary, using a and a to indicate the values true and false for variable A, respectively. A qualitative network further associates with its digraph a set of qualitative influences, describing probabilistic relationships between the variables [Wellman, 1990] . A qualitative influence associated with an arc A ! B expresses how the values of node A influence the probabilities of the values of node B. A positive qualitative influence, for example, of A on B, denoted that observing higher values for node A makes higher values for node B more likely, regardless of any other influences on B, that is, for any combination of values x for the set X of parents of B other than A. The '+' in S + (A; B) is termed the influence's sign. A negative qualitative influence S , and a zero qualitative influence S 0 , are defined analogously. If the influence of node A on node B is non-monotonic or unknown, we say that it is ambiguous, denoted S ? (A; B). The set of influences of a qualitative probabilistic network exhibits various properties [Wellman, 1990] . The symmetry property states that, if S - (A; B), then also S - (B; A), ?g. The transitivity property asserts that a sequence of qualitative influences along a chain that specifies at most one incoming arc per node, combine into a single influence with the -operator from Table 1. The composition property asserts that multiple influences between two nodes along parallel chains combine into a single influence with the -operator. Table 1: The - and -operators. A qualitative network further captures qualitative synergies between three or more nodes; for details we refer to [Druzdzel For inference with a qualitative network, an efficient algorithm is available [Druzdzel & Henrion, 1993 ] . The basic idea of the algorithm is to trace the effect of observing a node's value on the other nodes in the network by message passing between neighbouring nodes. For each node, a node sign is determined, indicating the direction of change in the node's probability distribution occasioned by the new observation given all previously observed node values. Initially, all node signs equal '0'. For the newly observed node, an appropriate sign is entered, that is, either a '+' for the observed value true or a ' ' for the value false. Each node receiving a message updates its node sign and subsequently sends a message to each neighbour whose sign needs updating. The sign of this message is the -product of the node's (new) sign and the sign of the influence it traverses. This process is repeated throughout the network, building on the properties of sym- metry, transitivity, and composition of influences. Since each node can change its sign at most twice, once from '0' to `+' or ' ', and then only to `?', the process visits each node at most twice and is therefore guaranteed to halt. 3 Context-independent signs Context-specific information cannot be represented explicitly in a qualitative probabilistic network, but is hidden in the net- work's qualitative influences. If, for example, the influence of a node A on a node B is positive for one combination of values for the set X of B's parents other than A, and zero for all other combinations of values for X , then the influence of A on B is positive by definition. The zero influences are hidden due to the fact that the inequality in the definition of qualitative influence is not strict. We present an example illustrating such hidden zeroes. R P Figure 1: The qualitative surgery network. Example 1 The qualitative network from Figure 1 represents a highly simplified fragment of knowledge in oncology; it pertains to the effects and complications to be expected from treatment of oesophageal cancer. Node L models the life expectancy of a patient after therapy; the value l indicates that the patient will survive for at least one year. Node T models the therapy instilled; we consider surgery, modelled by t, and no treatment, modelled by t, as the only alternatives. The effect to be attained from surgery is a radical resection of the oesophageal tumour, modelled by node R. After surgery a life-threatening pulmonary complication, modelled by node may result; the occurrence of this complication is heavily influenced by whether or not the patient is a smoker, modelled by node S. We consider the conditional probabilities from a quantified network representing the same knowledge. We would like to note that these probabilities serve illustrative purposes although not entirely unrealistic, they have not been specified by domain experts. The probability of attaining a radical resection upon surgery is Pr(r there can be no radical resection, we have Pr(r j t From these probabilities we have that node T indeed exerts a positive qualitative influence on node R. The probabilities of a pulmonary complication occurring and of a patient's life expectancy after therapy are, respectively, From the left table, we verify that both T and S exert a positive qualitative influence on node P . The fact that the influence of T on P is actually zero in the context of the value s for node S, is not apparent from the influence's sign. Note that this zero influence does not arise from the probabilities being zero, but rather from their having the same value. From the right table we verify that node R exerts a positive influence on node L; the qualitative influence of P on L is negative. The previous example shows that the level of representation detail of a qualitative network can result in information hid- ing. As a consequence, unnecessarily weak answers may result upon inference. For example, from the probabilities involved we know that performing surgery on a non-smoker has a positive influence on life expectancy. Due to the conflicting reasoning chains from T to L in the qualitative network, how- ever, entering the observation t for node T will result in a '?' for node L, indicating that the influence is unknown. We recall from the definition of qualitative influence that the sign of an influence of a node A on a node B is independent of the values for the set X of parents of B other than A. A '?' for the influence of A on B may therefore hide the information that node A has a positive influence on node B for some combination of values of X and a negative influence for another combination. If so, the ambiguous influence is non-monotonic in nature and can in fact be looked upon as specifying different signs for different contexts. We present an example to illustrate this observation. Figure 2: The qualitative cervical metastases network. Example 2 The qualitative network from Figure 2 represents another fragment of knowledge in oncology; it pertains to the metastasis of oesophageal cancer. Node L represents the location of the primary tumour that is known to be present in a patient's oesophagus; the value l models that the tumour resides in the lower two-third of the oesophagus and the value l expresses that the tumour is in the oesophagus' upper one- third. An oesophageal tumour upon growth typically gives rise to lymphatic metastases, the extent of which are captured by node M . The value of M indicates that just the local and regional lymph nodes are affected; m denotes that distant lymph nodes are affected. Which lymph nodes are local or regional and which are distant depends on the location of the tumour in the oesophagus. The lymph nodes in the neck, or cervix, for example, are regional for a tumour in the upper one-third of the oesophagus and distant otherwise. Node C represents the presence or absence of metastases in the cervical lymph nodes. We consider the conditional probabilities from a quantified network representing the same knowledge; once again, these probabilities serve illustrative purposes only. The probabilities of the presence of cervical metastases in a patient are Pr(c) l l From these probabilities we have that node L indeed has a negative influence on node C. The influence of node M on C, however, is non-monotonic: The non-monotonic influence hides a '+' for the value l of node L and a ' ' for the context l. From the two examples above, we observe that context-specific information about influences that is present in the conditional probabilities of a quantified network cannot be represented explicitly in a qualitative probabilistic network: upon abstracting the quantified network to the qualitative net- work, the information is effectively hidden. 4 Context-specificity and its exploitation The level of representation detail of a qualitative probabilistic network enforces influences to be independent of specific contexts. In this section we present an extension to the basic formalism of qualitative networks that allows for associating context-specific signs with qualitative influences. In Section 4.1, the extended formalism is introduced; in Section 4.2, we show, by means of the example networks from the previous section, that exploiting context-specific information can prevent unnecessarily weak results upon inference. 4.1 Context-specific signs Before introducing context-specific signs, we define a notion of context for qualitative networks. Let X be a set of nodes, called the context nodes. A context c X for X is a combination of values for a subset Y X of the set of context nodes. we say that the context is empty, denoted ; we say that the context is maximal. The set of all possible contexts for X is called the context set for X and is denoted CX . To compare different contexts for the same set of context nodes X , we use an ordering on contexts: for any two combinations of values c X and c 0 respectively, we say that c X and c 0 X specify the same combination of values for Y 0 . A context-specific sign now basically is a sign that may vary from context to context. It is defined as a function ?g from a context set CX to the set of basic signs, such that for any two contexts c X and c 0 with c X > c 0 X we have that, if -(c 0 0g. For abbreviation, we will write -(X) to denote the context-specific sign - that is defined on the context set CX . Note that the basic signs from regular qualitative networks can be looked upon as context-specific signs that are defined by a constant function. In our extended formalism of qualitative networks, we assign context-specific signs to influences. We say that a node A exerts a qualitative influence of sign -(X) on a node B, denoted is the set of parents of B other than A, iff for each context c X for X we have that combination of values c X y for X; such combination of values c X such combination of values c X Note that we take the set of parents of node B other than A for the set of context nodes; the definition is readily extended to apply to arbitrary sets of context nodes, however. Context-specific qualitative synergies can be defined analogously. A context-specific sign -(X) in essence has to specify a basic sign from f+; ; 0; ?g for each possible combination of values in the context set CX . From the definition of -(X), however, we have that it is not necessary to explicitly indicate a basic sign for every such context. For example, consider an influence of a node A on a node B with the set of context nodes Eg. Suppose that the sign -(X) of the influence is defined as The function -(X) is uniquely described by the signs of the smaller contexts whenever the larger contexts are assigned the same sign. The function is therefore fully specified by The sign-propagation algorithm for probabilistic inference with a qualitative network, as discussed in Section 2, is easily extended to handle context-specific signs. The extended algorithm propagates and combines basic signs only. Before a sign is propagated over an influence, it is investigated whether or not the influence's sign is context-specific. If so, the currently valid context is determined from the available observations and the basic sign specified for this context is propa- gated; if none of the context nodes have been observed, then the sign specified for the empty context is propagated. 4.2 Exploiting context-specific signs In Section 3 we presented two examples showing that the influences of a qualitative probabilistic network can hide context-specific information. Revealing this hidden information and exploiting it upon inference can be worthwhile. The information that an influence is zero for a certain context can be used, for example, to improve the runtime of the sign-propagation algorithm because propagation of a sign can be stopped as soon as a zero influence is encountered. More importantly, however, exploiting the information can prevent conflicting influences arising during inference. We illustrate this observation by means of an example. Example 3 We reconsider the qualitative surgery network from Figure 1. Suppose that a non-smoker is undergoing surgery. In the context of the observation s for node S, propagating the observation t for node T with the basic sign- propagation algorithm results in the sign '?' for node L: there is not enough information present in the network to compute a non-ambiguous sign from the two conflicting reasoning chains from T to L. We now extend the qualitative surgery network by assigning the context-specific sign -(S), defined by to the influence of node T on node P , that is, we explicitly include the information that non-smoking patients are not at risk for pulmonary complications after surgery. The thus extended network is shown in Figure 3(a). We now reconsider our non-smoking patient undergoing surgery. Propagating the observation t for node T with the extended sign- propagation algorithm in the context of s results in the sign (0 L: we find that surgery is likely to increase life expectancy for the patient. R P (a) (b) Figure 3: A hidden zero revealed, (a), and a non-monotonicity captured, (b), by a context-specific sign. In Section 3 we not only discussed hidden zero influ- ences, but also argued that positive and negative influences can be hidden in non-monotonic influences. As the initial '?'s of these influences tend to spread to major parts of a network upon inference, it is worthwhile to resolve the non-monotonicities involved whenever possible. Our extended formalism of qualitative networks provides for effectively capturing information about non-monotonicities, as is demonstrated by the following example. Example 4 We reconsider the qualitative cervical metastases network from Figure 2. We recall that the influence of node M on node C is non-monotonic since ml) and In the context l, therefore, the influence is positive, while it is negative in the context l. In the extended network, shown in Figure 3(b), this information is captured explicitly by assigning the sign -(L), defined by to the influence of node M on node C. 5 Context-specificity in real-life networks To get an impression of the context-specific information that is hidden in real-life qualitative probabilistic networks, we # influences with sign -: ALARM oesophagus Table 2: The numbers of influences with '+', ` ', '0' and `?' signs for the qualitative ALARM and oesophagus networks. computed qualitative abstractions of the well-known ALARM- network and of the network for oesophageal cancer. The ALARM-network consists of 37, mostly non-binary, nodes and 46 arcs; the number of direct qualitative influences in the abstracted network - using the basic definition of qualitative influence - therefore equals 46. The oesophagus network consists of 42, also mostly non-binary, nodes and 59 arcs. Table summarises for the two abstracted networks the numbers of direct influences with the four different basic signs. The numbers reported in Table 2 pertain to the basic signs of the qualitative influences associated with the arcs in the networks' digraphs. Each such influence, and hence each associated basic sign, covers a number of maximal contexts. For a qualitative influence associated with the arc A ! B, the number of maximal contexts equals 1 (the empty context) node B has no other parents than A; otherwise, the number of maximal contexts equals the number of possible combinations of values for the set of parents of B other than A. For every maximal context, we computed the proper (context- specific) sign from the original quantified network. Table 3 summarises the number of context-specific signs covered by the different basic signs in the two abstracted networks. From the table we have, for example, that the 17 qualitative influences with sign '+' from the ALARM network together cover different maximal contexts. For 38 of these contexts, the influences are indeed positive, but for 21 of them the influences are actually zero. # cX with sign total 72 64 44 28 218 # cX with sign total Table 3: The numbers of contexts c X covered by the '+', ` ', '0' and `?' signs and their associated context-specific signs, for the qualitative ALARM and oesophagus networks. For the qualitative ALARM-network, we find that 35% of the influences are positive, 17% are negative, and 48% are ambiguous; the network does not include any explicitly specified zero influences. For the extended network, using context-specific signs, we find that 32% of the qualitative influences are positive, 31% are negative, 20% are zero, and 17% remain ambiguous. For the qualitative oesophagus network, we find that 54% of the influences are positive, 21% are nega- tive, and 25% are ambiguous; the network does not include any explicit zero influences. For the extended network, using context-specific signs, we find that 46% of the qualitative influences are positive, 22% are negative, 10% are zero, and 22% remain ambiguous. We observe that for both the ALARM and the oesophagus network, the use of context-specific signs serves to reveal a considerable number of zero influences and to substantially decrease the number of ambiguous influences. Similar observations were made for qualitative abstractions of two other real-life probabilistic networks, pertaining to Wilson's disease and to ventricular septal defect, respectively. We conclude that by providing for the inclusion of context-specific information about influences, we have effectively extended the expressive power of qualitative probabilistic networks. 6 Extension to enhanced networks The formalism of enhanced qualitative probabilistic networks introduces a qualitative notion of strength of influences into qualitative networks. We briefly argue that the notions from the previous sections can also be used to provide for the inclusion and exploitation of context-specific information about such strengths. In an enhanced qualitative network, a distinction is made between strong and weak influences by partitioning the set of all influences into two disjoint subsets in such a way that any influence from the one subset is stronger than any influence from the other subset; to this end a cut-off value is used. For example, a strongly positive qualitative influence of a node A on a node B, denoted S ++ (A; B), expresses that for any combination of values x for the set X of parents of B other than A; a weakly positive qualitative influence of A on B, denoted S for any such combination of values x. The sign '+ ? ' is used to indicate a positive influence whose relative strength is am- biguous. Strongly negative qualitative influences S , and weakly negative qualitative influences S , are defined anal- a negative influence whose relative strength is ambiguous is denoted S ? . Zero qualitative influences and ambiguous qualitative influences are defined as in regular qualitative probabilistic networks. Renooij &Van der Gaag (1999) also provide extended definitions for the - and -operators to apply to the double signs. These definitions cannot be reviewed without detailing the enhanced formalism, which is beyond the scope of the present paper; it suffices to say that the result of combining signs is basically as one would intuitively expect. Our notion of context-specific sign can be easily incorporated into enhanced qualitative probabilistic networks. A context-specific sign now is defined as a function ?g from a context set CX to the extended set of basic signs, such that for any two contexts c X and c 0 X we have that, if the sign is strongly positive for c 0 must be strongly positive for c X , if the sign is weakly positive for c 0 must be either weakly positive or zero for c X , and if it is ambiguously positive for may be (strongly, weakly or ambiguously) pos- itive, or zero for c X . Similar restrictions hold for negative signs. Context-specific signs are once again assigned to in- fluences, as before. For distinguishing between strong and weak qualitative influences in an enhanced network, a cut-off value has to be chosen in such a way that, basically, for all strong influences of a node A on a node B we have that j Pr(b j contexts x, and for all weak influences we have that j Pr(b j ax) Pr(b j ax)j for all such contexts. If, for a specific cut-off value , there exists an influence of node A on node B for which there are contexts x and x 0 with ax)j > and signs of ambiguous strength would be introduced into the enhanced network, which would seriously hamper the usefulness of exploiting a notion of strength. A different cut-off value had better be cho- sen, by shifting towards 0 or 1. Unfortunately, may then very well end up being 0 or 1. The use of context-specific information about qualitative strengths can now forestall the necessity of shifting the cut-off value, as is illustrated in the following example. R P Figure 4: Context-specific sign in an enhanced network. Example 5 We reconsider the surgery network and its associated probabilities from Example 1. Upon abstracting the network to an enhanced qualitative network, we distinguish between strong and weak influences by choosing a cut-off value of, for example, We then have that a pulmonary complication after surgery strongly influences life ex- pectancy, that is, S (P; L). For this cut-off value, however, the influence of node T on node P is neither strongly positive nor weakly positive; the value therefore does not serve to partition the set of influences in two distinct subsets. To ensure that all influences in the network are either strong or weak, the cut-off value should be either 0 or 1. For the influence of node T on node P , we observe that, for 0:46, the influence is strongly positive for the value s of node S and zero for the context s. By assigning the context-specific sign -(S) defined by to the influence of node T on node P , we explicitly specify the otherwise hidden strong and zero influences. The thus extended network is shown in Figure 4. We recall from Example 3 that for non-smokers the effect of surgery on life expectancy is positive. For smokers, however, the effect could not be unambiguously determined. From the extended net-work in Figure 4, we now find the effect of surgery on life expectancy for smokers to be negative: upon propagating the observation t for node T in the context of the information s for node S, the sign results for node L. Conclusions We extended the formalism of qualitative probabilistic networks with a notion of context-specificity. By doing so, we enhanced the expressive power of qualitative networks. While in a regular qualitative network, zero influences as well as positive and negative influences can be hidden, in a net-work extended with context-specific signs this information is made explicit. Qualitative abstractions of some real-life probabilistic networks have shown that networks indeed can incorporate considerable context-specific information. We further showed that incorporating the context-specific signs into enhanced qualitative probabilistic networks that include a qualitative notion of strength renders even more expressive power. The fact that zeroes and double signs can be specified context- specifically allows them to be specified more often, in gen- eral. We showed that exploiting context-specific information about influences and about qualitative strengths can prevent unnecessary ambiguous node signs arising during inference, thereby effectively forestalling unnecessarily weak results. --R Efficient reasoning in qualitative probabilistic networks. Elicitation of probabilities for belief net- works: combining qualitative and quantitative informa- tion Enhancing QPNs for trade-off resolution Fundamental concepts of qualitative probabilistic networks. On the role of context-specific independence in probabilistic inference --TR Probabilistic reasoning in intelligent systems: networks of plausible inference The computational complexity of probabilistic inference using Bayesian belief networks (research note) Fundamental concepts of qualitative probabilistic networks Building Probabilistic Networks On the Role of Context-Specific Independence in Probabilistic Inference Pivotal Pruning of Trade-offs in QPNs Qualtitative propagation and scenario-based scheme for exploiting probabilistic reasoning --CTR Jeroen Keppens, Towards qualitative approaches to Bayesian evidential reasoning, Proceedings of the 11th international conference on Artificial intelligence and law, June 04-08, 2007, Stanford, California

context-specific independence;qualitative reasoning;probabilistic reasoning;non-monotonicity

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Accelerating filtering techniques for numeric CSPs.

Search algorithms for solving Numeric CSPs (Constraint Satisfaction Problems) make an extensive use of filtering techniques. In this paper we show how those filtering techniques can be accelerated by discovering and exploiting some regularities during the filtering process. Two kinds of regularities are discussed, cyclic phenomena in the propagation queue and numeric regularities of the domains of the variables. We also present in this paper an attempt to unify numeric CSPs solving methods from two distinct communities, that of CSP in artificial intelligence, and that of interval analysis.

Introduction In several fields of human activity, like engineering, science or business, people are able to express their problems as constraint problems. The CSP (Constraint Satisfaction Problem) schema is an abstract framework to study algorithms for solving such constraint problems. A CSP is defined by a set of variables, each with an associated domain of possible values and a set of constraints on the variables. This paper deals more specifically with CSPs where the constraints are numeric nonlinear relations and where the domains are continuous domains (numeric CSPs). * Corresponding author. E-mail addresses: ylebbah@univ-oran.dz (Y. Lebbah), olhomme@ilog.fr (O. Lhomme). This paper is an extended version of [31]. see front matter 2002 Elsevier Science B.V. All rights reserved. In general, numeric CSPs cannot be tackled with computer algebra systems: there is no algorithm for general nonlinear constraint systems. And most numeric algorithms cannot guarantee completeness: some solutions may be missed, a global optimum may never be found, and, sometimes a numeric algorithm even does not converge at all. The only numeric algorithms that can guarantee completeness-even when floating-point computations are used-are coming either from the interval analysis community or from the AI community (CSP). Unfortunately, those safe constraint-solving algorithms are often less efficient than non-safe numeric methods, and the challenge is to improve their efficiency. The safe constraint-solving algorithms are typically a search-tree exploration where a filtering technique is applied at each node. Improvement in efficiency is possible by finding the best compromise between a filtering technique that achieves a strong pruning at a high computational cost and another one that achieves less pruning at a lower computational cost. And thus, a lot of filtering techniques have been developed. Some filtering techniques take their roots from numerical analysis: the main filtering technique used in interval analysis [37] is an interval variation of Newton iterations. (See [24,28] for an overview of such methods.) Other filtering techniques originate from artificial intelligence: the basic filtering technique is a kind of arc-consistency filtering [36] adapted to numeric CSPs [17, 26,32]. Higher-order consistencies similar to k-consistency [21] have also been defined for numeric CSPs [25,32]. Another technique from artificial intelligence [19,20] is to merge the constraints concerning the same variables, giving one "total" constraint (thanks to numerical analysis techniques) and to perform arc-consistency on the total constraints. Finally, [6,45] aim at expressing interval analysis pruning as partial consistencies, bridging the gap between the two families of filtering techniques. All the above works address the issue of finding a new partial consistency property that can be computed by an associated filtering algorithm with a good efficiency (with respect to the domain reductions performed). Another direction, in the search of efficient safe algorithms, is to try to optimize the computation of already existing consistency techniques. Indeed, the aim of this paper is to study general methods for accelerating consistency techniques. The main idea is to identify some kinds of regularity in the dynamic behavior of a filtering algorithm, and then to exploit those regularities. A first kind of regularities we exploit is the existence of cyclic phenomena in the propagation queue of a filtering algorithm. A second kind of regularities is a numeric regularity: when the filtering process converges asymptotically, its fixed point often can be extrapolated. As we will see in the paper, such ideas, although quite general, may lead to drastic improvements in efficiency for solving numeric CSPs. The paper focus on numeric continuous problems, but the ideas are more general and may be of interest also for mixed discrete and continuous problems, or even for pure discrete problems. The paper is organized in two main parts. The first part (Section 2) presents an overview of numeric CSPs; artificial intelligence works and interval analysis works are presented through a unifying framework. The second part consists of the next two sections, and presents the contribution of the paper. Section 3 introduces the concept of reliable transformation, and presents two reliable transformations that exploit two kinds of regularities occurring during the filtering process: cyclic phenomena in the propagation queue and numeric regularities of the domains of the variables. Section 4 discusses related works. 2. Numeric CSPs This section presents numeric CSPs in a slightly non-standard form, which will be convenient for our purposes, and will unify works from interval analysis and constraint satisfaction communities. A numeric CSP is a triplet #X , D,C# where: . X is a set of n variables x 1 , . , x n . . denotes a vector of domains. The ith component of D, D i , is the domain containing all acceptable values for x i . . denotes a set of numeric constraints. denotes the variables appearing in C j . This paper focuses on CSPs where the domains are intervals: D # {[a, The following notation is used throughout the paper. An interval [a, b] such that a > b is an empty interval. A vector of domains D such that a component D i is an empty interval will be denoted by #. The lower bound, the upper bound and the midpoint of an interval D i (respectively interval vector D) are respectively denoted by D i , D i , and (respectively D, D, and m( D)). The lower bound, the upper bound, the midpoint, the inclusion relation, the union operator and the intersection operator are defined over interval vectors; they have to be interpreted componentwise. For instance D means #D 1 , . , D n #; D #D means D # i #D i for all i # 1, . , n; D #D # means #D # A k-ary constraint C k-ary relation over the real numbers, that is, a subset of R k . 2.1. Approximation of projection functions The algorithms used over numeric CSPs typically work by narrowing domains and need to compute the projection-denoted #C j ,x i( D) or also # j,i( D)-of a constraint over the variable x i in the space delimited by D . The projection # j,i( D) is defined as follows. . If x i / . If x i # the projection is defined by the set of all elements we can find elements for the k - 1 remaining variables of # . (1) Usually, such a projection cannot be computed exactly due to several reasons, such as: (1) the machine numbers are floating point numbers and not real numbers so round-off errors occur; (2) the projection may not be representable as floating-point numbers; (3) the computations needed to have a close approximation of the projection of only one given constraint may be very expensive; (4) the projection may be discontinuous whereas it is much easier to handle only closed intervals for the domains of the variables. Thus, what is usually done is that the projection of a constraint over a variable is approximated. Let #C j ,x i( D) or also # j,i( D) denote such an approximation. In order to guarantee that all solutions of a numeric CSP can be found, a solving algorithm that uses D) needs that # D) includes the exact projection. We will also assume in the rest of the paper that # j,i( D) satisfies a contractance property. Thus we have: D) hides all the problems seen above. In particular, it allows us not to go into the details of the relationships between floating point and real numbers (see for example [2] for those relationships) and to consider only real numbers. It only remains to build such a # j,i . Interval analysis [37] makes it possible. 2.1.1. Interval arithmetic Interval arithmetic [37], on which interval analysis is built, is an extension of real arithmetic. It defines the arithmetic functions {+,-,#,/} over the intervals with simple set extension semantics. Notation. To present interval arithmetic, we will use the following convention to help the reading: x, y will denote real variables or vectors of real variables and X,Y will denote interval variables or vectors of interval variables. Distinction between a scalar variable and a vector of variables will be clear from the context. With this notation, an arithmetic function #,/} over the intervals is defined by: Thanks to the monotonicity property of arithmetic operators # , X#Y can be computed by considering the bounds of the intervals only. Let X,Y # X,X] , and Y] , the arithmetic operators are computed on intervals as follows: # Y. 2.1.2. Interval extension of a real function For an arbitrary function over the real numbers, it is not possible in general to compute the exact enclosure of the range of the function [29]. The concept of interval extension has been introduced by Moore: the interval extension of a function is an interval function that computes outer approximations on the range of the function over a domain. Different interval extensions exist. Let f be a function over the real numbers defined over the variables x 1 , . , x n , the following interval extensions are frequently used: the natural interval extension of a real function f is defined by replacing each real operator by its interval counterpart. It is easy to see that contains the range of f , and is thus an interval extension. Example 1 (Natural extension of x 2 The natural extension of x 2 2. The natural extension of x 2 the Taylor interval extension of a real function f , over the interval vector X, is defined by the natural extension of a first-order Taylor development of f [42]: f nat #f # . (2) The intuition why is an interval extension is given in a footnote. 2 Example 2 (Taylor extension of x - x 2 ). Let The Taylor extension of The Taylor extension gives generally a better enclosure than the natural extension on small intervals. 3 Nevertheless, in general neither give the exact range of f . For example, let 2] , we have: [-1,3] , whereas the range of f over X= [0, is [3/4, 3] . 2 The Taylor interval extension comes from a direct application of the mean value theorem: Let f be a real function defined over [a, be continuous and with a continuous derivative over [a, be two points in [a, . Then, there exists # between x 1 and x 2 such that #) # is unknown, but what can be done is to replace it by an interval that contains it, and to evaluate the natural extension of the resulting expression. Thus we know that #( [a, As this is true for every x 1 and x 2 in [a, , we can replace x 1 by the midpoint of [a, by an interval that contains it. This leads to #( [a, #( [a, (2) is the generalization for vectors of the above result. 3 The Taylor extension has a quadratic convergence, whereas the natural extension has a linear convergence; see for example [42]. 2.1.3. Solution function of a constraint To compute the projection # j,i( D) of the constraint C j on the variable x i , we need to introduce the concept of solution function that expresses the variable x i in terms of the other variables of the constraint. For example, for the constraint x z, the solution functions are: y, f Assume a solution function is known that expresses the variable x i in terms of the other variables of the constraint. Thus an approximation of the projection of the constraint over x i given a domain D can be computed thanks to any interval extension of this solution function. Thus we have a way to compute # D). Nevertheless, for complex constraints, there may not exist such an analytic solution for example, consider x log( The interest of numeric methods as presented in this paper is precisely for those constraints that cannot be solved algebraically. Three main approaches have been proposed: . The first one exploits the fact that analytic functions always exist when the variable to express in terms of the others appears only one time in the constraint. This approach simply considers that each occurrence of a variable is a different new variable. In the previous example this would give: log( x( That way, it is trivial to compute a solution function: it suffices to know the inverse of basic operators. In our example, we obtain f log( x( 2) ) and f An approximation of the projection of the constraint over x i can be computed by intersecting the natural interval extensions of the solution functions for all occurrences of x i in C j . For the last example, we could take # log( X)#exp -X . Projection functions obtained by this way will be called # nat in this paper. . The second idea uses the Taylor extension to transform the constraint into an interval linear constraint. The nonlinear equation nat #f m( X). Now consider that the derivatives are evaluated over a box D that contains X. D is considered as constant, and let c D). The equation becomes: nat #f #( D) This is an interval linear equation in X, which does not contain multiple occurrences. The solution functions could be extracted easily. But, instead of computing the solution functions of the constraint without taking into account the other constraints, we may prefer to group together several linear equations in a squared system. Solving the squared interval linear system allows much more precise approximations of projections to be computed. (See the following section.) Projection functions obtained by this way are called # Tay . For example, consider the constraint x log( by using the Taylor form on the box D, we obtain the following interval linear equation log( c) that is: log( c) - c/D. The unique solution function of this 1- dimensional linear equation is straightforward: X =-B/A. . A third approach [6] does not use any analytical solution function. Instead, it transforms the constraint C 1, . , k. The mono-variable constraint C j,l on variable x is obtained by substituting their intervals for the other variables. The projection # j,j l is computed thanks to C j,l . The smallest zero of C j,l in the interval under consideration is a lower bound for the projection of C j over x j . And the greatest zero of C j,l is an upper bound for that projection. Hence, an interval with those two zeros as bounds gives an approximation of the projection. Projection functions computed in that way are called # box . In [6], the two extremal zeros of C j,l are found by a mono-variable version of the interval Newton method. 4 Another problem is that the inverse of a nonmonotonic function is not a function over the intervals. For example the range of the inverse of the function for an interval Y is the union of Y ). It is possible to extend interval arithmetic in order to handle unions of intervals. A few systems have taken this approach [26,44]. Nevertheless, this approach may lead to a highly increasing number of intervals. The two other approaches more commonly used consist of computing the smallest interval encompassing a union of or to split the problem in several sub-problems in which only intervals appear. 2.2. Filtering algorithm as fixed point algorithms A filtering algorithm can generally be seen as a fixed point algorithm. In the following, an abstraction of filtering algorithms will be used: the sequence {D k } of domains generated by the iterative application of an operator Op : -# I( R) n (see Fig. 1). The operator Op of a filtering algorithm generally satisfies the following three properties Op( D) #D (contractance). . Op is conservative; that is, it cannot remove solutions. . D # #Op( D) (monotonicity). Under those conditions, the limit of the sequence {D k }, which corresponds to the greatest fixed point of the operator Op, exists and is called a closure. We denote it by #Op( D). A fixed point for Op may be characterized by a property lc-consistency, called a local consistency, and alternatively #Op( D) will be denoted by # lc( D). The algorithm achieving filtering by lc-consistency is denoted lc-filtering. A CSP is said to be lc-satisfiable if lc- filtering of this CSP does not produce an empty domain. 4 The general (multi-variable) interval Newton method is briefly presented in Section 2.3. Fig. 1. Filtering algorithms as fixed point algorithms. Consistencies used in numeric CSPs solvers can be categorized in two main classes: arc-consistency-like consistencies and strong consistencies. 2.3. Arc-consistency-like consistencies Most of the numeric CSP systems (e.g., BNR-prolog [40], Interlog [13,16], CLP(BNR) [5], PrologIV [15], UniCalc [3], Ilog Solver [27] and Numerica [46] compute an approximation of arc-consistency [36] which will be named 2B-consistency in this paper. 5 2B-consistency states a local property on a constraint and on the bounds of the domains of its variables (B of 2B-consistency stands for bound). Roughly speaking, a constraint C j is 2B-consistent if for any variable x i in the bounds D i and D i have a support in the domains of all other variables of C j (w.r.t. the approximation given by # ). 2B-consistency can be defined in our notation as: 2B-consistent if and only if A filtering algorithm that achieves 2B-consistency can be derived from Fig. 1 by instantiating Op as in Operator 1. Note the operator Op 2B applies on the same vector D all the # j,i( D) operators. Operator 1 (2B-consistency filtering operator). Op j,1( D), . , # Fig. 2 shows how projection functions are used by a 2B-consistency filtering algorithm to reduce the domains of the variables. Depending on the projection functions used, we obtain different 2B-filtering algorithms Op nat The operator Op nat will denote Op 2B with # nat . It abstracts the filtering algorithm presented in [5,17,32]. There are two main differences between our abstraction and the implementations. (1) In classic implementations, projection functions are applied sequentially and not all on the same domain. In the abstraction (and in our non-classic 5 We have a lot of freedom to choose # j,i( D), so the definition of 2B-consistency here abstracts both 2B- consistency in [32] and box-consistency in [6]. Fig. 2. 2B-filtering on the constraint system {x 2 implementations) they are applied on the same domain. This has the drawback of increasing the upper bound of the complexity, but has the advantage of generating much more "regular" sequences of domains. (See Section 3.2.) (2) Implementations always applied an AC3-like optimization [36]. It consists of applying at each iteration only those projection functions that may reduce a domain: only the projection functions that have in their parameters a variable whose domain has changed are applied. For the sake of simplicity, AC3-like optimization does not appear explicitly in this algorithm schema. Op box This operator denotes Op 2B that uses # box . It abstracts the filtering algorithm presented in [6,45]. Differences with our abstraction are the same as above. Op Tay This operator denotes Op 2B that uses # Tay . It abstracts the interval Newton method [2,37]. The interval Newton method controls in a precise way the order in which projection functions are computed. It is used for solving squared nonlinear equation systems such as 0}. The interval Newton method replaces the solving of the nonlinear squared system by the solving of a sequence of interval linear squared systems. Each linear system is obtained by evaluating the interval Jacobi matrix over the current domains, and by considering the first-order Taylor approximation of the nonlinear system. The resulting interval linear system is typically solved by the interval Gauss-Seidel method. The Gauss-Seidel method associates each constraint C i with the variable x i (after a possible renaming of variables), and loops while applying only the projection functions # i,i . To summarize, the main differences with our abstraction are that, in an implementation, the partial derivatives are recomputed periodically and not at each step, and that the Gauss-Seidel method does not apply all the projection functions. A more realistic implementation of the Interval Newton method would correspond to Operator 2 as follows. 6 Operator 2 (Interval Newton operator). Op A i,i endfor Note also that, in general, the Gauss-Seidel method does not converge towards the solution of the interval linear system, but it has good convergence properties for diagonally-dominant matrices. So, in practice, before solving the linear system, a preconditioning step is achieved that transforms the Jacobi matrix into a diagonally dominant matrix. Preconditioning consists of multiplying the interval linear equation A # by a matrix M , giving the new linear system . The matrix M is typically the inverse of the midpoint matrix of A. A nice property of the interval Newton operator is that in some cases, it is able to prove the existence of a solution. When Op Tay( D) is a strict subset of D, Brouwer's fixed-point theorem applies and states existence and unicity of a solution in D (cf. [38]). 2.4. Strong consistencies The idea of constraint satisfaction is to tackle difficult problems by solving easy-to- solve sub-problems: the constraints taken individually. It is often worth to have a more global view, which generally leads to a better enclosure of the domains. This is why strong consistencies have been proposed for solving CSP [21,22]. Their adaptation to numeric CSPs is summarized in this section. Interval analysis methods such as Op Tay extensively use another kind of global view: the preconditioning of the Jacobi matrix. Nevertheless, the need for strong consistencies, although less crucial with interval analysis methods, may appear for very hard problems such as [43]. Strong consistencies have first been introduced over discrete CSPs (e.g., path- consistency, k-consistency [21] )-consistency [22]), and then over numeric CSPs 6 The for-loop corresponds to only one iteration of the Gauss-Seidel method and not to the complete solving of the interval linear system, which in practice is not useful [24]. (3B-consistency [32] and kB-consistency [33]). kB-consistency is the adaptation consistency over numeric CSP. Filtering k)-consistency is done by removing from each domain values that can not be extended to k variables. kB-consistency ensures that when a variable is instantiated to one of its two bounds, then the CSP is |k -1|B-satisfiable. we refer to Operator 1. More generally, as given in Definition 2, consistency ensures that when a variable is forced to be close to one of its two bounds (more precisely, at a distance less than w), then the CSP is |k - 1|B( w)-satisfiable. For simplest presentation, w)-consistency refers to 2B-consistency. (kB( w)-consistency). We say that a CSP #X , D,C# is w)-consistent if and only if: #( D, i, w) is |k - 1|B( w)-satisfiable, and #( D, i, w) is |k - 1|B( w)-satisfiable, where #( D, i, w) (respectively #( D, i, w)) denotes #X , D # , C# where D # is the same domain as D except that D i is replaced by D (respectively D i is replaced by D The direct filtering operator Op kB( w) underlying the w)-consistency uses a kind of proof by contradiction: the algorithm tries to increase the lower bound D i by proving that the closure by |k - 1|B( w)-consistency of #D 1 , . , [ +w] , . , D n # is not empty and tries to decrease the upper bound in a symmetric way. 3B-consistency filtering algorithms, used for example in Interlog, Ilog Solver or Numerica, can be derived from Fig. 1 by instantiating operator Op to Op 3B as defined in Operator 3. Operator 3 w)-consistency filtering operator: Op the filtering operator Op kB( w)( P), with k # 3, is defined as follows: Op being computed as follows: do while D # do while D # do endfor Fig. 3 shows how w)-filtering uses 2B-filtering. Fig. 3. 3B( w)-filtering on the constraint system {x 2 Implementations using this schema may be optimized considerably, but we do not need to go into details here. The reader is referred to [32] for the initial algorithm, and to [12] which studies the complexity of an unpublished implementation we used for years (see for example [30]) and that is more efficient than the algorithm published in [32]. The algorithm that achieves box-consistency is closely related to 3B-consistency. Indeed, box-consistency can be seen as a kind of one-way 3B-consistency limited to one constraint. The reader can found in [14] a theoretical comparison between box-consistency and 3B-consistency. 3. Acceleration of filtering techniques The question of choosing the best filtering algorithm for a given constraint system is an open problem. Some preliminary answers may come from the observation that the above fixed point algorithms suffer from two main drawbacks, which are tightly related: . the existence of "slow convergences", leading to unacceptable response times for certain constraint systems; . "early quiescence" [17], i.e., the algorithm stops before reaching a good approximation of the set of possible values. The focus of this paper is on the first drawback. Its acuteness varies according to the Op operator: Op nat Due to its local view of constraints, Op nat often suffers from early quiescence, but its simplicity makes it the most efficient operator to compute, and many problems are best solved by this filtering operator (e.g., Moreaux problem [46]). At first sight, one could think that slow convergence phenomena do not occur very often with Op nat . It is true that early quiescence of Op nat is far more frequent than slow convergence. However, Op nat is typically interleaved with a tree search (or is called from inside another higher-order filtering algorithm). During this interleaved process, slow convergence phenomena may occur and considerably increase the required computing time. Op box The comments above remain true for Op box , although it may take more time to be computed and may perform some stronger pruning in some cases. Op Tay The interval Newton operator, on the one hand, may have a very efficient behavior. It may have an asymptotically quadratic convergence when it is used near the solution. In our experience, quadratic convergence is essential to compute precise roots of nonlinear systems of equations. On the other hand, far from the solution, the Jacobi matrix has a great chance of being singular, which typically leads to the "early quiescence" problem. Hence Op Tay does not have really slow convergence problems, but it needs expensive computation since the preconditioning of the Jacobi matrix needs to compute an inversion of its midpoint matrix. On some problems like Moreaux problem [46] with huge dimension n # 320, Op Tay is very expensive, whereas by Op nat the solution is found quickly. Op w)-consistency filtering algorithms may perform a very strong pruning, making the tree-search almost useless for many problems. For example, we have tried w)-filtering over the transistor problem [41,43]. It finds the unique solution, without search, in the same cpu time as the filtering search method used in [41]. We have also tried w)-filtering over the benchmarks listed in [45]. They are all solved without search (only p choice points are made when the system has Unfortunately, most of the time slow convergence phenomena occur during a w)-filtering. The different filtering algorithms are thus complementary and the more robust way to solve a problem is probably to use several of them together. In the fixed point schema of Fig. 1, the operator Op would be the result of the composition of some operators above. In the remainder of this section, we focus on the problem of slow convergence that occurs in Op nat and Op kB( w) . The observation of many slow convergences of those algorithms led us to notice that some kinds of "regularity" often exist in a slow convergence phenomenon. Our intuition was that such regularities in the behavior of algorithms could be exploited to optimize their convergence. As seen in Section 2, the filtering algorithms are abstracted through a sequence of interval vectors. Accelerating a filtering algorithm thus consists in transforming that sequence into another sequence, hoping it converges faster. In numerical analysis, engineers use such transformation methods. Unfortunately, they cannot be sure of the reliability of their results. But this does not change the essence of usual floating-point computation: unreliability is everywhere! For filtering techniques, the completeness of the results must be guaranteed, or in other words, no solution of the CSP can be lost. Thus the question of reliability becomes crucial. This leads us to define a reliable transformation. Definition 3 (Reliable transformation). Let {S n } be a sequence that is complete for a set of solutions Sol: #k,Sol # S k . Let A be a transformation and let {T n A( {S n }). A is a reliable transformation for {S n } w.r.t. Sol if and only if The practical interest of a reliable transformation is directly related to its ability to accelerate the greatest number of sequences. Acceleration of a sequence is traditionally defined in terms of improvement of the convergence order of the sequence. Convergence order characterizes the asymptotic behavior of the sequences. (See Section 3.2 for a formal definition of the convergence order.) In addition to convergence order, some practical criteria may be of importance, like, for example, the time needed to compute a term of a sequence. To build a reliable transformation that accelerates the original sequence, we will exploit some regularities in the sequence. When we detect a regularity in the filtering sequence, the general idea is to assume that this regularity will continue to appear in the following part of the sequence. The regularities that we are looking for are those which allow computations to be saved. A first kind of regularity that we may want to exploit is cyclicity. Section 3.1 summarizes a previous work based on that idea. Another kind of regularity, that can be caught by extrapolation methods, is then developed in Section 3.2. 3.1. A previous work: dynamic cycle simplification This subsection summarizes a previous work [34,35], built on the idea that there is a strong connection between the existence of cyclic phenomena and slow convergence. More precisely, slow convergence phenomena move very often into cyclic phenomena after a transient period (a kind of stabilization step). The main goal is to dynamically identify cyclic phenomena while executing a filtering algorithm and then to simplify them in order to improve performance. This subsection is more especially dedicated to the acceleration of Op nat and Op box algorithms, . a direct use of those accelerated algorithms also leads to significant gain in speed for w)-filtering algorithms since they typically require numerous computations of . this approach could be generalized to identify cyclic phenomena in w)-filtering algorithms. Considering the application of Op 2B over D i , there may exist several projection functions that perform a reduction of the domain of a given variable. As Op 2B performs an intersection, and since domains are intervals, there may be 0, 1 or 2 projection functions of interest for each variable. (One that gives the greatest lower bound, one that gives the lowest upper bound.) Call these projection functions relevant for D i , and denote by R i the set of those relevant projection functions for D i . Thus we have 2B( D i ); that is, if we know in advance all the R i , we can compute # 2B( D) more efficiently by applying only relevant projection functions. This is precisely the case in a cyclic phenomenon. We will say we have a cyclic phenomenon of period p when: #i <N, R i+p =R i , where N is a "big" number. Now, consider R i and R i+1 . If a projection function is in R i+1 , this is due to the reduction of domains performed by some projection functions in R i . We will say that f # R j depends on g # R i , where j > i , denoted by g # f if and only if g #= f and g computes the projection over a variable that belongs to The dependency graph is the graph whose vertices are pairs #f, i#, where f # R i , and arcs are dependency links. (See Fig. 4(a).) If we assume that we are in a cyclic phenomenon, then the graph is cyclic. (See Fig. 4(b) where - denotes all the steps i such that i mod According to this assumption, two types of simplification can be performed: . Avoid the application of non-relevant projection functions. . Postpone some projection functions: a vertex #f, i# which does not have any successor in the dynamic dependency graph corresponds to a projection function that can be postponed. Such a vertex can be removed from the dynamic dependency graph. Applying this principle recursively will remove all non-cyclic paths from the graph. For instance, in graph (b) of Fig. 4, all white arrows will be pruned. When a vertex is removed, the corresponding projection function is pushed onto a stack. (The removing order must be preserved.) Then, it suffices to iterate on the simplified cycle until a fixed point is reached, and, when the fixed point has been reached, to evaluate the stacked projection functions. The transformation that corresponds to the above two simplifications together is clearly a reliable transformation. It does not change the convergence order, but is in general an accelerating transformation. In [34] first experimental results are reported, gains in Fig. 4. Dynamic dependency graphs. efficiency range from 6 to 20 times faster for 2B-filtering and w)-filtering. More complete experiments have been performed in [23], but, for the sake of simplicity, only the first simplification (applying only relevant projection functions) has been tried. Different combinations of several improvements of 2B-filtering are tested. For all problems, the fastest combination uses cycle simplification. Ratio in CPU time varies from 1 to 20 compared with the same combination without cycle simplification. 3.2. Extrapolation The previous section aims at exploiting cyclicity in the way projection functions are applied. The gain is in the computation of each term of {D n }, but the speed of convergence of D n is unchanged. Now we address how to accelerate the convergence of {D n }. {D n } is a sequence of intervals. Numerical analysis provides different mathematical tools for accelerating the convergence of sequences of real numbers. Extrapolation methods are especially interesting for our purposes, but {D n } is a sequence of interval vectors and there does not exist any extrapolation method to accelerate interval sequences. Nevertheless an interval can be seen as two reals and D can be seen as a 2-column matrix of reals. The first column is the lower bounds, and the second the upper bounds. Thus we can apply the existing extrapolation methods. The field of extrapolation methods, for real number sequences, is first summarized; for a deeper overview see [10]. Then we will show how to use extrapolation methods for accelerating filtering algorithms. 3.2.1. Extrapolation methods Let {S n } .) be a sequence of real numbers. A sequence {S n } converges if and only if it has a limit S: lim n# S We say that the numeric sequence {S n } has the order r # 1 if there exist two finite constants A and B such that 7 A# lim # B. A quadratic sequence is a sequence which has the order 2. We say that a sequence is linear if lim The convergence order enables us to know exactly the convergence speed of the sequence. For example [8], for linear sequences with we obtain a significant number every 2500 iterations. Whereas, for sequences of order 1.01, the number of significant numbers doubles every 70 iterations. These examples show the interest of using sequences of order r > 1. Accelerating the convergence of a sequence {S n } amounts of applying a transformation A which produces a new sequence {T n }: {T n } 7 For more details, see [10]. As given in [10], in order to present some practical interest, the new sequence {T n } must exhibit, at least for some particular classes of convergent sequences {S n }, the following properties: (1) {T n } converges to the same limit as {S n }: lim n# T (2) {T n } converges faster than {S n }: lim These properties do not hold for all converging sequences. Particularly, a universal transformation A accelerating all converging sequences cannot exist [18]. Thus any transformation can accelerate a limited class of sequences. This leads us to a so-called kernel 8 of the transformation which is the set of convergent A well-known transformation is the iterated # 2 process from Aitken [1] which gives a sequence {T n } of nth term The kernel of # 2 process is the set of the converging sequences which have the form Aitken's transformation has a nice property [10]: it transforms sequences with linear convergence into sequences with quadratic convergence. We can apply the transformation several times, leading to a new transformation. For example, we can apply # 2 twice, giving # 2( {S n })). Many acceleration transformations (G-algorithm, #-algorithm, # -algorithm, overholt-process, .) are multiple application of transformations. See [11] and [9] for attempts to build a unifying framework of transformation. Scalar transformations have been generalized to the vectoral and matrix cases. Two kinds of optimization for filtering algorithms are now given. The first one makes a direct use of extrapolation methods and leads to a transformation which is not reliable. The second one is a reliable transformation. 3.2.2. Applying extrapolation directly Let {D n } be a sequence generated by a filtering algorithm. We can naively apply extrapolation method directly on the main sequence {D n }. The experimental results given in the rest of the paper are for scalar extrapolations, which consider each element of the matrix-each bound of a domain- independently of the others. For example, the scalar process uses for each bound of domain the last three different values to extrapolate a value. 8 The definition of the kernel given here considers only converging sequences. Accelerating directly the convergence of {D n } can dramatically boost the convergence, as illustrated in the following problem: 1000] , y # [0, 1000] , z # [0, #] . The following table shows the domain of the variable t in the 278th, 279th, 280th and 281st iterations of 3B-filtering (after a few seconds on a Sun Sparc 5). The precision obtained is it t 278 [3.14133342842583 . , 3.14159265358979 . .] By applying Aitken's process on the domains of the iterations 278, 279 and 280, we obtain the domain below. The precision of this extrapolated domain is 10 -14 . Such precision has not been obtained after 5 hours of the 3B-filtering algorithm without extrapolation. [3.14159265358977 . , 3.14159265358979 . .] Let's take another example: Table 1 shows the domain of the variables x and y in the first, second and third iterations of 5B( w)-filtering. The precision obtained is about 10 -6 . Table 5B( w)-filtering on the problem above Iteration domains for x and y By applying the # 2 process on the domain of the iterations 1, 2 and 3, we obtain the domains below. The precision of this extrapolated domain is 10 -19 . Such a precision has not been obtained after many hours of the w)-filtering algorithm without extrapolation. [-8.93e-19,-8.85e-19] This result is not surprising since we have the following proposition: Theorem 1 (Convergence property of Aitken's process [7]). If we apply # 2 on some which converges to S and if we have: lim then the sequence # converges to S, and more quickly Note that, in the solution provided by Aitken's process, we have a valid result for x , but not for y . This example shows that extrapolation methods can lose solutions. The extrapolated sequence may or may not converge to the same limit as the initial sequence. This anomaly can be explained by the kernel of the transformation: when the initial sequence belongs to the kernel, then we are sure that the extrapolated sequence converges to the same limit. Furthermore, intuition suggests that, if the initial sequence is "close" to the kernel then there are good hopes to get the same limit. However, it may be the case that the limits are quite different. This is cumbersome for the filtering algorithms which must ensure that no solution is lost. We propose below a reliable transformation that makes use of extrapolation. 3.2.3. Reliable transformation by extrapolation The reliable transformation presented in this section is related to the domain sequences generated by w)-filtering algorithms. For the sake of simplicity, we will only deal with 3B( w)-filtering, but generalisation is straightforward. This transformation is reliable thanks to the proof-by-contradiction mechanism used in 3B( w)-algorithm: it tries to prove-with a 2B-filtering-that no solution exists in a subpart of a domain. If such a proof is found, then the subpart is removed from the domain, else the subpart is not removed. The point is that we may waste a lot of time trying to find a proof that does not exist. If we could predict with good probability that such a proof does not exist, we could save time in not trying to find it. Extrapolation methods can do the job. The idea is simply that if an extrapolated sequence converges to a 2B-satisfiable CSP (which can be quickly known), then it is probably a good idea not to try to prove the 2B-unsatisfiability. This can be done by defining a new consistency, called consistency, that is built upon the existence of a predicate 2B-predict( D) that predicts 2B-satisfiability. (P2B stands for 2B based on Prediction.) Definition 4 (P2B-consistency). A CSP #X , D,C# is P2B-consistent if and only if it is 2B-consistent or 2B-predict(D) is true. Op Op Fig. 5. P 2B-consistency filtering schema. may use extrapolation methods, for example the # 2 process. Thus, the prediction 2B-predict(D) may be wrong, but from the Proposition 1 we know that a filtering algorithm by P2B-consistency cannot lose any solutions. Proposition 1. # D). The proof is straightforward from the definition. A filtering algorithm that achieves P2B-consistency can be a fixed point algorithm where Op is as defined in Fig. 5. The main difference with Op 2B is, before testing for 2B- satisfiability, to try in the function 2B-predict, to predict 2B-satisfiability by extrapolation methods. Following that idea, the algorithm schema for Fast-3B( w)-consistency can be modified, as given in Operator 4. (We may obtain in the same way the algorithm schema for Fast-kB( w)-consistency. It needs a P -kB operator that applies an extrapolator over the domains generated by the kB operator.) Operator 4. Let the filtering operator Op Fast-3B( w) is defined as follows: Op being computed as follows: do while D # do while D # do endfor The following proposition means that this algorithm schema allows acceleration methods to be applied while keeping the completeness property of filtering algorithms. We thus have a reliable transformation. Proposition 2 (Completeness). Fast-kB( w)-algorithm does not lose any solutions. The proof is built on the fact that a domain is reduced only when we have a proof- by |k - 1|B( w)-satisfiability and without extrapolation-that no solution exists for the removed part. Table w)-filtering results over some benchmarks Problem nbr-#( Fast-kB) nbr-#( kB) time( Fast-kB) time( kB) brown caprasse 0.46 0.60 chemistry 0.61 0.69 neuro-100 0.53 0.66 The counterpart of this result is that improvements in efficiency of w)-filtering compared with w)-filtering may be less satisfactory than improvement provided by direct use of extrapolation. Another counterpart is that the greatest fixed point of Op Fast-kB( w) is generally greater than the greatest fixed point of Op kB( w) . In practice the overhead in time has always been negligible and the improvement in efficiency may vary from 1 to 10. Table 2 compares Fast-3B-filtering with 3B-filtering over some problems taken from [23,46]. It gives the ratios in time ( time( Fast-kB) time( kB) ) and in number of projection function calls ( nbr-#( Fast-kB) nbr-#( kB) ) for the two algorithms. 4. Related works Two methods commonly used for solving numeric CSPs can be seen as reliable transformations: preconditioning and adding redundant constraints. 4.1. Preconditioning in the interval Newton operator Numeric CSPs allow general numeric problems to be expressed, without any limitation on the form of the constraints. In numerical analysis, many specific cases of numeric CSPs have been studied. The preconditioning of squared linear systems of equations is among the most interesting results for its practical importance. We say that a linear system of equation is near 1. In practice, a well conditioned system is better solved than an ill conditioned one. Preconditioning methods transform the system to a new system A # has the same solution but is better conditioned than the first system. Solving A # better precision and more reliable computations than solving the original system. A classic preconditioning method consists of multiplying the two sides of the system by an approximate inverse M of A. Thus we have A # =MA and b # =Mb. In interval analysis, the interest of preconditioning is not reliability, which already exists in interval methods, but precision and convergence. As already presented in Section 2.3, preconditioning is a key component of the interval Newton method. Experimental results (for example see [28,45]) show the effectiveness of preconditioning for solving squared nonlinear systems of equations. Many theoretical results can be found in [2,28,38,39]. 4.2. Redundant constraints A classic reliable transformation is the adding of some redundant constraints to the original constraint system. This approach is very often used for discrete CSPs to accelerate the algorithms. It is not the case for interval analysis methods over numeric CSPs, since they exploit the fact that the system is square. For artificial intelligence methods over numeric CSPs, Benhamou and Granvilliers [4] propose to add some redundant polynomial constraints that are automatically generated by a depth-bounded Groebner bases algorithm. 5. Conclusion and perspectives Our aim in this paper was to accelerate existing filtering algorithms. That led us to the concept of reliable transformation over the filtering algorithms, which preserves completeness of the filtering algorithms. Two kinds of reliable transformation have been proposed. They exploit some regularities in the behavior of the filtering algorithms. The first one is based on cyclic phenomena in the propagation queue. The second one is an extrapolation method: it tries to find a numeric equation satisfied by the propagation queue and then solves it. A first perspective is to detect other kinds of regularities and to exploit them. A reliable transformation always has some intrinsic limitations; for example, logarithmic sequences cannot be accelerated by extrapolation methods. However, in that case, the cyclic phenomena simplification may improve the running time. Thus, combining different reliable transformations to try to accumulate the advantages of each transformation may be of high interest. Finally, a direction of research that could be fruitful comes from the remark that algorithms are designed with efficiency and simplicity in mind only. Regularity is never considered as an issue. Perhaps it is time to consider it as an issue, and to try to make more regular the existing algorithms in order to exploit their new regularities. Acknowledgements We would like to thank Christian Bliek, Michel Rueher and Patrick Taillibert for their constructive comments on an early draft of the paper, and Kathleen Callaway for a lot of English corrections. This work has been partly supported by the Ecole des Mines de Nantes. --R On Bernoulli's numerical solution of algebraic equations Introduction to Interval Computations Automatic generation of numerical redundancies for non-linear constraint solving Applying interval arithmetic to real CLP(intervals) revisited Algorithmes d'Acc-l-ration de la Convergence: -tude Num-riques Derivation of extrapolation algorithms based on error estimates Extrapolation Methods A general extrapolation procedure revisited Improved bounds on the complexity of kB-consistency Constraint logic programming on numeric intervals A note on partial consistencies over continuous domains solving techniques Interlog 1.0: Guide d'utilisation Constraint propagation with interval labels Arc consistency for continuous variables Local consistency for ternary numeric constraints A sufficient condition for backtrack-bounded search Consistances locales et transformations symboliques de contraintes d'intervalles Global Optimization Using Interval Analysis Consistency techniques for continuous constraints Constraint reasoning based on interval arithmetic: The tolerance propagation approach ILOG Solver 4.0 Continuous Problems Computational Complexity and Feasibility of Data Processing and Interval Computations Acceleration methods for numeric CSPs Consistency techniques for numeric CSPs Dynamic optimization of interval narrowing algorithms Boosting the interval narrowing algorithm Consistency in networks of relations Interval Analysis Interval Methods for Systems of Equations A simple derivation of the Hansen-Bliek-Rohn-Ning-Kearfott enclosure for linear interval equations Extending prolog with constraint arithmetic on real intervals A constraints satisfaction approach to a circuit design problem Computer Methods for the Range of Functions Experiments using interval analysis for solving a circuit design problem Hierarchical arc consistency applied to numeric constraint processing in logic programming Solving polynomial systems using branch and prune approach Modeling Language for Global Optimization --TR A sufficient condition for backtrack-bounded search Constraint propagation with interval labels Constraint reasoning based on interval arithmetic Arc-consistency for continuous variables CLP(intervals) revisited A derivation of extrapolation algorithms based on error estimates Solving Polynomial Systems Using a Branch and Prune Approach Acceleration methods of numeric CSPc Synthesizing constraint expressions A Constraint Satisfaction Approach to a Circuit Design Problem A Note on Partial Consistencies over Continuous Domains --CTR Yahia Lebbah , Claude Michel , Michel Rueher, Using constraint techniques for a safe and fast implementation of optimality-based reduction, Proceedings of the 2007 ACM symposium on Applied computing, March 11-15, 2007, Seoul, Korea Yahia Lebbah , Claude Michel , Michel Rueher, A Rigorous Global Filtering Algorithm for Quadratic Constraints, Constraints, v.10 n.1, p.47-65, January 2005

extrapolation methods;interval analysis;propagation;acceleration methods;filtering techniques;numeric constraint satisfaction problem;strong consistency;interval arithmetic;nonlinear equations;pruning

604379

A Simulation Study of Decoupled Vector Architectures.

Decoupling techniques can be applied to a vector processor, resulting in a large increase in performance of vectorizable programs. We simulate a selection of the Perfect Club and Specfp92 benchmark suites and compare their execution time on a conventional single port vector architecture with that of a decoupled vector architecture. Decoupling increases the performance by a factor greater than 1.4 for realistic memory latencies, and for an ideal memory system with zero latency, there is still a speedup of as much as 1.3. A significant portion of this paper is devoted to studying the tradeoffs involved in choosing a suitable size for the queues of the decoupled architecture. The hardware cost of the queues need not be large to achieve most of the performance advantages of decoupling.

Introduction Recent years have witnessed an increasing gap between processor speed and memory speed, which is due to two main reasons. First, technological improvements in cpu speed have not been matched by similar improvements in memory chips. Second, the instruction level parallelism available in recent processors has increased. Since several instructions are being issued at the same processor cycle, the total amount of data requested per cycle to the memory system is much higher. These two factors have led to a situation where memory chips are on the order of 10 to a 100 times slower than cpus and where the total execution time of a program can be greatly dominated by average memory access time. Current superscalar processors have been attacking the memory latency problem through basically three main types of techniques: caching, multithreading and decoupling (which, sometimes, may appear together). Cache-based superscalar processors reduce the average memory access time by placing the working set of a program in a faster level in the memory hierarchy. Software and hardware techniques such as [5, 23] have been devised to prefetch data from high levels in the memory hierarchy to lower levels (closer to the cpu) before the data is actually needed. On top of that, program transformations such as loop blocking [16] have proven very useful to fit the working set of a program into the cache. Recently, address and data prediction receive much attention as a potential solution for indirectly masking memory latency [21]. Multithreaded processors [1, 30] attack the memory latency problem by switching between threads of computations so that the amount of parallelism exploitable aug- ments, the probability of halting the cpu due to a hazard decreases, the occupation of the functional units increases and the total throughput of the system is improved. While each single thread still pays latency delays, the cpu is (presumably) never idle thanks to this mixing of different threads of computation. Decoupled scalar processors [27, 25, 18] have focused on numerical computation and attack the memory latency problem by making the observation that the execution of a program can be split into two different tasks: moving data in and out of the processor and executing all arithmetic instructions that perform the program com- putations. A decoupled processor typically has two independent processors (the address processor and the computation processor) that perform these two tasks asynchronously and that communicate through architectural queues. Latency is hidden by the fact that usually the address processor is able to slip ahead of the computation processor and start loading data that will be needed soon by the computation processor. This excess data produced by the address processor is stored in the queues, and stays there until it is retrieved by the computation processor. Vector machines have traditionally tackled the latency problem by the use of long vectors. Once a (memory) vector operation is started, it pays for some initial (po- tentially long) latency, but then it works on a long stream of elements and effectively amortizes this latency across all the elements. Although vector machines have been very successful during many years for certain types of numerical calculations, there is still much room for improvement. Several studies in recent years [24, 8] show how the performance achieved by vector architectures on real programs is far from the theoretical peak performance of the machine. In [8] it is shown how the memory port of a single-port vector computer was heavily underutilized even for programs that were memory bound. It also shows how a vector processor could spend up to 50% of all its execution cycles waiting for data to come from memory. Despite the need to improve the memory response time for vector architectures, it is not possible to apply some of the hardware and software techniques used by scalar processors because these techniques are either expensive or exhibit a poor performance in a vector context. For example, caches and software pipelining are two techniques that have been studied [17, 19, 28, 22] in the context of vector processors but that have not been proved useful enough to be in widespread use in current vector machines. The conclusion is that in order to obtain full performance of a vector processor, some additional mechanism has to be used to reduce the memory delays (com- ing from lack of bandwidth and long latencies) experienced by programs. Many techniques can be borrowed from the superscalar microprocessor world. In this paper we focus on decoupling, but we have also explored other alternatives such as multithreading [12] and out-of-order execution [13]. The purpose of this paper is to show that using decoupling techniques in a vector processor [11], the performance of vector programs can be greatly improved. We will show how, even for an ideal memory system with zero latency, decoupling provides a significant advantage over standard mode of operation. We will also present data showing that for more realistic latencies, decoupled vector architectures perform substantially better than non-decoupled vector architectures. Another benefit of JOURNAL OF SUPERCOMPUTING 3 decoupling is that it also allows to tolerate latencies inside the processor, such as functional unit and register crossbar latencies. This paper is organized as follows. Section 2 describes both the baseline and decoupled architectures studied throughout this paper. In section 3 we discuss our simulation environment and the benchmark programs used in the experiments presented. Section 4 provides a background analysis of the performance of a traditional vector machine. In section 5 we detail the performance of our decoupled vector proposal. Finally, section 6 presents our conclusions and future lines of work. 2. Vector Architectures and Implementations This study is based on a traditional vector processor and numerical applications, primarily because of the maturity of compilers and the availability of benchmarks and simulation tools. We feel that the general conclusions will extend to other vector applications, however. The decoupled vector architecture we propose is modeled after a Convex C3400. In this section we describe the base C3400 architecture and implementation (henceforth, the reference architecture), and the decoupled vector architecture (generically referred to as DVA). The main implication of the election of a C3400 is that this study is restricted to the class of vector computers having one memory port and two functional units. It is also important to point out that we used the output of the Convex compilers to evaluate our decoupled architecture. This means that the proposal studied in this paper is able to execute in a fully transparent manner an already existing instruction set. 2.1. The Reference Architecture The Convex C3400 [7] consists of a scalar unit and an independent vector unit (see fig. 1). The scalar unit executes all instructions that involve scalar registers S registers), and issues a maximum of one instruction per cycle. The vector unit consists of two computation units (FU1 and FU2) and one memory accessing unit (MEM). The FU2 unit is a general purpose arithmetic unit capable of executing all vector instructions. The FU1 unit is a restricted functional unit that executes all vector instructions except multiplication, division and square root. Both functional units are fully pipelined. The vector unit has 8 vector registers which hold up to 128 elements of 64 bits each. The eight vector registers are connected to the functional units through a restricted crossbar. Pairs of vector registers are grouped in a register bank and share two read ports and one write port that links them to the functional units. The compiler is responsible for scheduling vector instructions and allocating vector registers so that no port conflicts arise. @ Fetch Decode unit S-regs A-regs R-XBAR W-XBAR Figure 1. The reference vector architecture modeled after a Convex C3400. 2.2. The Decoupled Vector Architecture The decoupled vector architecture we propose uses a fetch processor to split the incoming, non-decoupled, instruction stream into three different decoupled streams (see fig. 2). Each of these three streams goes to a different processor: the address processor (AP), that performs all memory accesses on behalf of the other two processors, the scalar processor (SP), that performs all scalar computations and the vector processor (VP), that performs all vector computations. The three processors communicate through a set of implementational queues and proceed independently. This set of queues is akin to the implementational queues that can be found in the floating point part of the R8000 microprocessor[15]. The main difference of this decoupled architecture with previous scalar decoupled architectures such as the ZS- 1 [26], the MAP-200 [6], PIPE [14] or FOM [4], is that it has two computational processors instead of just one. These two computation processors, the SP and the VP, have been split due to the very different nature of the operands on which they work (scalars and vectors, respectively). The fetch processor fetches instructions from a sequential, non-decoupled instruction stream and translates them into a decoupled version. The translation is such that each processor can proceed independently and, yet, synchronizes through the communication queues when needed. For example, when a memory instruction that loads register v5 is fetched by the FP, it is translated into two pseudo-instructions: a load instruction, which is sent to the AP, that will load data into the vector load data queue (VLDQ, queue no. 1 in fig. 2), and a qmov instruction, sent to the VP, that dictates a move operation between the VLDQ and the final destination register v5. It is important to note that the qmov's generated by the FP are not JOURNAL OF SUPERCOMPUTING 5 (2) @ (1) Figure 2. The decoupled vector architecture studied in this paper. Queue names: (1) vector load data queue -VLDQ, (2) vector store data queue -VSDQ, (3) address load queue -ALQ, (4) address store queue -ASQ, (5) scalar load data queue -SLDQ, (6) scalar store data queue -SSDQ, Scalar-Address Control Queues, Vector-Address Control Queue, (10/11) Scalar-Vector Control Queues "instructions" in the real sense, i.e., they do not belong to the programmer visible instruction set. These qmov opcodes are hidden inside the implementation. Note that the total hardware added to the original reference architecture shown in figure 1 consists only of the communication queues and a private decode unit for each one of the three processors. The resources inside each processor are the same in the decoupled vector architecture and in the reference architecture. It is worth noting, though, that while most queues added are scalar queues and, therefore, require a small amount of extra area, the VLDQ and VSDQ hold full vector registers (queues 2 and 3 in Fig. 2). That is, each slot in these queues is equivalent to a normal vector register of 128 elements, thus requiring 1Kb of storage space. One of the key points in this architecture will be to achieve good performance with relatively few slots in these two queues. The address processor performs all memory accesses, both scalar and vector, as well as all address computations. Scalar memory accesses go first through a scalar cache that holds only scalar data. Vector accesses do not go through the cache and access main memory directly. There is only one pipelined port to access memory that has to be shared by all memory accesses. The address processor inserts load instructions into the Address Load Queue (ALQ) and store instructions into the Address Store Queue (ASQ). Stores stay in the queue until their associated data 6 ROGER ESPASA AND MATEO VALERO shows up either in the output queue of the VP (the vector store data queue - VSDQ), or in the output queue of the SP (the scalar store data queue -SSDQ). When either a load or a store becomes ready, i.e, it has no dependencies and its associated data, if necessary, is present, it is sent over the address bus as soon as it becomes available. In the case of having both a load and a store ready, the AP always gives priority to loads. To preserve the sequential semantics of a program, the address processor needs to ensure a safe ordering between the memory instructions held in the ALQ and ASQ. All memory accesses are processed in two steps: first, their associated "memory region" is computed. Second, this region is used to disambiguate the memory instruction against all previous memory instructions still being held in the address queues of the AP. Using this disambiguation information, a dependency scoreboard is maintained. This scoreboard ensures that (1) all loads are executed in-order, (2) all stores are executed in-order and (3) loads can execute before older stores if their associated memory regions do not overlap at all. When dependences are found, the scoreboard guarantees that loads and stores will be performed in the original program order so that correctness is guaranteed. A "memory region" is defined by a 5-tuple: h@ are the start and end addresses, resp., of a consecutive region of bytes in memory, and vl, vs, and sz are the vector length, vector stride and access granularity needed by vector memory operations. The end address, @ 2 , is computed as @ 1 sz. For scalar memory accesses, vl is set to 1 and vs to 0. For the special case of gathers and scatters, which can not be properly characterized by a memory region, @ 1 is set to 0 and @ 2 is set to 2 so that the scoreboard will find a dependence between a gather/scatter and all previous and future memory instructions. The vector processor performs all vector computations. The main difference between the VP and the reference architecture is that the VP has two functional units dedicated to move data in and out of the processor. This two units, the QMOV units, are able to move data from the VLDQ data queue (filled by AP) into the vector registers and move data from the registers into the VADQ (which will be drained by AP sending its contents to memory). We have included two QMOV units instead of one because otherwise the VP would be paying a high overhead in some very common sequences of code, when compared to the reference architecture. The set of control queues connecting the three processors, queues 7-11 in Fig. 2, are needed for those instructions that have mixed operands. The most common case is in vector instructions, that can have a scalar register as a source operand (i.e., mul v0,s3 -? v5). Other cases include mixed A- and S- register instructions, vector gathers that require an address vector to be sent to the AP, and vector reductions that produce a scalar register as a result. JOURNAL OF SUPERCOMPUTING 7 3. Methodology 3.1. Simulation Environment To asses the performance benefits of decoupled vector architectures we have taken a trace driven approach. The Perfect Club and Specfp92 programs have been chosen as our benchmarks [3]. The tracing procedure is as follows: the Perfect Club programs are compiled on a Convex C3480 [7] machine using the Fortran compiler (version 8.0) at optimization level -O2 (which implies scalar optimizations plus vectorization). Then the executables are processed using Dixie [9], a tool that decomposes executables into basic blocks and then instruments the basic blocks to produce four types of traces: a basic block trace, a trace of all values set into the vector length register, a trace of all values set into the vector stride register and a trace of all memory references (actually, a trace of the base address of all memory references). Dixie instruments all basic blocks in the program, including all library code. This is especially important since a number of Fortran intrinsic routines (SIN, COS, EXP, etc.) are translated by the compiler into library calls. This library routines are highly vectorized and tuned to the underlying architecture and can represent a high fraction of all vector operations executed by the program. Thus it is essential to capture their behavior in order to accurately model the execution time of the programs. Once the executables have been processed by Dixie, the modified executables are run on the Convex machine. This runs produce the desired set of traces that accurately represent the execution of the programs. This trace is then fed to two different simulators that we have developed: the first simulator is a model of the Convex C34 architecture and is representative of single memory port vector computers. The second simulator is an extension of the first, where we introduce decoupling. Using these two cycle-by-cycle simulators, we gather all the data necessary to discuss the performance benefits of decoupling. 3.2. The benchmark programs Because we are interested in the benefits of decoupling for vector architectures, we selected benchmark programs that are highly vectorizable (- 70%). From all programs in the Perfect and Specfp92 benchmarks we chose the 10 programs that achieve at least 70% vectorization. Table 1 presents some statistics for the selected Perfect Club and Specfp92 programs. Column number 2 indicates to what suite each program belongs. Column 3 presents the total number of memory accesses, including vector and scalar and load and store accesses. Next column is the total number of operations performed in vector mode. Column 5 is the number of scalar instructions executed. The sixth column is the percentage of vectorization of each program. We define the percentage of vectorization as the ratio between the number of vector operations and the total number of operations performed by the program. Finally column seven presents the average vector length used by vector instructions, and is the ratio of vector operations over vector instructions. Table 1. Basic operation counts for the Perfect Club and programs (Columns 3-5 are in millions). Mem Vect Scal % avg. Program Suite Ops Ops Ins Vect VL hydro2d Spec 1785 2203 23 99.0 101 arc2d Perf. 1959 2157 flo52 Perf. 706 551 su2cor Spec 1561 1862 66 95.7 125 bdna Perf. 795 889 128 86.9 81 trfd Perf. 826 438 156 75.7 22 dyfesm Perf. 502 298 108 74.7 21 The most important thing to remark from table 1 is that all our programs are memory bound when run on the reference machine. If we take column labeled "Vect Ops" and divide it by 2 we get the minimum number of cycles required to execute all vector computations on the two vector functional units available. Comparing now column "Mem Ops" against the result of our division we will see that the bottleneck for all these programs is always the memory port. That is, the absolute minimum execution time for each of these programs is determined by the total amount of memory accesses it performs. This remark is worth keeping in mind, since, as following sections will show, even if the memory port is the bottleneck for all programs, its usage is not always as good as one would intuitively expect. 4. Bottlenecks in the Reference Architecture First we present an analysis of the execution of the ten benchmark programs when run through the reference architecture simulator. Consider only the three vector functional units of our reference architecture (FU2, FU1 and MEM). The machine state can be represented with a 3-tuple that represents the individual state of each one of the three units at a given point in time. For example, the 3-tuple hFU2; FU1;MEM i represents a state where all units are working, while represents a state where all vector units are idle. The execution time of a program can thus be split into eight possible states. Figure 3 presents the splitting of the execution time into states for the ten benchmark programs. We have plotted the time spent in each state for memory latencies of 1, 20, 70, and 100 cycles. From this figure we can see that the number of cycles where the programs proceed at peak floating point speed (states low. The number of cycles in these states changes relatively little as the memory latency increases, so the fraction of JOURNAL OF SUPERCOMPUTING 9 swm25620006000 Execution cycles hydro2d10003000 arc2d10003000 nasa710003000 < , > Execution cycles dyfesm5001500 Figure 3. Functional unit usage for the reference architecture. Each bar represents the total execution time of a program for a given latency. Values on the x-axis represent memory latencies in cycles. fully used cycles decreases. Memory latency has a high impact on total execution time for programs dyfesm, and trfd and flo52, which have relatively small vector lengths. The effect of memory latency can be seen by noting the increase in cycles spent in state h ; ; i. The sum of cycles corresponding to states where the MEM unit is idle is quite high in all programs. These four states correspond to cycles where the memory port could potentially be used to fetch data from memory for future computations. Figure 4 presents the percentage of these cycles over total execution time. At latency 70, the port idle time ranges between 30% and 65% of total execution time. All 10 benchmark programs are memory bound when run on a single port vector machine with two functional units. Therefore, these unused memory cycles are not the result of a lack of load/store work to be done. 5. Performance of the DVA In this section we present the performance of the decoupled vector architecture versus the reference architecture (REF). We first start by ignoring all latencies of the functional units inside the processor and concentrate on the study of the effects of main memory latency (sections 5.1-5.5). This study will determine the most swm256 hydro2d arc2d flo52 nasa7 su2cor tomcatv bdna trfd dyfesm2060 Idle Memory port% 170 Figure 4. Percentage of cycles where the memory port was idle, for 4 different memory latencies. cost-effective parameters that achieve the highest performance. Then we proceed to consider the effect of arithmetic functional units and register crossbar latencies in execution time (section 5.6). We will first show that decoupling tolerates very well memory latencies and is also useful for tolerating the smaller latencies inside the processor. We will start by defining a DVA architecture with infinite queues and no latency delays -the Unbounded DVA, or UDVA for short- that we will compare to the reference architecture. Then we will introduce limitations into the UDVA, such as branch misprediction penalties, limited queue sizes and real functional unit laten- cies, step by step to see the individual effect of each restriction. After all these steps we will reach a realistic version of the DVA - the RDVA- that will be compared against the REF and UDVA machines. 5.1. UDVA versus REF The Unbounded DVA architecture (UDVA) is a version of the decoupled architecture that has all of its queues set to a very large value (128 slots) and no latency delays. Moreover, a perfect branch prediction model is assumed. The I-cache is not modeled in any of the following experiments, since our previous data indicates a very low pressure on the I-cache [10]. All arithmetic functional units, both scalar an vector, have a 1 cycle latency. The vector register file read and write crossbars have no latency and there is no startup penalty for vector instructions. The benefits of decoupling can be seen in fig. 5. For each program we plot the total execution time of the UDVA and the REF architectures when memory latency is varied between 1 and 100 cycles. In each graph we also show the minimum absolute execution time that can theoretically be achieved (curve "IDEAL", along the bottom of each graph). To compute the IDEAL execution time for a program we use the total number of cycles con- JOURNAL OF SUPERCOMPUTING 11 swm2565060cycles x flo5210cycles cycles dyfesm10cycles REF UDVA IDEAL Figure 5. UDVA versus Reference architecture for the benchmark programs. sumed by the most heavily used vector unit (FU1, FU2, or MEM). Thus, in IDEAL we essentially eliminate all data and memory dependences from the program, and consider performance limited only by the most saturated resource across the entire execution. The overall results suggest two important points. First, the DVA architecture shows a clear speedup over the REF architecture even when memory latency is just 1 cycle. Even if there is no latency in the memory system, the decoupling produces a similar effect as a prefetching technique, with the advantage that the AP knows which data has to be loaded (no incorrect prefetches). The second important point is that the slopes of the execution time curves for the reference and the decoupled architectures are substantially different. This implies that decoupling tolerates long memory delay much better than current vector architectures. Memory latency in cycles1.21.6 hydro2d arc2d su2cor bdna trfd dyfesm Figure 6. Speedup of the DVA over the Reference architecture for the benchmark programs Overall, decoupling is helping to minimize the number of cycles where the machine is halted waiting for memory. Recall from section 4 that the execution time of the program could be partitioned into eight different states. Decoupling greatly reduces the cycles spent in state h ; ; i. To summarize the speedups obtained, fig. 6 presents the speedup of the DVA over the REF architecture for each particular value of memory latency. Speedups (at latency 100) range from a 1.32 for TOMCATV to a 1.70 for DYFESM. 5.2. Reducing IQ length The first limitation we introduce into the UDVA is the reduction of the instruction queues that feed the three computational processors (AP, SP, VP). In this section we look at the slowdown experienced by the UDVA when the size of the APIQ, SPIQ and VPIQ queues is reduced from 128 instructions to 32, 16, 8 and 4 instructions only. In order to reduce the amount of simulation required, we have chosen to fix the value of the memory latency parameter at 50 cycles. As we have seen in the previous section, the UDVA tolerates very well a wide range of memory latencies. Thus we expect this value of 50 cycles to be quite representative of the full 1-100 latency range. The size of the instruction queues is very important since it gives an upper bound on the occupation of all the queues in the system. For example, it determines the maximum number of entries that we can have waiting in the load address queues. Figure 7 presents the slowdown with respect to the UDVA for our ten benchmarks when the three instruction queues are reduced to 32, 16, 8 and 4 slots. From fig. 7 we can see that the performance for 128-, 32- and 16-entries instruction queues is virtually the same for all benchmarks. From these numbers, we decided to set the IQ length to 16 entries for the rest of experiments presented in this paper. This size is in line with the typical instruction queues found in current microprocessors [31]. JOURNAL OF SUPERCOMPUTING 13 swm256 hydro2d arc2d flo52 nasa7 su2cor tomcatv bdna trfd dyfesm1.021.06 Figure 7. Slowdown experienced by UDVA when reducing the IQ size. swm256 hydro2d arc2d flo52 nasa7 su2cor tomcatv bdna trfd dyfesm1.011.03 Slowdown Figure 8. Slowdown due to branch mispredictions for three models of speculation. 5.3. Effects of branch prediction In this section we look at the negative effects introduced by branch mispredictions. The branch prediction mechanism evaluated is a direct-mapped BTB holding for each entry the branch target address and a 2-bit predictor (the predictor found in [20]). We augmented the basic BTB mechanism with an 8-deep return stack (akin to the one found in [2]). We evaluated the accuracy of the branch predictor for a 64 entries BTB. The accuracy varies a lot across the set of benchmarks. Programs FLO52 and NASA7 come out with the worst misprediction rate (around 30%) while TOMCATV has less than 0.4% of mispredicted branches. Nonetheless, the misprediction rate is rather high for a set of programs that are considered to have an "easy" jumping pattern (numerical codes tend to be dominated by DO-loops). This is due to the combination of two facts: first, vectorization has reduced the absolute total number of branches performed by the programs in an unbalanced way. The number of easily- predictable loop branches has been diminished by a factor that is proportional to the vector length (could be as high as 128) while the difficult branches found in the remaining scalar portion of the code are essentially the same. The second factor is that we are using a very small BTB compared to what can be found in current superscalar microprocessors, where a typical BTB could have up to 4096 entries [29]. Although the prediction accuracy is not very good, the impact of mispredicted branches on total execution time is very small. Figure 8 presents the slowdown 14 ROGER ESPASA AND MATEO VALERO due to mispredicted branches relative to the performance of the architecture from section 5.2. Since the prediction accuracy was not very high, we tested the benefit that could be obtained by being able to speculate across several branches. In fig. 8 the bars labeled "u=1" correspond to an architecture that only allows one unresolved branch. Bars labeled "u=2" and "u=3" correspond to being able to speculate across 2 and 3 branches respectively. A first observation is that the impact of mispredicted branches is rather low. See how while FLO52 has a 30% misprediction rate, the total impact of those mispredicted branches is below 0.5%. A second observation is that while speculating across multiple branches provides some benefits, specially for DYFESM, its cost is certainly not justified. The simplicity of only having one outstanding branch to be resolved is a plus for vector architectures. All the simulations in the following sections have been performed using a 64-entry BTB and allowing only 1 unresolved branch. 5.4. Reducing the vector queues length 5.4.1. Vector Load Data Queue This section will look at the usage of the vector load data queue. The goal is to determine a queue size that achieves almost the same performance as the 128-slot queue used in the previous sections and yet minimizes as much as possible hardware costs. Figure 9 presents the distribution of busy slots in the VLDQ for the benchmark programs. For each program we plot three distributions corresponding to three different memory latency values. Each bar in the graphs represents the total number of cycles that the VLDQ had a certain number of busy slots For example, for trfd at latency 1, the VLDQ was completely empty (zero busy slots) for around 500 millions of cycles. From fig. 9 we can see that it is not very common to use more than 6 slots. Except for swm256 and tomcatv, 6 slots are enough to cover around 85-90% of all cycles. When latency is increased from 1 cycle to 50 and 100 cycles, the graphs show a shift of the occupation towards higher number of slots. As an example, consider programs arc2d, nasa7 and su2cor. For 1 cycle memory latency, these programs have typically 2-3 busy slots. When latency increases, these three programs show an increase in the total usage of the VLDQ, and they typically use around 4-5 slots. As expected, the longer the memory latency, the higher the number of busy slots, since the memory system has more outstanding requests and, therefore, needs more slots in the queue. The execution impact of reducing the VLDQ size can be seen in fig. 10. As expected from the data seen in fig. 9, reducing the queue size to 16 or 8 slots is not noticeable for most programs. Going down to 4 slots affects mostly NASA7 and BDNA but the impact is less than a 1%. Further reducing the VLDQ to 2 slots would start to hurt performance although not very much. The worst case would be again for NASA7 with around a 4% impact. As we have already discussed, 2 slots is clearly a lower bound on the size of the VLDQ to accommodate most memory JOURNAL OF SUPERCOMPUTING 15 swm256500cycles arc2d100300500 flo52100200cycles tomcatv100cycles dyfesm100300cycles Figure 9. Busy slots in the VLDQ for the benchmark programs for three different memory latency values. bound loops. Reducing that queue to 1 slot would stop most of the decoupling effect present in the architecture. Looking at all the data presented in this section we decided to pick a 4 slots VLDQ. All following sections use this size for the VLDQ. 5.4.2. Vector Store Data Queue The usage of the vector store data queue presents a very different pattern from the VLDQ. Recall that the AP always tries to give priority to load operations in front of stores. This has the effect of putting much more pressure on the VSDQ, which can, at some points, become full (even with 128 slots!). This situation is not has unusual as it may seem. As long as the AP encounters no dependencies between a load and a store and as long swm256 hydro2d arc2d flo52 nasa7 su2cor tomcatv bdna trfd dyfesm1.021.06 Figure 10. Slowdown due to reducing the VLDQ size (relative to section 5.3). as there are loads to dispatch, no stores will be retrieved from the VSDQ and sent to memory. Thus the occupation of the VSDQ is much higher than that of the VLDQ. Figure 11 presents the distribution of busy slots in the VSDQ for the benchmark programs. As we did for the load queue, we plot three distributions corresponding to three different memory latency values. Each bar in the graphs represents the total number of cycles that the VSDQ had a certain number of busy slots. To make the plots more clear, we have pruned some of the graphs. Next to the name of each program we indicate the quartile amount being shown. For example, the full set of data is shown for nasa7 (q=100%) while the bars in the hydro2d graph only present 95% of all available data (the rest of the data set was to small to be seen on the plot). To compensate for this loss of information, each graph also includes the maximum value that the X axis took for that particular program. Again, for hydro2d, the graph shows that the maximum occupation of the VSDQ reached 118 slots, although the X-axis of the plot only goes up to 50. Note that for 6 of the programs, at some point the queue was completely filled (128 full slots), although the most common occupation ranges between slots. For the other 4 programs, the occupation of the queue is bounded. For bdna, with a maximum occupation of 34 slots in the queue and su2cor (maximum 23), the bounding is mostly due to their high percentage of spill code. Each time a vector load tries to recover a vector from the stack previously spilled by a store, the AP detects a dependency a needs to update the contents of memory draining the queue. This heavily limits the amount of old stores that are kept in the VSDQ. Programs trfd and dyfesm are qualitatively different. These two program simply don't decouple very well. Program dyfesm has a recurrence that forces the three main processors, the AP, the SP and the VP to work in lock step, thus typically allowing only a maximum of 1 full slot. Program trfd has at its core a triangular matrix decomposition. The order in which the matrix is accessed makes each iteration of the main loop dependent on some of the previous iterations, which causes a lot of load-store dependencies in the queues. These dependencies are resolved, as JOURNAL OF SUPERCOMPUTING 17 (q=99.9%)400800cycles hydro2d (q=95%)400800 arc2d (q=94.2%)1000 cycles 28 78 128 su2cor (q=99.9%)400800 tomcatv (q=100%)50cycles trfd (q=100%)5000 1 dyfesm (q=99.9%)200600cycles Figure 11. Busy slots in the VSDQ for the benchmarks for three different memory latency values. in the case for spill code, by draining the queue and updating memory. Thus, the queue does never reach a large occupation. The execution impact of reducing the VSDQ size can be seen in fig. 12. The bars show that the amount of storage in the VSDQ is not very important to performance. This is mostly due to the fact that we are in a single-memory port environment. No matter how we reorder loads and stores between themselves, every single store will have to be performed anyway. Thus, sending a store to memory at the point where its data is ready or later does not change much the overall computation rate. From all the data presented in this section we selected the 4-slot VSDQ for all following experiments. swm256 hydro2d arc2d flo52 nasa7 su2cor tomcatv bdna trfd dyfesm1.001.02 Figure 12. Slowdown due to reducing the VSDQ queues size (relative to section 5.4.1). swm256 hydro2d arc2d flo52 nasa7 su2cor tomcatv bdna trfd dyfesm1.00Slowdown Figure 13. Slowdown due to reducing the Scalar queues size (relative to section 5.4.2). 5.5. Reducing the scalar queues length In this section we will look at the impact of reducing the size of the various scalar queues in the system. Looking back to fig. 2 we will be reducing queues numbered 3-8 and 10-11 from 128 slots down to 16 slots. Queue number 9, the VACQ, VP- to-AP Control queue, will be reduced from 128 slots to just 1 slot. Note that this queue holds one full vector register used in gather/scatter operations. The size is chosen because it is reasonably close to what modern out-of-order superscalar processors have in their queues [31]. The impact of those reductions can be seen in fig. 13. Overall, using an 8 entry queue for all the scalar queues is enough to sustain the same performance as the 128 entry queues. Even for small 2 entry queues, the slowdown is around 1.01 for only 3 programs, dyfesm, bdna and nasa7. Nonetheless, it has to be beard in mind that our programs are heavily vectorized. A small degradation in performance on the scalar side is tempered by the small percentage of scalar code present in our benchmarks. In order to make a safe decision we took a 16 entries queue for all the scalar queues present in the architecture. JOURNAL OF SUPERCOMPUTING 19 Table 2. Latency parameters for the vector and scalar functional units. Parameters Latency Scal Vect (int/fp) vector startup - 1 read x-bar - 2 add 1/2 6 mul 5/2 7 logic/shift 1/2 4 div 34/9 20 sqrt 34/9 20 5.6. Effects of functional unit latencies In this section we will look at the effects of latencies inside the computation processors of our architecture. So far, all models simulated had all of their functional units using a 1 cycle latency and the vector registers read/write crossbars were modeled as if it was free to go through them. This section will proceed in three steps. First we will add to our architecture the latencies of the vector functional units. Table 2 shows the values chosen. In a second step, we will add a penalty of 1 cycle of vector startup for each vector operation. In a third step we will add 2 cycles of vector read crossbar latency and then we will add 2 cycles of vector write crossbar latencies. In the last step, we will set the latencies of the scalar units to those also shown in table 2. Figure 14 shows as a set of stacked bars, the degradation in performance as each of the aforementioned effects is added. The bar at the bottom, labeled "vect. lat" represents the slowdown relative to section 5.5. The following bar, labeled "startup", is the slowdown with respect to the performance of the "vect. lat" bar. Similarly for each of the following bars. Thus, the total height of each bar is the combined slowdown of all these effects. Figure 14 shows two different behaviors. For seven out of the ten programs, latencies have a very small impact (below 5%). This is due to the fact that decoupling is not only good for tolerating memory latencies but, in general, it helps in covering up the latencies inside the processor. On the other hand, two programs, trfd and dyfesm show slowdowns as bad as 1.11 and 1.15, respectively. The behavior of these two programs is not surprising given what we already saw in section 5.4.1. Both trfd and dyfesm have difficulties in decoupling because of the inter-iteration dependences of trfd and the recurrences in dyfesm. As we saw, both of them only achieve a very small occupation of the vector load data queue which indicates a bad degree of decoupling. If we couple this fact with the swm256 hydro2d arc2d flo52 nasa7 su2cor tomcatv bdna trfd dyfesm1.05Slowdown scal. lat. r xbar startup vect. lat. Figure 14. Slowdown due to modeling arithmetic unit latencies and vector pipeline crossbars. (relative to section 5.5). relatively low vector lengths of trfd and dyfesm, we see that any cycle added to the vector dependency graph is typically enlarging the critical path of the program. Going into the detailed breakdown, Fig. 14 shows that the vector functional unit latencies have the highest impact of all latencies added in this section. It is worth noting, though, that the order in which the latencies are added might have some impact on the relative importance of each individual category. Nonetheless, since the vector latencies units are the largest of the latencies added and since the programs are highly vectorized they are the group that most likely will impact performance, as fig. 14 confirms. The startup penalty is only seen in programs trfd and dyfesm, where its impact is less than a 1%. The vector register file read/write crossbar latencies have an impact on all programs, except, again, swm256 and tomcatv. Typically both latencies have the same amount of impact, being between 0.5 and 1 percentage points for the most vectorized codes and around 2-3 percentage points in the less vectorized trfd and dyfesm. The scalar latencies have a low impact on all programs, partly due to them being shorter than the vector ones, partly due to the small fraction of scalar code and partly due to scalar latencies masked under other vector latencies. We decided to compare the impact of functional unit latencies in the reference machine and in the DVA machine. To do so, we simulate a reference machine with no latencies at all and a reference machine with the standard latencies and compute the resulting slowdowns. Then we compare these slowdowns to the slowdowns of Fig. 14, which we just presented. The result of the comparison can be seen in Fig. 15. The results show that in all cases the effect of functional unit latencies is much worse in the in-order reference machine that in the decoupled machine. Since decoupling introduces some form of dynamic scheduling, it can hide latencies that were previously in the critical path by performing memory loads in advance. JOURNAL OF SUPERCOMPUTING 21 swm256 hydro2d arc2d flo52 nasa7 su2cor tomcatv bdna trfd dyfesm1.051.15 Slowdown UDVA REF Figure 15. Comparison of functional unit latency impact between the UDVA and REF machines. 5.7. RDVA versus REF With the data presented in the last section, we have reached a realistic implementation of the originally proposed UDVA. This realistic version will be referred to as RDVA and its main parameters are as follows: all instruction queues and scalar queues are 16 entries long. The address queues in the AP are also entries long. The latencies used in the functional units and in the read/write crossbars of the register file are those shown in table 2. The VLDQ and VSDQ each have 4 slots, and the control queue connecting the VP and the AP has a single slot. The branch prediction mechanism is a 64 entry BTB with at most 1 unresolved branch being supported. In this section we will re-plot a full-scale comparison of UDVA, RDVA and REF for several latencies. Figure 16 presents the data for the three architectures when memory latency is varied from 1 cycle to 100 cycles. For almost all programs, the difference between the UDVA and RDVA is rather small, and their slopes are relatively parallel. For swm256, the difference is almost 0. For programs hydro2d, tomcatv, arc2d and su2cor, the slowdown between RDVA and UDVA is less than (respectively, it is 1.029, 1.031, 1.037 and 1.044). Programs flo52, bdna and nasa7 have a higher slowdown and, moreover, the slope of the curve for the RDVA performance starts diverging from the UDVA at high values of latency. Finally, dyfesm and trfd, as seen in previous sections, take a significant performance hit when going from the UDVA to the RDVA. 6. Summary and Future Work In this paper we have described a basic decoupled vector architecture (DVA) that uses the principles of decoupling to hide most of the memory latency seen by vector processors. 22 ROGER ESPASA AND MATEO VALERO swm2565060cycles x flo5210cycles x cycles x dyfesm10cycles x REF RDVA UDVA IDEAL Figure 16. Comparison of REF, UDVA and RDVA execution times for several latencies. The DVA architecture shows a clear speedup over the REF architecture even when memory latency is just 1 cycle. This speedup is due to the fact that the AP slips ahead of the VP and loads data in advance, so that when the VP needs its input operand they are (almost) always ready in the queues. Even if there is no latency in the memory system, this "slipping" produces a similar effect as a prefetching technique, with the advantage that the AP knows which data has to be loaded (no incorrect prefetches). Thus, the partitioning of the program into separate tasks helps in exploiting more parallelism between the AP and VP and translates into an increase in performance, even in the absence of memory latency. Moreover, as we increase latency, we see how the slopes of the curves of the execution time of the benchmarks remain fairly stable, whereas the REF architecture is much more sensitive to the increase in memory latency. When memory latency is set to JOURNAL OF SUPERCOMPUTING 23 50 cycles, for example, speedups of the RDVA over the REF machine are in the range 1.18-1.40, and when latency is increased to 100 cycles, speedups go as high as 1.5. We have seen that this speed improvements can be implemented with a reasonable cost/performance tradeoff. Section 5.4 has shown how the length of the queues does not need to be very large to allow for the decoupling to take place. A vector load queue of four slots is enough to achieve a high fraction of the maximumperformance obtainable by an infinite queue. On the other side, the vector store queue does not need to be very large. Our experiments varying the store queue length indicate that a store queue of two elements achieves almost the same performance as one with sixteen slots. The ability to tolerate very large memory latencies will be critical in future high performance computers. In order to reduce the costs of high performance SRAM vector memory systems, they should be turned into SDRAM based memory sys- tems. This change, unfortunately, can significantly increase memory latency. At this point is where decoupling can come to rescue. As we have shown, up to 100 cycle latencies can be gracefully tolerated with a performance increase with respect to a traditional, in-order machine. Moreover, although in this paper we only look at the single processor case, the decoupling technique would also be very effective in vector multiprocessors to help reducing the negative effect of conflicts in the interconnection network and in the memory modules. The simulation results presented in this paper indicate that vector architectures can benefit from many of the techniques currently found in superscalar processors. Here we have applied decoupling, but other alternatives are applying multithreaded techniques to improve the memory port usage [12] and out-of-order execution together with register renaming [13]. Currently we are pursuing the latter approach. --R Performance Tradeoffs in Multithreaded Processors. The Perfect Club benchmarks: Effective performance evaluation of supercomput- ers Organization and architecture tradeoffs in FOM. A performance study of software and hardware data prefetching strategies. Functionally parallel architectures for array processors. CONVEX Architecture Reference Manual (C Series) Quantitative analysis of vector code. Dixie: a trace generation system for the C3480. Instruction level characterization of the Perfect Club programs on a vector computer. Decoupled vector architectures. Multithreaded vector architectures. PIPE: A VLSI Decoupled Architecture. Optimizing for parallelism and data locality. Cache performance in vector supercomputers. Memory Latency Effects in Decoupled Architectures. Software pipelining: An effective scheduling technique for VLIW machines. Branch prediction strategies and branch target buffer design. Value locality and load value prediction. Vector register design for polycyclic vector scheduling. Design and evaluation of a compiler algorithm for prefetching. Explaining the gap between theoretical peak performance and real performance for supercomputer architectures. Decoupled Access/Execute Computer Architectures. A Simulation Study of Decoupled Architecture Computers. Polycyclic vector scheduling vs. chaining on 1-port vector supercomputers The design of the microarchitecture of UltraSPARC-I Exploiting choice: Instruction fetch and issue on an implementable simultaneous multithreading processor. The Mips R10000 Superscalar Microprocessor. --TR A simulation study of decoupled architecture computers The ZS-1 central processor Software pipelining: an effective scheduling technique for VLIW machines Polycyclic Vector scheduling vs. Chaining on 1-Port Vector supercomputers Optimizing for parallelism and data locality Design and evaluation of a compiler algorithm for prefetching Designing the TFP Microprocessor A performance study of software and hardware data prefetching schemes Cache performance in vector supercomputers Explaining the gap between theoretical peak performance and real performance for supercomputer architectures Out-of-order vector architectures Vector register design for polycyclic vector scheduling Decoupled access/execute computer architectures The MIPS R10000 Superscalar Microprocessor Memory Latency Effects in Decoupled Architectures Performance Tradeoffs in Multithreaded Processors Decoupled vector architectures Multithreaded Vector Architectures Quantitative analysis of vector code --CTR Mostafa I. Soliman , Stanislav G. Sedukhin, Matrix bidiagonalization: implementation and evaluation on the Trident processor, Neural, Parallel & Scientific Computations, v.11 n.4, p.395-422, December Mostafa I. Soliman , Stanislav G. Sedukhin, Trident: a scalable architecture for scalar, vector, and matrix operations, Australian Computer Science Communications, v.24 n.3, p.91-99, January-February 2002

vector architectures;decoupling;instruction-level parallelism;memory latency

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Application-Level Fault Tolerance as a Complement to System-Level Fault Tolerance.

As multiprocessor systems become more complex, their reliability will need to increase as well. In this paper we propose a novel technique which is applicable to a wide variety of distributed real-time systems, especially those exhibiting data parallelism. System-level fault tolerance involves reliability techniques incorporated within the system hardware and software whereas application-level fault tolerance involves reliability techniques incorporated within the application software. We assert that, for high reliability, a combination of system-level fault tolerance and application-level fault tolerance works best. In many systems, application-level fault tolerance can be used to bridge the gap when system-level fault tolerance alone does not provide the required reliability. We exemplify this with the RTHT target tracking benchmark and the ABF beamforming benchmark.

Introduction In a large distributed real-time system, there is a high likelihood that at any given time, some part of the system will exhibit faulty behavior. The ability to tolerate this behavior must be an integral part of a real-time system. Associated with every real-time application task is a deadline by which all calculations must be completed. In order to ensure that deadlines are met, even in the presence of failures, fault tolerance must be employed. In this paper we consider fault tolerance at two separate levels, system-level and application-level. System-Level Fault Tolerance encompasses redundancy and recovery actions within the system hardware and software. While system hardware includes the computing elements and I/0 (network) sub-system, the system software includes the operating system and components such as the scheduling and allocation algorithms, check- pointing, fault detection and recovery algorithms. For example, in the event of a failed processing unit, the component of the system responsible for fault tolerance would take care of rescheduling the task(s) which had been executing on the faulty node, and restarting them on a good node from the previous checkpoint. Application-Level Fault Tolerance encompasses redundancy and recovery actions within the application software. Here various tasks of the application may communicate in order to learn of faults and then provide recovery services, making use of some data-redundancy. In certain situations, we nd that fault tolerance at the application-level can greatly augment the overall fault-tolerance of the system. For example, if a task's checkpoint is very large, application-level fault tolerance can help mask a fault while the system is moving the large checkpoint and restarting the task on another node. N-Modular Redundancy is a well-known fault tolerance technique. A number of identical copies of the software are run on separate machines, the output from all of them is compared, and the majority decision is used [1]. This technique however, involves a large amount of redundancy and is thus costly. The recovery block approach combines elements of checkpointing and backup alternatives to provide recovery from hard failures [2]. All tasks are replicated but only a single copy of each task is active at any time. If a computer hosting an active copy of a task fails, the backup is executed. The task may be completely restarted (which increases the chances of a deadline miss) or else executed from its most recent checkpoint [4]. The later option requires that the active copy of the task periodically copy (checkpoint) its state to its backups. This can entail a large amount of overhead, especially when the state information to be transferred is large. Such is the case with the applications that we are dealing with. Another common technique is the use of less precise (i.e., approximate) results [3], obtained by operating on a much smaller data set, using the same algorithm. A data set can be chosen such that a su-ciently accurate result can be obtained with a greatly reduced execution time. A smaller data set is chosen either by prioritizing the data set or by reducing the granularity. Examples of such applications are target tracking and image processing, where it is better to have less precise results on time, rather than precise results too late or not at all. Our recovery technique caters to applications that exhibit data- parallelism, involves a large data set and can make do with a less precise result for a short period of time. Our approach makes use of facets of the recovery block technique and employs reduced precision state information and results in order to tolerate faults. We employ a certain degree of redundancy within each of the parallel processes. The application as a whole is able to make use of that redundancy in the event of a fault to ensure that the required level of reliability is achieved. We consider only failures that render a process' results erroneous or inaccessible. In the case of such a fault, the redundant element's less precise results are used instead of those from the failed process. In this way, our technique can provide a high degree of reliability with only a small computational overhead in certain applications. Section 2 introduces the RTHT and ABF benchmarks that will be used to demonstrate our technique. In Section 3 we describe in detail our application-level fault tolerance technique. Section 4 analyzes the eectiveness of this technique when used in conjunction with each of the benchmarks, and Section 5 concludes the paper. 2. The Benchmarks Each of the benchmarks has the form shown in Figure 1. There are multiple, parallel application processes, which are fed with input data from a source - in this case, a source process which simulates a radar system or an array of sonar sensors. When the parallel computations are complete, the results are output to a sink process, simulating system display or actuators. Our technique is concerned with the ability to withstand faults at the parallel processes. Sink collects the results. random noise. Source generates input data consisting of real points and Application processes perform computations in order to track the targets, or form beams. Process 1 Process 2 Process 3 Process 4 Source Sink Figure 1. Software architecture of both the RTHT and ABF benchmarks. 2.1. The RTHT Target Tracking Benchmark The Honeywell Real-Time Multi-Hypothesis Tracking (RTHT) Benchmark [6, 7], is a general-purpose, parallel, target-tracking benchmark. The purpose of this benchmark is to track a number of objects moving about in a two-dimensional coordinate plane, using data from a radar system. The data is noisy, consisting of false targets and clutter, along with the real targets. The original, non-fault-tolerant application consists of two or more processes running in parallel, each working on a distinct subset of the data from the radar. Periodically, frames of data arrive from the radar, or source process in this case, and are split among the processes for computation of hypotheses. Each possible track has an associated hypothesis which includes a gure of likelihood, representing how likely it is to be a real track. A history of the data points and a covariance matrix are used in generating up-to-date likelihood values. For every frame of radar data, each parallel process performs the following steps: Creation of new hypotheses for each new data point it receives, 2) Extension of existing hypotheses, making use of the new radar data and the existing covariance Participation in system-wide compilation or ranking of hypotheses, led by a Root application process, and 4) Merging of its own list of hypotheses with the system-wide list that resulted from the compilation step. The deadline of one frame's calculations is the arrival of the next frame. By evaluating the performance of the original, non-fault-tolerant, benchmark when run in conjunction with our RAPIDS real-time system simulator [9], it became apparent that despite the inherent system-level fault tolerance in the simulated system, the benchmark still saw a drastic degradation of tracking accuracy as the result of even a single faulty node. Even if the benchmark task was successfully reassigned to a good node after the fault, the chances that it had already missed a deadline were high. This was in part due to the overheads associated both with moving the large process checkpoint over the network and with restarting such a large process. Once the process had missed the deadline, it was unable to take part in the compilation phase and had to start all over again and begin building its hypotheses anew. This took time, and caused a temporary loss of tracking reliability of up to ve frames. Although better than a non-fault-tolerant system, in which that process would simply have been lost, it was still not as reliable as desired. We decided to address two points, in order to improve the performance of the benchmark in the presence of faults: 1) The overhead involved with moving such a large checkpoint and 2) A source of hypotheses for the process to start with after restart. Our measure of reliability is the number of real targets successfully tracked by the application (within a su-cient degree of accuracy) as a fraction of the exact number of real targets that should have been tracked. To simplify this calculation, the number of targets is kept constant and no targets enter or leave the system during the simulation. 2.2. The ABF Beam Forming Benchmark The Adaptive Beam Forming (ABF) Benchmark [8] is a simulation of the real-time process by which a submarine sonar system interprets the periodic data received from a linear array of sensors. In particular, the goal is to distinguish signals from noise and to precisely identify the direction from which a signal is arriving, across a specied range of frequencies. In this implementation, the application receives periodic samples of data as if from the linear sensor array. The data is generated so that it contains four reference beams, or signals, arriving from distinct locations in a 180-degree eld of view, along with random noise. The application itself consists of several application processes, each attempting to locate beams at a distinct subset of the specied frequency range. Frames of data for each frequency are \scattered" periodically from the source process. Output, in the form of one beam pattern per frequency, is \gathered" by the sink process. Figure 2 depicts a typical beam pattern output, shown here at frame 18, frequency 250Hz, with reference beams at -20, -60, 20 and 60 degrees. Each application process performs calculations according to the following loop of pseudo-code, for each frame of input. for_each ( frequency ) { Update dynamic weights. for_each ( direction of arrival) { Search for signal, blocking out interference from other directions and frequencies. Magnitude Direction of Arrival (Angle) - degrees Figure 2. Typical beam pattern output. For each frequency, the process rst updates a set of weights that are dynamically modied from frame to frame. Applying these weights to the input samples has the eect of forming a beam which emphasizes the sound arriving from each direction. The process searches in each possible direction (-90 to 90 degrees) for incoming signals. The granularity of this direction is directly related to the number of sensors. In addition, at the start of a run, there is an initialization period in which the weights are set to some initial values, and then 15 to 20 frames are necessary to \learn" precisely where the beams are. It is evident that this sort of application faces reliability problems similar to those of the RTHT benchmark. If a processing element fails, all output for those frequencies is lost during the down time, and when the lost task is nally replaced by the system, it has to go through a startup period all over again. Here, too, the data sets of these processes are very large, creating a considerable overhead if checkpointing is employed. To avoid the delay associated with this overhead, be able to maintain full output during the fault, and provide quick restart after the fault, application-level fault tolerance must be employed. We evaluate the quality of the ABF output with two tests applied to the resultant beam pattern. In the Placement Test we check whether the direction of arrival of the beam has been detected within a certain tolerance. In the Width Test the aim is to determine how accurately the beam has been detected by measuring the width of the beam, in degrees, at 3db down from the peak. A beam that passes both tests is considered to be correctly detected. 3. Implementation of Application-Level Fault Tolerance Our technique uses redundancy in the form of extra work done by each process of the application. Each process takes, in addition to its own distinct workload, some portion of its neighbor's workload, as shown in Figure 3. The process then tracks beams or targets for both its own work and overlaps part of its neighbor's, but makes use of the redundant information only in case this neighbor becomes faulty. We now explain brie y how the data set is divided, how the application might learn of faults, and how it would recover from them. P3Process 1 Process 2 Process 3 Process 424 (2) (1) Frame of data arrives here, at each node. Time Figure 3. Architecture of both benchmarks with application-level fault tolerance. 3.1. Division of Load The extent of duplication between two neighboring nodes will greatly aect the level of reliability which can be achieved. Duplication arises from the way we divide the data set among the parallel processing nodes. First, each frame of data is divided as evenly as possible among the nodes. The section of the process that takes on this set of data is the primary task section, P i . Then we assign each node, n i some additional work: part of its neighbor, n i 1 's, primary task. The section of the process that takes on this set of data is the secondary task section, S i . In other words, The primary task section, P i , refers to the calculations which node n i carries as part of the original application. The secondary task section, S i , refers to the calculations which node n i carries out as a backup for its neighbor, hosts the secondary corresponding to the primary running on the highest numbered node. The secondary section, S i , will be kept in synchronization with the primary P i 1 . 3.2. Detection of Faults There are two ways in which fault detection information can reach the various application processes. In the rst, the system informs the application of a faulty node, and the second is through specic timeouts at the phase of the application where communication is expected. The former would typically incur the cost of periodic polling, while the latter could result in late detection of the fault. Although the exact integration of application-level fault tolerance would vary depending on the fault detection technique chosen, the eectiveness of our technique should not. 3.3. Fault Recovery If, at a deadline prior to that of the frame, node n i is discovered to be faulty and is unable to output any results, then node n i+1 which is serving as its backup will send as output S i+1 's data in place of the data that n i is unable to supply. In the meantime, the system will be working on replacing or restarting the process that was interrupted by the fault. In fact, the system's job here is made easier by the fact that if the process has to be restarted on another node, the process data segment no longer needs to be moved. When the process is rescheduled, it will make use of the information maintained by its secondary on its behalf in order to pick up where it left o before the fault. This way, the application fault tolerance is able to work in conjunction with the system fault tolerance. This will help even in the case of transient faults, in that the application-level fault tolerance allows more leeway to postpone the restarting of the process on another node, in the hope that the fault will soon disappear. 3.4. Extension to a higher level of redundancy Our technique guarantees the required reliability in the presence of one fault but could also withstand two or more simultaneous failures depending on which nodes are hit by the faults. For example, in a six-node system if the nodes running processes 1, 3, and 5 fail, the technique would still be able to achieve the required reliability. Of course, this is contingent on the assumption that the processes on the faulty nodes are transferred to a safe node and restarted by the beginning of the next frame. 3.5. Benchmark Integration Specics We next discuss specic details regarding the application of our technique to each of the benchmarks. 3.5.1. RTHT benchmark In the RTHT Benchmark, the \unit of redundancy" is the hypothesis. That is, each secondary task section creates and extends some fraction of the total number of hypotheses created and extended by the process for which it is secondary. The amount of secondary redundancy is expressed as a percentage of the number of hypotheses extended by the primary. Redundancy is implemented in the following way: At the beginning of each frame, the source process broadcasts the input radar data, and hypotheses are created and extended as before, with the exception that additionally the secondary extends a percentage of those extended by the corresponding primary. The secondary section is kept in synchronization with primary P i 1 via the compilation process, which in this case is again a process-level broadcast communication, so that no extra communication is necessary. If node n i is discovered to be faulty and is unable to participate in the compilation of that frame, then node n i+1 which is serving as its backup will make use of S i+1 's data in the compilation process in place of the data that n i is unable to supply. When the process is rescheduled, it will make use of the hypotheses extended by the secondary on its behalf so as to pick up where it left o. This information is obtained from the secondary process by way of compilation - the newly rescheduled process merely listens in on the compilation process and copies those hypotheses which have been extended by its secondary. 3.5.2. ABF benchmark There are two ways in which we have integrated application-level fault tolerance with the ABF Benchmark. They dier in the manner in which the secondary abbreviates the calculations of the primary so as to obtain a full set of results. The methods are: The Limited Field of View (Limited FOV) Method in which the secondary looks for beams at every frequency as in the primary, however it searches only a subsection of the primary's eld of view (divided into one or more segments). Ideally the secondary will place these \windows" at directions in which beams are known to be arriving. We impose a minimum width of these windows, due to the fact that if an individual window is too narrow, the output could always (perhaps erroneously) pass the width-based quality test, described in section 2. The amount of redundancy is expressed as the percentage of the eld of view searched by the secondary. The Reduced Directional Granularity Method in which the secondary looks for beams at every frequency and in every direction, but with a reduced granularity of direction. The amount of redundancy is expressed as a percentage of the original granularity computed by the primary. Both techniques serve to reduce the computational time of the secondary task set, while maintaining useful system output. In addition, the two techniques may be employed concurrently in order to further reduce the computational time required by the secondary task. To implement either variation of the technique, the input frame of data is scattered a second time from the source to the application processes. This is time - rotated, so that each process receives the input data of the process for which it is a secondary. Each process rst carries out its primary computational tasks, and then carries out its secondary task. At the frame's deadline, if a process is detected to be down, the sink will gather output from the non-faulty processes, including the backup results from the process that is secondary to the one that is faulty. In the event of an application process being restarted after a fault, it will receive the current set of weights from its secondary in order to jump-start its calculations. Some synchronization between primary and secondary is required in the Limited FOV Method. It is a small, periodic communication in which either the sink process or the primary itself tells the secondary at what frequencies and directions it is detecting beams. Such synchronization is not necessary for the Reduced Granularity Method. 4. Results 4.1. The RTHT Benchmark When applied to the RTHT benchmark, we found that only a small amount of redundancy between the primary and secondary sections is necessary in order to provide a considerable amount of fault tolerance. Furthermore, the increase in system resource requirements, even after including overheads of the technique's implementation, is minimal compared to that of other techniques, in achieving the same amount of reliability. These points are demonstrated in Figures 4, 5, and 6. Each run contains 30 targets which remain in the system until the end of the simulation (the 30th frame), as well as some number of false alarms. The case when only system-level fault tolerance exists corresponds to the case when the secondary extends 0% of the primary hypotheses. In Figure 4 we see the number of targets which are successfully tracked, when we have just two application processes and a fault occurs at frame 15. (In this case there were roughly alarms per frame of data.) In this run, 15% redundancy allows us to track all of the real targets, despite the fault. We can attribute the fact that a small amount of redundancy can have a great eect on the tracking stability, Number of Targets Tracked Frame Number Secondary extends 15% of primary hyps Secondary extends 10% of primary hyps Secondary extends 5% of primary hyps Secondary extends 0% of primary hyps Figure 4. Tracking accuracy, in number of real targets tracked for a given percentage of redundancy to the fact that the hypotheses which are being extended by the secondary are the ones most likely to be real targets. At the beginning of the compilation phase, each application process sorts its hypotheses, placing the most likely at the head of the list for compilation. Thus, at the beginning of the next frame, each application process and its secondary begin extending those hypotheses with the highest chance of being real targets. To rene this point, Figure 5 shows the average percentage of redundancy required for a given number of application processors and a single fault, as before. The amount required shows a gradual decrease as we add more processors. We can attribute this to the fact that the chance of a single process containing a high percentage of the real targets decreases as processors are added. In addition, a proportionately small load is imposed on the processor by the computation of the secondary task set, as seen in Figure 6. This can be attributed to the fact that a hypothesis whose position and velocity are known precisely, does not take as much time to extend compared to those hypotheses which are less well- known. And since the most likely hypotheses are generally the most well-known and are the hypotheses which the secondary extends, the amount of processor time taken to execute the secondary task is proportionally much smaller. Percentage of secondary overlap required Number of Application Processors Figure 5. Average minimum percentage of secondary overlap required to miss no targets despite one node being faulty. 4.2. ABF Benchmark Results When we integrate application-level fault tolerance with the ABF benchmark, we nd that only a small amount of redundancy is necessary to ensure complete masking of single frame faults. With either variation (reduced granularity or limited FOV method) we see that a secondary redundancy of 33% is adequate to provide complete and accurate results in the faulty frame and the following frames (after the faulty process is restarted). If we combine the two techniques, we see an even further reduction in the computational eort imposed by the secondary in order to mask the fault. We have not taken additional network overhead and/or latency into account in gures of overhead - they refer solely to computational overhead. Network overhead will depend greatly on the medium used. In particular, a shared medium would allow the secondary to \snoop" on the primary's input and output, eliminating the need for additional communication. All results were obtained by running simulations with 75 sensors and four reference input beams for 50 frames. There are two application processors, and a fault occurs in one of them at frame 30. Results are presented and discussed for three redundancy methods: the Limited FOV method, the Reduced Granularity method and a Combined method (a combination of the rst two). The quality of the results is assessed by totalling the number of beams that were tracked successfully. Here, there are four input beams at each frequency and 32 frequencies { making 128 of Secondary Execution time to Primary Percentage of Secondary Overlap Figure 6. Ratio of time taken to compute the secondary hypothesis to the time to compute the primary hypothesis versus the percentage of secondary overlap. beams in all. As an example, Figure 7 presents the results for several runs of the ABF benchmark while utilizing the Limited FOV redundancy method alone, with a single processor fault occurring at frame 30 and lasting one frame. We see that a 30% overlap is adequate to preserve all beam information within the system despite the loss of one processor in frame 30. We have tabulated the results for all three methods in Table 1. 4.2.1. ABF Results: Limited FOV Alone As we see in Table 1, roughly 30% secondary overlap is adequate to provide full masking of the fault. The computational overhead imposed by the secondary is about 30%. In addition, Figure 8 shows the rather linear increase in overhead as we increase the fraction of overlap. Table 1. Amount of secondary overhead imposed by various redundancy methods, each of which is capable of fully masking a single fault. Redundancy Technique Secondary Overlap Computational Overhead Reduced Granularity 33% 35% Limited FOV 30% 30% Combined - 30%FOV,50%Granularity 15% 17% Total number of beams detected Time 30% secondary 20% secondary 10% secondary 0% secondary Figure 7. The number of beams correctly tracked in each frame, for the given levels of redundancy, for the Limited Field of View Method. A single process experiences a fault of duration one frame, at frame 30. Associated with this technique however, is a potential dependence on the number of beams detected in the system, as described earlier. In order to ensure that the width test applied to the output can fail, we impose a minimum window-width. This minimum width dictates that for a given amount of overlap, there is a maximum number of windows in which the secondary may search for beams. If there are more beams than the maximum number of windows then some may be missed by the secondary search, depending on the direction of arrival. However, the system designer can lessen the likelihood of this occurring by carefully choosing the amount of overlap allotted, and tuning the criteria with which areas will be searched by the secondary. 4.2.2. ABF Results: Reduced Granularity Alone Here, too, we see that, according to Table 1, operating the secondary at 33% of the granularity of the primary results in complete masking of the fault, and that this imposes a 35% overhead to the processing node. Figure 8 again shows a linear relationship between the computational overhead and the overlap, and indicates that the overhead of the method itself is a bit higher than that of the Limited FOV method. When considering the Reduced Granularity method, we see no dependence on the number of beams detected, although beams could be missed if their peaks were within a few degrees of each other, and the granularity were very coarse. of secondary exection time to primary Percentage of Secondary Field of View Overlap Reduced Granularity method Limited FOV method Limited FOV at 50% Granularity Limited FOV at 33% Granularity Figure 8. The ratio of secondary to primary execution time for the variations of application-level tolerance integrated with the ABF Benchmark versus the percentage of secondary eld of view overlap. 4.2.3. ABF Results: Combined methods When we combine these two techniques, we see the greatest reduction in computational overhead of the secondary task. As shown in Table 1, a 30% eld of view combined with a 50% granularity maintains the tracking ability similar to that of either one alone, yet cuts the computational overhead nearly in half. This reduction is illustrated in Figure 8, in the lower two curves, representing the overhead imposed as we vary the eld of view and make use of 50% and 33% granularity respectively. 5. Conclusions A high degree of fault tolerance may be obtained with a minimal investment of system resources in applications exhibiting data parallelism, such as the ABF and RTHT Benchmarks. It is achieved through a combination of application-level and system-level fault tolerance. A prioritized ordering within the data set, as in the RTHT benchmark, or a reduced granularity, as in the ABF benchmark, is made use of, to decrease the computational overhead of our technique. The processes in these benchmarks are very large, so that moving a checkpoint and restarting the task may take a signicant amount of time. The application-level fault tolerance is able to ensure that, despite the temporary loss of the task, the required reliability is maintained. Since the primary and secondary task sets are incorporated within a single application process, the primary is always executed rst and the secondary next. Once the primary has completed, it may alert the scheduler, indicating that the secondary need not be executed. It is useful, but not necessary, for the secondary to still be executed, as this allows it to be better synchronized with its primary counterpart. If a fault is detected, the priority of the secondary could be raised, to ensure that it will complete without missing its deadline, and provide the necessary data for compilation. This technique is a substantial improvement over complete system duplication, in that it does not require 100% system redundancy, but merely adds a small amount of load to the existing system in achieving the same amount of fault tolerance. It diers from the recovery block approach in that the secondary does not have to be cold-started, but is ready for execution when a failure of the primary is detected. In addition, the level of reliability may be varied by varying the amount of redundancy. In order to integrate such application-level fault tolerance, the designer will need to rst determine how to prioritize the data set and/or reduce the granularity in order to dene the secondary's dataset. Second, the designer should choose mechanisms by which the secondary gets the input data it needs, is able to output results when necessary, and is able to communicate with the primary for synchronization purposes. Naturally, some sort of fault detection will have to used as well. The designer must carefully weigh the overheads imposed by various methods to achieve fault tolerance and the quality of results that may be obtained from each. In conclusion, we believe that steps to integrate this technique into the application should be taken right from the early stages of the design in order for this approach to be most eective. Acknowledgments This eort was supported in part by the Defense Advanced Research Projects Agency and the Air Force Research Laboratory, Air Force Materiel Command, USAF, under agreement number F30602-96-1-0341, order E349. The government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the o-cial policies or endorsements, either expressed or implied, of the Defense Advanced Research Projects Agency, Air Force Research Laboratory, or the U. S. Government. --R System Structure for Software Fault Tolerance. Imprecise Computations. Using Passive replicates in Delta-4 to Provide Dependable Distributed Computing A Fault-Tolerant Scheduling Problem Implementation and Results of Hypothesis Testing from the C 3 I Parallel Benchmark Suite. Honeywell Technology Center. RAPIDS: A Simulator Testbed for Distributed Real-Time Systems --TR A fault-tolerant scheduling problem Reliable computer systems (2nd ed.) Implementation and Results of Hypothesis Testing from the C3I Parallel Benchmark Suite --CTR Osman S. Unsal , Israel Koren , C. Mani Krishna, Towards energy-aware software-based fault tolerance in real-time systems, Proceedings of the 2002 international symposium on Low power electronics and design, August 12-14, 2002, Monterey, California, USA

checkpointing;distributed real-time systems;target tracking;imprecise computation;fault tolerance;beam forming

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High Performance Computations for Large Scale Simulations of Subsurface Multiphase Fluid and Heat Flow.

TOUGH2 is a widely used reservoir simulator for solving subsurface flow related problems such as nuclear waste geologic isolation, environmental remediation of soil and groundwater contamination, and geothermal reservoir engineering. It solves a set of coupled mass and energy balance equations using a finite volume method. This contribution presents the design and analysis of a parallel version of TOUGH2. The parallel implementation first partitions the unstructured computational domain. For each time step, a set of coupled non-linear equations is solved with Newton iteration. In each Newton step, a Jacobian matrix is calculated and an ill-conditioned non-symmetric linear system is solved using a preconditioned iterative solver. Communication is required for convergence tests and data exchange across partitioning borders. Parallel performance results on Cray T3E-900 are presented for two real application problems arising in the Yucca Mountain nuclear waste site study. The execution time is reduced from 7504 seconds on two processors to 126 seconds on 128 processors for a 2D problem involving 52,752 equations. For a larger 3D problem with 293,928 equations the time decreases from 10,055 seconds on 16 processors to 329 seconds on 512 processors.

Introduction Subsurface flow related problems touch many important areas in to- day's society, such as natural resource development, nuclear waste underground storage, environmental remediation of groundwater contami- nation, and geothermal reservoir engineering. Because of the complexity of model domains and physical processes involved, numerical simulation play vital roles in the solutions of these problems. This contribution presents the design and analysis of a parallel implementation of the widely used TOUGH2 software package [9, 10] for numerical simulation of flow and transport in porous and fractured media. The contribution includes descriptions of algorithms and methods used in the parallel implementation and performance evaluation Present address: Department of Computing Science and High Performance Computing Center North, Ume-a University, SE-901 87 Ume-a, Sweden. c 2000 Kluwer Academic Publishers. Printed in the Netherlands. HIGH PERFORMANCE COMPUTATIONS FOR SUBSURFACE SIMULATIONS 3 for parallel simulations with up to 512 processors on a Cray T3E-900 on two real application problems. Although the implementation and analysis is made on Cray T3E, the use of the standard Fortran 77 programming language and the MPI message passing interface makes the software portable to any platform where Fortran 77 and MPI are available. The serial version of TOUGH2 (Transport Of Unsaturated Ground-water and Heat version 2) is now being used by over 150 organizations in more than 20 countries (see [11] for some examples). The major application areas include geothermal reservoir simulation, environmental remediation, and nuclear waste isolation. TOUGH2 is one of the official codes used in the US Department of Energy's civilian nuclear waste management for the evaluation of the Yucca Mountain site as a repository for nuclear wastes. In this context arises the largest and most demanding applications for TOUGH2 so far. Scientists at Lawrence Berkeley National Laboratory are currently developing a 3D flow model of the Yucca Mountain site, involving computational grids of 10 5 to blocks, and related coupled equations of water and gas flow, heat transfer and radionuclide migration in subsurface [3]. Considerably larger and more difficult applications are anticipated in the near future, with the analysis of solute transport, with ever increasing demands on spatial resolution and a comprehensive description of complex geolog- ical, physical and chemical processes. High performance capability of the TOUGH2 code is essential for these applications. Some early results from this project were presented in [5]. 2. The TOUGH2 Simulation The TOUGH2 simulation package solves mass and energy balance equations that describe fluid and heat flow in general multiphase, multicomponent systems. The fundamental balance equations have the following d dt Z Z Z where the integration is over an arbitrary volume V , which is bounded by the surface S. Here M (k) denotes mass for the k-th component, (water, gas, heat, etc), F (k) is the flux of fluids and heat through the surface, and q (k) is source or sink inside V . This is a general form. All flow and mass parameters can be arbitrary non-linear functions of the primary thermodynamic variables, such as density, pressure, saturation, etc. Given a computational geometry, space is discretized into many small volume blocks. The integral on each block becomes a variable; this leads naturally to the finite volume method, resulting in the following ordinary differential equations: dt is the volume of the block n, and Anm is the interface area bordering between blocks n; m and Fnm is the flow between them. Note that flow terms usually contain spatial derivatives, which are replaced by simple difference between variables defined on blocks n; m and divided by the distances between the block centers. See Figure 1 for an illustration. On the left-hand side, a 3-dimensional grid block is illustrated with arrows illustrating flow throw interface areas between neighboring grid blocks. On the left-hand side, two neighboring blocks m and n are illustrated by a 2-dimensional picture. Here, each block center is marked by a cross. Included are also the variables Vm and V n for volumes and Dm and D n for distance between grid block centers and the interface area. nm A F nm Figure 1. Space discretization and geometry data. Time is implicitly discretized as a first order difference equation: \Deltat nm where the vector x (t) consists of prime variables at time t. Flow and source/sink terms on the right hand side are evaluated at t + \Deltat for HIGH PERFORMANCE COMPUTATIONS FOR SUBSURFACE SIMULATIONS 5 Initialization and setup do Time step advance do Newton iteration Calculate the Jacobian matrix Solve linear system do do output Figure 2. Sketch of main loops for the TOUGH2 simulation. numerical stability for the multi-phase problems. This lead to coupled nonlinear algebraic equations, which are solved using Newton's method. 3. Computational Procedure The main solution procedures can be schematically outlined as in Figure 2. After reading data and setting up the problem, the time consuming parts are the main loops for time stepping, Newton iteration, and the iterative linear solver. At each time step, the nonlinear discretized coupled algebraic equations are solved with the Newton method. Within each Newton iteration, the Jacobian matrix is first calculated by numerical differentiation. The implicit system of linear equations is then solved using a sparse linear solver with preconditioning. After several Newton iterations, the convergence is checked by a control parameter, which measures the maximum component of the residual in the Newton iterations. If the Newton iterations converge, the time will advance one more time step, and the process repeats until the pre-defined total time is reached. If the Newton procedure does not converge after a preset max- Newton-iteration, the current time step is reduced (usually by half) and the Newton procedure is tried for the reduced time step. If converged, the time will advance; otherwise, time step is further reduced and another round of Newton iteration follows. This procedure is repeated until convergence in the Newton iteration is reached. The system of linear equation is usually very ill-conditioned, and requires very robust solvers. The dynamically adjusted time step size is the key to overcome the combination of possible convergence problems for the Newton iteration and the linear solver. For this highly dynamic 6 ELMROTH, DING AND WU system, the trajectory is very sensitive to variations in the convergence parameters. Computationally, the major part (about 65%) of the execution time is spent on solving the linear systems, and the second major part (about 30%) is the assembly of the Jacobian matrix. 4. Designing the Parallel Implementation The aim of this work is to develop a parallel prototype of TOUGH2, and to demonstrate its ability to efficiently solve problems significantly larger than problems that have previously been solved using the serial version of the software. The problems should be larger both in the number of blocks and the number of equations per block. The target computer system for this prototype version of the parallel TOUGH2 is the 696 processor Cray T3E-900 at NERSC, Lawrence Berkeley National Laboratory. In the following sections, we give an overview of the design of the main steps, including grid partitioning, grid block reordering, assembly of the Jacobian matrix, and solving the linear system, as well as some further details about the parallel implementation. 4.1. Grid Partitioning and Grid Block Reordering Given a finite domain as described in Section 2, we will in the following consider the dual mesh (or grid), obtained by representing each block (or volume element) by its centroid and by representing the interfaces between blocks by connections. (The words blocks and connections are used in consistency with the original TOUGH2 documentation [10].) The physical properties for blocks and their interfaces are represented by data associated with blocks and connections, respectively. In TOUGH2 the computational domain is defined by the set of all connections given as input data. From this information, an adjacency matrix is constructed, i.e., a matrix with a non-zero entry for each element (i; j) where there is a connection between blocks i and j. In the current implementation the value 1 is always used for non-zero elements, but different weights may be used. The adjacency matrix is stored in a compressed row format, called CRS format, which is a slight modification of the Harwell-Boeing format. See, e.g., [2] for descriptions of CRS and Harwell-Boeing formats. The actual partitioning of the grid into p almost equal-sized parts is performed using three different partitioning algorithms, implemented in the METIS software package version 4.0 [8]. The three algorithms are here denoted the K-way, the VK-way and the Recursive partitioning algorithm, in consistency with the METIS documentation. K-way is multilevel version of a traditional graph partitioning algorithm that minimizes the number of edges that straddle the partitions. VK-way is a modification of K-way that instead minimizes the actual total communication volume. Recursive is a recursive bisection algorithm which objective is to minimize the number of edges cut. After partitioning the grid on the processors, the blocks (or more specifically, the vector elements and matrix rows associated with the blocks) are reordered by each processor to a local ordering. The blocks for which a processor computes the results are denoted the update set of that processor. The update set can be further partitioned into the internal set and the border set. The border set consists of blocks with an edge to a block assigned to another processor and the internal set consists of all other blocks in the update set. Blocks not included in the update set but needed (read only) during the computations defines the external set. Figure 3 illustrates how the blocks can be distributed over the pro- cessors. (The vertices of the graph represent blocks and the edges represent connections, i.e., interface areas between pairs of blocks.) Table I shows how the blocks are classified in the update and the external sets and how the update sets are further divided into internal and border sets. In the table, the elements are placed in local order and the global numbering illustrates the reordering.11314135 Processor 0 Processor 2Processor 11 Figure 3. A grid partitioning on 3 processors. Table I. Example of block distribution and local ordering for the Internal, Border, and External sets. (Internal Processor 0: (7; Processor 1: (2; 3 j Processor 2: (6; 14 j 5; 10; 13 k12; 4; In order to facilitate the communication of elements corresponding to border/external blocks, the local renumbering of the nodes is made in a particular way. All blocks in the update set precede the blocks in the external set, and in the update set, all internal blocks precede the border blocks. Finally, the external blocks are ordered internally with blocks assigned to a specific processor placed consecutively. One possible ordering is given as an example in Table I. For processor 0 in this example, the grid blocks numbered 7 and 11 are internal blocks, i.e., these blocks are updated by processor 0 and there are no dependencies between these blocks and blocks assigned to other processors. The grid blocks 8 and 12 are border blocks for processor 0, i.e., the blocks are updated by processor 0 but there are dependencies to blocks assigned to other processors. Finally, blocks 1, and 13 are external blocks for processor 0, i.e., these blocks are not updated by processor 0 but data associated with these blocks are needed read-only during the computations. The amount of data that a processor is to send and receive during the computations are approximately proportional to the number of border and external blocks, respectively. The consecutive ordering of the external blocks that reside on each processor makes it possible to receive data corresponding to these blocks into appropriate vectors without use of buffers and with no need for further reordering, provided that the sending processor has access to the ordering information. However, it is not possible in general to order the border blocks so that transformations can be avoided when sending, basically because some blocks in the border set may have to be sent to more than one processor. 4.2. Jacobian Matrix Calculations A new Jacobian matrix is calculated once for each Newton step, i.e., several times for each Time step of the algorithm. In the parallel al- gorithm, each processor is responsible for computing the rows of the HIGH PERFORMANCE COMPUTATIONS FOR SUBSURFACE SIMULATIONS 9 Jacobian matrix that correspond to blocks in the processor's update set. All derivatives are computed numerically. The Jacobian matrix is stored in the Distributed Variable Block Row format (DVBR) [7]. All matrix blocks are stored row wise, with the diagonal blocks stored first in each block row. The scalar elements of each matrix block are stored in column major order. The use of dense matrix blocks enable use of dense linear algebra software, e.g., optimized level 2 (and level subproblems. The DVBR format also allows for a variable number of equations per block. Computation of the elements in the Jacobian matrix is basically performed in two phases. The first phase consists of computations relating to individual blocks. At the beginning of this phase, each processor already holds the information necessary to perform these calculations. The second phase include all computations relating to interface quan- tities, i.e., calculations using variables corresponding to pairs of blocks. Before performing these computations, exchange of relevant variables is required. For a number of variables, each processor sends elements corresponding to border blocks to appropriate processors, and it receives elements corresponding to external blocks. 4.3. Linear Systems The non-symmetric linear systems to be solved are generally very ill-conditioned and difficult to solve. Therefore, the parallel implementation of TOUGH2 is made so that different iterative solvers and preconditioners easily can be tested. All results presented here have been obtained using the stabilized bi-conjugate gradient method (BICGSTAB) [14] in the Aztec software package [7], with 3 \Theta 3 Block Jacobi scaling and a domain decomposition based preconditioner with possibly overlapping subdomains, i.e., Additive Schwarz (see, e.g., [13]), using the ILUT [12] incomplete LU factorization. The domain decomposition based procedure can be performed with different levels of overlapping, and for the case the procedure turns into another variant of Block Jacobi preconditioner. In order to distinguish the 3 \Theta 3 Block Jacobi scaling from the full subdomain Block Jacobi scaling obtained by chosing in the domain decomposition preconditioning procedure, we will refer to the former as the Block Jacobi scaling and the latter as the domain decomposition based preconditioner, though both are of course preconditioners. As an illustration of the difficulties arising in these linear systems, we would like to mention a very small problem from the Yucca Mountain simulations mentioned in the Introduction. This non-symmetric problem includes 45 blocks, 3 equations per block, and 64 connections. When solving the linear system, the Jacobian matrix is of size 135 \Theta 135 with 1557 non-zero elements. For the first Jacobian generated (in the first Newton step of the first Time step), i.e., the matrix involved in the first linear system to be solved, the largest and smallest singular values are 2:48\Theta10 32 and 2:27\Theta10 \Gamma12 , respectively, giving the condition number 1:1 \Theta 10 44 . By applying block Jacobi scaling, where each block row is multiplied by the inverse of its 3 \Theta 3 diagonal block, the condition number is significantly reduced. The scaling reduces the largest singular value to 7:69 \Theta 10 3 and the smallest is increased to 9:83 \Theta 10 \Gamma5 , altogether reducing the condition number to 7:8 \Theta 10 7 . This is, however, still an ill-conditioned problem. Therefore, the domain decomposition based preconditioner with incomplete LU factorization mentioned above is applied after the block Jacobi scaling. This procedure has shown to be absolutely vital for convergence on problems that are significantly larger. 4.4. Parallel Implementation In this section, we outline the parallel implementation by describing the major steps in some important routines. In all, the parallel TOUGH2 includes about 20,000 lines of Fortran code (excluding the METIS and Aztec packages) in numerous subroutines using MPI for message passing [6]. However, in order to understand the main issues in the parallel implementation, it is sufficient to focus on a couple of routines. Of course, several other routines are also modified compared to the serial version of the software, but these details would only be distracting. Cycit Initially, processor 0 reads all data describing the problem to be solved, essentially in the same way as in the serial version of the software. Then, all processors call the routine Cycit which contains the main loops for time stepping and Newton iterations. This routine also initiates the grid partitioning and data distribution. The partitioning described in Section 4.1 defines how the input data should be distributed on the processors. The distribution is performed in several routines called from Cycit. There are five categories of data to be distributed and possibly reordered. Vectors with elements corresponding to grid blocks are distributed according to the grid partitioning and reordered to the local order with Internal, Border, and External elements as described in Section 4.1. Vectors with elements corresponding to connections are distributed and adjusted to the local grid block numbering after each HIGH PERFORMANCE COMPUTATIONS FOR SUBSURFACE SIMULATIONS 11 processor have determined which connections are involved in its own local partition. Vectors with elements corresponding to sinks and sources are replicated in full before each processor extracts and reorders the parts needed. There are in addition a number of scalars and small vectors and matrices that are fully replicated, i.e., data structures which sizes do not depend on the number of grid blocks or connections. Finally, processor 0 constructs the data structure for storing the Jacobian matrix and distributes appropriate parts to the other processors. This include all integer vectors defining the matrix structure but not the large array for holding the floating point numbers for the matrix elements. As the problem is distributed, the time stepping procedure begins. A very brief description of the routine Cycit is given in Figure 4. In this description, lots of details have been omitted for clarity, and calls have been included to a couple of routines that require further description. ExchangeExternal The routine ExchangeExternal is of particular interest for the parallel implementation. The main loop of this routine is outlined in Figure 5. When called by all processors with a vector and a scalar noel as argu- ments, an exchange of vector elements corresponding to external grid blocks is performed between all neighboring processors. The parameter noel is the number of vector elements exchanged per external grid block. Some additional parameters that defines the current partition, e.g., information about neighbors etc, need also to be passed to the routine, but we have for clarity chosen not to include them in the figure. Though some details are omitted, we have chosen to include the full MPI syntax (using Fortran interface) for the communication primitives. The routine pack, called by ExchangeExternal, copies appropriate elements from vector into a consecutive work array. The external elements for a given processor are specified by sendindex. We remark that the elements can be stored directly into the appropriate vector when received (since external blocks are ordered consecutively for each neighbor), whereas the border elements to be sent need to be packed into a consecutive work space before they are sent. Note that we use the nonblocking MPI routines for sending and receiving data. With use of blocking routines we would have had to assure that all messages are sent and received in an appropriate order to avoid deadlock. When using nonblocking primitives, the sends and receives can be made in arbitrary order. A minor inconvenience with use of the nonblocking routines is that the work space used to store elements to be sent need to be large enough to store all elements a processor is to send to all its neighbors. Cycit(.) Initialization, grid partitioning, data distribution etc Set up first time step and the first Newton step number of secondary variables (in PAR) per grid block while Time ! EndTime while not Newton converged call Multi(.) Newton converged = result from convergence test if Newton converged Update primary variables Increment Time, define new time step and set else if Iter - MaxIter Solve linear system call Eos(.) call ExchangeExternal(PAR, num sec vars) if Iter ? MaxIter or Physical properties out of range time step has been decreased too many times Stop execution Print message about failure to solve problem else Reduce time step call Eos(.) call ExchangeExternal(PAR, num sec vars) while while Figure 4. Outline of the routine Cycit, executed by all processors. ExchangeExternal(vector, noel) do neighbors call MPI IRECV(vector(recvstart), rlen, MPI DOUBLE PRECISION, proc, tag+myid, MPI COMM WORLD, req(2*i-1), ierr) call pack(i, vector, sendindex, slen, work(iw), noel) call MPI ISEND(work(iw), slen, MPI DOUBLE PRECISION, proc, tag+proc, MPI COMM WORLD, req(2*i), ierr) do call MPI WAITALL(2*num neighbors, req, stat, ierr) Figure 5. Outline of the routine ExchangeExternal. When simultaneously called by all processors it performs an exchange of noel elements per external grid block for the data in vector. The routine Multi is called to set up the linear system, i.e., the main part of the computations in Multi is for computing the elements of the Jacobian matrix. Computationally, Multi performs three major steps. First it performs all computations that depend on individual grid blocks. This is followed by computations of terms arising from sinks and sources. So far all computations can be made independently by all processors. The last computational step in Multi is for interface quantities, i.e., computations involving pairs of grid blocks. Before performing this last step, and exchange of external variables is required for the vectors X (primary variables), DX (the last increments in the Newton process), DELX (small increments of the X values, used to calculate incremental parameters needed for the numerical calculation of the derivatives), 14 ELMROTH, DING AND WU and R (the residual). The number of elements to be sent per external grid block equals the number of equations per grid block, for all four vectors. This operation is performed by calling ExchangeExternal before performing the computations involving interface quantities. Eos3 and other Eos routines The thermophysical properties of fluid mixtures needed in assembling the governing mass and energy balance equations are provided by a routine called Eos (Equations of state). The main task for the Eos routine is to provide values for all secondary (thermophysical) variables as functions of the primary variables, though it also performs some additional important tasks (see [10], pages 17-26 for details). Several Eos routines are available for TOUGH2, and new Eos routines will become available. However, Eos3 is the only one that have been used in this parallel implementation. In order to provide maximum flexibility, we strive to minimize the number of changes that needs to be done to the Eos routine when moving from the serial to the parallel implementation. This have been done by organizing data and assigning appropriate values to certain variables before calling the Eos routine. In the current parallel implementation, the Eos3 routine from the serial code can be used unmodified, with the exception of some statements. Though, this still needs to be verified in practice, we believe that the current parallel version of TOUGH2 can handle also other Eos routines, with the only exception being some write statements needing adjustments. 4.5. Cray T3E-The Target Parallel System The parallel implementation of TOUGH2 is made portable through use of the standard Fortran 77 programming language and the MPI Message Passing Interface for interprocessor communication. The development and analysis, however, have been performed on a 696 processor Cray T3E-900 system. The T3E is a distributed memory computer, each processor has its own local memory. Together with some network interface hardware, the processor (known as Digital EV-5 or Alpha) and local memory form a Processing Element (PE), is sometimes called a node. All 696 PEs are connected by a network arranged in a 3-dimensional torus. See, e.g., [1] for details about the performance of the Cray T3E system. 5. Performance Analysis Parallel performance evaluation have been performed for a 2D and a 3D real application problem arising in the Yucca Mountain nuclear waste site study. Results have been obtained for up to 512 processors of the Cray T3E-900 at NERSC, Lawrence Berkeley National Laboratory. The linear systems have been solved using BICGSTAB with 3 \Theta 3 Block Jacobi scaling and a domain decomposition based preconditioner with the ILUT incomplete LU factorization. Different levels of overlapping have been tried for this procedure, though all results presented are for non-overlapping tests, which in general have shown to give good performance. The stopping criteria used for the linear solver is denote the residual and the right hand side, respectively. Both test problems require simulated times of 10 4 to 10 5 years, which would require a significant execution time also with good parallel performance and a large number of processors. In order to investigate the parallel performance, we have therefore limited the simulated time to 10 years for the 2D problem and 0:1 year for the 3D problem, which still require enough time steps to perform the analysis of the parallel performance. A shorter simulated time will of course give the initialization phase unproportionally large impact on the performance figures. The initialization phase is therefore excluded from the timings. Tests have been performed using the K-way, the VK-way, and the Recursive partitioning algorithms in METIS. As we will see later, different orderings of the grid blocks lead to variations in the time discretization following from the unstructured nature of the problem. This in turn lead to variations in the number of time steps required and thereby in the total amount of work performed. By trying all three partitioning algorithms and chosing the one that leads to the best performance for each problem and number of processors, we reduce these somewhat "artificial" performance variations resulting from differences in the number of time steps required. For all results presented, we indicate which partitioning algorithm have been used. 5.1. Results for 2D and 3D Real Application Problems The 2D problem consists of 17,584 blocks, 3 components per block and 43,815 connections between blocks, giving in total 52,752 equations. The Jacobian matrix in the linear systems to be solved for each Newton step is of size 52,752 \Theta 52,752 with 946,926 non-zero elements. The topmost graph in Figure 6 illustrates the reduction in execution time for increasing number of processors. The execution time is Number of processors Execution time 2h 5m 4s 2m 6s Number of processors Figure 6. Execution time and parallel speedup on the 2D problem for 2, 4, 8, 16, 32, 64, and 128 processors on the Cray T3E-900. HIGH PERFORMANCE COMPUTATIONS FOR SUBSURFACE SIMULATIONS 17 reduced from 7504 seconds (i.e., 2 hours, 5 minutes, and 4 seconds) on two processors to 126 seconds (i.e., 2 minutes and 6 seconds) on 128 processors. The parallel speedup for the 2D problem is presented in the second graph of Figure 6. Since the problem cannot be solved on one processor with the parallel code the speedup is normalized to be 2 on two pro- cessors, i.e., the speedup on p processors is calculated as 2T 2 =T p , where denote the wall clock execution time on 2 and p processors, respectively. For completeness we also report that the execution time for the original serial code is 8245 seconds on the 2D problem. The 3D problem consists of 97,976 blocks, 3 components per block and 396,770 connections between blocks, giving in total 293,928 equa- tions. The Jacobian matrix in the linear systems to be solved for each Newton step is of size 293,928 \Theta 293,928 with 8,023,644 non-zero elements. The topmost graph in Figure 7 illustrates the reduction in execution time for the 3D problem for increasing number of processors. Memory and batch system time limits prohibits tests on less than 16 processor. Results are therefore presented for 16, 32, 64, 128, 256, and 512 pro- cessors. The execution time is significantly reduced as the number of processors is increased, all the way up to 512 processors. It is reduced from 10055 seconds (i.e., 2 hours, 47 minutes, and 35 seconds) on processors to 329 seconds (i.e., 5 minutes and 29 seconds) on 512 processors. The ability to efficiently use larger number of processors is even better illustrated by the speedup shown in the second graph of the Figure 7. The speedup is defined as 16T 16 =T p since performance result are not available for smaller number of processors. The results clearly demonstrate very good parallel performance up to very large number of processors for both problems. We observe speedups up to 119.1 on 128 processors for the 2D problem and up to 489.3 on 512 processors for the 3D problem. When repeatedly doubling the number of processors from 2 to 4, from 4 to 8, etc, up to 128 processors for the 2D problem, we obtain the speedup factors 1.58, 2.85, 2.19, 1.91, 1.96, and 1.62. For the 3D problem, the corresponding speedup factors when repeatedly doubling the number of processors from 16 to 512 processors are 2.70, 2.28, 1.95, 1.50, and 1.69. As 2.00 would be the ideal speedup each time the number of processors is doubled, the speedup, e.g., 2.70 and 2.28 for the 3D problem are often called super-linear speedup. We will present the explanations for this in later sections. Number of processors Execution time 2h 47m 35s 5m 29s Number of processors Figure 7. Execution time and parallel speedup on 3D problem for 16, 32, 64, 128, 256 and 512 processors on the Cray T3E-900. HIGH PERFORMANCE COMPUTATIONS FOR SUBSURFACE SIMULATIONS 19 Overall the parallel performance is very satisfactory, and we complete this analysis by providing some insights and explaining the super-linear speedup. 5.2. An Unstructured Problem In the ideal case, the problem can be evenly divided among the processors not only with approximately the same number of internal grid blocks per processor, but also roughly the same number of external blocks per processor. Our problems, however, are very unstructured, which means that the partitioning can not be made even in these both aspects. This leads, for example, to imbalances between the number of external elements per processor when the internal blocks are evenly dis- tributed. For the 3D problem on 512 processors, the average number of external grid blocks is 234 but the maximum number of external blocks for any processor is 374. It follows that at least one processor will have 60% higher communication volume than the average processor (assuming the communication volume to be proportional to the number of external blocks). Note here that the average number of internal grid blocks is 191 for the same case. This means that the average processor actually has more external blocks that internal blocks. Finally, the average number of neighboring processors is 12.59 and the maximum number of neighbors for any processor is 25. Altogether this indicates that the communication pattern is irregular and that the amount of communication is becoming significant both in terms of number of messages and total communication volume. At the same time, the amount of computations that can be performed without external elements is becoming fairly small. Despite of these difficulties, the parallel implementation shows ability to efficiently use a large number of processors: every half second wall clock time, on 512 processors, a new linear system of size 293,928 \Theta 293,928 with 8,023,644 non-zero elements is generated and solved. This includes the time for the numerical differentiation for all elements of the Jacobian matrix, the 3 \Theta 3 Block Jacobi scaling for each block row, the ILUT factorization for the domain decomposition based preconditioner, and a number of BICGSTAB iterations. 5.3. Analysis of Work Load Variations Several issues need to be considered when analyzing the performance as the number of processors is increased. First, the sizes of the individual tasks to be performed by the different processors is decreased, giving an increased communication to computation ratio, and the relative load imbalance is also likely to increase. In addition, we may find variations in how the time discretization is performed (the number of time steps) and the number of iterations in the Newton process and the linear solver. In order to conduct a more detailed study, we present a summary of iteration counts and timings for the two test problems in Table II. The table shows the average number of Newton iterations per time step, and the average number of iterations in the linear solver per time step and per Newton step, as well as the total number of time steps, Newton iterations, and iterations in the linear solver. We recall that the linear system solve is the most time consuming operation and the computation of the Jacobian matrix is the second largest time consumer. Both of these operations are performed once for each Newton step. For both problems we note that some variations occur in the time discretization when the problem is solved on different number of pro- cessors. Similar behavior has been observed, for example, when using different linear solvers in the serial version of TOUGH2. The variations in time discretization leads to variations both in the number of time steps needed and the number of Newton iterations required. Notably, the 4 processors execution on the 2D problem requires 15% more time steps, 35% more Newton steps, and 91% more iterations in the linear solver compared to the execution on 2 processors. This increase of work fully explains the low speedup on 4 processors. Similar variations in the amount of work also contribute to a very good speedup for some cases. However, the figures in Table II alone do not fully explain the super-linear speedup observed for some cases. We will therefore continue our study by looking at the performance of the linear solver. Before doing that, however, we show some examples that motivates this continued study, i.e., cases where the speedup actually is higher than we would expect from looking at iteration counts only. For example, on the 2D problem the speedup on 8 processors is 12:4% larger than maximum expected (i.e., 8.99 vs. 8.00), but compared to the execution on two processors, the 8 processor execution actually requires slightly more Newton iterations and iterations in the linear solver. The number of time steps is the same for both tests. When doubling the number of processors from 8 to 16, we see another factor of 2.19 in speedup, even though the reduction in number of Newton iterations and iterations in the linear solver is only 2.4% and 9.6%, respectively. The speedup on the 3D problem from 16 to processors (2.70) and from 32 to 64 processors (2.28) is also higher than what would be expected by looking at Table II alone. So far, we can summarize the following observations for the two problems. The unstructured nature of the problem naturally leads to variations in the work load between different tests. This alone explains Table II. Iteration counts and execution times for the 2D and 3D test problems. 2D problem Partitioning algorithm VK VK K Rec. Rec. Rec. K Rec. #Time steps 104 120 104 104 104 94 94 103 Total #Newton iterations 645 869 669 653 663 697 620 637 #Newton iter./Time step 6.20 7.24 6.43 6.28 6.38 7.41 6.60 6.18 Total #Lin. solv. iterations 8640 16528 10934 9888 11011 11282 11894 19585 #Lin. solv. iter./Newton step 13.40 19.02 16.34 15.14 16.61 18.46 19.18 30.75 #Lin. solv. iter./Time step 83.1 137.1 105.1 95.1 105.9 120.0 126.5 190.1 Time spent on Lin. solv. Time spent on other Total time 3D problem Partitioning algorithm K Rec. K K Rec. Rec. #Time steps 154 149 143 137 185 166 Total #Newton iterations 632 606 585 561 708 646 #Newton iter./Time step 4.10 4.07 4.09 4.09 3.83 3.89 Total #Lin. solv. iterations 8720 10275 9357 10362 14244 14487 #Lin. solv. iter./Newton step 13.80 16.96 15.99 18.47 20.12 22.43 #Lin. solv. iter./Time step 56.6 69.0 65.4 75.6 77.0 87.3 Time spent in Lin. solv. Time spent on other Total execution time 22 ELMROTH, DING AND WU some of the speedup anomalies observed, but for a couple of cases, it is evident that are other issues to be investigated. We therefore continue this study by focusing on the performance of the linear solver and the preconditioner. 5.4. Performance of Preconditioner and Linear Solver A breakup of the speedup in one part for the linear solver (including preconditioner) and one for all other computations (mainly assembly of the Jacobian matrix) is presented for both problems in Figure 8. The figure illustrates that the super-linear speedup for the whole problem follows from super-linear speedup of the linear solver. Note that the results presented are for the total time spent on these parts, i.e., a different number of linear systems to be solved or a difference in the number of iterations required to solve a linear system affects these numbers. The speedup of the "other parts" is close to p for all tests on both problems, and this is also an indication that this part of the computation may show good performance also for larger number of processors. The slight decrease on 256 and 512 processors for the 3D problem is due to increased number of time steps. We conclude that the performance of "the other parts" is satisfying and that it needs no further explanations. We continue with the study of the super-linear speedup of the linear solver. 5.4.1. Effectiveness of the Preconditioner The preconditioner is crucial to the number of iterations per linear system solved. The domain decomposition based process is expected to become less efficient as the number of processors increases. The best effect of the preconditioner is expected when the whole matrix is used in the factorization, but in order to achieve good parallel performance, the size for the preconditioning operation on each processor is restricted to its local subdomain. On average, the matrix used in the preconditioning by each processor is n is the size of the whole (global) matrix and p is the number of processors. The reduced effectiveness follows naturally from the smaller subdomains, i.e., the decreased size of the matrices used in the preconditioner, since only diagonal blocks are used to calculate an approximate solution. The number of iterations required per linear system for the two test problems confirms this theory (see Table II). For both test problems the number of iterations required per linear system increases with the number of processors (with exceptions for going from 4 to 8 and 8 to processors for the 2D problem and 32 to 64 for the 3D problem). Number of processors Linear solver Other parts Number of processors Linear solver Other parts Figure 8. Breakup of speedup for the 2D and 3D problems in one part for the linear solver (marked "r") and one part for all other computations (marked "4"). The ideal speedup is defined by the straight line. We have also included the results for 256 processors on the 2D problem in Table II. We note an increase in the number of iterations per linear system by more than 50% compared to 128 processors. It is clear that the preconditioner does not perform a very good job when the number of processors is increased to 256. By introducing one level of overlapping (Additive Schwarz) in the domain decomposition based preconditioner on 256 processors, the number of iterations in the linear solver is reduced to the same order as for smaller number of proces- sors. This is however done at the additional cost for performing the overlapping and the overall time is roughly unchanged. Our main observation is that despite the overall increasing number of iterations in the linear solver for increasing number of processors, the speedup of the linear solver is higher than what would normally be expected up to a certain number of processors. The increased number of iterations per linear system for larger number of processors is obviously following from the reduced effectiveness of the preconditioner. In the following sections, we will conclude the performance analysis by investigating the parallel performance of the actual computations performed during the preconditioning and linear iteration processes. 5.4.2. Performance of the Preconditioner Another effect of the decreased sizes of the subdomains in the domain decomposition based preconditioner is that the total amount of work to perform the incomplete LU factorizations becomes significantly smaller as the number of processors is increased. For example, as the number of processor is doubled, the size of each processor's local matrix in the ILUT factorization is decreased by a factor of 4, on average from n p to n 2p \Theta n 2p . Hence will the amount of work per processor be reduced a factor between 2 and 8 depending on the sparsity structure. Hence, the amount of work per processor is reduced faster than we normally expect when we assume the ideal speedup to be 2. Figure 9 gives a further breakup of the speedup, now with the speedup for the linear solver separated into one part for the preconditioner (i.e., the ILUT factorization) and one for the other parts of the linear solver. The preconditioner shows a dramatic improvement of the performance as the number of processors increases, following naturally from the decreased work in the ILUT factorization. As we continue, this will turn out to be the single most important explanation for the sometimes super-linear speedup. Number of processors Other parts Preconditioner Linear solver excluding preconditioner Number of processors Other parts Preconditioner Linear solver excluding preconditioner Figure 9. Breakup of speedup for the 2D and 3D problems in one part for the linear solver excluding the ILUT factorization for the preconditioner (marked "r"), one part for ILUT factorization (marked "*") and one part for all other computations (marked "4"). The ideal speedup is defined by the straight line. 26 ELMROTH, DING AND WU 5.4.3. Performance of the Linear Iterations We have explained the super-linear speedup of the preconditioning part of the linear solver, presented in Figure 9, but we also observe only a modest speedup of the other parts of the computation. The low speedup for the other parts is partly explained by the increased number of iterations, as seen in the previous section. Increased communication to computation ratio and slightly increased relative load imbalance are other factors. Already the figures presented in Section 5.2 showing large number of external elements per processor and an imbalance in the number of external elements per processor presented for the 3D problem on 512 processors indicated that the communication to computation ratio would eventually become large. The performance obtained for the iteration process in the linear solver supports this observation. 5.4.4. Impacts on the Overall Performance In order to fully understand the total (combined) effect of the super-linear behavior of the preconditioner and the moderate speedup of the other parts of the linear solver, we now only need to investigate how large proportion is spent on preconditioning out of the total time required for solving the linear systems. This is illustrated in Table III. Table III. Time spent on preconditioning as percentage of total time spent in linear solver. 2D problem #Processors Percentage 83.5% 77.1% 73.9% 69.2% 61.6% 49.9% 36.4% 3D problem #Processors Percentage 88.4% 78.9% 73.0% 66.0% 58.6% 39.4% As the number of processors becomes large, the amount of time spent on preconditioning becomes small compared to the time spent on iterations, whereas the relation is the opposite for 2 processors on the 2D problem and for 16 processors on the 3D problem. For example, for 16 processors on the 3D problem 88.4% of the time in the linear solver was spent on the factorization for the preconditioner, whereas the corresponding number is only 39.4% for 512 processors. As long as the preconditioner consumes a large portion of the time, its super-linear speedup will have significant effects on the overall performance of the implementation. It is evident, that up to a certain number of processors, the super-linear speedup of the incomplete LU factorization in the domain decomposition based preconditioner is sufficient to give super-linear speedup for the whole application. As the number of processor becomes large, the factorization consumes a smaller proportion of the execution time, and hence, it's super-linear behavior has less impact on the overall per- formance. Instead, there are other issues that become more critical for large number of processors, such as the increased number of iterations in the linear solver. 6. Conclusions This contribution presents the design and analysis of a parallel prototype implementation of the TOUGH2 software package. The parallel implementation shows to efficiently use up to at least 512 processors of the Cray-T3E system. The implementation is constructed to have flexibility to use different linear solvers, preconditioners, and grid partitioning algorithms, as well as alternative Eos modules for solving different problems. Computational experiments on real application problems show high speedup for up to 128 processors on a 2D problem and up to 512 processors on a 3D problem. The results are accompanied by an analysis that explains the good parallel performance observed. It also explains some minor variations in performance following from the unstructured nature of the problem and some super-linear speedups following from decreased work in the preconditioning process. The results also illustrate the trade-off between the time spent on preconditioning and the effect of its result. With the objective to minimize the wall clock execution time, we note that, for these particular problems, smaller subdomains could be used, at least on small number of processors. We have seen some variations in performance in tests using three different partitioning algorithms. Some of these variations clearly follow from variations in the amount of work required, e.g., due to differences in the time discretization. Further analysis is required in order to determine whether these variations follow some particular pattern or if they are only a result from unpredictable circumstances. 28 ELMROTH, DING AND WU The problems we are targeting in the near future are larger both in terms of number of blocks and number of equations per block. Moreover should the simulation time be significantly longer. With increased problem size we expect to be able to efficiently use an even larger number of processors (if available), and longer simulations should not directly affect the parallel performance. Future investigations include studies of alternative non-linear solvers and further studies of the interplay between the time stepping proce- dure, the non-linear systems, and the linear systems. Evaluations of different linear solvers, preconditioners and parameter settings would be of general interest and may help to further improve the performance of this particular implementation. A related study of partitioning algorithms have recently been completed [4]. Acknowledgements We thank Karsten Pruess, the author of the original TOUGH2 software, for valuable discussions during this work, and the anonymous referees for constructive comments and suggestions. This work is supported by the Director, Office of Science, Office of Laboratory Policy and Infrastructure, of the U.S. Department of Energy under contract number DE-AC03-76SF00098. This research uses resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy. --R 'Aztec User's Guide. 'TOUGH User's Guide'. Domain Decomposition. --TR BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems Domain decomposition A parallel implementation of the TOUGH2 software package for large scale multiphase fluid and heat flow simulations Performance of the CRAY T3E multiprocessor

performance analysis;software design;grid partitioning;groundwater flow;preconditioners;iterative linear solvers

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Asynchronous Transfer Mode and other Network Technologies for Wide-Area and High-Performance Cluster Computing.

We review fast networking technologies for both wide-area and high performance cluster computer systems. We describe our experiences in constructing asynchronous transfer mode (ATM)-based local- and wide-area clusters and the tools and technologies this experience led us to develop. We discuss our experiences using Internet Protocol on such systems as well as native ATM protocols and the problems facing wide-area integration of cluster systems. We are presently constructing Beowulf-class computer clusters using a mix of Fast Ethernet and Gigabit Ethernet technology and we anticipate how such systems will integrate into a new local-area Gigabit Ethernet network and what technologies will be used for connecting shared HPC resources across wide-areas. High latencies on wide-area cluster systems led us to develop a metacomputing problem-solving environment known as distributed information systems control world (DISCWorld). We summarize our main developments in this project as well as the key features and research directions for software to exploit computational services running on fast networked cluster systems.

Introduction Cluster computing has become an important part of the high-end computing world. Many of the applications traditionally run on high-end supercomputers are now successfully run on computer clusters. In this paper we describe our experiences in building high performance computer clusters and in linking them together as clusters of clusters over wide areas. This approach towards an Australian "National Computer Room" was started in 1996 and led us to two important conclusions about wide-area computing. Firstly from a technical standpoint, the limitations of latency and of net-work reliability are likely to be more important than bandwidth limitations in the future and therefore successful wide-area systems need to be constructed accordingly. ondly, the political and administrative issues behind supercomputer resource location and ownership mean that in the future it will be attractive to loosely cluster compute resources nationally and even internationally. Institutions have strong reasons to wish to retain ownership and control of their own computer resources. Clusters of local re- sources, each of which itself may be a cluster of some sort, are therefore an attractive architecture to target for software development. In this article we try to divide cluster computing systems issues into those relevant to achieving high performance (typically on local area compute clusters) and those issues pertaining to the successful sharing of resources between institutions (typically wide-area issues). The size and utilisation of computer clusters varies enormously. The recent IEEE Workshop on Cluster Computing [3] demonstrated this as well as the different uses of the term cluster computing. We will use cluster in its broader sense to describe a collection of networked compute resources that may be heterogeneous in hardware and in operating systems software, but which are capable of cooperating in some way on a distributed application. This may be using any of the parallel computing software models such as PVM[8], MPI[25] or HPF[17] or through some higher granularity object or service based system like Globus[7], Legion[10], CORBA[27] or DISCWorld[16]. We use the term Beowulf-class cluster in the sense it now widely adopted to describe a cluster that is dedicated in some sense, and is not just a random collection of networked computers. The term "constellation" is now being adopted to describe a compute cluster that uses shared memory or bus based technology. We focus on clusters and Beowulf-class systems in describing our work in this paper. At the time of writing the watershed sizes for clusters are limited by the number of ports on commercially available network switches. For example, many groups report work on modest clusters of 8 or 16 processors. We continue to operate our prototype 8 node Beowulf system for student experiments. Modern switches are typically capable of interconnecting 16, 24 or even 48 compute nodes using Fast Ethernet at 100MBit/s bandwidth between all nodes. The limits in size from a single switch therefore might be used as the delimiting size for distinguishing large and small computer clusters. This number is at present larger than the number of processors that can be economically linked using shared memory technology. High end systems such as the ASCI systems and Tera computer architecture do achieve considerably higher number of nodes interlinked with shared memory, but can hardly be described as commodity economical architectures. Switches such as the 510 series from Intel allow switches to be stacked using high bandwidth links between individual switches to preserve the full point to point band-width between compute nodes. We have constructed a 120 node Beowulf class system using this flat switch architecture and dual processor nodes. The Intel 510 series is limited to linking 7 of these 24 port switches in this manner and this limit of 168 networked nodes represents well the limitations of current network technology. Fast Ethernet technology is used to link individual nodes, with Gigabit Ethernet uplinking used between the switch backplane and the file server node. Beowulf systems larger than this are being built, but must use some sort of hierarchical network architecture and are no longer "as commodity" in nature. It is likely that network technology will move forward with economics and technical improvements and that commodity switchgear will soon allow bigger systems. systems will therefore continue to encroach on the traditional supercomputing market area. Scalable software to run on such systems is available for some applications prob- lems, but our experience is that some effort is still needed to ensure the major contribution towards latency in parallel applications running on clusters does not remain the operating system kernel or communications software. Achieving low latency in sending messages between nodes in a Beowulf system appears to be a more important limitation at present than achieving high bandwidth. We describe our experiences in constructing ATM-based clusters in section 2. We attempt to summarise our metacomputing approach to cluster management in section 3 and describe how some of the latency and distributed computing problems, that are so important to wide-area clusters, can be tackled. We review some of the networking technologies actively used for cluster computing systems in section 4 and discuss their relative technical and economic advantages. In this section we describe some experiences from an experimental wide area system we ran from 1996 to 1998, employing local and wide area ATM technology to connect high performance computing clusters around Australia. The Research Data networks (RDN) Cooperative Research Centre (CRC) was set up as a joint venture involving our own University of Adelaide, the Australian National University in Canberra, Monash University in Melbourne, the University of Queensland and Telecom Australia (Tel- stra). The map shown in figure 1 shows the initial broadband network we were able to set up and use. It connected Adelaide, Melbourne, Sydney, Canberra and Brisbane; some of its latency and performance aspects are discussed below. We trialed links of 34Mbit/s and 155Mbit/s over various branches of the network and experimented with various combinations of: cross mounted long distance filesystems; video conferencing; simple shared whiteboard style collaborative software; and cross connected computer clusters. We describe the Asynchronous Transfer Mode (ATM) technology used for these experiments in section 4. The main features of our experiments were to analyse the effects of having relatively high wide area bandwidths available for cluster computing yet having perceptibly high latencies arising from the long distances. We were able to construct well-connected compute clusters having the peculiar bisection bandwidth and latency properties that arise from having part of the cluster at each geographically separate site. Although a number of wide area broadband networks have been built in the USA [4, 26], it is unusual to have a network that is fully integrated over very long distances rather than local area use of ATM technology. Telstra built the Experimental Broad-band Network (EBN) [21] to provide the foundation for Australian broadband application development. Major objectives of this were to provide a core network for service providers and customers to collaborate in the development and trial of new broadband applications, and to allow Telstra and developers to gain operational experience with public ATM-based broadband services. OCEAN Kilometres Network (EBN) Broadband Telstra's Experimental @KAH,Great Australian Bight Bass Strait Figure 1: Telstra's Experimental Broadband Network (EBN). The EBN is a 34Mbit/s ATM network connecting research and commercial partner sites. Figure shows how our prototype storage and processing hardware were arranged at both Adelaide and Canberra (separated by some 1200km) and connected by the EBN. The Adelaide and Canberra cells consisted of a number of DEC AlphaStations, interconnected locally by 155MBit/s ATM, and with the cells at the two sites connected across the EBN at 34MBit/s. We experimented extensively with cluster computing resources at Adelaide and Canberra in assessing the capabilities of a distributed high performance computing system operating across long distances. It is worth considering the fundamental limitations involved in these very long distance networks. Although we employ OC-3c (155MBit/s) multi-mode fibre for local area networking we were restricted to an E-3 (34MBit/s) interface card to connect Adelaide to Melbourne and hence to Canberra. In practice this was not a severe limitation as for realistic applications were were unable to fully utilise a 155MBit/s link over wide areas except in the most contrived circumstances. This would not be the case for a shared link however. The line-of-sight distances involved in parts of the network were: Adelaide/Melbourne ASX-1000 Switch Compute File Servers X-Terminals DEC Alpha SGI Power Challenge Workstation Multi-Mode Fibre, OC-3c (155Mbps) Farm Storage Works RAID Tape Library DEC Alpha Workstation Farm Storage Works RAID Canberra Network (EBN) Broadband Experimental Telstra's Adelaide Brisbane Sydney Melbourne at Adelaide and Canberra on the EBN Project Resources Gigaswitch Fileserver Figure 2: DHPC Project Hardware Resources at Adelaide and Canberra connected via Telstra's EBN. While traffic can be sent between sites via the 'ordinary' Internet, bandwidth-intensive experiments were carried out by specifically routing traffic over the EBN. 732 km. Consequently the effective network distances between Adelaide and the other cities are shown in table 1. The light-speed-limited latencies shown in table 1 are calculated on the basis of the vacuo light-speed (2.997810 5 kms -1 ). It should therefore be noted that this is a fundamental physics limitation and does not take into consideration implementation details. EBN City Network Distance from Adelaide (km) Light-speed-Limited Latency (ms) Melbourne 660 2.2 Table 1: Inter-city distances (from Adelaide) and Light-speed-limited latencies for the Experimental Broadband Network. We made some simple network-performance measurements using the Unix ping utility, which uses the Internet Control Message Protocol (ICMP), for various packet sizes which are initiated at Adelaide, and bounced back from a process running at other networked sites. These are shown in table 2. By varying the packet sizes sent it is possible to derive crude latency and bandwidth measurements. It should be emphasized these measurements are approximations of what is achievable and are only for comparison with the latency limits in table 1. The times in table 2 were averaged over 30 pings and represent a round-trip time. Measurements are all to a precision of 1 ms except for those for Syracuse which had a significant packet loss and variations that suggest an accuracy of 20 ms is appropriate. The ping measured latency between Adelaide and Canberra appears to be approx- Ping Mean Time Mean Time Mean Time Mean Time Packet Canberra Syracuse USA local machine local machine Size via EBN via Internet via ethernet via ATM switch (Bytes) (ms) (ms) (ms) (ms) 1008 2008 8008 22 441 Table 2: Approximate Performance Measurements using Ping. imately 15 ms. This is to be compared with the theoretical limit for a round-trip of 7.6 ms. The switch technology has transit delays of approximately 10 s per switch. Depending upon the exact number of switches in the whole system this could approach a measurable effect but is beyond the precision of ping to resolve. We believe that allowing for non-vacuo light-speeds in the actual limitation our measured latency is close (within better than a factor of two) to the best achievable. We believe that variations caused by factors such as the exact route the EBN takes, the slower signal propagation speed over terrestrial copper cables, the routers and switch overheads and small overheads in initiating the ping, all combined, satisfactorily explain the discrepancy in latencies. The EBN appears to provide close to the best reasonably achievable latency. Also of interest is the bandwidth that can be achieved. The actual bandwidth achieved by a given application will vary depending upon the protocols and buffering layers and other traffic on the network but these ping measurements suggest an approximate value of 2 8kB/(22 - 22.7MBit/s. This represents approximately 84% of the 27MBit/s of bandwidth available to us, on what was an operational network. The Unix utility ttcp was useful for determining bandwidth performance measurements which are outwith the resolution possible with ping. A typical achievable band-width between local machines on the operational 155MBit/s fibre network is 110.3 MBit/s compared with a typical figure on local 10MBit/s ethernet of 6.586MBit/s Both these figures are representative of what was a busy network with other user traffic on them. In 1998 and 1999 an additional link connecting Australia and Japan was made available to us with a dedicated 1MBit/s of bandwidth available for experimentation. We carried out some work in collaboration with the Real World Computing Partner- ship(RWCP) [28] in Japan and found that there was an effective latency of around 200ms single trip between Adelaide and Tsukuba City in Japan over this ATM net-work All these experiences suggest that the major limitations for wide area cluster computing in the future are more likely to be from latency limitations rather than from bandwidth limitations. The Telstra Experimental Broadband network was closed at the end of 1998 and we have had an alternative ATM network supplied by Optus Pty Ltd made available to us in 1999. This network has similar latency characteristics but has the advantage that we can apply to burst up to a full 155MBit/s between Adelaide and Canberra at will. In practice, we have found that 8MBit/s is entirely adequate for our day to day wide-area cluster computing operations. It is interesting to reflect on our experiences in setting up and using these net- works. Staff at Telstra's research laboratories were instrumental in setting up the initial circuits and allocating bandwidth to our experiments. As the trials proceeded this became a well automated process with little human intervention required except when we needed to split bandwidth allocations between video, cluster and file system circuits for example. One of the problems with the experimental set up was that only permanent virtual circuits (PVCs) were available to us and that the network configuration (on the routing hosts) needed to be changed (manually) each time a reconfiguration was required. One of the promises of the ATM standard is that of Switched Virtual Circuits (SVCs) which could be manipulated in software to reallocate bandwidth to different applications. Although we were able to experiment with pseudo SVCs using the proprietary facilities of our ATM switch gear, it was not possible to enable proper switch virtual circuits across different vendors' switch gear at the different sites in our collaboration. To our knowledge this is still not available on wide area ATM networks and is, we believe a disappointing limitation of this technology in practice. We expended considerable time and effort in carrying out these experiments, which was particularly difficult when we had to reconfigure routers and operating systems and driver software every time we needed to carry out a different experiment. ATM technology has matured since the start of our experiments but we believe it will not be the technology of choice at the cluster computing level. ATM is likely to still find a place as a network backbone technology, certainly for long distance networks and perhaps even still in local area networks. Driver support and administration tools for ATM based cluster computing do not appear to be forthcoming and consequently we believe it much more likely that Internet Protocols implemented on top of ATM or other technologies perhaps will be much more useful for cluster computing. 3 Metacomputing Cluster Management Our vision at the start of our wide area network trials was of a "National Computer Room". We imagined this as a network and software management infrastructure that would enable institutions to share access to scarce high-performance computing re- sources. This seemed important given the scarcity of supercomputing systems in Australia in particular. We have now revised this vision however. While there are still some areas in which a computer sharing mechanism is important, the whole supercomputing industry has continued to be shaken up. Global reduction in the number of supercomputer vendors still in business can be attributed to a number of factors, but not least of these is the greater availability of cluster computing resources. The need for institutions to own and control their own resources appears to be a strong one, and relatively cheap cluster computing systems enables this phenomena. We believe therefore that although there is still a need for software to enable compute resource sharing, the resources are more likely to be clusters themselves and this affects the characteristics of software and applications that can integrate them. Software such as Netsolve, Globus, Legion, Ninf and our own DISCWorld system are all aimed at enabling the use of distributed computing resources for applications. Our favoured approach is to encapsulate applications (including parallel ones) as services that can run in a metacomputing environment and we have focussed our efforts in to building Java middleware to allow this. Other approaches have taken experiences from the parallel computing era to allow wide area systems to behave as a more tightly coupled parallel system running multi processor applications across the wide area net- works. The choice is a matter of granularity and of how to group parts of the application together. The wide area parallel approach is attractive for its simplicity in that the tools and technologies from parallel computing can be redeployed with only software re-engineering efforts. The metacomputing approach recognizes that networks (espe- cially wide-area ones will remain unreliable, and that long distance latencies will not improve (short of some new physics breakthrough) and that the fundamental distributed computing problems need to be considered when building wide-area cluster systems. Distributed Information Systems Control World (DISCWorld) [16] is a metacomputing model or framework with a series of prototype systems developed to date. The basic unit of execution in the DISCWorld is that of a service. Services are pre-written software components or applications. These are either written in Java or are legacy codes that have been provided with a Java wrapper. Users can compose a number of services together to form a complex processing request. Jobs can be scheduled across the participating nodes [19]. An example DISCWorld application is a land planning system [5], where a client application requires access to land titles information at one site, and digital terrain map data at another, and aerial photography or satellite imagery stored at another site. DISCWorld itself does not build on parallel computing technology, but can embed parallel programs as services. A support module known as JUMP provides integration with message-passing parallel programs which might run on a conventional supercomputer as an or on a cluster[13]. An environment such as DISCWorld is ideal for use across a cluster or a distributed network of clusters. The nodes within the cluster can be used as federated hosts connected via a high-speed, dedicated network. Some or all of the nodes can be used as a parallel processor farm, running those services which are implemented as parallel programs. We employed both these approaches in our multi cluster experiments across our ATM network. The ability to make intelligent decisions on where to schedule services in the DISCWorld environment, in order to minimize either execution time or total resource cost relies on the ability to characterise both the services and the nodes in the environment. Fast Networking Technologies In this section we review the major networking technologies and describe their roles for wide area and high-performance cluster computing. We discuss Asynchronous Transfer Mode (ATM); Fast Ethernet; Gigabit Ethernet; Scalable Coherent Interconnect (SCI); and Myrinet technologies. Each of these technologies may have differing roles to play in wide-area and cluster computing over the next few years. Table 3 summarises the networking technologies we consider here. The data in the table has been combined from our own benchmarking experiments as well as reports in the literature [6, 22, 23, 29]. It can be seen that for problems which do not require gigabit-order bandwidth Fast Ethernet seems a logical choice. If gigabit-order band-width is required, the decision is not so clear-cut: there is a complex trade-off between the bandwidth and latency attainable, the approximate cost per node, and whether the technology is considered commodity. When we use the term commodity, we refer to the fact that the technology is available to the mass market, and is available from more than a few specialist vendors. Technology Theoretical Measured TCP/IP Latency Approx Cost Bandwidth Bandwidth Per Node ($US) Ethernet 10MBit/s 6.58MBit/s 1.2ms 80 Fast Ethernet 100MBit/s 68MBit/s 1ms 150 ATM 155MBit/s 110.3MBit/s 1ms 2500 Gigabit Ethernet 1000MBit/s 950MBit/s 12ms 1200 SCI 1600MBit/s 106MBit/s 4s 1400 Myrinet 1200MBit/s 1147MBit/s 117s 1700 Table 3: Interconnection network technologies characteristics as measured and also reported in the open literature. We were able to trial our ATM network of clusters using sponsorship from the Australian Commonwealth government. Table 3 shows that ATM is still not an especially economic choice. It is our belief that Gigabit Ethernet technology will be widely adopted and that it will be correspondingly driven down in price. 4.1 ATM Asynchronous Transfer Mode (ATM) [1] is a collection of communications protocols for supporting integrated data and voice networks. ATM was developed as a standard for wide-area broadband networking but also finds use as a scalable local area networking technology. ATM is a best effort delivery system - sometimes known as bandwidth- on-demand, whereby users can request and receive bandwidth dynamically rather than at a fixed predetermined (and paid for) rate. ATM guarantees the cells transmitted in a sequence will be received in the same order. ATM technology provides cell-switching and multiplexing and combines the advantages of packet switching, such as flexibility and efficiency of intermittent traffic, with those of circuit switching, such as constant transmission delay and guaranteed capacity. ATM uses a point-to-point, full-duplex transmission medium, and provides connection-oriented protocols. In addition to supporting native ATM protocols such as ATM Adaption Layer (AAL) 3/4 and 5, the use of LANE allows the ATM network to be viewed as part of a TCP/IP network (to allow multicast and broadcast). There have been a number of studies that consider the use of the AALs for high-speed interconnections in parallel computing [20]. We employ a 155MBit/s local-area ATM network [14, 15] as well as a 34MBit/s broadband network (EBN). ATM allows guaranteed bandwidth reservation across a circuit (a link or a number of links with defined end-points. There are three different types of bandwidth reser- vation: constant bit rate (CBR); variable bit rate (VBR); and available bit rate (ABR). CBR is used when the traffic between sites is constant and will rarely, if ever change in bandwidth requirements. VBR is used to characterise traffic that has a mean bandwidth requirement that can change slightly. Traffic along a VBR circuit can operate at the cir- cuit's maximum bandwidth only for short amounts of time; intermediate switches may drop cells that exceed the VBR parameters. Finally ABR exists for bursty traffic which cannot be easily characterised. When ABR traffic is sent it uses any available link ca- pacity; if there is not enough link capacity for the traffic the ABR traffic is queued until the buffers fill, in which case cells are dropped. ATM has been popular with the Telecommunications Industry [21] for broadband networks, where CBR and VBR traffic is used for dedicated (statically- and dynamically- allocated) customer links (for voice and constant-rate data). As typical data is characterised as bursty, it does not make sense to reserve bandwidth in a CBR or even VBR capacity. ABR, which uses any available bandwidth on an ATM link, must be used to avoid wasting capacity. There have not been many sites that have adopted ATM as a local area network. The major factor prohibiting widespread adoption is the cost of ATM switches and of ATM interface cards for individual nodes. 4.2 Other Notable Networking Technologies High Performance Parallel Interface (HiPPI) is a point-to-point link that uses twisted-pair copper cables to connect hosts via crossbar switches. HiPPI gets its name because data is transmitted in parallel; the connecting cables have 50 cores: are used to transmit data, one bit per line. The standard allows for transmission rates of 800MBit/s and 1600MBit/s but the maximum length of copper cables is 25 metres. A serial version of HiPPI is available, using fibre optic media, which allows a maximum distance of 10km between ends. HiPPI has been successfully used for networking supercomputer systems, but now seems likely to be overtaken by other cheaper technologies. Fibre Channel (FC), defined by the Fibre Channel Standard, is a circuit-switching and also packet-switching technology that allows transmission at multiple rates. Data is sent in frames of 2148 bytes (2048 bytes of data). FC is successfully used in interconnecting hosts but is also likely to be overtaken by the economics of other technologies. 4.3 Internet or Native Technology Protocols Many networking technologies have proprietary communications protocols, such as ATM Adaptation Layers (AAL). While these proprietary protocols provide optimized communication, there is usually a trade-off for code portability. Using proprietary protocols can also be fraught with pitfalls. The majority of users do not wish to make every optimisation to their code, or because users are working with heterogeneous mixtures of networking hardware, the custom protocols are not widely used. For example, experiments with ATM's AAL5 uncovered a bug in the implementation that the manufacturer's engineers said could only be fixed by purchasing a much faster machine [20]. We experimented with implementing message passing communications software on raw ATM Adaptation layers and conclude that since this approach has not been widely supported by ATM vendors, this will not be a feasible approach in the long term. We conclude from our experiments that using a well-known, standard protocol such as IP is the easiest, and most portable approach. Most manufacturers provide for the encapsulation of IP packets within their proprietary protocols (such as ATM's LAN Emulation). IP promises to be around a long time with the advent of IPv6, which features IP tunneling, allowing standard IP traffic to be encapsulated within IPv6 packets[18]. In summary, we have found that while 10MBit/s desktop connections are still very common, most current machines are now capable of effectively utilising a 100MBit/s connection. For clusters of workstations with medium bandwidth requirements 100MBit/s Fast Ethernet is a viable and affordable solution. If high bandwidth is required then a trade-off between the cost and performance is necessary: SCI provides the best latency but the measured bandwidth is nowhere near as high as Myrinet; Myrinet has the largest bandwidth of all the systems we consider, and the latency is still under a millisecond. If a larger latency can be tolerated, and bandwidth is not critical, then we believe the open standards of Gigabit Ethernet may be an appropriate choice. We reiterate that we believe the economics of large markets will ensure Gigabit Ethernet technology becomes widespread for cluster computing. 5 Summary and Conclusions In this article we have reviewed a number of fast networking technologies for cluster computing and related our experiences with ATM in particular. Preliminary experiments and experiences now being reported leads us to believe that the combination of Fast Ethernet and Gigabit Ethernet will be the chosen route for most general purpose Beowulf class cluster computing systems. We believe that the technical criteria being fairly finely balanced with only factor of two advantages for one technology over an- other, that the economics of the mass market will dictate which technology becomes most widespread. This of course is a feedback phenomena as cheaper switches and network cards will be adopted even more widely. We believe ATM may still have a role to play in wide area cluster computing systems but it is preferable that it be transparent to cluster users and cluster system soft- ware. Internet protocol will surely be implemented on top of ATM and users and cluster software will continue to interface to that. We believe there are still many interesting developments to be made in software to manage wide area clusters as well as high-performance clusters. It seems important to recognise that will message passing level technology can play a useful part in high performance systems, there is some considerable research still to be done to address the latency and reliability issues for wide area cluster systems. 6 Acknowledgements Thanks to J.A.Mathew for assistance is carrying out some of the measurements reported in this work. It is also a pleasure to thank all those who helped in conducting trials of the EBN: K.J.Maciunas, D.Kirkham, S.Taylor, M.Rezny, M.Wilson and M.Buchhorn. --R Available at http://www. CASA Project. A Comparison of High Speed LANs The Grid: Blueprint for a New Computing In- frastructure PVM: Parallel Virtual Machine A Users' guide and Tutorial for Networked Parallel Computing Gigabit Ethernet Alliance. IEEE Standards On-line IEEE Standards On-Line Geographic Information Systems Applications on an ATM-Based Distributed High Performance Computing System DISCWorld: An Environment for Service-Based Metacomputing High Performance Fortran Forum (HPFF). Internet Engineering Task Force. Scheduling in Metacomputing Systems. Using ATM In Distributed Applications. Telstra's Experimental Broadband Network. A comparison of two Gigabit SAN/LAN technolo- gies: Scalable Coherent Interface and Myrinet An Assessment of Gigabit Ethernet as Cluster Interconnect. ATM Performance Characteristics on Distributed High Performance Computers. Message Passing Interface Forum. Applications and Enabling Technology for NYNET Upstate Corridor. Object Management Group. 3Com Corporation. --TR PVM: Parallel virtual machine The grid DISCWorld Myrinet Geographic Information Systems Application on an ATM-based Distributed High Performance Computing System Geostationary-satellite imagery applications on distributed, high-performance computing Legion-a view from 50,000 feet An Assessment of Gigabit Ethernet as Cluster Interconnect

DISCWorld;gigabit ethernet;ATM;fast ethernet;metacomputing;cluster computing

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Piecewise Self-Similar Solutions and a Numerical Scheme for Scalar Conservation Laws.

The solution of the Riemann problem was a building block for general Cauchy problems in conservation laws. A Cauchy problem is approximated by a series of Riemann problems in many numerical schemes. But, since the structure of the Riemann solution holds locally in time only, and, furthermore, a Riemann solution is not piecewise constant in general, there are several fundamental issues in this approach such as the stability and the complexity of computation.In this article we introduce a new approach which is based on piecewise self-similar solutions. The scheme proposed in this article solves the problem without the time marching process. The approximation error enters in the step for the initial discretization only, which is given as a similarity summation of base functions. The complexity of the scheme is linear. Convergence to the entropy solution and the error estimate are shown. The mechanism of the scheme is introduced in detail together with several interesting properties of the scheme.

Introduction . Self-similarity of the Cauchy problem for one dimensional conservation laws, with Riemann initial data has been the basis of various schemes devised for general initial value problems, Glimm [9] and Godunov [10], for example. The self-similarity of the Riemann problem is the property that the solution is a function of the self-similarity variable x=t. In other words the solution is constant along the self-similarity lines x The basic idea of the Godunov scheme for a general initial value problem is to approximate the initial data by a piecewise constant function and then apply the self-similarity structure to the series of Riemann problems. There are two basic issues we have to consider immediately in this kind of ap- proach. First, since the self-similarity for a piecewise constant solution holds locally in time only, the structure of the Riemann problem can be applied for a small time period. In other words the scheme is not free from the CFL condition and, hence, the scheme can march just a little amount of time every time step and it costs computation time. Furthermore, since rarefaction waves appear immediately, the solution is not piecewise constant anymore. So a numerical scheme contains a process which rearranges the rarefaction wave into a piecewise constant function every time step. The numerical viscosity enters in this process and tracking down the behavior of the scheme becomes extremely hard. IMA, University of Minnesota, Minneapolis, MN 55455-0436 (yjkim@ima.umn.edu). LeVeque [15] considers a large time step technique based on the Godunov method for the genuinely nonlinear problem. In the scheme the CFL number may go beyond 1, and it is even possible to solve the propagation of a simple wave in a single step, for the given final time T ? 0. However the scheme handles interactions between waves incorrectly if the CFL number is so large. One way to avoid the rearranging process is to consider a modified equation, where h and u 0 approximate f and v 0 respectively. Dafermos [6] considers a polygonal approximation h f , i.e., h is a continuous, piecewise linear function. In the case the exact solution of (1.4) is piecewise constant. So the method does not require a rearranging process and, hence, it does not introduce numerical viscosity and the error is controlled by taking the polygonal approximation h. In this approach the exact behavior of the solution can be monitored more closely and we may get a more detailed understanding of the numerical scheme based on this approach. This idea has been developed in Holden and Holden [11], and it has been extended to multi-dimensional problems in Holden and Risebro [13] and to systems of conservation laws in Holden, Lie and Risebro [12]. In particular we refer Bressan [2], [3] for systems. This front tracking method has been developed, especially by the Norwegian School, as a computational tool. Lucier [17] approximates the actual flux f by a piecewise parabolic function h and achieves a second order scheme. In the case the initial data v 0 (x) is approximated by a piecewise linear function u 0 and the solution remains piecewise linear. The difference between the solutions of the original problem (1.1) and the modified problem (1.4) is estimated by Since the linear approximation is of second order, he achieves a second order scheme for a fixed time t ? 0. If we want to design a numerical scheme which represents the exact solution, we have to find a way to choose the grid points correctly. If they are simply fixed, it is clear that the scheme can not represent the exact solution and, hence, we need to rearrange the solution to fit the solution to the fixed grid points. So it is natural to consider moving mesh method, see Miller [18]. In Lucier [17] the moving mesh method is used to find the exact solution of (1.4), where mesh points move along characteristics. Another option is not to use any grid point. In numerical schemes based on the front tracking method we mentioned earlier grid points are used just for the initial discretization. The scheme we consider in this article does not use any grid point, neither. This article has two goals. The first one is to introduce the mathematical idea which is behind the piecewise self-similar solutions. The second one is to demonstrate how to implement the idea into a numerical scheme and show properties of the scheme. From the study of the Burgers equation [14] it is easily observed that the primary structure which dominates the evolution is a saw-tooth profile. In fact it is a series of N-waves and eventually the solution evolves to a single N-wave, see Liu and Pierre [16]. The starting point of our scheme is to use this structure as the unit of the scheme. This scheme has several unique properties that other schemes based on piecewise constant functions do not have. Piecewise Self-similar Solutions 3 Suppose that u(x; t) is a special solution of (1.1) which is a function of the self-similarity variable x=t. Then the self-similarity profile (or the rarefaction wave), easily derived from (1.1). It is natural to expect that characteristic lines pass through the origin, i.e., they are compatible with self-similarity lines (1.3). The piecewise self-similarity initial profile is considered in the sense that Note that the time index t k can be a negative number. In this article we show that the solution of (1.1) with piecewise self-similarity initial profile has such a structure for all t ? 0, i.e., and give the explicit formula for this kind of solutions under several situations. First we consider a convex flux with positive wave speed, where f is locally Lipschitz continuous. The convexity of the flux f 00 (u) 0 is to get the explicit formula 'g' of the self-similarity profile such that f 0 and the self-similarity profile (1.7) can be written as Note that the equality is included for the second derivative of the flux in (H) and, hence, the monotonicity of f 0 is not strict and g is not exactly the inverse function of f 0 and g(f 0 (u)) 6= u in the case. In this approach we may include a piecewise linear flux of the front tracking method, see Remark 6.4. In section 3 we consider a piecewise self-similar solution which can be written as a self-similarity summation (or simply S-summation), of finite number of base functions. We give definitions for the S-summation and base functions in the section and show that u(x; (x) is the solution of (1.1) with initial data u Theorem 3.6. We consider u as an approximation of the solution v with the original initial data v 0 . Then the L 1 contraction theory of conservation laws implies It is the estimate corresponding to the error estimate (1.5), which does not have the time dependent term anymore. It is natural to expect that the error increases in time if the flux is changed. In our approach we use the original flux and the error decreases in time. The convergence of the scheme is now clear (see Theorem 3.6, Corollary 3.7). Note that the self-similarity summation (1.9) represents only special kind of piecewise self-similar profiles (1.6), which have positive indexes t k ? 0 and are ordered appropriately, i.e., c if an b n ::: a 2 b 2 ::: a 1 b 1 . 4 Y.-J. Kim The self-similarity summation is coded for a numerical scheme successfully in Section 4. This scheme has several unique properties. First it does not require a time marching procedure. So the complexity of the scheme is of order O(N ), not O(N 2 ). Second it captures the shock place very well even if small number of base functions (or mesh points) are used, Figure 4.3. In Figure 4.5 it is clearly observed that the solution with finer mesh always passes through bigger artificial shocks and this property provides a uniform a posteriori error estimate of the numerical approximation. Since it does not introduce numerical viscosity at all, we may get very good resolution for an inviscid problem. Our scheme also distinguishes physical shocks and artificial ones clearly. Table 4.1 shows the time when the physical shock appears. In Section 5 we generalize the method. For a general convex flux case, the method is applied through the transformations (5.1) and (5.3). If the flux has inflection points, then the scheme becomes considerably complicate and it is beyond the purpose of this article. But, if the flux has only one inflection point, for example, then we can easily apply the scheme through a similar transformation (5.4). Dafermos considers a flux with a single inflection point through generalized characteristics. The flux of the Buckley-Leverett equation satisfies this condition. The flux which appears in thin film flows (see Bertozzi, Munch and Shearer [1]), also belongs to this category. Figure 5.3 shows the strength of our scheme over the upwind scheme in this case. The scheme is not good enough for a short time behavior t !! 1 since the initial error controlled efficiently. To resolve the situation we add an extra structure to base functions in Section 6. Using these base functions we can approximate the initial data with second order accuracy and still solve the exact solution for the modified initial datum without the time discretization. Furthermore, a general piecewise self-similarity profile (1.6) can be written in terms of self-similarity summation of these modified base functions. 2. Self-similarity of conservation laws. Consider one dimensional scalar conservation laws, where the flux f is locally Lipschitz continuous. For a nonlinear flux f(u) the solution may have a singularity and hence the solution is considered in the weak sense with the entropy admissibility condition : ~ for any number ~ u lying between a conservation law is from the fact that a rescaled function, Piecewise Self-similar Solutions 5 is also the solution of (2.1) if and only if the initial profile u 0 (x) satisfies It is clear that the Riemann initial condition, ae satisfies (2.4) and, hence, u(x; is a function of the self-similarity variable, The structure of a Riemann solution is given in Figure 2.1 together with characteristic lines. Note that a self-similarity line R is not a characteristic line and the solution is constant along it. This is a special property of Riemann problem and it is not expected in a general situation. If the solution is constant along a line, it is natural to assume that the line is a characteristic line and it is the starting point of our scheme. x st x (a) Characteristic lines (b) Self-similarity lines Fig. 2.1. Let f 0 (u+ lines are different from characteristic lines. Even though the solution is constant along self-similarity lines. If the total mass of the initial data u 0 (x) is finite, Z then the relation (2.4) cannot be satisfied since the transformation u does not preserve the total mass. So the solution cannot be a function of self-similarity variable x=t. In the following we consider techniques to achieve the Riemann solution like self-similarity for general Cauchy problems. In the Godunov method the space is discretized into small intervals and the initial function u 0 is approximated by a step function which takes the cell average over those intervals. Then the problem can be considered as a sequence of Riemann problems and the structure of a Riemann solution holds locally in time and space. The scheme finds the cell average of the solution after a small amount of time using the self-similarity structure of Riemann solutions. It is fair to say that this method is more 6 Y.-J. Kim focused on the structure of the Riemann initial data which makes the self-similarity of the problem rather than the self-similarity itself. As a result the method takes cell-averages every time step and loses the accessibility to the exact solution. In the front tracking method the nonlinear flux f(u) is approximated by a continuous function hm (u) which is linear between points f k example, and then the initial datum is approximated by a piecewise constant function by taking these values, not cell averages. Then every discontinuity propagates as an admissible shock of the modified problem, in the sense of entropy condition (2.2) until it may possibly collide to other shocks. We may say that the self-similarity of the original problem (2.1) has been modified to get it fit to the piecewise constant functions. In this approach the exact solution of the modified problem is accessible and, hence, the method can be employed as an analytical tool as well as a computational one. Now we suggest a new approach which keeps the self-similarity globally in time. Suppose that characteristic lines of the solution u(x; t) pass through the origin. Then we have Since the right hand side diverges as t ! 0, we consider the initial datum as the profile at a given time simplest case of L 1 initial datum of the kind is Characteristic lines of this initial profile are given in Figure 2.2. Non-vertical characteristics pass through the point (0; \Gammat 0 ) and there is a region in which characteristic lines overlap with each other. The solution is given by finding the shock characteristic correctly. In this case the shock characteristic is not a straight line and the solution is not a function of x=(t+ t 0 ). Even though the solution is a function of in the region Since the shock speed s 0 (t) satisfies the Rankine-Hugoniot jump condition, the shock place s(t) can be decided by its integral form. On the other hand, if the convexity of the flux f is assumed, we may consider the self-similarity profile g such that f 0 x. In the case on (0; s(t)) and we can find the shock place s(t) easily from the equal area rule, (2. Piecewise Self-similar Solutions 7 x Fig. 2.2. Characteristic lines of a self-similarity solution are similar to self-similarity lines. The main difference is that the shock characteristic is not a straight line anymore. Since the conservation law (2.1) does not depend on the x variable explicitly, we may translate the initial data (2.9) in the x direction. We can also consider initial data which consist of finite number of these structures. A simple case is where centers c k and shock places s k satisfy The time index t k in (2.12) decides the slope of the initial profile and they can be chosen differently. Condition (2.13) implies that all profiles in (2.12) are separated. If not, the simple summation in (2.12) breaks down the self-similarity structure we want to keep. In Section 3 we consider a self-similarity summation which preserves it. Figure 2.3 shows characteristic lines for initial data (2.12) with 4. In this case tracking down a shock is more complicate and (2.11) is not enough for the purpose since waves interact to each other. x Fig. 2.3. Shock characteristics of (2.12) interact together and make a bigger shock. 3. Piecewise self-similar solutions. In this section we give the definition of the self-similarity summation and show that the exact solution of (2.1) is given as an S-summation. Notations in this section are directly converted into a numerical scheme in Section 4. In this section we consider a flux under the hypothesis, 8 Y.-J. Kim In this case the self-similarity profile 'g' is the profile which satisfies f 0 As it is mentioned earlier g is not exactly the inverse function of f 0 since the monotonicity of f 0 is not strict. We also assume f 0 this section for our convenience, and it implies that the solution is actually assumed to be positive under (H). The result of this section are generalized in Section 5. 3.1. Base functions. As it is mentioned earlier, the self-similarity profile represents the asymptotic behavior of the conservation law (1.1). The function, ae g serves as a base function in this article. A base function has the self-similarity profile over the interval between the center 'c' and the shock place `s'. The area (or the mass) enclosed by the x-axis and the base function is given by c Z s\Gammacg(x=t)dx =: m(t; c; s): It is convenient to consider the mass m as the fourth index of the base function, say m;t;c;s (x), or any three of them as an index set. In any case we consider it under the assumption that indexes m; t; c; s satisfy the relation (3.3). So if any three of them are given, the fourth one is decided by the relation. Consider a Cauchy problem, It is clear from (2.10) that the solution u(\Delta; t) has the self-similarity profile with time index t+t 0 between the original center c 0 and a new shock place s(t). Since the initial total mass m 0 should be preserved, the solution of (3.4) is where the shock place decided by the relation (3.3). Remark 3.1. If we take a ffi-function as the initial datum, for example the solution is given by u(x; t 0 (x). So the slope of the base function represents the time of the evolution starting from the ffi-function like initial data, and that is why we take index t for the base function. Remark 3.2. For the Burgers case, the self-similarity profile is given as the identity function, x. In the case (3.3) gives following relations, Remark 3.3. The rescaling (2.3) does not preserve the total mass. So it can not measure the invariance property for L 1 solutions of conservation laws. For the Burgers equation we consider Piecewise Self-similar Solutions 9 where the rescaling preserves the total mass. We can easily check that variables are invariant under the rescaling after the translation t. These variables are called self-similarity variables for L 1 Cauchy problems, and the Burgers equation is transformed to We can easily check that Bm0 ;t 0 =1;c0=0 (i) is an admissible steady state of the equation and hence w(i; is the solution of (3.9). If we transform the variables back to u; t; x, then we get u(x; This is another way to show (3.5). In this example we can see that the approach with piecewise self-similar solutions captures the self-similarity of the general Cauchy problems exactly. For a detailed study for the transformed problem (3.9) we refer [14]. 3.2. Self-similarity Summation. Since the solution of (3.4) is given by (3.5), we can easily guess that is the solution of the conservation law with initial data if all the supports of the base functions in (3.10) are disjoint. But it is not usually the case since the support of a base function expands in time. The self-similarity summation(or simply S-summation), is to handle the case that supports of base functions overlap with each other. The definition is given inductively in the following. (x). Suppose that (x) is well defined and supp(B c . Suppose there exists a point j ? c j such that Under the assumption of (3.13), the left hand side of (3.14) is monotone in j and, hence, such a point is unique. If there is no such a point we say that the S-summation (3.12) is not defined. If there exists such a point ae g Base functions are ordered by centers c k and then the S-summation is given from the right hand side. It is because of the positiveness assumption for the wave speed, (H). If the order of the summation is changed, the result is different. So the S-summation is not associative. Remark 3.4. If the time indexes are identical, t then we can show the S-summation (3.12) is well defined. If then, since the self-similarity profile g is an increasing function, we have g has values of g (3.15), the inequality (3.13) is satisfied for all j 2 R. Furthermore the left hand side of (3.14) has value for and diverges to 1 as j !1. So there exists a point j satisfying (3.14) and the S-summation is well defined. Remark 3.5. We may consider j as the j-th shock generated by the base function Bm j . Suppose that j i.e., the j-th shock caught up the (j-1)-th shock. The definition (3.15) implies that the self-similarity profile g disappears. We can easily check that we will get the same S-summation (3.15) if we remove the (j-1)-th base function and modify m j by adding . This property represents the irreversibility of the conservation laws. Theorem 3.6. Suppose that the flux f(u) satisfies Hypothesis (H). If the self-similarity summation (x) is well defined, then u(x; +t;ck (x) is also well defined and it is the solution of (1.1) with initial data u 0 . If v(x; t) is the entropy solution of (1.1) with initial data Proof. The proof is completed through inductive arguments. Suppose that +t;ck (x) is the solution with the initial condition (x). It is assumed that u j (x; (x) is well defined and we let u j (x; t) be the solution of (1.1) with this initial data. Let the shock characteristic given by the j-th base function, i.e., j for the j in (3.13,3.14). If x ? j (t), then it is clear that u j (x; since characteristics on the right hand side of do not interact with it because f 0 (u) 0. since the vertical characteristics starting in the region x do not touch shock characteristics moving to the right hand side. The characteristic passing through a point (x; is a straight line connecting and, hence, u . Since the total mass is preserved, the shock place should satisfy +t;ck (x) from the definition of the S-summation and the first part of the proof is completed. The second part (3.16) is simply the L 1 contraction theory for conservation laws. In the proof we employ the theory of characteristics (see [8], ch. 11). The error estimate (3.16) implies that the initial error decreases in time, and the solution with modified initial data is obtained exactly in a single step for any given time t ? 0. The scheme has ideal properties for the study of asymptotic behavior. Now we consider u (x) as an approximation of L 1 initial data v 0 . Let a partition be the set of centers. Its norm is defined Piecewise Self-similar Solutions 11 by j. There can be many ways to discretize the initial data. To guarantee the convergence of the scheme, we need the existence of ffi; L ? 0 such that where a constant " ? 0 is given. An example of such a discretization is given in Section 4.1. The convergence of the scheme satisfying (3.17) is clear from (3.16). Corollary 3.7. (Convergence) The scheme of the self-similarity summation (x) with initial discretization u satisfying (3.17) converges to the entropy solution v(x; t) with initial data as Remark 3.8. Now we consider the S-summation between two base functions, Figure 3.1. It gives a good example to figure out the meaning of the S-summation. Furthermore, in the numerical computation, we can possibly compare only two base functions each time and, hence, it is worth to consider it in detail. If these two base functions are separated, s then the shock place of the definition (3.15) is simply Z implies that two base functions are merged, i.e.,K Suppose that s 1 is far away and the shock place is guaranteed to be between s 2 and s 1 . Then (3.18) can be written as Z c1 Z The solution of (3.19) has a special meaning in the coding. We define as an operator between two base functions, Bm2 := . Note that in the definition of the operator we do not use the information m 1 at all. We just assume it is big enough and, hence, ! s 1 . This operator is used in Section 4 to check if two adjacent base functions are merged or not. For the Burgers case, implies that, in Figure 3.1, Trapezoid has the same area as Triangle Ac 1 , and the relation (3.19) can be written as an algebraic relation, The operator ' ' between two base functions is now given explicitly, c 2+s 2\Gamma2c 2 s2 which is the solution of the algebraic relation (3.20) with We consider this operator in Section 4.1 again. A ff Fig. 3.1. The equal area rule gives the shock place when two base functions interact together. 4. Coding Strategy. In this section we show how the self-similarity summation can be implemented into a numerical scheme. To see what is really happening in each step it is helpful to consider a specific example. For that purpose we consider the Burgers equation, The result of the scheme is compared with the Godunov scheme. 4.1. Implementation. Here we introduce a grid-less scheme based on the self-similarity (a) The Equal Area Rule (b) Base Functions with overlaps Fig. 4.1. Initial data are approximated by a piecewise self-similarity profile. It turns out to be an S-summation of base functions. Step 1. (Initial discretization) The first step is to design a method to approximate the initial datum v 0 (x) by a self-similarity summation u 0 (x) which satisfies (3.17). Consider n base functions B[k]; Each element B[k] consists of two members B[k]:m; B[k]:c, which represent the area and the center of the base function. We use identical time index t hence, we do not need a member for the time index. Piecewise Self-similar Solutions 13 0 be a cell average approximation of v 0 with steps of mesh size profiles with time index t 0 ? 0 which pass through the left end points of the constant parts of the step function v " Figure 4.1 (a). Let B[k]:c be the x-intercept of the k-th self-similarity profile from the right hand side and B[k]:m be the area enclosed by x-axis, , and the k-th and the profiles. This discretization is well defined only if B[n]:c ! ::: ! B[1]:c. To achieve it a small initial time index t 0 should be chosen depending on the initial data. Since the initial self-similarity profile of the example (4.1) is a line with the slope 1=t 0 and the slope of the initial data is bounded by v x (x; we have to take t 0 ! 1. In Figure 4.1 the initial data in (4.1) has been discretized using 10 base functions, base functions (b) have some overlaps and the self-similarity summation (a) has a saw-tooth profile. The size of the triangle like areas added and subtracted by the approximation is proportional to " 2 and the total number them is proportional to 1=". So we have jjv Theorem 3.6 says u(x; is the solution with the modified initial data u 0 . So the rest of the scheme is focused on how to display the given solution. Even if it is possible to follow the inductive arguments of the definition, we will get serious complexity in the coding if behind shocks capture the front ones, i.e., . In the case the S-summation is not changed even if two base functions are merged, Remark 3.5, and hence we do the merging process first. From now on the corresponding time index is t for each k. Step 2. (Merging) The operator ' ' between two base functions defined by (3.19) for the general case or by (3.21) for the Burgers case plays the key role here. Suppose that there is no contact between shocks for k Then we can easily check that the k-th shock in (3.15) is given by 1. Suppose that In the case j 6= B[j] B[j \Gamma 1] in general. Even though it implies j ? j \Gamma1 and the self-similarity profile of the (j-1)-th base function B[j \Gamma 1] disappears, Remark 3.5. In the case these two base functions B[j] and B[j \Gamma 1] can be combined, i.e., put remove then rearrange the array B[\Delta] from is the number of base functions left after the previous step. Since the combined base function may take over another one again, we decrease the index j if j 6= 2. If (4.2) does not hold, we increase index j. We continue this procedure from Note that there is no base function B[0] and we use B[1] B[0] := B[1]:s in (4.2) for given by the relation (3.3).(Step 2 is complete.) If there are no base functions merged together, there will be of (4.2). If m base functions are merged, then base functions are left and the maximum number of the comparison (4.2) is n In Figure 4.2 (b) base functions at time are displayed after the merging process. There were 50 base functions initially (a) and 38 of them are left after the merging step. It means that small base functions has been merged together and made a big base function. The big base function can be considered as an accumulation of small artificial shocks in some sense and it represents the physical shock. 14 Y.-J. Kim0.050.150.25 (a) 50 Initial Base functions (b) 38 Base functions left at Fig. 4.2. The initial base function with slope 1=t 0 has slope 1=(t0 + t) at time t without area change. After the merging process, Step 2, some of the base functions are merged together and make a big base function which represents a physical shock. Now we are ready to display the solution. Suppose that base functions B[j]; are left after the merging step. Let Then the right and the left hand side limits are given by, So to display the solution it is enough to plot the points Between these point the solution has the self-similarity profile. So if we connect these points with self-similarity profile with time index B[j]:c, we get the solution. In Figure 4.3 solutions are displayed using different number of base functions. We can clearly see that the solution converges as the number of base functions is increased.0.050.150.250.350.45 (a) Initial Discretization (b) Solutions at Fig. 4.3. Three S-summations using 10,40 and 160 base functions. The solution finds the shock correctly even if a very rough initial discretization is given. A solution with finer mesh passes through artificial shocks. 4.2. Comparison with Godunov. A typical way to discretize the initial data is taking the cell average, Figure 4.4 (a). The Godunov scheme solves Riemann problems between each cells for a short amount of time \Deltat and then repeat the process until it Piecewise Self-similar Solutions 15 reaches a given time t ? 0. In Figure 4.4 (b) we can see that the numerical solution converges to the same limit as the S-summation, Figure 4.3 (b), as \Deltax ! 0.0.050.150.250.350 (a) Data Discretization (b) Solutions at Fig. 4.4. Three approximations by Godunov using 1=160. The scheme is convergent to the same limit of the S-summation. We can observe that numerical solutions are separated near the shock and it is hard to guess where the limit is from a single computation. Remark 4.1. (Computation time) Let N be the number of mesh points. Then the number of operations for the S-summation is of order N since the time marching process is not required, Theorem 3.6. The number of operations is almost independent from the final time t ? 0. On the other hand the Godunov scheme has operations of and the situation becomes worse if the final time t is increased.0.220.260.30.34 Fig. 4.5. A magnification of Figure 4.3 (b) near the physical shock shows that self-similarity solutions with finer mesh passes through the middle of artificial shocks. Remark 4.2. (Error estimate) We can clearly see that the exact solution v of limit of the S-summation) always passes though the artificial shocks of self-similarity solutions, Figure 4.3 (b). This property makes it possible to get a uniform a posteriori error estimate. Figure 4.5 is a magnification of Figure 4.3 (b). There are couple of other things we can observe here. First the sizes of artificial shocks decrease in time with order of O(1=(t We can also observe that, even if we use small number of base functions, we can get the physical shock very closely. Remark 4.3. (Shock Appearance Time) In a numerical scheme the solution is approximated by piecewise continuous functions and hence it is hard to see if a Table Shock Appearance time. The exact solution with initial data (4.1) blows up at 1. The time of shock appearance can be measured by counting the base functions after the merging step. initial number of base functions the time when the number is decreased by 2 200 T=1.0015 discontinuity represents the physical shock or not. In our scheme, as we can see from Figure 4.2, the accumulation of base functions represents the physical shock. So if a base function is merged by its behind one in the sense (4.2), we may conclude that a physical shock has appeared. The physical shock appears at time in the example since min(@ x v 0 \Gamma1. We can easily check that if (4.2) happens around the time. Table 4.1 shows the time when the number of initial base functions decreases. 5. General cases. The self-similarity summation has been considered under Hypothesis (H). In this section we generalize it under Hypothesis (H1) and (H2). 5.1. General convex flux. We consider L 1 initial function u 0 which is uniformly bounded, say \GammaA u 0 (x) B. Then the solution of (2.1) is always bounded, Consider a convex flux, If the flux satisfies f 00 (u) 0, we may change the variable get an equation f satisfies (H1). Note that we include the equality in (H1) and a piecewise linear flux can be considered. We can easily check that a new flux, satisfies the hypothesis (H) and h 0 be the solution of We can easily check that is the solution with the original flux f and initial data u 0 . Since u \GammaA, the solution v(x; t) is positive. Now we are in the exactly same situation as in the previous sections except the structure of the initial data. The initial data v(\Delta; 0) is not L 1 anymore. To handle the situation we consider two special base functions with infinite mass, Piecewise Self-similar Solutions 17 These base functions handles the transformation u A. Note that the speed of the shock connecting the state our case. The Self-similarity summation including these two base functions can be defined in a similar way. We omit the detail. Figure 5.1 shows how the self-similar solution evolves for the Burgers case. In the figure even the solution with very rough initial discretization with only 16 base functions represents the asymptotic behavior very correctly. -0.4 (a) Data Discretization (b) Solutions at Fig. 5.1. Three S-summations are displayed using 16,64 and 256 base functions. It handles sign changing solutions correctly. This figure shows the time convergence to an inviscid N-wave. 5.2. Flux without convexity. Consider a flux with a single inflection point, Then, under the change of variables, the problem (2.1) is transformed to Then the new flux h satisfies and A is not the lower bound of the solution u(x; t) in general, we can not expect v 0. So in this case we have to consider positive part and negative part together. It is possible since h 0 (u) is monotone on (\Gamma1; 0) and (0; 1) respectively. All we have to do is to consider negative base functions together with the positive ones. Since the wave speed h 0 (u) is positive, the self-similarity summation is defined from the right hand side as in the previous cases. Example 5.1. Consider an inviscid thin film flow in [1], where the initial datum is compactly supported supp(u 0 has a single inflection point under the transformation (5.4), we get the flux It satisfies which is not exactly same as (5.5) but has the opposite direction in the inequalities. We do the self-similarity summation from the left hand side instead of changing the space variable using \Gammax. Now the original problem (5.6) is transformed into In this case the self-similarity profile (2.8) is given by, and the corresponding base functions are, ae The initial data v 0 (x) converges to \GammaA as x ! \Sigma1 and we need to consider two base functions with infinite mass, Note again that, in our example (5.6), the infinite state is \Gamma1=3 and the shock speed is Numerical solutions of (5.6) with initial data, are in Figure 5.2. The first picture shows the initial data and the self-similarity summation using 200 base functions. A part of it has been magnified with numerical approximations of upwind scheme in the second picture. We can clearly see that the solution of upwind scheme converges to the self-similarity summation. This example shows that the self-similarity summation gives a very accurate resolution using small number of mesh points. Furthermore, since it gives the solution without time marching procedure, computational time is a lot smaller. 5.3. Flux with the space dependence. Since the self-similarity of the problem (2.1) depends on the fact that the flux depends on the solution only we have no clue how to generalize our scheme to a problem with a general space dependent though, if the space dependence is given by the equation is transformed to under the change of variable R x1=a(s)ds and our scheme can be applied. Piecewise Self-similar Solutions 190.10.30.50.7 (a) initial data and S-summation at t=6 (b) comparison with upwind Fig. 5.2. Flux is Picture (a) shows the initial data and the self-similarity summation at shows that upwind converges to the self-similarity summation. 200 base functions are used in the S-summation and 800 and 4,000 meshes are used in upwind scheme. Since the self-similarity of hyperbolic conservation laws is the one-dimensional property, it should be possible to expand the scheme to multi-dimension problems. Consider a 2-dimensional problem, with a velocity vector field satisfying Cvetkovic and Dagans [5] suggest space variables y 1 dy 1 dj which transform (5.14) to Problem (5.16) can be considered as a set of one-dimensional problems and, hence, the complexity of the scheme for it is of order O(N 2 ). Since the transformation (5.15) also has the complexity of O(N 2 ), we eventually get a scheme of O(N 2 ) for a two-dimensional problem. In this approach each channel of the velocity vector field is considered separately and, hence, it seems useful to channel problems. 6. Second order approximation. The scheme introduced in the previous sections solves the problem exactly with modified initial data, and the size of the initial error decreases in time. Even though the scheme is not good enough for the short time behavior since the error generated by the initial discretization can be huge. Here we add an extra structure to base functions and make the initial data discretization to be second order. In this way we can handle general self-similarity solutions (1.7). 6.1. Modified base functions. The base function considered in the previous sections has three indexes, say m; t; c. In this section we introduce two more indexes, h and t. Note that there are two time indexes t and which play different roles. We assume 1. For the simplicity we consider under the hypothesis (H). It can be easily generalized as we did in Section 5. To figure out the structure of the new base function B h; t m;t;c (x), we consider and Let g be the self-similarity profile, f 0 As an intermediate step we define t;c (x) first. For defined by and, for it is defined by The constant c is the center of the top self-similarity profile with time index t and the constant x is the x-coordinate of the intersection point between two self-similarity profiles with index t an t. We can easily see from (6.2) that c ! x for t ? 0 and t;c (x) is well defined for since the corresponding domain is empty. For Now we introduce the index m ? 0 which decides the support of the base function. c be the solution of, Z c For it always has a solution. For t 0 it has a solution only R c t;c (x)dx. The base function is now defined by t;c The self-similarity summation among these base functions can be similarly defined using the profile g \Delta in the domain c ! x ! x and the profile g for x ! x. We omit the detail. We may consider the base function (3.2) as a special case of (6.7) with 6.2. Initial discretization and the exact solution. Suppose the initial function be a partition of the interval [A; B]. We can approximate v 0 with self-similarity profiles over interval which is second order. For the Burgers case it is simply a piecewise linear approximation. The approximation u 0 can be written as Piecewise Self-similar Solutions 21 R Initially the supports of base functions are disjoint and, hence, the self-similarity summation is the usual summation. The exact solution of the conservation law with initial data (6.8) is We still consider the exact solution and the contraction theory implies Remark 6.1. The initial discretization (6.8) is trivial in comparison with Step 1 in Section 4.1. It is an additional advantage when the modified base function is used in a numerical scheme. Even though this additional structure may cause extra complexity when it is used as an analytical tool. Remark 6.2. (Piecewise Constant Data) In many cases initial data are given as piecewise constant functions from the beginning. In the case an initial datum can be considered as a summation of base functions with In Figure 6.1 we consider the Burgers case (4.1) using base functions B h;1 m;t;c (x). We can clearly see that these approximations represent the shock place very well. Unlike the previous case, the solution with finer mesh always passes though the constant parts of coarse (a) Data Discretization (b) Solutions at Fig. 6.1. The S-summation for the modified base functions (6.7) with constant, piecewise self-similar solution. In the figure 3 summations are displayed together using base functions. We can observe that finer one always passes the constant parts. Remark 6.3. (Singular Initial Data) If singular initial data are given, then extra mesh points are usually introduced to capture the effect of the singularity of the data. But, since our method handles initial data individually, extra mesh points are not needed. In Figure 6.2 the Burgers equation is solved with singular initial data (a). We use 6 modified base functions with Remark 6.4. (Front Tracking) It is possible to consider the front tracking method in terms of the self-similarity summation. Consider an L 1 solution of the Burgers equation bounded by 0 u(x; t) 1. Let h(u) be the polygonal approximation of the flux with the partition f0; 1=n; :::; 1g. So h 0 (u) is a step function, 22 Y.-J. Kim0.050.150.250.350 (a) Singular initial data (b) Solutions at Fig. 6.2. The scheme does not require extra meshes to handle singular initial data (a). In the S-summation every datum is handled exactly by a base function. Only 6 base functions solves this example. and the self-similarity profile g(x) is also a step function, So the values of g(x) are the breaking points of the flux h(u). We can approximate the given initial data v 0 by taking a cell average, not just breaking points. Then the initial discretization u 0 can be written in a from of (6.8) with 1. This is a simplified version of the front tracking method under Hypothesis (H). 7. Conclusion. The basic idea of the method introduced in this article is to approximate the solution of a conservation law by a self-similarity summation of base functions. In that approach we get the exact solution in the class of functions. This method can be easily converted into a numerical scheme and the complexity of the scheme is of order N , not N 2 since no time marching procedure is needed. Convergence of the scheme is now a trivial matter, Theorem 3.6 and Corollary 3.7. The method can be used as an analytical tool. In fact the author is preparing an article on asymptotic behavior of scalar conservation laws through this method. Various issues appear when we apply this idea to other cases, systems or convection-diffusion equations. The author does not have a good understanding for these cases yet. Acknowledgement The author would like to thank Professor A. E. Tzavaras. He gave the author the motivation and valuable remarks for this work. The author also would like to thank people in IMA for all the discussions and supports. --R On the partial difference equations of mathematical physics Polygonal approximations of solutions of the initial value problem for a conservation law Regularity and large time behaviour of solutions of a conservation law without convexity Hyperbolic conservation laws in continuum physics Solutions in the large for nonlinear hyperbolic systems of equations A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics On scalar conservation laws in one dimension. An unconditionally stabl method for the Euler equations A method of fractional steps for scalar conservation laws without the CFL condition Diffusive N-waves and Metastability in Burgers equation Large time step shock-capturing techniques for scalar conservation laws A moving mesh numerical method for hyperbolic conservation laws --TR

characteristics;gridless scheme;front tracking;self-similarity

604557

Asymptotic Size Ramsey Results for Bipartite Graphs.

We show that $\lim_{n\to\infty}\hat r(F_{1,n},\dots,F_{q,n},F_{q+1},\dots,F_{r})/n$ exists, where the bipartite graphs $F_{q+1},\dots,F_r$ do not depend on $n$ while, for $1\le i\le q$, $F_{i,n}$ is obtained from some bipartite graph $F_i$ with parts $V_1\cup V_2=V(F_i)$ by duplicating each vertex $v\in V_2$ $(c_v+o(1))n$ times for some real $c_v>0$.In fact, the limit is the minimum of a certain mixed integer program. Using the Farkas lemma we show how to compute it when each forbidden graph is a complete bipartite graph, in particular answering the question of Erdos, Faudree, Rousseau, and Schelp [Period.\ Math.\ Hungar., 9 (1978), pp. 145--161], who asked for the asymptotics of $\hat r(K_{s,n},K_{s,n})$ for fixed $s$ and large $n$. Also, we prove (for all sufficiently large $n$) the conjecture of Faudree, Rousseau, and Sheehan in [Graph Theory and Combinatorics, B. Bollobas, ed., Cambridge University Press, Cambridge, UK, 1984, pp. 273--281] that $\hat r(K_{2,n},K_{2,n}) =18n-15$.

Introduction . Let be an r-tuple of graphs which are called for- bidden. We say that a graph G arrows any r-colouring of E(G), the edge set of G, there is a copy of F i of colour i for some i 2 [r] := rg. We denote this arrowing property by G ! The (ordinary) Ramsey number asks for the minimum order of such G. Here, however, we deal exclusively with the size Ramsey number which is the smallest number of edges that an arrowing graph can have. Size Ramsey numbers seem hard to compute, even for simple forbidden graphs. For example, the old conjecture of Erd}os [6] that ^ r(K 1;n ; K 3 recently been disproved in [16], where it is shown that ^ any xed 3-chromatic graph F . (Here, K m;n is the complete bipartite graph with parts of sizes m and n; Kn is the complete graph of order n.) This research initiated as an attempt to nd the asymptotics of ^ r(K 1;n xed graph F . The case treated in [17] (and [16] deals with What can be said if F is a bipartite graph? Faudree, Rousseau and Sheehan [11] proved that for every m 9 if n is su-ciently large (depending on m) and stated that their method shows that ^ r(K 1;n ; K 2;2 They also observed that K s;2n arrows the is a cycle of order 2s; hence ^ r(K 1;n ; C 2s ) 2sn. Let P s be the path with s vertices. Lortz and Mengersen [14] showed that conjectured that this is sharp for any s 4 provided n is su-ciently large, that is, Supported by a Research Fellowship, St. John's College, Cambridge. Part of this research was carried out during the author's stay at the Humboldt University, Berlin, sponsored by the German Academic Exchange Service (DAAD). y DPMMS, Centre for Mathematical Sciences, Cambridge University, Cambridge CB3 0WB, Eng- land, O.Pikhurko@dpmms.cam.ac.uk The conjecture was proved for 4 s 7 in [14]. Size Ramsey numbers ^ (and in some papers F 1 is a small star) are also studied in [9, 5, 2, 3, 8, 10, 13, 12] for example. It is not hard to see that, for xed s This follows, for example, by assuming that s considering K v1 s e. The latter graph has the required arrowing property. Indeed, for any r-colouring, each vertex of V 2 is incident to at least s edges of same colour; hence there are at least v 2 monochromatic K s;1 -subgraphs and some S 2 V1 s appears in at least rtn such subgraphs of which at least tn have same colour. Here we will show that the limit lim exists if each forbidden graph is either a xed bipartite graph or a subgraph of K s;btnc which 'dilates' uniformly with n (the precise denition will be given in Section 2). In particular, tends to a limit for any xed bipartite graph F . The limit value can in fact be obtained as the minimum of a certain mixed integer program (which does depend on n). We have been able to solve the MIP when each F i;n is a complete bipartite graph. In particular, this answers a question by Faudree, Rousseau and Schelp [9, Problem B] who asked for the asymptotics of Working harder on the case we prove (for all su-ciently large n) the conjecture of Faudree, Rousseau and Sheehan [11, Conjecture 15] that where the upper bound is obtained by considering K 3;6n fortunately, the range on n from (1.3) is not specied in [11], although it is stated there that ^ r(K 2;2 ; K 2;2 where the upper bound follows apparently from K 6 Unfortunately, our MIP is not well suited for practical calculations and we were not able to compute the asymptotics for any other non-trivial forbidden graphs; in particular, we had no progress on (1.1). But we hope that the introduced method will produce more results: although the MIP is hard to solve, it may well be possible that, for example, some manageable relaxation of it gives good lower or upper bounds. Our method does not work if we allow both vertex classes of forbidden graphs to grow with n. In these settings, in fact, we do not know the asymptotics even in the simplest cases. For example, the best known bounds on seem to be r < 3 (Erd}os and Rousseau [10]). Our theorem on the existence of the limit can be extended to generalized size Ramsey problems; this is discussed in Section 5. 2. Some Denitions. We decided to gather most of the denitions in this section for quick reference. We assume that bipartite graphs come equipped with a xed bipartition V embeddings need not preserve it. We denote v i For A V 1 ASYMPTOTIC SIZE RAMSEY RESULTS 3 where F (v) denotes the neighbourhood of v in F . (We will write (v), etc., when the encompassing graph F is clear from the context.) Clearly, in order to determine F (up to an isomorphism) it is enough to have This motivates the following denitions. A weight f on a set V (f) a sequence (f A ) A22 V (f) of non-negative reals. A bipartite graph F agrees with f if V 1 A sequence of bipartite graphs (Fn ) n2N is a dilatation of f (or dilates f ) if each Fn agrees with f and jF A (Of course, the latter condition is automatically true for all A Clearly, e(Fn so we call e(f) the size of f . Also, the order of f is and the degree of x 2 V (f) is A3x Clearly, For example, given t 2 R>0 , the sequence (K s;dtne ) n2N is the dilatation of k s;t , where the symbol k s;t will be reserved for the weight on [s] which has value t on [s] and zero otherwise. (We assume that V 1 (K s;dtne It is not hard to see that any sequence of bipartite graphs described in the abstract is in fact a dilatation of some weight. We write F f if for some bipartition V there is an injection such that for any A V 1 dominated by a vertex of V 2 is This notation is justied by the following trivial lemma. Lemma 2.1. Let (Fn ) n2N be a dilatation of f . If F f , then F is a subgraph of Fn for all su-ciently large n. Otherwise, which is denoted by F 6 f , no Fn contains F . Let f and g be weights. Assume that v(f) v(g) by adding new vertices to letting g be zero on all new sets. We write f g if there is an injection (g) such that This can be viewed as a fractional analogue of the subgraph relation F G: h embeds how much of F A V 2 mapped into G B . The fractional -relation enjoys many properties of the discrete one. For example, A3x fA A3x The following result is not di-cult and, in fact, we will implicitly prove a sharper version later (with concrete estimates of ), so we omit the proof. Lemma 2.2. Let (Fn ) n2N and (Gn ) n2N be dilatations of f and g respectively. implies that for any > 0 there is n 0 such that Fn Gm for any n n 0 )n. Otherwise, which is denoted by f 6 g, there is > 0 and n 0 such that Fn 6 Gm for any n n 0 and m (1 An r-colouring c of g is a sequence (c A1 ;:::;A r ) of non-negative reals indexed by r-tuples of disjoint subsets of V (g) such that c The i-th colour subweight c i is dened by V c The analogy: to dene an r-colouring of G, it is enough to dene, for all disjoint how many vertices of G A1 [[Ar are connected, for all i 2 [r], by colour i precisely to A i . Following this analogy, there should have been the equality sign in (2.2); however, the chosen denition will make our calculations less messy later. 3. Existence of Limit. Let r q 1. Consider a sequence Assume that F i does not have an isolated vertex (that is, x We say that a weight g arrows F (denoted by g ! F) if for any r-colouring c of g we have F i c i for some i 2 [r]. Dene The denition (3.1) imitates that of the size Ramsey number and we will show that these are very closely related indeed. However, we need a few more preliminaries. considering k a;b which arrows F if, for example, su-ciently large, cf. (1.2). Let l be an integer greater lg. Proof. Let > 0 be any real smaller than d 0 . Let g ! F be a weight with v(g) > l and To prove the theorem, it is enough to construct g 0 ! F with We have d(x) (^r(F) the weight g 0 on V (g) n fxg by We claim that g 0 arrows F. Suppose that this is not true and let c 0 be an F-free r-colouring of g 0 . We can assume that ASYMPTOTIC SIZE RAMSEY RESULTS 5 Dene c by c Anfxg d i where we denote A if g 0 A > 0, and The reader can check that c is an r-colouring of g. By the assumption on g, we have F i c i for some i 2 [r]. But this embedding cannot use x because for we have d c i A d i A d i is too small, see (2.1). But c i;A c 0 i , which is the desired contradiction. Hence, to compute ^ r(F) it is enough to consider F-arrowing weights on only. Lemma 3.2. There exists r(F). (And we call such a weight extremal.) Proof. Let gn ! F be a sequence with V (gn ) L such that e(g n ) approaches r(F). By choosing a subsequence, assume that V (g n ) is constant and exists for each A 2 2 L . Clearly, remains to show that g ! F. Let c be an r-colouring of g. Let - be the smallest slack in inequalities (2.2). Choose su-ciently large n so that jg n;A gA j < - for all A 2 2 L . We have that is, c is a colouring of gn as well. Hence, F i c i for some i, as required. Now we are ready to prove our general theorem. The proof essentially takes care of itself. We just exploit the parallels between the fractional and discrete universes, which, unfortunately, requires messing around with various constants. Theorem 3.3. Let be a dilatation of f i , i 2 [q]. Then, for all su-ciently large n, In particular, the limit lim exists. Proof. Let We will prove that By Lemma 3.2 choose an extremal weight g on L. Dene a bipartite graph G as follows. Choose disjoint from each other (and from L) sets G A with jG A 6 OLEG PIKHURKO In G we connect x 2 L to all of G A if x 2 A. These are all the edges. Clearly, A22 L A22 L as required. Hence, it is enough to show that G has the arrowing property. Consider any r-colouring c : E(G) ! [r]. For disjoint sets fy (jC 0; otherwise, that is, c is an r-colouring of g. Hence, F i c i for some i 2 [r]. Suppose that i 2 [q]. By the denition, we nd appropriate We aim at proving that F i;n G i , where G i G is the colour-i subgraph. Partition F A so that . This is possible for any A: if w for at least one B, then B22 L B22 L i;n j: and we have Hence, we can extend h to the whole of V mapping [ h(A)B W A;B injectively into B. Suppose that i 2 [q+1; r]. The relation F i c i means that there exist appropriate L. We view h as a partial embedding of F i into G i and will extend h to the whole of V Take consecutively y There is B i L such that c i;B i > 0 and h((y)) . The inequality c i;B i > 0 implies that there are disjoint B j 's, j 2 [r] n fig, such that c B1 ;:::;B r > 0. Each vertex in CB1 ;:::;B r is connected by colour i to the whole of h((y)). The inequality c B1 ;:::;B r > 0 means that jCB1 ;:::;B r we can always extend h to y. Hence, we nd an F i -subgraph of colour i in this case. Thus the constructed graph G has the desired arrowing property, which proves the upper bound. ASYMPTOTIC SIZE RAMSEY RESULTS 7 As the lower bound, we show that, for all su-ciently large n, Suppose on the contrary that we can nd an arrowing graph G contradicting (3.4). Let L V (G) be the set of vertices of degree at least d 0 n=2 in G. From d 0 njLj=4 < e(G) < ln it follows that jLj 4l=d 0 . For A 2 2 L , dene We have A22 L A22 L Thus there is an F-free r-colouring c of g. We are going to exhibit a contradictory r-colouring of E(G). For each choose any disjoint sets CB1 ;:::;B r G B (indexed by r-tuples of disjoint sets partitioning B) such that they partition G B and This is possible because For colour the edge fx; yg by colour j. All the remaining edges of G (namely, those lying inside L or inside V (G) n L) are coloured with colour 1. There is i 2 [r] such that G i G, the colour-i subgraph, contains a forbidden subgraph. Suppose that be an embedding. If n is large, then which implies that h(V 1 for All other wA;B 's are set to zero. For A B22 L wA;B (jF A For that is, h (when restricted to V (f i )) and w demonstrate that f i c i , which is a contradiction. Suppose that i 2 [q +1; r]. Let V 1 consists of those vertices which are mapped by This is a legitimate bipartition of F i because any colour-i edge of G connects L to V (G)nL. Let y together with c B1 ;:::;B r > 0 shows that F i g i . This contradiction proves the theorem. 4. Complete Bipartite Graphs. Here we will compute asymptotically the size Ramsey number if each forbidden graph is a complete bipartite graph. More precisely, we show that in order to do this it is enough to consider only complete bipartite graphs having the arrowing property. Theorem 4.1. Let r 2 and q 1. Suppose that we are given t there exist Proof. Let us rst describe an algorithm nding extremal s and t. Some by-product information gathered by our algorithm will be used in the proof of the ex- tremality of k s;t ! F. Choose l 2 N bigger than which is the same deni- tion of l as that before Lemma 3.1. We claim that l > , where 1). Indeed, take any extremal f ! F without isolated vertices. The proof of Lemma 3.1 implies that d(x) t 0 for any necessarily v(f) > , which implies the claim. For each integer s 2 [+1; l], let t 0 s > 0 be the inmum of t 2 R such that k s;t ! F. Also, let s be the set of all sequences a non-negative integers with a For a sequence a = (a and a set A of size a consist of all sequences r ) of sets partitioning A with jA We claim that t 0 s is sol(L s ), the extremal value of the following linear program a2s w a over all sequences (w a ) a2s of non-negative reals such that a2s w a a i 1 The weight k s;t does not arrow F for t < sol(L s ). To prove this, let an r-colouring c of k s;t by s a other c's are zero. It is indeed a colouring: a2s a a2s We have k s i For example, for , we have a2s a a2s w a s a2s w a s BS ASYMPTOTIC SIZE RAMSEY RESULTS 9 Also, K s some Suppose that the claim is not true and we can nd an F-free r-colouring c of k s;t . By the denition, c A1 ;:::;A c so we can pick x 2 A j and set c A1 ;:::;A increasing c :::;A j nfxg;:::;A i [fxg;::: by c. Clearly, c remains an F-free colouring. Thus, we can assume that all the c's are zero except those of the form c A , A 2 [s] a for some a 2 s . Now, retracing back our proof of Claim 1, we obtain a feasible solution w a a larger objective function, which is a contradiction. The claim is proved. Thus, t 0 is an upper bound on ^ r(F). Let us show that in fact We rewrite the denition of ^ r(F) so that we can apply the Farkas Lemma. The proof of the following easy claim is left to the reader. weights g on L such that there do not exist non-negative reals a with the following properties a A22 L a A c A t L Let g be any feasible solution to the above problem. By the Farkas Lemma there exist xA 0, A 2 2 L , and y i;S 0, , such that A a y i;S < A22 L We deduce that xA 0 (and hence considering (4.2) for some A with jA For each A with a := jAj > repeat the following. Let (w a ) a2a be an extremal solution to L a . For each a 2 a , take the average of (4.2) over all A 2 A a , multiply it by w a , and add all these equalities together to obtain the following. a2a a wAxA a2a w a a a y i;S y i;S a2a a w a a s i a y i;S a2a a w a a i a (In the last inequality we used (4.1).) Substituting the obtained inequalities on the xA 's into (4.3) we obtain y i;S < A22 L gA s As the y i;S 's are non-negative, some of these variables has a larger coe-cient on the right-hand side. Let it be y i;S . We have AS gA A22 L A jAj: The last inequality follows from the fact that for any integer a > , we have 1=t 0 a a=m u , which in turn follows from the denition of m u . Hence, e(g) > m u as required. Corollary 4.2. Let r q 1, t such that t i s i for be an integer sequence with Let l 2 N be larger than lim lim In other words, in order to compute the limit in Corollary 4.2 it is su-cient to consider only complete bipartite graphs arrowing Fn . It seems that there is no simple general formula, but the proof of Theorem 4.1 gives an algorithm for computing ^ r(F). The author has realized the algorithm as a C program calling the lp solve library. (The latter is a freely available linear programming software, currently maintained by Michel Berkelaar [4].) The reader is welcome to experiment with our program; its source can be found in [15]. For certain series of parameters we can get a more explicit expression. First, let us treat the case when only the rst forbidden graph dilates with n. We can assume that t scaling n. Theorem 4.3. Let 2. Then for any s s 1). Proof. The Problem L s has only one variable w s s 0 ;s 2 1;:::;s r 1 . Trivially, t 0 s and the theorem follows. In the case s we obtain the following formula (with a little bit of algebra). Corollary 4.4. For any s have r ASYMPTOTIC SIZE RAMSEY RESULTS 11 Another case with a simple formula for r(F) is without loss of generality we can assume that t Theorem 4.5. Let 2. Then for any s; s with s s s with a Proof. Let a 2 N> and let (w a ) a2a be an extremal solution to L a . (Where we obviously dene s Excluding the constant indices in w a , we assume that the index set a consists of pairs of integers (a 1 ; a 2 ) with a 1 +a Clearly, (w 0 is also an extremal solution, where w 0 (w Thus we can assume that w If w then we can set w increasing w ba 0 =2c;da 0 =2e and w da 0 =2e;ba 0 =2c by c. The easy inequality s s s a 0 b s a 0 b 1 implies inductively that the left-hand side of (4.1) strictly decreases while the objective function a2a w a does not change, which clearly contradicts the minimality of w. Now we deduce that, for any extremal solution (w a ) a2a , we have w unless moreover, it follows that necessarily w ba 0 =2c;da 0 which proves the theorem. The special case r = 2 of Theorem 4.5 answers the question of Erd}os, Faudree, Rousseau and Schelp [9, Problem B], who asked for the asymptotics of ^ r(K s;n ; K s;n ). Unfortunately, we do not think that the formula (4.6) can be further simplied in this case. Finally, let us consider the case of Theorem 4.5 in more detail. It is routine to check that Theorem 4.5 implies that ^ O(1). But we are able to show that (1.3) holds for all su-ciently large n, which is done by showing that a (K 2;n ; K 2;n )-arrowing graph with 18n can have at most 3 vertices of degree at least n for all large n. Theorem 4.6. There is n 0 such that, for all n > n 0 , we have 18n 15 and K 3;6n 5 is the only extremal graph (up to isolated vertices). Proof. For Gn be a minimum (K 2;n ; K 2;n )-arrowing graph. We know that e(Gn ) 18n 15 so l us assume Ln [18]. su-ciently large n. Suppose on the contrary that we can nd an increasing subsequence (n i ) i2N with l n i 4 for all i. Choosing a further subsequence, assume that Ln does not depend on i and that exists for any A 2 2 L . The argument of Lemma 3.2 shows that the weight g on L arrows We have It is routine to check that at 0 a > for any a 2 [4; 18]. The inequality (4.4) implies that, for some jAj > 4 or A 6 S. Let J be the set of those j 2 L with g fx;y;jg > 0. We have Consider the 2-colouring c of g obtained by letting c disjoint It is easy to check that neither c 1 nor c 2 contains A22 L Afx;yg c i;A < (5 3:5)=2 (Recall that d g (x) 1 for all x 2 L.) This contradiction proves Claim 1. Thus, jLn j 3 for all large n. By the minimality of Gn , spans no edge and each x 2 sends at least 3 edges to Ln . (In particular, jLn Thus, disregarding isolated vertices, implies that m 6n 5, which proves the theorem. Remark. We do not write an explicit expression for n 0 , although it should be possible to extract this from the proof (with more algebraic work) by using the estimates of Theorem 3.3. 5. Generalizations. If all forbidden graphs are the same, then one can generalize the arrowing property in the following way: a graph G (r; s)-arrows F if for any r-colouring of E(G) there is an F-subgraph that receives less than s colours. Clearly, in the case we obtain the usual r-colour arrowing property G This property was rst studied by Ekeles, Erd}os and Furedi (as reported in [7, Section 9]); the reader can consult [1] for references to more recent results. Axenovich, Furedi and Mubayi [1] studied the generalized arrowing property for bipartite graphs in the situation when F and s are xed, G = K n;n , and r grows with n. We s) to be the minimal size of a graph which (r; s)-arrows F . Our technique extends to the case when r and s are xed whilst F grows with n (i.e., is a dilatation). Namely, it should be possible to show the following. be a dilatation of a weight f and let r s be xed. Then the limit us denote it by We have weights g such that for any r-colouring c of g there is s such that cS f , where c where the sum is taken over all disjoint A We omit the proof as the complete argument would not be short and it is fairly obvious how to proceed. Also, one can consider the following settings. Let F i be a family of graphs, i 2 [r]. We any r-colouring of E(G), there is such that we have an F-subgraph of colour i. The task is to compute the minimum size of a such G. Again, we believe that our method extends to this case as well. But we do not provide any proof, so we do not present this as a theorem. ASYMPTOTIC SIZE RAMSEY RESULTS 13 Acknowledgements . The author is grateful to Martin Henk, Deryk Osthus, and Gunter Ziegler for helpful discussions. --R On size Ramsey number of pathes lp solve 3.2 A class of size Ramsey problems involving stars The size-Ramsey number of trees The size Ramsey number of trees with bounded degree Size Ramsey results for paths versus stars Asymptotic size Ramsey results for bipartite graphs --TR

farkas lemma;mixed integer programming;bipartite graphs;size Ramsey number

604630

Tensor product multiresolution analysis with error control for compact image representation.

A class of multiresolution representations based on nonlinear prediction is studied in the multivariate context based on tensor product strategies. In contrast to standard linear wavelet transforms, these representations cannot be thought of as a change of basis, and the error induced by thresholding or quantizing the coefficients requires a different analysis. We propose specific error control algorithms which ensure a prescribed accuracy in various norms when performing such operations on the coefficients. These algorithms are compared with standard thresholding, for synthetic and real images.

Introduction Multiresolution representations of data, such as wavelet basis decompositions, are a powerful tool in several areas of application like numerical simulation, statistical estimation and data compression. In such applications, one typically exploits the ability of these representations to describe the involved functions with high accuracy by a very small set of coe-cients. A pivotal concept for a rigorous analysis of their performance is nonlinear approximation: a function f is \compressed" by its partial expansion f N in the wavelet basis, which retains only the N largest contributions in some prescribed metric X. The most commonly used metric for thresholding images is other norms can be considered. For a given function f , the rate of best N-term approximation is the largest r such that N behaves as O(N r ) as N tends to +1. In many instances a given rate is equivalent to some smoothness property on the function f (see e.g. [11] for a survey on such results). Several recent works have demonstrated that such a rate is also re ected in the performance of adaptive methods in the above mentioned applications: see [6] for numerical simulation, [13, 12] for statistical estimation and [14, 7, 9] for data compression, as well as the celebrated image coding algorithms proposed in [22, 21]. In the case of applications involving images, such as compression and denoising, the main limitation to a high rate of best N-term approximation is caused by the presence of edges, since the numerically signicant coe-cients at ne scales are essentially those for which the wavelet support is intersected by such discontinuities. In particular, this limits the e-ciency of high order wavelets due to their large supports. From a mathematical point of view, this is re ected by the poor decay - in O(N 1=2 ) - of the best wavelet N-term approximation L 2 error for a \sketchy image where is a bounded domain with a smooth boundary. This re ects the fact that this type of approximation essentially provides local isotropic renement near the edges. Improving on this rate through a better choice of the representation has motivated the recent development of ridgelets and curvelets in [4], which are bases and frames having some anisotropic features, resulting in the better rate O(N 1 ). Another possible track for such improvements is oered by multiresolution representations which incorporate a specic adaptive treatment of edges. Tools of this type have been introduced in the early 90's by Ami Harten (see e.g. [16, 19, 17]) as a combination of wavelet analysis and of numerical procedures introduced by the same author in the context of shock computations, the so-called essentially non-oscillatory (ENO) and subcell resolution (ENO- reconstruction techniques (see e.g. [18, 20]). Formally speaking, the multiresolution representations proposed by Harten have the same structure as the standard wavelet trans- forms: a sequence v L of data sampled at resolution 2 L is transformed into (v where the v 0 corresponds to a sampling at the coarsest resolution and each sequence d k represents the intermediate details which are necessary to recover v k from v k 1 . However, the main dierence is that the basic interscale decomposition/reconstruction process which connects v k with (v k 1 ; d k ) is allowed to be nonlinear. As we shall see in image examples, the nonlinear process allows a better adapted treatment of singularities, in the sense that they do not generate so many large detail coe-cients as in standard wavelet transforms. On the other hand, it raises new problems in terms of stability which need to be addressed in order to take full advantage of this gain in sparsity. Let us explain this point in more detail. In most practical applications, the multiscale representation (v processed into a new (^v d L ) which is close to the original one, in the sense that for some prescribed discrete norm k k, where the accuracy parameters are chosen according to some criteria specied by the user. Such processing corresponds for example to quantization in the context of data compression or simple thresholding in the context of statistical estimation, adaptive numerical simulation or nonlinear approximation. Applying the inverse transform to the processed representation, we obtain a modied sequence ^ v L which is expected to be close to the original discrete set v L . In order for this to be true, some form of stability is needed, i.e., we must require that lim l In the case of linear multiresolution representations, the stability properties can be precisely analyzed in terms of the underlying wavelet system: for example if this system constitutes a Riesz basis for L 2 , stability can be re-expressed as a norm equivalence between v L and (v constants that do not depend on L. However, these techniques are no more applicable in the setting of non-linear representations which cannot anymore be thought as a change of basis. In the nonlinear case, stability can be ensured by modifying the algorithm for the direct transform in such a way that the error accumulated in processing the values of the multi-scale representation remains under a prescribed value. This idea was introduced by Harten for one dimensional algorithms in [16, 3, 1]. The aim of this paper is to introduce and analyze two-dimensional multiresolution processing algorithms that ensure stability in the above sense, and to test them on image data. We consider here the tensor product approach, which is also used in most wavelet compression algorithms. While this approach limitates the compactness of the representation for edges which are not horizontal or vertical, it inherits the simplicity of the one dimensional techniques in any dimension. This allows us to develop our algorithms in the same spirit as in the one dimensional case, yet pointing out new ways of truncating the data and proving explicit a-priori error bounds in the L 1 , L 2 and L 1 metric, in terms of the quantization and thresholding parameters. Let us mention that closely related algorithms have been recently introduced in [5] based on a non-linear extension of the so-called lifting scheme. The paper is organized as follows: we recall in x2 the discrete framework for multiresolution introduced by Harten, and we focus on two specic cases corresponding to point-value and cell-average discretizations for which we recall the linear, ENO and ENO-SR reconstruction techniques. The tensor-product error control algorithms are discussed in x3, where we also discuss the sharpness of the error bounds and the connexions with standard wavelet thresholding algorithms. These strategies are then practically tested on images in x4. In a rst set of experiments, we test the reliability and sharpness of our a-priori error bounds for a prescribed target accuracy. In a second set of experiments, we apply an alternate processing strategy based on a-posteriori error bounds in order to further reduce the number of preseved coe-cients while remaining within this prescribed accuracy. In the last set of experiments, we compare the compression performances of linear and nonlinear multiresolution decomposi- tions, using either error control or standard thresholding, on geometric and real images. The results can roughly be summarized as follows: nonlinear decompositions clearly outperform standard linear wavelet decompositions for geometric images, with an inherent limitation due to the use of a tensor product strategy, but brings less improvement for real images due to the presence of additional texture. This raises both perspectives of developping appropriate non-tensor product representations and of separating the geometric and textural information in the image in order to take more benet from these new representations. Acknowledgment: we are grateful to the anonymous reviewers for their constructive suggestions in improving the paper. 2. Harten's framework for Multiresolution 2.1. The general framework The discrete multiresolution framework introduced by Harten essentially relies on two pro- cedures: decimation and prediction. From a purely algebraic point of view, decimation and prediction can be considered simply as interscale operators connecting linear vector spaces, that represent in some way the dierent resolution levels (k increasing implies more res- olution), i.e., (a) (b) While the decimation D k 1 k is always assumed to be linear, we do not x this constraint on the prediction P k . The basic consistency property that these two operators have to satisfy is is the identity operator in V k 1 , which in particular implies that D k 1 k has full rank. If v the vectors derived by iterative decimation the prediction error. Clearly v k and (v contain the same information, but e k expressed as a vector of V k contains some redundancy since the consistency relation (2) implies that e k is an element of the null space, (D k 1 Keeping the algebraic viewpoint, we can remove this redundancy by introducing an operator G k which computes the coordinates of e k in a basis of N (D k 1 k ), and another that recovers the original redundant description of e k from its non-redundant part, i.e., such that e These ingredients allow us to write a purely algebraic description of the direct and inverse multiresolution transform as follows: These algorithms which connect v L with its multiscale representation (v can be viewed as a simple change of basis in the case where the prediction operator is linear. F F @ @ @R @ @ @I Figure 1: Denition of transfer operators In practice, the construction of the operators D k 1 k 1 is based on two fundamental tools: discretization and reconstruction. The discretization operator D k acts from a non-discrete function space F onto the space V k and yields discrete information at the resolution level specied by a grid X k . It is required that D k be a linear operator. The reconstruction operator R k , on the other hand acts from V k to F and produces an approximation to a function F from its discretized values: the function R k D k f . A basic consistency requirement of the framework is that Given sequences of discretization and reconstruction operators satisfying (5), it is then possible to dene the decimation and prediction operators according to This denition is schematically described in gure 1. Observe that there seems to be an explicit dependence of D k 1 k on R k but it is easy to prove that the decimation operator is in fact totally independent of the reconstruction process whenever the sequence of discretization is nested, i.e., it satises This property implies, in essence, that all the information contained in the discretized data at a given resolution level is also included in the next (higher) one. For a nested sequence of discretization, the description of D k 1 k as R k D k 1 , is just a formal one (useful in some contexts), and in practice one does not resort to the reconstruction sequence to decimate discrete values. The description of the prediction step as P k opens up a tremendous number of possibilities in designing multiresolution schemes. The reconstruction process is the step. The subband ltering algorithms associated to biorthogonal wavelet decompositions correspond to particular cases where this process is linear. In contrast, nonlinear reconstruction operators will lead naturally to nonlinear multiresolution representations which cannot anymore be thought as a change of basis. For the sake of completeness and ease of reference, in the remainder of this section we give a very brief description in one dimension of more restricted frameworks associated to point-value and cell-average discretizations, together with the corresponding linear, ENO and ENO-SR reconstruction operators. The left-out details in the entire section (and much more) can be found in [1, 17, 2, 3]. multiresolution representations have also recently been proposed in [5] based on a non-linear extension of the lifting scheme. In particular, the nonlinear ENO and ENO-SR representations that will be used in this paper could be introduced with the lifting terminology in place of the above general framework in which they were originally in- troduced. On the other hand the error control strategies that we develop in x3 are specically adapted to the restricted frameworks of point-value and cell-average discretizations (and different from the general \synchronization" strategy of [5]). These restricted frameworks have a particularly simple interpretation within Harten's general framework. 2.2. Point-value multiresolution analysis in 1D Let us consider a set of nested grids: where N 0 is some xed integer. Consider the point-value discretization is the space of sequences of dimension we obtain (D Notice that N (D k 1 can be dened as follows: A reconstruction procedure for the discretization operator dened by (8) is given by any operator R k such that which means that (R k therefore, (R k should be a continuous function that interpolates the data f k on X k . Let us change slightly the notation and denote by I k (x; such an interpolatory reconstruction of the data f k . The prediction operator can be computed as follows: and the direct and inverse transforms (3) and (4) take the following simple form: Notice that in the point-value framework, the detail coe-cients are simply interpolation errors at the odd points of the grid that species the level of resolution. A natural way of dening a linear interpolation operator is as follows: some integer m > 0 being xed, we consider the unique polynomial p i of degree 2m 1 such that p i simply dene I This choice coincides with the so-called Lagrange interpolatory wavelet transform. Note that as m increases, the interpolation process has higher order accuracy, i.e. the details d k i will be smaller if f is smooth on [x k 1 ]. On the other hand, the intervals [x k 1 larger with m so that a singularity will aect more detail coe-cients. Non-linear essentially non-oscillatory (ENO) interpolation techniques, which were rstly introduced in [20], circumvent this drawback: the idea is to replace in (15) the polynomial by a polynomial p selected among fp in order to avoid the in uence of the singularity. The selection process is usually made by picking the \least oscillatory" polynomial using numerical information on the divided dierences of f at the points x k 1 . In the present paper we have been using the so called hierachical selection process which is detailed in [1]. Once this selection is made, we thus dene I Such a process still produces large details d k when a singularity is contained in the interval In order to reduce further the interpolation error, subcell resolution methods were introduced in [18] as an elaboration of ENO interpolation. The idea is rst to detect the possible presence of singularities by agging those i such that p i.e. such that the selection process tends to escape the interval [x k 1 such i+1 intersect at a single point a 2 [x k 1 we identify this point as the singularity and replace in (16) the polynomial p i by the piecewise polynomial function which coincides with p Note that such a process is better tted to localize the singularities of the rst order, i.e. jumps in f 0 , rather than the discontinuities of f . Such discontinuities, which correspond to edges in image processing, are better treated in the cell-average framework that we now describe. 2.3. Cell-average multiresolution analysis in 1D With the same nested grids structure of last section, we dene the discretization is the space of absolutely integrable functions in [0; 1]. It is su-cient to consider weighted averages since these contain information on f over [0; 1]. Thus, V k is the space of sequences with N k components. Additivity of the integral leads to the following decimation step: thus, the prediction error satises and the operators G k and E k can be dened as follows A reconstruction operator for the discretization in (17) is any operator R k satisfying (D (R k That is, R k k (x) has to be a function in L 1 ([0; 1]) whose mean value on the ith cell coincides with In one dimension, the simplest way to construct R k is via the \primitive function". Dene the sequence fF k i g on the k-th grid as The function F (x) is a primitive of f(x), and the sequence fF k corresponds to a discretization by point-values of F (x) on the k-th grid. Let us denote by I k (x; F k ) an interpolatory reconstruction of F (x), and dene (R k dx I k (x; F k It is easy to see that D k R (R k With these denitions the direct transform (3) and its inverse (4) can be described as follows: where d dx The linear, ENO and ENO-SR techniques for cell-averages are simply derived from the corresponding techniques in the point-value framework, using the above primitivation. Note that jumps in f get transformed into jumps in F 0 , so that the ENO-SR process is now well tted to localize the discontinuities. Remark 2.1 In practice, the primitive function is used as a design tool and it is never computed explicitly: All calculations are done directly on the discrete cell-values ([16, 1]). In particular, for linear techniques, we obtain the biorthogonal wavelets decompositions corresponding to the case where the dual scaling function ~ ' is the box function [0;1] (if in addition we have for the accuracy parameter, we obtain nothing but the Haar system). 3. MR-based compression schemes with error-control Multiresolution representations lead naturally to data-compression algorithms. Probably the simplest data compression procedure is truncation by thresholding, which amounts to setting to zero all detail coe-cients which fall below a prescribed, possibly level dependent, threshold, Thresholding is used primarily to reduce the \dimensionality" of the data. A more elaborate procedure, which is used to reduce the digital representation of the data is quantization, which can be modeled by where round [] denotes the integer obtained by rounding. For example, if jd k then we can represent i by an integer which is not larger than 32 and commit a maximal error of 4. Observe that jd k and that in both cases While the thresholding procedure is usually applied only to the scale coe-cients, the quantization process is also applied to the coarse level representation. After the application of a particular compression strategy, such as truncation or quantiza- tion, we obtain a compressed multiresolution representation M d L g, where represents the compression parameters, i.e., applying truncation by thresholding). Obviously, M f is close to M g. Applying the inverse multiresolution transform to the compressed representation, we obtain ^ f , an approximation to the original signal f L . The fundamental question is that of estimating, and thus being able to control, the error k f L k in some prescribed norm. It often happens that we are given a target accuracy, i.e., a maximum allowed deviation , thus our goal is to obtain a compressed representation M f such that We can formulate this goal in various ways : f L k from the errors kd k f L k from the thresholds k . For linear multiscale representations corresponding to wavelet decompositions, error estimates of the type we are looking for are typically derived by using the stability properties of the underlying wavelet system. In the nonlinear framework, an error-control strategy was rst proposed by Harten in [16] to directly accomplish (29), namely modify the direct transform in such a way that the modication allows us to keep track of the cumulative error and truncate accordingly. Let us explain in a nutshell the idea of the error-control algorithm (EC henceforth). As a rst step, the sequences f 0 are computed from f L by iterative application of the decimation operator. Then, we start at the coarsest level and dene ^ f 0 by applying some perturbation process (thresholding or quantization) on is now to dene the processed details ^ d k and processed kth scale representation ^ f k in an intertwined manner from coarse to ne scales: for L, the processed details ^ represent a perturbation of the prediction error involving the processed data at the coarser scale ^ while the sequence of processed data is simultaneously computed according to ^ d k . In the following, we shall focus on the point-value and cell-average frameworks for which there is a rather natural specic way to dene the processed details ^ d k . In one dimension, the strategy can be schematically described as follows: d L g Here given by the one-dimensional decimation operator correspnding to the chosen framework, i.e. 2i for point-values and cell-averages. On the other hand [ stands for the kth step of the 1D error control algorithms presented in [1], which we recall here for the sake of clarity: ~ ~ (a) (b) Figure 2: algorithms in 1d. (a) Point-values. (b) Cell-averages A key point is that the values ^ f k are precisely the values computed by the corresponding inverse multiresolution transform M 1 at each resolution level, i.e. (14) in the point-value framework and (25) in the cell-average framework. In other words: A second key point is that, in both algorithms, the coe-cients ~ coincide with the true prediction errors at odd points e k (i.e., the scale coe-cients in the direct encoding) only when In the EC algorithm, the details need to be dened in such a way that, after compression takes place, the error accumulated at each renement step, i.e., jj can be controlled. To get a direct control on jj f k jj, the details ~ must contain relevant information on the error committed in predicting the true values at the kth level, i.e. k , from the computed In the point-value framework, there is no error at even points, and only the prediction errors at the odd points needs to be controlled. It is not hard to deduce from gure 2-(a) (see also [1]) that one has In the cell average setting, the compressed details ^ d k are dened by applying the processing strategy on the half-dierences ~ adjacent points. This is su-cient to ensure control on the prediction errors at each location on a given resolution level, because from gure 2-(b) one can easily deduce that [1] Relations (31) in the point-value framework, and (32) in the cell-average framework, express the compression error at the kth level in terms of the compression error at the previous level plus a quantity that is directly related to the thresholds k . These basic relations lead to the one-dimensional error-bounds in [1]. product EC algorithms in two dimensions Needs a small paragraph on why we choose tensor product (simple, everybody else does that etc,,,) The modied algorithms for the direct and inverse multiresolution transforms we shall describe below can be viewed as two-dimensional tensor product extensions of the one dimensional algorithms in [16, 1]. For each resolution level, one acts with the one-dimensional decimation operator on each row of the 2d array ij , and computes intermediate values, say 2 . This array, which has N k rows but only N k 1 columns, is decimated again column by column with the one-dimensional operator to obtain These values are stored until the bottom level is reached. Then computed values ^ 2 and processed details ^ d k are computed in an intertwined manner using only the one-dimensional algortihms described in the previous section. In this context, and similarly to the tensor-product wavelet transform, we obtain three types of details ^ d k (3). To be more explicit, we give next a schematic description of the typical 2D algorithm for the EC-direct transform, Algorithm 1: Modied Direct transform The intermediate values f k are discarded once the algorithm concludes, and the outcome of the EC-direct transform is d d d L (3)g The inverse multiresolution transform can be described using the 1d operator inver1d, which denotes the k step of algorithms (14) in the point-value framework and (25) in the cell average framework. Algorithm 2: Inverse transform As in the 1D case, the inverse transform (decoding) satises It should be remarked that, due to the intertwined structure of the error control algorithm, the details, ~ will be dierent if one acts rst on the rows or rst on the columns. In any case, and even though the computed details might be dierent, they are of the same order of magnitud. 1.1 Explicit Error Bounds for the EC strategies in two dimensions We shall consider the following norms: In what follows we shall see that it is possible to estimate the error between the original signal f L and the signal obtained from decoding its compressed representation, i.e. ^ f L from either the dierences jj ^ or the threshold parameters k . Proposition 3.2 Given a discrete sequence f L , with the modied direct transform for the pointvalue framework in 2-d (Algorithm 1) we obtain a multiresolution representation M M f L such that if we apply the inverse transform (Algorithm 2) we obtain ^ f L satisfying: (jjj ~ where Proof: From the 1d relations (31) we obtain hence i;j (2) and i;j (2) Then (j and we obtain (34). Since N (j which proves (35) and (36). Corollary 3.3 Consider the error control multiresolution scheme described in proposition 3.2, and a processing strategy for the detail coe-cients such that we obtain that In particular, if we assume that jj we obtain Proposition 3.5 Given a discrete sequence f L , with the modied direct transform for the cell average framework in 2-d (Algorithm 1) we obtain a multiresolution representation M M f L such that if we apply the inverse transform (Algorithm 2) we obtain ^ f L satisfying: Proof: As a consequence of the relations in (32) we obtain and i;j (2) i;j (2) i;j (2) i;j (2) i;j (2) and i;j (2) Hence max(j ~ i;j (2) i;j (2) i;j (2) thus and we deduce (40). Also from (44), we deduce 4j and hence we have proved (41). To prove (42), we obtain from (43) and d hence 2and the pronof is concluded. Corollary 3.6 Consider the error control algorithm for the cell average framework described in proposition 3.5 and a processing strategy for the detail coe-cients such that 3: Then In particular, if we assume that jj we obtain 3.4 Remarks and comparison with standard thresholding The results of Propositions 3.2 and 3.5 can be viewed as a-posteriori bounds on the compression error, since they involve the ~ d k which themselves depend on the processing strategies applied at the coarser levels. In practice, this becomes important; the a-posteriori bound can be evaluated at the same time the compression process is taking place, and the a-posteriori bound coincides in some cases with the exact compression error ((34), (35), (36) and (42)). On the other hand, Corollary 3.3 and 3.6 provide a-priori bounds on the compression error. Consider for example the L 2 -error in the cell-average framework and assume, for simplicity, that f 0 . According to Corollary 3.6, we can ensure an error of order by choosing a sequence and requiring that the processed details ^ 3: The truncation strategies given by (26) and (27) both satisfy (37). Note, however, that with either one of these two strategies, the error bound jj ~ is already not sharp since it corresponds to the worst case scenario, where all the dierences j( ~ are close to k , while in practice these dierences are often zero, or much smaller than k . Therefore, we expect in this case that the a-priori bound in is over-estimated and much larger than the a-posteriori bound given by Proposition 3.5, which in this particular case is precisely the compression error k This fact is examined in detail in the next section, where it is used as the starting point for the design of new truncation strategies. Another important issue is how to choose the dependence in k of the truncation parameter k in order to optimize the compression process. By \optimize", we mean here to minimize the number of resulting parameters for a given prescribed error. This number corresponds to the number of non-discarded coe-cients in the case of thresholding, and to the total number of bits in the case of quantization. In the case of linear multiresolution representations associated to wavelet decompositions, the answer to this question is now well understood: the threshold or truncation parameters should be normalized in accordance with the error norm that one is targeting. More precisely, if the detail coe-cients d represent the expansion of a function in a wavelet basis and if one is interested in controlling the L p norm with a minimal number of parameters, then these coe-cients should be perturbed according to d d k L d d jk k L p where is xed independently of . We refer to [11] and [7] for such type of results. It is easily seen that in the case of two-dimensional cell-average multiresolution, this corresponds to taking in Corollary 3.6. This suggests to use similar normalizations in the setting of nonlinear multiresolution algorithms with error control, although there is no guarantee now that such a strategy will be optimal in the above sense. This issue will be revisited in the next section through numerical testing. As an example consider again the L 2 -error in the cell-average discretization. In the linear case, an optimal choice for k is of the form According to Corollary 3.6, we can take ensure a global L 2 error less than by taking . However, as already remarked, we may expect that the L 2 error is actually much less than , so that one can still lower the number of compression parameters by raising while the a-posteriori L 2 -error bound, which in this case is precisely the exact remains below the tolerance . The fact that the error-control strategy permits to monitor the compression error at each resolution level, through the a-posteriori bounds, allows us to design new processing strategies that aim at reducing as much as possible the number of resulting parameters, while keeping, at the same time, the total compression error below a specied tolerance. An example of such strategy is given in the next section. --R A Surprisingly E Nonlinear Wavelet Transforms for Image Coding via Lifting Scheme Adaptive wavelet algorithms for elliptic operator equations - Convergence rate to appear in Math Tree approximation and optimal encoding. Biorthogonal bases of compactly supported wavelets. On the importance of combining wavelet-based nonlinear approximation with coding strategies Ten Lectures on Wavelets. Nonlinear approximation. Unconditional Bases are Optimal Bases for Data Compression and for Statistical Estimation Wavelet shrinkage: asymptotia Analysis of low bit rate image coding. Analyses Multir Discrete multiresolution analysis and generalized wavelets. Multiresolution representation of data II: General framework. ENO schemes with subcell resolution. Multiresolution representation of cell-averaged data Uniformly high order accurate essentially non-oscillatory schemes III An image multiresolution representation for lossless and lossy compressio. Embedded image coding using zerotrees of wavelet coe-cient --TR Uniformly high order accurate essentially non-oscillatory schemes, 111 ENO schemes with subcell resolution Multiresolution representation of data Multiresolution Based on Weighted Averages of the Hat Function I Multiresolution Based on Weighted Averages of the Hat Function II Adaptive wavelet methods for elliptic operator equations --CTR F. Arndiga , R. Donat , P. Mulet, Adaptive interpolation of images, Signal Processing, v.83 n.2, p.459-464, February S. Amat , J. Ruiz , J. C. Trillo, Compression of color image using nonlinear multiresolutions, Proceedings of the 5th WSEAS International Conference on Signal Processing, Robotics and Automation, p.11-15, February 15-17, 2006, Madrid, Spain S. Amat , J. C. Trillo , P. Viala, classical multiresolution algorithms for image compression, Proceedings of the 5th WSEAS International Conference on Signal Processing, Robotics and Automation, p.1-5, February 15-17, 2006, Madrid, Spain Sergio Amat , S. Busquier , J. C. Trillo, Nonlinear Harten's multiresolution on the quincunx pyramid, Journal of Computational and Applied Mathematics, v.189 n.1, p.555-567, 1 May 2006

stability;multi-scale decomposition;non linearity;tensor product

604689

Open-loop video distribution with support of VCR Functionality.

Scalable video distribution schemes have been studied for quite some time. For very popular videos, open-loop broadcast schemes have been devised that partition each video into segments and periodically broadcast each segment on a different channel. Open-loop schemes provide excellent scalability as the number of channels required is independent of the number of clients. However, open-loop schemes typically do not support VCR functions. We will show for open-loop video distribution how, by adjusting the rate at which the segments are transmitted, one can provide VCR functionality. We consider deterministic and probabilistic support of VCR functions: depending on the segment rates chosen, the VCR functions are supported either 100% of the time or with very high probability. For the case of probabilistic support of PLAY and Fast-forward (FF) only, we model the reception process as a semi-Markov accumulation process. We are able to calculate a lower bound on the probability of successfully executing FF actions.

Introduction 1.1 Classication VoD systems can be classied in open{loop systems [12] and closed{loop systems [10,16]. In general, open loop VoD systems partition each video into smaller pieces called segments and transmit each segment on a separate channel at its assigned transmission rate. Those channels may be logical, implemented with an adequate multiplexing. All segments are transmitted periodically and indenitely. The rst segment is transmitted more frequently than later segments because it is needed rst in the playback. In open{loop systems there is no feedback from the client to the server, and transmission is completely one{way. In closed-loop systems, on the other hand, there is a feedback between the client and the server. Closed{loop systems generally open a new unicast/multicast stream each time a client or a group of clients issues a re- quest. To make better use of the server and network resources, client requests are batched and served together with the same multicast stream. Open-loop systems use segmentation in order to reduce the network bandwidth requirements, which makes them highly scalable because they can provide Near Video on Demand (NVoD) services at a xed cost independent of the number of users. In this paper, we will show how open-loop NVoD schemes can support VCR functions, which are dened as follows: PLAY Play the video at the basic video consumption rate, b; PAUSE Pause the playback of the video for some period of time; SF/SB Slow forward/Slow backward: Playback the video at a rate equal to some period of time. We have Y S < FF/FB Fast forward/Fast backward: Playback the video at a rate equal to some period of time. We have X F > 1. 1.2 Related Work Most VoD systems do not support VCR functions. It is assumed that users are passive and keep playing the video from the beginning until the end without issuing any VCR function. However support of VCR functions makes a VoD service much more attractive. Most research on interactive VoD focuses on closed-loop schemes [1,6,15,13]. To support VCR functions such as Fast-Forward all these schemes serve the client who issues a FF command via a dedicated unicast transmission, referred to as contingency channel. When the client returns into PLAY state, (s)he joins again the multicast distribution. It is obvious that such a solution is not very scalable since it requires separate contingency channels and also explicit interaction with the central server. Thus, open-loop schemes are particularly well suited when: a) the number of users grows large, or b) the communication medium has no feedback channel, which is the case in satellite or cable broadcast systems. Very little work has been done to support VCR functions in open-loop VoD schemes [2,4,8,14]. Except for the paper by Fei et al. [8], all the other schemes only consider PAUSE or discrete jumps in the video. Fei et al. propose a scheme called \staggered broadcast" and show how it can be used together with what they call \active buer" management to provide limited interac- tivity. In staggered broadcast, the whole video of duration L is periodically transmitted on N channels at the video consumption rate b. Transmission of the video on channel i starts t later than channel i 1. Depending on the buer content and the duration of the VCR action, the VCR action may be possible or not. In the case that the VCR action is not possible, it is approximated by a so-called discontinuous interactive function where the viewing jumps to the closest (with respect to the intended destination of the interaction) point of the video that allows the continuous playout after the VCR action has been executed. The big dierence between the related work and our scheme is that up to now, the support of VCR functions either required a major extension of the transmission scheme (e.g. contingency channels) or was very restricted (e.g. staggered broadcast). We will demonstrate the feasibility of deterministic support of VCR functions in open-loop VoD systems by increasing the transmission rate of the dierent segments. While this idea looks very straightforward, it has been, to the best of our knowledge, never proposed before. The rest of the paper is organized as follows. We rst describe the so-called tailored transmission scheme, then discuss how to adapt this scheme to support VCR functions. For the case of PLAY and FF user interactions we develop an analytical model that allows the computation of a lower bound on the probability that a user interaction can be successfully executed and then provide some quantitative results. The paper ends with a brief conclusion. 2.1 Introduction Many dierent open-loop NVoD schemes have been proposed in the literature; for a survey see [12]. These schemes typically dier in the way a video is partitioned into segments and can be classied mainly in three categories: Schemes that partition the video in dierent length segments and transmit each segment at the basic video consumption rate [9,19]; Schemes that partition the video in equal-length segments and decrease the transmission rate of each segment with increasing segment number [3]; Hybrid schemes that combine the two above methods [14,20]. In the following, we will present in more detail the scheme called tailored transmission scheme that was proposed by Birk and Mondri [3] and is a generalization of many of the other open-loop NVoD schemes previously described. 2.2 Tailored Transmission Scheme The base version of the tailored transmission scheme works as follows. A video is partitioned into N equal-length segments. Each segment is transmitted periodically and repeatedly on its own channel. A client who wants to receive a video starts by listening to one, more, or all channels and records these segments. We shall need the following notation: s i denote the time the client starts recording segment denote the time the client has entirely received segment denote the time the client starts viewing segment r i denote the transmission rate of segment i [bits/sec]; D denote the segment size [bits]; b denote the video consumption rate [bits/sec]. To assure the continuous playout of the video we require that each segment is fully received before its playout starts, i.e. v i w i . Given a segment size the transmission rate r i of segment i must satisfy the following condition to assure a continuous playout of the video: (v If the client starts recording all segments at the same time, i.e. s and Mondri have shown (Lemma 1 in [3]) that the transmission rate will be minimal and is given as r min (w Without loss of generality, we may assume that t is the duration of a segment. Then, r min and the total server transmission bandwidth is R min Figure 1 illustrates the tailored transmission scheme for the case of minimal transmission rates. The client starts receiving all segments at time t 0 . The shaded areas for each segment contain exactly the content of that segment as received by the client who started recording at time t 0 . A client is not expected to arrive at the starting point of a segment; instead a client begins recording at whatever point (s)he arrives at, and stores the data for later consumption. Therefore, the startup latency of is the scheme corresponds to the segment duration D=b.00000011111100011100000000111111110011001111000001111111111000011111111000001111111111 Segments Time Client joins Fig. 1. An example of the tailored transmission scheme with minimal transmission rates. 3 How to Support VCR Functions Given the base scheme of the tailored transmission with its minimal transmission rates, we will show how to adapt (increase) the segment transmission rates to support VCR functions. To convey the main idea, we will limit ourselves rst to the case where the only VCR function possible is FF. In fact, the FF is the only VCR action that \accelerates" the consumption of the video, which possibly can lead to a situation where the consumption of the video gets ahead of the reception of the video. We present a solution that makes sure that any FF command issued can be successfully executed. The other user interactions such as SF, SB, FB or PAUSE can be accommodated by buering at the client side. From now on, we therefore consider only two states: PLAY and FF. We make the following two central assumptions: The client has enough disk storage to buer the contents of a large portion of the video; The client has enough network access and disk I/O bandwidth to start receiving the N segments at the same time. The trend for terminal equipments appears to be that more and more storage capacity is available. Actually, there already exist products that meet the above assumptions. An example is the digital video recorder by TiVo [18] that can store up to 60 hours of MPEG II video and, connected to a satellite feed, can receive transmissions at high data-rates. However, for the case where the assumption on storage does not hold, we also know how to support VCR functions: the idea will remain the same, only the individual segment transmission rates required will be higher. The scheme we propose may be adapted to this situation. Note that the trade-o between the storage capability of the client and segment transmission rates for the case of NVoD has already been explored by Birk and Mondri [3]. 3.1 Deterministic Support Whenever a client issues a FF command, the video is viewed at a playout rate than the normal rate, i.e. the consumption of the video occurs at a rate equal to X F b and each segment will be consumed after D bXF units of time instead of D=b units of time in case of PLAY. As a consequence, the viewing times of all segments not yet viewed will be \advanced" in time. To obtain a deterministic guarantee that every FF command issued during the viewing of a video can be executed, we consider the worst case scenario where the client views the whole video in FF. i denote the time the client starts viewing segment i, given that (s)he has viewed segments mode. We can compute the v FF i as If the client starts recording all segments at the same time, i.e. s compute, similar to (2), the transmission rate r FF i that allows unrestricted FF interactions as r FF (v FF XF D 1 The playout and therefore VCR actions do not start before segment 1 has been entirely received; we therefore have v FF If we assume that t D=b, the expression simplies to r FF 3.2 Probabilistic Support for FF In the previous subsection, we have computed the minimal transmission rates r FF i such that all the FF interactions issued can be realized. We have considered the worst case scenario where the client views the whole video in FF mode. While a client might do so, we think that it is much more likely that the viewing of a video will alternate between PLAY and FF modes (and possibly other VCR actions). We will in the following use a model for the viewing behavior where a user strictly alternates between PLAY and FF. We refer to this behavior as S-FF (for Simple FF). Our goal is to support FF interactions with high probability while transmitting each segment at a rate lower than r FF . To this purpose we dene the rates r I as follows: The server transmits the segments i Ng at a rate r I where A is the rate increase factor, with 1 A X F , and r min i is computed in (2); The server transmits segment 1 at rate r I still because the playout does not start before segment 1 has been entirely received. Analytical Model for the S-FF Model In this section we will compute a closed-form lower-bound on the probability that a segment is successfully consumed by the client. Segment i is successfully consumed by the client if segment available to him/her before the consumption of segment i has been completed; otherwise we will say that the consumption of segment i has failed. A failure is resolved once the next segment is entirely available to the client. It is worth pointing out that failures may occur both in mode PLAY and in mode FF, as shown on Figure 2. We will assume that the client alternates between both modes of consump- tion. More precisely, we introduce two independent renewal sequences of rvs fS P (n)g n and the fS FF (n)g n , where S P (n) and S FF (n) will represent the duration of the n-th PLAY and FF periods, respectively. F F Playout limit in FF mode in P mode time bits Consumption curve Fig. 2. Failures occurring in PLAY and FF modes. For modeling purposes, and also because we believe this assumption corresponds to a reasonable behavior of the client, we will assume that the remaining duration of a PLAY or FF period when a failure occurs is resumed when the next segment is available to the client. This corresponds, for instance, to the situation where the client wants to reach a particular point in the video or avoid a particular scene, regardless of the failures that (s)he may encounter while viewing the video. In order to ensure a probabilistic support for FF (cf. Section 3.2) recall that segment i is transmitted at rate r I Therefore, the i-th segment will be entirely available to the client at time w The continuous playout of segment i requires that at the viewing time v i , this segment has been entirely received, that is v i w i . Segment will fail if this inequality does not hold. The continuous playout of the video requires that all segments be on time, namely, Recall that v since the client cannot start viewing the rst segment before it has been entirely received. The number L of segments on time is given by where 1A stands for the indicator function of the event A, from which we deduce the mean number of segments on time Denote by R(t) the number of bits of the video which have been consumed by the client in [v 1 Computing P(v i w i ) in closed-form for all i is not an easy task. Indeed, it is related to computing the distribution of the length of a busy period in a uid queue fed by a Markov-Modulated Rate Process. In the present paper, we will content ourselves with the derivation of an elementary lower bound. To derive this lower bound, we consider the semi-Markov accumulation process which is constructed as follows: during a PLAY period Q(t) continuously increases with the rate b and during a FF period it continuously increases with the rate bX F . More precisely, for t > 0, with T n := By convention T By construction of Q(t) and R(t) it is obvious that (see Figure 3) Observe that both processes fR(t); t w 1 g and fQ(t); t w 1 g would be identical in the absence of failures. We see from (6) and the denition (5) that which implies that Hence, cf. (4), For the transmission scheme we described in Section 3.2, the segment arrival times are given by w but the analysis above actually holds for any reception schedule of segments given by a sequence fw Failure SFF (2) SP (2) Fig. 3. Comparison of Q(t) and R(t). In Section 5, we present results for determining P(Q(T ) < x), for any T and x. These results are actually obtained for any semi-Markov accumulation process with a nite state-space (see Section 5.2). When S P (n) and S FF (n) are exponentially distributed random variables (rvs) with respective means 1= P and 1= FF , we can apply the formulas in Section 5.3. First, use (16) with use the formulas for q ij (x) (the density of the distribution of Q, conditionally on the start/end states) with and . The probabilities P(Q(T are then obtained by numerical integration. 5 Semi-Markov Accumulation Process In this section, we develop a framework for evaluating the workload distribution generated in a given time-interval by a semi-Markov accumulation process with an arbitrary (but nite) state-space. After dening the process (Section 5.1), we show that the Laplace transforms of the sought distributions satisfy the linear system of equations (15). Finally, we apply the formula to the case of a two-state continuous-time Markov process (Section 5.3), where the Laplace transform can be inverted to obtain the density of the distribution. 5.1 Denition We rst construct formally the accumulation process from a semi-Markov process. Kg be a nite state-space. Let be a sequence of i.i.d. rvs, for each fZ(n)g n be a homogeneous discrete-time Markov chain on the state-space The semi-Markov process fX(t); t 0g is dened jointly with a sequence of jump times as with nonnegative rv. The accumulation process Q(t) is such that while the process X(t) is in state i, Q(t) accumulates at a constant rate r i . Formally, fQ(t); t 0g is constructed as follows: set This construction is illustrated in Figure 4. The upper part shows the evolution of the discrete-time Markov chain Z(n), and of X(t). The lower part displays Q(t) as a function of the jump times T n . The accumulation rates are such that 5.2 Distribution of Q(t) Let Q i;r (T ) denote the quantity accumulated in [0; T ) given that and T In other words, the process X starts in state i with a residual time r in this state. Similarly, denote Q i;S i (T ) the same quantity, but given that distributed according to the total sojourn time distribution (i.e., as if a transition to state i had occurred at time 0). Depending on the problem to be solved, one may be interested in the distribution of Q i;S i (T ) or that of Q S i is the forward recurrence time of . The latter corresponds to the case where the semi-Markov process fX(t)g Fig. 4. Construction of the accumulation process is stationary. The common procedure for computing these distributions is to compute that of Q i;r (T ) for an arbitrary r, and then integrate with respect to the proper distribution. We are therefore interested in the distribution of Q i;r (T ), jointly with that of X(T ), namely P(Q i;r (T ) We shall actually compute the Laplace-Stieltjes Transform (LST) The computation below may be seen as a generalization of the analysis developed by Cox and Miller in [5, x9.3] for alternating renewal processes (i.e. First, if r T , then no jump occurs before time T , and since In that case, On the other hand, if r < T , then at least one jump occurs in the time-interval conditioning on the state reached after the rst jump (i.e. Z(1)) then using the stationarity and independence of the underlying sequences, we have We now compute the Laplace transform of ^ T () with respect to T . With the help of (10)-(11), we obtain re r dT dT A relation involving only the rvs Q i;S i (T ) is obtained from (12) by integrating both sides with respect to r, considered to be distributed as S i . Let S i (r) denote the distribution function of S i and let S be its LST. Introduce also the notation dT Then, we have dT This is a system of linear equations from which the required Laplace transforms can be computed. To see this better, dene the matrices diag (S diag denotes the mm diagonal matrix with elements Then, (14) rewrites as K=L The matrix I SP is invertible because the spectral radius of SP is less than This follows by application of a standard bound on the spectral radius ([11, Cor. 6.1.5]): (SP) )j. This is less than one in the specied domain, from well known properties of Laplace transforms. Once the matrix K is computed, other initial conditions of the process fX(t); may be investigated. For instance, if the residual sojourn time in state i is r, then the distribution is obtained using (12), that is If the residual sojourn time in state i is given by e the forward recurrence time of S i (in other words, if fX(t); t 0g is stationary), then integrating (12) gives, with obvious notation f dT Remark: A simple extension of this derivation shows that the accumulation process may be generalized by replacing the constant-rate process by any stationary process with independent increments. The formulas above hold with the term \r replaced by some i () characteristic of the process (see [7, Eq. (7.3') p. 419]). For instance, for the Poisson process with rate r, process with drift r and variance 2 , 5.3 Application to a Two-State Markov Accumulation Process In this section, we address the case of a two-state, continuous-time Markov process, with innitesimal generator (T ) denote the quantity accumulated during the interval [0; T ) when accumulation rates in states 1 and 2 are r 1 and r 2 , respectively. In distribution, we have Computing the distribution of Q r1 ;r 2 (T ) is therefore reduced to computing the distribution of Q 1;0 (T ), which is the visit time in state 1 during the interval apply formulas of Section 5.3. We assume that the residual time in the initial state has the same distribution as the total sojourn time. Observe that due to the memoryless property of the exponential distribution, S i and e have the same distribution. The relevant matrices are: Using (15), we obtain The last step is to invert the Laplace transform K ij (; ) with respect to and . From the denition (13), this will give the density of the distribution of Q(T ). The inversion can be performed using general rules and tables for Laplace transforms (see e.g. [17]). Inverting with respect to is straightforward, because we have a rational function of degree 1 in . We obtain: dq P(Q For the inversion with respect to , we use in particular the following properties a I 1 aT aT a I 1 aT (i.e. inverse of the Laplace transform g(s) at point t), I n () is the modied Bessel function of the rst kind and order n (see e.g. [17, p. 7]) and - a (t) is the Dirac function at point a. for x 0. We nally nd, with s x s x In order to obtain the distribution functions P(Q(T the Laplace transforms K ij (; )= should be inverted. This leads to more involved series which shall not be reproduced here. 6 Numerical Results We have applied the bound in (7) to a video of length 2 hours = 7200 sec. We have varied the segment size from 200 sec to 800 sec. The number N of segments varies inversely from 36 to 9. The playout factor for FF is This is a standard value for VCRs, also used in other papers. We consider two dierent duration ratios (that is: PLAY periods last 2 times, resp. 5 times longer than FF periods). The parameters chosen are detailed in Table 1. We have displayed in this table the average \natural" consumption rate of the video, given by FF Table Parameters of the numerical experiments. 1= P 1= FF A b N =b In order to compare the performance of our scheme for videos of dierent lengths, we have measured the probability of success: The results should depend on how the natural rate b N compares to the rate increase factor A. If b N =b < A, then the law of large numbers will force the \natural" consumption curve Q(t) (and therefore R(t)) to lie below the playout limit with large probability. Note that this eect may be long to appear if b N =b is close to A. If b N =b > A, then the converse eect appears. In that case, it also turns out that the actual curve R(t) records a large number of failures. Another eect may kick in: the probability that a failure occurs within segment may depend on i. First, the time between w 1 and w 2 (=D (2=A 1)=b) is smaller than the typical inter-arrival time between segments w This may give a signicant advance of data, and with few (large) segments, may result in a large success probability. On the other hand, when b N =b < A, the rst segments tend to be vulnerable to uctuations in the consumption rate and have a smaller success probability. But if b N =b > A, the rst segments are more likely to be played out without failures than later ones. The results are reported in Figure 5. The curve for 1= exhibits the poorest performance. This was expected, since b N =b > A in this case. Note however that the accuracy of the bound is not good for small Probability of success Segment duration Play 60, FF=30, A=1.9 Play 45, FF=9, A=1.4 Play 60, FF=30, A=1.8 Play 60, FF=30, A=1.7 Play 45, FF=9, A=1.3 Fig. 5. Lower bounds on the probability of success E [L]=N for segments of the whole video. values of the segment length (see Table 2), and that the probability of success is actually larger than 80%. The other curves exhibit a probability of success larger than 85% for 1= (which is just slightly larger than b N =b), and larger than 95% for the three other sets of parameters. The curves with almost coincide. The experiments show that choosing a parameter A only slightly larger than the expected consumption rate of the user, coupled with su-ciently large segment sizes, achieves a very reasonable success probability. The accuracy of the bound (7) is not good in relative terms, as demonstrated in Table 2. In this table, the bound is compared with values obtained by simulating a million replications of a playout of the entire video. The relative accuracy improves when D increases; this is explained by the fact that the law of large numbers has more eects when segments are longer. The accuracy is however su-cient to assess the e-ciency of the rate increase technique, and may be used to optimize the parameter A, in a compromise between the probability of success and the bandwidth requirements. Such an optimization is outside the scope of this paper. Table Comparison of the lower bound on the success probability (B) with simulations (S); 1= 9. 200 0.9602 0.9837 200 0.5226 0.8099 300 0.9794 0.9898 300 0.6132 0.8110 700 0.9989 0.9993 700 0.8716 0.9021 Conclusions We have shown how by increasing the segment transmission rates for the tailored transmission scheme one can provide either deterministic or probabilistic support of user interactions. Since the FF action is the most \challenging" one to support, we restricted our analysis to a viewing behavior where only PLAY and FF are allowed. We rst derived deterministic guarantees for satisfying all possible FF actions. Since the deterministic guarantees were based on the pessimistic assumption that the user watches the whole video from start to end in FF mode, we then dened a model for the viewing behavior (S-FF model) that consists of the user alternating between the PLAY and the FF modes. For the S-FF model, we derived an analytic expression for a lower bound on the success probability. The reception of the segments is modeled as a semi-Markov accumulation process that allows the computation of the amount of video data received. While supporting VCR functions (and in particular FF) requires an increase in the segment transmission rates, our results indicate that this increase remains \moderate". The analytical results obtained for the S-FF are still pessimistic ones in the sense that a user who executes not only PLAY and FF but also actions such as PAUSE of SF will reduce the rate at which the video is consumed compared to the case of the S-FF model. In future extensions of this research, we shall exploit the theoretical formalism for accumulation processes that we have developed in this paper in order to handle various user's behavior and other VCR functions. --R Providing unrestricted VCR functions in multicast video-on-demand servers The role of multicast communication in the provision of scalable and interactive video-on-demand service Tailored transmissions for e-cient near-video-on- demand service The Theory of Stochastic Processes. Channel allocation under batching and VCR control in video-on-demand systems Stochastic Processes. Providing interactive functions through active client bu Supplying instantaneous video-on-demand services using controlled multicast Matrix Analysis. The split and merge protocol for interactive video on demand. Multicast delivery for interactive video-on-demand service Multicast with cache (Mcache): An adaptive zero-delay Video-on-Demand service Schaum's Outline of Theory and Problems of Laplace Transforms. Pyramid broadcasting for video on demand service. --TR Matrix analysis Channel allocation under batching and VCR control in video-on-demand systems The Split and Merge Protocol for Interactive Video-on-Demand Providing Interactive Functions through Active Client-Buffer Management in Partitioned Video Multicast VoD Systems An Efficient Implementation of Interactive Video-on-Demand Supplying Instantaneous Video-on-Demand Services Using Controlled Multicast Multicast Delivery for Interactive Video-On-Demand Service Providing Unrestricted VCR Functions in Multicast Video-on-Demand Servers

interactivity;semi-Markov accumulation process;stochastic bounds;video on demand

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Challenges of component-based development.

It is generally understood that building software systems with components has many advantages but the difficulties of this approach should not be ignored. System evolution, maintenance, migration and compatibilities are some of the challenges met with when developing a component-based software system. Since most systems evolve over time, components must be maintained or replaced. The evolution of requirements affects not only specific system functions and particular components but also component-based architecture on all levels. Increased complexity is a consequence of different components and systems having different life cycles. In component-based systems it is easier to replace part of system with a commercial component. This process is however not straightforward and different factors such as requirements management, marketing issues, etc., must be taken into consideration. In this paper we discuss the issues and challenges encountered when developing and using an evolving component-based software system. An industrial control system has been used as a case study.

Overview ABB is a global electrical engineering and technology company, serving customers in power generation, transmission and distribution, in industrial automation products, etc. The ABB group is divided into companies, one of which, ABB Automation Products AB, is responsible for development of industrial automation products. The automation products encompass several families of industrial process-control systems including both software and hardware. The main characteristics of these products are reliability, high quality and compatibility. These features are results of responses to the main customers requirements: The customers require stable products, running around the clock, year after year, which can be easily upgraded without impact on the existing process. To achieve this, ABB uses a component-based system approach to design extendable and flexible systems. The Advant Open Control System (OCS) (ABB, 2000) is component-based to suit different industrial applications. The range includes systems for Power Utilities, Power Plants and Infrastructure, Pulp and Paper, Metals and Minerals, Petroleum, Chemical and Consumer Industries, Transportation systems, etc. An overview of the Advant system is shown in Figure 1. Business System Information Management Station Operator Station Process Controller Process Controller Process Controller Figure 1. An overview of the conceptual architecture of the Advant open control system. Advant OCS performs process control and provides business information by assembling a system of different families of Advant products. Process information is managed at the level of process controllers. The process controllers are based on a real-time operating system and execute the control loops. The Operator Station (OS) and Information Management Station (IMS) gather and product information, while the business system provides analysis information for optimization of the entire processes. Advant products use standard and proprietary communication protocols to satisfy real-time requirements. Advant OCS therefore includes information management functions with real-time insight into all aspects of the process controlled. Advant Information Management has an SQL-based relational database accessible to resident software and all connected computers. Historical data acquisition reports, versatile calculation packages and an application programming interface (API) for proprietary and third party applications are examples of the functionality provided. Advant components have access to process, production and quality data from any Process Control unit in a plant or in an Intranet domain. Designing with Reuse Designing with reuse of existing components has many advantages (Sommerville, 1996). The software development time can be reduced and the reliability of the products increased. These were important prerequisites for the Advant OCS development. Advant OCS products can be assembled in many different configurations for use in various branches of industry. Specific systems are designed with the reuse of Advant OCS products and other external products. This means customers get a tailor-made system that meets their needs. External products and components can be used together with the Advant OCS due to the openness of the system. For example a satellite communication component, which is used to transmit data from the offshore station to the supervision system inland, can be integrated with the Advant OCS. The Advant system architecture is designed for reuse. Different products such as Operator and Information Management Stations are used as system components in assembling complete systems. The two operator station versions, Master OS and MOD OS are used in building different types of operator applications. Scalability Advant OCS can be configured in a multitude of ways, depending on the size and complexity of the process. The initial investment can consist of stand-alone process controllers and, optionally, local operator stations for control and supervision of separate machines and process sections. Subsequently, several process controllers can be interconnected and, together with central operator and information management stations build up a control network. Several control networks can be interconnected to give a complete plant network which can share centrally located operator, information and engineering workplaces. Openness The system is further strengthened by the flexibility to add special hardware and software for specific applications such as weighing, fixed- and variable-speed motor drives, safety systems and product quality measurements and control in for example the paper industry. Second- and third party administrative, information, and control can also be easily incorporated. Cost-effectiveness The step-by-step expansion capability of Advant OCS allows users to add new functionality without making existing equipment obsolete. The system's self-configuration capability eliminates the need for engineers to enter or edit topology descriptions when new stations are physically installed. New units can be added while the system is in full operation. With Advant OCS, system expansion is therefore easy and cost-effective. Reusable Components The Advant OCS products are component based to minimize the cost of maintenance and development. Figure 2 shows the component architecture of the operator station assembled from components. Figure 2. The operator station is assembled from Operator Station Functional Components Object Management OMF OS-Base functions Real-time operating System User Interface Component Library Standard Operating System components. The operator station consists of a specific number of functional components and of a set of standard Advant components. These components use the User Interface System (UIS) component. Object Management Facility (OMF) is a component which handles the infrastructure and data management. OMF is similar to CORBA (OMG, 2000) in that it provides a distributed object model with data, operation and event services. The UxBase component provides drivers and other specific operating system functions. Helper classes for strings, lists, pointers, maps and other general-purpose classes are available in the C++_complib library component. The components are built upon operating systems, one, a standard system(such as Unix or Windows), and the other a proprietary real-time system. To illustrate different aspects of component-based development and maintenance, we shall further look at two components: Object Management Facility (OMF), a business type of component with a high-level of functionality and a complex internal structure; - C++_complib is a basic and a very general library component. Object Management Facility (OMF) OMF (Nbling et al., 1999) is object-oriented middle-ware for industrial process automation. It encapsulates real-time process control entities of almost every conceivable description into objects that can be accessed from applications running on different platforms, for example Unix and Windows NT. Programming interfaces are available for many languages such as C, C++, Visual Basic, Java, Smalltalk and SQL while interfaces to the IEC 1131-3 (IEC, 1992) process control languages are under development. OMF is also adapted to Microsoft Component Object Model (COM) via adapters and another component called OMF COM aware. The adapters for OPC (OLE for Process Control) (OPC, 1998) and OLE Automation are also implemented. Thanks to all these software interfaces, OMF makes process and production data available to the majority of computer programmers and users i.e. even to those not necessarily involved in the industrial control field. For instance, it is easy to develop applications in Microsoft Word, Excel and Access to access process information. OMF has been developed for demanding real-time applications, and incorporates features, such as real-time response, asynchronous communications, standing queries and priority scheduling of data transfers. On one side OMF provides industry-standard interfaces to software applications, and on the other, it offers interfaces to many important communication protocols in the field including MasterNet, MOD DCN, TCP/IP and Fieldbus Foundation. These adapters make it possible to build homogeneous control systems out of heterogeneous field equipment and disparate system nodes. OMF reduces the time and cost of software development by providing frameworks and tools for a wide range of platforms and environments. These utilities are well integrated into their respective surroundings, allowing developers to retain the tools and utilities they prefer to work with. C++_complib C++_complib is a class library that contains general-purpose classes, such as containers, string management classes, file management classes, etc. The C++_complib library was developed when no standard libraries, such as were available on the market. The main purpose of this library was to improve the efficiency and quality, and promote the uniform usage of the basic functions. C++_complib is not a component according to the definition in (Szyperski, 1998), where a component is a unit of composition deployed independently of the product. However, in a development process C++_complib is treated in a very similar way as binary components with some restrictions, such dynamic configuration. Experience The Advant system is a successful system and the main reasons for its success are its component-based architecture giving flexibility, robustness, stability and compatibility, and effective build and integration procedures. This type of architecture is similar to product line architectures (Bass et al., 1999). Some case studies (Bosch, 1999) have shown that product-line architectures are successfully applied in small- and medium-sized enterprises although there exists a number of problems and challenges issues (organization, training, information distribution, product variants, etc. The Advant experience shows that applying of product-line architectures can be successful for large organizations. However, the cost of achieving these features has been high. To suit the requirements of an open system, new ABB products have always to be backward compatible. It would have been easier to develop a new system that not required being compatible with the previous systems. A guarantee that the system is backward compatible is a warranty that an existing system will work with new products and this makes the system trustworthy. Development with large components which are easy to reuse increases the efficiency significantly as compared with reusing a smaller component that could have been developed in-house at the same cost as its purchase price. Advant OCS products are examples of large components which have been used to assemble process automation systems. 3 Different Reuse Challenges Component generality and efficiency Reuse principles place high demands on reusable components. The components must be sufficiently general to cover the different aspects of their use. At the same time they must be concrete and simple enough to serve a particular requirement in an efficient way. Developing a reusable component requires three to four times more resources than developing a component, which serves a particular case (Szyperski, 1998). The fact that the requirements of the components are usually incomplete and not well understood (Sommerville, 1996) brings additional level of complexity. In the case of C++_complib, the situation was simpler, because the functional requirements were clear. It was relatively easy to define the interface, which was used by different components in the same way. The situation was more complicated with complex components, such as OMF. Although the basic concept of component functionality was clear, the demands on the component interface and behavior were different in different components and products. Some components required a high level of abstraction, others required the interface to be on a more detailed level. These different types of requirements have led to the creation of two levels of components: OMF base, including all low-level functions, and OMF framework, containing only a higher level of functions and with more pre-defined behavior and less flexibility. In general, requirements for generality and efficiency at the same time lead to the implementation of several variants of components which can be used on a different abstraction level. In some specific cases, a particular solution must be provided. This type of solution is usually beyond the object-oriented mechanisms, since such components are on the higher abstraction level. System Evolution Long-life products are most often affected by evolution of different kinds: - Evolution of system requirements, functional and non- functional. A consequence of a continually competitive market situation is a demand for continually improved system performance. The systems controlling and servicing business, industrial, and other processes should permanently increase the efficiency of these processes, improve the quality of the products, minimize the production and maintenance costs etc. - Evolution of technology related to different domains. The advance of technology in the different fields in which software is used requires improved software. The improvements may require a completely new approach to or new functions in software. - Evolution of technology used in software products. Evolution in computer hardware and software technology is so fast that an organization manufacturing long-life and complex products must expect significant technology changes during the product life cycle. From the reliability and risk point of view, such organizations prefer not to use the latest technology, but because of the demands of a highly competitive market, are forced to adopt new technology as it appears. The often unpredictable changes which must be made in products cause delivery delays and increased production costs. - Evolution of technology used for the product development. As in the case of products themselves, new technology and tools used in the development process appear frequently on the market. Manufacturers are faced with a dilemma - to adopt the new technology and possibly improve the development process at the risk of short term higher costs (for training and migration), or to continue using the existing technology and thereby miss an opportunity to lower development costs in the long run. - Evolution of society. Changes in society (for example environmental requirements, or changes in the relations between countries - as in the EU) can have a considerable impact on the demands on products (for example new standards, new currency, etc.) and on the development process (relations between employers and employees, working hours, etc. Business Changes. We face changes in government policies, business integration processes, deregulation, etc. These changes have an impact on the nature of business, resulting, for examples, in a preference for short-term planning rather than long-term planning and more stringent time-to-market requirements. Organizational Changes. Changes in society and business have direct effects on business organizations. We can see a globalization process, more abrupt changes in business operations and a demand for more flexible structures and management procedures, just- in-time deliveries of resources, services and skills. These changes require another, fast and flexible approach to the development process. All these changes have a direct or indirect impact on the product life cycle. The ability to adapt to these changes becomes the crucial factor in achieving business success (Brown, 2000). In the following sections we discuss some of these changes and their consequences in the development process and product life cycle. Evolution of Functional Requirements The development of reusable components would be easier if functional requirements did not evolve during the time of development. As a result of new requirements for the products, new requirements for the components will be defined. The more reusable a component is, the more demands are placed on it. A number of the requirements coming from different products, may be the same or very similar, but this is not necessarily the case for all requirements passed to the components. This means that the number of requirements of reusable components grow faster than of particular products or of a non-reusable piece of software. The relation between component requirements and the requirements from the products is expressed with the following equation: RC0 denotes direct requirements of the component, Rpi requirements of the products Pi , ai impact factors to the component and RC is the total number of the component requirements. To satisfy these requirements the components must be updated more rapidly and the new versions must be released more frequently than the products using them. The process of the change of components is more dynamic in the early stage of the components lives. In that stage the components are less general and cannot respond to the new requirements of the products without being changed. In later stages, their generality and adaptability increase, and the impact of the product requirements become less significant. In this period the products benefit from combinatorial and synergy effects of components reuse. In the last stage of its life, the components are getting out-of-date, until they finally become obsolete, because of different reasons: Introduction of new techniques, new development and run-time platforms, new development paradigms, new standards, etc. There is also a higher risk that the initial component cohesion degenerates when adding many changes, which in turn requires more efforts. This process is illustrated in Figure 3. The first graph shows the growing number of requirements for certain products and for a component being used by these products. The number of requirements of a common component grows faster in the beginning, saturates in the period [t0- t1], and grows again when the component features become inadequate. Some of the product requirements are satisfied with new releases of products and components, which are shown as steps on the second graph. The component implements the requirements by its releases, which normally precede the releases of the product if the requirements originated from the product requirements. Accumulated Requirements Component Product P2 Product P1 t-0 t-1 Time Requirements satisfied in the releases Component Product P2 Product P1 t-0 t-1 Time Figure 3. To satisfy the requirements the reusable component must be modified more often in the beginning of their life. Indeed this was the case with both components we are analyzing here: New functions and classes were required from C++_complib, and new adapters and protocol support were required from OMF. The development time for these components was significantly shorter than for products: While new versions of a product are typically released every six months, new versions of components are released as least twice as often. After several years of intensive development and improvement, the components became more stable and required less effort for new changes. In that period the frequency of the releases has been lowered, and especially the effort has been significantly lower. New efforts for further development of components appeared with migration of products on different platforms and newer platforms versions. Although the functions of the products and components did not changed significantly a considerable amount of work was done on the component level. Migration Between Different Platforms During their several years of development, Advant products have been ported to different platforms. The reasons for this were the customer requirement, that the products should run on specific platforms, and general trends in the growing popularity of certain operating systems. Of course, at the same time, new versions and variants of the platform already used appeared, supporting new, better and cheaper hardware. The Advant products have migrate through different platforms: Starting on Unix HP-UX 8.x and continuing trough new releases (HP-UX 9.x, 10.x ), they have been ported to other Unix platforms, such as Digital Unix, and also to complete different platforms, such as Open VMS and Windows NT family (NT 3.5, NT 4.0 and Windows 2000). The products have been developed and maintained in parallel. The challenge with this multi-platform development was to keep the compatibility between the different variants of the products, and to maintain and improve them with the minimal efforts. As an important part of the reuse concept was to keep the high-level components unchanged as far as possible, it was decided to encapsulate the differences between operating systems in low-level components. This concept works, however, only to some extent. The minimal activity required for each platform is to rebuild the system for that platform. To make it possible to rebuild the software on every platform, standard-programming languages C and C++ have been used. Unfortunately, different implementations of the C++ standard in different compilers, caused problems in the code interpretation and required the rewriting of certain parts of the code. To ensure that standard system services are available on all platforms, the POSIX standard has been used. POSIX worked quite well on different Unix platforms, but much less so on Windows NT. The second level of compatibility problem was Graphical User Interface (GUI). The main dilemma was whether to use exactly the same GUI on every platform, or to use the standard "look and feel" GUI for each platform. This question applied particularly on NT in relation to Unix platforms. Experience has shown that it is not possible to give a definitive answer. In some cases it was possible to use the same GUI and the same graphical packages, but in general, different GUIs were implemented. The main work regarding the reuse of code on different platforms was performed on low-level components, such as UxBase and OMF. While UxBase provides different low-level packages for every platform (for example different drivers), OMF capsulated the differences directly in the code using conditional compilation. OMF itself is designed in such a way that it was possible to divide the code into two layers. One layer is specific for each operating system, and the other layer, with the business logic, is implemented for all of the supported platforms. Reuse issues on different platforms for C++_complib were easier, strictly the package contains general algorithms, which are not depending on specific operating system. Some problems appeared however, related to different characteristics of compilers on different platforms. Compatibility One of the most important factors for successful reusability is the compatibility between different versions of the components. A component can be replaced easily or added in new parts of a system if it is compatible with its previous version. The compatibility requirements are essential for Advant products, since smooth upgrading of systems, running for many years, is required. Compatibility issues are relative simple when changes introduced in the products are of maintenance and improvement nature only. Using appropriate test plans, including regression tests, functional compatibility can be tested to a reasonable extent. More complicated problems occur when new changes introduced in a reusable component eliminate the compatibility. In such a case, additional software, which can manage both versions, must be written. A typical example of such an incompatible change, is a change in the communication protocol between OMF clients and servers. All different versions of OMF must be able to talk to each other to make the system flexible and open. It is possible to have different combinations of operating systems and versions of OMF and it still works. This has been solved with an algorithm that ensures the transmission of correct data format. If two OMF nodes have the same version, they talk in their native protocol. If an old OMF node talks with a new, the new OMF is responsible for converting the data to the new format, this being designated RMIR ("receiver makes it right"). If a new OMF sends data to an older, the older OMF can not convert the data since it is unaware of the new protocol. In this case the newer OMF must send in the old protocol format, SMIR ("sender makes it right"). This algorithm builds on that fact all machines know about each other and that they also know what protocol they talk. However, if an OMF-based node does not know of the other node then it can always send in a predefined protocol referred to as well known format. All nodes do recognize this protocol and can translate from it. This algorithm minimizes the number of data conversions between the nodes. In the case of C++_complib the problems with compatibility were somewhat different. New demands on the same classes and functions appeared because of new standards and technology. One example is the use of C++ templates. When the template technology became sufficiently mature, the new requirements were placed for C++_complib: All the classes were to be re-implement as template classes. The reason for this was the requirement for using basic classes in a more general and efficient way. Another example is Unicode support in addition to ASCII- support. These new functions were added by new member- functions in the existing classes and by adding new classes using the inheritance mechanism for reusing the already existing classes. The introduction of the same functions in different format have led to additional efforts in reusing them. In most of the cases the old format has been replaced by new one, with help of simple tools built just for this purpose. In some other cases, due to non-proper planning and prioritizing the time-to-market requirements, both old and new formats have been used in the same source modules which have led to lower maintainability and to some extend to lower quality of the products. Development Environment When developing reusable components several dimensions of the development process must be considered: - Support for development of components on different platforms; - Support for development of different variants of components for different products; - Support for development and maintenance of different versions of components for different product versions. Independent development of components and products. To cope with these types of problems, it is not sufficient to have appropriate product architecture and component design. Development environment support is also essential. The development environment must permit an efficient work in the project - editing, compiling, building, debugging and testing. Parallel and distributed development must also be supported, because the same components are to be developed and maintained at the same time on different platforms. This requires the use of a powerful Configuration Management (CM) tool, and definition of an advanced CM-process. The CM process support exists on two levels. First on the source-code level, where source-code files are under version management and binary files are built. The second level is the product integration phase. The product built must contain a consistent set of the component versions. For example, Figure 4 shows an inconsistent set of components. The product version P1-V2 uses the component versions C1-V2 and C2-V2. At the same time the component version C1-V2 uses the component version C2-V1, an older version. Integrating different versions of the same component may cause unpredictable behavior of the product. C2 Version V1 C2 Version V2 Figure 4. An example of inconsistent component integration. Another important aspect of CM in developing reusable components is Change Management. Change management keeps track of changes on the logical level, for example error reports, and manages their relations with implemented physical changes (i.e. changes of documentation, source code, etc. Because change requests (for example functional requirements or error reports) come from different products, it is important to register information about the source of change requests. It is also important to relate a change request from one product to other products. The following questions must be answered: What impact can the implemented change have on other products? If an error appears in one product, does it appear in other products? Possible implications must be investigated, and if necessary, the users of the products concerned must be informed. The development environment designated Software Development Environment (SDE) (Crnkovic, 1997) is used in developing Advant products. It is an internally-built program package which encapsulates different tools, and provides support for parallel development. The CM tool, based on RCS (Tichy, 1985), provides support for all CM disciplines, such as change management, works pace management, build management, etc. SDE runs on different platforms, with slightly modified functions. For example, the build process is based on Makefiles and autoconf on Unix platforms, while Microsoft Developer Studio with additional Project Settings is used on Windows NT. The main objective of SDE is to keep the source-code in one place under version control. Different versions of components are managed using baselines, and change requests. Change requests are also under version control, which gives a possibility of acquiring information useful for project follow-up, for every change from registration to implementation and release (Crnkovic and Willfr, 1998). Independent Component Development Component development independent of the products gives several advantages. The functions are broken down in smaller entities that are easier to construct, develop and maintain. The independent component development facilitates distributed development, which is common in large enterprises. Development of components independently of product or other component development introduces also a number of problems. The component and product test become more difficult. On the component level, a proper test environment must be built, which often must include a number of other components or even maybe the entire product. Another problem is the integration and configuration problem. A situation shown on Figure 4 must be avoided. When it is about complex products, it is impossible to manually track dependencies between the components, but a tool support for checking consistency must exist. In the Advant development the components were treated as separate products even if they were developed within the enterprise. To have this approach helped when third party components were used since they all were managed in a uniform way. Every component contained a file called import file that included a specification of all component versions used to build the component. When the final product was assembled from the components, the import file has been used for integration and checking if the consistent sets of the components have been selected. The development environment, based on make, was set up to use the import files and the common product structure. All released components were stored in the product structure for availability to others. Another structure was used during development of a component. The component was exported to the product structure when the development was finished. Using this approach it was shown that the architecture design plays a crucial role. A good architecture with clear and distinguish relations between components facilitate the development process. The whole development process is complex and requires organized and planned support, which is essential for efficient and successful development of reusable components and of applications using these. The Maintenance Process The maintenance process is also complex, because it must be handled on different levels: On the system level, where customers report their problems, on the product level, where errors detected in a specific product version are reported, and finally on the component level, where the fault is located. The modification of the component can have an impact on other components and other products, which can lead to an explosion of new versions of different products which already exist in several versions. To minimize this cumbersome process, ABB adopted a policy of avoiding the generation of and supply of specific patches to selected customers. Instead, revised products incorporating sets of patches were generated and delivered to all customers with maintenance contracts, to keep customer installations consistent. The relations between components, products and systems must be carefully registered to make possible the tracing of errors on all levels. A systematic use of Software Configuration Management has a crucial role in the maintenance process. To support the maintenance process, Advant products and component specifications together with error reports are stored in several classes of repositories (see Figure 5). External Customers Customer complaints Direct Action Report to Customers PMR Direct Action PMR Released products Beta release PPR PPR Development CR Figure 5. Different levels of error report management On the highest level, the repository managing customers reports (CCRP) makes it possible for service personnel to provide customers with prompt support. Information saved on this level is customer and product oriented. Reports indicating a product problem are registered in the product maintenance report repository (PMR) where all known problems related to products and components are filed. Also, product structure information is stored on this level. The product structure, showing dependencies between products and components provides product and component developers with assistance in relating error reports to the source of the problem, on both product and component level. A similar error management process is defined for products in the beta phase i.e. not yet released. All of the problems identified in this phase (typically by test groups) are registered in the form of pre-release problem reports (PPR). These problems are either solved before the product is released, or are reclassified as product error reports and saved in PMR. Any change applied in code or documentation is under change control, and each change is initiated by a Change Request. If a change required comes from an error report, a Change Request will be generated from a PMR. When a change made in a component is tested and verified, the action description is exported to the correlating PMR, propagated to the products involved and finally returned to the customer via the CCRP repository. This procedure is not unique to component-based development. It is a means of managing complex products and of maintaining many products. What is specific to the component-based approach is the mapping between products and components and the management of error reports on product and component level, the most difficult part of the management. In this case the entire procedure is localized on the PMR level, i.e. product level. On the customer side, information with the highest priority is related to products and customers. On the development level, all changes registered are related primarily to components. Information about both products and components is stored on the development level. Error management on this level is the most complex. An error may be detected in a specific product version, but may also be present in other products and other product versions. The error may be discovered in one component, but it can be present in different versions of that component. The problem can be solved in one component version, but it also may be necessary to solve it in several. The revised component versions are eventually subsequently integrated in new versions of one or several products. This multidimensional problem (many error reports, impact on different versions of components and products, the solution included in different components and product versions) is only partially managed automatically, as many steps in the process require direct human decisions (for example a decision if a solution to a problem will or will not be included in the next product release). Although the whole procedure is carefully designed and rigorously followed, it has happened on occasions that unexpected changes have been included, and that changes intended for inclusion were absent from new product releases. For more details of the maintenance process see (Kajko-Mattson, 1999a, Kajko-Mattson, 1999b)). Another important subject is the maintenance of external components. It has been shown that external components must be treated in the same way as internal components. All known errors and the complete error management process for internal and external components are treated in similar way. The list of known, and corrected errors in external components is important for developers, product managers and service people. The cost of maintaining components, even those maintained by others, must be taken into consideration. Integrating Standard Components In recent years the demands of customers on systems have changed. Customers require integration with standard technologies and the use of standard applications in the products they buy. This is a definite trend on the market but there is little awareness of the possible problems involved. An improper use of standard components can cause severe problems, especially in distributed real-time and safety-critical systems, with long-period guarantees. In addition to these new requirements, time-to-market demands have become a very important factor. These factors and other changes in software and hardware technology (Aoyama, 1998) have introduced a new paradigm in the development process. The development process is focused now on the use of standard and de-facto standard components, outsourcing, COTS and the production of components. At the same time, final products are no longer closed, monolith systems, but are instead component-based products that can be integrated with other products available on the market. This new paradigm in the development process and marketing strategy has introduced new problems and raised new questions (McKinney, 1999): - The development process has been changed. Developers are now not only designers and programmers, they are also integrators and marketing investigators. Are the new development methods established? Are the developers properly educated? - What are the criteria for the selection of a component? How can we guarantee that a standard component fulfills the product requirements? - What are the maintenance aspects? Who is responsible for the maintenance? What can be expected of the updating and upgrading of components? How can we satisfy the compatibility and reliability requirements? - What is the trend on the market? What can we expect to buy not only today but also on the day we begin delivering our product? When developing a component, how can we guarantee that the "proper" standard is used? Which standard will be valid in five, ten years? All these questions must be considered before beginning a component-based development project. Josefsson (Josefsson, 1999) presents certain recommendations to the component integrator for use as guidelines: Test the imported component in the environment where it is to run and limit the practical number of component suppliers to minimize the compatibility problems. Make sure that the supplier is evaluated before a long-term agreement is signed. The focus of development environment support should be transferred from the edit-build-test cycle to the component integration-test cycle. Configuration management must give more consideration to run-time phase (Larsson and Crnkovic, 1999). Replacing Internal Components with Standard Components In the middle of the eighties, ABB Advant products were completely proprietary systems with internally developed hardware, basic and application software. In the beginning of the nineties, standard hardware components and software platforms were purchased while the real-time additions and application software were developed internally. The system is now developed further using components based on new, standard technologies. During this development, further new components become available on the market. ABB faced this issue more than once. At one point in time, it was necessary to abandon the existing solutions in a favor of new solutions based on existing components and technologies. To illustrate the migration process we discuss the possibility of replacing OMF and C++_complib with standard components. Experience from these examples showed that it is easier to replace a component if the replacement process is made in small incremental steps. Allowing the new component to coexist with the old one makes it easier to be backward compatible and the change will be smooth. Replacing OMF with DCOM Moving from a UNIX based system to a system based on Windows NT had serious effect on the system architecture. Microsoft components using a new object model were available, namely COM/DCOM (Box, 1998). DCOM has functionality similar to that of OMF and this became a new issue when DCOM was released. Should ABB continue to develop its proprietary OMF or change to a new standard component? The problem was that DCOM did not have all the functionality of OMF and vice versa. The domains overlap only partially. A subscription of data with various capabilities can be made in OMF, and this subscription functionality is not supported by DOCM. On the other hand, DCOM can create objects when they are required and not like OMF where objects are created before the actual use of them. Both technologies support object communication and in this area it is easier to replace OMF with DCOM. If the decision was made to continue with OMF, all the new components that run on top of COM could not be used, which would drastically reduce the possibilities of integration with other, third-party components. On the other hand, it would require considerable work to make the current system run on top of COM. This was the dilemma of COM vs. OMF. To begin with, OMF was adapted to COM with an adapter designated OMF COM aware. This functionality helped COM developers access OMF objects and vice versa. However, this solution to the problem using two different object models was not optimal since it added overhead in the communication. Nor was it possible to match the data types one to one, which made the solution limited. A decision was taken to build the new system on COM technologies with proprietary extensions adding the functions missing from COM. All communication with the current system was to be through the OMF COM. This solution made it easy to remove the old OMF and replace it with COM in small steps over time. Adapters are very useful when a new component is to used in parallel with an existing one (Rine et al., 1999). More adapters to other systems such as Orbix(CORBA) and Fieldbus Foundation were constructed. If the external systems have similar data types it is fairly straightforward to build a framework for adapters where the parts that take care of the proprietary system can be reused. New systems can be accessed by adding a server and client stub to the adapter framework. To be able to build functional adapters between two middleware components it is important to have the capability to create remote calls dynamically. For instance the Dynamic Invocation Interface (DII) in CORBA can be used. If the middleware does not have this possibility it might be possible to generate code automatically that takes care of the different types of calls which are going to be placed through the adapter to the other system. Replacing C++_complib with STL To switch from C++_complib to STL (Austern, 1999) was much easier because STL covers almost all the C++_complib functions and provides additional functionality. Still, much work reminded to be done, since all the code using C++_complib had to be changed to be able to use STL instead. The decision was taken to continue using both components and to use STL whenever new functionality was added. After a time the use of old components was reduced and the internal maintenance cost reduced. In some cases in the same components both libraries were used, which gave some disadvantages, especially in the maintenance process. Managing Evolution of Standard Components Use of standard components implies less control on them (Larsson and Crnkovic, 1999, Larsson and Crnkovic, 2000, Cook and Dage, 1999), especially if the components are updated at run-time. A system of components is usually configured once only during the build-time when known and tested versions of components are used. Later, when the system evolves with new versions of components, the system itself has no mechanism to detect if new components have been installed. There might be a check that the version of replacement component is at least the same as or newer than the original version. This approach prevents the system from using old components, but it does not guarantee its functionality when new components are installed. Applying ideas from configuration management, such as version and change management, in managing components is an approach which can be used to solve some of the problems. A certain level of configuration control will be achieved when it is possible to identify components with their versions and dependencies to other components. Information about a system can be placed under version control for later retrieval. This makes it possible to compare different baselines of a system configuration. To manage dependencies, a graphic representation of the configuration is introduced. The graphs are then placed under version control. This information can be used to predict which components will be affected by a replacement or installation of a new component. It is generally difficult to identify components during run-time and to obtain their version information. When the components are identified it is possible to build graphs of dependencies, which can be represented in various ways and placed under configuration control (Larsson, 2000). To improve the control of external components, they can be placed under change management to permit the monitoring of changes and bugs. Instead of attaching source code files to change requests, which is common in change management, the name and version of the component can be used to track changes. When a problem report is analysed, the outcome can be a change request for each component involved. Each such change request can contain a list of all the changed source files or a description of the patches if the component is external. Patches from the component vendor must be stored to permit recreation of the same configuration later. In cases where the high quality of products must be assured, the enterprise developing products must have special, well-defined relations to the component vendors for the support and maintenance. 5 Conclusion We have presented the ABB Advant Control Systems (OCS) as a successful example of the development of a component-based system. The success of these systems on the market has been primarily the result of appropriate functionality and quality. Success in development, maintenance and continued improvement of the systems has been achieved by a careful architecture design, where the main principle is the reuse of components. The reuse orientation provides many advantages, but it also requires systematic approach in design planning, extensive development, support of a more complex maintenance process, and in general more consideration being given to components. It is not certain that an otherwise successful development organization can succeed in the development of reusable components or products based on reusable components. The more a reusable component is developed, the more complex is the development process, and more support is required from the organization. Even when all these requirements are satisfied, it can happen that there are unpredictable extra costs. One example illustrate this: In the early stage of the ABB Advant OCS development, insufficient consideration was given to Windows NT and ABB had to pay the price for this oversight when it suddenly became clear that Windows NT would be the next operating platform. The new product versions on the new platform have been developed by porting the software from the old platform, but the costs were significantly greater than if the design had been done more independent from the first platform. Another problem we have addressed, is the question of moving to new technologies which require the re-creation of the components or the inclusion of standard components available on the market. In both cases it can be difficult to keep or achieve the same functionality as the original components had. However, it seems that the process of replacing proprietary components by standard components available from third parties is inevitable and then it is important to have a proper strategy for migrating from old components to the new ones. 6 --R New Age of Software Development: How component-based Software Engineering Changes the way of Software Development Generic Programming and STL Third Product Line Practice Report Experience with Change-oriented SCM Tools Programvarukomponenter i praktiken Maintenance at ABB (I): Software Problem Administration Processes Maintenance at ABB (II): Change execution processes Applying Configuration Management Techniques to Component-Based Systems Licentiate Thesis Dissertation 2000-007 New Challenges for Configuration Management Component Configuration Management Impact of Commercial Off-The-Shelf (COTS) Software on the Interface Between systems and Software Engineering Using Adapters to Reduce Interaction Complexity in reusable Component-Based Software Development Software Engineering --TR RCSMYAMPERSANDmdash;a system for version control Component software Highly reliable upgrading of components Product-line architectures in industry Impact of commercial off-the-shelf (COTS) software on the interface between systems and software engineering Using adapters to reduce interaction complexity in reusable component-based software development Large-Scale, Component Based Development Software Engineering Experience with Change-Oriented SCM Tools Change Measurements in an SCM Process New Challenges for Configuration Management Maintenance at ABB (I) Maintenance at ABB (II) --CTR Ade Azurat, Mechanization of invasive software composition in F-logic, Proceedings of the 2007 annual Conference on International Conference on Computer Engineering and Applications, p.89-94, January 17-19, 2007, Gold Coast, Queensland, Australia Lin Chin-Feng , Tsai Hsien-Tang , Fu Chen-Su, A logic deduction of expanded means-end chains, Journal of Information Science, v.32 n.1, p.5-16, February 2006

architecture;commercial components;component-based development;development environment;reuse

605725

Data parallel language and compiler support for data intensive applications.

Processing and analyzing large volumes of data plays an increasingly important role in many domains of scientific research. High-level language and compiler support for developing applications that analyze and process such datasets has, however, been lacking so far.In this paper, we present a set of language extensions and a prototype compiler for supporting high-level object-oriented programming of data intensive reduction operations over multidimensional data. We have chosen a dialect of Java with data-parallel extensions for specifying a collection of objects, a parallel for loop, and reduction variables as our source high-level language. Our compiler analyzes parallel loops and optimizes the processing of datasets through the use of an existing run-time system, called active data repository (ADR). We show how loop fission followed by interprocedural static program slicing can be used by the compiler to extract required information for the run-time system. We present the design of a compiler/run-time interface which allows the compiler to effectively utilize the existing run-time system.A prototype compiler incorporating these techniques has been developed using the Titanium front-end from Berkeley. We have evaluated this compiler by comparing the performance of compiler generated code with hand customized ADR code for three templates, from the areas of digital microscopy and scientific simulations. Our experimental results show that the performance of compiler generated versions is, on the average 21% lower, and in all cases within a factor of two, of the performance of hand coded versions.

Introduction Analysis and processing of very large multi-dimensional scientic datasets (i.e. where data items are associated with points in a multidimensional attribute space) is an important component of science and engineering. Examples of these datasets include raw and processed sensor data from satellites [27], output from hydrodynamics and chemical transport simulations [23], and archives of medical images[1]. These datasets are also very large, for example, in medical imaging, the size of a single digitized composite slide image at high power from a light microscope is over 7GB (uncompressed), and a single large hospital can process thousands of slides per day. Applications that make use of multidimensional datasets are becoming increasingly important and share several important characteristics. Both the input and the output are often disk-resident. Applications may use only a subset of all the data available in the datasets. Access to data items is described by a range query, namely a multidimensional bounding box in the underlying multidimensional attribute space of the dataset. Only the data items whose associated coordinates fall within the multidimensional box are retrieved. The processing structures of these applications also share common characteristics. However, no high-level language support currently exists for developing applications that process such datasets. In this paper, we present our solution towards allowing high-level, yet ecient programming of data intensive reduction operations on multidimensional datasets. Our approach is to use a data parallel language to specify computations that are to be applied to a portion of disk-resident datasets. Our solution is based upon designing a prototype compiler using the titanium infrastructure which incorporates loop ssion and slicing based techniques, and utilizing an existing run-time system called Active Data Repository [8, 9, 10]. We have chosen a dialect of Java for expressing this class of computations. We have chosen Java because the computations we target can be easily expressed using the notion of objects and methods on objects, and a number of projects are already underway for expressing parallel computations in Java and obtaining good performance on scientic applications [4, 25, 36]. Our chosen dialect of Java includes data-parallel extensions for specifying collection of objects, a parallel for loop, and reduction variables. However, the approach and the techniques developed are not intended to be language specic. Our overall thesis is that a data-parallel framework will provide a convenient interface to large multidimensional datasets resident on a persistent storage. This research was supported by NSF Grant ACR-9982087, NSF Grant CCR- 9808522, and NSF CAREER award ACI-9733520. Conceptually, our compiler design has two major new ideas. First, we have shown how loop ssion followed by interprocedural program slicing can be used for extracting important information from general object-oriented data-parallel loops. This technique can be used by other compilers that use a run-time system to optimize for locality or communication. Second, we have shown how the compiler and the run-time system can use such information to eciently execute data intensive reduction computations. Our compiler extensively uses the existing run-time system ADR for optimizing the resource usage during execution of data intensive applications. ADR integrates storage, retrieval and processing of multidimensional datasets on a parallel machine. While a number of applications have been developed using ADR's low-level API and high performance has been demonstrated [9], developing applications in this style requires detailed knowledge of the design of ADR and is not suitable for application programmers. In comparison, our proposed data-parallel extensions to Java enable programming of data intensive applications at a much higher level. It is now the responsibility of the compiler to utilize the services of ADR for memory management, data retrieval and scheduling of processes. Our prototype compiler has been implemented using the titanium infrastructure from Berkeley [36]. We have performed experiments using three dierent data intensive application templates, two of which are based upon the Virtual Microscope application [16] and the third is based on water contamination studies [23]. For each of these templates, we have compared the performance of compiler generated versions with hand customized versions. Our experiments show that the performance of compiler generated versions is, on average, 21% lower and in all cases within a factor of two of the performance of hand coded versions. We present an analysis of the factors behind the lower performance of the current compiler and suggest optimizations that can be performed by our compiler in the future. The rest of the paper is organized as follows. In Section 2, we further describe the charactestics of the class of data intensive applications we target. Background information on the run-time system is provided in Section 3. Our chosen language extensions are described in Section 4. We present our compiler processing of the loops and slicing based analysis in Section 5. The combined compiler and run-time processing for execution of loops is presented in Section 6. Experimental results from our current prototype are presented in Section 7. We compare our work with existing related research eorts in Section 8 and conclude in Section 9. 2 Data Intensive Applications In this section, we rst describe some of the scientic domains which involve applications that process large datasets. Then, we describe some of the common characteristics of the applications we target. Data intensive applications from three scientic areas are being studied currently as part of our project. Analysis of Microscopy Data: The Virtual Microscope [16] is an application to support the need to interactively view and process digitized data arising from tissue specimens. The Virtual Microscope provides a realistic digital emulation of a high power light microscope. The raw data for such a system can be captured by digitally scanning collections of full microscope slides under high power. At the basic level, it can emulate the usual behavior of a physical microscope including continuously moving the stage and changing magnication and focus. Used in this manner, the Virtual Microscope can support completely digital dynamic telepathology. Water contamination studies: Environmental scientists study the water quality of bays and estuaries using long running hydrodynamics and chemical transport simulations [23]. The chemical transport simulation models reactions and transport of contaminants, using the uid velocity data generated by the hydrodynamics simulation. The chemical transport simulation is performed on a dierent spatial grid than the hydrodynamics simulation, and also often uses signicantly coarser time steps. To facilitate coupling between these two simulation, there is a need for mapping the uid velocity information from the hydrodynamics grid, averaged over multiple ne-grain time steps, to the chemical transport grid and computing smoothed uid velocities for the points in the chemical transport grid. Satellite data processing: Earth scientists study the earth by processing remotely-sensed data continuously acquired from satellite-based sensors, since a signicant amount of Earth science research is devoted to developing correlations between sensor radiometry and various properties of the surface of the Earth [9]. A typical analysis processes satellite data for ten days to a year and generates one or more composite images of the area under study. Generating a composite image requires projection of the globe onto a two dimensional grid; each pixel in the composite image is computed by selecting the \best" sensor value that maps to the associated grid point. Data intensive applications in these and related scientic areas share many common characteristics. Access to data items is described by a range query, namely a multidimensional bounding box in the underlying multidimensional space of the dataset. Only the data items whose associated coordinates fall within the multidimensional box are retrieved. The basic computation consists of (1) mapping the coordinates of the retrieved input items to the corresponding output items, and (2) aggregating, in some way, all the retrieved input items mapped to the same output data items. The computation of a particular output element is a reduction operation, i.e. the correctness of the output usually does not depend on the order in which the input data items are aggregated Another common characteristic of these applications is their extremely high storage and computational requirements. For example, ten years of global coverage satellite data at a resolution of four kilometers for our satellite data processing application Titan consists of over 1.4TB of data [9]. For our Virtual Microscope application, one focal plane of a single slide requires over 7GB (uncompressed) at high power, and a hospital such as Johns Hopkins produces hundreds of thousands of slides per year. Similarly, the computation for one ten day composite Titan query for the entire world takes about 100 seconds per processor on the Maryland sixteen node IBM SP2. The application scientists typically demand real-time responses to such queries, therefore, e-cient execution is extremely important. 3 Overview of the Run-time System Our compiler eort targets an existing run-time infrastructure, called the Active Data Repository (ADR) [9] that integrates storage, retrieval and processing of multidimensional datasets on a parallel machine. We give a brief overview of this run-time system in this section. Processing of a data intensive data-parallel loop is carried out by ADR in two phases: loop planning and loop execution. The objective of loop planning is to determine a schedule to e-ciently process a range query based on the amount of available resources in the parallel machine. A loop plan species how parts of the nal output are computed. The loop execution service manages all the resources in the system and carries out the loop plan generated by the loop planning service. The primary feature of the loop execution service is its ability to integrate data retrieval and processing for a wide variety of applications. This is achieved by pushing processing operations into the storage manager and allowing processing operations to access the buer used to hold data arriving from the disk. As a result, the system avoids one or more levels of copying that would be needed in a layered architecture where the storage manager and the processing belong in dierent layers. A dataset in ADR is partitioned into a set of (logical) disk blocks to achieve high bandwidth data retrieval. The size of a logical disk block is a multiple of the size of a physical disk block on the system and is chosen as a trade-o between reducing disk seek time and minimizing unnecessary data transfers. A disk block consists of one or more objects, and is the unit of I/O and communication. The processing of a loop on a processor progresses through the following three phases: (1) Initialization { output disk blocks (possibly replicated on all processors) are allocated space in memory and initialized, (2) Local Reduction { input disk blocks on the local disks of each processor are retrieved and aggregated into the output disk blocks, and (3) Global Combine { if necessary, results computed in each processor in phase 2 are combined across all processors to compute nal results for the output disk blocks. ADR run-time support has been developed as a set of modular services implemented in C++. ADR allows customization for application specic processing (i.e., mapping and aggregation functions), while leveraging the commonalities between the applications to provide support for common operations such as memory management, data retrieval, and scheduling of processing across a parallel machine. Customization in ADR is currently achieved through class inheritance. That is, for each of the customizable services, ADR provides a base class with virtual functions that are expected to be implemented by derived classes. Adding an application-specic entry into a modular service requires the denition of a class derived from an ADR base class for that service and providing the appropriate implementations of the virtual functions. Current examples of data intensive applications implemented with ADR include Titan [9], for satellite data processing, the Virtual Microscope [16], for visualization and analysis of microscopy data, and coupling of multiple simulations for water contamination studies [23]. 4 Java Extensions for Data Intensive Computing In this section, we describe a dialect of Java that we have chosen for expressing data intensive computations. Though we propose to use a dialect of Java as the source language for the compiler, the techniques we will be developing will be largely independent of Java and will also be applicable to suitable extensions of other languages, such as C, C++, or Fortran 90. 4.1 Data-Parallel Constructs We borrow two concepts from object-oriented parallel systems like Titanium [36], HPC++ [5], and Concurrent Aggregates [11]. Interface Reducinterface f *Any object of any class implementing this interface is a reduction variable* public class VMPixel f char colors[3]; void Initialize() f *Aggregation Function* void Accum(VMPixel Apixel, int avgf) f public class VMPixelOut extends VMPixel implements Reducinterface; public class VMScope f static int static int Ydimen = . ; static *Data Declarations* static static static new VMPixel[VMSlide]; public static void main(String[] args) f int lowend)/subsamp]; foreach(p in Outputdomain) f *Main Computational Loop* foreach(p in querybox) f Fig. 1. Example Code Domains and Rectdomains are collections of objects of the same type. Rect- domains have a stricter denition, in the sense that each object belonging to such a collection has a coordinate associated with it that belongs to a pre-specied rectilinear section of the domain. The foreach loop, which iterates over objects in a domain or rectdomain, and has the property that the order of iterations does not in uence the result of the associated computations. Further, the iterations can be performed in parallel. We also extend the semantics of foreach to include the possibility of updates to reduction variables, as we explain later. We introduce a Java interface called Reducinterface. Any object of any class implementing this interface acts as a reduction variable [18]. The semantics of a reduction variable is analogous to that used in version 2.0 of High Performance Fortran (HPF-2) [18] and in HPC++ [5]. A reduction variable has the property that it can only be updated inside a foreach loop by a series of operations that are associative and commutative. Furthermore, the intermediate value of the reduction variable may not be used within the loop, except for self-updates. 4.2 Example Code Figure 1 outlines an example code with our chosen extensions. This code shows the essential computation in the virtual microscope application [16]. A large digital image is stored in disks. This image can be thought of as a two dimensional array or collection of objects. Each element in this collection denotes a pixel in the image. Each pixel is comprised of three characters, which denote the color at that point in the image. The interactive user supplies two important pieces of information. The rst is a bounding box within this two dimensional box, which implies the area within the original image that the user is interested in scanning. We assume that the bounding box is rectangular, and can be specied by providing the x and y coordinates of two points. The rst 4 arguments provided by the user are integers and together, they specify the points lowend and hiend. The second information provided by the user is the subsampling factor, an integer denoted by subsamp. The subsampling factor tells the granularity at which the user is interested in viewing the image. A subsampling factor of 1 means that all pixels of the original image must be displayed. A subsampling factor of n means that n 2 pixels are averaged to compute each output pixel. The computation in this kernel is very simple. First, a querybox is created using specied points lowend and hiend. Each pixel in the original image which falls within the querybox is read and then used to increment the value of the corresponding output pixel. There are several advantages associated with specifying the analysis and processing over multidimensional datasets in this fashion. The programs can specify the computations assuming a single processor and at memory. It also assumes that the data is available in arrays of object references, and is not in persistent storage. It is the responsibility of the compiler and run-time system to locate individual elements of the arrays from disks. Also, it is the responsibility of the compiler to invoke the run-time system for optimizing resource usage. 4.3 Restrictions on the Loops The primary goal of our compiler will be to analyze and optimize (by performing both compile-time transformations and generating code for ADR run-time system) foreach loops that satisfy certain properties. We assume standard semantics of parallel for loops and reductions in languages like High Performance Fortran (HPF) [18] and HPC++ [5]. Further, we require that no Java threads be spawned within such loop nests, and no memory locations read or written to inside the loop nests may be touched by another concurrent thread. Our compiler will also assume that no Java exceptions are raised in the loop nests and the iterations of the loop can be reordered without changing any of the language semantics. One potential way of enabling this can be to use bound checking optimizations [25]. 5 Compiler Analysis In this section, we rst describe how the compiler processes the given data-parallel data intensive loop to a canonical form. We then describe how inter-procedural program slicing can be used for extracting a number of functions which are passed to the run-time system. 5.1 Initial Processing of the Loop Consider any data-parallel loop in our dialect of Java, as presented in Section 4. The memory locations modied in this loop are only the elements of collection of objects, or temporary variables whose values are not used in other iterations of the loop or any subsequent computations. The memory locations accessed in this loop are either elements of collections or values which may be replicated on all processors before the start of the execution of the loop. For the purpose of our discussion, collections of objects whose elements are modied in the loop are referred to as left hand side or lhs collections, and the collections whose elements are only read in the loop are considered as right hand side or rhs collections. The functions used to access elements of collections of objects in the loop are referred to as subscript functions. Denition 1 Consider any two lhs collections or any two rhs collections. These two collections are called congruent i The subscript functions used to access these two collections in the loop are identical. The layout and partitioning of these two collections are identical. By identical layout we mean that elements with the identical indices are put together in the same disk block for both the collections. By identical partitioning we mean that the disk blocks containing elements with identical indices from these collections reside on the same processor. Consider any loop. If multiple distinct subscript functions are used to access rhs collections and lhs collections and these subscript functions are not known at compile-time, tiling output and managing disk accesses while maintaining high reuse and locality is going to be a very di-cult task for the run-time system. In particular, the current implementation of ADR does not support such cases. Therefore, we perform loop ssion to divide the original loop into a set of loops, such that all lhs collections in any new loop are congruent and all rhs collections are congruent. We now describe how such loop ssion is performed. Initially, we focus on lhs collections which are updated in dierent statements of the same loop. We perform loop ssion, so that all lhs collections accessed in any new loop are congruent. Since we are focusing on loops with no loop-carried dependencies, performing loop ssion is straight-forward. An example of such transformation is shown in Figure 2, part (a). We now focus on such a new loop in which all lhs collections are congruent, but not all rhs collections may be congruent. For any two rhs accesses in a loop that are not congruent, there are three possibilities: 1. These two collections are used for calculating values of elements of dierent lhs collections. In this case, loop ssion can be performed trivially. 2. These two collections Y and Z are used for calculating values of elements of the same lhs collection. Such lhs collection X is, however, computed as follows: such that, op i op j . In such a case, loop ssion can be performed, so that the element X(f(i)) is updated using the operation op i with the values of Y (g(i)) and Z(h(i)) in dierent loops. An example of such transformation is shown in Figure 2, part (b). 3. These two collections Y and Z are used for calculating values of the elements of the same lhs collection and unlike the case above, the operations used are not identical. An example of such a case is In this case, we need to introduce temporary collection of objects to copy the collection Z. Then, the collection Y and the temporary collection can be accessed using the same subscript function. An example of such transformation is shown in Figure 2, part (c). After such a series of loop ssion transformations, the original loop is replaced by a series of loops. The property of each loop is that all lhs collections are accessed with the same subscript function and all rhs collections are also accessed with the same subscript function. However, the subscript function for accessing the lhs collections may be dierent from the one used to access rhs collections. (a) foreach (p in box) f foreach (p in box) f foreach (p in box) f (b) foreach (p in box) f foreach (p in box) f foreach (p in box) f (c) foreach (p in box) f foreach (p in box) f foreach (p in box) f Fig. 2. Examples of Loop Fission Transformations O 1 [S L (r)] op Om [S L (r)] op Fig. 3. A Loop In Canonical Form 5.1.1 Discussion Our strategy of performing loop ssion so that all lhs collections are accessed with the same subscript function and all rhs collections are accessed with the same subscript function is clearly not the best suited for all classes of applications. Particularly, for stencil computations, it may result in several accesses to each disk block. However, for the class of data intensive reductions we have focused on, this strategy works extremely well and simplies the later loop execution. In the future, we will incorporate some of the techniques from parallel database join operations for loop execution, which will alleviate the need for performing loop ssion in all cases. 5.1.2 Terminology After loop ssion, we focus on one individual loop at a time. We introduce some notation about this loop which is used for presenting our solution. The terminology presented here is illustrated by the example loop in Figure 3. The domain over which the iterator iterates is denoted by R. Let there be n rhs collection of objects read in this loop, which are denoted by I Similarly, let the lhs collections written in the loop be denoted by O Further, we denote the subscript function used for accessing right hand side collections by SR and the subscript function used for accessing left hand side collections by S L . int Fig. 4. Slice for Subscript Function (left) and for Aggregation Function (right) Given a point r in the range for the loop, elements S L (r) of the output collections are updated using one or more of the values I 1 and other scalar values in the program. We denote by A i the function used for creating the value which is used later for updating the element of the output collection O i . The operator used for performing this update is op i . 5.2 Slicing Based Interprocedural Analysis We are primarily concerned with extracting three sets of functions, the range function R, the subscript functions SR and S L , and the aggregation functions Similar information is often extracted by various data-parallel Fortran compilers. One important dierence is that we are working with an object-oriented language (Java), which is signicantly more di-cult to ana- lyze. This is mainly because the object-oriented programming methodology frequently leads to small procedures and frequent procedure calls. As a result, analysis across multiple procedures may be required in order to extract range, subscript and aggregation functions. We use the technique of interprocedural program slicing for extracting these three sets of functions. Initially, we give background information on program slicing and give references to show that program slicing can be performed across procedure boundaries, and in the presence of language features like polymorphism, aliases, and exceptions. 5.2.1 Background: Program Slicing The basic denition of a program slice is as follows. Given a slicing criterion (s; x), where s is a program point in the program and x is a variable, the program slice is a subset of statements in the programs such that these statements, when executed on any input, will produce the same value of the variable x at the program point s as the original program. The basic idea behind any algorithm for computing program slices is as follows. Starting from the statement p in the program, we trace any statements on which p is data or control dependent and add them to the slice. The same is repeated for any statement which has already been included in the slice, until no more statements can be added in the slice. ADR Pt outpoint(2); ADR Pt lowend(2); int return outpoint ; void Accumulate(ADR Box current block, ADR Box current tile, ADR Box querybox) f current block.intersect(querybox); ADR Pt inputpt(2); ADR Pt outputpt(2); int for for if (project(inputpt, outputpt, current tile)) f Output[outputpt].Accum(VScope[inputpt], Fig. 5. Compiler Generated Subscript and Aggregation Functions Slicing has been very frequently used in software development environments, for debugging, program analysis, merging two dierent versions of the code, and software maintenance and testing. A number of techniques have been presented for accurate program slicing across procedure boundaries [32]. Since object-oriented languages have been frequently used for developing production level software, signicant attention has been paid towards developing slicing techniques in the presence of object-oriented features like object references, polymorphism, and more recently, Java features like threads and exceptions. Harrold et al. and Tonnela et al. have particularly focused on slicing in the presence of polymorphism, object references, and exceptions [17, 28]. Slicing in the presence of aliases and reference types has also been addressed [3]. 5.2.2 Extracting Range Function We need to determine the rhs and lhs collection of objects for this loop. We also need to provide the range function R. The rhs and lhs collection of objects can be computed easily by inspecting the assignment statements inside the loop and in any functions called inside the loop. Any collection which is modied in the loop is considered a lhs collection, and any other collection touched in the loop is considered a rhs collection. For computing the domain, we inspect the foreach loop and look at the domain over which the loop iterates. Then, we compute a slice of the program using the entry of the loop as the program point and the domain as the variable. 5.2.3 Extracting Subscript Functions The subscript functions SR and S L are particularly important for the run-time system, as it determines the size of lhs collections written in the loop and the rhs disk blocks from each collection that contributes to the lhs collections. The function S L can be extracted using slicing as follows. Consider any statement in the loop which modies any lhs collection. We focus on the variable or expression used to access an element in the lhs collection. The slicing criterion we choose is the value of this variable or expression at the beginning of the statement where the lhs collection is modied. The function SR can be extracted similarly. Consider any statement in the loop which reads from any rhs collection. The slicing criterion we use is the value of the expression used to access the collection at the beginning of such a statement. Typically, the value of the iterator will be included in such slices. Suppose the iterator is p. After rst encountering p in the slice, we do not follow data-dependencies for p any further. Instead, the functions returned by the slice use such iterator as the input parameter. For the virtual microscope template presented in Figure 1, the slice computed for the subscript function S L is shown at the left hand side of Figure 4 and the code generated by the compiler is shown on the left hand side of Figure 5. In the original source code, the rhs collection is accessed with just the iterator therefore, the subscript function SR is the identity function. The function receives the coordinates of an element in the rhs collection as parameter (iterpt) from the run-time system and returns the coordinates of the corresponding lhs element. Titanium multidimensional points are supported by ADR as a class named ADR Pt. Also, in practice, the command line parameters passed to the program are extracted and stored in a data-structure, so that the run-time system does not need to explicitly read args array. 5.2.4 Extracting Aggregation Functions For extracting the aggregation function A i , we look at the statement in the loop where the lhs collection O i is modied. The slicing criterion we choose is the value of the element from the collection which is modied in this statement, at the beginning of this statement. The virtual microscope template presented in Figure 1 has only one aggregation function. The slice for this aggregation function is shown in Figure 4 and the actual code generated by the compiler is shown in Figure 5. The function Accum accessed in this code is obviously part of the slice, but is not shown here. The generated function iterates over the elements of a disk block and applies aggregation functions on each element, if that element intersects with the range of the loop and the current tile. The function is presented as a parameter of current block (the disk block being processed), the current tile (the portion of lhs collection which is currently allocated in memory), and querybox which is the iteration range for the loop. Titanium rectangular domains are supported by the run-time as ADR Box. Further details of this aggregation function are explained after presenting the combined compiler/run-time loop processing. 6 Combined Compiler and Run-time Processing In this section we explain how the compiler and run-time system can work jointly towards performing data intensive computations. 6.1 Initial Processing of the Input The system stores information about how each of the rhs collections of objects I i is stored across disks. Note that after we apply loop ssion, all rhs collections accessed in the same loop have identical layout and partitioning. The compiler generates appropriate ADR functions to analyze the meta-data about collections I i , the range function R, and the subscript function SR , and compute the list of disk blocks of I i that are accessed in the loop. The domain of each rhs collection accessed in the loop is SR R. Note that if a disk block is included in this list, it is not necessary that all elements in this disk block are accessed during the loop. However, for the initial planning phase, we focus on the list of disk blocks. We assume a model of parallel data intensive computation in which a set of disks is associated with each node of the parallel machine. This is consistent with systems like IBM SP and cluster of workstations. Let the set denote the list of processors in the system. Then, the information computed by the run-time system after analyzing the range function, the input subscript function and the meta-data about each of the collections of objects I i is the . For a given input collection I i and a processor is the set of disk blocks b that contain data for collection I i , is resident on a disk connected to processor intersects with SR R. Further, for each disk block b ijk belonging to the set B ij , we compute the information D(b ijk ), which denotes the subset of the domain SR R which is resident on the disk block b. Clearly the union of the domains covered by all selected disk blocks will cover the entire area of interest, or in formal terms, 6.2 Work Partitioning One of the issues in processing any loop in parallel is work or iteration parti- tioning, i.e., deciding which iterations are executed on which processor. The work distribution policy we use is that each iteration is performed on the owner of the element read in that iteration. This policy is opposite to the owner computes policy [19] which has been commonly used in distributed memory compilers, in which the owner of the lhs element works on the iteration. The rationale behind the approach is that the system will not have to communicate blocks of the rhs collections. Instead, only replicated elements of the lhs collections need to be communicated to complete the computation. Note that the assumptions on the nature of loops we have placed requires replacing an initial loop by a sequence of canonical loops, which may also increase net communication between processors. However, we do not consider it to be a problem for the set of applications we target. 6.3 Allocating Output Buers and Strip Mining The distribution of rhs collections is decided before performing the processing, and we have decided to partition the iterations accordingly. We now need to allocate buers to accumulate the local contributions to the nal lhs objects. We use run-time analysis to determine the elements of output collections which are updated by the iterations performed on a given processor. This run-time analysis is similar to the one performed by run-time libraries for executing irregular applications on distributed memory machines. Any element which is updated by more than one processor is initially replicated on all processors by which it is updated. Several dierent strategies for allocation of buers have been studies in the context of the run-time system [9]. Selecting among these dierent strategies for the compiler generated code is a topic for future research. The memory requirements of the replicated output space are typically higher than the available memory on each processor. Therefore, we need to divide the replicated output buer into chunks that can be allocated on the main memory of each processor. This is the same issue as strip mining or tiling used for improving cache performance. We have so far used only a very simple strip mining strategy. We query the run-time system to determine the available memory that can be allocated on a given processor. Then, we divide the lhs space into blocks of that size. Formally, we divide the lhs domain S L R into a set of smaller domains (called strips) fS 1 g. Since each of the lhs collection of objects in the loop is accessed through the same subscript function, the same strip mining is done for each of them. In performing the computations on each processor, we will iterate over the set of strips, allocate that strip for each of the n output collections, and compute local contributions to each strip, before allocating the next strip. To facilitate this, we compute the set of rhs disk blocks that will contribute to each strip of the lhs. 6.4 Mapping Input to the Output We use subscript functions SR and S L for computing the set of rhs disk blocks that will contribute to each strip of the lhs as indicated above. To do this we apply the function S L (S 1 R ) to each D(b ijk ) to obtain the corresponding domain in the lhs region. These output domains that each disk block can contribute towards are denoted as OD(b ijk ). If D(b ijk ) is a rectangular domain and if the subscript functions are monotonic, OD(b ijk ) will be a rectangular domain and can easily be computed by applying the subscript function to the two extreme corners. If this is not the case, the subscript function needs to be applied on each element of D(b ijk ) and the resulting OD(b ijk ) will just be a domain and not a rectangular domain. Formally, we compute the sets L jl , for each processor j and each output strip l, such that 6.5 Actual Execution The computation of sets L il marks the end of the loop planning phase of the run-time system. Using this information, now the actual computations are performed on each processor. The structure of the computation is shown in Figure 6. In practice, the computation associated with each rhs disk block and retrieval of disk blocks is overlapped, by using asynchronous I/O operations. We now explain the aggregation function generated by the compiler for the virtual microscope template presented in Figure 1, shown in Figure 5 on the right hand side. The accumulation function output by the compiler captures the Foreach element part of the loop execution model shown in Figure 6. The For each output strip S l : Execute on each Processor Allocate and initialize strip S l for O Foreach Read blocks b ijk disks Foreach element e of D(b ijk ) If the output pt. intersects with S l Evaluate functions A Global reduction to nalize the values for S l Fig. 6. Loop Execution on Each Processor run-time system computes the sets L jl as explained previously and invokes the aggregation function in a loop that iterates over each disk block in such a set. The current compiler generated code computes the rectangular domain D(b ijk ) in each invocation of the aggregation function, by doing an intersection of the current block and query block. The resulting rectangular domain is denoted by box. The aggregation function iterates over the elements of box. The conditional if project() achieves two goals. First, it applies subscript functions to determine the lhs element outputpt corresponding to the rhs element inputpt. Second, it checks if outputpt belongs to current tile. The actual aggregation is applied only if outputpt belongs to current tile. This test represents a signicant source of ine-ciency in the compiler generated code. If the tile or strip being currently processed represents a rectangular rhs region and the subscript functions are monotonic, then the intersection of the box and the tile can be performed before the inner loop. This is in fact done in the hand customization of ADR for virtual microscope [16]. Performing such an optimization automatically is a topic for future research and beyond the scope of our current work. 7 Current Implementation and Experimental Results In this section, we describe some of the features of the current compiler and then present experimental results comparing the performance of compiler generated customization for three codes with hand customized versions. Our compiler has been implemented using the publicly available Titanium infrastructure from Berkeley [36]. Our current compiler and run-time system only implement a subset of the analysis and processing techniques described in this paper. Two key limitations are as follows. We can only process codes in which the rhs subscript function is the identity function. It also requires that the domain over which the loop iterates is a rectangular domain and all subscript functions are monotonic. Titanium language is an explicitly parallel dialect of Java for numerical computing. We could use the Titanium front-end without performing any modications. Titanium language includes Point, RectDomain, and foreach loop which we required for our purposes. The concept of reducinterface is not part of Titanium language, but no modications to the parser were required for this purpose. Titanium also includes a large number of additional directives which we did not require, and has signicantly dierent semantics for foreach loops. We have used three templates for our experiments. VMScope1 is identical to the code presented in Figure 1. It models a virtual microscope, which provides a realistic digital emulation of a microscope, by allowing the users to specify a rectangular window and a subsampling factor. The version VMScope1 averages the colors of neighboring pixels to create a pixel in the output image. VMScope2 is similar to VMScope1, except for one important dierence. Instead of taking the average of the pixels, it only picks every subsamp th element along each dimension to create the nal output image. Thus, only memory accesses and copying is involved in this template, no computations are performed. Bess models computations associated with water contamination studies over bays and estuaries. The computation performed in this application determines the transport of contaminants, and accesses large uid velocity data-sets generated by a previous simulation. Virtual Microscope (averaging)103050701 2 4 8 # of processors Execution Time Compiled Original Fig. 7. Comparison of Compiler and Hand Generated Versions for VMScope1 These three templates represent data intensive applications from two important domains, digital microscopy and scientic simulations. The computations and data accesses associated with these computations are relatively simple and can be handled by our current prototype compiler. Moreover, we had access to hand coded ADR customization for each of these three templates. This allowed us to compare the performance of compiler generated versions against hand coded versions whose performance had been reported in previously published work [16, 23]. Our experiments were performed using the ADR run-time system ported on a cluster of dual processor 400 MHz Intel Pentium nodes connected by gigabit ethernet. Each node has 256MB main memory and GB of internal disk. Experiments were performed using 1, 2, 4 and 8 nodes of the cluster. ADR run-time system's current design assumes a shared nothing architecture and does not exploit multiple CPUs on the same node. Therefore, only one processor on each node was used for our experiments. The results comparing the performance of compiler generated and hand customized VMScope1 are shown in Figure 7. A microscope image of 19; 760 15; 360 pixels was used. Since each pixel in this application takes 3 bytes, a total of 910 MB are required for storage of such an image. A query with a bounding box of size 10; 00010; 000 with a subsampling factor of 8 was used. The time taken by the compiler generated version ranged from 73 seconds on 1 processor to 13 seconds on 8 processors. The speedup on 2, 4, and 8 processors was 1.86, 3.32, and 5.46, respectively. The time taken by the hand coded version ranged from 68 seconds on 1 processor to 8.3 seconds on 8 processors. The speedup on 2, 4, and 8 processors was 2.03, 4.09, and 8.2, respectively. Since the code is largely I/O and memory bound, slightly better than linear speedup is possible. The performance of compiler generated code was lower by 7%, 10%, 25%, and 38% on 1, 2, 4, and 8 processors respectively. From this data, we see that the performance of compiler generated code is very close to the hand coded one on the 1 processor case, but is substantially lower on the 8 processor case. We carefully compared the compiler generated and hand coded versions to understand these performance factors. The two codes use dierent tiling strategies of the lhs collections. In the hand coded version, an irregular strategy is used which ensures that each input disk block maps entirely into a single tile. In the compiler version, a simple regular tiling is used, in which each input disk block can map to multiple tiles. As shown in Figure 5, the compiler generated code performs an additional check in each iteration, to determine if the lhs element intersects with the tile currently being processed. In comparison, the tiling strategy used for the hand coded version ensures that this check always returns true, and therefore does not need to be inserted in the code. But, because of the irregular tiling strategy, an irregular mapping is required between the bounding box associated with each disk block and the actual coordinates on the allocated output tile. This mapping needs to be carried out after each rhs disk block is read from the memory. The time required for performing such mapping is proportional to the number of rhs disk blocks processed by each processor for each tile. Since the output dataset is actually quite small in our experiments, the number of rhs disk blocks processed by each processor per tile decreases as we go to larger congurations. As a result, the time required for this extra processing reduces. In comparison, the percentage overhead associated with extra checks in each iteration performed by the compiler generated version remains unchanged. This dierence explains why the compiler generated code is slower than the hand coded one, and why the dierence in performance increases as we go to larger number of processors. Virtual Microscope (subsampling)103050 # of processors Execution Time Compiled Original Fig. 8. Comparison of Compiler and Hand Generated Versions for VMScope2 The results comparing the performance of compiler generated and hand coded VMScope2 are shown in Figure 8. This code was executed on the same dataset and query as used for VMScope1. The time taken by the compiler generated version ranged from 44 seconds on 1 processor to 9 seconds on 1 processor. The hand coded version took 47 seconds on 1 processor and nearly 5 seconds on 8 processors. The speedup of the compiler generated version was 2.03, 3.31, and 4.88 on 2, 4, and 8 processors respectively. The speedup of the hand coded version was 2.38, 4.98, 10.0 on 2, 4, and 8 processors respectively. A number of important observations can be made. First, though the same query is executed for VMScope1 and VMScope2 templates, the execution times are lower for VMScope2. This is clearly because no computations are performed in VMScope2. However, a factor of less than two dierence in execution times shows that both the codes are memory and I/O bound and even in VMScope1, the computation time does not dominate. The speedups for hand coded version of VMScope2 are higher. This again is clearly because this code is I/O and memory bound. The performance of the compiler generated version was better by 6% on 1 processor, and was lower by 10% on 2 processors, 29% on 4 processors, and 48% on 8 processors. This dierence in performance is again because of the dierence in tiling strategies used, as explained previously. Since this template does not perform any computations, the dierence in the conditionals and extra processing for each disk block has more signicant eect on the overall performance. In the 1 processor case, the additional processing required for each disk block becomes so high that the compiler generated version is slightly faster. Note that the hand coded version was developed for optimized execution on parallel systems and therefore is not highly tuned for sequential case. For the 8 processor case, the extra cost of conditional in each iteration becomes dominant for the compiler generated version. Therefore, the compiler generated version is almost a factor of 2 slower than the hand coded one. Bays and Estuaries Simulation System2060100140 # of processors Execution Time Compiled Original Fig. 9. Comparison of Compiler and Hand Generated Versions for Bess The results comparing performance of compiler generated and hand coded version for Bess are shown in Figure 9. The dataset comprises of a grid with 2113 elements and 46,080 time-steps. Each time-step has 4 4-byte oating point numbers per grid element, denoting simulated hydrodynamics parameters previously computed. Therefore, the memory requirements of the dataset are 1.6 GB. The Bess template we experimented with performed weighted averaging of each of the 4 values for each column, over a specied number of time-steps. The number of time-steps used for our experiments was 30,000. The execution times for the compiler generated version ranged from 131 seconds on 1 processor to 11 seconds on 8 processors. The speedup on 2, 4, and 8 processors was 1.98, 5.53, and 11.75 respectively. The execution times for the hand coded version ranged from 97 seconds on 1 processor to 9 seconds on 8 processors. The speedup on 2, 4, and 8 processors was 1.8, 5.4, and 10.7 respectively. The compiler generated version was slower by a factor of 25%, 19%, 24%, and 19% on 1, 2, 4, and 8 processors respectively. We now discuss the factors behind the dierence in performance of compiler generated and hand coded Bess versions. As with both the VMScope versions, the compiler generated Bess performs checks for intersecting with the tile for each pixel. The output for this application is very small, and as a result, the hand coded version explicitly assumes a single output tile. The compiler generated version cannot make this assumption and still inserts the conditionals. The amount of computation associated with each iteration is much higher for this application. Therefore, the percentage overhead of the extra test is not as large as the VMScope templates. The second important dierence between the compiler generated and hand coded versions is how averaging is done. In the compiler generated code, each value to be added is rst divided by the total number of values which are being added. In comparison, the hand coded version performs the summation of all values rst, and then performs a single division. The percentage overhead of this is independent of the number of processors used. We believe that the second factor is the dominant reason for the dierence in performance of two versions. This also explains why the percentage dierence in performance remains unchanged as the number of processors is increased. The performance of compiler generated code can be improved by performing the standard strength reduction optimization. However, the compiler needs to perform this optimization interprocedurally, which is a topic for future work. As an average over these three templates and 1, 2, 4, and 8 processor congu- rations, the compiler generated versions are 21% slower than hand coded ones. Considering the high programming eort involved in managing and optimizing disk accesses and computations on a parallel machine, we believe that a 21% slow-down from automatically generated code will be more than acceptable to the application developers. It should also be noted that the previous work in the area of out-of-core and data intensive compilation has focused only on evaluating the eectiveness of optimizations, and not on any comparisons against hand coded versions. Our analysis of performance dierences between compiler generated and hand coded versions has pointed us to a number of directions for future research. First, we need to consider more sophisticated tiling strategies to avoid large performance penalties associated with performing extra tests during loop ex- ecution. Second, we need to consider more advanced optimizations like inter-procedural code motion and interprocedural strength reduction to improve the performance of compiler generated code. 8 Related Work Our work on providing high-level support for data intensive computing can be considered as developing an out-of-core Java compiler. Compiler optimizations for improving I/O accesses have been considered by several projects. The PASSION project at Northwestern University has considered several dierent optimizations for improving locality in out-of-core applications [6, 20]. Some of these optimizations have also been implemented as part of the Fortran D compilation system's support for out-of-core applications [29]. Mowry et al. have shown how a compiler can generate prefetching hints for improving the performance of a virtual memory system [26]. These projects have concentrated on relatively simple stencil computations written in Fortran. Besides the use of an object-oriented language, our work is signicantly dierent in the class of applications we focus on. Our techniques for executions of loops are particularly targeted towards reduction operations, whereas previous work has concentrated on stencil computations. Our slicing based information extraction for the runtime system allows us to handle applications which require complex data distributions across processors and disks and for which only limited information about access patterns may be known at compile-time. Many researchers have developed aggressive optimization techniques for Java, targeted at parallel and scientic computations. javar and javab are compilation systems targeting parallel computing using Java [4]. Data-parallel extensions to Java have been considered by at least two other projects: Titanium [36] and HP Java [7]. Loop transformations and techniques for removing redundant array bounds checking have been developed [12, 25]. Our eort is also unique in considering persistent storage, complex distributions of data on processors and disks, and the use of a sophisticated runtime system for optimizing resources. Other object-oriented data-parallel compilation projects have also not considered data residing on persistent storage [5, 11, 30]. Program slicing has been actively used for many software engineering applications like program based testing, regression testing, debugging and software maintenance over the last two decades [34]. In the area of parallel compi- lation, slicing has been used for communication optimizations by Pugh and Rosser [31] and for transforming multiple levels of indirection by Das and Saltz [15]. We are not aware of any previous work on using program slicing for extracting information for the runtime system. Several research projects have focused on parallelizing irregular applications, such as computational uid dynamics codes on irregular meshes and sparse matrix computations. This research has demonstrated that by developing run-time libraries and through compiler analysis that can place these runtime calls, such irregular codes can be compiled for e-cient execution [2, 21, 24, 35]. Our project is related to these eorts in the sense that our compiler also heavily uses a runtime system. However, our project is also signicantly dierent. The language we need to handle can have aliases and object references, the applications involve disks accesses and persistent storage, and the runtime system we need to interface to works very dierently. Several runtime support libraries and le systems have been developed to support e-cient I/O in a parallel environment [13, 14, 22, 33]. They also usually provide a collective I/O interface, in which all processing nodes cooperate to make a single large I/O request. Our work is dierent in two important ways. First, we are supporting a much higher level of programming by involving a compiler. Second, our target runtime system, ADR, also diers from these systems in several ways. The computation is an integral part of the ADR framework. With the collective I/O interfaces provided by many parallel I/O systems, data processing usually cannot begin until the entire collective I/O operation completes. Also, data placement algorithms optimized for range queries are also integrated as part of the ADR framework. 9 Conclusions In this paper we have addressed the problem of expressing data intensive computations in a high-level language and then compiling such codes to e-ciently manage data retrieval and processing on a parallel machine. We have developed data-parallel extensions to Java for expressing this important class of applications. Using our extensions, the programmers can specify the computations assuming that there is a single processor and at memory. Conceptually, our compiler design has two major new ideas. First, we have shown how loop ssion followed by interprocedural program slicing can be used for extracting important information from general object-oriented data-parallel loops. This technique can be used by other compilers that use a run-time system to optimize for locality or communication. Second, we have shown how the compiler and run-time system can use such information to e-ciently execute data intensive reduction computations. This technique for processing such loops is independent of the source language. These techniques have been implemented in a prototype compiler built using the Titanium front-end. We have used three templates, from the areas of digital microscopy and scientic simulations, for evaluating the performance of this compiler. We have compared the performance of compiler generated code with the performance of codes developed by customizing the run-time system ADR manually. Our experiments have shown that the performance of compiler generated codes is, on the average, 21% slower than the hand coded ones, and in all cases within a factor of 2. We believe that these results establish that our approach can be very eective. Considering the high programming eort involved in managing and optimizing disk accesses and computation on a parallel machine, we believe that a 21% slow-down from automatically generated code will be more than acceptable to the application developers. It should also be noted that the previous work in the area of out-of-core and data intensive compilation has focused only on evaluating the eect of optimizations, and not on any comparisons against hand coded versions. Further, we believe that by considering more sophisticated tiling strategies and other optimizations like interprocedural code motion and strength reduction, the performance of the compiler generated codes can be further improved. Acknowledgments We are grateful to Chialin Chang, Anurag Acharya, Tahsin Kurc, Alan Sussman and other members of the ADR team for developing the run-time system, developing hand customized versions of applications, helping us with the ex- periments, and for many fruitful discussions we had with them during the course of this work. --R Angelo De- marzo Interprocedural compilation of irregular applications for distributed memory machines. A prototype Java restructing compiler. Distributed pC A model and compilation strategy for out-of-core data parallel programs A customizable parallel database for multi-dimensional data Infrastructure for building parallel database systems for multi-dimensional data Alan Suss- man Concurrent aggregates (CA). The Vesta parallel Input/Output characteristics of Scalable Parallel Applications. Paul Havlak The Virtual Microscope. High Performance Fortran Forum. Compiling Fortran D for MIMD distributed-memory machines Improving the performance of out-of-core computations Compiling global name-space parallel loops for distributed execution Coupling multiple simulations via a high performance customizable database system. Exploiting spatial regularity in irregular iterative applications. Automatic compiler-inserted i/o prefetching for out-of-core applications NASA Goddard Distributed Active Archive Center (DAAC). Compiler support for out-of- core arrays on parallel machines object-oriented languages Iteration space slicing and its application to communication optimization. Speeding up slicing. A survey of program slicing techniques. --TR Dynamic slicing in the presence of unconstrained pointers Compiling Fortran D for MIMD distributed-memory machines object-oriented languages Speeding up slicing A model and compilation strategy for out-of-core data parallel programs Interprocedural compilation of irregular applications for distributed memory machines Input/output characteristics of scalable parallel applications Index array flattening through program transformation The Vesta parallel file system Automatic compiler-inserted I/O prefetching for out-of-core applications Flow insensitive C++ pointers and polymorphism analysis and its application to slicing Iteration space slicing and its application to communication optimization Reuse-driven interprocedural slicing Concurrent aggregates (CA) Passion Distributed Memory Compiler Design For Sparse Problems Compiling Global Name-Space Parallel Loops for Distributed Execution Titan Improving the Performance of Out-of-Core Computations Infrastructure for Building Parallel Database Systems for Multi-Dimensional Data Exploiting spatial regularity in irregular iterative applications Compiler support for out-of-core arrays on parallel machines

data intensive applications;data parallel language;run-time support;compiler techniques

605734

Parallel two level block ILU Preconditioning techniques for solving large sparse linear systems.

We discuss issues related to domain decomposition and multilevel preconditioning techniques which are often employed for solving large sparse linear systems in parallel computations. We implement a parallel preconditioner for solving general sparse linear systems based on a two level block ILU factorization strategy. We give some new data structures and strategies to construct a local coefficient matrix and a local Schur complement matrix on each processor. The preconditioner constructed is fast and robust for solving certain large sparse matrices. Numerical experiments show that our domain based two level block ILU preconditioners are more robust and more efficient than some published ILU preconditioners based on Schur complement techniques for parallel sparse matrix solutions.

Introduction High performance computing techniques, including parallel and distributing computa- tions, have undergone a gradual maturation process in the past two decades and are now moving from experimental laboratory studies into many engineering and scientific appli- cations. Although shared memory parallel computers are relatively easy to program, the most commonly used architecture in parallel computing practices is that of distributed Technical Report No. 305-00, Department of Computer Science, University of Kentucky, Lexington, KY, 2000. This research was supported in part by the U.S. National Science Foundation under grants CCR- 9902022 and CCR-9988165, in part by the University of Kentucky Center for Computational Sciences and the University of Kentucky College of Engineering. y E-mail: cshen@cs.uky.edu. z E-mail: jzhang@cs.uky.edu. URL: http://www.cs.uky.edu/-jzhang. memory computers, using MPI or PVM for message passing [17, 20]. Even on shared memory parallel computers, the use of MPI for code portability has made distributed programming style prevalent. As a result, developing efficient numerical linear algebra algorithms specifically aiming at high performance computers becomes a challenging issue [9, 10]. In many numerical simulation and modeling problems, the most CPU consuming part of the computations is to solve some large sparse linear systems. It is now accepted that, for solving very large sparse linear systems, iterative methods are becoming the method of choice, due to their more favorable memory and computational costs, comparing to the direct solution methods based on Gaussian elimination. One drawback of many iterative methods is their lack of robustness, i.e., an iterative method may not yield an acceptable solution for a given problem. A common strategy to enhance the robustness of iterative methods is to exploit preconditioning techniques. However, most robust preconditioners are derived from certain type of incomplete LU factorizations of the coefficient matrix and their efficient implementations on parallel computers are a nontrivial challenge. A recent trend in parallel preconditioning techniques for general sparse linear systems is to use ideas from domain decomposition concepts in which a processor is assigned a certain number of rows of the linear system to be solved. For discussions related to this point of view and comparisons of different domain decomposition strategies, see [3, 19, 34] and the references therein. A simple parallel preconditioner can be derived using some simple parallel iterative methods. Commonly used parallel preconditioners in engineering computations are point or block Jacobi preconditioners [4, 36]. These preconditioners are easy to implement, but are not very efficient, in the sense that the number of preconditioned iterations required to solve realistic problems is still large [35]. A more sophisticated approach to parallel preconditioning is to use domain decomposition and Schur complement strategies for constructing parallel preconditioners [34]. Preconditioners constructed from this approach may be scalable, i.e., the number of preconditioned iterations does not increase rapidly as the number of processors increases. Some techniques in this class include various distributed Schur complement methods for solving general sparse linear systems developed in [2, 5, 28, 27]. For sparse matrices arising from (finite difference) discretized partial differential equations (PDEs), a level set technique can usually be employed to extract inherent parallelism from the discretization schemes. If an ILU(0) factorization is performed, then the forward and backward triangular solves associated with the preconditioning can be parallelized within each level set. This approach seems most suitable for implementations on shared memory machines with a small number of processors [11]. For many realistic problems with unstructured meshes, the parallelism extracted from the level set strategy is inadequent. Furthermore, ILU(0) preconditioner may not be accurate enough and the subsequent preconditioned iterations may converge slowly or may not converge at all. Thus, higher accuracy preconditioners have been advocated by a few authors for increased robustness [8, 21, 37, 45, 24, 30]. However, higher accuracy preconditioners usually means that more fill-in entries are kept in the preconditioners and the couplings among the nodes are increased as well [24]. The increased couplings reduce inherent parallelism and new ordering techniques must be employed to extract parallelism from higher accuracy preconditioners. In addition to standard domain decomposition concepts, preconditioning techniques designed specifically to target parallel computers include sparse approximate inverse and multilevel treatments [1, 7, 14, 39, 40]. Although claimed as inherently parallel precon- ditioners, efficient sparse approximate inverse techniques that can be run respectfully on distributed parallel computers are scarce [9]. Recently, a class of high accuracy preconditioners that combine the inherent parallelism of domain decomposition, the robustness of ILU factorization, and the scalability potential of multigrid method have been developed [30, 31]. The multilevel block ILU preconditioners (BILUM and BILUTM) have been tested to show promising convergence rate and scalability for solving certain problems. The construction of these preconditioners are based on block independent set ordering and recursive block ILU factorization with Schur complements. Although this class of preconditioners contain obvious parallelism within each level, their parallel implementations have not yet been reported. In this study, we mainly address the issue of implementing the multilevel block ILU preconditioners in a distributed environment using distributed sparse matrix template [26]. The BILUTM preconditioner of Saad and Zhang [31] is modified to be implemented as a two level block ILU preconditioner on distributed memory parallel architectures (PBILU2). We used Saad's PSPARSLIB library 1 with MPI as basic communication routines. Our PBILU2 preconditioner is compared with one of the most favorable Schur complement based preconditioners of [27] in a few numerical experiments. This article is organized as follows. In Section 2 some background on block independent set ordering and the BILUTM preconditioner is given. In Section 3, we outline the distributed representations of general sparse linear systems. In Section 4, we discuss the construction of a preconditioner (PBILU2) based on two level block ILU factorization. Numerical experiments with a comparison of two Schur complement based preconditioners for solving various distributed linear systems are presented in Section 5 to demonstrate the merits of our two level block ILU preconditioner. Concluding remarks and comments on future work are given in Section 6. Independent Set and BILUTM Most distributed sparse matrix solvers rely on classical domain decomposition concept to partition the adjacency graph of the coefficient matrix. There are a few graph partitioning 1 The PSPARSLIB library is available online from http://www.cs.umn.edu/Research/arpa/p sparslib/psp-abs.html. algorithms and software packages available [16, 18, 22]. Techniques to extract parallelism from incomplete LU factorizations, such as BILUM and BILUTM, usually relay on the fact that many rows of a sparse matrix can be eliminated simultaneously at a given stage of Gaussian elimination. A set consisting of such rows is called an independent set [13]. For large scale matrix computations, the degree of parallelism extracted from traditional (point) independent set ordering is inadequent and the concept of block independent set is proposed [30]. Thus a block independent set is a set of groups (blocks) of unknowns such that there is no coupling between unknowns of any two different groups (blocks) [30]. Various heuristic strategies for finding point independent sets may be extended to find a block independent set with different properties [30]. A simple and usually efficient strategy is the so-called greedy algorithm, which groups the nearest nodes together. Considering a general sparse linear system of the form where A is an unstructured real-valued matrix of order n. The greedy algorithm (or other graph partitioners) is used to find a block independent set from the adjacency graph of the matrix A. Initially, the candidate nodes for a block include all nodes corresponding to each row of the matrix A. Given a block size k, the greedy algorithm starts from the first node, groups the nearest k neighboring nodes, and drops the other nodes which are linked to any of the grouped k nodes into the vertex cover set. Here the vertex cover set is a set of nodes that have at least one link to at least one node of at least one block of the block independent set. The process can be repeated for a few times until all the candidate nodes have gone either into one of the independent blocks or into the vertex cover set. (If the number of remaining candidate nodes is less than k, all of them are put in the vertex cover set, and the meaning of the vertex cover set is then generalized to cover this case.) For detailed algorithm descriptions, see [30]. We remark that it is not necessary that all independent blocks have the same number of nodes [33]. They are chosen to have the same cardinality for the sake of load balance in parallel computations and for the sake of easy programming. In parallel implementations, a graph partitioner, similar to the greedy algorithm just described, is first invoked to partition the adjacency graph of A. Based on the resulting partitioning, the matrix A and the corresponding right hand side and the unknown vectors b and x are distributed to the individual processors. Suppose a block independent set with a uniform block size k has been found and the matrix A is symmetrically permuted into a two by two block matrix of the form where P is a permutation matrix. diagonal matrix of dimension ks, where s is the number of uniform blocks of size k. The blocks are usually dense if k is small. But they are sparse if k is large. In the implementation of BILUM, an exact inverse technique is used to compute B \Gamma1 by inverting each small independently. As it is noted in [33], such direct inversion strategy usually produces dense inverse matrices even if the original blocks are highly sparse with large size. There have been several sparsification strategies proposed to maintain the sparsity of B \Gamma1 [33]. In addition, sparse approximate inverse based multilevel block ILU preconditioners have been proposed in [43]. In this article, we employ an ILU factorization strategy to compute a sparse incomplete LU factorization of B. The approach is similar to the one used for BILUTM [31]. The construction of BILUTM preconditioner is based on a restricted ILU factorization of (2) with a dual dropping strategy (ILUT) [31]. This multilevel block ILU preconditioner (BILUTM) not only retains the robustness and flexibility of ILUT, but also is more powerful than ILUT for solving some difficult problems and offers inherent parallelism that can be exploited on parallel or distributed architectures. 3 Distributed Sparse Linear System and SLU Preconditioner A distributed sparse linear system is a collection of sets of equations that assigned to different processors. The parallel solution of a sparse linear system begins with partitioning the adjacency graph of the coefficient matrix A. Based on the resulting partitioning, the data is distributed to processors such that pairs of equations-unknowns are assigned to the same processor. A type of distributed matrix data structure based on subdomain decomposition concepts has been proposed in [29, 26], also see [28]. Based on these concepts, after the matrix A is assigned to each processor, the unknowns in each processor are divided into three types: (1) interior unknowns that are coupled only with local equations; (2) local interface unknowns that are coupled with both nonlocal (external) and local equations; and (3) external interface unknowns that belong to other subdomains and are coupled with local equations. The submatrix assigned to a certain processor, say, processor i, is split into two parts: the local matrix A i , which acts on the local variables, and an interface matrix X i , which acts on the external variables. Accordingly, the local equations in a given processor can be written as The local matrix is reordered in such a way that the interface points are listed last after the interior points. Then we have a local system written in a block format !/ where N i is the indices of subdomains that are neighbors to the reference subdomain i. It is exactly the set of processors that the reference processor needs to communicate with to receive information. The a part of the product X i y i;ext which reflects the contribution to the local equation from the neighboring subdomain j. The sum of these contributions is the result of multiplying X i by the external interface unknowns, i.e., The preconditioners which are built upon this distributed data structure for the original matrix will not form an approximation to the global Schur complement explicitly. Some of such domain decomposition based preconditioners are exploited in [27]. The simplest one is the additive Schwarz procedure, which is a form of block Jacobi (BJ) iteration, where the blocks refer to submatrices associated with each subdomains, i.e., Even though it can be constructed easily, block Jacobi preconditioning is not robust and is inefficient, comparing to other Schur complement type preconditioners. One of the best among these Schur complement based preconditioners is SLU which is the distributed approximate Schur LU preconditioner [27]. The preconditioning to the global matrix A is defined in terms of a block LU factorization which involves a solve with the global Schur complement system at each preconditioning step. Incomplete LU factorization is used in SLU to approximate the local Schur complements. Numerical results reported in [27] show that this Schur (I)LU preconditioner demonstrates superior scalability performance over block Jacobi preconditioner and is more efficient than the latter in terms of parallel run time. 4 A Class of Two Level Block Preconditioning Techniques PBILU2 is a two level block ILU preconditioner based on the BILUTM techniques described in [31]. As we noted before, BILUTM offers a good parallelism and robustness due to its large size of block independent set. The graph partitioner in BILUTM is the greedy algorithm for finding a block independent set [30, 31]. 4.1 Distributed matrix based on block independent set In our implementation, the block size of the block independent set must be given before the search algorithm starts. The choice of the block size k is based on the problem size and the density of the coefficient matrix A. The choice of k may also depend upon the number of available processors. Assume that a block independent set with a uniform block size k has been found, and the coefficient matrix A is permuted into a block form as in The "small" independent blocks are then divided into several groups according to the number of available processors. For the sake of load balance on each processor, each group holds approximate the same number of independent blocks. (The numbers of independent blocks in different groups may differ at most by 1.) At the same time, the global vector of unknowns x is split into two subvectors . The right hand side vector b is also conformally split into subvectors f and g. Such a reordering leads to a block systemB Bm Fm um where m is the number of processors used in the computation. Each block diagonal contains several independent blocks. Note that the submatrix F i has the same row numbers with the block submatrix B i . The submatrices E and C are also divided into m parts according to the load balance criterion in order to have approximately the same amount of loads in each processor. E i and C i also have the same row numbers. Those submatrices are assigned to the same processor i. u i and y i are the local part of the unknown vector, and f i and g i are the local part of the right hand side vectors. They are partitioned and assigned to a certain processor i at the same time when the matrix is distributed. When this processor-data assignment is done, each processor holds several rows of the equations. The local system of equations in processor i can be written as: where u is some part of the unknown subvector: which acts on sub-matrices another part of the unknown subvector: acts on F i and C i . (Only B i acts on the completely local vector u i .) We take u i and y i as the local unknowns, but they are not completely interior vectors. So preconditioners based on this type of block independent set ordering domain decomposition is different from the straightforward domain decomposition based on a rowwise strip partitioning as (3) used in [27]. An obvious difference between the partitionings (3) and (5) is that in (5), the action of F i is not completely local, while it is local in (3). However, since the nature of the submatrices different in the two decomposition strategies, it is not easy to say which one is better at this stage. 4.2 Derivation of Schur complement techniques A key idea in domain decomposition techniques is to develop preconditioners for the global system (1) by exploiting methods that approximately solve the Schur complement system in parallel. A parallel construction of the PBILU2 preconditioner based on block independent set domain decomposition for computing an approximation to the global Schur complement will be described. For deriving the global Schur complement, another parts of the coefficient matrix A need to be partitioned and sent to a certain processor. We rewrite the reordered coefficient matrix in the system (2) Bm Fm Thus there are two ways to partition the submatrix E: one is to partition E by rows and the other is by columns. That is Those submatrices which will also be assigned to the processor i, have the same number of columns as the block diagonal submatrix and the same number of rows as the submatrix C. Remark 4.1 Here we clear a potential confusion for the two representations of the sub-matrix E. The row partitioning of E in (4) is used for representing the "local" matrix A i in the form of (5), which is different from the local matrix A i in (3) and will be kept throughout the computational process. The column partitioning of E in (6) is just for the convenience of computing the Schur complement matrix in parallel. The column partitioning of E is not kept after the construction of the Schur complement matrix. In most cases, the submatrix E is small and highly sparse, if B is large. Consider a block LU factorization of (2) in the form of I 0 !/ where S is the global Schur complement: Now, suppose we can invert B by some means, we can rewrite Equation (8) as: FmC A Each processor can compute one component of the sum independently, then partitions the rows of the submatrix M each part of the rows must be conformal to each submatrix C m. They are then scattered to other processors. (There is a global communication needed for scattering.) Finally, the local part of the Schur complement matrix can be constructed independently in each processor. The simplest implementation for this approach to constructing the distributed Schur complement matrix in an incomplete LU factorization is to use a parallel block restricted IKJ version of Gaussian elimination, similar to the sequential algorithm used in BILUTM [31]. This method can decrease communications among processors and offers flexibility in controlling the amount of fill-in during the ILU factorization. 4.3 Parallel restricted Gaussian elimination BILUTM is a high accuracy preconditioner based on incomplete LU factorization. It utilizes a dual dropping strategy of ILUT to control the computational and storage (memory) costs [24]. Its implementation is based on the restricted IKJ version of Gaussian elimi- nation, discussed in detail in [31]. In the remaining part of this subsection, we outline a parallel implementation of the restricted IKJ version of Gaussian elimination, using the distributed data structure discussed in the previous subsection. On the ith processor, a local submatrix is formed based on those submatrices assigned to this processor and an ILU factorization on this local matrix will be performed. 2 This local matrix in processor i looks like In PBILU2, the "local" matrix A i means it is stored in the local processor i. It does not necessarily mean that A i acts only on interior unknowns. Note that the submatrix - C i has the same size as the submatrix C in the Equation (7). In the submatrix - only the elements corresponding to the nonzero entries of the submatrix may not be zero, the others are zero elements. Recalling the permuted matrix in (2) and in the left hand side of (7), if we let the submatrices C the submatrix - C i is then obtained. We perform a restricted Gaussian elimination on the local matrix (10). This is a slightly different elimination procedure. First we perform an (I)LU factorization (Gaussian elimination) to the upper part of the local matrix, i.e., to the submatrix (B i F i ). We then continue the Gaussian elimination to the lower part (M i - but the elimination is only performed with respect to nonzero (and the accepted fill-in) entries of the submatrix M i . The entries in - are modified accordingly. When performing these operations on the lower part, the upper part of the matrix is only accessed, but not modified, see Figure 1. F Not processed, not accessed Processed, not accessed Accessed, not modified Not accessed, not modified Accessed, not modified Figure 1: Illustration of the restricted IKJ version of Gaussian elimination. Here the submatrices respectively, as in Equation (10). After this is done, three kinds of submatrices are formed, which will be used in later iterations: 1. The upper part of the matrix after the upper part Gaussian elimination is (UB i so we have L \Gamma1 2. The upper part of the matrix has also been performed an (I)LU factorization for the block diagonal submatrix B i so that B i - LB i . Thus we have We can extract the submatrices LB i and UB i from the upper part of the factored local matrix for later use. 3. In the restricted factorization of the lower part of A i , we obtain a new reduced submatrix, which is represented by ~ C i and will form a piece of the global Schur complement matrix. In fact, this submatrix ~ ~ Note that B \Gamma1 F i and the factor matrix L \Gamma1 F i is already available after the factorization of the upper part of A i . So can be computed in processor i by first solving for an auxiliary matrix Q i in UB i followed by a matrix-matrix multiplication of M i Q i . However, this part of the computation is done implicitly in the restricted IKJ Gaussian elimination process, in the sense that all computations in constructing a piece of the Schur complement matrix S, ~ processor i, is done by a restricted ILU factorization on the lower part of the local matrix. In other words, ~ C i is formed without explicit linear system solve or matrix-matrix multiplication. For detailed computational procedure, see [31]. Considering Equation (9) for Schur complement computation, it can be rewritten in the form of: This computation can be done in parallel, thanks to the block diagonal structure of B. If the Gaussian elimination is an exact factorization, the global Schur complement matrix can be formed by summing all these submatrices ~ together. That is Each submatrix ~ parts (use the same partitionings as the original submatrix C), and corresponding parts are scattered to relevant processors. After receiving and summing all of those parts of submatrices which have been scattered from different processors, the "local" Schur complement matrix ~ S i is formed. Here the "local" means some rows of the global Schur complement that are held in a given processor. Note that ~ Remark 4.2 The restricted IKJ Gaussian elimination yields a block (I)LU factorization of the local matrix (10) in the form of I i !/ However, the submatrices L \Gamma1 are no longer needed in later computations and are discarded. This strategy saves considerable storage space and is different from the current implementation of SLU in the PSPARSLIB library [25, 29, 27]. 4.4 Induced global preconditioner It is possible to develop preconditioners for the global system (1) by exploiting methods that approximately solve the reduced system (8). These techniques are based on reordering the global system into a two by two block form (2). Consider the block LU factorization in the Equation (7), This block factored matrix can be preconditioned by an approximate LU factorization such as I 0 where ~ S is an approximation to the global Schur complement matrix S, formed in (12). Therefore, a global preconditioning operation induced by a Schur complement solve is equivalent to solving LU y f by a forward solve with L and a backward substitution with U . The computational procedure would consist of the following three steps (with ~ g being used as an auxiliary 1. Compute the Schur complement right hand side ~ 2. Approximately solve the reduced system ~ 3. Back substitution for the u variables, i.e., solve Each of these steps can be computed in parallel in each processor with some communications and boundary information exchange among the processors. As our matrix partitioning approach is different from the one used in [27], it needs some communications among processors while computing the global Schur complement right hand side ~ g in each processor. It is easy to see that ~ ~ mC A =B @ mC A \GammaB @ So each of the local Schur complement right hand side can be computed in this way: ~ We rewrite the approximate reduced (Schur complement) system ~ ~ ~ where the submatrix X ij is a boundary matrix which acts on external variables y There are numerous ways to solve this reduced system. One option considered in [27] starts by replacing ~ by an approximate system of the form in which - S i is the local approximation to the local Schur complement matrix ~ This formulation can be viewed as a block Jacobi preconditioned version of the Schur complement system (13). The above system is then solved by an iterative accelerator such as GMRES which requires a solve with - S i at each step. In our current implementation, an ILUT factorization of ~ S i is performed for the purpose of block Jacobi preconditioning. The third step in the Schur complement preconditioning can be performed without any problem. Since B is block diagonal, the solution of can be computed in parallel at each iteration step. In each processor i, we have y, and we actually solve LB i y, as the factors LB i are available. Here we need to exchange boundary information among the processors, since not all components of y required by F i is in processor i. 5 Numerical Experiments In numerical experiments, we compared the performance of the previously described PBILU2 preconditioner and the distributed Schur complement LU (SLU) preconditioner of [27] for solving a few sparse matrices from discretized two dimensional convection diffusion problems, and from application problems in computational fluid dynamics. The computations were carried out on a 32 processor (200 MHz) subcomplex of a (64 processor) HP Exemplar 2200 (X-Class) supercomputer at the University of Kentucky. It has 8 super nodes interconnected by a high speed and low latency network. Each super node has 8 processors attached to it. This supercomputer has a total of 16 GB shared memory and a theoretical operation speed at 51 GFlops. We used the MPI library for interprocessor communications. The other (major) parts of the code are mainly written in Fortran 77 programming language, with a few C routines for handling dynamic allocations of memory. Many of the communication subroutines and the SLU preconditioner code were taken from the PSPARSLIB library [25]. In all tables containing numerical results, "n" denotes the dimension of the matrix; "nnz" represents the number of nonzeros in the sparse matrix; "np" is the number of processors used; "iter" is the number of preconditioned FGMRES iterations (outer it- erations); "F-time" is the CPU time in seconds for the preconditioned solution process with FGMRES; "P-time" is the total CPU time in seconds for solving the given sparse matrix, starting from the initial distribution of matrix data to each processor from the master processor (processor 0). P-time does not include the graph partitioning time and initial permutation time associated with the partitioning, which were done sequentially in processor 0. Thus, P-time includes matrix distribution, local reordering, preconditioner construction, and iteration process time (F-time). "S-ratio" stands for the sparsity ratio, which is the ratio between the number of nonzeros in the preconditioner and the number of nonzeros in the original matrix A. k is the block size used in PBILU2, "p" is the number of nonzeros allowed in each of the L and U factors of the ILU factorizations, - is the drop tolerance. p and - have the same meaning as those used in Saad's ILUT [24]. Both preconditioners use a flexible variant of restarted GMRES (FGMRES) [23] to solve the original linear system since this accelerator permits a change in the preconditioning operation at each step, which is our current case, since we used an iterative process for approximately solving the Schur complement matrix in each outer FGMRES iteration. The size of the Krylov subspace was set to be 50. The linear systems were formed by assuming that the exact solution is a vector of unit. Initial guess was some random vectors with components in (0; 1). The convergence was achieved when the 2-norm residual of the approximate solution was reduced by 6 orders of magnitude. We used an inner-outer iteration process. The maximum number of outer preconditioned FGMRES iterations was 500. The inner iteration to solve the Schur complement system used GMRES(5) (without restart) with a block Jacobi type preconditioner. The inner iteration was stopped when the 2-norm residual of the inner iteration was reduced by 100, or the number of the inner iterations was greater than 5. 5.1 5-POINT and 9-POINT matrices We first compared the parallel performance of different preconditioners for solving some 5-POINT and 9-POINT matrices. The 5-POINT and 9-POINT matrices were generated by discretizing the following convection diffusion equation on a two dimensional unit square. Here Re is the so-called Reynolds number. The convection coefficients were chosen as p(x; exp(\Gammaxy). The right hand side function was not used since we generated artificial right hand sides for the sparse linear systems as stated above. The 5-POINT matrices were generated using the standard central difference discretization scheme. The 9-POINT matrices were generated using a fourth order compact difference scheme [15]. These two types of matrices have been used to test BILUM and other ILU type preconditioners in [30, 31, 41]. Most comparison results for parallel iterative solvers report CPU timing results and iteration numbers. However, in general, it is difficult to make a fair comparison for two different preconditioning algorithms without listing the resource costs to achieve the given results. Since the accuracy of the preconditioners is usually influenced by the fill-in entries kept, the memory (storage) cost of a preconditioner is an important indicator of the efficiency of a preconditioner. Preconditioners that use more memory space are, in general, faster than those that use less memory space. A good preconditioner should not use too much memory space and still achieve fast convergence. To this end, we report in this paper the number of preconditioned iterations, the parallel CPU time for the preconditioned solution process, the parallel CPU time for the entire computational process, and the sparsity ratio. We first chose Re = 0 and for the 5-POINT matrix. The block size was chosen as 200, and the dropping parameters were chosen as . For SLU, we used one level overlapping among the subdomains, as suggested in [27]. The test results are listed in Table 1. We found that our PBILU2 preconditioner is faster than the SLU preconditioner of [27] for solving this problem. PBILU2 takes a smaller number of iterations to converge than SLU did. The convergence rates of both PBILU2 and SLU are not strongly affected by the number of processors employed, which indicates a good scalability with respect to the parallel system for these two preconditioners. Moreover, PBILU2 took much less parallel CPU time than SLU and needed only about a half of the memory space consumed by SLU to solve this matrix. (See Remark 4.2 for an explanation on the difference in storage space for PBILU2 and SLU. 3 ) We also tested the same matrix with a smaller value of In this case, we report two test cases with SLU: one level overlapping of subdomains and nonoverlapping of subdomains. The test results are listed in Table 2. In our experiments, we found 3 The sparsity ratios for PBILU2 and SLU were measured for all storage spaces used for storing the preconditioner. It may be the case that some of these storage spaces for SLU could be released. However, the sparsity ratios for SLU reported in this article were based on the SLU code distributed in the PSPARSLIB library version 3.0 and was downloaded from http://www.cs.umn.edu/Research/arpa/p sparslib/psp-abs.html in November 1999. Table 1: 5-POINT matrix: One level overlapping for SLU. Preconditioner np iter F-time P-time S-ratio PBILU2 SLU 34 34.22 54.37 12.02 PBILU2 Table 2: 5-POINT matrix: One level overlapping (nonoverlapping for results in brackets) for SLU. Preconditioner np iter F-time P-time S-ratio SLU 44 (50) 31.47 51.59 9.15 (9.25) that overlapping or nonoverlapping of subdomains do not make much difference in terms of parallel run time. (Only parallel CPU timings for the overlapping cases are reported in Table 2.) This observation is in agreement with that made in [27]. However, the overlapping version of SLU converged faster than the nonoverlapping version. Ironically, the nonoverlapping version has a slightly larger sparsity ratio. This is because the storage space for the preconditioner is primarily determined by the dual dropping parameters p and - . The overlapping makes the local submatrix look larger, thus reduces the sparsity ratio which is relative to the number of nonzeros of the coefficient matrix. We remark that PBILU2 is again seen to converge faster and to take less parallel run time than SLU, overlapped or nonoverlapped, to solve this 5-POINT matrix using the given parameters. Since the costs and performance of overlapping and nonoverlapping SLU are very close, we only report results with overlapping version of SLU in the remaining numerical tests. Comparing the results of Tables 1 and 2, we see that a higher accuracy PBILU2 preconditioner (using larger p) performed better than a lower accuracy PBILU2 in terms of iteration counts and parallel run time. The higher accuracy one, of course, takes more memory space to store. The SLU preconditioner with overlapping has been tested in [27]. It was compared with some other preconditioners such as BJ (block Jacobi), SI ("pure" Schur complement iteration) and SAPINV (distributed approximate block LU factorization with sparse approximate etc. The numerical experiments in [27] showed that SLU retains its superior performance over BJ and SI preconditioners and can be comparable with Schur complement preconditioning (with local Schur complements inverted by SAPINV). How- ever, our parallel PBILU2 preconditioner is shown to be more efficient than SLU. Here we explain the communication cost of PBILU2 with some experimental data corresponding to the numerical results in Table 2. For example, for the total parallel computation time (P-time) is 18:03 seconds for PBILU2, the communication time for constructing the Schur complement matrix is only 1:45 seconds. So the communication for constructing PBILU2 in this case only costs about 8:04% of the total parallel computation time. For 4, the total parallel computation time (P-time) is 114:25 seconds, the communication time is 0:72 second. The communication time for constructing PBILU2 is only about 0:63% of the total parallel computation time. So the cost for communication in constructing the PBILU2 preconditioner is not high. We also used larger n and varied Re to generate some larger 5-POINT and 9-POINT matrices. The comparison results are given in Tables 3 and 4. These results are comparable with the results listed in Tables 1 and 2. However, the parallel run time (P-time) for SLU in Tables 3 and 4 increased dramatically (more than tripled) when the number of processors increased from 24 to 32. The results for a 5-POINT matrix of is given in Table 5. Once again, we see that PBILU2 performed much better than SLU. Furthermore, the scalability of SLU is degenerating for this test problem. The number of iterations is 13 when 4 processors were used. It increased to 22 when processors were used. For our PBILU2 preconditioner, the number of iterations is almost constant 12 when the number of processors increased from 4 to 32. The very large P-time results for SLU, especially for that the distribution of a large amount of data on this given parallel computer using the partitioning strategy in SLU may present some problems. Another set of tests were run for solving the 9-POINT matrix with . The parallel iteration times (F-time) with respect to different number of processors for both PBILU2 and SLU are plotted in Figure 2. Once again, PBILU2 solved this 9-POINT matrix faster than SLU did. In Figure 3, the numbers of preconditioned FGMRES iterations of PBILU2 and SLU are compared with respect to the number of Table 3: 9-POINT matrix: \Gamma4 . One level overlapping for SLU. Preconditioner np iter F-time P-time S-ratio SLU 19 16.76 46.66 4.50 PBILU2 SLU 19 24.31 151.24 4.51 Table 4: 5-POINT matrix: \Gamma4 . One level overlapping for SLU Preconditioner np iter F-time P-time S-ratio Table 5: 5-POINT matrix: One level overlapping for SLU. Preconditioner np iter S-time P-time S-ratio k SLU 19 41.80 101.68 7.11 SLU 22 35.22 1630.28 7.10 time number of processors dash line - SLU iteration solid line - PBILU2 iteration Figure 2: Comparison of parallel iteration time (F-time) for the PBILU2 and SLU preconditioners for solving a 9-POINT matrix with Parameters used are processors employed, to solve the same 9-POINT matrix. Figure 3 indicates that the convergence rate of PBILU2 improved as the number of processor increased, but the convergence rate of SLU deteriorated as the number of processors increased. We summarize the comparison results in this subsection with the 5-POINT and 9-POINT matrices from the finite difference discretized convection diffusion problems. From the above tests, it can be seen that PBILU2 needs less than half of the storage space required for SLU, when the parameters are chosen comparably. With more storage space consumed by SLU, PBILU2 still outperformed SLU with a faster convergence rate and less parallel run time. Meanwhile, we can see that as the number of processors increases, the parallel CPU time decreases, the number of iterations is not affected significantly for PBILU2. 5.2 FIDAP matrices This set of test matrices were extracted from the test problems provided in the FIDAP package [12]. 4 As many of these matrices have small or zero diagonals, they are difficult to solve with standard ILU preconditioners [42]. We tested more than 31 FIDAP matrices for both preconditioners. We found that PBILU2 can solve more than twice as many FIDAP matrices as SLU does. In out tests, PBILU2 solved 20 FIDAP matrices and SLU solved 9. These tests show that our parallel two level block ILU preconditioner is more 4 These matrices are available online from the MatrixMarket of the National Institute of Standards and Technology at http://math.nist.gov/MatrixMarket. number number of processors dash line - SLU iteration solid line - PBILU2 iteration Figure 3: Comparison of the numbers of preconditioned FGMRES iterations for the PBILU2 and SLU preconditioners for solving a 9-POINT matrix with Parameters used are robust than the SLU preconditioner. Our approach has also shown its merits in terms of its smaller construction and parallel solution costs, smaller memory cost, smaller number of iterations, compared with SLU preconditioner. For the sake of brevity, we only listed results for three representative large test matrices in Tables 6 and 7, and in Figure 4. Note that "-" in Table 6 means that the preconditioned iterative method did not converge or the number of iterations is greater than 500. We varied the parameters of fill-in (p) and drop tolerance (-) in Table 6 for both preconditioners and adjusted the size of block independent set for the PBILU2 approach. PBILU2 is clearly shown to be more robust than SLU to solve this FIDAP matrix. FIDAP035 is a matrix larger than FIDAPM29. In this test, we have also adjusted the fill-in and drop tolerance parameters (p; -) from (50; SLU and PBILU2. The test results for PBILU2 with convergence are reported in Table 7. Note that very small - values are required for the ILU factorizations. It seems difficult for SLU to converge for this test problem for the parameter pairs listed in Table 7 and other parameter pairs tested. So no SLU results are listed in Table 7. Figure 4 shows the parallel iteration time (F-time) with respect to the number of processors for PBILU2 to solve the FIDAP019 matrix. We see that the parallel iteration time decreased as the number of processors increased, which demonstrates a good speedup for solving an unstructured general sparse matrix. Even for some of the FIDAP matrices that both PBILU2 and SLU converge, PBILU2 usually shows superior performance over SLU in terms of the number of iterations and the Table Preconditioner np p - iter F-time P-time S-ratio Table 7: FIDAP035 matrix, Preconditioner np p - k iter F-time P-time S-ratio 28 1.69 4.13 3.96 iteration time number of processors solid line: PBILU2 iteration Figure 4: Parallel iteration time (F-time) of PBILU2 for the FIDAP019 matrix with Parameters used for PBILU2 were 900. The sparsity ratio was approximately 3:45. Table 8: Flat10a matrix, Preconditioner np k iter F-time P-time S-ratio Table 9: Flat30a matrix, Preconditioner np k iter S-ratio sparsity ratio. 5.3 Flat matrices The Flat matrices are from fully coupled mixed finite element discretization of three dimensional Navier-Stokes equations [4, 44] 5 . Flat10a means that the matrix is from the first Newton step of the nonlinear iterations, with 10 elements in each of the x and y coordinate directions, and 1 element in the z coordinate direction. There is only one element in the z coordinate direction because of the limitation on the computer memory used to generate these matrices. The same explanation holds for the Flat30a matrix, which uses 30 elements in each of the x and y coordinate directions. These matrices were generated to keep the variable structural couplings in the Navier-Stokes equations, so they may have "nonzero" entries that actually have a "numerical" zero value. Note that these two matrices are actually symmetric, since they are from the first Newton step where the velocity vector is set to be zero. However, this symmetry information is not utilized in our computation. We see from Tables 8 and 9 that PBILU2 was able to solve these two CFD matrices with small - values. These two matrices were difficult for SLU to converge. The small sparsity ratios reflect our previous remark that the two Flat matrices have many numerical zero entries, which are ignored in the thresholding based ILU factorization, but are counted towards the sparsity ratio calculations. Matrices from fully coupled mixed finite element discretizations of Navier-Stokes equations are notoriously difficult to solve with preconditioned iterative methods [6, 44]. Standard ILU type preconditioners tend to fail or produce unstable factorizations, unless 5 The Flat matrices are available from the second author. the variables are orders properly [44]. The suitable orderings are not difficult to implement in sequential environments [6, 44]. It seems, however, a nontrivial task to perform analogous orderings in a parallel environment. 6 Concluding Remarks and Future Work We have implemented a parallel two level block ILU preconditioner based on a Schur complement preconditioning. We discussed the details on the distribution of "small" independent blocks to form a subdomain in each processor. We gave a computational procedure for constructing a distributed Schur complement matrix in parallel. We compared our parallel preconditioner, PBILU2, with a scalable parallel two level Schur LU preconditioner published recently. Numerical experiments show that PBILU2 demonstrates good scalability in solving large sparse linear systems on parallel computers. We also found that PBILU2 is faster and computationally more efficient than SLU in most of our test cases. PBILU2 is also efficient in terms of memory consumption, since it uses less memory space than SLU to achieve better convergence rate. The FIDAP and Flat matrices tested in Sections 5.2 and 5.3 have small or zero main diagonal entries. The poor convergence performance of both PBILU2 and SLU is mainly due to the instability associated with ILU factorizations of these matrices. Diagonal thresholding strategies [32, 38] can be employed in PBILU2 to exclude the rows with small diagonals from the submatrix B, so that its ILU factorization will be stable. The parallel implementation of diagonally thresholded PBILU2 will be investigated in our future study. We plan to extend our parallel two level block ILU preconditioner to truly parallel multilevel block ILU preconditioners in our future research. 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Distributed Schur complement techniques for general sparse linear systems. Domain decomposition and multi-level type techniques for general sparse linear systems Design of an iterative solution module for a parallel sparse matrix library (P SPARSLIB). BILUM: block versions of multielimination and multilevel ILU preconditioner for general sparse linear systems. BILUTM: a domain-based multilevel block ILUT preconditioner for general sparse matrices Diagonal threshold techniques in robust multi-level ILU preconditioners for general sparse linear systems Enhanced multilevel block ILU preconditioning strategies for general sparse linear systems. Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. High performance preconditioning. Parallel computation of incompressible flows in materials processing: numerical experiments in diagonal preconditioning. Application of sparse matrix solvers as effective preconditioners. A multilevel dual reordering strategy for robust incomplete LU factorization of indefinite matrices. A parallelizable preconditioner based on a factored sparse approximate inverse technique. A sparse approximate inverse for parallel preconditioning of sparse matrices. Preconditioned iterative methods and finite difference schemes for convection-diffusion Preconditioned Krylov subspace methods for solving nonsymmetric matrices from CFD applications. Sparse approximate inverse and multilevel block ILU preconditioning techniques for general sparse matrices. Performance study on incomplete LU preconditioners for solving linear systems from fully coupled mixed finite element discretization of 3D Navier-Stokes equations Use of iterative refinement in the solution of sparse linear systems. --TR A comparison of domain decomposition techniques for elliptic partial differential equations and their parallel implementation High performance preconditioning Application of sparse matrix solvers as effective preconditioners Partitioning sparse matrices with eigenvectors of graphs Introduction to parallel computing A flexible inner-outer preconditioned GMRES algorithm Towards a cost-effective ILU preconditioner with high-level fill Domain decomposition Parallel computation of incompressible flows in materials processing Parallel Multilevel series <i>k</i>-Way Partitioning Scheme for Irregular Graphs Developments and trends in the parallel solution of linear systems Preconditioned iterative methods and finite difference schemes for convection-diffusion Distributed Schur Complement Techniques for General Sparse Linear Systems A Priori Sparsity Patterns for Parallel Sparse Approximate Inverse Preconditioners Sparse approximate inverse and multilevel block ILU preconditioning techniques for general sparse matrices Enhanced multi-level block ILU preconditioning strategies for general sparse linear systems Scalable Parallel Computing Numerical Linear Algebra for High Performance Computers Computer Solution of Large Sparse Positive Definite A Multilevel Dual Reordering Strategy for Robust Incomplete LU Factorization of Indefinite Matrices Iterative Methods for Sparse Linear Systems --CTR Chi Shen , Jun Zhang, A fully parallel block independent set algorithm for distributed sparse matrices, Parallel Computing, v.29 n.11-12, p.1685-1699, November/December Jun Zhang , Tong Xiao, A multilevel block incomplete Cholesky preconditioner for solving normal equations in linear least squares problems, The Korean Journal of Computational & Applied Mathematics, v.11 n.1-2, p.59-80, January Chi Shen , Jun Zhang , Kai Wang, Distributed block independent set algorithms and parallel multilevel ILU preconditioners, Journal of Parallel and Distributed Computing, v.65 n.3, p.331-346, March 2005

schur complement techniques;parallel preconditioning;BILUTM;sparse matrices;domain decomposition

606070

Global optimization approach to unequal sphere packing problems in 3D.

The problem of the unequal sphere packing in a 3-dimensional polytope is analyzed. Given a set of unequal spheres and a polytope, the double goal is to assemble the spheres in such a way that (i) they do not overlap with each other and (ii) the sum of the volumes of the spheres packed in the polytope is maximized. This optimization has an application in automated radiosurgical treatment planning and can be formulated as a nonconvex optimization problem with quadratic constraints and a linear objective function. On the basis of the special structures associated with this problem, we propose a variety of algorithms which improve markedly the existing simplicial branch-and-bound algorithm for the general nonconvex quadratic program. Further, heuristic algorithms are incorporated to strengthen the efficiency of the algorithm. The computational study demonstrates that the proposed algorithm can obtain successfully the optimization up to a limiting size.

Introduction The optimization of the packing of unequal spheres in a 3-dimensional polytope is ana- lyzed. Given a set of unequal spheres and a polytope, the objective is to assemble them in such a way that (1) the spheres do not overlap with each other and (2) the sum of the volumes of the packed spheres is maximized. We note that the conventional 2-dimensional and 3-dimensional packing problems (also called the bin-packing problem), which have been extensively studied [2, 7, 11], are fundamentally di#erent from the problem considered in the present work. The unequal sphere packing problem has important applications in automated radio- surgical treatment planning [12, 14]. Stereotactic radiosurgery is an advanced medical technology for treating brain and sinus tumors. It uses the Gamma knife to deliver a set of extremely high dose ionizing radiations, called "shots", to the target tumor area [13]. In good approximation, these shots can be considered as solid spheres. For large or irregular target regions, multiple shots are used to cover di#erent parts of the tumor. However, this procedure usually results in (1) large dose inhomogeneities, due to the overlap of the di#erent shots, and (2) the delivery of large amount of dose to normal tissue arising from enlargement of the treated region when two or more shots overlap. Optimizing the number, the position, and the individual sizes of the shots can significantly reduce both the inhomogeneities and the dose to normal tissue while simultaneously achieving the required coverage. Unfortunately, since the treatment planning process is tedious, the quality of the protocol depends heavily on the experience of the users. There- fore, an automated planning process is desired. To achieve this goal, Wang and Wu et al. [12, 14] mathematically formulated this planning problem as the packing of spheres into a 3D region with a packing density greater than a certain given level. This packing problem was proved to be NP-complete and an approximate algorithm was proposed [12]. In this work we formulate the question as a nonconvex quadratic optimization and present solution methods based on the branch-and-bound technique. Let K be the number of di#erent radii of the spheres in the given set and r k 1, ., K) the corresponding radii. There are L available spheres for each radius. Therefore, the total number of the spheres in the set is KL. Here, we use a single value of L for simplicity of the presentation. However, this model can be easily modified for the case where di#erent numbers L k of spheres are available for di#erent radii r k and the total number of spheres in the given set is Let the polytope be given by { (x, y, z) # R 3 L be the maximum number of the spheres to be packed. We designate variables as the location of sphere i in a packing. For each sphere in the packing, a radius has to be assigned. The variables t ik 1, ., K) are used to handle this task: sphere i has radius r k , With these preliminaries, the optimization can be formulated as follows. .3 s.t. # a 2 Constraints (1) and (3) respectively ensure that no two spheres overlap with each other and that each sphere is centered within the polytope. Constraints (4) and (5) guarantee that at most one radius is chosen for each sphere, i.e., if t then the sphere i is packed with radius r k , and if t then the sphere i with radius r k is not packed. Together with (3), (4) and (5), constraints (2) state that the distance between the center of a sphere and the boundary of the polytope is at least as large as the radius of that sphere. Hence, (2) and (3) force all the spheres to be packed inside the polytope. . By (3), constraint (2) can be rewritten as for each i, m. (6) Since the right-hand side of (6) is nonnegative, (6) implies (3). Moreover, the binary 0-1 variables t ik in (5) can be replaced by the inequalities t ik (t ik - Note that t ik # 1 is implied by the constraint (4). These steps allow restatement of Problem (P1) as follows. Note that the constant 4# in the original objective function is omitted here. The numbers of the variables and the quadratic constraints are (3+K)L and L(L-1)+LK, respectively. The quadratic function in each constraint (7) is neither convex nor concave. There exist several algorithms [1, 9] that have been developed for solving the general nonconvex quadratic program (NQP in short). A NQP can be transformed into a semidefinite programming problem (SDP) with an additional rank-one constraint. Dropping the rank-one constraint, one obtains a SDP relaxation problem, which is the tightest relaxation among all others [3]. Hence, (P2) can be solved by using a branch-and-bound type method based on the SDP relaxation. However, due to the relative slowness and the instability of the SDP software, this conceptual algorithm is impractical even for problem having a small size, since it requires many iterations of the SDP relaxation. Commonly, methods of solution for the NQP class are designed through linear programming (LP) relaxation, an approach known as the reformulation-linearization technique [10]. Under the assumption that additional box constraints for the variables are new nonlinear constraints are generated from each pair of the box constraints in order to construct a relaxed convex area of the original nonconvex feasible region. The nonlinear terms in each nonlinear constraint are accordingly replaced by new variables. The result of this recasting is a linear program. Based on this technique of linearization, Al-khayyal et al. [1] proposed a rectangular branch-and-bound algorithm to solve a class of quadratically constrained programs. Raber [9] proposed another branch-and-bound algorithm for the same problem based on the use of simplices as the partition elements and the use of an underestimate a#ne function, whose value at each vertex of the simplex agrees with that of the corresponding nonconvex quadratic function. This work [9] demonstrated that the simplicial algorithm often has a better performance over the rectangular algorithm with respect to the computational time. Interestingly, Raber [9] mentioned that both the simplicial and the rectangular algorithms exhibit poor performance for a packing problem : giving the experimental details. It is obvious that their packing problem is similar to ours, but has much simpler structures for the constraints. In this paper, we examine a customization of Raber's simplicial branch-and-bound algorithm [9] tailored to our problem. It is well known that the underlying simplicial sub-division is a key factor influencing the quality of the relaxation, therefore, the e#ciency of the branch-and-bound algorithm. The investigation of the structure of the optimization suggests (i) an e#cient simplicial subdivision and (ii) di#erent underestimations of the nonconvex functions. Based on these observations, three variants of the algorithm have been constructed. The discrete nature of the packing enables a heuristic design for obtaining good feasible solutions, an outcome which leads to savings on both computational time and memory size. The remainder of this paper is organized as follows. In Section 2 we derive the LP relaxation of the problem with respect to the simplicial subdivision. The simplicial branch- and-bound algorithm is presented in Section 3. Section 4 gives the heuristic algorithms and Section 5 presents three variations of the previous algorithm based on the use of special structures. Section 6 reports the computational results of the proposed branch- and-bound algorithm. The conclusions of the work are presented in Section 7. Programming Relaxation The construction of the LP relaxation of Problem (P2) is the same as the one developed in [9]. For the sake of a complete description, we outline this procedure below. the transpose of a vector a. First, we write Problem (P2) in the following form. d d ik # R n , and c # R n is the coe#cient vector of the objective d correspond to the constraints (7) and (8), respectively. Av # b represents all linear constraints of (9), (10) and (11). Furthermore, the matrix Q il can be specified as follows. il O O where O is a matrix having zero for all entries with appropriate size, . . . 1 . -1 . 1 . -1 . corresponds to the coe#cients of in (7) for i and l, and . r 2 . r 1 r K r 2 r K . r 2 . r 2 . r 1 r K r 2 r K . r 2 K . corresponds to those of (t 11 , ., t 1K , ., t L1 , ., t LK ) T in (7) for i and k. Similarly, R 3L-3L and d ik # R KL-KL can be written as follows: O . O. 0 . O d m be the number of the linear constraints in (P2). Denote the polytope defined by these linear constraints as m-n and b # R - . To construct the LP relaxation problem, we need to represent the matrix Q il (resp. by the sum of a positive semidefinite matrix C il (resp. negative semidefinite matrix D il (resp. # D ik ). Usually, a spectrum decomposition achieves this goal. However, we do not need to perform such a task, since the matrices Q il and possess special structures that give the decomposition immediately. It is readily seen that the decompositions O il O O O and satisfy the desired property. Now we consider how to construct our linear programming relaxation problem. Let be an n-simplex (U #S #), where v i are its vertices. Then Let W S # R n-n be a matrix which consists of columns (v each point v in S can be represented as Through substitution of v in (P2') by (14), we obtain the following equivalent problem. s.t. d By replacing Q il and with (12) and (13), the quadratic term of the left-hand side of each constraint of (15) and (16) is divided into a convex and a concave functions by replacing Q il and with (12) and (13), respectively. The relaxation of the above problem is constructed by ignoring the convex part and replacing the concave part with a linear underestimate function. For such an underestimation, we use the convex envelope of a concave function f with respect to the simplex S, which is an a#ne function whose value at each vertex of S coincides with that of f . More precisely, for the quadratic constraints (15) we have where (v is the convex envelope of (W S #) T D il W S # with respect to S. In a similar fashion, for the quadratic constraints (17), we have d d where (v is the convex envelope of (W S #) T # D ik W S # with respect to S. Obviously, an upper bound of the objective function of (P3) can be obtained by solving the following LP relaxation problem. d 3 The Simplicial Branch-and-Bound Algorithm The simplicial branch-and-bound algorithm is presented in this section. As mentioned above, we use the algorithm in [9] as our prototype. However, two heuristics designed for obtaining feasible solutions are embedded. The branching operation is carried out by dividing the current simplex S into two simplices. Let v i # and v j # be two vertices of S satisfying where #a# 2 denotes the 2-norm of a vector a. Define The simplex S is split into two simplices and The splitting of the simplices has the property that for each nested sequence {S q } of simplices, For further details, see Horst [4, 5]. The resulting algorithm is presented below. Branch and Bound Algorithm : Step 1. Start Heuristic-1 to calculate a possible feasible solution v f and the objective function value f(v f ). If successful, set Step 1.1. Let Construct a simplex S 0 which contains the polytope U . Set Solve the problem (LPR) S0 . If the problem is infeasible, then the original problem has no solution, stop. Otherwise, let the optimal solution be and the optimal value be -(S 0 ). Set is a feasible solution of (P2), stop. Otherwise, start Heuristic-2 with v 0 to calculate a feasible solution v # 0 and the value f(v # 0 ). If LB < f(v # 0 ), then set Step 1.2. If (UB - LB)/UB #, then stop. k and S 2 k according to (25) and (28). Set k }. For Step 2.1. Solve the problem (LPR) S j . If it is infeasible, set let the optimal solution be k and the optimal value be -(S j Step 2.2. If v j k is a feasible solution of (P2), set Step 2.3. If v j k is not feasible, then run Heuristic-2 with v j k to calculate a feasible solution (v j and the value f((v j Otherwise, select a simplex - -(S). Set S 2. The details of Heuristic-1 and Heuristic-2 will be given in Section 4. The parameter # gives the tolerance of the solution obtained by the algorithm. We call the solution obtained from the above algorithm an #-optimal solution. The convergence of the algorithm is guaranteed as follows. Theorem 1 ([6, 8, 9]) If the algorithm generates an infinite sequence {v k }, then every accumulation point v # of this sequence is an #-optimal solution of Problem (1). The algorithms Heuristic-1 and Heuristic-2 are described in this section. Recall that Heuristic-1 finds a feasible solution at the beginning of the branch-and-bound algorithm and Heuristic-2 generates a feasible solution from an infeasible solution obtained from a relaxation subproblem. First, we give the algorithm of Heuristic-1. The basic idea is to place as many spheres as possible having relatively large radii in the polytope. Let C be a list of the given set of KL candidate spheres, which are ordered such that r 1 # r KL . Consider a 3-D triangle defined by four constraints arbitrarily chosen from those of the polytope P . Designate triangles. Fixing a P i , the algorithm starts by picking a sphere from the top of C. Then it checks whether the sphere can be located at one of the four corners of the triangle P i . This procedure is continued for the rest of the spheres on the list until four spheres are placed or the search of the spheres in the list is exhausted. All the spheres packed have to satisfy (1)that they touch exactly three sides of the triangle, (2) that no mutual intersection occurs between each pair of the spheres (see Figure 1), and (3) that they satisfy all other constrains on P . After obtaining the initial packing, the algorithm attempts to insert more spheres between each pair of the spheres in P i without violating the packing condition (see Figure 2). Phase I Figure 1: Figure 2: Step 1. Determine the center of the sphere l so that it is tangent to three planes corresponding to the three constraints of P s . If the sphere l satisfies the constraints of polytope P and does not overlap with the spheres 1, ., k - 1, then pack the sphere l and set Step 2. If (i)k = L or (ii) l = KL and k > 0, go to Step 5. Step 3. If stop. Otherwise set l=1 and go to Step 1. Step 4. Set l to Step 1. Phase II Step 5. If Otherwise, for each pair of the packed spheres i, j, set l = 1, repeat Steps 6 and 7. Step 6. If l > KL stop. Otherwise set Step 7. Locate the center of the sphere l at M . If it satisfies the constraints P and does not overlap with the spheres 1, ., k, then pack the sphere l and set go to Step 6. By the termination of the algorithm if k > 0 a feasible packing is obtained. Next, we describe Heuristic-2. Let be the solution of a relaxation subproblem. Suppose that it is not feasible. Then either t # ik is not integral or some spheres overlap with each other if the radius is decided by t # each i, i.e., use r k as the radius of the sphere i (see Figure 3). In the following, we fix the centers of these spheres and determine their radii so that they do not mutually overlap (see Figure 4). Note that for a triplet means that no sphere is placed there. Let r 1 > - > r K . Figure 3: Figure 4: Heuristic-2 Step 1. Set Step 2. If the sphere l centered at satisfies the polytope constraints and does not overlap with spheres 1, ., l - 1, then set t # (the sphere l centered at l , z # l ) has radius r k ), go to Step 4. Step 3. If k < K, set 2. Otherwise, set t # lk (no sphere is centered at l , z # l )). Step 4. If l < L, set l 2. Otherwise, stop. 5 The Improvement of the Algorithm In this section we discuss the special structures of the optimization and present the improvements on the previous algorithm based on these structures. First, we describe how to construct the initial simplex. 5.1 Generating the Initial Simplex To start the branch-and-bound algorithm, we need an initial simplex S 0 which contains the polytope U . It is generated by the following algorithm. Generate Initial Simplex: Step 1. Choose a nondegenerate vertex v 0 of the polytope U . Step 2. Construct a matrix A # R n-n from the n binding constraints Step 3. Compute Step 4. Set S Remark 1: The n constraints which decide the nondegenerate vertex v 0 can be selected as follows. For each i we choose three constraints from (9) and all constraints of (11). It is not di#cult to see that n coe#cient vectors from these constraints are linearly independent. Set these linear inequality constraints as linear equations, i.e., The above system of linear equations determines an unique solution which can be considered as v 0 . Remark 2: The vertex v 0 has KL zero entries, which are determined by (31). Further- more, if an equation in (30) is replaced by (- #, then the solution obtained maintains t k. Repeating this process for all equations in (30) yields 3L solutions, which we index as v 1 , ., v 3L . All the other solutions generated by replacing are denoted by v 3L+1 , ., v n . More precisely, we have and 5.2 Splitting a Simplex in a Di#erent Way It is well known that in the simplicial branch-and-bound paradigm both the computational time and the usage of the memory grow extremely fast as the dimension of the polytope U increases. When a simplex S is split into two simplices according to (25) - (28), the length of is the longest among all other pairs of the vertices of the simplex S. If the vertex v 0 is not replaced by the vertex v M , then the matrix W S 1 is the same as W S except the column v i # - v 0 , which is replaced by the column v M - v 0 . Similarly, all columns in the matrix W S 2 are the same as W S except the column v j # - v 0 , which is replaced by v M - v 0 . Recall that the entries (i, are zeros in the matrices (12). Hence, the coe#cients of # j in # Sil (#) of (22), only depends on the first 3L coordinates of v 0 , values for the first 3L coordinates, then # Sil (#) remains unchanged in the constraints (22) of the subproblem with respect to the simplices S j (j = 1, 2), and the quality of the relaxation would not be significantly improved. Such a splitting leads to the computation of subproblems which provide neither good lower nor useful upper bounds. To avoid this situation, we choose v i # , v j # such that (v is the maximum among all other Consequently, the coordinates corresponding to t ik will not be considered. One negative e#ect of this choice is that the quality of the relaxation of the constraint necessarily improved. However, since the linear constraint is always considered, this defect is not expected to be serious. Accordingly, the number of subproblems can be decreased in the whole branch-and-bound process. There is another potential di#culty with the convergence of the algorithm which must be addressed. Let - (v are vertices of S q }. Since we divide the simplex to minimize the largest value (v vertices, it is possible that a nested sequence {S q } of simplices with - does not satisfy (29). In this case, we are not guaranteed that the accumulation point of an infinite sequence obtained by the algorithm will be an optimal solution. 5.3 Another Decomposition of the Matrix Q il As shown in Section 2, to make the underestimation of the quadratic function of the left-hand side of the quadratic constraint (7), we use the decomposition of (12). From Replacing t 2 ik by t ik in (7) could result in a di#erent matrix il O O . . r 2 . r 1 r k r 2 r k . r 2 and A spectral decomposition of the matrix Q # il provides positive semidefinite and negative semidefinite matrices which are di#erent from those given in (12). Following a procedure similar to that in Section 2, we have where C # il and D # il are positive semidefinite and negative semidefinite, respectively. For example, if only two magnitudes of radii r 1 and r 2 are considered, then C # t il and D # t R KL-KL can be constructed as follows. Let 2 . Then O il O O where . r 2(r 2+3r 2+s) . r 2(r 2+3r 2+s) . r 1 r 2 r 2 . r 2 . r 2 . r 1 r 2 r 2 and il . It can be shown that the eigenvalues of matrix C # t il are nonnegative (resp. nonpositive). Hence, the two matrices have the desired properties. Consequently, another LP relaxation can be constructed based on the new spectral decomposition of that remains the same here, so there is no change in its decomposition. 5.4 Another Form of the Relaxation In this section we focus on a di#erent form of relaxation of problem (P2). Let us omit the quadratic constraints (8), which are corresponding to the 0 - 1 condition of t ik . Furthermore, we relax the condition that two spheres can not overlap with each other. Here the distance between two spheres only needs to be greater than a small positive value -#. More precisely, consider the problem s.t. Remark. One can take the smallest value among all radii given in the set as the magnitude of - #. Since only the variables are quadratic in (P4), the convex envelope of the concave function -(x i - x l can be determined by the first 3L coordinates of the vertices of the corresponding simplex as given above. This implies that it is su#cient to construct simplices in the 3L-dimensional space. Let be the initial simplex in the 3L-dimensional space that contains the polytope P . The ith entry of v # j is identical to that of v j for # be defined similarly as Q il and A, respectively. Let A # x and A # t be the submatrices of A # , which are corresponding to we have the following quadratic program. s.t. negative semidefinite matrix, the convex envelope of the quadratic term (v Therefore, we obtain the relaxation of (P4) as follows. Since the constraints (8) are ignored and the constraints (7) are relaxed as (42) in the above problem, the quality of this relaxation may be inferior to the previous methods. However, the dimension of the simplices kept in the memory is only 3L. Therefore, the total memory used may be far smaller than that required in the other methods. 6 The Computational Study In this section we discuss details of the implementation of the algorithm and report the experimental results. Reoptimization : Suppose that S is the simplex at a branching node. The simplex S is divided into S 1 and S 2 by (25) to (28). If the vertex v 0 is not replaced by v M , then only columns of matrices W S 1 and W S 2 , are replaced by v M - v 0 , respectively. All the other columns remain unchanged. Therefore, in this case, when two new subproblems are generated by splitting S into S 1 and S 2 , we could use the information of the optimal solution of the LP relaxation problem associated with S. By the construction of the LP relaxation problem with respect to S 1 and S 2 , only those coe#cients corresponding to # i # and # j # in the constraints (22) and (23) are changed due to the substitutions of the column v M - v 0 in W S 1 and W S 2, respectively. Consequently, by changing these coe#cients, we could use the reoptimization technique to solve the LP relaxation problem starting from the optimal solution corresponding to the simplex S. Test Problems : The test problems are generated as follows. First, construct a simplex with vertices (0, 0, 10), (10, 0, 0), (0, 10, 0) and (10, 10, 10). Calculate the maximum inscribed sphere of the simplex (the radius is 2.88675). Then randomly generate m points on the surface of the sphere. Construct tangent planes {(x, y, z) | a which pass each of those m points respectively. Let y, z) | a be the halfspace containing the inscribed sphere. The intersection of H i with the simplex is the 3-dimensional polytope in which the sphere packing problem is considered. Hence, the total number of the linear constraints of the polytope is In our test, m was set at 4. Di#erent pairs of radii of spheres that we used are shown in Tables 1-2. Table 1: Combinations of radii for C r radius 6 1.00 0.50 Table 2: Combinations of radii for C r radius The computational experiments were conducted on a DEC Alpha 21164 Workstation (600MHz). We used CPLEX 6.5.1 as an LP solver for the relaxation problems. The limits of the memory and the computational time were set at 512MB and 3600 seconds, respectively. Besides the prototype algorithm given in Section 3, three variations were also implemented. They were (1) Algorithm NSS, which uses the simplex splitting given in Section 5.2, (2) Algorithm NMD, which uses the matrix decomposition given in Section 5.3, and (3) Algorithm XYZ, which uses the simplex in the XYZ space given in Section 5.4. For each algorithm, we tested 5 instances for each and each pair C r of radii from the Table 1. Since the other three algorithms reached either the limit of memory or the limiting computational time for most of the instances for present the results of Algorithm XYZ. The data shown in Tables 4-7 and Table 10 represent the average of the results. The legends used in the tables are given in Table 3. From Tables 4-7 we observe that the computational time grows drastically as the number of spheres packed is increased, since the dimension of the problem is (3 This outcome is consistent with the observation in [1, 9] that the computational time Table 3: Legends used in the tables legend meaning L the maximum number of the spheres packed C r the combination of the radii #.LP the number of linear programs solved #.T the number of cases terminated caused by the limit of CPU time (3600 seconds) #.E the number of cases terminated when #.M the number of cases terminated caused by the limit of memory (512 MB) Time CPU time (seconds) UB-LB the ratio of the values (UB - LB) and UB, where UB and LB is the upper and lower bounds of the objective function value, respectively Algorithm ORG the prototype algorithm given in Section 3 Algorithm NSS the algorithm using the simplex splitting given in Section 5.2 Algorithm NMD the algorithm using the spectral decomposition given in Section 5.3 Algorithm XYZ the algorithm using the simplex in the XYZ space in Section 5.4 increases exponentially as the dimension increases. There is no big di#erence between Algorithm ORG and Algorithm NMD, in the sense of their gross behaviors. Algorithm NSS solves fewer LPs than the two previously mentioned algorithms do, therefore, it needs less time. This leads to the solution of a greater number of instances without violating the limits on either memory or time. Since the simplex splitting is based on the length of the first 3L coordinates, it avoids solving unnecessary subproblems. Among all the algorithms, Algorithm XYZ shows the best performance. All instances for are solved except the case C It even successfully obtained the solutions for about half of the instances for 5. Two reasons for this behavior are considered. First, since the dimension of simplices is only 3L, much less memory is needed to keep the information on the simplices. Second, since the simplices involve only the coordinates of variables (x, y, z), the simplex splitting has the favorable characteristic of Algorithm NSS that fewer LPs are needed to be solved. It should be noted that (i) the dimension of the instances solved in the previous papers [1, 9] range up to 16 and (ii) the computational time and memory demand of the simplicial branch-and-bound algorithm usually increases exponentially with the dimension of the problem. Therefore, a modest increase in the dimension could put the solution out of range. In contrast, for the sphere packing problem discussed above, the dimension of the instances solved by Algorithm XYZ extended up to Hence, we conclude that Algorithm XYZ gains e#ciency, since it takes advantage of the intrinsic structure of the problem. Next, let us focus on Algorithm ORG and Algorithm XYZ and consider the influence of the polytope P on their performance. Five instances of the polytope are generated randomly and they have same number of constraints. Taking the results are shown in Tables 8-9. It is observed that even if L and C r are identical, the comparative behaviors of the algorithms are very sensitive to the shape of the polytope. This behavior arises since (i) the quality of solutions generated by the heuristics depends on the shape of the polytope and (ii) the initial simplex S 0 depends on the polytope. For a skinny polytope, the volume of S 0 \ P 0 would be large, a fact which means that the algorithms copiously waste e#ort on solving LPs defined on this zone before reaching the optimal solution. Table shows the results when three values of radii are considered, i.e., that (i) the value of K e#ects the number of the quadratic constraints in (8), and (ii) increasing the number of the quadratic constraints enhances the di#culty of the problem. Comparing this case with the results for (Table 7), we observed that there is no big change in the numbers of LPs solved. The algorithm is less sensitive to the value of K than to the size of of L. This occurs, since the numbers of both the quadratic constraints (7) and (8) grow with increasing of L. Finally, all of the algorithms exhibited reduced performance when C namely, when the di#erences of the radii are large. Conclusions In this paper, we considered the optimization of unequal sphere packing and demonstrated the improvements over the existing simplicial branch-and-bound algorithm through advantageous use of the intrinsic structure of the problem. Specially, the computational study showed that the improved Algorithm XYZ could solve instances with much larger size. We observed that optimal solutions were found for many instances when the algorithm reached the limitations of either computational time or memory. This signals that Table 4: Results of Algorithm ORG Time #.E #.T #.M the algorithm spends a large amount of e#ort verifying the optimality of the solution. In turn, this behavior indicates that (1) developing an improved method of relaxation is necessary for solving problems with a larger size, (2) when the algorithm is terminated, the solution obtained then can be of high quality, and (3) a better heuristic method, which could start with a feasible solution obtained from the branch-and-bound algorithm, would be very important for obtaining the approximate solution to larger problems. Table 5: Results of Algorithm NMD Time #. E #. T #.M Table Results of Algorithm NSS Table 7: Results of Algorithm XYZ Time #.E #.T #.M Table 8: The results of Algorithm ORG Time Table 9: The results of Algorithm XYZ Time Table 10: The results of Algorithm XYZ Time #.E #.T #.M --R A relaxation method for nonconvex quadratically constrained quadratic programs Semidefinite programming relaxations for nonconvex quadratic programs On generalized bisection of n-simplices Handbook of Global Optimization Global Optimization - Deterministic Approaches SIAM Journal on Comput- ing Global Optimization in Action A simplicial branch-and-bound method for solving nonconvex all- quadratic programs A new reformulation-linearization technique for bilinear programming problems A strip-packing algorithm with absolute performance bound 2 Packing of unequal spheres and automated radiosurgical treatment plan- ning Physics and dosimetry of the gamma knife --TR On three-dimensional packing On generalized bisection of <italic>n</italic>-simplices A Strip-Packing Algorithm with Absolute Performance Bound 2 A Simplicial Branch-and-Bound Method for Solving Nonconvex All-Quadratic Programs

LP relaxation;nonconvex quadratic programming;simplicial branch-and-bound algorithm;heuristic algorithms;unequal sphere packing problem

606457

Computing iceberg concept lattices with TITANIC.

We introduce the notion of iceberg concept lattices and show their use in knowledge discovery in databases. Iceberg lattices are a conceptual clustering method, which is well suited for analyzing very large databases. They also serve as a condensed representation of frequent itemsets, as starting point for computing bases of association rules, and as a visualization method for association rules. Iceberg concept lattices are based on the theory of Formal Concept Analysis, a mathematical theory with applications in data analysis, information retrieval, and knowledge discovery. We present a new algorithm called TITANIC for computing (iceberg) concept lattices. It is based on data mining techniques with a level-wise approach. In fact, TITANIC can be used for a more general problem: Computing arbitrary closure systems when the closure operator comes along with a so-called weight function. The use of weight functions for computing closure systems has not been discussed in the literature up to now. Applications providing such a weight function include association rule mining, functional dependencies in databases, conceptual clustering, and ontology engineering. The algorithm is experimentally evaluated and compared with Ganter's Next-Closure algorithm. The evaluation shows an important gain in efficiency, especially for weakly correlated data.

Introduction Since its introduction, Association Rule Mining [1], has become one of the core data mining tasks, and has attracted tremendous interest among data mining researchers and practitioners. It has an elegantly simple problem statement, that is, to find the set of all subsets of items (called itemsets) that frequently occur in many database records or transactions, and to extract the rules telling us how a subset of items influences the presence of another subset. The prototypical application of associations is in market basket analysis, where the items represent products and the records the point-of-sales data at large grocery or departmental stores. These kinds of database are generally sparse, i.e., the longest frequent itemsets are relatively short. However there are many real-life datasets that very dense, i.e., they contain very long frequent itemsets. It is widely recognized that the set of association rules can rapidly grow to be unwieldy, especially as we lower the frequency requirements. The larger the set of frequent itemsets the more the number of rules presented to the user, many of which are redundant. This is true even for sparse datasets, but for dense datasets it is simply not feasible to mine all possible frequent itemsets, let alone to generate rules between itemsets. In such datasets one typically finds an exponential number of frequent itemsets. For example, finding long itemsets of length or 40 is not uncommon [2]. In this paper we show that it is not necessary to mine all frequent itemsets to guarantee that all non-redundant association rules will be found. We show that it is sufficient to consider only the closed frequent itemsets (to be defined later). Further, all non-redundant rules are found by only considering rules among the closed frequent itemsets. The set of closed frequent itemsets is a lot smaller than the set of all frequent itemsets, in some cases by 3 or more orders of magnitude. Thus even in dense domains we can guarantee completeness, i.e., all non-redundant association rules can be found. The main computation intensive step in this process is to identify the closed frequent itemsets. It is not possible to generate this set using Apriori-like [1] bottom-up search methods that examine all subsets of a frequent itemset. Neither is it possible to mine these sets using algorithms for mining maximal frequent patterns like MaxMiner [2] or Pincer-Search [9], since to find the closed itemsets all subsets of the maximal frequent itemsets would have to be examined. We introduce CHARM, an efficient algorithm for enumerating the set of all closed frequent itemsets. CHARM is unique in that it simultaneously explores both the itemset space and transaction space, unlike all previous association mining methods which only exploit the itemset search space. Furthermore, CHARM avoids enumerating all possible subsets of a closed itemset when enumerating the closed frequent sets. The exploration of both the itemset and transaction space allows CHARM to use a novel search method that skips many levels to quickly identify the closed frequent itemsets, instead of having to enumerate many non-closed subsets. Further, CHARM uses a two-pronged pruning strategy. It prunes candidates based not only on subset infrequency (i.e., no extensions of an infrequent are tested) as do all association mining methods, but it also prunes candidates based on non-closure property, i.e., any non-closed itemset is pruned. Finally, CHARM uses no internal data structures like Hash-trees [1] or Tries [3]. The fundamental operation used is an union of two itemsets and an intersection of two transactions lists where the itemsets are contained. An extensive set of experiments confirms that CHARM provides orders of magnitude improvement over existing methods for mining closed itemsets, even over methods like AClose [14], that are specifically designed to mine closed itemsets. It makes a lot fewer database scans than the longest closed frequent set found, and it scales linearly in the number of transactions and also is also linear in the number of closed itemsets found. The rest of the paper is organized as follows. Section 2 describes the association mining task. Section 3 describes the benefits of mining closed itemsets and rules among them. We present CHARM in Section 4. Related work is discussed in Section 5. We present experiments in Section 6 and conclusions in Section 7. Association Rules The association mining task can be stated as follows: Let I = f1; 2; ; mg be a set of items, and let ng be a set of transaction identifiers or tids. The input database is a binary relation - I T . If an item i occurs in a transaction t, we write it as (i; t) 2 -, or alternately as i-t. Typically the database is arranged as a set of transaction, where each transaction contains a set of items. For example, consider the database shown in Figure 1, used as a running example throughout this paper. Here I = fA; C; D;T ; Wg, and 6g. The second transaction can be represented as fC-2; D-2; W -2g; all such pairs from all transactions, taken together form the binary relation -. A set X I is also called an itemset, and a set Y T is called a tidset. For convenience we write an itemset C; Wg as ACW , and a tidset f2; 4; 5g as 245. The support of an itemset X , denoted (X), is the number of transactions in which it occurs as a subset. An itemset is frequent if its support is more than or equal to a user-specified minimum support (minsup) value, i.e., if (X) minsup. An association rule is an expression X 1 are itemsets, and X 1 \ ;. The support of the rule is given as (i.e., the joint probability of a transaction containing both X 1 and X 2 ), and the confidence as (i.e., the conditional probability that a transaction contains X 2 , given that it contains rule is frequent if the itemset rule is confident if its confidence is greater than or equal to a user-specified minimum confidence (minconf) value, i.e, p minconf. The association rule mining task consists of two steps [1]: 1) Find all frequent itemsets, and 2) Generate high confidence rules. Finding frequent itemsets This step is computationally and I/O intensive. Consider Figure 1, which shows a bookstore database with six customers who buy books by different authors. It shows all the frequent itemsets with and CDW are the maximal-by-inclusion frequent itemsets (i.e., they are not a subset of any other frequent itemset). be the number of items. The search space for enumeration of all frequent itemsets is 2 m , which is exponential in m. One can prove that the problem of finding a frequent set of a certain size is NP-Complete, by reducing it to the balanced bipartite clique problem, which is known to be NP-Complete [8, 18]. However, if we assume that there is a bound on the transaction length, the task of finding all frequent itemsets is essentially linear in the database size, since the overall complexity in this case is given as O(r n 2 l ), where is the number of transactions, l is the length of the longest frequent itemset, and r is the number of maximal frequent itemsets. A C D T W A C D W A C T W C D W A C T W A C D T W541 ALL FREQUENT ITEMSETS W, CW A, D, T, AC, AW CD, CT, ACW 100% 50% (3) AT, DW, TW, ACT, ATW Itemsets Support CTW, CDW, ACTW Items Transcation Jane Austen Agatha Christie Sir Arthur Conan Doyle P. G. Wodehouse Mark Twain Figure 1: Generating Frequent Itemsets Generating confident rules This step is relatively straightforward; rules of the form X 0 p are generated for all frequent itemsets X (where minconf. For an itemset of size k there are potentially confident rules that can be generated. This follows from the fact that we must consider each subset of the itemset as an antecedent, except for the empty and the full itemset. The complexity of the rule generation step is thus O(s 2 l ), where s is the number of frequent itemsets, and l is the longest frequent itemset (note that s can be O(r 2 l ), where r is the number of maximal frequent itemsets). For example, from the frequent itemset ACW we can generate 6 possible rules (all of them have support of 4): A 1:0 C, and CW 0:8 A. 3 Closed Frequent Itemsets In this section we develop the concept of closed frequent itemsets, and show that this set is necessary and sufficient to capture all the information about frequent itemsets, and has smaller cardinality than the set of all frequent itemsets. 3.1 Partial Order and Lattices We first introduce some lattice theory concepts (see [4] for a good introduction). Let P be a set. A partial order on P is a binary relation , such that for all x; the relation is: 1) Reflexive: x x. 2) Anti-Symmetric: x y and y x, implies y. x z. The set P with the relation is called an ordered set, and it is denoted as a pair (P; ). We write x < y if x y and x 6= y. be an ordered set, and let S be a subset of P . An element u 2 P is an upper bound of S if s u for all is a lower bound of S if s l for all s 2 S. The least upper bound is called the join of S, and is denoted as S, and the greatest lower bound is called the meet of S, and is denoted as S. If also write x _ y for the join, and x ^ y for the meet. An ordered set (L; ) is a lattice, if for any two elements x and y in L, the join x _ y and exist. L is a complete lattice if S and S exist for all S L. Any finite lattice is complete. L is called a join semilattice if only the join exists. L is called a meet semilattice if only the meet exists. Let P denote the power set of S (i.e., the set of all subsets of S). The ordered set (P(S); ) is a complete lattice, where the meet is given by set intersection, and the join is given by set union. For example the partial orders (P(I); ), the set of all possible itemsets, and (P(T ); ), the set of all possible tidsets are both complete lattices. The set of all frequent itemsets, on the other hand, is only a meet-semilattice. For example, consider Figure 2, which shows the semilattice of all frequent itemsets we found in our example database (from Figure 1). For any two itemsets, only their meet is guaranteed to be frequent, while their join may or may not be frequent. This follows from AC AW AT CD CT CW DW TW ACW ACT ACTW A (D x 2456) Figure 2: Meet Semi-lattice of Frequent Itemsets the well known principle in association mining that, if an itemset is frequent, then all its subsets are also frequent. For example, is frequent. For the join, while AC _ ACDW is not frequent. 3.2 Closed Itemsets Let the binary relation - I T be the input database for association mining. Let X I, and Y T . Then the mappings define a Galois connection between the partial orders (P(I); ) and (P(T ); ), the power sets of I and T , respec- tively. We denote a (X; t(X)) pair as X t(X), and a (i(Y Figure 3 illustrates the two mappings. The mapping t(X) is the set of all transactions (tidset) which contain the itemset X , similarly i(Y ) is the itemset that is contained in all the transactions in Y . For example, t(ACW In terms of individual elements x2X t(x), and i(Y y2Y i(y). For example The Galois connection satisfies the following properties (where X;X 1 For example, for 245 2456, we have Let S be a set. A function c : P(S) 7! P(S) is a closure operator on S if, for all X;Y S, c satisfies the following properties: subset X of S is called closed if I and Y T . Let c it (X) denote the composition of the two mappings Dually, let c ti (Y are both closure operators on itemsets and tidsets respectively. We define a closed itemset as an itemset X that is the same as its closure, i.e., (X). For example the itemset ACW is closed. A closed tidset is a tidset For example, the tidset 1345 is closed. The mappings c it and c ti , being closure operators, satisfy the three properties of extension, monotonicity, and idempotency. We also call the application of i - t or t - i a round-trip. Figure 4 illustrates this round-trip starting with an itemset X . For example, let then the extension property says that X is a subset of its closure, since c it we conclude that AC is not closed. On the other hand, the idempotency property say that once we map an itemset to the tidset that contains TRANSACTIONS ITEMS Y Figure 3: Galois Connection it ITEMS TRANSACTIONS Figure 4: Closure Operator: Round-Trip it, and then map that tidset back to the set of items common to all tids in the tidset, we obtain a closed itemset. After this no matter how many such round-trips we make we cannot extend a closed itemset. For example, after one round-trip for AC we obtain the closed itemset ACW . If we perform another round-trip on ACW , we get c it (ACW For any closed itemset X , there exists a closed tidset given by Y , with the property that (conversely, for any closed tidset there exists a closed itemset). We can see that X is closed by the fact that then plugging thus X is closed. Dually, Y is closed. For example, we have seen above that for the closed itemset ACW the associated closed tidset is 1345. Such a closed itemset and closed tidset pair X Y is called a concept. Figure 5: Galois Lattice of Concepts Figure Frequent Concepts A concept X 1 Y 1 is a subconcept of X 2 Y 2 , denoted as Let B(-) denote the set of all possible concepts in the database, then the ordered set (B(-); ) is a complete lattice, called the Galois lattice. For example, Figure 5 shows the Galois lattice for our example database, which has a total of concepts. The least element is the concept C 123456 and the greatest element is the concept ACDTW 5. Notice that the mappings between the closed pairs of itemsets and tidsets are anti-isomorphic, i.e., concepts with large cardinality itemsets have small tidsets, and vice versa. 3.3 Closed Frequent Itemsets vs. All Frequent Itemsets We begin this section by defining the join and meet operation on the concept lattice (see [5] for the formal proof): The set of all concepts in the database relation -, given by (B(-); ) is a (complete) lattice with join and meet given by For the join and meet of multiple concepts, we simply take the unions and joins over all of them. For example, consider the join of two concepts, (ACDW 45) _ (CDT On the other hand their meet is given as, (ACDW Similarly, we can perform multiple concept joins or meets; for example, (CT 1356)_ We define the support of a closed itemset X or a concept X Y as the cardinality of the closed tidset closed itemset or a concept is frequent if its support is at least minsup. Figure 6 shows all the frequent concepts with tidset cardinality at least 3). The frequent concepts, like the frequent itemsets, form a meet-semilattice, where the meet is guaranteed to exist, while the join may not. Theorem 1 For any itemset X , its support is equal to the support of its closure, i.e., PROOF: The support of an itemset X is the number of transactions where it appears, which is exactly the cardinality of the tidset t(X), i.e., it (X))j, to prove the lemma, we have to show that Since c ti is closure operator, it satisfies the extension property, i.e., t(X) c ti Thus t(X) t(c it (X)). On the other hand since c it is also a closure operator, X c it (X), which in turn implies that due to property 1) of Galois connections. Thus This lemma states that all frequent itemsets are uniquely determined by the frequent closed itemsets (or frequent concepts). Furthermore, the set of frequent closed itemsets is bounded above by the set of frequent itemsets, and is typically much smaller, especially for dense datasets (where there can be orders of magnitude differences). To illustrate the benefits of closed itemset mining, contrast Figure 2, showing the set of all frequent itemsets, with Figure 6, showing the set of all closed frequent itemsets (or concepts). We see that while there are only 7 closed frequent itemsets, there are 19 frequent itemsets. This example clearly illustrates the benefits of mining the closed frequent itemsets. 3.4 Rule Generation Recall that an association rule is of the form X 1 Its support equals its confidence is given as )j. We are interested in finding all high support (at least minsup) and high confidence rules (at least minconf). It is widely recognized that the set of such association rules can rapidly grow to be unwieldy. The larger the set of frequent itemsets the more the number of rules presented to the user. However, we show below that it is not necessary to mine rules from all frequent itemsets, since most of these rules turn out to be redundant. In fact, it is sufficient to consider only the rules among closed frequent itemsets (or concepts), as stated in the theorem below. Theorem 2 The rule X 1 is equivalent to the rule c it (X 1 PROOF: It follows immediately from the fact that the support of an itemset X is equal to the support of its closure c it (X), i.e., (X)). Using this fact we can show that There are typically many (in the worst case, an exponential number of) frequent itemsets that map to the same closed frequent itemset. Let's assume that there are n itemsets, given by the set S 1 , whose closure is C 1 and m itemsets, given by the set S 2 , whose closure is C 2 , then we say that all n m 1 rules between two non-closed itemsets directed from S 1 to S 2 are redundant. They are all equivalent to the rule C 1 . Further the m n 1 rules directed from S 2 to S 1 are also redundant, and equivalent to the rule C 2 . For example, looking at Figure 2 we find that the itemsets D and CD map to the closed itemset CD, and the itemsets W and CW map to the closed itemset CW . Considering rules from the former to latter set we find that the rules D 3=4 CD 3=4 are all equivalent to the rule between closed itemsets CD 3=4 CW . On the other hand, if we consider the rules from the latter set to the former, we find that W 3=5 ! D are all equivalent to the rule CW 5=6 CD. We should present to the user the most general rules (other rules are more specific; they contain one or more additional items in the antecedent or consequent) for each direction, i.e., the rules D 3=4 !W and W 3=5 0:6). Thus using the closed frequent itemsets we would generate only 2 rules instead of 8 rules normally generated between the two sets. To get an idea of the number of redundant rules mined in traditional association mining, for one dataset (mushroom), at 10% minimum support, we found 574513 frequent itemsets, out of which only were closed, a reduction of more than 100 times! 4 CHARM: Algorithm Design and Implementation Having developed the main ideas behind closed association rule mining, we now present CHARM, an efficient algorithm for mining all the closed frequent itemsets. We will first describe the algorithm in general terms, independent of the implementation details. We then show how the algorithm can be implemented efficiently. This separation of design and implementation aids comprehension, and allows the possibility of multiple implementations. CHARM is unique in that it simultaneously explores both the itemset space and tidset space, unlike all previous association mining methods which only exploit the itemset space. Furthermore, CHARM avoids enumerating all possible subsets of a closed itemset when enumerating the closed frequent sets, which rules out a pure bottom-up search. This property is important in mining dense domains with long frequent itemsets, where bottom-up approaches are not practical (for example if the longest frequent itemset is l, then bottom-up search enumerates all 2 l frequent subsets). The exploration of both the itemset and tidset space allows CHARM to use a novel search method that skips many levels to quickly identify the closed frequent itemsets, instead of having to enumerate many non-closed subsets. Further, CHARM uses a two-pronged pruning strategy. It prunes candidates based not only on subset infrequency (i.e., no extensions of an infrequent itemset are tested) as do all association mining methods, but it also prunes branches based on non-closure property, i.e., any non-closed itemset is pruned. Finally, CHARM uses no internal data structures like Hash-trees [1] or Tries [3]. The fundamental operation used is an union of two itemsets and an intersection of their tidsets. A ACDT ACDTW CD AC ACD CT ACT CDW AD AT AW CW ACW ADT ADW ATW ACTW ACDW Figure 7: Complete Subset Lattice Consider Figure 7 which shows the complete subset lattice (only the main parent link has been shown to reduce clutter) over the five items in our example database (see Figure 1). The idea in CHARM is to process each lattice node to test if its children are frequent. All infrequent, as well as non-closed branches are pruned. Notice that the children of each node are formed by combining the node by each of its siblings that come after it in the branch ordering. For example, A has to be combined with its siblings C; D;T and W to produce the children AC;AD;AT and AW . A sibling need not be considered if it has already been pruned because of infrequency or non-closure. While a lexical ordering of branches is shown in the figure, we will see later how a different branch ordering (based on support) can improve the performance of CHARM (a similar observation was made in MaxMiner [2]). While many search schemes are possible (e.g., breadth-first, depth-first, best-first, or other hybrid search), CHARM performs a depth-first search of the subset lattice. 4.1 CHARM: Algorithm Design In this section we assume that for any itemset X , we have access to its tidset t(X), and for any tidset Y we have access to its itemset i(Y ). How to practically generate t(X) or i(Y ) will be discussed in the implementation section. CHARM actually enumerates all the frequent concepts in the input database. Recall that a concept is given as a closed itemset, and Y = t(X) is a closed tidset. We can start the search for concepts over the tidset space or the itemset space. However, typically the number of items is a lot smaller than the number of transactions, and since we are ultimately interested in the closed itemsets, we start the search with the single items, and their associated tidsets. U U it it it it it it it it ITEMS TRANSACTIONS ITEMS TRANSACTIONS ITEMS TRANSACTIONS ITEMS TRANSACTIONS Figure 8: Basic Properties of Itemsets and Tidsets 4.1.1 Basic Properties of Itemset-Tidset Pairs be a one-to-one mapping from itemsets to integers. For any two itemsets X 1 and X 2 , we say defines a total order over the set of all itemsets. For example, if f denotes the lexicographic ordering, then itemset AC < AD. As another example, if f sorts itemsets in increasing order of their support, then AD < AC if support of AD is less than the support of AC. Let's assume that we are processing the branch X 1 t(X 1 ), and we want to combine it with its sibling X 2 t(X 2 ). That is X 1 X 2 (under a suitable total order f ). The main computation in CHARM relies on the following properties. 1. If t(X 1 Thus we can simply replace every occurrence of X 1 with further consideration, since its closure is identical to the closure of In other words, we treat as a composite itemset. 2. If t(X 1 Here we can replace every occurrence of X 1 with occurs in any transaction, then X 2 always occurs there too. But since t(X 1 generates a different closure. 3. If t(X 1 In this we replace every occurrence of X 2 with produces a different closure, and it must be retained. 4. If In this case, nothing can be eliminated; both X 1 and X 2 lead to different closures. Figure 8 pictorially depicts the four cases. We see that only closed tidsets are retained after we combine two itemset- tidset pairs. For example, if the two tidsets are equal, one of them is pruned (Property 1). If one tidset is a subset of another, then the resulting tidset is equal to the smaller tidset from the parent and we eliminate that parent (Properties 2 and 3). Finally if the tidsets are unequal, then those two and their intersection are all closed. Example Before formally presenting the algorithm, we show how the four basic properties of itemset-tidset pairs are exploited in CHARM to mine the closed frequent itemsets. A x 1345 Figure 9: CHARM: Lexicographic Order A x 1345 Figure 10: CHARM:Sorted by Increasing Support Consider Figure 9. Initially we have five branches, corresponding to the five items and their tidsets from our example database (recall that we used To generate the children of item A (or the pair A 1345) we need to combine it with all siblings that come after it. When we combine two pairs the resulting pair is given as In other words we need to perform the intersection of corresponding tidsets whenever we combine two or more itemsets. When we try to extend A with C, we find that property 2 is true, i.e., t(C). We can thus remove A and replace it with AC. Combining A with D produces an infrequent set ACD, which is pruned. Combination with T produces the pair ACT 135; property 4 holds here, so nothing can be pruned. When we try to combine A with W we find that t(A) t(W ). According to property 2, we replace all unpruned occurrences of A with AW . Thus AC becomes ACW and ACT becomes ACTW . At this point there is nothing further to be processed from the A branch of the root. We now start processing the C branch. When we combine C with D we observe that property 3 holds, i.e., t(C) t(D). This means that wherever D occurs C always occurs. Thus D can be removed from further consideration, and the entire D branch is pruned; the child CD replaces D. Exactly the same scenario occurs with T and W . Both the branches are pruned and are replaced by CT and CW as children of C. Continuing in a depth-first manner, we next process the node CD. Combining it with CT produces an infrequent itemset CDT , which is pruned. Combination with CW produces CDW and since property 4 holds, nothing can be removed. Similarly the combination of CT and CW produces CTW . At this point all branches have been processed. Finally, we remove CTW 135 since it is contained in ACTW 135. As we can see, in just 10 steps we have identified all 7 closed frequent itemsets. 4.1.2 CHARM: Pseudo-Code Description Having illustrated the workings of CHARM on our example database, we now present the pseudo-code for the algorithm itself. The algorithm starts by initializing the set of nodes to be examined to the frequent single items and their tidsets in Line 1. The main computation is performed in CHARM-EXTEND which returns the set of closed frequent itemsets C. CHARM-EXTEND is responsible for testing each branch for viability. It extracts each itemset-tidset pair in the current node set Nodes (X i t(X i ), Line 3), and combines it with the other pairs that come after it (X j t(X j ), Line 5) according to the total order f (we have already seen an example of lexical ordering in Figure 9; we will look at support based ordering below). The combination of the two itemset-tidset pairs is computed in Line 6. The routine CHARM-PROPERTY tests the resulting set for required support and also applies the four properties discussed above. Note that this routine may modify the current node set by deleting itemset-tidset pairs that are already contained in other pairs. It also inserts the newly generated children frequent pairs in the set of new nodes NewN . If this set is non-empty we recursively process it in depth-first manner (Line 8). We then insert the possibly extended itemset X, of X i , in the set of closed itemsets, since it cannot be processed further; at this stage any closed itemset containing X i has already been generated. We then return to Line 3 to process the next (unpruned) branch. The routine CHARM-PROPERTY simply tests if a new pair is frequent, discarding it if it is not. It then tests each of the four basic properties of itemset-tidset pairs, extending existing itemsets, removing some subsumed branches from the current set of nodes, or inserting new pairs in the node set for the next (depth-first) step. CHARM (- I T , minsup): 1. 2. CHARM-EXTEND (Nodes, C) CHARM-EXTEND (Nodes, C): 3. for each X i t(X i ) in Nodes 4. 5. for each X j t(X j ) in Nodes, with f(j) > f(i) 7. CHARM-PROPERTY(Nodes, NewN) 8. if NewN 9. is not subsumed CHARM-PROPERTY (Nodes, NewN): 10. if (jYj minsup) then 11. if t(X i 12. Remove X j from Nodes 13. Replace all X i with X 14. else if 15. Replace all X i with X 16. else if 17. Remove X j from Nodes 18. Add XY to NewN 19. else if 20. Add XY to NewN Figure 11: The CHARM Algorithm 4.1.3 Branch Reordering We purposely let the itemset-tidset pair ordering function in Line 5 remain unspecified. The usual manner of processing is in lexicographic order, but we can specify any other total order we want. The most promising approach is to sort the itemsets based on their support. The motivation is to increase opportunity for non-closure based pruning of itemsets. A quick look at Properties 1 and 2 tells us that these two situations are preferred over the other two cases. For Property 1, the closure of the two itemsets is equal, and thus we can discard X j and replace X i with . For Property 2, we can still replace X i with . Note that in both these cases we do not insert anything in the new nodes! Thus the more the occurrence of case 1 and 2, the fewer levels of search we perform. In contrast, the occurrence of cases 3 and 4 results in additions to the set of new nodes, requiring additional levels of processing. Note that the reordering is applied for each new node set, starting with the initial branches. Since we want t(X i that we should sort the itemsets in increasing order of their support. Thus larger tidsets occur later in the ordering and we maximize the occurrence of Properties 1 and 2. By similar reasoning, sorting by decreasing order of support doesn't work very well, since it maximizes the occurrence of Properties 3 and 4, increasing the number of levels of processing. Example Figure 10 shows how CHARM works on our example database if we sort itemsets in increasing order of support. We will use the pseudo-code to illustrate the computation. We initialize Nodes = fA 1345; D 2456; T in Line 1. At Line 3 we first process the branch A 1345 (we set in Line 4); it will be combined with the remaining siblings in Line 5. AD is not frequent and is pruned. We next look at A and T ; since t(A) 6= t(T ), we simply insert AT in NewN . We next find that t(A) t(W ). Thus we replace all occurrences of A with AW (thus which means that we also change AT in NewN to ATW . Looking at A and C, we find that t(A) t(C). Thus AW becomes ACW in NewN becomes ACTW . At this point CHARM-EXTEND is invoked with the non-empty NewN (Line 8). But since there is only one element, we immediately exit after adding ACTW 135 to the set of closed frequent itemsets C (Line 9). When we return, the A branch has been completely processed, and we add to C. The other branches are examined in turn, and the final C is produced as shown in Figure 10. One final note; the pair CTW 135 produced from the T branch is not closed, since it is subsumed by ACTW 135, and it is eliminated in Line 9. 4.2 CHARM: Implementation Details We now describe the implementation details of CHARM and how it departs from the pseudo-code in some instances for performance reasons. Data Format Given that we are manipulating itemset-tidset pairs, and that the fundamental operation is that of intersecting two tidsets, CHARM uses a vertical data format, where we maintain a disk-based list for each item, listing the tids where that item occurs. In other words, the data is organized so that we have available on disk the tidset for each item. In contrast most of the current association algorithms [1, 2, 3] assume a horizontal database layout, consisting of a list of transactions, where each transaction has an identifier followed by a list of items in that transaction. The vertical format has been shown to be successful for association mining. It has been used in Partition [16], in (Max)Eclat and (Max)Clique [19], and shown to lead to very good performance. In fact, the Vertical algorithm [15] was shown to be the best approach (better than horizontal) when tightly integrating association mining with database systems. The benefits of using the vertical format have further been demonstrated in Monet [12], a new high-performance database system for query-intensive applications like OLAP and data mining. Intersections and Subset Testing Given the availability of vertical tidsets for each itemset, the computation of the tidset intersection for a new combination is straightforward. All it takes is a linear scan through the two tidsets, storing matching tids in a new tidset. For example, we have The main question is how to efficiently compute the subset information required while applying the four properties. At first this might appear like an expensive operation, but in fact in the vertical format, it comes for free. When intersecting two tidsets we keep track of the number of mismatches in both the lists, i.e., the cases when a tid occurs in one list but not in the other. Let m(X 1 ) and m(X 2 ) denote the number of mismatches in the tidsets for itemsets and . There are four cases to consider: For t(A) and t(D) from above, and as we can see, t(A) 6= t(D). Next consider which shows that t(A) t(W ). Thus CHARM performs support, subset, equality, and inequality testing simultaneously while computing the intersection itself. Eliminating Non-Closed Itemsets Here we describe a fast method to avoid adding non-closed itemsets to the set of closed frequent itemsets C in Line 9. If we are adding a set X, we have to make sure that there doesn't exist a set C such that X C and both have the same support (MaxMiner [2] faces a similar problem while eliminating non-maximal itemsets). Clearly we want to avoid comparing X with all existing elements in C, for this would lead to a O(jCj 2 ) complexity. The solution is to store C in a hash table. But what hash function to use? Since we want to perform subset checking, we can't hash on the itemset. We could use the support of the itemsets for the hash function. But many unrelated subsets may have the same support. CHARM uses the sum of the tids in the tidset as the hash function, i.e., This reduces the chances of unrelated itemsets being in the same cell. Each hash table cell is a linked list sorted by support as primary key and the itemset as the secondary key (i.e., lexical). Before adding X to C, we hash to the cell, and check if X is a subset of only those itemsets with the same support as X. We found experimentally that this approach adds only a few seconds of additional processing time to the total execution time. Optimized Initialization There is only one significant departure from the pseudo-code in Figure 11. Note that if we initialize the Nodes set in Line 1 with all frequent items, and invoke CHARM-EXTEND then, in the worst case, we might perform n(n 1)=2 tidset intersections, where n is the number of frequent items. If l is the average tidset size in bytes, the amount of data read is l n (n 1)=2 bytes. Contrast this with the horizontal approach that reads only l n bytes. It is well known that many itemsets of length 2 turn out to be infrequent, thus it is clearly wasteful to perform To solve this performance problem we first compute the set of frequent itemsets of length 2, and then we add a simple check in Line 5, so that we combine two items I i and I j only if I i [ I j is known to be frequent. The number of intersections performed after this check is equal to the number of frequent pairs, which is in practice closer to O(n) rather than O(n 2 ). Further this check only has to be done initially only for single items, and not in later stages. We now describe how we compute the frequent itemsets of length 2 using the vertical format. As noted above we clearly cannot perform all intersections between pairs of frequent items. The solution is to perform a vertical to horizontal transformation on-the-fly. For each item I , we scan its tidset into memory. We insert item I in an array indexed by tid for each T 2 t(I). For example, consider the tidset for item A, given as We read the first tid insert A in the array at index 1. We also insert A at indices 3; 4 and 5. We repeat this process for all other items and their tidsets. Figure 12 shows how the inversion process works after the addition of each item and the complete horizontal database recovered from the vertical tidsets for each item. Given the recovered horizontal database it is straightforward to update the count of pairs of items using an upper triangular 2D array. Add C Add D Add W Add T Add A246246246 A A A A A A A A A A A A A A A A A A Figure 12: Vertical-to-Horizontal Database Recovery Memory Management For initialization CHARM scans the database once to compute the frequent pairs of items (note that finding the frequent items is virtually free in the vertical format; we can calculate the support directly from an index array that stores the tidset offsets for each item. If this index is not available, computing the frequent items will take an additional scan). Then, while processing each initial branch in the search lattice it needs to scan single item tidsets from disk for each unpruned sibling. CHARM is fully scalable for large-scale database mining. It implements appropriate memory management in all phases as described next. For example, while recovering the horizontal database, the entire database will clearly not fit in memory. CHARM handles this by only recovering a block of transactions at one time that will fit in memory. Support of item pairs is updated by incrementally processing each recovered block. Note that regardless of the number of blocks, this process requires exactly one database scan over the vertical format (imagine k pointers for each of k tidsets; the pointer only moves forward if the tid is points to belongs to the current block). When the number of closed itemsets itself becomes very large, we cannot hope to keep the set of all closed itemsets C in memory. In this case, the elimination of some non-closed itemsets is done off-line in a post-processing step. Instead of inserting X in C in Line 9, we simply write it to disk along with its support and hash value. In the post-processing step, we read all close itemsets and apply the same hash table searching approach described above to eliminate non-closed itemsets. Since CHARM processes each branch in the search in a depth-first fashion, its memory requirements are not substantial. It has to retain all the itemset-tidsets pairs on the levels of the current left-most branches in the search space. Consider 7 for example. Initially is has to retain the tidsets for fAC;AD;AT ; AWg, fACD;ACT ; ACWg, fACDTWg. Once AC has been processed, the memory requirement shrinks to fAD;AT ;- fADTWg. In any case this is the worst possible situation. In practice the applications of subset infrequency and non-closure properties 1, 2, and 3, prune many branches in the search lattice. For cases where even the memory requirement of depth-first search exceed available memory, it is straightforward to modify CHARM to write temporary tidsets to disk. For example, while processing the AC branch, we might have to write out the tidsets for fAD;AT ; AWg to disk. Another option is to simply re-compute the intersections if writing temporary results is too expensive. 4.3 Correctness and Efficiency Theorem 3 (correctness) The CHARM algorithm enumerates all closed frequent itemsets. PROOF: CHARM correctly identifies all and only the closed frequent itemsets, since its search is based on a complete subset lattice search. The only branches that are pruned as those that either do not have sufficient support, or those that result in non-closure based on the properties of itemset-tidset pairs as outlined at the beginning of this section. Finally CHARM eliminates the few cases of non-closed itemsets that might be generated by performing subsumption checking before inserting anything in the set of all closed frequent itemsets C. Theorem 4 (computational cost) The running time of CHARM is O(l jCj), where l is the average tidset length, and C is the set of all closed frequent itemsets. PROOF: Note that starting with the single items and their associated tidsets, as we process a branch the following cases might occur. Let X c denote the current branch and X s the sibling we are trying to combine it with. We prune the X s branch if t(X c 1). We extend X c to become Finally a new node is only generated if we get a new possibly closed set due to properties 3 and 4. Also note that each new node in fact represents a closed tidset, and thus indirectly represents a closed itemset, since there exists a unique closed itemset for each closed tidset. Thus CHARM performs on the order of O(jCj) intersections (we confirm this via experiments in Section 6; the only extra intersections performed are due to case where CHARM may produce non-closed itemsets like CTW 135, which are eliminated in Line 9). If each tidset is on average of length l, an intersection costs at most 2 l. The total running time of CHARM is thus 2 l jCj or O(l jCj). Theorem 5 (I/O cost) The number of database scans made by CHARM is given as O( jCj is the set of all closed frequent itemsets, I is the set of items, and is the fraction of database that fits in memory. PROOF: The number of database scans required is given as the total memory consumption of the algorithm divided by the fraction of database that will fit in memory. Since CHARM computes on the order of O(jCj) intersections, the total memory requirement of CHARM is O(l jCj), where l is the average length of a tidset. Note that as we perform intersections the size of longer itemsets' tidsets shrinks rapidly, but we ignore such effects in our analysis (it is thus a pessimistic bound). The total database size is l jIj, and the fraction that fits in memory is given as l jIj. The number of data scans is then given as (l jCj)=( l Note that in the worst case jCj can be exponential in jIj, but this is rarely the case in practice. We will show in the experiments section that CHARM makes very few database scans when compared to the longest closed frequent itemset found. 5 Related Work A number of algorithms for mining frequent itemsets [1, 2, 3, 9, 10, 13, 16, 19] have been proposed in the past. Apriori [1] was the first efficient and scalable method for mining associations. It starts by counting frequent items, and during each subsequent pass it extends the current set of frequent itemsets by one more item, until no more frequent itemsets are found. Since it uses a pure bottom-up search over the subset lattice (see Figure 7), it generates all 2 l subsets of a frequent itemset of length l. Other methods including DHP [13], Partition [16], AS-CPA [10], and DIC [3], propose enhancements over Apriori in terms of the number of candidates counted or the number of data scans. But they still have to generate all subsets of a frequent itemset. This is simply not feasible (except for very high support) for the kinds of dense datasets we examine in this paper. We use Apriori as a representative of this class of methods in our experiments. Methods for finding the maximal frequent itemsets include All-MFS [8], which is a randomized algorithm, and as such not guaranteed to be complete. Pincer-Search [9] not only constructs the candidates in a bottom-up manner like Apriori, but also starts a top-down search at the same time. Our previous algorithms (Max)Eclat and Max(Clique) [19, 17] range from those that generate all frequent itemsets to those that generate a few long frequent itemsets and other subsets. MaxMiner [2] is another algorithm for finding the maximal elements. It uses novel superset frequency pruning and support lower-bounding techniques to quickly narrow the search space. Since these methods mine only the maximal frequent itemsets, they cannot be used to generate all possible association rules, which requires the support of all subsets in the traditional approach. If we try to compute the support of all subsets of the maximal frequent itemsets, we again run into the problem of generating all 2 l subsets for an itemset of length l. For dense datasets this is impractical. Using MaxMiner as a representative of this class of algorithms we show that modifying it to compute closed itemsets renders it infeasible for all except very high supports. CD CT AT AD AC A A AT Find Generators Compute Figure 13: AClose Algorithm: Example AClose [14] is an Apriori-like algorithm that directly mines closed frequent itemsets. There are two main steps in AClose. The first is to use a bottom-up search to identify generators, the smallest frequent itemsets that determines a closed itemset via the closure operator c it . For example, in our example database, both c it c it only A is a generator for ACW . All generators are found using a simple modification of Apriori. Each time a new candidate set is generated, AClose computes their support, pruning all infrequent ones. For the remaining sets, it compares the support of each frequent itemset with each of its subsets at the previous level. If the support of an itemset matches the support of any of its subsets, the itemset cannot be a generator and is thus pruned. This process is repeated until no more generators can be produced. The second step in AClose is to compute the closure of all the generators found in the first step. To compute the closure of an itemset we have to perform an intersection of all transactions where it occurs as a subset, i.e., the closure of an itemset X is given as c it is a tid. The closures for all generators can be computed in just one database scan, provided all generators fit in memory. Nevertheless computing closures this way is an expensive operation. Figure 13 shows the working of AClose on our example database. After generating candidate pairs of items, it is determined that AD and DT are not frequent, so they are pruned. The remaining frequent pairs are pruned if their support matches the support of any of their subsets. AC;AW are pruned, since their support is equal to the support of A. CD is pruned because of D, CT because of T , and CW because of W . After this pruning, we find that no more candidates can be generated, marking the end of the first step. In the second step, AClose computes the closure of all unpruned itemsets. Finally some duplicate closures are removed (e.g., both AT and TW produce the same closure). We will show that while AClose is much better than Apriori, it is uncompetitive with CHARM. A number of previous algorithms have been proposed for generating the Galois lattice of concepts [5, 6]. These algorithms will have to be adapted to enumerate only the frequent concepts. Further, they have only been studied on very small datasets. Finally the problem of generating a basis (a minimal non-redundant rule set) for association rules was discussed in [18] (but no algorithms were given), which in turn is based on the theory developed in [7, 5, 11]. 6 Experimental Evaluation We chose several real and synthetic datasets for testing the performance of CHARM. The real datasets are the same as those used in MaxMiner [2]. All datasets except the PUMS (pumsb and pumsb*) sets, are taken from the UC Irvine Machine Learning Database Repository. The PUMS datasets contain census data. pumsb* is the same as pumsb without items with 80% or more support. The mushroom database contains characteristics of various species of mushrooms. Finally the connect and chess datasets are derived from their respective game steps. Typically, these real datasets are very dense, i.e., they produce many long frequent itemsets even for very high values of support. These datasets are publicly available from IBM Almaden (www.almaden.ibm.com/cs/quest/demos.html). We also chose a few synthetic datasets (also available from IBM Almaden), which have been used as benchmarks for testing previous association mining algorithms. These datasets mimic the transactions in a retailing environment. Usually the synthetic datasets are sparse when compared to the real sets, but we modified the generator to produce longer frequent itemsets. Avg. Record Length # Records Scaleup DB Size chess 76 37 3,196 31,960 connect 130 43 67,557 675,570 mushroom 120 23 8,124 81,240 pumsb* 7117 50 49,046 490,460 pumsb 7117 74 49,046 490,460 Table 1: Database Characteristics Table also shows the characteristics of the real and synthetic datasets used in our evaluation. It shows the number of items, the average transaction length and the number of transactions in each database. It also shows the number of records used for the scaleup experiments below. As one can see the average transaction size for these databases is much longer than conventionally used in previous literature. All experiments described below were performed on a 400MHz Pentium PC with 256MB of memory, running RedHat Linux 6.0. Algorithms were coded in C++. 6.1 Effect of Branch Ordering Figure 14 shows the effect on running time if we use various kinds of branch orderings in CHARM. We compare three ordering methods - lexicographical order, increasing by support, and decreasing by support. We observe that decreasing order is the worst. On the other hand processing branch itemsets in increasing order is the best; it is about a factor of 1.5 times better than lexicographic order and about 2 times better than decreasing order. Similar results were obtained for synthetic datasets. All results for CHARM reported below use the increasing branch ordering, since it is the best. Time per Closed (sec) Minimum Support (%) chess Decreasing Lexicographic Increasing48121695.596.597.5Time per Closed (sec) Minimum Support (%) connect Decreasing Lexicographic Increasing135715253545Time per Closed (sec) Minimum Support (%) mushroom Decreasing Lexicographic Increasing0.51.52.53.54.55.5949698Time per Closed (sec) Minimum Support (%) pumsb Decreasing Lexicographic Time per Closed (sec) Minimum Support (%) pumsb* Decreasing Lexicographic Increasing Figure 14: Branch Ordering100100001e+06657585Number of Elements Minimum Support (%) chess Frequent Closed Maximal1001000095.596.597.5Number of Elements Minimum Support (%) connect Frequent Closed Maximal100100001e+0615253545Number of Elements Minimum Support (%) mushroom Frequent Closed Maximal101000949698Number of Elements Minimum Support (%) pumsb Frequent Closed Maximal101000100000455565Number of Elements Minimum Support (%) pumsb* Frequent Closed Maximal Figure 15: Set Cardinality481260708090 Longest Freq Scans Minimum Support (%) chess chessLF chessDBI chessDBL1357995.596.597.5Longest Freq Scans Minimum Support (%) connect connectLF connectDBI Longest Freq Scans Minimum Support (%) mushroom mushroomLF mushroomDBI mushroomDBL1357949698Longest Freq Scans Minimum Support (%) pumsb pumsbLF pumsbDBI pumsbDBL261014455565Longest Freq Scans Minimum Support (%) pumsb* pumsb*LF pumsb*DBI pumsb*DBL Figure Longest Frequent Item-set vs. Database Scans (D- BI=Increasing, DBL=Lexical Order) 11001000030507090Total Time (sec) Minimum Support (%) chess AClose CMaxMiner Charm Total Time (sec) Minimum Support (%) connect AClose CMaxMiner Charm Total Time (sec) Minimum Support (%) mushroom AClose CMaxMiner Charm MaxMiner0.1101000758595Total Time (sec) Minimum Support (%) pumsb AClose CMaxMiner Charm Total Time (sec) Minimum Support (%) pumsb* AClose CMaxMiner Charm MaxMiner10100000.40.81.2Total Time (sec) Minimum Support (%) AClose CMaxMiner Charm MaxMiner1010000.20.611.4 Total Time (sec) Minimum Support (%) AClose CMaxMiner Charm MaxMiner101000012Total Time (sec) Minimum Support (%) AClose CMaxMiner Charm MaxMiner Figure 17: CHARM versus Apriori, AClose, CMaxMiner and MaxMiner 6.2 Number of Frequent, Closed, and Maximal Itemsets Figure 15 shows the total number of frequent, closed and maximal itemsets found for various support values. It should be noted that the maximal frequent itemsets are a subset of the closed frequent itemsets (the maximal frequent itemsets must be closed, since by definition they cannot be extended by another item to yield a frequent itemset). The closed frequent itemsets are, of course, a subset of all frequent itemsets. Depending on the support value used the set of maximal itemsets is about an order of magnitude smaller than the set of closed itemsets, which in turn is an order of magnitude smaller than the set of all frequent itemsets. Even for very low support values we find that the difference between maximal and closed remains around a factor of 10. However the gap between closed and all frequent itemsets grows more rapidly. For example, for mushroom at 10% support, the gap was a factor of 100; there are 558 maximal, 4897 closed and 574513 frequent itemsets. 6.3 CHARM versus MaxMiner, AClose, and Apriori Here we compare the performance of CHARM against previous algorithms. MaxMiner only mines maximal frequent itemsets, thus we augmented it by adding a post-processing routine that uses the maximal frequent itemsets to generate all closed frequent itemsets. In essence we generate all subsets of the maximal itemsets, eliminating an itemset if its support equals any of its subsets. The augmented algorithm is called CMaxMiner. The AClose method is the only extant method that directly mines closed frequent itemsets. Finally Apriori mines only the frequent itemsets. It would require a post-processing step to compute the closed itemsets, but we do not add this cost to its running time. Figure 17 shows how CHARM compares to the previous methods on all the real and synthetic databases. We find that Apriori cannot be run except for very high values of support. Even in these cases CHARM is 2 or 3 orders of magnitude better. Generating all subsets of frequent itemsets clearly takes too much time. AClose can perform an order of magnitude better than Apriori for low support values, but for high support values it can in fact be worse than Apriori. This is because for high support the number of frequent itemsets is not too much, and the closure computing step of AClose dominates computation time. Like Apriori, AClose couldn't be run for very low values of support. The generator finding step finds too many generators to be kept in memory. CMaxMiner, the augmented version of MaxMiner, suffers a similar fate. Generating all subsets and testing them for closure is not a feasible strategy. CMaxMiner cannot be run for low supports, and for the cases where it can be run, it is 1 to 2 orders of magnitude slower than CHARM. Only MaxMiner was able to run for all the values of support that CHARM can handle. Except for high support values, where CHARM is better, MaxMiner can be up to an order of magnitude faster than CHARM, and is typically a factor of 5 or 6 times better. The difference is attributable to the fact that the set of maximal frequent itemsets is typically an order of magnitude smaller than the set of closed frequent itemsets. But it should be noted that, since MaxMiner only mines maximal itemsets, it cannot be used to produce association rules. In fact, any attempt to calculate subset frequency adds a lot of overhead, as we saw in the case of CMaxMiner. These experiments demonstrate that CHARM is extremely effective in efficiently mining all the closed frequent itemsets, and is able to gracefully handle very low support values, even in dense datasets. 6.4 Scaling Properties of CHARM Figure shows the time taken by CHARM per closed frequent itemset found. The support values are the same as the ones used while comparing CHARM with other methods above. As we lower the support more closed itemsets are found, but the time spent per element decreases, indicating that the efficiency of CHARM increases with decreasing support. Figure 19 shows the number of tidset intersections performed per closed frequent itemset generated. The ideal case in the graph corresponds to the case where we perform exactly the same number of intersections as there are closed frequent itemsets, i.e., a ratio of one. We find that for both connect and chess the number of intersections performed by CHARM are close to ideal. CHARM is within a factor of 1.06 (for chess) to 2.6 (for mushroom) times the ideal. This confirms the computational efficiency claims we made before. CHARM indeed performs O(jCj) intersections. Figure shows the number of database scans made by CHARM compared to the length of the longest closed frequent itemset found for the real datasets. The number of database scans for CHARM was calculated by taking the sum of the lengths of all tidsets scanned from disks, and then dividing the sum by the tidset lengths for all items in the database. The number reported is pessimistic in the sense that we incremented the sum even though we may have space in memory or we may have scanned the tidset before (and it has not been evicted from memory). This effect is particularly felt for the case where we reorder the itemsets according to increasing support. In this case, the most frequent itemset ends up contributing to the sum multiple times, even though its tidset may already be cached (in memory). For this reason, we also show the number of database scans for the lexical ordering, which are much lower than those for the sorted case. Even with these pessimistic estimates, we find that CHARM makes a lot fewer database scans than the longest frequent itemset. Using lexical ordering, we find, for example on pumsb*, that the longest closed itemset is of length 13, but CHARM makes only 3 database scans.0.0010.10.0001 0.001 Time per Closed (sec) #Closed Frequent Itemsets (in 10,000's) Real Datasets chess connect mushroom pumsb pumsb* Figure 18: Time per Closed Frequent #Intersections per Closed #Closed Frequent Itemsets (in 10,000's) Real Datasets ideal chess connect mushroom pumsb pumsb* Figure 19: #Intersections per Closed Total Time (sec) Number of Transactions (in 100,000's) Synthetic Datasets Figure 20: Size Scaleup on Synthetic Total Time (sec) Replication Factor Real Datasets Figure 21: Size Scaleup on Real Datasets Finally in Figures 20 and 21 we show how CHARM scales with increasing number of transactions. For the synthetic datasets we kept all database parameters constant, and increased the number of transactions from 100K to 1600K. We find a linear increase in time. For the real datasets we replicated the transactions from 2 to 10 times. We again find a linear increase in running time with increasing number of transactions. Conclusions In this paper we presented and evaluated CHARM, an efficient algorithm for mining closed frequent itemsets in large dense databases. CHARM is unique in that it simultaneously explores both the itemset space and tidset space, unlike all previous association mining methods which only exploit the itemset space. The exploration of both the itemset and tidset space allows CHARM to use a novel search method that skips many levels to quickly identify the closed frequent itemsets, instead of having to enumerate many non-closed subsets. An extensive set of experiments confirms that CHARM provides orders of magnitude improvement over existing methods for mining closed itemsets. It makes a lot fewer database scans than the longest closed frequent set found, and it scales linearly in the number of transactions and also is also linear in the number of closed itemsets found. Acknowledgement We would like to thank Roberto Bayardo for providing us the MaxMiner algorithm, as well as the real datasets used in this paper. --R Fast discovery of association rules. Efficiently mining long patterns from databases. Dynamic itemset counting and implication rules for market basket data. Introduction to Lattices and Order. Formal Concept Analysis: Mathematical Foundations. Incremental concept formation algorithms based on galois (concept) lattices. Familles minimales d'implications informatives resultant d'un tableau de donnees binaires. Discovering all the most specific sentences by randomized algorithms. A new algorithm for discovering the maximum frequent set. Mining association rules: Anti-skew algorithms Implications partielles dans un contexte. An effective hash based algorithm for mining association rules. Discovering frequent closed itemsets for association rules. Integrating association rule mining with databases: alternatives and implications. An efficient algorithm for mining association rules in large databas- es Scalable algorithms for association mining. Theoretical foundations of association rules. New algorithms for fast discovery of association rules. --TR An algorithm for insertion into a lattice: application to type classification Mining association rules between sets of items in large databases An incremental concept formation approach for learning from databases Approximate inference of functional dependencies from relations On automatic class insertion with overloading On the inference of configuration structures from source code Efficiently mining long patterns from databases Reengineering class hierarchies using concept analysis Design of class hierarchies based on concept (Galois) lattices Efficient mining of association rules using closed itemset lattices A fast algorithm for building lattices Mining frequent patterns with counting inference Levelwise Search and Borders of Theories in Knowledge Discovery Automatic Structuring of Knowledge Bases by Conceptual Clustering Efficient Discovery of Functional Dependencies and Armstrong Relations Conceptual Information Systems Discussed through in IT-Security Tool Discovering Frequent Closed Itemsets for Association Rules CEM - A Conceptual Email Manager Conceptual Knowledge Discovery and Data Analysis Conceptual Knowledge Discovery in Databases Using Formal Concept Analysis Methods Fast Algorithms for Mining Association Rules in Large Databases Merging Inheritance Hierarchies for Database Integration iO2 - An Algorithmic Method for Building Inheritance Graphs in Object Database Design Towards an Object Database Approach for Managing Concept Lattices Mining Minimal Non-redundant Association Rules Using Frequent Closed Itemsets TOSCANA - a Graphical Tool for Analyzing and Exploring Data Intelligent Structuring and Reducing of Association Rules with Formal Concept Analysis Mining Ontologies from Text --CTR Jean Diatta, A relation between the theory of formal concepts and multiway clustering, Pattern Recognition Letters, v.25 n.10, p.1183-1189, July 2004 Alain Casali , Rosine Cicchetti , Lotfi Lakhal, Extracting semantics from data cubes using cube transversals and closures, Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining, August 24-27, 2003, Washington, D.C. Mei-Ling Shyu , Shu-Ching Chen , Min Chen , Chengcui Zhang, A unified framework for image database clustering and content-based retrieval, Proceedings of the 2nd ACM international workshop on Multimedia databases, November 13-13, 2004, Washington, DC, USA Bradley J. Rhodes, Taxonomic knowledge structure discovery from imagery-based data using the neural associative incremental learning (NAIL) algorithm, Information Fusion, v.8 n.3, p.295-315, July, 2007 S. Ben Yahia , T. Hamrouni , E. Mephu Nguifo, Frequent closed itemset based algorithms: a thorough structural and analytical survey, ACM SIGKDD Explorations Newsletter, v.8 n.1, p.93-104, June 2006 Xiaodong Liu , Wei Wang , Tianyou Chai , Wanquan Liu, Approaches to the representations and logic operations of fuzzy concepts in the framework of axiomatic fuzzy set theory II, Information Sciences: an International Journal, v.177 n.4, p.1027-1045, February, 2007 Gerd Stumme, Off to new shores: conceptual knowledge discovery and processing, International Journal of Human-Computer Studies, v.59 n.3, p.287-325, September

closure systems;algorithms;formal concept analysis;knowledge discovery;lattices;database analysis

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Action graphs and coverings.

An action graph is a combinatorial representation of a group acting on a set. Comparing two group actions by an epimorphism of actions induces a covering projection of the respective graphs. This simple observation generalizes and unifies many well-known results in graph theory, with applications ranging from the theory of maps on surfaces and group presentations to theoretical computer science, among others. Reconstruction of action graphs from smaller ones is considered, some results on lifting and projecting the equivariant group of automorphisms are proved, and a special case of the split-extension structure of lifted groups is studied. Action digraphs in connection with group presentations are also discussed.

Introduction With a group G acting on a set Z we can naturally associate, relative to a subset (di)graph called the action (di)graph. Its vertices are the elements of the set Z, with adjacencies being induced by the action of the elements of S on Z. The denition adopted here is such that a connected action (di)graph corresponds to a Schreier coset (di)graph, with \repeated generators" and semiedges allowed. However, to think of an action (di)graph actually as a Schreier coset (di)graph is much too rigid in many instances. For similar concepts dealing with (di)graphs and group actions see [1, 2, 3, 8, 12, 17, 18, 21, 22, 28, 43, 45]. Some of them, although conceptually dierent, bare the same name [45], and some of them, quite close to our denition, are referred to by a variety of other names [1, 3]. It seems that the term action (di)graph should be attributed to T. Parsons [43]. For a computer implementation of (a variant of) action graphs see [44]. Supported in part by \Ministrstvo za znanost in tehnologijo Slovenije", proj.no. J1-0496-99. Group actions are compared by morphisms. The starting observation of this paper is that an epimorphism between two actions invokes a covering projection of the respective action graphs. Surprisingly enough, this simple result does not seem to have been explicitly stated so far, although there are many well-known special cases with numerous applications. For instance, it is generally known that Schreier coset (di)graphs are actually covering (di)graphs, and that a Schreier coset (di)graph is regularly covered by its corresponding Cayley (di)graph. These facts are commonly used as background results in the theory of group presentations [12, 13, 28, 31, 32, 14, 48, 52], and have recently been applied in the design and analysis of interconnection networks and parallel architectures [1, 2, 3, 22], among others. Coverings of Cayley graphs are frequently employed to construct new graphs with variuos types of symmetry and other graph-theoretical properties [9, 19, 34, 47] as well as to prove that a subgroup cannot have genus greater than the group itself [7, 19]. As for the maps on surfaces [4, 5, 10, 16, 19, 20, 25, 26, 27, 30, 35, 36, 40, 41, 46], one of the many combinatorial approaches to this topic is by means of a Schreier representation [25, 40]. Such a representation is actually a certain action graph in disguise, and homomorphisms of maps correspond to covering projections of the respective action graphs. Some important basic facts can be elegantly derived along these lines. We here give a unied approach to all these diverse topics, and in addition, we derive certain results which appear to be new. Section 2 is preliminary. Action graphs are introduced formally in Section 3, and covering projections induced by morphisms of actions in Section 4. Further basic properties of such coverings are discussed in Sections 5 and 6. In Section 7 we brie y consider automorphism groups of action graphs. Section 8 is devoted to lifting and projecting automorphisms, with focus on the equivariant group. In Section 9 we determine the group of covering transformations, and apply some results of [35] to obtain conditions for a natural splitting of a lifted group of map automorphisms (valid also if the map homomorphism is not valency preserving [35, 36]). In Section 10 we treat action digraphs in connection with group presentations. The lifting problem along a regular covering (of graphs as well as of general topological spaces) is reduced to a question about action digraphs. Preliminaries: Group actions, Graphs and Coverings By an ordered pair (Z; G) we denote a group G acting on the right on a nonempty set Z. (For convenience we omit the dot sign indicating the action.) A morphism of actions is an ordered pair function and : G ! G 0 is a homomorphism such that (u Morphisms are composed on the left. Left actions and their morphisms are dened similarly. Morphisms of the form (; id) : (Z; are called equivariant, and morphisms of the form (id; are called invariant. Invariant epimorphisms formalize the intuitive notion of \groups, acting in the same way on a given set". We say that an action ( ~ covers an action (Z; G) whenever there exists an epimorphism (; This terminology is justied by the fact that the cardinality j 1 (z)j depends just on the orbit of G to which z 2 Z belongs. A covering of actions can be decomposed into an equivariant covering followed by an invariant covering where the action of ~ G on Z is dened by z ~ Proposition 2.1 There exists a covering ( ~ of transitive actions if and only if there exists, for a xed chosen ~ b 2 ~ Z, a group such that q( ~ Z. The corresponding onto mapping of sets is then given by ~ b;b ( ~ b~g) := bq(~g). In particular, two transitive actions are isomorphic if and only if there exists an isomorphism between the respective groups mapping a stabilizer onto a stabilizer. Example 2.2 Let H H 0 G and K/G. The group G acts by right multiplication on the set of right cosets HjG. Similarly, the quotient group G=K acts on the set of right cosets H 0 KjG. There is an obvious covering of actions (HjG; In particular, the regular action (G; G) r of G on itself by right multiplication covers any transitive action of G=K. Example 2.3 There is an equivariant isomorphism representing a transitive action of a group G as an action on the cosets of a stabilizer. Moreover, all transitive and faithful quotient actions of G can be treated in a similar fashion. Indeed, a conjugacy class C of subgroups in G determines the action of G=core(C) on the cosets of an element of C. This action is transitive and faithful. Conversely, Q be a group epimorphism and let (Z; Q) be transitive and faithful. Dene the action (Z; G) such that (id; is an invariant covering, and let G Q b be a stabilizer of (Z; G). Let jG; G) be the standard representation of (Z; G). Then jG; Q) is the standard representation of (Z; Q). It follows that (Z; Q) determines a conjugacy class C Q in G with core(C Q q. Thus, the isomorphism classes of transitive and faithful actions of quotient groups of G are in natural correspondence with conjugacy classes of subgroups in G. Moreover, covering satisfying is a morphism. That is, such a covering exists if and only if G Q b is contained in a conjugate subgroup of G Q 0 b . See Examples 4.7 and 4.8 for an application. By Aut(Z; G) we denote the automorphism group of (Z; G). An automorphism of G is called admissible whenever there exists a bijection on Z such that G). The group of admissible automorphisms is denoted by AdmZ G. By Aut(Z)G we denote the equivariant group of the action, formed by all bijections on Z for which (; id) 2 Aut(Z; G). If (Z; G) is transitive, then Aut(Z)G can be computed explicitly relative to a point of reference b 2 Z as . Also, the left action of Aut(Z)G on Z is xed-point free, and is transitive if and only G acts with a normal stabilizer. Hence if G is, in addition, faithful, then Aut(Z)G is regular if and only if G is regular. In this case, (Z; G) is essentially the right multiplication (G; G) r , whereas (Aut(Z)G ; Z) is essentially the left multiplication (G; G) l . A graph is an ordered 4-tuple are disjoint nonempty sets of darts and vertices, respectively, beg is an onto mapping which assigns to each dart x its initial vertex beg x, and inv : D ! D is an involution which interchanges every dart x and its inverse x x. For notational convenience we use beg and inv just as symbolic names denoting the actual concrete functions. The terminal vertex end x of a dart x is the initial vertex of x 1 . The orbits of beg are called edges. An edge is called a semiedge if x and it is called a link otherwise. Walks are dened as sequences of darts in the obvious way. By we denote the set of all walks and the set of all u-based closed walks of a graph X, respectively. By recursively deleting all consecutive occurrences of a dart and its inverse in a given walk we obtain its reduction. Two walks with the same reduction are called homotopic. The naturally induced operation in set of all reduced u-based closed walks denes the fundamental group morphism of graphs . For convenience we write are the appropriate restrictions. Graph morphisms are composed on the left. A graph called a covering projection if, for every vertex ~ X , the set of darts with ~ u as the initial vertex is bijectively mapped onto the set of darts with the initial vertex p(~u). The graph X is called the base graph and ~ X the covering graph. By b we denote the bre over the vertex u and the dart x of X, respectively. A morphism of covering projections is an ordered pair (f; ~ f) of graph morphisms f ~ f . An equivalence of covering projections p and 0 of the same base graph is a morphism of the the form (id; ~ f is graph isomorphism. Equivalence of covering projections dened on the same covering graph is dened similarly. An automorphism of is of course a pair of automorphisms ( ~ f . The automorphism ~ f is called a lift of f , and f the projection of ~ f . In particular, all lifts of the identity automorphism form the group CT(p) of covering transformations. If the covering graph (and hence the base graph) is connected, then CT(p) acts semiregularly on vertices and on darts of ~ X. The covering projection of connected graphs is regular whenever CT(p) acts regularly on each bre. There exists an action of the set of walks W on the vertex-set of ~ dened by ~ u W is the unique lift of W such that beg u. In other words, we have (~u u. The mapping ~ u W denes a bijection b beg W ! b end W . Homotopic walks have the same action. In particular, W u and u have the same action on b u . The walk- action implies that coverings (of connected graphs) can be studied from a purely combinatorial point of view [35]. A voltage space on a connected graph dened by an action of a voltage group on a set F , called the abstract bre, and by an assignment : D ! such that x This assignment extends to a homomorphism carrying the same voltage. The group Loc called the local group at the vertex u. As the graph is assumed connected, the local groups at distinct vertices are conjugate subgroups, and if any of them is transitive we call such a voltage space locally transitive. With every voltage space on a connected graph inv) we can associate a covering X. The graph as the vertex-set and ~ as the dart- set. The incidence function is beg(x; and the switching involution inv is given by (x; The covering graph is connected if and only if the voltage space is locally transitive. In particular, the Cayley voltage space (;; ), where acts on itself by right multiplication, gives rise to a regular covering. Conversely, each covering of a connected base graph is associated with some voltage space, and each regular covering is associated with a Cayley voltage space (;; ), 3 Action graphs Let G be a (nontrivial) group acting (on the right) on a nonempty set Z and let S G be a Cayley set, that is, ; S. With the triple (Z; G; S) we naturally associate the action graph G; We shall actually need to consider Cayley multisets, that is, S has repeated elements (where for each s 2 S the elements s and s 1 have the same multiplicity). Our denition of a graph must then be extended accordingly. Example 3.1 The action graph of a group G, acting on a one-element set relative to a Cayley (multi)set S, is called a monopole and denoted by mnp(S). Example 3.2 The graph Act(HjG; G; S) is the Schreier coset graph Sch(G; H;S). By taking we get the Cayley graph Cay(G; S) and a monopole, respectively. Example 3.3 Let be permutations of a nite set Z. By representing each of these permutations pictorially in the obvious way we obtain the action graph for the permutation group relative to the symmetrized generating (multi)set f 1 g. Example 3.4 A nite oriented map M is a nite graph, cellularly embedded into a closed orientable surface endowed with a global orientation. Let D be the dart- set of the embedded graph, L its dart-reversing involution, and R the local rotation which cyclicaly permutes the darts in their natural order around vertices consistently with the global orientation. In studying the combinatorial properties of such a map we only need to consider the permutation group hR; Li on D (together with the generating set fR; Lg), and consequently, its Schreier representation. See Jones and Singerman [25]. Equivalently, we only need to consider the action graph of the group hR; Li acting on D relative to the Cayley set fR; R 1 ; Lg. This graph, here denoted by Map(D;R;L) or sometimes by Act(M ), is also known as the truncation of the map [40]. More generally, maps on all compact surfaces can be viewed combinatorially in terms of certain permutation groups and their generators, as shown by Bryant and Singerman [10]. For closed surfaces, the associated action graphs correspond to graph encoded maps of Lins [30]. A walk W in the action graph Act(Z; G; S) denes a word w(W the alphabet S. Conversely, if w 2 S is a word, then W (z; w) denotes the (set of starting at z 2 Z and determined by w. The fact that in the case of repeated generators there is no bijective correspondence between words over S and walks, rooted at a chosen vertex, is a minor technical di-culty which we usually (but not always) can ignore. The action graph is connected if and only if S generates a transitive subgroup of G. Without loss of generality we can in most cases assume that the action is transitive and that the Cayley (multi)set generates the group. Morphisms arising from coverings of actions A mapping Z ~ is a morphism of action graphs G; ~ only if z (~s)); (1) Zg is a collection of mappings ~ z (~s) and ( ~ z This follows directly from the denition of a graph morphism. In particular, morphisms arising naturally from coverings of actions can be viewed as graph covering projections (by taking the Cayley (multi)sets to correspond bijectively in a natural way). We state this formally as a theorem, and list some of the well-known special cases and applications. Theorem 4.1 Let (; be a covering of actions and let ~ G be a Cayley (multi)set. Consider S) as a multiset in a bijective correspondence with ~ S. Then the induced graph morphism p G; ~ G; S) is a covering projection of action graphs. Proof. The mapping S satises (1) and (2). Hence S is a graph morphism. Since it is onto with qj ~ S a bijection, it is a covering projection. Example 4.2 Let the action be transitive. Then the mapping G; Sch(G; G b ; S), where c(b g, is an equivariant isomorphism. Example 4.3 Let H H 0 G. Then g, is an equivariant covering projection. Example 4.4 Let K / G. Then is an invariant 1-fold covering projection, and hence an invariant isomorphism. Example 4.5 Let (id; G) be an invariant covering of actions. Then Act(Z; G; S) is invariantly isomorphic to Act(Z; G; q(S)). Thus, in studying action graphs we may restrict to faithful actions by taking similar result dealing with isomorphisms of Cayley digraphs can be found in [22]. Example 4.6 A homomorphism ~ of oriented maps is a morphism of the underlying graphs which extends to a mapping between the supporting surfaces. Topologically it corresponds to a branched covering with possible singularities in face-centres, edge-centres and vertices. Combinatorially we have a mapping of the respective dart-sets : ~ D such that (~x ~ This together with ~ R 7! R denes a covering of actions and consequently, a covering projection of action graphs Map( ~ Example 4.7 Recall from Example 2.3 that transitive and faithful actions of quotient groups of G can be \modeled by conjugacy classes of subgoups" in G. It follows that there exists a covering projection arising from actions (where only if G Q b is contained in a conjugate subgroup of G Q 0 b . A special case is essentially considered in [50]. The situation as described above is encountered in the theory of maps and hypermaps. Example 4.8 Oriented maps and their homomorphisms can be modeled by conjugacy classes within triangle groups, see Jones and Singerman [25]. The idea extends to all maps [10] and even hypermaps [27]. 5 Structure-preserving morphisms Nonisomorphic actions can give rise to isomorphic graphs, as shown by Examples 4.4, 4.5 and by Example 5.1 below. Also, isomorphic actions can have isomorphic graphs with no graph isomorphism arising from an isomorphism of actions, see Example 5.2. Example 5.1 The triangular prism is a Cayley graph for the groups S 3 and ZZ 6 . It is also an action graph for the group S 4 , obtained by representing S 4 as the subgroup of S 6 generated by permutations (12)(45)(36), (23)(56)(14) and (13)(46)(25). See also Example 10.1. Example 5.2 Take a Cayley graph Cay(G; S) where the generating set S is not a CI-set. Then there is a generating Cayley set T with Cay(G; S) that no automorphism of G maps S onto T . In view of these remarks we note the following. When considering an action graph as having a certain structure arising from the action, we are actually considering the induced equivariant covering Act(Z; G; G; ~ G; S) is structure-preserving if there exists a mapping of the monopoles mnp( ~ together with mnp( ~ is a morphism of covering projections. In other words, does not depend on the S. For example, coverings arising from coverings of actions are structure-preserving, with mnp( ~ Proposition 5.3 Let G; ~ G; S) be a structure-preserving covering, where ~ S and S are generating (multi)sets and G is faithful. Then this covering arises from a covering of actions. Proof. By induction we have (~z ~ ~ Z and any choice of generators ~ s S. Let ~ s 1 ~ As is onto and G faithful, we have (~s 1 Hence extends to a homomorphism, as required. Example 5.4 A covering projection of action graphs Map( ~ such that ~ R 7! R and ~ arises from a covering of actions h ~ hence represents a homomorphism of the respective maps. Although coverings of action graphs, even isomorphisms, in general do not arise from actions nor are at least structure-preserving, we may still ask the following. Let Act( ~ G; ~ G; S) be covering projections. Is there an action-structure for the graph X (respectively, ~ X) such that these projections arise from coverings of actions? Theorem 5.5 Let G; ~ be a covering projection. Then ther exists an action graph structure for X such that + is equivalent to a covering arising from actions if and only if there exists a covering projection which makes the following diagram G; ~ y y commutative. In this case the action-structure for X can be chosen in such a way that the respective covering is equivariant. Proof(Sketch). If + is equivalent to a projection arising from a covering of actions, then such a decomposition clearly exists. Conversely, let Z be the vertex-set of X. One can show easily that z ~ g), ~ z 2 1 (z), is a well dened action of ~ G on Z, with G) an equivariant covering of actions. The projection G; ~ G; ~ S) is equivalent to + . We can interpret Theorem 5.5 by saying that a quotient of a Schreier coset graph decomposing the natural projection onto a monopole is again a Schreier coset graph of the same group. Stated in this form, the result is due to Siran and Skoviera [50]. We now turn to the second question above. We assume that (Z; G) is transitive and that S is a generating Cayley (multi)set. We may also assume that the covering projection G; S) is given by means of a voltage space G; S). Theorem 5.6 With the notation and assumptions above there exists an action graph structure for Cov(F; ; ) such that the natural projection completes the diagram G; S) y y Moreover, if G is faithful then p is equivalent to a covering arising from actions. Proof. The derived graph has Z F as the vertex-set and Z S F as the dart- set, with the incidence function and the switching involution being, respectively, By the unique walk lifting, the collection of closed walks in Act(Z; G; S) representing the orbits of s 2 S lift to a collection of closed walks representing a permutation of Z F which we denote by s . This permutation is dened by (z; i) is a bijection and that s s . Let ~ Sg. The action graph Act(Z F; ~ G; ~ has Z S as the vertex-set and Z F ~ S as the dart-set. The incidence is given by beg(z; and the switching involution is inv(z; The mappings id on vertices and (z; s; i) 7! (z; dene an isomorphism G; ~ S). This induces a projection Act(Z F; ~ G; ~ G; S) equivalent to p , and the natural projection Act(ZF; ~ G; ~ induces an equivalent natural projection S) making the required diagram commutative. The last statement in the theorem follows by Proposition 5.3. Example 5.7 Let a connected graph Cov(F; ; ) be a covering of the action graph Map(D;R;L) associated with a map M , and let Cov(F; ; ) inherit the action- structure as in Theorem 5.6. Since L is an involution, Cov(F; ; ) is the action graph Map(D F M , and the covering projection essentially arises from the map homomorphism ~ M !M , by Example 5.4. This shows that homomorphisms of oriented maps can be studied just by considering coverings of associated action graphs. Compare with the discussion on Schreier representations of maps in [41]. We dene a map homomorphism ~ M to be regular if Map( ~ Map(D;R;L) is a regular covering. See also [36]. Finally, let us brie y consider the following question. What is the necessary and su-cient condition for a connected graph as in Theorem 5.6 to be the Cayley graph of the group ~ First of all, s 1 G (z;i) if and only if s 1 F for each z 2 Z. Thus, assuming that G is faithful and the covering is regular, the answer to the above question is the following: any closed walk with trivial voltage must correspond to a relation s 1 and in this case, all closed walks corresponding to this word must have trivial voltage. In particular, a regular covering of a Cayley graph as in Theorem 5.6 is the required Cayley graph if and only if the following holds: whenever a closed walk has trivial voltage then all closed walks corresponding to the respective word must have trivial voltage. This condition is equivalent to saying that the equivariant group lifts, see Section 8 and [35]. Example 5.8 Let ~ M !M be a regular homomorphism of oriented maps, where M is a regular map (see Section 7). Since the action graph Act( ~ M ) can be reconstructed from the action graph Act(M) as in Theorem 5.6 (see Theorem 6.1), it follows that ~ M is a regular map if and only if AutM lifts (see Section 7 and Example 8.9, and also [36]). 6 Reconstruction Let be a covering of transitive actions, let ~ G be a generating Cayley (multi)set and S) a (multi)set in bijective correspondence with ~ S. We would like to reconstruct the action graph Act( ~ G; ~ S) from Act(Z; G; S) in terms of voltages. This can be done by means of a canonical voltage space ( ~ G; ) (relative to (; q)) on Act(Z; G; S), with voltages on darts dened by the rule The derived covering graph Cov( ~ G; ) has vertex-set ~ Z and the dart-set ~ Z. The incidence function beg : ~ V is given by the projection beg(z; s; ~ z), and the switching involution is inv(z; s; ~ (s). The corresponding local group is Its action on ~ Z is, modulo relabeling, the same as the action of W b on the vertex bre over b in G; ). But W b and q 1 (G b ) also act on the bre b b in Act( ~ G; ~ S). In fact, is an invariant covering of actions. Theorem 6.1 With the notation above, the component C(b; ~ b) of Cov( ~ G; the vertex (b; ~ b), consists exactly of all vertices of the form (z; ~ z), (~z), and the restriction G; S) is equivalent to G; ~ G; S). (If ~ G is a permutation group, then the action graph structure imposed on C(b; ~ b) as in Theorem 5.6 coincides with Act( ~ G; ~ S).) Next, all restrictions of G; G; S) to its connected components are equivalent to G; ~ G; S) if and only if the restrictions of the action of q 1 (G b ) on all its orbits have the same conjugacy class of stabilizers. In particular, let the action of ~ G be such that no stabilizer is properly contained in another stabilizer (say, the group is nite). Then all restrictions of p to its connected components are equivalent to p ;q if and only if q( ~ Proof. Let (z; ~ u) be in the same component as (b; ~ b), vertices in this component are of the required form. If (z; ~ z) and (u; ~ u) have the same label ~ Hence no two vertices are labeled by the same label. Moreover, all labels from ~ Z actually appear since ~ S generates ~ G and ~ G is transitive on ~ Z. The bre over b in C(b; ~ b) is labelled by the orbit of Loc b on ~ Z which is precisely It follows that the actions of W b on bres over b in C(b; ~ b) and G; ~ are essentially the same. Hence G; S) and G; ~ G; S) are equivalent. The explicit graph isomorphism which establishes this equivalence is (z; ~ z) 7! ~ z on vertices and (z; s; ~ z) 7! (~z; ~ s) on darts. If ~ G is faithful, then the action graph structure imposed on C(b; ~ b) as in Theorem 5.6 obviously coincides with Act( ~ G; ~ S). Clearly, all restrictions of p to the components of Cov( ~ G; ) are equivalent if and only if the induced actions of Loc its orbits in ~ Z are equivari- antly isomorphic, that is, all of these actions must have the same conjugacy class of stabilizers in Loc b . Observe that Loc b G ~ z for ~ ~ G ~ Now, if all stabilizers of ~ G are contained in Loc b , then all the above actions of do have the same conjugacy class of stabilizers. Conversely, the fact that each ~ G ~ is also the stabilizer of some point ~ z 2 1 (b) implies ~ G ~ z ~ G ~ u . Since the action of ~ G is not pathological, we have equality, and hence each stabilizer of ~ G must be contained in Loc This condition can be further rephrased as follows. Since q is onto and ~ G ~ z , ~ z 2 ~ Z are conjugate subgroups, we have G. Thus, q( ~ Conversely, if q( ~ then q( ~ G ~ . This completes the proof. Example 6.2 The action graph Act( ~ G; ~ S) can be reconstructed from Act(Z; G; S) by taking just any connected component of the derived covering, for instance, when ~ G (or G) acts with a normal stabilizer. A special case is encountered when reconstructing Cayley graphs form Cayley graphs of quotient groups [19]. Example 6.3 Consider the dihedral group 1i with subgroups Then Sch(G; H;S) equivariantly covers Sch(G; H 0 ; S). Since not all conjugates of H, the graph Sch(G; H;S) cannot be reconstructed by taking just any connected component of the derived covering Cov(HjG; G; ). In this particular case we can also apply the Burnside-Frobenius counting lemma. Namely, the action of H 0 on the cosets of H has two orbits, whereas the derived covering is 9-fold. 7 Automorphisms We henceforth assume that actions are transitive and that Cayley (multi)sets generate the groups in question. A bijective self-mapping + of Z +ZS is an automorphism of Act(Z; G; S) if and only if (z; a collection of bijective self-mappings of S satisfying (z Studying general automorphisms, even the subgroup of structure-preserving ones, is di-cult. Next in the line is the subgroup Aut S (Z; G) of extends to (or just is) an automorphism of G. Proposition 5.3 implies Proposition 7.1. Proposition 7.1 Let a transitive action (Z; G) be faithful, and let S G be a generating Cayley (multi)set. Then each automorphism of Act(Z; G; S) of the form is an action-automorphism. Let Aut S denote the subgroup of S-automorphisms of G, and let Adm S its S-admissible subgroup formed by automomorphisms which preserve S as well as the conjugacy class of stabilizers of G (as a set). Of course, + is an action-automorphism of Act(Z; G; S) if and only if (; Z G. In particular, we can identify Eq(Z)G , the equivariant group of automorphisms of Act(Z; G; S), and the equivariant group of the action, Aut(Z)G . It is easy to see that the projection Aut S (Z; is a group epimorphism with kernel Aut(Z)G . We state this observation formally. Theorem 7.2 The group Aut S (Z; G) of action-automorphisms of Act(Z; G; S) is isomorphic to an extension of the equivariant group Aut(Z)G by Adm S Z G. While Aut(Z)G is isomorphic to NG (G b )=G b , the identication of Adm S Z G and the extension itself might not be easy. Here is a simple and well-known example. Example 7.3 In view of Proposition 7.1 there are two kinds of automorphism of Cayley graphs. Those for which the mapping of darts is not constant at all points, and those which are action-automorphisms. Let + be an action-automorphism. a (g), for some a 2 G and 2 Adm S G. Moreover, the assignment (a; is an isomorphism G Aut S (G; G) r . Hence the group of action-automorphisms Aut S (G; G) r of Cay(G; S) is isomorphic to a subgroup in the holomorph of G. The group Aut S (G; G) r can as well be characterized as the normalizer of the left regular representation f a;id j a 2 Gg of G within the full automorphism group of Cay(G; S) [18]. Example 7.4 From the very denition it follows that the automorphism group AutM of an oriented map M can be identied with the equivariant group Aut(D) hR;Li [25]. Example 7.5 An oriented map is called regular if its automorphism group acts transitively (and hence regularly) on the dart-set of the underlying graph. Recall that the equivariant group acts regularly if and only if the group itself is regu- lar. Therefore, a map is regular if and only if Map(D;R;L) is a Cayley graph for the group hR; Li [25, 40]. Since any Map(D;R;L) is equivariantly covered by every map is a (branched) regular quotient of some regular map [25, 26, 46]. The idea extends to maps on bordered surfaces [5]. As a last remark, let us ask what are the necessary and su-cient conditions for a transitive faithful action (Z; G) to extend to a group of automorphisms of G; S). This problem is of interest [45] (however, action graphs in [45] dier from ours), and di-cult in general. But let us assume that the induced automorphism group preserves the natural structure as a covering of mnp(S). Then the answer is trivial. Proposition 7.6 Let the action (Z; G) be transitive and faithful, and let S G be a generating Cayley (multi)set. Then T g, is a group of (action) automorphisms of Act(Z; G; S) if and only if S is a union of conjugacy classes. In particular, T (G) acts as a subgroup of the equivariant group if and only if the action graph is a Cayley graph Cay(G; S) and G is abelian. Other types of symmetries of action graphs will not be discussed here. Arc- transitivity of Cayley digraphs is considered, for instance, in [2, 22]. 8 Lifting and projecting automorphisms G; ~ G; S) be a covering arising from transitive actions, where ~ S and are generating Cayley (multi)sets. The problem of lifting automorphisms has recently obtained considerable attention [4, 5, 6, 9, 15, 23, 24, in various contexts. From the general theory we infer that the lifting condition, expressed in terms of the canonical voltages valued in ~ G, reads: An automorphism lifts along p ;q if and only if there exists ~ z 2 b b such that, for each W 2 W b , G ~ b if and only if W 2 ~ G ~ However, we here explore the possibility that the lifting condition be expressed without the usual explicit reference to mappings of closed walks and their voltages, but, rather, expressed in terms of certain subgroups of G and ~ G. To this end we have, in spite of the fact that coverings arising from actions are somewhat peculiar, restrict our considerations either to a very special class of action-automorphisms (the equivariant group), or else to action-automorphisms with addtional requirements imposed on the covering of actions (Example 8.2). We also note that the lifts of action-automorphisms, although structure-preserving, need not be action- automorphisms. Example 8.1 Let ~ G be faithful. Then, by Proposition 7.1, a lift of an action- automorphism of Act(Z; G; S) is an action-automorphism of Act( ~ G; ~ S),. Example 8.2 Suppose that a covering of actions is equivariant. Then the conclusion as in the previous example holds, too. Moreover, from (3) we easily derive that an action-automorphism lifts if and only if there exists g 2 G such that g. (An alternative direct proof avoiding (3) is readily at hand and is left to the reader.) In particular, action-automorphisms do lift along the equivariant covering Cay(G; G; S). We now focus our attention on the equivariant group Eq(Z)G of Aut(Z; G; S). That one can derive a reasonable lifting condition in terms of subgroups of ~ G and G is not surprising because with equivariant automorphisms we could as well consider just coverings of actions. Theorem 8.3 The equivariant group Eq(Z)G lifts along G; ~ G; S) if and only if q(N ~ G ~ b )) intersects every coset of G b within NG (G b ), and the lifted group is then a subgroup of the equivariant group Eq( ~ G . In particu- lar, if the covering projection is regular, then Eq(Z)G lifts if and only if NG (G b ) q(N ~ Proof. We present a proof which avoids the reference to (3). Clearly, a lift of an equivariant automorphism is equivariant. Consider a pair of equivariant automorphisms of the covering graph and of the base graph, respectively. Their action on vertices is given, relative to ~ b and G ~ b , and c (b be a lift of c + c id, that is, let ~ ~ We easily get q(~c) 2 G b c. Conversely, if c and ~ c satisfy this condition, then ~ ~ c ~ c id is a lift of c id. The lifting condition can now be expressed as: for each c 2 NG (G b ) there exists ~ c 2 N ~ such that q(~c) 2 G b c. The claim follows. The covering projection p ;q is regular if and only q 1 G ~ b ), by Theorem 9.1 below. This implies G b q(N ~ G ~ b )), and hence the lifting condition now obviously reduces to NG (G b ) q(N ~ G ~ b )). The alternative form follows because contains Ker q. Example 8.4 Let the group ~ G act with a normal stabilizer. Then the equivariant group lifts. In particular, Eq(Z)G lifts along G; ~ G; S). Example 8.5 Let ~ M !M be a homomorphism of oriented maps. If ~ M is a regular map, then AutM lifts [36]. Example 8.6 The lift of the equivariant group Eq(Z)G along a regular covering projection p ;q is isomorphic to q 1 (NG (G b ))= ~ G ~ b . Proposition 8.7 Let the covering G; ~ Cay(G; S) be regular. If the equivariant group of Cay(G; S) lifts, then Act( ~ G; ~ S) is isomorphic to the Cayley graph Cay( ~ G ~ S). Proof. By Theorem 8.3 we have q(N ~ G. Thus N ~ coset of Ker q. Since Ker q q 1 Theorem 9.1 below), N ~ contains Ker q. Hence N ~ G, and the proof follows. Example 8.8 Consider a regular covering G; ~ ~ G is faithful. Then Act( ~ G; ~ S) is isomorphic to the Cayley graph Cay( ~ G; ~ S) if and only if the equivariant group of Cay(G; S) lifts, by Example 8.4 and Proposition 8.7. In view of the lifting condition (3) we may rephrase this as in Section 5. Example 8.9 Let ~ M be a regular homomorphism, where M is a regular map. If AutM lifts, then ~ M is also a regular map. In view of Example 8.5 we obtain the if and only if statement of Example 5.8. See also [20, 36]. Let us now consider projecting automorphisms of Act( ~ G; ~ along p ;q . An automorphism ~ projects whenever each vertex-bre is mapped onto some vertex- bre and each dart-bre is mapped onto some dart-bre. If the covering is regular, then an automorphism projects if and only if it normalizes the group of covering transformations. This is actually a theorem of Macbeath [37] which holds for general topological coverings as well as in our combinatorial context. A similar result for digraphs is proved in [39]. Note that projections of action-automorphisms are structure-preserving but need not be action-automorphisms. One particular instance when such projections are indeed action-automorphisms is when the covering is equivariant. The case when G acts faithfully is another. In general, the following holds. Proposition 8.10 Suppose that an action-automorphism ~ ~ of Act( ~ G; ~ projects along G; ~ G; S). Then the projected automorphism is an action-automorphism of Act(Z; G; S) if and only Ker q is invariant for ~ . Proof. Let be the projected automorphism. We know that is dened q 1 on S. Now extends to an automorphism of G if and only if Equivalently, we must have Ker q, that is, ~ Theorem 8.11 An action-automorphism ~ of Act( ~ G; ~ projects along G; ~ G; S) if and only if there exists ~ G such that ~ and ~ Proof. First of all, if an action-automorphism is vertex-bre preserving, then it is also dart-bre preserving. Moreover, it is enough to require that just one vertex-bre is mapped to a bre. The proof of this fact is left to the reader. It follows that ~ ~ projects if and only if ~ maps Writing ~ ~ g and taking into account that the stabilizers are conjugate subgroups, we obtain the desired result. Example 8.12 An action-automorphism ~ of Act( ~ G; ~ projects along G; ~ only if ~ (Ker Ker q, and the projection is necessarily an action-automorphism. Theorem 8.13 The group Eq( ~ G projects along G; ~ G; S) if and only if N ~ (equivalently, q(N ~ )). The projected group is a subgroup of Eq(Z)G . For regular coverings, the condition simplies to Proof. Clearly, the projection of an equivariant automorphism is equivariant. By Theorem 8.11, an automorphism ~ c id, where ~ c~g and ~ c 2 G ~ b , projects if and only if there exists ~ g 2 ~ G such that ~ b ~ and ~ This is equivalent to saying that N ~ intersects the coset ~ should intersect every coset of ~ G ~ b within N ~ implying that N ~ should contain N ~ claimed. The alternative form is also evident. direct proof similar to the proof of Theorem 8.3 is left to the reader.) The rest follows by Theorem 9.1. Example 8.14 Let G act with a normal stabilizer. Then the group Eq( ~ G projects. In particular, it projects along G; ~ Example 8.15 Let ~ M be a homomorphism of oriented maps, where M is a regular map. Then Aut ~ projects [36]. Corollary 8.16 Let the group Eq( ~ G project along G; ~ G; S). Then Act(Z; G; S) is isomorphic to the Cayley graph Cay(G=GZ ; S). Proof. By Theorem 8.13 we have G. Hence G b is normal in G, and the proof follows. Example 8.17 Let ~ M be a homomorphism of oriented maps, where ~ M is a regular map. In view of Example 8.15 and Corollary 8.16 the group Aut ~ projects if and only if M is also a regular map. (The homomorphism itself must then be regular, see Example 9.3.) Corollary 8.18 Let the covering projection G; ~ G; S) be regular. Then the equivariant group Eq(Z)G lifts and the equivariant group Eq( ~ G projects if and only if q(N ~ In this case, Eq(Z)G lifts to Eq( ~ G and Eq( ~ G projects onto Eq(Z)G . Proof. By Theorems 8.3 and 8.13 we must have q(N ~ and the claim follows. The last statement is evident as well. Example 8.19 The projection G; ~ Cay(G; S) is regular (see Example 9.3). The left regular representation of G lifts to the left regular representation of ~ G, and hence the latter projects onto the former. In particular, if ~ a homomorphism of regular maps, then Aut M lifts to Aut ~ M and Aut ~ projects 9 The structure of lifted groups Theorem 9.1 Let a covering projection G; ~ G; S) arise from a covering of transitive actions, where ~ S and are generating Cayley (multi)sets. Choose ~ b 2 ~ Z and base-points. Then: (a) is a subgroup of Eq( ~ G . G ~ b g is isomorphic to (q 1 G ~ b . (c) The covering projection p is G ~ b ]-fold, and is regular if and only if ~ Proof. The statement (a) is obvious. Since ~ G is transitive on ~ Z we can explicitly calculate the elements of Aut( ~ G relative to ~ b as ~ ( ~ b ~ G ~ b . Now b. Consequently, ~ c 2 q 1 (G b ), giving (b). The rst statement of (c) follows from the fact that the covering is connected. Indeed, the group q 1 acts transitively on the bre 1 (b) and has ~ G ~ b as its stabilizer. The covering is regular, by denition, if CT(p) is transitive on 1 )g. But this holds if and only if q 1 part (c) follows. Example 9.2 Let H H 0 G, and let be the corresponding covering projection, where g. Then is isomorphic to (H 0 \N(H))=H. The covering projection is [H and is regular if and only if H / H 0 . Example 9.3 A covering G; ~ G; S) is always regular. Hence a homomorphism ~ of oriented maps, where ~ M is a regular map, must be a regular homomorphism. In particular, homomorphisms between regular maps are regular [36]. A lifted group of automorphisms is an extension of the group of covering transformations by the respective group of the base graph. This extension is di-cult to analyze in general [9, 20, 33, 35, 36, 51]. We end this section by considering the case when the equivariant group lifts as a split extension along a covering projection arising from actions. As the equivariant group acts without xed points, each orbit of an arbitrary complement to CT(p) within the lifted group intersects each bre in at most one point (thus forming an invariant transversal). This is equivalent with the requirement that the covering projection be reconstructed by means of a voltage space for which the distribution of voltages is well behaved relative to the action of the equivariant group. The claim takes a particularly nice form whenever the covering projection is regular [33, 35]. A straightforward application of these considerations to regular homomorphisms of oriented maps gives Theorem 9.4. A direct proof in terms of voltages associated with angles of the map can be found in [36]. By we denote the set of all walks with endvertices in a subset of vertices Theorem 9.4 Let ~ M be a regular homomorphism of oriented maps, and let be an orbit (or a union of orbits) of a dart in M relative to AutM . Then AutM lifts as a spit extension of CT (the lift of the identity automorphism) if and only if the covering projection of action graphs Act( ~ can be reconstructed by means of a Cayley voltage space (CT; CT; ) such that the set of walks fW 2 in Act(M) is invariant for the action of AutM . Moreover, the extension is a direct product if and only if Act( ~ can be reconstructed by a Cayley voltage space (CT; CT; ) such that each of the sets fW 2 is invariant for the action of AutM . Proof. The map automorphism group corresponds to the equivariant group in the associated action graph, and the lift of the identity corresponds to the group of covering transformations for the regular covering projection of the respective action graphs. The theorem follows by applying Theorems 9.1 and 9.3, and Corollaries 9.7 and 9.8 of [35]. Generators and relations a nonempty antisymmetric subset, that is, ! S \ is either empty or else all of its elements are of order 2. With (Z; G) and ! S we associate the action digraph Act(Z; G; with the vertex-set Z and the arc-set Z ! S , where in the case of action graphs it is sometimes necessary to consider the set ! S as a multiset. The underlying graph of Act(Z; G; is the action graph act(Z; G; ~ G; ). (Note that involutory loops collapse to semiedges.) Omitting formal basic denitions we only mention that epimorphisms of actions give rise to covering projections of action digraphs. The study of group presentations involves a variety of techniques, see [13, 31, 32, 48] and the references therein. Action digraphs can often provide much insight to a formal algorithmic approach, and were (in disguise) at least partially present in the original works of Reidemeister and Schreier. Choose a spanning tree in Act(Z; G; ~ S), where (Z; G) is transitive and ~ generates G. Each cotree arc gives rise to a unique fundamental closed walk based at z 2 Z, and the set of all such walks generates the set of all closed walks at z, up to reduction. Thus, if ~ C is a set of labels bijectively associated with all the cotree arcs, which evaluate to words in ( ~ dened by the fundamental closed walks rooted at z 2 Z, then each element of G z , expressed as a word in ( ~ can be written as a word in ( ~ . This is done by trailing the closed walk associated with a given word in ( ~ simultaneously keeping track of the labels in ~ when traversing a cotree arc. The process is known as the rewriting process relative to z 2 Z. A variant of the Schreier-Reidemeister theorem now states the following. Ri be a presentation of G, and let RewR be the set of (reduced) words in ( ~ obtained from all the relators in R by a rewriting process relative to all z 2 Z. Then the stabilizer G b has the presentation h ~ C; RewRi. Many of the generators and relators obtained by this method can be redundant. However, sophisticated techniques for simplifying the presentation do exist in certain cases [12, 13, 32, 48, 52]. An action digraph Act(Z; G; ~ S), where ~ S generates G, obviously determines the group G=GZ up to isomorphism (also if G is not transitive). Suppose that the action digraph is nite. Denote by ~ 1 the generators of the stabilizer G b 1 , expressed as words in ( associated with fundamental closed walks at b 1 relative to a spanning tree in the appropriate component of Act(Z; G; ~ S). By repeating this process on Act(Znfb 1 on, we can recursively construct a generating set for the pointwise stabilizer GZ . Hence if G is faithful, we can nd a presentation of G. If j ~ then the number of generators of GZ obtained in this way can amount up to (n 1)m!+ 1. Thus, the method is not practical unless one can detect enough many redundant generators at each step, or has su-cient control over the recursive construction of the generators. As for the improvements which allow eective computer implementation we refer to [11, 12, 48] and the references therein. Example 10.1 We leave to the reader to check that the following three permutations in the symmetric a subgroup which is isomorphic to S 4 . Despite the remarks above, Act(Z; G; ~ proves useful in gathering at least partial information about the dening relations, particularly when its underlying graph is highly asymmetric with special structure. The idea is to use graph-theoretical properties of Act(Z; G; ~ S) to derive such information. The following example is taken from [34]. Example 10.2 Consider the alternating group A n , where n 11 is odd, and the generators analysis of the action digraph shows that in the Cayley graph the cycles of girth-length, which is 6, arise essentially from the relation (ab 1 that cycles of length n arise essentially from the obvious relations a This information is crucial in proving that the above Cayley graph is 1/2-transitive. What we have discussed so far can be applied to a problem encountered with lifts of automorphisms. Let be a covering projection of connected graphs (or even more general topological spaces, see [33]), given by means of a voltage space acts faithfully on F . A necessary condition (also su-cient if the covering is regular) for an automorphism to lift is that the set of all closed paths with trivial voltage be invariant under its action [33, 35]. In order to test this eectively (assuming of course, that the covering has nite number of folds and that the fundamental group of X is nitely generated) we need the generators of the kernel of : b ! , expressed in terms of a generating set ~ S of b , save for those cases where ad-hoc techniques apply. One possibility is to consider the auxiliary regular covering X. The required generators of Ker are then obtained by projecting the generators of ( ~ b; Cov), where ~ b 2 b b . However, this requires the construction of Cov(;; ), which is not always appropriate. A better alternative is to consider the fundamental group b acting on by right multiplication (). The stabilizer of this action is Ker , and so the required generators can be found by means of a spanning tree in Act(; b ; ~ S)). It is close to rst constructing the coset representatives of Ker , that is, nding a closed path (rooted at the based point) with voltage , for all 2 , and then applying the Schreier method. Another possibility is to consider the action of b on the abstract bre F given by i (). The kernel Ker is then equal to the pointwise stabilizer of this action. Thus, the required generators can be found recursively by considering the action digraph Act(F; b ; ~ S). A similar problem is to construct a generating set for the trivial voltage paths with endpoints in an orbit of a given group A of automorphisms of X. Namely, a necessary and su-cient condition for A to lift along a regular covering projection as a special kind of split extension of CT(p) (one with an invariant transversal [33, 35]) is that the set of paths as above be invariant for the action of A (recall Theorem 9.4). Suppose that X is a graph and a vertex orbit. Introduce a new vertex B not in X, and connect B with all the vertices in Moreover, extend the voltage assignment so that these new edges carry the trivial voltage. The required generating set of walks is obtained in the same way as before by considering the extended graph with B as the base point. Note that the group A has the required type of lift if and only if, viewed as the stabilizer of B in the extended graph, lifts along the extended covering projection. (The idea clearly extends to nite CW-complexes.) The preceding discussion is summarized in the following theorem. Theorem 10.3 With notation and assumptions above, the problem whether a given group A of automorphisms of a graph X (or a more general topological space) lifts along a regular covering projection given by a faithful voltage space can be tested in time, proportional to the number of generators of A, multiplied by the time required for the construction of Cay(; ( ~ and its spanning tree (or multiplied by the time required for the construction of Act(F; b ; ~ S) and nding the generators of the pointwise stabilizer). If X is a graph (or a nite CW-complex), a similar statement holds for the problem whether a given group of automorphisms of X lifts as a split extension of CT(p) with an invariant transversal. --R On group graphs and their fault tolerance A. group theoretic model for symmetric Group action graphs and parallel architectures Homological coverings of graphs Some applications of graph contractions Conjugacy graphs with an application to imbedding metric graphs Foundations of the theory of maps on surfaces with boundary Cambridge University Press Construction of de Generators and relations for discrete groups Isomorphism classes of concrete graphs coverings Routage uniformes dans les graphes sommet-transitifs On the full automorphism group of a graph Graph homomorphisms: structure and symmetry Isomorphisms and automorphisms of coverings Graph covering projections arising from Theory of maps on orientable surfaces Theory Ser. Group actions Isomorphism classes of graph bundles Theory Ser. Combinatorial group theory: Presentations of groups in terms of generators and relations On a theorem of Hurwitz Automorphisms of groups and isomorphisms of Cayley digraphs Vega version 0.2 quick reference manual and Vega graph gallery Automorphism groups of covering graphs Computation with Graph coverings and group liftings Surfaces and planar discontinuous groups --TR Topological graph theory On group graphs and their fault tolerance A Group-Theoretic Model for Symmetric Interconnection Networks Group action graphs and parallel architectures Lifting map automorphisms and MacBeath''s theorem Isomorphisms and automorphisms of graph coverings Which generalized Petersen graphs are Cayley graphs? Graph covering projections arising from finite vector spaces over finite fields Automorphism groups of covering graphs Group actions, coverings and lifts of automorphisms Isomorphism Classes of Concrete Graph Coverings Constructing 4-valent <inline-equation> <f> <fr><nu>1</nu><de>2</de></fr></f> </inline-equation>-transitive graphs with a nonsolvable automorphism group Lifting graph automorphisms by voltage assignments Strongly adjacency-transitive graphs and uniquely shift-transitive graphs --CTR Tomaz Pisanski , Thomas W. Tucker , Boris Zgrabli, Strongly adjacency-transitive graphs and uniquely shift-transitive graphs, Discrete Mathematics, v.244 n.1-3, p.389-398, 6 February 2002 Aleksander Malni , Roman Nedela , Martin koviera, Regular homomorphisms and regular maps, European Journal of Combinatorics, v.23 n.4, p.449-461, May 2002

regular map;voltage group;cayley graph;lifting automorphisms;covering projection;action graph;schreier graph;group action;group presentation

606529

Letter graphs and well-quasi-order by induced subgraphs.

Given a word w over a finite alphabet and a set of ordered pairs of letters which define adjacencies, we construct a graph which we call the letter graph of w. The lettericity of a graph G is the least size of the alphabet permitting to obtain G as a letter graph. The set of 2-letter graphs consists of threshold graphs, unbounded-interval graphs, and their complements. We determine the lettericity of cycles and bound the lettericity of paths to an interval of length one. We show that the class of k-letter graphs is well-quasi-ordered by the induced subgraph relation, and that it has a finite set of minimal forbidden induced subgraphs. As a consequence, k-letter graphs can be recognized in polynomial time for any fixed k.

Introduction In graph theory, a reflexive and transitive relation is called a quasi-order. A quasi-order - on X is a well-quasi-order if for any infinite sequence a 1 ; a there are indices such that a i - a j . Equivalently, X contains no infinite strictly decreasing sequences and no infinite antichains. Yet another equivalent characterization of well-quasi-orders is that every nonempty subset of X has a nonzero finite number of minimal elements (cf. [9, 12]). By the famous Graph Minor Theorem of N. Robertson and P. D. Seymour, the graph minor relation is a well-quasi-order on the class of all graphs. This, however, is not true for the more restrictive relations such as the topological minor (or homeomorphic embeddabil- ity), the subgraph, and the induced subgraph relations. It is therefore of interest to identify restricted classes of graphs which are well-quasi-ordered by these relations. For example, the class of all trees is well-quasi-ordered by the topological minor relation, according to a well-known theorem of J. B. Kruskal [11]. G. Ding has proved that a subgraph ideal (i.e., a class of graphs closed under taking subgraphs) is well-quasi-ordered by the subgraph relation if and only if it contains at most finitely many graphs C n and F n (C n being the cycle on vertices, and F n the path on n vertices with two pendant edges attached to each of its endpoints). Concerning the induced subgraph relation - i that we shall consider here, the following is known. P. Damaschke [3] has proved that P 4 -reducible graphs (i.e., graphs in which all induced paths on four vertices are vertex-disjoint) are well-quasi-ordered by - i . G. Ding has proved that the following classes of graphs are well-quasi-ordered by the class of graphs G such that for some R ' V (G) with jRj - r, the graph G \Gamma R has matroidal number at most three [6], ffl any subgraph ideal which is well-quasi-ordered by the subgraph relation [5]. In [3] and [5], several further classes of graphs defined by excluding a finite set of forbidden induced subgraphs have been shown well-quasi-ordered by - i . In this paper we present another family of induced-subgraph ideals which are well-quasi- ordered by - i . Given a word w over a finite alphabet and a set of ordered pairs of letters which define adjacencies, we construct a graph which we call the letter graph of w. The lettericity of a graph G is the least size of alphabet permitting to obtain G as a letter graph. In Section 3 we state some basic properties of k-letter graphs. The class of 2-letter graphs is described completely in Section 4: it is composed of threshold graphs, unbounded-interval graphs, and their complements. In Section 5 we determine the lettericity of cycles and paths (the latter only to within an interval of length one) and show that for large n there are n-vertex graphs whose lettericity exceeds 0:707 n. In Section 6 we show that the class of k-letter graphs is well-quasi-ordered by - i and has a finite set of minimal forbidden induced subgraphs. As a consequence, for any fixed k the class of k-letter graphs can be recognized in polynomial time. Definitions and notation Our graphs are undirected and simple. We write x -G y if x and y are adjacent vertices of G. As a set of pairs, the adjacency relation in V (G) is denoted by AdjG . The complement of a graph G is denoted by G. If A is a set of graphs we write A for the set fG ; G 2 Ag. The disjoint union of G 1 and G 2 is denoted by G 1 and the disjoint union of n copies of G is denoted by nG. As usual, K n denotes the complete graph on n vertices, K p;q the complete bipartite graph on p + q vertices, P n the path on n vertices, and C n the cycle of length n. The vertex set of P n is The vertex set of C n is f0; If A is a set of graphs closed under taking induced subgraphs we denote by Obs(A) the set of obstructions or minimal forbidden induced subgraphs for A (i.e., the minimal elements of the complement of A quasi-ordered by the induced subgraph relation). The isomorphism relation among graphs is denoted by - =. By z(G) we denote the cochromatic number of G, which is the minimum cardinality of a partition of V (G) into subsets that are either a clique or an independent set. Let \Sigma be a finite alphabet and \Sigma the set of all words over \Sigma (i.e., the free monoid generated by \Sigma under concatenation). For a word its reverse. If A is a set of words we write A R for Ag. Let P ' \Sigma 2 be a fixed set of ordered pairs of symbols from \Sigma. To each word its letter graph G(P; w) in the following way: The vertices of G(P; w) are naturally labelled with the symbols of w. Example 1 Take abcabc. The corresponding letter graph G(P; w) is shown in Fig. 1 where vertex i is labelled with s i . In this case, G(P; w) is the 6-cycle C 6 . a b c a b c Figure 1: C 6 as a 3-letter graph Denote G \Sigma (P); Thus G k is the set of all graphs that are letter graphs over some alphabet of size k, and l(G) is the least alphabet size that suffices to represent G as a letter graph. The graphs from G k will be called k-letter graphs, and l(G) the lettericity of G. Example 1 shows that l(C 6 ) - 3. 3 Some properties of k-letter graphs First we restate the definition of k-letter graphs in purely graph-theoretic terms. Proposition 1 A graph G is a k-letter graph if and only if 1. there is a partition such that each V i is either a clique or an independent set in G, and 2. there is a linear ordering L of V (G) such that for each pair of indices 1 - the intersection of AdjG with V i \Theta V j is one of (a) (d) ;. Proof: If G is a k-letter graph then be the different symbols from \Sigma that actually appear in w. Define If a i a clique, otherwise it is an independent set. Let L be the order induced on the vertices of V (G) by the linear ordering of their labels in w, and 1 - i distinguish four cases: (a) a i a only if xLy, so AdjG " (b) a i a (c) a i a In this case x - y for all x (d) a i a 2 P: In this case x 6- y for all x Conversely, let G be a graph on n vertices which satisfies conditions 1 and 2. Take Number the vertices of G so that v 1 We claim that the mapping v i 7! i is an isomorphism from G to First assume that x -G y. If must be a clique in G, so a i a hence l -H m. If i 6= j we distinguish four cases corresponding to those in condition 2: In this case a i a j 2 P. As x L y, we have l ! m and hence l -H m. In this case a j a i 2 P. As x L \Gamma1 y, we have l ? m and hence l -H m. In this case a i a This case is impossible because by assumption, (x; y) 2 AdjG " Now assume that l -H m and, w.l.o.g., that l ! m. Then a i a y. If then is a clique in G, so x -G y. If i 6= j we distinguish three cases corresponding to those in the definition of P: As x L y, it follows that x -G y. As y L \Gamma1 x, it follows that x -G y. In this case x -G y for all x Corollary 1 Let G be a k-letter graph. Then V (G) can be partitioned into p - k sets each of which is either a clique or an independent set in G, such that for each pair of indices 1 - the family of neighborhoods N j all a chain of subsets of V j . Proof: Let L be the linear order on V (G) described in Proposition 1. Pick x; y that x L y. If AdjG " then z L x, so z L y and y -G z, hence N j (x) ' N j (y). If AdjG " In all four cases, one of N j (x), N j (y) is a subset of the other. 2 Next we list some simple observations without proof. be a bijection, extended to \Sigma 1 as a homomorphism. Proposition 2 (i) G(f(P); (ii) Corollary 2 (i) G \Sigma 2 (P). Proposition 3 (i) If only if G If z is a (not necessarily contiguous) subword of w then G(P; z) is an induced subgraph of G(P; w). Hence the set G \Sigma (P) is closed under taking induced subgraphs, and therefore has a characterization with forbidden induced subgraphs. The same is true for G k . Thus lettericity is a monotone parameter w.r.t. the induced subgraph relation. 4 2-letter graphs By Proposition 1, 2-letter graphs are bipartite, split, or cobipartite graphs. In this section we characterize cobipartite 2-letter graphs as unbounded-interval graphs, and split 2-letter graphs as threshold graphs. We also show how our representation helps enumerate the nonisomorphic n-vertex graphs in these classes. For a fixed set of pairs P write this is an equivalence relation in the set \Sigma n of words of length n over \Sigma. 4.1 Unbounded-interval graphs An unbounded-interval graph is the intersection graph of a family of intervals of infinite length on the real line. We denote the set of unbounded-interval graphs by U . Unbounded-interval graphs are studied in [10]. Complements of unbounded-interval graphs are studied in [4]. Example 2 Let I Fig. 2). The intersection graph of these four intervals is the path P 4 , which is therefore an unbounded- interval graph. I 1 I 3 I 2 I 423 I 1 I 2 I 3 I 4 Figure 2: A family of unbounded intervals whose intersection graph is P 4 . The following characterization of unbounded-interval graphs can be found in [10]: Theorem 1 For a graph G, the following assertions are equivalent: (ii) G is triangulated and G is bipartite, (iii) G has no induced subgraphs isomorphic to K 3 , C 4 , or C 5 , RR;RLg. In (iv), vertices corresponding to intervals unbounded on the left (resp. right) are labelled a Fig. 2 shows the example be the word obtained by reversing w and swapping L's and R's. Let ! be a rewrite relation defined by It turns out that the reflexive-transitive closure of ! in \Sigma n coincides with the equivalence relation - defined at the beginning of the section. This fact is used in [10] to show that the number of nonisomorphic n-vertex unbounded-interval graphs is 2 4.2 Threshold graphs A graph G is called threshold if there is a labelling f of its vertices by nonnegative inte- gers, and an integer threshold t such that a set X ' V (G) is independent if and only if t. We denote the set of threshold graphs by T . Threshold graphs were introduced by Chv'atal and Hammer in [1] where the following theorem is proved (see also [2], [7]): Theorem 2 For a graph G, the following assertions are equivalent: (ii) G has no induced subgraphs isomorphic to C 4 , C 4 , or P 4 , are the degrees of the nonisolated vertices of G, is the set of all vertices of degree y, then x is adjacent to y iff Here we characterize threshold graphs as 2-letter graphs. Theorem 3 CSg. Proof: Consider a word w 2 \Sigma , partitioned into blocks of successive C's and S's: By changing the last letter of w if necessary, we can assume that the last nonempty block of w has length at least two. As both C and S have identical sets of left neighbors in P , such change does not affect G. Let D i be the set of vertices of G corresponding to the i-th block of S's in w, and Dm\Gammai the set of vertices corresponding to the i-th block of C's where m is the total number of nonempty blocks in the subword C q . It is straightforward to verify that: vertices within D i have identical degree, say distinct vertices are adjacent iff m. By Theorem 2(iii), G is a threshold graph. Conversely, let G be a threshold graph. Partition V (G) into D 0 , D 1 , . , Dm as described in Theorem 2(iii), and let d dm=2e . It is straightforward to verify that G Let ! be a rewrite relation defined by It is easy to see that the reflexive-transitive closure of ! in \Sigma n coincides with the equivalence relation - defined at the beginning of the section. From this it follows immediately that the number of nonisomorphic n-vertex threshold graphs is 2 4.3 An overview of 2-letter graphs Theorem 4 G Proof: Table 1 gives an overview of the possible classes of 2-letter graphs over induced subgraphs, and their census. As K the theorem follows. 2 elements elements of number of pairwise nonisomorphic of P G \Sigma (P) Obs(G \Sigma (P)) n-vertex graphs in G \Sigma (P) aa ab U K 3 Table 1: 2-letter graphs (p and q denote nonnegative integers). Corollary 3 All graphs on four or fewer vertices are 2-letter graphs. Proof: According to Theorem 2(ii), all graphs on four or fewer vertices except C and C 4 are threshold graphs. As C 4 the claim follows from Theorem 4. 2 Corollary 4 Proof: From Theorem 4 and Table 1 it follows that the graphs not in G 2 have at least one induced subgraph in each of the sets fC g. Checking all 27 combinations and discarding redundant ones we see that such graphs contain at least one of the following seven sets of induced subgraphs: fC g. Thus a minimal forbidden induced subgraph for G 2 can have at most 3 vertices. 2 Corollary 5 2-letter graphs can be recognized in polynomial time. Proof: This follows from Theorem 4 because each of the classes T , U , U has a polynomial-time recognition algorithm. 2 5 Lettericity of some n-vertex graphs In this section we consider the lettericity of cycles, paths, and perfect matchings. By a counting argument we show that for large n there are n-vertex graphs whose lettericity exceeds 0:707 n. 5.1 Cycles Call an independent set S in C n tight if ng for some . If a 2 \Sigma gives rise to an independent set S of size three or more in G(P; w) then: (i) S is tight, (iii) the labels of the two vertices of G(P; w) which have both neighbors in S are distinct. Proof: (i) Let R be a maximal run of consecutive vertices of C n which are not in S. If R has two or more vertices then the labels of the two vertices of S adjacent to one of the endpoints of R must be the leftmost and the rightmost a's in w. Hence there is at most one such run, meaning that S is tight. (ii) If S contains more than three vertices, it is tight by (i). W.l.g. assume that 0; 2; 4; 6 2 S. Then in w, the label of 1 (which is adjacent to 0 and 2, but not adjacent to 4 or must be between the labels of 0; 2 and 4; 6, while the label of 3 must be between the labels of 2; 4 and 0; 6. As this is impossible, (iii) By (i) and (ii), S is tight and has three vertices. W.l.g. assume that 4g. If the vertices 1 and 3 are labelled the same, say b, these five vertices correspond to a subword ababa of w where the left b is the label of 3 and forces ba 2 P, while the right b is the label of 1 and forces ab 2 P. But then 1 and 3 would have degree three or more. It follows that vertices 1 and 3 must be labelled differently. 2 Theorem 5 Let Proof: First we prove that at least b n+4c letters are needed to obtain C n . Let C n different letters. As n - 4, the largest clique in C n is of size 2. From Lemma 1(ii) it follows that each letter appears at most three times in w. Therefore dn=3e, so the assertion is proved. If It remains to show that in the latter two cases letters do not suffice. a) Assume that w is a word consisting of k different letters whose letter graph is C 3k . By Lemma 1(ii), each letter gives rise to an independent set of size three. By Lemma 1(i) and (iii), the vertices of C 3k must be (cyclically) labelled a 1 1 a 3 k a 2 2 a 3 3 a 3 k a 1 1 where superscripts distinguish the three occurrences of each letter. It remains to see how these symbols could be arranged linearly in w. As a 3 k is adjacent to a 1 1 and a 2 2 is adjacent to a 2 1 and a 3 1 , it follows that a 2 must be between a 1 1 and a 3 1 in w. W.l.g. assume that the arrangement of these symbols in w is a 1 1 a 3 1 . By induction on i it can be shown that a 1 precedes a 2 which precedes a 3 in w, and also that a 1 precedes a 1 Hence a 1 1 precedes all three occurrences of a k in However, being adjacent to exactly two of the corresponding vertices this is impossible. As before, assume that w is a word consisting of k different letters whose letter graph is C . This is only possible if of the letters give rise to an independent set of size three, and the remaining letter, say a rise to either a clique or an independent set of size two. In case of a clique, an independent set bordering on it must have the intervening two vertices labelled the same, contrary to Lemma 1(iii). So a 1 gives rise to an independent set of size two. By Lemma 1(i) and (iii), the only possible way to label (cyclically) the vertices of C 3k\Gamma1 is a 1 1 a 3 k a 1 3 a 3 k a 1 1 where superscripts distinguish different occurrences of each letter. It remains to see how these symbols could be arranged linearly in w. Similarly as in the case a) we can establish that a 1 precedes a 2 which precedes a 3 i in w, for that a 1 precedes a 1 Hence a 1 precedes all three occurrences of a k in w. However, being adjacent to exactly one of the corresponding vertices this is impossible. It remains to construct C n using no more than b n+4c letters. We distinguish three cases w.r.t. n mod 3. In all three cases, the alphabet is a) k a 2 k a 3 k\Gamma2 where superscripts are added for easier reference. Write t i\Gamma2 . Then it is easy to check that G(P; w) is the cycle k a 1 k of length 3k + 1. k a 2 k a 3 k\Gamma3 . As be- fore, i\Gamma2 . Then it is easy to check that G(P; w) is the cycle k a 1 k a 2 of length 3k. For construction is shown in Fig. 1 (with a c) k a 2 k a 3 k\Gamma4 . Write again t i\Gamma2 . Then it is easy to check that G(P; w) is the cycle k a 1 k a 2 k\Gamma2 a 1 k\Gamma2 of length 3k \Gamma 1. 2 5.2 Paths . If a 2 \Sigma gives rise to an independent set S of size three or more in G(P; w) then S is of one of the following types: (a) f1; 3; (b) (c) (d) Proof: Similar to that of Lemma 1. 2 Theorem 6 b n+1c - Proof: For the upper bound, we show how to construct P n using no more than b n+4c letters. We distinguish two cases w.r.t. n mod 3. a) a 1 k a 2 k a 3 k\Gamma2 where superscripts are added for easier reference. Write . Then it is easy to check that G(P; w) is the path t k a 1 k of length 3k + 1. By Theorem 5, C n+1 can be constructed using letters. The same then goes for P n as it is an induced subgraph of C n+1 . For the lower bound, let P n different letters. Lemma 2 implies that at most one letter can appear four times in w, while the rest can appear three times at most. Therefore n - 4 Conjecture: If n - 3 then 5.3 Maximum lettericity of n-vertex graphs Let l(n) denote the maximum lettericity of an n-vertex graph. Clearly, 2. As l(G) - z(G), the maximum cochromatic number of an n-vertex graph (which is known to be of order n= log n [8]) constitutes a lower bound for l(n). But this is a poor bound: we have seen that the lettericity of paths and cycles on n vertices is about n=3 which is much larger than n= log n when n is large. It is also easy to see that and n=2. By a counting argument we now improve this bound to l(n) ? 0:707 n, provided that n is large enough. Theorem 7 For each ff ! there is an N such that for all n ? N there are n-vertex graphs G with l(G) ? ff n. Proof: Assume that l(G) - ff n for all graphs G on n vertices. Write our assumption, all graphs on n vertices are k-letter graphs. There are 2 ( n) labelled graphs on n vertices. Over a k-letter alphabet, there are k 2 pairs of letters, 2 k 2 sets of pairs of letters, k n words of length n, and at most n! possible labellings of a graph on n vertices, hence there are no more than n! labelled k-letter graphs on n vertices. Therefore Taking base 2 logarithms we have n: this is impossible when n is large. 2 As for a simple upper bound, Proposition 3(i) implies that l(n) - 2. It is also not difficult to see that l(n) - 6 k-letter graphs and well-quasi-order By deleting a vertex the lettericity of a graph can decrease by more than one: for example, 2. We need an upper bound on the extent of this decrease. Proof: Let g. Let a i 1 be the labels of the neighbors of v in w. Take \Sigma are new symbols, and P l ; a 0 l ; a j a l 2 rg. Denote by w 0 the word obtained from w by replacing the labels a i of the neighbors of v by a 0 Theorem 8 The class G k of k-letter graphs is well-quasi-ordered by the induced subgraph relation. Proof: Fix an alphabet \Sigma of cardinality k and a set of pairs P ' \Sigma 2 . By Higman's Lemma [9, Thm. 4.4], \Sigma is well-quasi-ordered by the (not necessarily contiguous) subword relation. Clearly, z is a subword of w if and only if G(P; z) is an induced subgraph of G(P; w), hence G \Sigma (P) is well-quasi-ordered by the induced subgraph relation. As G k is a union of finitely many sets of the form G \Sigma (P) (one for each of the 2 k 2 possible P's) the conclusion follows. 2 Theorem 9 The sets of obstructions Obs(G \Sigma (P)) and Obs(G k ) are finite. As Obs(G \Sigma (P)) is an antichain, Theorem 8 implies that it is finite. Finiteness of Obs(G k ) is proved in the same way. 2 Corollary 6 The graphs from G \Sigma (P) and G k are recognizable in polynomial time. Proof: The relation H - i G is decidable in time O(n m ) where jV (H)j. For fixed H this is polynomial in n. Thus by Theorem 9, checking that H 6- i G for is given is a polynomial-time recognition algorithm for G \Sigma (P) (resp. G k ). 2 Note that the proof of Corollary 6 is nonconstructive as the specification of the algorithm given there is incomplete: the finite sets of obstructions for G \Sigma (P) and G k that are used by the algorithm are, in general, unknown. 7 Conclusion We conclude by listing some open problems. Problem 1. Design efficient algorithms to recognize k-letter graphs for small fixed values of k. Problem 2. What is the time complexity of finding the lettericity of a given graph? Problem 3. Find the maximal possible lettericity of an n-vertex graph, and the corresponding extremal graphs. Acknowledgements The author is indebted to Bojan Mohar and Toma-z Pisanski for helping out with this paper (in particular, Toma-z suggested that k-letter graphs should be recognizable in polynomial time, and Bojan pointed out Theorem 7). He also wishes to thank the referees for their careful reading of the paper and valuable suggestions. --R Aggregation of inequalities in integer programming Induced subgraphs and well-quasi-ordering Covering the edges with consecutive sets Subgraphs and well-quasi-ordering Stable sets versus independent sets Algorithmic Graph Theory and Perfect Graphs Some extremal results in cochromatic and dichromatic theory Ordering by divisibility in abstract algebras Intersection graphs of halflines and halfplanes The theory of well-quasi-ordering: a frequently discovered concept --TR Intersection graphs of halflines and halfplanes Induced subgraphs and well-quasi-ordering Subgraphs and well-quasi-ordering Stable sets versus independent sets

induced subgraph relation;lettericity;well-quasi-order

606695

Stability Analysis of Second-Order Switched Homogeneous Systems.

We study the stability of second-order switched homogeneous systems. Using the concept of generalized first integrals we explicitly characterize the "most destabilizing" switching-law and construct a Lyapunov function that yields an easily verifiable, necessary and sufficient condition for asymptotic stability. Using the duality between stability analysis and control synthesis, this also leads to a novel algorithm for designing a stabilizing switching controller.

Introduction . We consider the switched homogeneous system: _ are homogeneous functions (with equal degree of homogeneity), and Co denotes the convex hull. An important special case reduces to a switched linear system. Switched systems appear in many elds of science ranging from economics to electrical and mechanical engineering [15][18]. In particular, switched linear systems were studied in the literature under various names, e.g., polytopic linear dieren- tial inclusions [4], linear polysystems [6], bilinear systems [5], and uncertain linear systems [20]. is an equilibrium point of (1.1). Analyzing the stability of this equilibrium point is di-cult because the system admits innitely many solutions for every initial value 1 . Stability analysis of switched linear systems can be traced back to the 1940's since it is closely related to the well-known absolute stability problem [4][19]. Current approaches to stability analysis include (i) deriving su-cient but not necessary and su-cient stability conditions, and (ii) deriving necessary and su-cient stability conditions for the particular case of low-order systems. Popov's criterion, the circle criterion [19, Chapter 5] and the positive-real lemma [4, Chapter 2] can all be considered as examples of the rst approach. Many other su-cient conditions exist in the literature 2 . Nevertheless, these conditions are su-cient but not necessary and su-cient and are known to be rather conservative conditions. Far more general results were derived for the second approach, namely, the particular case of low-order linear switched systems. The basic idea is to single out the \most unstable" solution ~ x(t) of (1.1), that is, a solution with the following property: If ~ x(t) converges to the origin then so do all the solutions of (1.1). Then, all that is left to analyze is the stability of this single solution (see, e.g., [3]). Department of Theoretical Mathematics, Weizmann Institute of Science, Rehovot, Israel 76100. (holcman@wisdom.weizmann.ac.il). y (Corresponding author) Department of Electrical Engineering-Systems, Tel Aviv University, Israel 69978. (michaelm@eng.tau.ac.il). An analysis of the computational complexity of some closely related problems can be found in [2] 2 See, for example, the recent survey paper by Liberzon and Morse [11] D. HOLCMAN AND M. MARGALIOT Pyatnitskiy and Rapoport [16][17] were the rst to formulate the problem of nd- ing the \most unstable" solution of (1.1) using a variartional approach. Applying the maximum principle, they developed a characterization of this solution in terms of a two-point boundary value problem. Their characterization is not explicit but, never- theless, using tools from convex analysis they proved the following result. Let be the collection of all the q-sets of linear functions fA xg for which (1.1) is asymptotically stable, and denote the boundary 3 of by @. Pyatnitskiy and Rapoport proved that if fA 1 then the \most unstable" solution of (1.1) is a closed trajectory. Intuitively, this can be explained as follows. If fAx; Bxg 2 then, by the denition of , ~ x(t) converges to the origin; if fAx; Bxg is unbounded. Between these two extremes, that is, when fAx; Bxg 2 @, ~ x(t) is a closed solution. This leads to a necessary and su-cient stability condition for second and third-order switched linear systems [16][17], however, the condition is a nonlinear equation in several unknowns and, since solving this equation turns out to be di-cult, it cannot be used in practice. Margaliot and Langholz [14] introduced the novel concept of generalized rst integrals and used it to provide a dierent characterization of the closed trajectory. Unlike Pyatnitskiy and Rapoport, the characterization is constructive and leads, for second-order switched linear systems, to an easily veriable necessary and su-cient stability condition. Furthermore, their approach yields an explicit Lyapunov function for switched linear systems. In the general homogeneous case, the functions f i () are nonlinear functions, and therefore, the approaches used for switched linear systems cannot be applied. Filippov derived a necessary and su-cient stability condition for second-order switched homogeneous systems. However, his proof uses a Lyapunov function that is not constructed explicitly. In this paper we combine Fillipov's approach with the approach developed by Margaliot and Langholz to provide a necessary and su-cient condition for asymptotic stability of second-order switched homogeneous systems. We construct a suitable explicit Lyapunov function and derive a condition that is easy to check in practice. A closely related problem is the stabilization of a several unstable systems using switching. This problem has recently regained new interest with the discovery that there are systems that can be stabilized by hybrid controllers whereas they cannot be stabilized by continuous state-feedback [18, Chapter 6]. To analyze the stability of (1.1), we synthesize the \most unstable" solution ~ x(t) by switching between several asymptotically stable systems. Designing a switching controller is equivalent to synthesizing the \most stable" solution by switching between several unstable systems. These problems are dual and, therefore, a solution of the rst is also a solution of the second. Consequently, we use our stability analysis to develop a novel procedure for designing a stabilizing switching controller for second-order homogeneous systems. The rest of this paper is organized as follows. Section 2 includes some notations and assumptions. Section 3 develops the generalized rst integral which will serve as our main analysis tool. Section 4 analyzes the sets and @. Section 5 provides an explicit characterization of the \most destabilizing" switching-law. Section 6 presents an easily veriable necessary and su-cient stability condition. Section 7 describes a new algorithm for designing a switching controller. The nal section summarizes. 3 The set is open [17] SECOND-ORDER SWITCHED HOMOGENEOUS SYSTEMS 3 2. Notations and Assumptions. For > 1, let that is, the set of homogeneous functions of degree . We denote by E the set of Consider the system _ . Transforming to polar coordinates we get _ where R() and A() are homogeneous functions of degree +1 in the variables cos() and sin(). Following ([9], Chapter III), we analyze the stability of (2.1) by considering two cases. If A() has no zeros then the origin is a focus and (2.1) yields: R is periodic in with period 2, and h :=2 R 2R(u) If A has zeros say, then the line = is a solution of (2.1) (the origin is a node) and along this line r(t) Hence, if ES := ff 2 asymptotically stableg, then ES ES N , where 4 has no zeros and sgn(h) 6= sgn(A)g ES N such that Given f its dierential at x by (Df)(x) := The dierential's norm is jj(Df)(x)jj := sup h2R vector norm on R 2 . The distance between two functions f is dened by [10]: (jjf Note that (E ; d(; )) is a Banach space and that in the topology induced by d(; ) the set ES is open. For simplicity 5 , we consider the dierential inclusion (1.1) with 2: _ Given an initial condition x 0 , a solution of (2.3) is an absolutely continuous function x(t), with almost all t. Clearly, there is 4 Here F stands for focus and N for node 5 Our results can be easily generalized to the case q > 2 4 D. HOLCMAN AND M. MARGALIOT an innite number of solutions for any initial condition. To dierentiate the possible solutions we use the concept of a switching-law. Definition 2.1. A switching-law is a piecewise constant function : [0; [0; 1]. We refer to the solution of _ as the solution corresponding to the switching-law . The solution x(t) 0 is said to be uniformly 6 locally asymptotically stable if: Given any > 0, there exists -() > 0 such that every solution of (2.3) with There exists c > 0 such that every solution of (2.3) satises lim Since f and g are homogeneous, local asymptotic stability of (2.3) implies global asymptotic stability. Hence, when the above conditions hold, the system is uniformly globally asymptotically stable (UGAS). Definition 2.2. A set P R 2 is an invariant set of (2.3) if every solution x(t), with Definition 2.3. We will say that singular if there exists an invariant set that does not contain an open neighborhood of the origin. We assume from here on that Assumption 1. The set (x) is not singular. The role of Assumption 1 will become clear in the proof of Lemma 5.4 below. Note that it is easy to check if the assumption holds by transforming the two systems _ f (x) and _ to polar coordinates and examining the set of points where _ for each system. For example, if there exists a line l that is an invariant set for both _ l is an invariant set of (2.3) and Assumption 1 does not hold. To make the stability analysis nontrivial, we also assume Assumption 2. For any xed 2 [0; 1], the origin is a globally asymptotically stable equilibrium point of _ 3. The Generalized First Integral. If the system _ is Hamiltonian [8] then it admits a classical rst integral, that is, a function H(x) which satises H(x(t)) H(x(0)) along the trajectories of (3.1). In this case, the study of (3.1) is greatly simplied since its trajectories are nothing but the contours const. In particular, it turns out that the rst integral provides a crucial analysis tool for switched linear systems [14]. The purpose of this section is to extend this idea to the case where f 2 ES and, therefore, (3.1) is not Hamiltonian. dv dx2 are both homogeneous functions of degree and, therefore, the ratio f2 (x1 ;x2 ) is a function of v only which we denote by (v). Hence, along the trajectories of (3.1): dv x1 , that is, e R dv const. Thus, we dene 6 The term \uniform" is used here to describe uniformity with respect to switching signals SECOND-ORDER SWITCHED HOMOGENEOUS SYSTEMS 5 the generalized rst integral of (3.1) by 2k where L(v) := R dv and k is a positive integer. Note that we can write substituting Let S be the collection of points where H(x not dened or not continuous then, by construction, H along the trajectories of (3.1). If classical rst integral of the system. In general, how- ever, S 6= ;. Nevertheless, this does not imply that H cannot be used in the analysis of (3.1). Consider, for example, the case where S is a line and a trajectory x(t) of (3.1) can cross S but not stay on S. Then, H(x(t)) will remain constant except perhaps at a crossing time 7 where its value can \jump". Thus, a trajectory of the system is a concatenation of several contours of H . This motivates the term generalized rst integral. To clarify the relationship between the trajectories of _ and the contours const, we consider an example. Example 1. Consider the system _ _ Here (3.2) yields 2k , and using In this case 0g. It is easy to verify that l 1 is an invariant set of (3.3), that is, x(t) \ l the trivial trajectory that starts and stays on l 1 ). Furthermore, it is easy to see that a trajectory of (3.3) cannot stay on the line l 2 . Fig. 3.1 shows the trajectory x(t) of (3.3) for x Fig. 3.2 displays H(x(t)) as a function of time. It may be seen that H(x(t)) is a piecewise constant function that attains two values. Note that the \jump" in H(x(t)) occurs when x is, when x(t) 2 S. 4. The Boundary of Stability. Let be the set of all pairs (f ; g) for which (2.3) is UGAS. In this section we study and its boundary @. Our rst result, whose proof is given in the appendix, is an inverse Lyapunov theorem. Lemma 4.1. If (f ; g) 2 , then there exists a C 1 positive-denite function such that for all x Furthermore, V (x) is positively homogeneous of degree one 8 . Lemma 4.2. is an open cone. Proof. Let (f ; g) 2 . Clearly, (cf ; cg) 2 for all c > 0. Hence, is a cone. 7 That is, a time t 0 such that x(t 0 8 That is, 6 D. HOLCMAN AND M. MARGALIOT Fig. 3.1. The trajectory of (3.3) for x 0 Fig. 3.2. H(x(t)) as a function of time. To prove that is open, we use the common Lyapunov function V from Lemma 4.1. Denote is a closed curve encircling the origin. Hence, there exists a < 0 such that for all x 2 rV (x)f(x) < a and rV (x)g(x) < a If ~ f 2 ES and ~ are such that su-ciently small, then for all x 2 It follows from the homogeneity of V , ~ f , and ~ g that ( ~ f ; ~ 5. The Worst-Case Switching Law. In this section we provide two explicit characterizations of the switching-law that yields the \most unstable" solution of (2.3). SECOND-ORDER SWITCHED HOMOGENEOUS SYSTEMS 7 be the generalized rst integral of _ Definition 5.1. Dene the worst-case switching-law (WCSL) by: We denote so the solution corresponding to WCSL satises _ h(x). Note that WCSL is a state-dependent switching-law and that since or respectively, that is, the vertices of . Furthermore, it is easy to see that h(x) is homogeneous of degree . Intuitively, WCSL can be explained as follows. Consider a point x where f (x) and g(x) are as shown in Fig. 5.1. A solution of _ follows the contour H f const, whereas, a solution of _ contour going further away from the origin. In this case, rH f (x)g(x) > 0 so WCSL is which corresponds to setting _ \pushes" the trajectory away from the origin as much as possible. const Fig. 5.1. Geometrical explanation of WCSL when rH f (x)g(x) > 0 Note that the denition of WCSL using (5.1) is meaningful only for x since rH f (x) is not dened for x 2 S. However, extending the denition of WCSL to any x 2 R 2 is immediate since x 2 S implies one of two cases. In the rst case, x 2 l, where l is a line in R 2 which is an invariant set of _ (with c < 0 since f is asymptotically stable) so clearly WCSL must use g. In the second case, the trajectory of _ so the value of the switching-law on the single point x can be chosen arbitrarily. We expect WCSL to remain unchanged if we swap the roles of f and g. Indeed, this is guaranteed by the following lemma whose proof is given in the Appendix. 8 D. HOLCMAN AND M. MARGALIOT Lemma 5.2. For all x 2 D where sgn() is the sign function. We can now state the main result of this section. Theorem 5.3. (f ; g) 2 @ if and only if the solution corresponding to WCSL is closed 9 . Proof. Denote the solution corresponding to WCSL by x(t) and suppose that x(t) is closed. Let be the closed curve is the smallest time such that x(T that using the explicit construction of (x) (see (5.1)) we can easily dene explicitly as a concatenation of several contours of H f (x) and H g (x). Note also that the switching between _ takes place at points x where rH f (see (5.1)), that is, when g(x) and f (x) are collinear. Hence, the curve has no corners. We dene the function V (x) by V that is, the contours of V are obtained by scaling (see [1]). The function V (x) is positively homogeneous (that is, for any c 0: V radially unbounded, and dierentiable on R 2 n f0g. Note that both f (x) and g (x) belong to E . We use V (x) to analyze the stability of the perturbed system _ (x)g. Consider the derivative of V along the trajectories of _ _ t. If at some x, V (x) corresponds to a contour H f then rV and, by the denition of WCSL (see (5.1)), rV (x)g(x) 0 so _ corresponds to a contour H g so rV any < 0 we have _ since this holds for for all x and all (t) 2 [0; 1], we get that for < 0: On the other hand, for > 0 and _ since this holds for all x, _ x (x) admits an unbounded solution for > 0. The derivations above hold for arbitrarily small and, therefore, For the opposite direction, assume that (f ; g) 2 @, and let x(t) be the solution corresponding to WCSL, that is, x(t) satises _ To prove that x(t) is a closed trajectory we use the following Lemma, whose proof appears in the Appendix. 9 We omit specifying the initial condition, because the fact that h(x) is homogeneous implies that, if the solution starting at some x 0 is closed, then all solutions are closed. SECOND-ORDER SWITCHED HOMOGENEOUS SYSTEMS 9 Lemma 5.4. If (f ; g) 2 @ then the solution corresponding to WCSL rotates around the origin. Thus, for a given x 0 6= 0, there exists t 1 > 0 such that x(t), with since h(x) is homogeneous, we get: x(nt 1 We consider two cases. If c > 1 then x(t) is unbounded and using the homogeneity of h(x) we conclude that 0 is a (spiral) source. It follows from the theory of structural stability (see, e.g., [10]) that there exists an > 0 such that for all g) with the origin is a source of the perturbed dynamical system _ This implies that (f ; g) 62 @ which is a contradiction. If c < 1 then x(t) converges to the origin and, by the construction of WCSL, so does any other solution, so (f ; g) 2 which is again a contradiction. Hence, that is, x(t) is closed. The characterization of WCSL using the generalized rst integrals leads to a simple and constructive proof of Theorem 5.3. However, to actually check whether the solution corresponding to WCSL is closed, a characterization of WCSL in polar coordinates is more suitable. Representing (2.3) in polar coordinates, we get _ r _ cos sin sin r cos r If (f ; g) 2 @ then WCSL yields a closed solution. By using the transformation necessary) we may always assume that this solution rotates around the origin in a counter clockwise direction, that is, _ that this implies that if at some point x the trajectories of one of the systems are in the clockwise direction, then WCSL will use the second system. Hence, determining WCSL is non-trivial only at points where the trajectories of both systems rotate in the same direction and we assume from here on that both rotate in a clockwise direction (note that this explains why in Lemma 5.2 it is enough to consider x 2 D). so F is a parameterization of the set of directions in for which _ > 0. For any (r; ) we dene the switching-law _ r _ that is, is the switching-law that selects, among all the directions which yield _ > 0, the direction that maximizes d ln r d . Using (5.4), we get (cos sin )j (r; ) sin cos )j (r; ) Let (cos sin )j (r; ) sin cos )j (r; ) so that along the trajectory corresponding to : 1 r _ m. Note that since f and g are homogeneous, D. HOLCMAN AND M. MARGALIOT It is easy to verify that the function q(y) := ay+b(1 y) (where c and d are such that the denominator is never zero) is monotonic and, therefore, (r; ) in (5.6) is always 0 or 1 and m(r; ) in (5.7) is always one of the two values: (cos sin sin cos (cos sin sin cos respectively. The next lemma, whose proof is given in the Appendix, shows that the switching- law is just the worst-case switching-law . Lemma 5.5. The switching-law yields a closed solution if and only if yields a closed solution. Let I := Z 2m()d d d where (r(t); (t)) is the solution corresponding to the switching-law , and T is the time needed to complete a rotation around the origin. This solution is closed if and only ln(r(T Combining this with Lemma 5.5 and Theorem 5.3 we immediately obtain Theorem 5.6. (f ; g) 2 @ if and only if I = 0. It is easy to calculate I numerically and, therefore, Theorem 5.6 provides us with a simple recipe for determining whether (f ; g) 2 @. However, note that we assumed throughout that the closed solution of the system rotates in a counter clockwise di- rection. Thus, to use Theorem 5.6 correctly, I has to be compute twice: First, for the original system, and then for the transformed system r value by I 0 ). (f ; g) 2 @ if and only if max(I ; I 0 In this way, we nd whether the system has a closed trajectory, rotating around the origin in a clockwise or counter clockwise direction. The following example demonstrates the use of Theorem 5.6. Example 2. (Detecting the boundary of stability). Consider the system: _ where It is easy to verify that f 2 ES N 3 and since g 0 have . The problem is to determine the smallest k > 0 such that (f(x); g k (x)) 2 @ . Transforming to polar coordinates we get: sin 3 2 cos 3 cos sin 2 cos SECOND-ORDER SWITCHED HOMOGENEOUS SYSTEMS 11 Fig. 5.2. I(k) as a function of k so sin 3 2 cos 3 cos sin 2 cos and (cos sin )j (r; ) sin cos )j (r; ) d where F () includes 0 if ( sin cos )j 0 Note that although j is a function of both r and , the integrand in (5.11) is a function of (and k) but not of r. We calculated I(k) numerically for dierent values of k. The results are shown in Fig. 5.2. The value k for which I(k (to four-digit accuracy) and it may be seen that for k < We repeated the computation for the transformed system and found that there exists no closed solution rotating around the origin in a clockwise direction. Hence, the system (5.9) and (5.10) is UGAS for all k 2 [0; k ) and unstable for all k > k . The WCSL (see (5.6)) for Fig. 5.3 depicts the solution of the system given by (5.9) and (5.10) with 1:3439, WCSL (5.12), and x It may be seen that the solution is a closed trajectory, as expected. Note that this trajectory is not convex which implies that the Lyapunov function used in the proof of Theorem 5.3 (see (5.3)) is not convex. This 12 D. HOLCMAN AND M. MARGALIOT -0.4 Fig. 5.3. The solution of (5.9) and (5.10) for is a phenomenon that is unique to nonlinear systems. For switched linear systems the closed trajectory is convex and, therefore, so is the Lyapunov function V that yields a su-cient and necessary stability condition [14]. 6. Stability Analysis. In this section we transform the original problem of analyzing the stability of (2.3) to that of detecting the boundary of stability @. The later problem was solved in section 5. Given a new homogeneous function h k (x) with the following properties: (1) h 0 g. One possible example that satises the above is: Consider the switched homogeneous system _ The absolute stability problem is: nd the smallest k > 0, when it exists, such that Noting that and k1 k2 for all k 1 < k 2 , we immediately obtain the following result. Lemma 6.1. The system (2.3) is UGAS if and only if k > 1. Thus, we can always transform the problem of analyzing the stability of a switched dynamical system to an absolute stability problem. We already know how to solve the latter problem for second-order homogeneous systems. To illustrate this consider the following example. Example 3. Consider the system (2.3) It is easy to verify that f (x) and g(x) belong to ES 3 and that both Assumptions 1 and 2 are satised. SECOND-ORDER SWITCHED HOMOGENEOUS SYSTEMS 13 To analyze the stability of the system we use Lemma 6.1. Dening we must nd the smallest k such that (f ; h k ) 2 @ . We already calculated k in Example 2 and found that k 1. Hence, the system (2.3) with f and g given in (6.2) is UGAS. 7. Designing a Switching Controller. In this section we employ our results to derive an algorithm for designing a switching controller for stabilizing homogeneous systems. To be concrete, we focus on linear systems rather than on the general homogeneous case. Hence, consider the system _ where K 1 and K 2 are given matrices that represent constraints 10 on the possible controls. We would like to design a stabilizing state-feedback controller that satises the constraint u(t) 2 U for all t. We assume that for any xed matrix K 2 the matrix A is strictly unstable and, therefore, a linear controller not stabilize the system. However, it is still possible that a switching controller will stabilize the system and designing such a controller (if one exists) is the purpose of this section. Roughly speaking, we are trying to nd a switching-law that yields an asymptotically stable solution of _ each matrix in is strictly unstable. Using the transformation t, we see that such a solution exists if and only if this switching-law yields an unstable solution of _ Clearly, every matrix in is asymptotically stable. Hence, we obtain the main result of this section. Theorem 7.1. Let be WCSL for the system _ x be the corresponding solution. There exists a switching controller that asymptotically stabilizes (7.1) if and only if ~ x is unbounded and, in this case, stabilizing controller. Note that Theorem 7.1 provides an algorithm for designing a stabilizing switching controller whenever such a controller exists. We already solved the problem of analyzing ~ x for second-order systems. Example 4. (Designing a stabilizing switching controller). Consider the system (7.1) with is a constant. It is easy to verify that for any xed K 2 the unstable and ,therefore, no linear controller the system. Therefore, we design a switching controller. By Theorem 7.1 we must analyze the stability of the switched system (6.1) with x: example, by the physical limitations of the actuators 14 D. HOLCMAN AND M. MARGALIOT Transforming _ to polar coordinates we get: _ r _ (cos sin )r sin whereas _ becomes: _ r _ sin )r sin Clearly, the solutions of both these systems always rotate in a counter clockwise direction > 0 for all ) and, therefore, for all , we have where (cos sin ) sin It is easily veried that m 1 only if tan 0. Hence, WCSL is: and Z Z 3=2 Computing numerically we nd that the value of k for which I = 0 is k Hence, there exists a switching controller that asymptotically stabilizes (7.1) and (7.2) if and only if k > 6:98513 and is a stabilizing controller. Fig. 7.1 depicts the trajectory of the closed-loop system given by (7.1) and (7.2) with the switching controller (7.3), and x As we can see, the system is indeed asymptotically stable. 8. Summary . We presented a new approach to stability analysis of second-order switched homogeneous systems based on the idea of generalized rst integrals. Our approach leads to an explicit Lyapunov function that provides an easily veriable necessary and su-cient stability condition. Using our stability analysis, we designed a novel algorithm for constructing a switching controller for stabilizing second-order homogeneous systems. The algorithm determines whether the system can be stabilized using switching, and if the answer is a-rmative, outputs a suitable controller. Interesting directions for further research include the complete characterization of the boundary of stability @ and the study of higher-order switched homogeneous systems. Acknowledgments . We thank the anonymous reviewers for many helpful comments SECOND-ORDER SWITCHED HOMOGENEOUS SYSTEMS 15 Fig. 7.1. Trajectory of the closed-loop system with the switching controller with x 0 Appendix . Proof of Lemma 4.1. The existence of a common Lyapunov function V 0 (x) follows from Theorem 3.1 in [13] (see also [12]). However, V 0 is not necessarily homo- geneous. Denote is a closed curve encircling the origin. We dene a new function V (x) by V that is, the contours of V are obtained by scaling (see [1]). V (x) is dierentiable on R 2 n f0g, positively homogeneous of order one, and radially unbounded. For any x 2 we have rV using the homogeneity of V (x) and f (x) this holds for any x 2 R 2 n f0g. Similarly, rV (x)g(x) < 0 for all x 2 R 2 n f0g. Proof of Lemma 5.2. Let and . These two vectors form an orthonormal basis of R 2 and, therefore, and (rH g (x)) For any x 2 D we have a 1 > 0 and since rH f (x) (rH g (x)) is orthogonal to f (x) (g(x)), we also have b 2 > 0. Substituting in (8.1) yields sgn(a 2 which is just (5.2). Proof of Lemma 5.4. The system _ homogeneous and we can represent it in polar coordinates as in Eq. (2.1). If the solution corresponding to WCSL follows the line l := then the solution follows the line l to the origin. However, by the denition of WCSL this is only possible if both the solutions of _ coincide with the line l. D. HOLCMAN AND M. MARGALIOT Thus, the line l is an invariant set of the system which is a contradiction to Assumption 1. If R() 0, then we get a contradiction to Assumption 2. Hence, A() 6= 0 for all 2 [0; 2] and, therefore, there exists c > 0 such that A() > c or A() < c for all 2 [0; 2]. Thus, the solution rotates around the origin. Proof of Lemma 5.5. Suppose that WCSL yields a closed trajectory ~ x(t) that rotates around the origin in a counter clockwise direction ( _ > 0). Assume that at some point x along this trajectory, Note that by the denition of the generalized rst integral: rH f any x 2 R 2 n S. This implies that rH f so (8.2) yields Let be the polar coordinates of x. Since ~ rotates around the origin in a counter clockwise direction, and satises _ at x, we have ( sin cos on the other hand, ( sin cos then by the denition of (see Eq. (5.6)), only if: (cos sin sin cos (cos sin sin cos Simplifying, we see that (8.4) is equivalent to f 1 which is just Eq. (8.3), hence, we proved that and only if --R Set invariance in control A survey of computational complexity results in systems and control Stability of planar switched systems: The linear single input case Linear Matrix Inequalities in System and Control Theory A converse Lyapunov theorem for a class of dynamical systems which undergo switching Stability conditions in homogeneous systems with arbitrary regime switching Stability of Motion Basic problems in stability and design of switched systems A smooth converse Lyapunov theorem for robust stability A converse Lyapunov theorem for nonlinear switched systems Necessary and su-cient conditions for absolute stability: The case of second-order systems Control Using Logic-Based Switching Criteria of asymptotic stability of di An Introduction to Hybrid Dynamical Systems Nonlinear Systems Analysis Nonquadratic Lyapunov functions for robust stability analysis of linear uncertain systems --TR

absolute stability;robust stability;hybrid control;hybrid systems;switched linear systems

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Computing the Smoothness Exponent of a Symmetric Multivariate Refinable Function.

Smoothness and symmetry are two important properties of a refinable function. It is known that the Sobolev smoothness exponent of a refinable function can be estimated by computing the spectral radius of a certain finite matrix which is generated from a mask. However, the increase of dimension and the support of a mask tremendously increase the size of the matrix and therefore make the computation very expensive. In this paper, we shall present a simple and efficient algorithm for the numerical computation of the smoothness exponent of a symmetric refinable function with a general dilation matrix. By taking into account the symmetry of a refinable function, our algorithm greatly reduces the size of the matrix and enables us to numerically compute the Sobolev smoothness exponents of a large class of symmetric refinable functions. Step-by-step numerically stable algorithms are given. To illustrate our results by performing some numerical experiments, we construct a family of dyadic interpolatory masks in any dimension, and we compute the smoothness exponents of their refinable functions in dimension three. Several examples will also be presented for computing smoothness exponents of symmetric refinable functions on the quincunx lattice and on the hexagonal lattice.

Introduction . A d d integer matrix M is called a dilation matrix if the condition holds. A dilation matrix M is isotropic if all the eigen-values of M have the same modulus. We say that a is a mask on Z d if a is a nitely supported sequence on Z d such that 1. Wavelets are derived from renable functions via a standard multiresolution technique. A renable function is a solution to the following renement equation 2Z d where a is a mask and M is a dilation matrix. For a mask a on Z d and a dd dilation matrix M , it is known ([2]) that there exists a unique compactly supported distributional solution, denoted by M a throughout the paper, to the renement equation (1.1) such that ^ a where the Fourier transform of f 2 L 1 (R d ) is dened to be Z R d f(x)e ix dx; 2 R d and can be naturally extended to tempered distributions. When the mask a and dilation matrix M are clear from the context, we write instead of M a for simplicity. Symmetric multivariate wavelets and renable functions have proved to be very useful in many applications. For example, 2D renable functions and wavelets have been widely used in subdivision surfaces and image/mesh compression while 3D renable functions have been used in subdivision volumes, animation and video processing, etc. The research was supported by NSERC Canada under Grant G121210654 and by Alberta Innovation and Science REE under Grant G227120136. y Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1. E-mail: bhan@math.ualberta.ca, URL: http://www.ualberta.ca/bhan Han For a compactly supported function in R d , we say that the shifts of are stable if for every 2 R d , b . For a function 2 L 2 (R d ), its smoothness exponent is dened to be Z R d Smoothness is one of the most important properties of a wavelet system. Therefore, it is of great importance to have algorithms for the numerical computation of the smoothness exponent of a renable function. Let a be a mask and M be a dilation matrix. We denote k 1 the set of all polynomials of total degree less than k. By convention, 1 is the empty set. We say that a satises the sum rules of order k with respect to the lattice MZ d if 2MZ d 2MZ d Dene a new sequence b from the mask a by 2Z d the linear space of all nitely supported sequences on Z d . For a subset K of Z d , by '(K) we denote the linear space of all nitely supported sequences on Z d that vanish outside the set K. The transition operator T b;M associated with the sequence b and the dilation matrix M is dened by 2Z d (R d ) be a renable function with a nitely supported mask a and a dilation matrix M such that the shifts of are stable and a satises the sum rules of order k but not k + 1. Dene the set by \Z d and and dene the slightly smaller subspace V 2k 1 of '( b;M ) to be '( 2Z d When M is isotropic, it was demonstrated in [4, 5, 6, 10, 21, 23, 24, 26, 27, 33, 35] in various forms under various conditions that ) is the spectral radius of the operator T b;M acting on the nite dimensional T b;M -invariant subspace V 2k 1 of '( However, from the point of view of numerical computation, there are some diculties in obtaining the Sobolev smoothness exponent of a renable function via (1.7) by computing the quantity (T b;M due to the following considerations: Computing the Smoothness Exponent of a Symmetric Multivariate Renable Function 3 D1. It is not easy to nd a simple basis for the space V 2k 1 by a numerically stable procedure to obtain a representation matrix of T b;M under such a basis. Theoretically speaking, if some elements in a numerically found basis of V 2k 1 cannot satisfy the equality in (1.6) exactly, then it will dramatically change the spectral radius since in general T b;M has signicantly larger eigenvalues outside the subspace V 2k 1 . D2. When the dimension is greater than one and even when the mask has a relatively small support, in general, the dimensions of the spaces V 2k 1 and '( b;M are very large. For example, for a 3D mask with support [ 7; 7] 3 and sum rules of order 4, we have dim(V 7 dim('( This makes the numerical computation using (1.7) very expensive or even impossible. D3. In order to obtain the exact Sobolev smoothness exponent by (1.7), we have to check the assumption that the shifts of M a are stable which is a far from trivial condition to be veried. Fortunately, the di-culty in D1 was successfully overcome in Jia and Zhang [25], where they demonstrated that (T b;M ) is the largest value in modulus in the set consisting of all the eigenvalues of T b;M j '( excluding some known special eigenvalues. Note that '( b;M ) has a simple basis f- : b;M g, where - and - nfg. On the other hand, both symmetry and smoothness of a wavelet basis are very important and much desired properties in many applications. It is one of the purposes in this paper to try to overcome the di-culty in D2 for a symmetric renable function. We shall demonstrate in Algorithm 2.1 in Section 2 that we can compute the Sobolev smoothness exponent of a symmetric renable function by using a much smaller space than using the space '( b;M ). In Section 3, we shall see that for many renable functions, it is not necessary to directly verify the stability assumption since they are already implicitly implied by the computation. Therefore, the di-culty in D3 does not exist at all for many renable functions (almost all interesting known examples fall into this class). To give the reader some idea about how symmetry can be of help in computing the smoothness exponents of symmetric renable functions, we give the following comparison result in Table 1. See Section 2 for more detail and explanation of Table 1. Table The last two rows indicate the matrix sizes in computing the Sobolev smoothness exponents of symmetric renable functions using both the method in [25] and the method in Algorithm 2.1 in Section 2 in this paper. This table demonstrates that Algorithm 2.1 can greatly reduce the size of the matrix in computing the Sobolev smoothness exponent of a symmetric renable function. Mask 4D mask 3D mask 2D mask 2D mask 2D mask Symmetry full axes full axes hexagonal full axes hexagonal Dilation matrix 2I 4 2I 3 2I 2 Method in [25] 194481 24389 8911 5601 > 3241 Algorithm 2.1 715 560 756 707 294 Masks and renable functions with extremely large supports may be rarely used in real world applications. For a given mask which is of interest in applications, very often there are some free parameters in the mask and one needs to optimize the smoothness exponent of its renable function ([9, 12, 15, 17, 28, 30]). The e-cient algorithms 4 Bin Han proposed in this paper will be of help for such a smoothness optimization problem. On the other hand, a renable function vector satises the renement equation (1.1) with a matrix mask of multiplicity r. A matrix mask of multiplicity r is a sequence of r r matrices on Z d (Masks discussed in this paper correspond to are called scalar masks). Very recently, as demonstrated in [17], multivariate renable function vectors with short support and symmetry are of interest in computer aided geometric design (CAGD) and in numerical solutions to partial dierential equations. Let M be the quincunx dilation matrix (the fourth dilation matrix in Table 1) and let a be a matrix mask of multiplicity 3 with support [ masks of order 1 discussed in [17] are examples of such masks which often have many free parameters and are useful in CAGD). In order to compute the Sobolev smoothness exponent of its renable function vector with such a small mask, without using symmetry, we found that one has to deal with a 1161 1161 matrix (also see [23]). As a consequence, even in low dimensions and for masks with small supports, it is very important to take into account the symmetry of a renable function (vector) in algorithms for the numerical computation of its smoothness exponent. Though for simplicity we only consider scalar masks here, results in this paper can be generalized to matrix masks and renable function vectors which will be discussed elsewhere. The structure of the paper is as follows. In Section 2, we shall present step by step numerically stable and e-cient algorithms for the numerical computation of the Sobolev smoothness exponent of a symmetric renable function. In addition, an algorithm for computing the Holder smoothness exponent of a symmetric renable function will be given in Section 2 provided that the symbol of its mask is nonnegative. In Section 3, we shall study the relation of the spectral radius of a certain operator acting on dierent spaces. Such analysis enables us to overcome the di-culty in D3 for a large class of masks. In Section 4, we shall apply the results in Sections 2 and 3 to several examples including renable functions on quincunx lattice and hexagonal lattice. We shall also present a C 2 3-interpolatory subdivision scheme in Section 4. Next, we shall generalize the well known univariate interpolatory masks in Deslauriers and Dubuc [8] and the bivariate interpolatory masks in [15] to any dimension. Finally, we shall use the results in Sections 2 and 3 to compute Sobolev smoothness exponents of interpolating renable functions associated with such interpolatory masks in dimension three. Programs for computing the Sobolev and Holder smoothness exponents of symmetric renable functions based on the Algorithms 2.1 and 2.5 in Section 2, which come without warranty and are not yet optimized with respect to user interface, can be downloaded at http://www.ualberta.ca/bhan. 2. Computing smoothness exponent using symmetry. In this section, taking into account the symmetry, we shall present an e-cient algorithm for the numerical computation of the Sobolev smoothness exponent of a symmetric multivariate renable function with a general dilation matrix. As the main result in this section, Algorithms 2.1 and 2.5 are quite simple and can be easily implemented, though their proofs and some notation are relatively technical. Before proceeding further, let us introduce some notation and necessary back- ground. Let N 0 denote all the nonnegative integers. For d for For (R d Computing the Smoothness Exponent of a Symmetric Multivariate Renable Function 5 For ed , where e j is the jth coordinate unit vector in R d . Let - 0 denote the sequence such that nf0g. For norm is dened to be kuk p := ( Let M be a d d dilation matrix and a be a mask on Z d . Dene the subdivision operator 2Z d lim kr S n do Let M be a dilation matrix and max be the spectral radius of M (When M is isotropic, then When a mask a satises the sum rules of order k but not k + 1, we dene the following important quantity: The above quantity p (a; M) plays a very important role in characterizing the convergence of a subdivision scheme in a Sobolev space and in characterizing the L p smoothness exponent of a renable function. The L p smoothness of f 2 L p (R d ) is measured by its L p smoothness exponent: constant C and for large enough positive integer n When 2, the above denition of 2 (f) agrees with the denition in (1.2). By generalizing the results in [4, 5, 6, 10, 12, 21, 24, 26, 27, 32, 33, 35] and references therein, we have a and the equality holds when the shifts of M a are stable and M is an isotropic dilation matrix. When M is a general dilation matrix and the shifts of M a are stable, as demonstrated in [5] for the case only have the estimate log a where min := min 16j6d being all the eigenvalues of M . As pointed out in [5], the usual Sobolev smoothness dened in (1.2) and (2.3) is closely related to isotropic dilations and anisotropic Sobolev spaces are needed in the case of an anisotropic dilation matrix. See [5] for more detail on this issue. So, to compute the Sobolev smoothness exponent of a renable function, we need to compute 2 (a; M) and therefore, to compute k (a; M; 2). It is the purpose of this section to discuss how to e-ciently compute k (a; M; 2) when a is a symmetric mask. Let be a nite subset of integer matrices whose determinants are 1. We say that is a symmetry group with respect to a dilation matrix M (see [13]) if forms a group under matrix multiplication and MM 1 2 for all 2 . Obviously, each element in a symmetry group induces a linear isomorphism on Z d . 6 Bin Han Let A d denote the set of all linear transforms on Z d which are given by and is a permutation on d is called the full axes symmetry group. Obviously, A d is a symmetry group with respect to the dilation matrix 2I d . It is also easy to check that A 2 is a symmetry group with respect to the quincunx dilation matrices and Another symmetry group with respect to 2I 2 is the following group which is called the hexagonal symmetry group: Such a group H can be used to obtain wavelets on the hexagonal planar lattice (that is, the triangular mesh). For a symmetry group and a sequence u on Z d , we dene a new sequence (u) as follows: where # denotes the cardinality of the set . We say that a mask a is invariant under if (a) = a. Obviously, for any sequence u, (u) is invariant under since When is a symmetry group with respect to a dilation matrix M , then a is invariant under implies that the renable function M a is also invariant under ; that is, M a a for all 2 . See Han [13] for detailed discussion on symmetry property of multivariate renable functions. We caution the reader that the condition MM 1 2 for all 2 cannot be removed in the denition of a symmetry group with respect to a dilation matrix M . For example, as a subgroup of A is not a symmetry group with respect to the quincunx dilation matrices,though it is a symmetry group with respect to the dilation matrix 2I 2 . So even when a mask a is invariant under such a group , the renable function M a with the quincunx dilation matrices may not be invariant under . Let Z d denote a subset of Z d such that for every 2 Z d , there exists a unique satisfying In other words, Z d is a complete set of representatives of the distinct cosets of Z d under the equivalence relation induced by on Z d . Taking into account the symmetry of a mask, now we have the following algorithm for the numerical computation of the important quantity 2 (a; M ). Algorithm 2.1. Let M be a d d isotropic dilation matrix and let be a symmetry group with respect to the dilation matrix M . Let a be a mask on Z d such that 1. Dene the sequence b as in (1.3). Suppose that b is invariant under the symmetry group and a satises the sum rules of order k but not k+1. The quantity 2 (a; M), or equivalently, k (a; M; 2), is obtained via the following procedure: Computing the Smoothness Exponent of a Symmetric Multivariate Renable Function 7 (a) Find a nite subset K of Z d such that and (b) Obtain a (#K ) (#K ) matrix T as follows: (c) Let (T ) consist of the absolute values of all the eigenvalues of the square matrix T counting multiplicity of its eigenvalues. Then 2 (a; M) is the smallest number in the following set o/ with positive multiplicity where by default log j det Mj 0 := 1 and Before we give a proof to Algorithm 2.1, let us make some remarks and discuss how to compute the set K and the quantities m (j) in Algorithm 2.1. Since the matrix T in Algorithm 2.1 has a simple structure, it is not necessary to store the whole matrix T in order to compute its eigenvalues and many techniques from numerical analysis (such as the subspace iteration method and Arnoldi's method as discussed in [34]) can be exploited to further improve the e-ciency in computing the eigenvalues of T . We shall not discuss such an issue here. One satisfactory set K can be easily obtained as follows: Proposition 2.2. Let K 0 0g. Recursively compute \ Z d . Then K satises all the conditions in (a) of Algorithm 2.1. Proof. Note that K j ( integer r. Therefore, there must exist j 2 N such that K Consequently, jg. The set O j can be ordered according to the lexicographic order. That is, 8 Bin Han order if For a d d matrix A and which is uniquely determined by x It is easy to verify that S(AB; are all the eigen-values of A, then ; 2 O j are all the eigenvalues of S(A; since S(A; j) is similar to S(B; when A is similar to B. Moreover, by comparing the Taylor series of the same function e x T Ay and e y T A T x . The quantities m (j); j 2 N 0 can be computed as follows. Proposition 2.3. Let be a symmetry group. Then In particular, when I d 2 , then m (2j Proof. For 2 N d be the sequence given by q that (#)[(q are linearly independent, we have When I d 2 , we observe that Therefore, since Note that m (j) depends only on the symmetry group and is independent of the dilation matrix M . When is a subgroup of the full axes symmetry group A d , then m (j) can be easily determined since the matrix S(; j) is very simple for every d . For example, Computing the Smoothness Exponent of a Symmetric Multivariate Renable Function 9 For the convenience of the reader, we list the quantities m (j) in Algorithm 2.1 for some well known symmetry groups in Table 2. In Table 2, the symmetry groups 2 and 2 are dened to be Table The quantities m (j); j 2 N 0 in Algorithm 2.1 for some known symmetry groups. Note that N in this table. 26 28 A A A A For a sequence u on Z d , its symbol is given by 2Z d For denote the dierence operator given by and := 1 d for 2Z d To prove Algorithm 2.1, we need the following result. Theorem 2.4. Let a be a nitely supported mask on Z d and let b be the sequence dened in (1.3). Let be a symmetry group with respect to a dilation matrix M . Suppose that b is invariant under . Then (T b;M and where T b;M is the transition operator dened in (1.4) and W k is the minimal T b;M - invariant nite dimensional space which is generated by ( -), 2 N d Proof. Since is a symmetry group with respect to the dilation matrix M and b is invariant under , for 2Z d 2Z d 2Z d Therefore, (T b;M Note that b j. By the Parseval identity, we have kr S n Z r S n (2) d Z S n b;M -() d From the denition of the transition operator, it is easy to verify that For a sequence u such that b u() > 0 for all 2 R d , we observe that [11]). From the fact that \ S n follows that is the minimal T b;M -invariant subspace generated by we have lim do lim do which completes the proof. Proof of Algorithm 2.1: Let K Kg. Then it is easy to check that both '(K) and ('(K)) are invariant under T b;M (see [14, Lemma 2.3]). Since a satises the sum rules of order k, then the sequence b, which is dened in (1.3), satises the sum rules of order 2k and V 2k 1 is invariant under T b;M (see [20, Theorem 5.2]), where 2Z d Computing the Smoothness Exponent of a Symmetric Multivariate Renable Function 11 1g. Let W k denote the linear space in Theorem 2.4. Observe that W k U 2k 1 V 2k 1 . By Theorem 2.4 and (1.7), we have k (a; M; Since b satises the sum rules of order 2k and b is invariant under , we have Therefore, we have spec(T b;M j denotes the set of all the eigenvalues of T counting multiplicity and the linear space ('(K))=U 2k 1 is a quotient group under addition. Note that U the quotient group ('(K))=U 2k 1 is isomorphic to U 1 =U 0 By [25, Theorem 3.2] or by the proof of Theorem 3.1 in Section 3, we know that for any all the eigenvalues of T b;M we used the assumption that M is isotropic. Since U j 1 =U j is a subgroup of we deduce that all the eigenvalues of T b;M fact, by duality, we can prove that for any without assuming that M is isotropic, where [ (q)](x) := q(M 1 x); q 2 2k 1 .) By duality, Note that f(- is a basis of ('(K)) and the matrix T is the representation matrix of the linear operator T b;M acting on ('(K)) under the basis g. This completes the proof. From the above proof, without the assumption that M is isotropic, we observe that k (a; M; 2) is the largest number in the set (T )nfjj where (T ) is dened in Algorithm 2.1 and [ (q)](x) := q(M 1 Since is a symmetry group with respect to the dilation matrix M , it is easy to see that g. In passing, we mention that the calculation of the Sobolev smoothness for a bivariate mask which is invariant under A 2 with the dilation matrix 2I 2 was also discussed by Zhang in [36]. When a mask has a nonnegative symbol, then we can also compute k (a; M;1) in a similar way (see [14, Theorem 4.1]). For complete- ness, we present the following algorithm whose proof is almost identical to that of Algorithm 2.1. Algorithm 2.5. Let M be a d d isotropic dilation matrix and let be a symmetry group with respect to the dilation matrix M . Let a be a mask on Z d such that 1. Suppose that a is invariant under the symmetry group , the symbol of a is nonnegative (i.e., ba() > 0 for all 2 R d ), and a satises the sum rules of order k but not k + 1. The quantity 1 (a; M), or equivalently, k (a; M;1), is obtained via the following procedure: (a) Find a nite subset K of Z d such that ag \ Z d and 12 Bin Han (b) Obtain a (#K ) (#K ) matrix T as follows: (c) Let (T ) consist of the absolute values of all the eigenvalues of the square matrix counting multiplicity of its eigenvalues. Then 1 (a; M) is the smallest number in the following set with positive multiplicity Moreover, without the assumption that the symbol of the mask a is nonnegative, (a; M) is equal to or less than the quantity obtained in (c). Cohen and Daubechies in [4] discussed how to estimate the smoothness exponent of a renable function using the Fredholm determinant theory. Matlab routines for computing smoothness exponents using the method in [25] were developed and described in [28]. When a mask has a nonnegative symbol, matlab routines for estimating the Holder smoothness exponent was developed and described in [1] where symmetry is not taken into account and eigenvectors have to be explicitly computed and to be checked whether they belong to the subspace V k 1 or not. 3. Relations among k (a; M; p); k 2 N 0 . In this section, we shall study the relations among k (a; M; p); k 2 N 0 . Using such relations we shall be able to overcome the di-culty in D3 in Section 1 in order to check the stability condition for certain renable functions. The main results in this section are as follows. Theorem 3.1. Let M be a dilation matrix. Let a be a nitely supported mask on Z d such that a satises the sum rules of order k with respect to the lattice MZ d . Let min := min 16j6d are all the eigenvalues of M . Then min and for all j 2 N 0 and 1 6 p 6 q 6 1. Consequently, We say that a mask a is an interpolatory mask with respect to the lattice MZ d nf0g. Let a and b be two nitely supported masks on Z d . Dene a sequence c by If c is an interpolatory mask with respect to the lattice MZ d , then b is called a dual mask of a with respect to the lattice MZ d and vice versa. Let be a continuous function on R d . We say that is an interpolating function nf0g. For discussion on interpolating renable Computing the Smoothness Exponent of a Symmetric Multivariate Renable Function 13 functions and interpolatory masks, the reader is referred to [7, 8, 9, 15, 16, 30, 31] and references therein. For a compactly supported function on R d , we say that the shifts of are linearly independent if for every 2 C d , b Clearly, if the shifts of are linearly independent, then the shifts of are stable. When is a compactly supported interpolating function, then the shifts of are linearly independent since 2Z d be the renable function with a nitely supported mask and the dilation matrix 2I d . A method was proposed in Hogan and Jia [19] to check whether the shifts of are linearly independent or not. However, there are similar di-culties as mentioned in D1 and D2 in Section 1 when applying such a method in [19]. In fact, the procedure in [19] is not numerically stable and exact arithmetic is needed. Also see [29] on stability. An iteration scheme can be employed to solve the renement equation (1.1). Start with some initial function 0 2 L p (R d ) such that b nf0g. We employ the iteration scheme Q n is the linear operator on L p (R d by 2Z d (R d This iteration scheme is called a subdivision scheme or a cascade algorithm ([2, 18]). When the sequence Q n converges in the space L p (R d ), then the limit function must be M a and we say that the subdivision scheme associated with mask a and dilation matrix M converges in the L p norm. It was proved in [14] that the subdivision scheme associated with the mask a and dilation matrix M converges in the L p norm if and only if 1 (a; M; p) < j det M j 1=p (By Theorem 3.1, we see that this is equivalent to references therein on convergence of subdivision schemes. Let be a renable function with a nitely supported mask a and a dilation matrix M . It is known that is an interpolating renable function if and only if the mask a is an interpolatory mask with respect to the lattice MZ d and the subdivision scheme associated with mask a and dilation M converges in the L1 norm (equivalently, 1 (a; M;1) < 1, see [14]). However, in general, it is di-cult to directly check the condition 1 (a; M;1) < 1. On the other hand, in order to check that is an interpolating renable function with a nitely supported interpolatory mask a and a d d dilation matrix M , it was known in the literature (for example, see [1, 29, 31, 34]) that one needs to check the following two alternative conditions: 1) is a continuous function (Often, one computes the Sobolev smoothness exponent of to establish that 2 () > d=2 and consequently is a continuous function). - is the unique eigenvector of the transition operator T a;M j '( corresponding to a simple eigenvalue 1. In the following, we show that if a is an interpolatory mask and 2 (a; M) > d=2, then 2) is automatically satised. In other words, for an interpolatory mask a with respect to the lattice MZ d , we show that 2 (a; M) > d=2 implies 1 (a; M) > 0. Consequently, 1 (a; M;1) < 1 and the corresponding subdivision scheme converges in the L1 norm and its associated renable function is indeed an interpolating renable function. Corollary 3.2. Let a be a nitely supported mask on Z d and M be a dilation matrix. Suppose that b is a dual mask of a with respect to the lattice MZ d and 14 Bin Han Then the shifts of M a are linearly independent and consequently stable. If M is isotropic and (3.2) holds, then p (a; M) > 0 implies that M a 2 L p (R d ) and a In particular, if 2 (a; M) > d=2 (or more generally p (a; M) > d=p for and a is an interpolatory mask with respect to the lattice MZ d , then the subdivision scheme associated with mask a and dilation M converges in the L1 norm and consequently M a is a continuous interpolating renable function. Proof. Let j. Dene a sequence c by is an interpolatory mask with respect to the lattice MZ d . By [12, Theorem 5.2] and using Young's inequality, when 1=p Note that and for some proper integers j and k. Therefore, j+k (c; M;1) 6 p (a;M) q (b;M) It follows from Theorem 3.1 that 1 (c; M;1) < 1 and therefore, the subdivision scheme associated with mask c and dilation M converges in the L1 norm. Conse- quently, M c is an interpolating renable function and so its shifts are linearly in- dependent. Note that b a (). Therefore, the shifts of M a must be linearly independent and consequently stable. Note that - is a dual mask of an interpolatory mask and for any 1 6 q 6 1, since . The second part of Corollary 3.2 follows directly from the rst part. The second part can also be proved directly. Since p (a; M) > d=p, by Theorem 3.1, we have for some proper integer k. By Theorem 3.1, we have 1 (a; M;1) < 1. So the subdivision scheme associated with the mask a and the dilation matrix M converges in the L1 norm and therefore, we conclude that M a is a continuous interpolating renable function. Let k be a nonnegative integer. We mention that if j (a; M; p) < for some positive integer j, then one can prove that the mask a must satisfy the sum rules of order at least k respect to the lattice MZ d . In order to prove Theorem 3.1, we need to introduce the concept of ' p -norm joint spectral radius. Let A be a nite collection of linear operators on a nite-dimensional normed vector space V . We denote kAk the operator norm of A which is dened to be g. For a positive integer n, A n denotes the Cartesian power of A: Computing the Smoothness Exponent of a Symmetric Multivariate Renable Function 15 and for 1 6 p 6 1, we dene 1=p For any 1 6 p 6 1, the ' p -norm joint spectral radius (see [6, 15, 24] and references therein on ' p -norm joint spectral radius) of A is dened to be Let E be a complete set of representatives of the distinct cosets of the quotient group Z d =MZ d . To relate the quantities k (a; M; p) to the ' p -norm joint spectral radius, we introduce the linear operator E) on ' 0 (Z d ) as follows: 2Z d For Proof of Theorem 3.1: Let m := j det M j. Let K and 2Z d Since a satises the sum rules of order k, by [20, Theorem 5.2], By [14, Theorem 2.5], we have Note that g. For any ; 2 N d 0 such that jj 6 jj < k, we have 2Z d (M) 2Z d 2Z d (M) Note that (M) Since a satises the sum rules of order k, we have 2Z d (M) 2Z d 2Z d a(M) (M) Han Thus, for ; 2 N d 0 such that jj 6 jj < k, we have 2Z d a(M) (M) 2Z d [r -]() It is evident that 2Z d [r -]() r -; Therefore, 2Z d On the other hand, for any jj 6 jj, 2Z d (M) 2Z d 2Z d [r -]() is dened in (2.10). Therefore, we have 0 g is a basis for W j , we have Note that the spectral radius of S(M min for all j 2 N. Therefore, we deduce that holds. By the denition of the ' p -norm joint spectral radius, using the Holder inequality, we have (see [14]) for all 1 6 p 6 q 6 1. This completes the proof. Computing the Smoothness Exponent of a Symmetric Multivariate Renable Function 17 4. Some examples of symmetric renable functions. In this section, we shall give several examples to demonstrate the advantages of the algorithms and results in Sections 2 and 3 on computing smoothness exponents of symmetric renable functions. Example 4.1. Let . The interpolatory mask a for the butter y scheme in [9] is supported on [ 3; 3] 2 and is given Then a satises the sum rules of order 4 and a is invariant under the hexagonal symmetry group H . By Proposition 2.2, we have #K and by computing the eigenvalues of the 11 11 matrix T in Algorithm 2.1, we have log Let be the renable function with mask a and the dilation matrix 2I 2 . So by Algorithm 2.1, 2 (a; 2I 2 ) 2:44077 > 1. Therefore, by Corollary 3.2, is an interpolating renable function and 2 2:44077. Note that the matrix size using the method in [25] which is much larger than the matrix size used in Algorithm 2.1. Example 4.2. Let family of bivariate interpolatory masks RS r (r 2 N) was given in Riemenschneider and Shen [31] (also see Jia [22]) such that RS r is supported on [1 the sum rules of order 2r with respect to the lattice 2Z 2 and RS r is invariant under the hexagonal symmetry group H . Using the fact that the symbol of RS r has the factor [(1 by taking out some of such factors, Jia and Zhang [25, Theorem 4.1] was able to compute the Sobolev smoothness exponents of r for the renable function with the mask RS r and the dilation matrix 2I 2 . Note that the mask RS 16 is supported on [ In fact, in order to compute 2 ( 16 ), the method in [25, Theorem 4.1] has to compute the eigenvalues of two matrices of size (without factorization, the matrix size used in [25] is 11719). Without using any factorization, for any mask a which is supported on [ 31; 31] 2 and is invariant under H , by Algorithm 2.1, we have #K 992. So, to compute 2 ( 16 ), we only need to compute the eigenvalues of a matrix of size 992. Han Example 4.3. Let be the quincunx dilation matrix. The interpolatory mask a is supported on [ 3; 3] 2 and is given Note that a satises the sum rules of order 4 with respect to the quincunx lattice MZ 2 and a is invariant under the full axes symmetry group A 2 with respect to the dilation matrix M . This example was discussed in [25] and belongs to a family of quincunx interpolatory masks in [16]. Let be the renable function with the mask a and dilation matrix M . By Algorithm 2.1, we have #K A 2 2:44792 > 1. Therefore, 2 2:44792. Note that the matrix to compute 2 () using method in [25] has size 481 (see [25]) which is much larger than the size 46 when using Algorithm 2.1. Note that the symbol of a is nonnegative. By Algorithm 2.5, we have #K A 2 Therefore, by Corollary 3.2, 1 using method in [25], the matrix size is 129 (see [25]) which is much larger than the size 13 in Algorithm 2.5. Example 4.4. Let . A family of quincunx interpolatory masks r (r 2 N) was proposed in [16] such that g r is supported on [ the sum rules of order 2r with respect to MZ 2 , is an interpolatory mask with respect to MZ 2 and is invariant under the full axes symmetry group A . Note that the mask in Example 4.3 corresponds to the mask g 2 in this family. Since the symbols of g r are nonnegative, the L1 smoothness exponents 1 ( r ) were computed in [16] for r is the renable function with mask g r and the dilation matrix M . Using Algorithm 2.5, we are able to compute 1 in Table 3. Table The L1 (Holder) smoothness exponent of the interpolating renable function r whose mask is gr . 5.71514 6.21534 6.70431 7.18321 7.65242 8.11171 8.56039 8.99752 A coset by coset (CBC) algorithm was proposed in [12, 16] to construct quincunx biorthogonal wavelets. Some examples of dual masks of g r , denoted by (g r ) s k , were constructed in [16, Theorem 5.2] and some of their Sobolev smoothness exponents were given in Table 4 of [16]. Note that the dual mask (g r ) s k is supported on [ k satises the sum rules of order 2k, has nonnegative symbol and is invariant under the full axes symmetry group A 2 . However, in the paper [16] we are unable to complete the computation in Table 4 in [16] due to the di-culty mentioned in D2 in Section 1. In fact, to compute 2 (a; M) for a mask supported on [ k; k] 2 , the set b;M dened Computing the Smoothness Exponent of a Symmetric Multivariate Renable Function 19 in (1.5) is given by For example, in order to compute 2 ((g 4 set b;M consists of 16321 points which is beyond our ability to compute the eigenvalues of a 16321 16321 matrix. We now can complete the computation using Algorithm 2.1 in Section 2. Note that the quincunx dilation M here is denoted by Q in Table 4 of [16]. By computation, and the rest of the computation is given in Table 4. Table Computing by Algorithm 2.1. The result here completes Table 4 of [16]. 3.01166 2.92850 2.90251 2.91546 In passing, we mention that if a nitely supported mask a on Z 2 is invariant under the full axes symmetry group A proved in Han [13] that all the renable functions with the mask a and any of the quincunx dilation matrices are the same function which is also invariant under the full axes symmetry group A . Also see [3, 4] on quincunx wavelets. For any primal (matrix) mask and any dilation matrix, the CBC algorithm proposed in [12] can be used to construct dual (matrix) masks with any preassigned order of sum rules. Example 4.5. Let be the dilation matrix in a 3-subdivision scheme ([28]). The interpolatory mask a is supported on [ 4; 4] 2 and is given Note that a satises the sum rules of order 6 with respect to the lattice MZ 2 and a is invariant under the hexagonal symmetry group H with respect to the dilation matrix M . By Algorithm 2.1, we have #K Han be the renable function with the mask a and the dilation matrix M . Therefore, by Corollary 3.2, is a C 2 interpolating renable function and 2 3:28036. By estimate, the matrix b;M using the method in [25] is greater than 361 which is much larger than the size 38 when using Algorithm 2.1. Note that the symbol of a is nonnegative. By Algorithm 2.5, we have #K Therefore, by Corollary 3.2, 1 Using method in [25], the matrix a;M is greater than 85 which is much larger than the size 11 in Algorithm 2.5. Since 2 C 2 , this example gives us a C 2 pinterpolatory subdivision scheme. In the rest of this section, let us present some examples in dimension three. By generalizing the proof of [15, Theorem 4.3], we have the following result. Theorem 4.6. Let d be the dilation matrix. For each positive integer r, there exists a unique dyadic interpolatory mask g d r in R d with the following properties: (a) g d r is supported on the set f2 (b) g d r is symmetric about all the coordinate axes; (c) g d r satises the sum rules of order 2r with respect to the lattice 2Z d . By the uniqueness, we see that each g d r in Theorem 4.6 is invariant under the full axes symmetry group A d . By the uniqueness of g d r in Theorem 4.6 again, we see that were the masks given in [8] and g 2 were the masks proposed in [15]. Moreover, the masks g d r can be obtained via a recursive formula without solving any equations. In the following, let us give some examples of the above interpolatory masks in dimension three. Let A 3 Clearly, if a is a mask invariant under the group A 3 , then it is totally determined by all the coe-cients a(); 2 Z 3 A 3 Example 4.7. The coe-cients of the interpolatory mask g 3 2 on the set Z 3 A 3 are given by other 2 Z 3 A 3 Then 2 satises the sum rules of order 4 and there are only 81 nonzero coe-cients in the mask g 3 . Let be the renable function with the mask g 3 2 and the dilation matrix 2I 3 . By Algorithm 2.1, we have #K A 3 Therefore, by Corollary 3.2, is an interpolating renable function and 2 () 2:44077. Note Algorithm 2.1 can greatly reduce the size of the matrix to compute 2 (g 3 Example 4.8. The coe-cients of the interpolatory mask g 3 3 on the set Z 3 A 3 are Computing the Smoothness Exponent of a Symmetric Multivariate Renable Function 21 given by A 3 Then 3 satises the sum rules of order 6 and it has 171 nonzero coe-cients. Let be the renable function with the mask g 3 3 and the dilation matrix 2I 3 . By Algorithm 2.1, we have #K A 3 1:5. Therefore, by Corollary 3.2, is an interpolating renable function and 2 () 3:17513. Note and #K A 3 101. Hence, Algorithm 2.1 can greatly reduce the size of the matrix to compute 2 (g 3 Let r be the renable function with the mask g 3 r (r 2 N) and the dilation matrix 2I 3 . The Sobolev smoothness exponents of r are presented in Table 5. By [15, Theorem 3.3] and [12, Theorem 5.1], we see that g 3 the optimal Sobolev smoothness and optimal order of sum rules with respect to the support of their masks. In general, the Algorithms 2.1 and 2.5 roughly reduce the size of the matrix to be 1=(#) of the number of points in b;M in (1.5). Note that # A Algorithms 2.1 and 2.5 are very useful in computing the smoothness exponents of symmetric multivariate renable functions. Table The Sobolev smoothness exponent of the renable function r whose mask is 2:44077 3:17513 3:79313 4:34408 4:86202 5:36283 5:85293 6:33522 6:81143 7:28260 Acknowledgments . The author is indebted to Rong-Qing Jia for discussion on computing smoothness of multivariate renable functions. The author thanks IMA at University of Minnesota for their hospitality during his visit at IMA in 2001. The author also thanks the referees for their helpful comments to improve the presentation of this paper and for suggesting the references [1, 4, 7, 18]. --R The IGPM Villemoes Machine Nonseparable bidimensional wavelet bases A new technique to estimate the regularity of re Symmetric iterative interpolation processes A butter y subdivision scheme for surface interpolation with tension control Sobolev characterization of solutions of dilation equations Spectral radius formulas for subdivision operators Analysis and construction of optimal multivariate biorthogonal wavelets with compact support Symmetry property and construction of wavelets with a general dilation matrix Multivariate re Optimal interpolatory subdivision schemes in multidimensional spaces Quincunx fundamental re Multivariate re Dependence relations among the shifts of a multivariate re Approximation properties of multivariate wavelets Characterization of smoothness of multivariate re Interpolatory subdivision schemes induced by box splines Spectral analysis of the transition operators and its applications to smoothness analysis of wavelets Smoothness of multiple re Spectral properties of the transition operator associated to a multivariate re On the regularity of matrix re Multivariate matrix re On the analysis of p 3-subdivision schemes Stability and orthonormality of multivariate re Multidimensional interpolatory subdivision schemes Simple regularity criteria for subdivision schemes The Sobolev regularity of re Computing the Sobolev regularity of re Wavelet analysis of re Properties of re --TR --CTR Peter Oswald, Designing composite triangular subdivision schemes, Computer Aided Geometric Design, v.22 n.7, p.659-679, October 2005 Avi Zulti , Adi Levin , David Levin , Mina Teicher, C2 subdivision over triangulations with one extraordinary point, Computer Aided Geometric Design, v.23 n.2, p.157-178, February 2006 Bin Han, Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix, Journal of Computational and Applied Mathematics, v.155 n.1, p.43-67, 01 June Bin Dong , Zuowei Shen, Construction of biorthogonal wavelets from pseudo-splines, Journal of Approximation Theory, v.138 n.2, p.211-231, February 2006 Bin Han, Vector cascade algorithms and refinable function vectors in Sobolev spaces, Journal of Approximation Theory, v.124 n.1, p.44-88, September

multivariate refinable functions;interpolating functions;eigenvalues of matrices;quincunx dilation matrix;smoothness exponent;regularity;symmetry

606717

Sparse approximate inverse smoothers for geometric and algebraic multigrid.

Sparse approximate inverses are considered as smoothers for geometric and algebraic multigrid methods. They are based on the SPAI-Algorithm [MJ. Grote, T. Huckle, SIAM J. Sci. Comput. which constructs a sparse approximate inverse M of a matrix A, by minimizing I - MA in the Frobenius norm. This leads to a new hierarchy of inherently parallel smoothers: SPAI-0, SPAI-1, and SPAI(). For geometric multigrid, the performance of SPAI-1 is usually comparable to that of Gauss-Seidel smoothing. In more difficult situations, where neither Gauss-Seidel nor the simpler SPAI-0 or SPAI-1 smoothers are adequate, further reduction of automatically improves the SPAI() smoother where needed. When combined with an algebraic coarsening strategy [J.W. Ruge, K. Stben, in: S.F. McCormick (Ed.), Multigrid Methods, SIAM, 1987, pp. 73-130] the resulting method yields a robust, parallel, and algebraic multigrid iteration, easily adjusted even by the non-expert. Numerical examples demonstrate the usefulness of SPAI smoothers, both in a sequential and a parallel environment.Essential advantages of the SPAI-smoothers are: improved robustness, inherent parallelism, ordering independence, and possible local adaptivity.

Introduction Multigrid methods rely on the subtle interplay of smoothing and coarse grid correction. Only their careful combination yields an e-cient multigrid solver for large linear systems, resulting from the discretization of partial dierential equations [7,14,15,27]. Standard smoothers for multigrid usually consist of a few steps of a basic iterative method. Here we shall consider smoothers that are based on sparse approximate inverses. Hence, starting from the linear system we let M denote a sparse approximation of A 1 . The corresponding basic iterative method is Since the approximate inverse M is known explicitly, each iteration step requires only one additional M v matrix-vector multiply; therefore, it is easy to parallelize and cheap to evaluate, because M is sparse. Recently, various algorithms have been proposed, all of which attempt to compute directly a sparse approximate inverse of A [5,9,17]. For a comparative study of various approximate inverse preconditioners we refer to Benzi and Tuma [6]. Approximate inverse techniques are also gaining in importance as smoothers for multigrid methods. First introduced by Benson and Frederickson [3,4], they were shown to be eective on various di-cult elliptic problems on unstructured grids by Tang and Wan [25]. Advantages of sparse approximate inverse smoothers over classical smoothers, such as damped Jacobi, Gauss-Seidel, or ILU, are inherent parallelism, possible local adaptivity, and improved robustness. Here we shall consider sparse approximate inverse (SPAI) smoothers based on the SPAI-Algorithm by Grote and Huckle [13]. The SPAI-Algorithm computes an approximate inverse M explicitly by minimizing I MA in the Frobenius norm. Both the computation of M and its application as a smoother are inherently parallel. Since an eective sparsity pattern of M is in general unknown a priori, the SPAI-Algorithm attempts to determine the most promising entries dynamically. This strategy has proved eective in generating preconditioners for many di-cult and ill-conditioned problems [1,13,24]. Moreover, it provides the means for adjusting the smoother locally and automatically, if necessary. Nevertheless, by choosing an a priori sparsity pattern for M , the computational cost can be greatly reduced. Possible choices include powers of A or A > A, as suggested by Huckle [16] or Chow [10]. Hence we shall investigate the following hierarchy of sparse approximate inverse smoothers: SPAI-0, SPAI-1, and SPAI("). For SPAI-0 and SPAI-1 the sparsity pattern of M is xed: M is diagonal for SPAI-0, whereas for SPAI-1 the sparsity pattern of M is that of A. For SPAI(") the sparsity pattern of M is determined automatically by the SPAI-Algorithm ([13]); the parameter " controls the accuracy and the amount of ll-in of M . As structured geometric grids are di-cult to use with complex geometries, application code designers often turn to very large unstructured grids. Yet the lack of a natural grid hierarchy prevents the use of standard geometric multi- grid. In this context, algebraic multigrid (AMG) is often seen as the most promising method for solving large-scale problems. The original AMG algo- rithm, rst introduced in the 1980's by Ruge and Stuben [22], uses the (simple) Gauss-Seidel iteration as a smoother, but determines the coarse "grid" space in a sophisticated way to improve robustness of the method. But if the iteration fails to converge, there is no automatic way to improve on the smoother. As an alternative, we investigate the usefulness of smoothers based on SParse Approximate Inverses (SPAI). Not only inherently parallel, their performance can also easily be adjusted, even by a non-expert. Thus we aim for a more general and inherently parallel algebraic multigrid method. In Section 2 we brie y review the SPAI-Algorithm and show how sparse approximate inverses are used as smoothers within a multigrid iteration. A heuristic Green's function interpretation underpins their eectiveness as smoo- thers. Rigorous results on the smoothing property of approximate inverses were proved in [8]; they are summarized in Section 2.4. Next, we present in Section 3 a detailed description of the algebraic coarsening strategy used ([22]), together with key components of the algorithm for e-cient implementation. Finally, in Section 4, we compare the performance of SPAI smoothing to that of Gauss-Seidel smoothing on various test problems, either within a geometric or an algebraic multigrid setting. smoothing 2.1 Classical smoothers Consider a sequence of nested grids, T On the nest mesh, we wish to solve the n n linear system by a multigrid method { for further details on multigrid see Hackbusch [14] and ([15], Section 10) or Wesseling [27]. A multigrid iteration results from the recursive application of a two-grid method. A two-grid method consists of 1 pre-smoothing steps on level ', a coarse grid correction on level ' 1, and 2 post-smoothing steps again on level '. This leads to the iteration for the error e (m) and r are prolongation and restriction operators, respectively, between T ' and T ' 1 , while S ' denotes the smoother. If nested nite element spaces with Galerkin discretization are used, the Galerkin product representation holds: Otherwise, one can still use (4) to dene the coarse-grid problem for given r and p. We shall always use x (0) our initial guess. The multigrid (V-cycle) iteration proceeds until the relative residual drops below a prescribed tolerance, kb A ' x (m) kbk < tol: (5) Then we calculate the average rate of convergence 1=m The expected multigrid convergence behavior is achieved if the number of multigrid iterations, m, necessary to achieve a xed tolerance, is essentially independent of the number of grid levels '. Typically, the smoother has the form where W ' approximates A ' and is cheap to invert | of course, W 1 ' is never computed explicitly. Let with D the diagonal, L the lower triangular part, and U the upper triangular part of A. Then, damped Jacobi smoothing corresponds to whereas Gauss-Seidel smoothing corresponds to In (8) the parameter ! is chosen to maximize the reduction of the high frequency components of the error. The optimal value, ! , is problem dependent and usually unknown a priori. Although Gauss-Seidel typically leads to faster convergence, it is more di-cult to implement in parallel, because each smoothing step in (9) requires the solution of a lower triangular system. If neither Jacobi nor Gauss-Seidel smoothing lead to satisfactory convergence, one can either resort to more sophisticated (matrix dependent) prolongation and restriction operators ([29]) or to more robust smoothers, based on incomplete LU (ILU) factorizations of A ' ([28]). Unfortunately, ILU-smoothing is inherently sequential and therefore di-cult to implement in parallel. It is also di-cult to improve locally, say near the boundary or a singularity, without aecting the ll-in everywhere in the LU factors. 2.2 SPAI-smoothers As an alternative to inverting W ' in (7), we propose to compute explicitly a sparse approximate inverse M of A, and to use it for smoothing { we drop the grid index ' to simplify the notation. This yields the SPAI-smoother, where M is computed by minimizing kI MAk in the Frobenius norm for a given sparsity pattern. In contrast to W 1 ' in (7), the matrix M in (10) is computed explicitly. Therefore, the application of S SPAI requires only matrix-vector multiplications, which are easy to parallelize; it does not require the solution of any upper or lower triangular systems. Moreover, the Frobenius norm naturally leads to inherent parallelism because the rows m > of M can be computed independently of one another. Indeed, since the solution of (11) separates into the n independent least-squares problems for the m > Here e k denotes the k-th unit vector. Because A and M are sparse, these least-squares problems are small. Since an eective sparsity pattern of M is unknown a priori, the original SPAI- Algorithm in [13] begins with a diagonal pattern. It then augments progressively the sparsity pattern of M to further reduce each residual r Each additional reduction of the 2-norm of r k involves two steps. First, the algorithm identies a set of potential new candidates, based on the sparsity of A and the current (sparse) residual r k . Second, the algorithm selects the most protable entries, usually less than ve entries, by computing for each candidate a cheap upper bound on the reduction in kr k k 2 . Once the new entries have been selected and added to m k , the (small) least-squares problem (12) is solved again with the augmented set of indices. The algorithm proceeds until each row m > of M satises where " is a tolerance set by the user; it controls the ll-in and the quality of the preconditioner M . A larger value of " leads to a sparser and less expensive approximate inverse, but also to a less eective smoother with a higher number of multigrid cycles. A lower value of " usually reduces the number of cycles, but the cost of computing may become prohibitive; moreover, a denser M results in a higher cost per smoothing step. The optimal value of " minimizes the total time; it depends on the problem, the discretization, the desired accuracy, and the computer architecture. Further details about the original SPAI-Algorithm can be found in [13]. In addition to SPAI("), we shall also consider the following two greatly sim- plied SPAI-smoothers with xed sparsity patterns: SPAI-0, where M is di- agonal, and SPAI-1, where the sparsity pattern of M is that of A. Both solve the least-squares problem (12), and thus minimize kI MAk in the Frobenius norm for the sparsity pattern chosen a priori. This eliminates the search for an eective sparsity pattern of M , and thus greatly reduces the cost of computing the approximate inverse. The SPAI-1 smoother coincides with the smoother of Tang and Wan [25]. For diagonal and can be calculated directly. It is simply given by a kk where a > k is the k-th row of A { note that M is always well-dened if A is nonsingular. In contrast to damped Jacobi, SPAI-0 is parameter-free. To summarize, we shall consider the following hierarchy of SPAI-smoothers, which all minimize kI MAk in the Frobenius norm for a certain sparsity pattern of M . diagonal and given by (14). SPAI-1: The sparsity pattern of M is that of A. SPAI("): The sparsity pattern of M is determined automatically via the SPAI- Algorithm [13]. Each row m > of M satises (13) for a given ". Any of these approximate inverses leads to the smoothing step We have found that in many situations, SPAI-0 and SPAI-1 yield ample smoothing. However, the added exibility in providing an automatic criterion for improving the smoother via the SPAI-Algorithm remains very useful. Indeed, both SPAI-0 and SPAI-1 can be used as initial guess for SPAI("), and thus be locally improved upon where needed by reducing ". For matrices with inherent (small) block structure, typical from the discretization of systems of partial dierential equations, the Block-SPAI-Algorithm ([2]) greatly reduces the cost of computing M . 2.3 Green's function interpretation Why do approximate inverses yield eective smoothers for problems which come from partial dierential equations? As the mesh parameter h tends to zero, the solution of the linear system, A h u tends to the solution of the underlying dierential equation, with appropriate boundary conditions. Here the matrix A h corresponds to a discrete version of the dierential operator L. Let y h k denote the k-th row of solves the linear system with e h k the k-th unit vector. As h ! 0, y h k tends to the Green's function Here L denotes the adjoint dierential operator and -(x x k ) the \delta- centered about x k . To exhibit the correspondence between y h k and we recall that L is formally dened by the identity for all u; v in appropriate function spaces. Equation (20) is the continuous counterpart to the relation From (17), (19) and (20) we conclude that Similarly, the combination of (16), (18), and (21) leads to the discrete counterpart of (22), Comparison of (22) and (23) shows that y h k corresponds to to G(x; x k ) as The k-th row, (m h of the approximate inverse, M h , solves (12), or equiva- lently min Hence m h k approximates the k-th column of A > , that is y h k in (18), in the (discrete) 2-norm for a xed sparsity pattern of m h k . The nonzero entries of usually lie in a neighborhood of x k : they correspond to mesh points x j close to x k . Therefore, after an appropriate scaling in inverse powers of h, we see that m h approximates G(x; x k ) locally in the (continuous) L 1268 3 For (partial) dierential operators, G(x; x k ) typically is singular at x k and decays smoothly, but not necessarily rapidly, with increasing distance jx x k j. Clearly the slower the decay, the denser M h must be to approximate well A 1 deciency of sparse approximate inverse preconditioners was also pointed out by Tang [24]. At the same time, however, it suggests that sparse approximate inverses, obtained by the minimization of kI MAk in the Frobenius norm, naturally yield smoothers for multigrid. Indeed to be eective, a preconditioner must approximate uniformly over the entire spectrum of L. In contrast, an eective smoother only needs to capture the high-frequency behavior of . Yet this high-frequency behavior corresponds to the singular, local behavior of G(x; x k ), precisely that which is approximated by m h k . To illustrate this fact, we consider the standard ve-point stencil of the discrete Laplacian on a 15 15 grid. In Figure 1 on the following page we compare A 1 with the Gauss-Seidel approximate inverse, (L and two explicit approximate inverses, SPAI-1 and SPAI(0:2). We recall that Gauss-Seidel, a poor preconditioner for this problem, remains an excellent smoother, because it captures the high-frequency behavior of A 1 . Similarly, SPAI-1 and SPAI(") yield local operators with, as we shall see, good smoothing property. Despite the resemblance between the Gauss-Seidel and the SPAI approximate inverses, we note the one-sidedness of the former, in contrast to the symmetry of the latter. Gauss-Seidel, SPAI (0.2) Fig. 1. Row 112 of the following operators: A 1 (top left), the Gauss-Seidel inverse computed with SPAI-1 (bottom left), and M computed with SPAI(0:2) (bottom right). 2.4 Theoretical Properties In contrast to the heuristic interpretation of the previous section, we shall now summarize some rigorous results ([8]) on the smoothing property of the simplest smoother: SPAI-0. Multigrid convergence theory rests on two fundamental conditions: the smoothing property ([15], Denition 10.6.3): any function with lim and the approximation property ([15], Section 10.6.3). In general, the smoothing and approximation properties together imply convergence of the two-grid method and of the multigrid W-cycle, with a contraction number independent of the level number '. Moreover, for symmetric positive denite prob- lems, both conditions also imply multigrid V-cycle convergence independent of ' { see Hackbusch ([15], Sect. 10.6) for details. The approximation property is independent of the smoother, S ' ; it depends only on the discretization the prolongation operator p, and the restriction operator r. In [15] the approximation property is shown to hold for a large class of discrete elliptic boundary value problems. For symmetric positive denite problems the smoothing property usually holds for classical smoothers, such as damped Jacobi, (symmetric) Gauss-Seidel, and incomplete Cholesky. In [8] the smoothing property (25) was shown to hold for the SPAI-0 smoother under reasonable assumptions on the matrix A. More precisely, for A symmetric and positive denite, the SPAI-0 smoother satises the smoothing property, either if A is weakly diagonally dominant, or if A has at most seven nonzero o-diagonal entries per row. Furthermore, the two diagonal smoothers SPAI-0 and damped Jacobi, with optimal relaxation parameter ! , lead to identical smoothers for the discrete Laplacian with periodic boundary conditions in any space dimension [8]. In this special situation, the parameter-free SPAI-0 smoother automatically yields a scaling of diag(A), which minimizes the smoothing factor; in that sense it is optimal. In more general situations, however, both smoothers dier because of boundary conditions, even with constant coe-cients on an equispaced mesh. Comparison of these two diagonal smoothers via numerical experiments showed that SPAI-0 is an attractive alternative to damped Jacobi [8]. Indeed, SPAI-0 is parameter-free and typically leads to slightly better convergence rates than damped Jacobi. 3 Algebraic Multigrid Multigrid (MG) methods are sensitive to the subtle interplay between smoothing and coarse-grid correction. When a standard geometric multigrid method is applied to di-cult problems, say with strong anisotropy, this interplay is disturbed because the error is no longer smoothed equally well in all direc- tions. Although manual intervention and selection of coarse grids can sometimes overcome this di-culty, it remains cumbersome to apply in practice to unstructured grids and complex geometry. In contrast, an algebraic multigrid approach compensates for the decient smoothing by a sophisticated choice of the coarser grids and the interpolation operators, which is only based on the matrix A ' . Many AMG variants exist, which dier in the coarsening strategy or the interpolation used { an introduction to various AMG methods can be found in ([26]). Following Ruge and Stuben [22], we now describe the algebraic coarsening strategy and interpolation operators, which we shall combine with the SPAI smoothers from Section 2.2 and use for our numerical experiments. 3.1 Coarsening strategy1020301020300.20.6 x y Anisotropic stencil:h 26 6 6 6 6 4 Fig. 2. The error after ve Gauss-Seidel smoothing steps for the problem described in Section 4.2 on a with 0:01. The smooth error component is aligned with the anisotropy, which can be read from the stencils. The fundamental principle underlying the coarsening strategy is based on the observation that interpolation should only be performed along smooth error components. For symmetric M-matrices, the error is smoothed well along large (negative) o-diagonal entries in the matrix A ([23]). Therefore, at each grid point p, we may identify among neighboring points q good candidates for interpolation, by comparing the magnitude of the corresponding entries a pq . This leads to the following relations between the point p and its neighbors q in the connectivity graph of the matrix A: Condition Notation Interpretation a pq max apr <0 ja pr j p ( q p (strongly) depends on q and a pq 6= 0 or q (strongly) in uences p a pq < weakly depends on q and a pq 6= 0 or q weakly in uences p The parameter controls the threshold, which discriminates between strong and weak connections; typically 0:25. With this denition, all positive o-diagonal entries are necessarily weak. The relations p ( q and p q are symmetric only if A is symmetric. Next, we dene the set of dependencies of a point p as the set of in uences of a point p as I and the set of weak dependencies of a point p as On every level, the coarsening strategy must divide P , the set of all points on that level, into two disjoint sets: C, the \coarse points", also present on the coarser level, and F , the \ne points", which are absent on the coarser level. The choice of C and F induces the C/F{splitting of Coarse grid correction heavily depends on accurate interpolation. Accurate interpolation is guaranteed if every F point is surrounded by su-ciently many strongly dependent C points. A typical conguration is shown in Figure 3. q3 strong dependency dependency C point F point Strong dependencies are indicated with solid arrows, while weak dependencies are represented by dashed arrows. C points are represented by solid circles, whereas F points are represented by dashed circles. Hence all the strong dependencies of point are C points; therefore q 2 and q 4 are good candidates for interpolating p. Fig. 3. Ideal coarsening conguration for interpolation The coarsening algorithm attempts to determine a C/F{splitting, which maximizes the F {to{C dependency for all F points (Coarsening Goal 1), with a minimal set C (Coarsening Goal 2). It is important to strike a good balance between these two con icting goals, as the overall computational eort depends not only on the convergence rate, but also on the amount of work per multigrid cycle. Clearly the optimal C=F -splitting minimizes total execution time. However, since the convergence rate is generally unpredictable, the coarsening algorithm merely attempts to meet Coarsening Goals 1 and 2 in a heuristic fashion. In doing so, its complexity must not exceed O(n log n) to retain the overall complexity of the multigrid iteration. 3.2 Coarse grid selection: a greedy heuristic To split P into C and F , every step of a greedy heuristic moves the most promising candidate from P into C, while forcing neighboring points into F . This procedure is repeated until all points are distributed. If every step requires at most O(log n) operations, and the complexity of all other computations does not exceed O(n log n), the desired overall complexity of O(n log n) is reached. The greedy heuristic described in [23] is based on the following two principles, which correspond to Coarsening Goals 1 and 2: (1) The most promising candidate, p, for becoming a C point, is that with the highest number of in uences jI p j. Then all in uences of p are added to F . This choice supports Coarsening Goal 1 because all F points will eventually have at least one strong C dependency. (2) To keep the number of C points low (Coarsening Goal 2), the algorithm should prefer C points near recently chosen F points; hence, these in u- ences are given a higher priority. Starting with all points as \undecided points", that is the algorithm proceeds by selecting from U the most promising C point with highest priority. The priority of any point p is dened by Equation re ects the preference in choosing the next C point for a point which in uences many previously selected F points. The key advantage of (26) is the possibility to update the priority locally and in O(1) time, which results in the desired overall complexity of O(n log n). We now summarize the Coarse Grid Selection algorithm: Algorithm 1 Coarse Grid Selection procedure for all set Priority(p) := jI p j end for all U := while U (1) select p 2 U with maximal P riority(p) for all q 2 D p (all dependencies of p) end for all for all q 2 I p (all in uences of p) for all r 2 D q (all dependencies of q) end for all end for all while procedure To implement steps (1), (2), and (3) e-ciently in O(1) time, we maintain a list Q of all points sorted by priority, together with the list I of point indices of Q. Moreover, a list of boundaries B of all priorities occurring in Q enables the immediate update of the sorted list Q. Figure 4 shows a possible segment of the lists Q, B, and I. 5 611 15index of Q position in Q priority(p) position in Q3141312 I index of B Fig. 4. The three lists Q, I, and B enable the e-cient implementation of the coarsening algorithm. During the set{up phase of the coarsening algorithm, B is computed and sorted by priority. Step (1) simply chooses the last element of Q. Steps (2) and (3) are implemented by exchanging a point, whose priority must be either incremented or decremented, with its left{ or rightmost neighbor in Q with that same priority; then its priority is adjusted, while Q remains sorted. Both B and I are updated accordingly. Following a suggestion of K. Stuben, we shall skip the second pass of the original coarsening algorithm in [22], which enforces even stronger F {to{C dependency, because of the high computational cost involved. 3.3 Interpolation The grid function u h , dened on the ner grid, is interpolated from the grid function uH , dened on the coarser grid C, as follows: Hence, values at C points are simply transfered from the coarser level, whereas values at F points are interpolated from C neighbors. The four dierent dependencies possible between any F point and its neighbors are shown in Figure 5. For \standard interpolation" (see [23]), the choice of the weights, w pq , for interpolating p, is based on the equation a pp e a pq e Indeed, if Ae ' 0, the smoothing eect is minimal, and the error e is declared \algebraically smooth" { see [23] for details. Clearly, we cannot interpolate p from surrounding F points, whereas weakly dependent C points are not included either, because of the rough nature of the error in that direction. Thus connections (q 1 and q 2 in Figure 5) are always ignored in the interpolation and the corresponding interpolation weights set to zero (w p;q 1 cancellation of the weak dependencies, the neglected entries of the weakly dependent neighbors are added to the diagonal. Hence equation (28) becomes ~ a pp e a pq e a Strong C dependencies, such as q 3 in Figure 5, cause no di-culty because the value of uH is available at that coarse grid location. Division of (29) by ~ a pp yields the weight a pq ~ a pp However, strong F dependencies, such as q 4 in Figure 5, are not available for interpolation and must rst be interpolated from C points, on which they strongly depend. To do so, we replace a qq by ~ a qq for every q 2 D q \ F , with ~ a For every point r 2 D q this yields the weight a pq ~ a pp a qr ~ a qq If a a point q is both a direct and an indirect neighbor of p, so that both (30) and (32) apply, the two weights are calculated separately and then added to each other. q4 strong dependency dependency C point F point q3 indirect interpolation direct interpolation ignored ignored Fig. 5. The four dierent dependencies possible between p and its neighbors. The algorithm described above determines the coarse grid levels only on the basis of A, and not on that of the approximate inverse M . In fact, the information contained in M can be used to determine coarse grid levels and interpolation weights, as suggested by Meurant [19,20]. 3.4 Measuring computational costs and memory requirements When comparing the performance of various smoothers, we cannot limit ourselves to comparing the number of multigrid iterations, but also need to estimate the additional amount of work due to the smoother. To do so, we calculate the total density ratio, M , of nonzero entries in M to those in A on all grid levels, 1 i ', where smoothing is applied: The additional amount of work due to the smoother is proportional to M . While rapidly reducing the number of points from one level to the next, the matrices A i must also remain reasonably sparse, as measured by For instance, as Galerkin coarse grid approximation enlarges the standard ve{ point stencil on the nest grid to nine{point stencils on subsequent levels, the resulting value of A for geometric multigrid is about 1.6. If semi{coarsening together with one-dimensional interpolation is used, A increases up to two. All the results presented in the following section were computed with a MATLAB implementation. We shall evaluate the e-ciency of the various approaches by comparing their respective values for M and A . 4 Numerical results To illustrate the usefulness and versatility of SPAI smoothing, we shall now consider various standard test problems. In all cases, the dierential equation considered is discretized on the nest level with standard nite dierences on an equispaced mesh. For geometric multigrid, we use a regularly rened sequence of equispaced grids, with a single unknown remaining at the center of the domain. For algebraic multigrid, the coarser levels are obtained by the Coarse Grid Selection Algorithm described in Section 3.2. With the denition of strong dependency from section 3.1, the algorithm proceeds until the number of grid points drops below twenty. The coarse grid operators are obtained via the Galerkin product formula (4), with . For geometric multigrid, p correspond to standard linear interpolation, whereas for AMG p is obtained as described in Section 3.3. We use a multigrid V-cycle iteration, with two pre- and two post-smoothing steps 2). The multigrid iteration proceeds until the relative residual satises the prescribed tolerance in (5), with 4.1 Rotating ow problem We rst consider the convection{diusion problem, in (0; 1)(0; 1), with u(x; on the boundary. Here u represents any scalar quantity advected by the rotating ow eld. For convection dominated << h, the linear systems cease to be symmetric and positive denite, so that these problems lie outside of classical multigrid theory. We use centered second-order nite dierences for the diusion, but discretize the convection with rst-order upwinding to ensure numerical stability. Table Geometric MG convergence rates for the rotating ow problem on a 128128 grid, for dierent values of . The symbol y indicates that the multigrid iteration diverges. Smoother Gauss-Seidel SPAI-0 SPAI-1 SPAI(0.3) SPAI(0.2) Table convergence rates q obtained with standard MG. All smoothers yield acceptable convergence rates in the diusion dominated case, with . For however, the multigrid iteration diverges with Gauss{Seidel or SPAI-0 smoothing. In contrast, the SPAI-1 smoother still yields a convergent method. The use of SPAI(0.3) smoothing accelerates convergence even further, while M increases up to 1.4 only. As we reduce the diusion even further down to only the SPAI(0.2) smoother yields a convergent iteration. Although the resulting value of M is quite high, the construction of SPAI(0.2) remains parallel and fully auto- matic. We remark that symmetric Gauss-Seidel smoothing ([27]) leads to a convergent multigrid iteration, yet this approach does not generalize easily to unstructured grids. Parallel results Since the SPAI-1 smoother is inherently parallel, it is straightforward to apply within a parallel version of geometric MG. The data is distributed among processors via domain decomposition, which is well{known to work e-ciently for a number of multigrid applications ([18]). The platform we shall use is the ETH{Beowulf cluster, which consists of 192 dual CPU Pentium III (500 processors. All nodes are connected via a 100 MBit/s and 1 GB/s switched network, while communication is done with MPI. We now apply our parallel multigrid implementation to the rotating Flow Problem (35) with On 128 nodes, the total execution{time is 156 seconds on the 40964096 grid. The time includes the set{up for the construction of the SPAI-1 smoother, which requires the solution of about sixteen million small (259) and independent least{squares problems. As shown in table 1, the use of a coarsest level, which consists only of a single mesh point, leads to a divergent multigrid iteration for increasing the resolution of the coarsest level up to 3232 mesh points, one obtains a convergent multigrid iteration. Table Scalability of parallel MG using SPAI-1. The problem size and the number of processors is increased by a factor of 4, while total time increases by 30% only. Gridsize 512512 4 10231023 Number of processors Total time (sec) 20 26 To obtain good speed{up with a parallel MG code, it is important to perform coarse grid agglomeration (see [21]) because of the loss of e-ciency on coarser grid levels. Although we have not implemented such an agglomeration strategy, our computations scale reasonably well as long as the problem size matches the size of the parallel architecture { see Table 2. Rotating ow problem: algebraic coarsening is clearly aligned with the ow direction. Larger dots correspond to C points on coarser levels. (b) Locally anisotropic diusion: semi-coarsening is apparent in the center of the domain. Fig. 6. Examples of algebraic coarsening for the two model problems considered. AMG results None of these approaches, however, is entirely satisfactory for vanishing vis- cosity. To overcome the lack of robustness for small , we now apply the algebraic coarsening strategy described in Section 3. Figure 6(a) displays the coarse levels selected by the algorithm. In Table 3 both SPAI-0 and SPAI-1 yield convergence without any particular tuning. 0:5, the SPAI(") Table AMG convergence results for the rotating ow problem for varying on a 128128 grid. A 2:8 3:4 4:2 Smoother q M q M q M Gauss{Seidel 0.14 | 0.38 | 0.81 | Table AMG convergence rates for the rotating ow problem with Smoother Gauss{Seidel SPAI-0 SPAI-1 SPAI(0.5) Gridsize A q M q M q M q M 128128 4.2 0.81 | 0.36 (0.1) 0.21 (1.0) 0.27 (0.4) 4.3 0.96 | 0.38 (0.1) 0.24 (1.0) 0.34 (0.4) smoother yields a compromise between the SPAI-0 and SPAI-1 smoothers: both the storage requirement and the convergence rate lie between those obtained with the xed sparsity patterns of SPAI-0 and SPAI-1. Lower values of " reduce the convergence rate even further. The poor convergence rates obtained with Gauss-Seidel could probably be improved, either by smoothing C points before F points ([23]) or by using symmetric Gauss{Seidel. The results in Table 4 demonstrate the robustness of SPAI smoothing. In- deed, as ! 0 all convergence rates obtained with the combined SPAI-AMG approach remain bounded. 4.2 Locally anisotropic diusion In this section we consider the locally anisotropic problem, with u(x; on the boundary. The diusion coe-cient (x; except inside the square [1=4; 3=4] [1=4; 3=4], where (x; y) is constant. In Table 5 we observe that geometric multigrid has di-culties for small values . Because of the unidirectional smoothing of the error, aligned with the strong anisotropy, standard (isotropic) interpolation fails. Table Locally anisotropic diusion: geometric MG convergence rates q for varying on a 128 128 grid. Smoother q M q M q M Gauss{Seidel AMG results AMG overcomes these di-culties by performing automatic semi{coarsening and operator dependent interpolation only in the direction of strong couplings, which correspond to smooth error components. It is well{known (e.g. [23]) that AMG solves such problems with little di-culty. The results in Table 6 verify this fact for acceptable densities A and M . Both densities could be lowered even further by dropping the smallest entries in the interpolation operators ([23]); we do not consider such truncated grid transfer operators here. Table AMG convergence results for locally anisotropic diusion on a 128 128 grid. Note that q, A , and M remain bounded as ! 0. A 2.84 2.94 2.94 Smoother q M q M q M Gauss{Seidel 0.14 | 0.18 | 0.18 | The convergence rates obtained with Gauss-Seidel and SPAI-1 are comparable and both below 0.2, while SPAI-0 results in slightly slower convergence. Overall the SPAI-1 smoother is the most e-cient smoother for this problem. Although further reduction of " results in even faster convergence, the approximate inverses become too dense and thus too expensive. Again, the results in Tables 6 and 7 demonstrate robust multigrid behavior, either as h ! 0, or as Table AMG convergence rates q for locally anisotropic diusion, with . Note that q; A , and M remain bounded as Gridsize 6464 128128 256256 A 2.89 2.94 2.95 Gauss-Seidel 0.12 | 0.18 | 0.22 | Concluding remarks Our results show that sparse approximate inverses, based on the minimization of the Frobenius norm, provide an attractive alternative to classical Jacobi or Gauss-Seidel smoothing. The simpler smoothers, SPAI-0 and SPAI-1, often provide ample smoothing, comparable to damped Jacobi or Gauss-Seidel. Nev- ertheless, situations such as convection dominated rotating ow, where SPAI-1 leads to a convergent multigrid iteration, unlike Gauss-Seidel, demonstrate the improved robustness. Our implementation of geometric multigrid combined with SPAI-1 smoothing enables the fast solution of very large convection- diusion problems on massively parallel architectures. By incorporating the SPAI smoothers into AMG ([22]), we obtain a exible, parallel, and algebraic multigrid method, easily adjusted to the underlying problem and computer architecture, even by the non-expert. It is very interesting to incorporate information available from the approximate inverses into the coarsening strategy and grid transfer operators, as suggested in [19,20]. The expected benet would include improved robustness and local adaptivity for these multigrid components as well. The authors are currently pursuing these issues and will report on them elsewhere in the near future. Acknowledgment We thank Klaus Stuben for useful comments and suggestions. --R An MPI implementation of the SPAI preconditioner on the T3E A block version of the SPAI preconditioner Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems Frequency domain behavior of a set of parallel multigrid smoothing operators A sparse approximate inverse preconditioner for the conjugate gradient method A comparative study of sparse approximate inverse preconditioners Approximate inverse preconditioners via sparse-sparse iterations A priori sparsity patterns for parallel sparse approximate inverse preconditioners Robustness and scalability of algebraic multigrid Parallel preconditioning with sparse approximate inverses Iterative Solution of Large Sparse Systems of Equations Approximate sparsity patterns for the inverse of a matrix and preconditioning Factorized sparse approximate inverse preconditionings: I. Numerical experiments with algebraic multilevel preconditioners A multilevel AINV preconditioner Parallel adaptive multigrid Toward an e Sparse approximate inverse smoother for multi- grid Introduction to algebraic multigrid An Introduction to Multigrid Methods On the robustness of ILU-smoothing Matrix prolongations and restrictions in a black-box multigrid solver --TR On the robustness of Ilu smoothing Matrix-dependent prolongations and restrictions in a blackbox multigrid solver Multigrid methods on parallel computersMYAMPERSANDmdash;a survey of recent developments Factorized sparse approximate inverse preconditionings I A Sparse Approximate Inverse Preconditioner for the Conjugate Gradient Method Parallel Preconditioning with Sparse Approximate Inverses Approximate Inverse Preconditioners via Sparse-Sparse Iterations Approximate sparsity patterns for the inverse of a matrix and preconditioning A comparative study of sparse approximate inverse preconditioners Toward an Effective Sparse Approximate Inverse Preconditioner A Priori Sparsity Patterns for Parallel Sparse Approximate Inverse Preconditioners Robustness and Scalability of Algebraic Multigrid Sparse Approximate Inverse Smoother for Multigrid Robust Parallel Smoothing for Multigrid Via Sparse Approximate Inverses Coarse-Grid Selection for Parallel Algebraic Multigrid --CTR Michele Benzi, Preconditioning techniques for large linear systems: a survey, Journal of Computational Physics, v.182 n.2, p.418-477, November 2002

robust smoothing;algebraic multigrid;approximate inverses;parallel multigrid

606722

A parallel solver for large-scale Markov chains.

We consider the parallel computation of the stationary probability distribution vector of ergodic Markov chains with large state spaces by preconditioned Krylov subspace methods. The parallel preconditioner is obtained as an explicit approximation, in factorized form, of a particular generalized inverse of the generator matrix of the Markov process. Graph partitioning is used to parallelize the whole algorithm, resulting in a two-level method.Conditions that guarantee the existence of the preconditioner are given, and the results of a parallel implementation are presented. Our results indicate that this method is well suited for problems in which the generator matrix can be explicitly formed and stored.

Introduction Discrete Markov chains with large state spaces arise in many applications, including for instance reliability modeling, queueing network analysis, large scale economic modeling and computer system performance evaluation. The stationary probability distribution vector of an ergodic Markov process with n \Theta n transition probability matrix P is the unique 1 \Theta n vector - which satisfies Letting the computation of the stationary vector reduces to finding a nontrivial solution to the homogeneous linear system 0. The ergodicity assumption means that P (and therefore A) is irreducible. Perron-Frobenius theory [9] guarantees that A has rank n \Gamma 1, and that the (one-dimensional) null space N (A) of A is spanned by a vector x with positive entries. Upon normalization in the ' 1 -norm, this is the stationary distribution vector of the Markov process. The coefficient matrix A is a singular M-matrix, called the generator of the Markov process. 3 The matrix A is nonsymmetric, although it is sometimes structurally symmetric. See [26] for a good introduction to Markov chains and their numerical solution. Due to the very large number n of states typical of many real-world applica- tions, there has been increasing interest in recent years in developing parallel algorithms for Markov chain computations; see [2], [5], [10], [17], [19], [24]. Most of the attention so far has focused on (linear) stationary iterative meth- ods, including block versions of Jacobi and Gauss-Seidel [10], [19], [24], and on iterative aggregation/disaggregation schemes specifically tailored to stochastic matrices [10], [17]. In contrast, little work has been done with parallel preconditioned Krylov subspace methods. Partial exceptions are [5], where a symmetrizable stationary iteration (Cimmino's method) was accelerated using conjugate gradients on a Cray T3D, and [19], where an out-of-core, parallel implementation of Conjugate Gradient Squared (with no precondi- tioning) was used to solve very large Markov models with up to 50 million states. The suitability of preconditioned Krylov subspace methods for solving Markov models has been demonstrated, e.g., in [25], although no discussion of parallelization issues is given there. In this paper we investigate the use of a parallel preconditioned iterative method for large, sparse linear systems in the context of Markov chain com- Strictly speaking, the generator matrix is We work with A instead of Q to conform to the familiar notation of numerical linear algebra. putations. The preconditioning strategy is a two-level method based on sparse approximate inverses, first introduced in [3]. However, due to the singularity of the generator matrix A, the applicability of approximate inverse techniques in this context is not obvious. That this is indeed possible is a consequence of the fact that A is a (singular) M-matrix. The paper is organized as follows. In section 2 we discuss the problem of pre-conditioning singular equations in general, and we establish a link between some standard preconditioners and generalized inverses. Sections 3-5 are devoted to AINV preconditioning for Markov chain problems, including a discussion of the parallel implementation and a theoretical analysis of the existence of the preconditioner. Numerical tests are reported in section 6, and some conclusions are presented in section 7. Preconditioning Markov chain problems In the Markov chain context, preconditioning typically amounts to finding an easily invertible nonsingular matrix M (the preconditioner) which is a good approximation to A; a Krylov subspace method is then used to solve preconditioning). Notice that even if A itself is singular, the preconditioner must be nonsingular so as not to change the solution set, i.e., the null space (A) of A. Preconditioners can be generated by means of splittings N , such as those used in stationary iterative methods including Jacobi, Gauss- Seidel, SOR and block versions of these schemes; see [26]. Also in this class are the popular incomplete LU (ILU) factorization preconditioners. ILU-type methods have been successfully applied to Markov chain problems by Saad [25] in a sequential environment. The existence of incomplete factorizations for nonsingular M-matrices was already proved in [20]; an investigation of the existence of ILU factorizations for singular M-matrices can be found in [11]. Incomplete factorization methods work quite well on a wide range of problems, but they are not easily implemented on parallel computers. For this and other reasons, much effort has been put in recent years into developing alternative preconditioning strategies that have natural parallelism while being comparable to ILU methods in terms of robustness and convergence rates. This work has resulted in several new techniques known as sparse approximate inverse preconditioners; see [7] for a recent survey and extensive references. Sparse approximate inverse preconditioners are based on directly approximating the inverse of the coefficient matrix A with a sparse matrix G - A \Gamma1 . The application of the preconditioner only requires matrix-vector products, which are easily parallelized. Until now, these techniques have been applied almost exclusively to nonsingular systems of equations b. The only exception seems to be [13], where the SPAI preconditioner [16] was used in connection with Fast Wavelet Transform techniques on singular systems stemming from discretizations of the Neumann problem for Poisson's equation. The application of approximate inverse techniques in the singular case raises several interesting theoretical and practical questions. Because the inverse of A does not exist, it is not clear what matrix G is an approximation of. It should presumably be some generalized inverse of A, but which one? Note that this question can be asked of M \Gamma1 for any preconditioner M - A. In [26], page 143, it is stated that M \Gamma1 should be an approximation of the group generalized inverse A ] , and that an ILU factorization A - U implicitly yields such an approximation: ( - . As we will see, this interpretation is not entirely correct and is somewhat misleading. The group inverse (see [12]) is only one of many possible generalized inverses. It is well known [21] that the group inverse plays an important role in the modern theory of finite Markov chains. However, it is seldom used as a computational tool, in part because its computation requires knowledge of the stationary distribution vector -. As it turns out, different preconditioners result (implicitly or explicitly) in approximations M \Gamma1 to different generalized inverses of A, which are typically not the group inverse A ] . Let us consider ILU preconditioning first. If A is a n\Theta irreducible, singular M-matrix, then A has the LDU factorization where L and U are unit lower and upper triangular matrices (respectively) and D is a diagonal matrix of rank (see [26]). Notice that L and U are nonsingular M-matrices; in particular, L \Gamma1 and U \Gamma1 have nonnegative entries. Define the matrix It can be easily verified that A \Gamma satisfies the first two of Penrose's four conditions [12]: The first identity states that A \Gamma is an inner inverse of A and the second that A \Gamma is an outer inverse of A. A generalized inverse satisfying these two conditions is called a (1; 2)-inverse of A or an inner-outer inverse. Another term that is found in the literature is reflexive inverse; see [12]. Because A \Gamma does not necessarily satisfy the third and fourth Penrose conditions, it is not the Moore-Penrose pseudoinverse A y of A in general. Because A y is obviously a (1; 2)-inverse, this kind of generalized inverse is non-unique. Indeed, there are infinitely many such (1; 2)-inverses in general. Each pair R, N of subspaces of IR n that are complements of the null space and range of A (respectively) uniquely determines a (1; 2)-inverse GN;R of A with null space N (GN;R and range R(GN;R see [12]. In the case of A \Gamma it is readily verified that denotes the i-th unit basis vector in IR n . It is easy to see that R is complementary to N (A) and N is complementary to R(A). The pseudoinverse A y corresponds to The (1; 2)-inverse A \Gamma is also different from the group inverse A ] , in general. This can be seen from the fact that in general AA \Gamma 6= A \Gamma A, whereas the group inverse always satisfies AA A. Also notice that for a singular irreducible M-matrix A the (1; 2)-inverse A nonnegative ma- trix, which is not true in general for either the group or the Moore-Penrose Let now A - U be an incomplete LDU factorization of A, with - unit lower triangular, - U - U unit upper triangular and - diagonal matrix with positive entries on the main diagonal. Then clearly Hence, an ILU factorization of A yields an implicit approximation to A \Gamma rather than to A ] . This can also be seen from the fact that ( - always nonnegative, which is not true of A ] . It is straightfoward to check that A \Gamma A is the oblique projector onto along N (A) and that AA \Gamma is the oblique projector onto along g. Therefore A \Gamma A has eigenvalues 0 with multiplicity 1, and 1 with multiplicity likewise for AA \Gamma . Hence, it makes good sense to construct preconditioners based on approximating A \Gamma (either implicitly or explicitly), since in this case most eigenvalues of the preconditioned matrix will be clustered around 1. Next we consider the approximate inverse preconditioner AINV; see [4], [6]. This method is based on the observation that if Z and W are matrices whose columns are A-biorthogonal, then W T diagonal matrix. When all the leading principal minors of A (except possibly the last one) are nonzeros, Z and W can be obtained by applying a generalized Gram-Schmidt process to the unit basis vectors e 1 . In this case Z and W are unit upper triangular. It follows from the uniqueness of the LDU factorization that U \Gamma1 and LDU is the LDU factorization of A. The diagonal matrix D is the same in both factorizations. An approximate inverse in factorized form W T can be obtained by dropping small entries in the course of the generalized Gram-Schmidt process. Similar to ILU, this incomplete inverse factorization is guaranteed to exist for nonsingular M - matrices [4]; see the next section for the singular M-matrix case. In either case, W T is a nonnegative matrix. Hence, this preconditioner can be interpreted as a direct (explicit) approximation to the (1; 2)-inverse A \Gamma of Lastly, we take a look at sparse approximate inverse techniques based on Frobenius norm minimization; see, e.g., [16] and [14]. With this class of meth- ods, an approximate inverse G is computed by minimizing the functional subject to some sparsity constraints. Here jj \Delta jj F denotes the Frobenius matrix norm. The sparsity constraints could be imposed a priori, or dynamically in the course of the algorithm. In either case, it is natural to ask what kind of generalized inverse is being approximated by G when A is a singular matrix. It can be shown that the Moore-Penrose pseudoinverse A y is the matrix of smallest Frobenius norm that minimizes jjI \Gamma AXjj F ; see for instance [23], page 428. Hence, in the singular case the SPAI preconditioner can be seen as a sparse approximate Moore-Penrose inverse of A. This is generally very different from the approximate (1; 2)-inverses obtained by either ILU or AINV. For instance, SPAI will not produce a nonnegative preconditioner in general. In the next section we restrict our attention to the AINV preconditioner and its application to Markov chain problems. 3 The AINV method for singular matrices The AINV preconditioner [4], [6] is based on A-biorthogonalization. This is a generalized Gram-Schmidt process applied to the unit basis vectors e i , n. In this generalization the standard inner product is replaced by the bilinear form h(x; Ay. This process is well defined, in exact arith- metic, if the leading principal minors of A are nonzero, otherwise some form of pivoting (row and/or column interchanges) may be needed. If A is a nonsingular M-matrix, all the leading principal minors are positive and the process is well defined with no need for pivoting. This is perfectly analogous to the LU factorization of A, and indeed in exact arithmetic the A-biorthogonalization process computes the inverses of the triangular factors of A. When A is a singular irreducible M-matrix, all the leading principal minors of A except the n-th one (the determinant of are positive, and the process can still be completed. In order to obtain a sparse preconditioner, entries (fill-ins) in the inverse factors Z and W less than a given drop tolerance in magnitude are dropped in the course of the computation, resulting in an incomplete process. The stability of the incomplete process for M-matrices was analyzed in [4]. In particular, if d i denotes the i-th pivot, i.e., the i-th diagonal entry of - D in the incomplete process, then (Proposition 3.1 in [4]) - Because for an M-matrix, no pivot breakdown can occur. Exactly the same argument applies to the case where A is an irreducible, singular M-matrix. In this case there can be no breakdown in the first steps of the incomplete A-biorthogonalization process, since the first leading principal minors are positive, and the pivots in the incomplete process cannot be smaller than the exact ones. And even if the n-th pivot - happened to be zero, it could simply be replaced by a positive number in order to have a nonsingular preconditioner. The argument in [4] shows that - d n must be a nonnegative number, and it is extremely unlikely that it will be exactly zero in the incomplete process. Another way to guarantee the nonsingularity of the preconditioner is to perturb the matrix A by adding a small positive quantity to the last diagonal entry. This makes the matrix a nonsingular M-matrix, and the incomplete A-biorthogonalization process can then be applied to this slightly perturbed matrix to yield a well defined, nonsingular preconditioner. In practice, how- ever, this perturbation is not necessary, since dropping in the factors typically has an equivalent effect (see Theorem 3 below). The AINV preconditioner has been extensively tested on a variety of symmetric and nonsymmetric problems in conjunction with standard Krylov subspace methods like conjugate gradients (for symmetric positive definite matrices) and GMRES, Bi-CGSTAB and TFQMR (for unsymmetric problems). The preconditioner has been found to be comparable to ILU methods in terms of robustness and rates of convergence, with ILU methods being somewhat faster on average on sequential computers. The main advantage of AINV over the ILU-type methods is that its application within an iterative process only requires matrix-vector multiplies, which are much easier to vectorize and to parallelize than triangular solves. Unfortunately, the computation of the preconditioner using the incomplete A- biorthogonalization process is inherently sequential. One possible solution to this problem, adopted in [8], is to compute the preconditioner sequentially on one processor and then to distribute the approximate inverse factors among processors in a way that minimizes communication costs while achieving good load balancing. This approach is justified in applications, like those considered in [8], in which the matrices are small enough to fit on the local memory of one processor, and where the preconditioner can be reused a number of times. In this case the time for computing the preconditioner is negligible relative to the overall costs. In the Markov chain setting, however, the preconditioner cannot be reused in general and it is imperative that set-up costs be minimized. Furthermore, Markov chain problems can be very large, and it is desirable to be able to compute the preconditioner in parallel. 4 The parallel preconditioner In the present section we describe how to achieve a fully parallel precon- ditioner. The strategy used to parallelize the preconditioner construction is based on the use of graph partitioning. This approach was first proposed in [3] in the context of solving nonsingular linear systems arising from the discretization of partial differential equations. The idea can be illustrated as follows. If p processors are available, graph partitioning can be used to decompose the adjacency graph associated with the sparse matrix A (or with A A is not structurally symmetric) in p subgraphs of roughly equal size in such a way that the number of edge cuts is approximately minimized. Nodes which are connected by cut edges are removed from the subgraphs and put in the separator set. By numbering the nodes in the separator set last, a symmetric permutation Q T AQ of A is obtained. The permuted matrix has the following structure: The diagonal blocks A correspond to the interior nodes in the graph decomposition, and should have approximately the same order. The off-diagonal blocks connections between the subgraphs, and the diagonal block A S the connections between nodes in the separator set. The order of A S is equal to the cardinality of the separator set and should be kept as small as possible. Note that because of the irreducibility assumption, each block A i is a nonsingular M-matrix, and each of the LDU factorizations A LDU be the LDU factorization of Q T AQ. Then it is easy to see that S is the inverse of the unit lower triangular factor of the Schur complement matrix In the next section we show that S is a singular, irreducible M-matrix, hence it has a well defined LDU factorization Likewise, U S is the inverse of the unit upper triangular factor of S. It is important to observe that L \Gamma1 and U \Gamma1 preserve a good deal of sparsity, since fill-in can occur only within the nonzero blocks. The matrix D is simply defined as note that all diagonal entries of D are positive except for the last one, which is zero. The (1; 2)-inverse D \Gamma of D is defined in the obvious way. Hence, we can write the (generally dense) generalized inverse (Q T AQ) \Gamma of as a product of sparse matrices L In practice, however, the inverse factors L \Gamma1 and U \Gamma1 contain too many nonzeros. Since we are only interested in computing a preconditioner, we just need to compute sparse approximations to L \Gamma1 and U \Gamma1 . This is accomplished as follows. With graph partitioning, the matrix is distributed so that processor P i holds A One of the processors, marked as P S , should also hold A S . Each processor then computes sparse approximate inverse factors - W i such that - using the AINV algorithm. Once this is done, each processor computes the product . Until this point the computation proceeds in parallel with no communication. The next step is the accumulation of the approximate Schur complement - This accumulation is done in steps with a fan-in across the processors. In the next section we show that although the exact Schur complement S is singular, the approximate Schur complement - S is a nonsingular M-matrix under rather mild conditions. As soon as - S is computed, processor P S computes a factorized sparse approximate using the AINV algorithm. This is a sequential bottleneck, and explains why the size of the separator set must be kept small. Once the approximate inverse factors of - are computed, they are broadcast to all remaining processors. (Actually, the preconditioner application can be implemented in such a way that only the - needs to be broadcast.) Notice that because only matrix-vector products are required in the application of the preconditioner, there is no need to explicitly. In this way, a factorized sparse approximate (1; 2)-inverse of Q T AQ is obtained. This is a two-level preconditioner, in the sense that the computation of the preconditioner involves two phases. In the first phase, sparse approximate inverses of the diagonal blocks A i are computed. In the second phase, a sparse approximate inverse of the approximate Schur complement - S is computed. this second step the preconditioner would reduce to a block Jacobi method with inexact block solves (in the terminology of domain decomposition methods, this is additive Schwarz with inexact solves and no overlap). It is well known that for a fixed problem size, the rate of convergence of this preconditioner tends to deteriorate as the number of blocks (subdomains) grows. Hence, assuming that each block is assigned to a processor in a parallel computer, this method would not be scalable. However, the approximate Schur complement phase provides a global exchange of information across the processors, acting as a "coarse grid" correction in which the "coarse grid" nodes are interface nodes (i.e., they correspond to vertices in the separator set). As we will see, this prevents the number of iterations from growing as the number of processors grows. As long as the cardinality of the separator set is small compared to the cardinality of the subdomains (subgraphs), the algorithm is scalable in terms of parallel efficiency. Indeed, in this case the application of the preconditioner at each step of a Krylov subspace method like GMRES or Bi-CGSTAB is easily implemented in parallel with relativeley little communication needed. 5 The approximate Schur complement In this section we investigate the existence of the approximate (1; 2)-inverse of the generator matrix A. The key role is played by the (approximate) Schur complement. First we briefly review the situation for the case where A is a nonsingular M-matrix. Assume A is partitioned as A 21 A 22 Then it is well-known that the Schur complement is also a nonsingular M-matrix; see, e.g., [1]. Moreover, the same is true of any approximate Schur complement provided that O - X 11 - A \Gamma1 , where the inequalities hold componentwise; see [1], page 264. In the singular case, the situation is slightly more complicated. In the following we will examine some basic properties of the exact Schur complement of a singular, irreducible M-matrix A corresponding to an ergodic Markov chain. Recall that is the irreducible row-stochastic transition probability matrix. be an irreducible row-stochastic matrix partitioned as Assume that A 21 A 22 Then the Schur complement S of A 11 in A is a singular, irreducible M-matrix with a one-dimensional null space. Proof: Consider the stochastic complement [22] \Sigma of P 22 in Note that I \Gamma P 11 is invertible since P is irreducible. From the theory developed in [22], we know that \Sigma is row stochastic and irreducible since P is. Consider now the Schur complement S of A 11 in A: 22 Clearly, S is an irreducible singular M-matrix. It follows (Perron-Frobenius theorem) that S has a one-dimensional null space. 2 The previous lemma is especially useful in cases where the exact Schur complement is used. In the context of preconditioning it is often important to know properties of approximate Schur complements. As shown in the previous section, graph partitioning induces a reordering and block partitioning of the matrix A in the form (1) where and We are interested in properties of the approximate Schur complement - obtained by approximating the inverses of the diagonal blocks A i with AINV: In particular, we are interested in conditions that guarantee that - S is a non-singular M-matrix, in which case the AINV algorithm can be safely applied to S, resulting in a well defined preconditioner. We begin with a lemma. Recall that a Z-matrix is a matrix with nonpositive off-diagonal entries [9]. Lemma 2 Let S be a singular, irreducible M-matrix. Let C - O; C 6= O be such that - S is a nonsingular M-matrix. Proof: Since S is a singular M-matrix, we have where ae(B) denotes the spectral radius of B. Let is a Z-matrix, we have that B - C \Gamma D. Therefore we can write the modified S as - We distinguish the two following simple cases: Assume first that S is a nonsingular M-matrix since it can be written as ae(B)I \Gamma (B \Gamma C) with C). The last inequality follows from the irreducibility of B and properties of nonnegative matrices; see [9], page 27, Cor. 1.5 (b). Assume, on the other hand, that B. Note that by assumption, at least one of the diagonal entries d ii of D must be positive. Let denote the largest such positive diagonal entry. Since B + ffiI is irreducible, similar to the previous case. It follows that - D) is a nonsingular M-matrix. Finally, if both D 6= O and C 6= D the result follows by combining the two previous arguments. 2 In the context of our parallel preconditioner, this lemma says that if the inexactness in the approximate inverses of the diagonal blocks A i results in an approximate Schur complement - S that is still a Z-matrix and furthermore if nonnegative and nonzero, then - S is a nonsingular M-matrix. The following theorem states sufficient conditions for the nonsingularity of the approximate Schur complement. B i and B \Lambdaj denote the i-th row and the j-th column of matrix B, respectively. Theorem 3 Let A 21 A 22 be a singular irreducible M-matrix, with A 11 2 IR m\Thetam . Assume that ~ A 11 is an approximation to A \Gamma1 11 such that O - ~ A A 11 . Furthermore, assume that there exist indices such that the following three conditions are satisfied: A the approximate Schur complement - ~ A A 12 is a nonsingular M-matrix Proof: From Lemma 1 we know that the exact Schur complement S of A 11 in A is a singular, irreducible M-matrix. Note that the approximate Schur complement - S induced by an approximation ~ A 11 of the block A \Gamma1 11 and the exact Schur complement S are related as follows: ~ A A \Gamma1)A with C - O since A 21 - O; A 12 - O; and A \Gamma1 A S is a Z-matrix by its definition. From the assumption that ~ A \Gamma16= A \Gamma1and that there exists at least one nonzero entry ff ij of ~ A 11 not equal to (in fact, strictly less than) the corresponding entry of A \Gamma1 11 , we see that if the corresponding row i of A 21 and column j of A 12 are both nonzero, there is a nonzero entry in C: The result now follows from Lemma 2. 2 Let us now apply these results to the preconditioner described in the previous section. In this case, A 11 is block diagonal and the AINV algorithm is used to approximate the inverse of each diagonal block separately, in parallel. The approximate Schur complement (2) is the result of subtracting p terms of the We refer to these terms as (Schur complement) updates. Each one of these updates is nonnegative and approximates the exact update C from below in the (entrywise) nonnegative ordering, since O - see [6]. Theorem 3 says that as long as at least one of these updates has an entry that is strictly less than the corresponding entry in C , the approximate Schur complement - S is a nonsingular M-matrix. In practice, these conditions are satisfied as a result of dropping in the approximate inversion of the diagonal blocks A i . It is nevertheless desirable to have rigorous conditions that ensure nonsingularity. The following theorem gives a sufficient condition for having a nonsingular approximate Schur complement as a consequence of dropping in AINV. Namely, it specifies conditions under which any dropping forces - S to be nonsingular. Note that the conditions of this theorem do not apply to the global matrix (1), since A 11 is block diagonal and therefore reducible. However, they can be applied to any individual Schur complement update for which the corresponding diagonal block A i is irreducible, making the result fairly realistic. Theorem 4 Let A 11 2 IR m\Thetam and the singular M-matrix A 21 A 22 be both irreducible. Let A be the LU factorization of A 11 . Assume that in each column of L 11 (except the last one) and in each row of U 11 (except the last one) there is at least one nonzero entry in addition to the diagonal one: and Denote by - Z 11 11 the factorized sparse approximate inverse of A 11 obtained with the AINV algorithm. Then the approximate Schur complement S is a nonsingular M-matrix provided that - Z 11 6= Z 11 and - Proof: First note that the two conditions for nonzero entries in L 11 and U 11 are implied by similar conditions for the entries in the lower and upper triangular parts of A 11 . Namely, it is easy to see that (barring fortuitous cancellation) and for 0: Here tril(B) and triu(B) denote the lower and upper triangular part of matrix B, respectively. These conditions are easier to check than the weaker ones on the triangular factors of A 11 . Conditions (3) and (4) imply that there is a path in the graph of L T 11 and there is a path in the graph of U 11 for Because the matrix is irreducible, then A 12 6= O and A 21 6= O. This means that there exist indices s such that rs 6= 0: The existence of the previously mentioned paths implies that where W 11 and Z 11 are the exact inverse factors of A 11 . The approximate inverse factors from the AINV algorithm satisfy [6] O - Z 11 Hence, the conditions of Theorem 3 are satisfied and the result is proved. 2 It is instructive to consider two extreme cases. If A 11 is diagonal, then the approximate Schur complement is necessarily equal to the exact one, and is therefore singular. In this case, of course, the conditions of the last theorem are violated. On the other hand if A 11 is irreducible and tridiagonal, its inverse factors are completely dense and by the last theorem it is enough to drop a single entry in each inverse factor to obtain a nonsingular approximate Schur complement. The purpose of the theory developed here is to shed light on the observed robustness of the proposed preconditioner rather than to serve as a practical tool. In other words, it does not seem to be necessary to check these conditions in advance. Indeed, thanks to dropping the approximate Schur complement was always found to be a nonsingular M-matrix in actual computations. 6 Numerical experiments In this section we report on results obtained with a parallel implementation of the preconditioner on several Markov chain problems. The underlying Krylov subspace method was Bi-CGSTAB [27], which was found to perform well for Markov chains in [15]. Our FORTRAN implementation uses MPI and dynamic memory allocation. The package METIS [18] was used for the graph partitioning, working with the graph of A + A T whenever A was not structurally symmetric. The test problems arise from real Markov chain applications and were provided by T. Dayar. These matrices have been used in [15] to compare different methods in a sequential environment. A description of the test problems is provided in Table 1 below. Here n is the problem size and nnz the number of nonzeros in the matrix. All the test problems are structurally nonsymmetric except ncd and mutex. Most matrices are unstructured. Tables 2-11 contain the test results. All runs were performed on an SGI Origin 2000 at Los Alamos National Laboratory (using up to 64 processors), except for those with matrices leaky, ncd and 2d which were performed on an Origin 2000 at the Helsinki University of Technology (using up to 8 processors). In all cases, the initial guess was a constant nonzero vector; similar results were obtained with a randomly generated initial guess. In the tables, "P-time" denotes the time to compute the preconditioner, "P-density" the ratio of the number of nonzeros in the preconditioner to the number of nonzeros in the matrix A, "Its" denotes the number of iterations needed to reduce the ' 2 - norm of the initial residual by eight orders of magnitude, "It-time" the time to perform the iterations, and "Tot-time" the sum of "P-time" and "It-time." All timings are in seconds. Furthermore, "Sep-size" is the cardinality of the Information on test problems. Matrix n nnz Application hard 20301 140504 Complete buffer sharing in ATM networks Multiplexing model of a leaky bucket 2d 16641 66049 A two-dimensional Markov chain model telecom 20491 101041 A telecommunication model ncd 23426 156026 NCD queueing network mutex 39203 563491 Resource sharing model qn 104625 593115 A queueing network separator set (i.e., the order of the Schur complement matrix) and "Avg- dom" the average number of vertices in a subdomain (subgraph) in the graph partitioning of the problem. The drop tolerance - in the AINV algorithm was the same at both levels of the preconditioner (approximate inversion of A i for approximate inversion of the approximate Schur complement S), except for the mutex problem (see below). Tables present results for the matrix hard, using three different values of the drop tolerance in the AINV algorithm. It can be seen that changing the value of - changes the density of the preconditioner and the number of itera- tions. However, the total timings are scarcely affected, especially if at least 8 processors are being used. See [8] for a similar observation in a different con- text. It is also clear from these runs that good speed-ups are obtained so long as the size of the separator set is small compared to the average subdomain size. As soon as the separator set is comparable in size to the average subdomain or larger, the sequential bottleneck represented by the Schur complement part of the computation begins to dominate and performance deteriorates. The number of iterations remains roughly constant (with a slight downward trend) as the number of processors grows. This is due to the influence of the approximate Schur complement. The same problem was also solved using Bi-CGSTAB with diagonal precondi- tioning. This required approximately 700 iterations and 16.4 seconds on one processor. If implemented in parallel, this method would probably give results only slightly worse than those obtained with AINV. A similar observation applies to matrices qn and mutex. On the other hand, diagonally preconditioned Bi-CGSTAB did not converge on the telecom problem. Hence, AINV is a more robust approach. Furthermore, the ability to reduce the number of iterations, and therefore the total number of inner products, is an advantage on distributed memory machines, on which inner products incur an additional penalty due to the need for global communication. Matrix hard, P-time 2.35 1.19 0.65 0.40 0.34 P-density 6.21 5.90 5.67 5.14 4.52 Its It-time 9.43 2.91 1.42 0.99 0.88 Tot-time 11.8 4.10 2.07 1.39 1.22 Sep-size 156 321 540 900 1346 Avg-dom 10073 5155 2470 1213 592 Table Matrix hard, P-time 1.19 0.60 0.35 0.22 0.21 P-density 3.10 3.02 2.95 2.78 2.52 Its 106 109 99 98 97 It-time 6.85 2.90 1.59 1.05 1.19 Tot-time 8.04 3.50 1.94 1.27 1.40 Table Matrix hard, P-time 0.73 0.38 0.22 0.15 0.14 P-density 1.39 1.36 1.34 1.28 1.19 Its 170 167 159 151 153 It-time 6.03 2.52 1.50 1.30 1.64 Tot-time 6.76 2.90 1.72 1.45 1.78 Results for matrices leaky and 2d are reported in Tables 5 and 6. These two matrices are rather small, so only up to 8 processors were used. Note that the speed-ups are better for 2d than for leaky. Also notice that the preconditioner is very sparse for leaky, but rather dense for 2d. Tables 7 and 8 refer to the telecom test problem. Here we found that very small values of - (and, consequently, very dense preconditioners) are necessary in order to achieve convergence in a reasonable number of iterations. This problem is completely different from the matrices arising from the solution of elliptic partial differential equations. Notice the fairly small size of the sep- Matrix leaky, P-time 0.26 0.17 0.09 P-density No. its 134 134 132 It-time 1.39 0.84 0.62 Tot-time 1.65 1.01 0.71 Sep-size 48 144 335 Avg-dom 4105 2028 990 Table Matrix 2d, P-time 0.38 0.16 0.09 P-density 8.60 7.12 6.54 No. its 33 36 37 It-time 1.46 0.56 0.38 Tot-time 1.84 0.72 0.47 Sep-size 129 308 491 Avg-dom 8256 4083 2018 Table Matrix telecom, P-time 24.9 10.0 3.35 1.21 1.05 1.37 P-density 167 116 70 44 28 Its 11 12 14 13 12 12 It-time 36.5 14.0 6.04 0.96 0.38 0.43 Tot-time 61.4 24.0 9.39 2.17 1.43 1.80 Sep-size 34 97 220 471 989 1603 Avg-dom 10229 5099 2534 1251 609 295 arator set, which causes the density of the preconditioner to decrease very fast as the number of processors (and corresponding subdomains) grows. As a result, speed-ups are quite good (even superlinear) up to 32 processors. For a sufficiently high number of processors, the density of the preconditioner be- Matrix telecom, P-time 7.04 3.78 1.37 0.61 0.49 0.56 P-density No. its It-time 59.4 27.0 12.8 2.18 1.36 1.71 Tot-time 66.4 30.8 14.2 2.79 1.86 2.27 Table Matrix P-time 1.42 0.69 0.31 P-density 4.13 2.65 1.90 No. its 292 288 285 It-time 17.0 8.45 6.38 Tot-time 18.4 9.14 6.69 Sep-size 3911 6521 12932 Avg-dom 9758 4226 1379 Table Matrix P-time 1.19 0.40 0.10 P-density 0.14 0.14 0.14 No. its It-time 1.64 1.19 1.51 Tot-time 2.83 1.59 1.61 Sep-size 13476 17749 20654 Avg-dom 12864 5363 2319 comes acceptable, and the convergence rate is the same or comparable to that obtained with a very dense preconditioner on a small number of processors. Tables results for matrices ncd and mutex, respectively. For the first matrix we see that the separator set is larger than the average subdomain already for nevertheless, it is possible to use effectively up to 8 processors. Matrix mutex exhibits a behavior that is radically different Matrix qn, P-time 4.12 2.37 2.26 2.82 P-density 1.27 1.23 1.18 1.14 No. its It-time 13.4 6.33 4.58 5.68 Tot-time 17.5 8.70 6.84 8.50 Sep-size 2879 6579 13316 20261 Avg-dom 50873 24511 11414 5273 from that of matrices arising from PDE's in two or three space dimensions. The separator set is huge already for 2. This is due to the fact that the problem has a state space (graph) of high dimensionality, leading to a very unfavorable surface-to-volume ratio in the graph partitioning. In order to solve this problem, we had to use two different values of - in the two levels of AINV; at the subdomain level we used when forming the approximate Schur complement we dropped everything outside the main diagonal, resulting in a diagonal - S. In spite of this, convergence was very rapid. Nevertheless, it does not pay to use more than processors. In Table 11 we report results with the largest example in our data set, qn. This model consists of a network of three queues, and is analogous to a three-dimensional problem. Because of the fairly rapid growth of the separator set, it does not pay to use more than processors. The test problems considered so far, although realistic, are relatively small. Hence, it is difficult to make efficient use of more than 16 processors, with the partial exceptions of matrices hard and telecom. To test the scalability of the proposed solver on larger problems, we generated some simple reliability problems analogous to those used in [2] and [5]; see also [26], page 135. These problems have a closed form solution. In Table 12 we show timing results for running 100 preconditioned Bi-CGSTAB iterations on a reliability problem of entries. This problem is sufficiently large to show the good scalability of the algorithm up to processors. We conclude this section on numerical experiments by noting that in virtually all the runs, the preconditioner construction time has been quite modest and the total solution time has been dominated by the cost of the iterative phase. Reliability model, P-time 9.53 4.86 2.49 1.31 0.90 0.90 P-density 4.05 4.02 3.97 3.90 3.80 3.70 It-time 138.8 70.5 37.2 16.8 9.47 7.96 Tot-time 148.3 75.4 39.7 18.1 10.4 8.86 Sep-size 542 1229 2186 3268 4919 7336 Avg-dom 124729 62193 30977 15421 7659 3792 Conclusions We have investigated the use of a parallel preconditioner for Krylov subspace methods in the context of Markov chain problems. The preconditioner is a direct approximation, in factorized form, of a (1; 2)-inverse of the generator matrix A, and is based on an A-biorthogonalization process. Parallelization is achieved through graph partitioning, although other approaches are also possible. The existence of the preconditioner has been justified theoretically, and numerical experiments on a parallel computer have been carried out in order to assess the effectiveness and scalability of the proposed technique. The numerical tests indicate that the preconditioner construction costs are modest, and that good scalability is possible provided that the amount of work per processor is sufficiently large compared to the size of the separator set. The method appears to be well suited for problems in which the generator can be explicitly formed and stored. Parallelization based on graph partitioning is usually effective, with the possible exception of problems with a state space of high dimensionality (i.e., a large state descriptor set). For such problems, a different parallelization strategy is needed in order to achieve scalability of the implementation. Acknowledgements . We would like to thank Professors M. Gutknecht and W. Sch-onauer for their kind invitation to take part in this commemoration of our friend and colleague R-udiger Weiss. We are indebted to Tu-grul Dayar for providing the test matrices used in the numerical experiments and for useful information about these problems, as well as for his comments on an early version of the paper. Thanks also to Carl Meyer for his valuable input on generalized inverses. --R The arithmetic mean method for finding the stationary vector of Markov chains A sparse approximate inverse preconditioner for the conjugate gradient method A parallel block projection method of the Cimmino type for finite Markov chains A sparse approximate inverse preconditioner for nonsymmetric linear systems A comparative study of sparse approximate inverse preconditioners Approximate inverse preconditioning in the parallel solution of sparse eigenproblems Nonnegative Matrices in the Mathematical Sciences (Academic Press Distributed steady state analysis using Kronecker algebra Incomplete factorization of singular M-matrices Generalized Inverses of Linear Transformations (Pitman Publishing Ltd. Fast wavelet iterative solvers applied to the Neumann problem A priory sparsity patterns for parallel sparse approximate inverse preconditioners Comparison of partitioning techniques for two-level iterative solvers on large Parallel preconditioning with sparse approximate inverses Asynchronous iterations for the solution of Markov systems A fast and high quality multilevel scheme for partitioning irregular graphs Distributed disk-based solution techniques for large Markov models An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix The role of the group generalized inverse in the theory of finite Markov chains the theory of nearly reducible systems Matrix Analysis and Applied Linear Algebra (SIAM Experimental studies of parallel iterative solutions of Markov chains with block partitions Preconditioned Krylov subspace methods for the numerical solution of Markov chains Introduction to the Numerical Solution of Markov Chains (Princeton University Press BiCGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems --TR Incomplete factorization of singular M-matrices Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems Iterative solution methods A Sparse Approximate Inverse Preconditioner for the Conjugate Gradient Method Parallel Preconditioning with Sparse Approximate Inverses A Sparse Approximate Inverse Preconditioner for Nonsymmetric Linear Systems A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs A comparative study of sparse approximate inverse preconditioners Matrix analysis and applied linear algebra Comparison of Partitioning Techniques for Two-Level Iterative Solvers on Large, Sparse Markov Chains A Priori Sparsity Patterns for Parallel Sparse Approximate Inverse Preconditioners --CTR Aliakbar Montazer Haghighi , Dimitar P. Mishev, A parallel priority queueing system with finite buffers, Journal of Parallel and Distributed Computing, v.66 n.3, p.379-392, March 2006 Ilias G. Maglogiannis , Elias P. Zafiropoulos , Agapios N. Platis , George A. Gravvanis, Computing the success factors in consistent acquisition and recognition of objects in color digital images by explicit preconditioning, The Journal of Supercomputing, v.30 n.2, p.179-198, November 2004 Nicholas J. Dingle , Peter G. Harrison , William J. Knottenbelt, Uniformization and hypergraph partitioning for the distributed computation of response time densities in very large Markov models, Journal of Parallel and Distributed Computing, v.64 n.8, p.908-920, August 2004 Michele Benzi, Preconditioning techniques for large linear systems: a survey, Journal of Computational Physics, v.182 n.2, p.418-477, November 2002

singular matrices;generalized inverses;Bi-CGSTAB;graph partitioning;AINV;parallel preconditioning;discrete Markov chains;iterative methods

606859

On Two Applications of H-Differentiability to Optimization and Complementarity Problems.

In a recent paper, Gowda and Ravindran (Algebraic univalence theorems for nonsmooth functions, Research Report, Department of Mathematics and Statistics, University of Maryland, Baltimore, MD 21250, March 15, 1998) introduced the concepts of H-differentiability and H-differential for a function f : Rn Rn and showed that the Frchet derivative of a Frchet differentiable function, the Clarke generalized Jacobian of a locally Lipschitzian function, the Bouligand subdifferential of a semismooth function, and the C-differential of a C-differentiable function are particular instances of H-differentials.In this paper, we consider two applications of H-differentiability. In the first application, we derive a necessary optimality condition for a local minimum of an H-differentiable function. In the second application, we consider a nonlinear complementarity problem corresponding to an H-differentiable function f and show how, under appropriate conditions on an H-differential of f, minimizing a merit function corresponding to f leads to a solution of the nonlinear complementarity problem. These two applications were motivated by numerous studies carried out for C1, convex, locally Lipschitzian, and semismooth function by various researchers.

Introduction In a recent paper [10], Gowda and Ravindran introduced the concepts of the H- differentiability and H-differential for a function f They showed that Fr'echet differentiable (locally Lipschitzian, semismooth, C-differentiable) functions are H-differentiable (at given x) with an H-differential given by frf(x)g (respec- tively, @f(x), @B f(x), C-differential). In their paper, Gowda and Ravindran investigated the injectivity of an H-differentiable function based on conditions on H-differentials. Also, in [25], H-differentials were used to characterize P(P 0 )- functions. In this paper, we consider two applications of H-differentiability. In the first application, we derive a necessary optimality condition for a local minimum of an H-differentiable real valued function. Specifically, we show in Theorem 3 that if x is a local minimum of such a function f , then an H-differential of f at x . In the second application, we consider a nonlinear complementarity problem NCP(f) corresponding to an H-differentiable function f such that By considering an NCP function \Phi associated with NCP(f) so that x solves NCP(f); and the corresponding merit function in this paper (see Sections 6,7, and 8), we show how, under appropriate P 0 (P, regularity)-conditions on an H-differential of f , finding local/global minimum of \Psi (or a 'stationary point' of \Psi) leads to a solution of the given nonlinear complementarity problem. Our results unify/extend various similar results proved in the literature for C 1 , locally Lipschitzian, and semismooth functions [1], [5], [6], [7], [8], [9], [11], [12], [13], [14]. 2. Preliminaries We regard vectors in R n as column vectors. We denote the inner-product between two vectors x and y in R n by either x T y or hx; yi. Vector inequalities are interpreted componentwise. For a set E ' R n , co E denotes the convex hull of E and co E denotes the closure of co E. For a differentiable function f : R n denotes the Jacobian matrix of f at x. For a matrix A, A i denotes the ith row of A. A function OE is called an NCP function if OE(a; 0: For the problem NCP(f ), we define APPLICATIONS OF H-DIFFERENTIABILITY 3 (2) and, by abuse of language, call \Phi(x) an NCP function for NCP(f ). We now recall the following definition and examples from Gowda and Ravindran [10]. Definition 1. Given a function f where\Omega is an open set in R n and x 2 \Omega\Gamma we say that a nonempty subset T (x ) (also denoted by T f (x )) of R m\Thetan is an H-differential of f at x if for every sequence fx k g '\Omega converging to x ; there exist a subsequence fx k j g and a matrix A 2 T (x ) such that We say that f is H-differentiable at x if f has an H-differential at x . A useful equivalent definition of an H-differential T (x ) is: For any sequence x k := all k, there exist convergent subsequences lim Remarks As noted by a referee, it is easily seen that if a function f R m is H-differentiable at a point x, then there exist a constant L ? 0 and a neighbourhood B(x; ffi) of x with Conversely, if condition (4) holds, then T (x) := R m\Thetan can be taken as an H- differential of f at x. We thus have, in (4), an alternate description of H-differentiability. But, as we see in the sequel, it is the identification of an appropriate H-differential that becomes important and relevant. any function locally Lipschitzian at x will satisfy (4). For real valued func- tions, condition (4) is known as the 'calmness' of f at x. This concept has been well studied in the literature of nonsmooth analysis (see [24], Chapter 8). As noted in [10], (i) any superset of an H-differential is an H-differential, (ii) H-differentiability implies continuity, and (iii) H-differentials enjoy simple sum, product and chain rules. We include the following examples from [10]. differentiable at x 2 R n , then f is H- differentiable with frf(x )g as an H-differential. Example 2 Let f locally Lipschitzian at each point of an open Let\Omega f be the set of all points in\Omega where f is Fr'echet differentiable. For denote the Bouligand subdifferential of f at x . Then, the (Clarke) generalized Jacobian [2] is an H-differential of f at x . Example 3 Consider a locally Lipschitzian function f that is semismooth at x 2\Omega [17], [20], [22]. This means for any sequence x k ! x , and for any V k 2 @f(x k ); Then the Bouligand subdifferential is an H-differential of f at x . In particular, this holds if f is piecewise smooth, i.e., there exist continuously differentiable functions Example 4 C-differentiable in a neighborhood D of x . This means that there is a compact upper semicontinuous multivalued mapping n\Thetan satisfying the following condition at any a 2 D: For Then, f is H-differentiable at x with T (x ) as an H-differential. See [21] for further details on C-differentiability. We recall the definitions of P 0 and P-functions (matrices). Definition 2. For a function f : R n ! R n , we say that f is a P 0 (P)-function if, for any x 6= y in R n , A matrix M 2 R n\Thetan is said to be a P 0 (P)-matrix if the function is a P 0 (P)-function or equivalently, every principle minor of M is nonnegative (respectively, positive [3]). APPLICATIONS OF H-DIFFERENTIABILITY 5 We note that every monotone (strictly monotone) function is a P 0 (P)-function. The following result is from [18] and [25]. Theorem 1 Under each the following function. (a) f is Fr'echet differentiable on R n and for every x 2 R n , the Jacobian matrix rf(x) is a P 0 (P)-matrix. (b) f is locally Lipschitzian on R n and for every x 2 R n , the generalized Jacobian @f(x) consists of P 0 (P)-matrices. (c) f is semismooth on R n (in particular, piecewise affine or piecewise smooth) and for every x 2 R n , the Bouligand subdifferential @B f(x) consists of P 0 (P)- matrices. (d) f is H-differentiable on R n and for every x 2 R n , an H-differential T f (x) consists of P 0 (P)-matrices. 3. Necessary optimality conditions in H-differentiable optimization In this section, we derive necessary optimality conditions for optimization problems involving H-differentiable functions. We first consider the H-differentiability of minimum/maximum of several H-differentiable functions. Theorem 2 For be H-differentiable at x with an H-differential R be defined by where I(x is H-differentiable at x with T f as an H-differential. Also, a similar statement holds if 'min' in (6) is replaced by 'max'. Proof. We prove the result for the min-function; the proof of the max-function is similar. Consider a sequence fx k g converging to x in R n . Then there exist l 2 ng and a subsequence fx k j g such that f(x k j We have f(x (by the continuity of f l and f ). Now because of the H- differentiability of f l at x , there is a subsequence of fx k j g, which we continue to write as fx k j g for simplicity, and a matrix A l 2 T f l (x ) such that which leads to that f is H-differentiable at x with (defined in (7)) as an H-differential. This completes the proof. Remark In the above theorem, we considered real valued functions. With obvious modifications, one can consider vector valued functions. See Example 8 for an Theorem 3 Suppose f : R n ! R and x is a local optimal solution of the problem min If f is H-differentiable at x and T (x ) is any H-differential, then Proof. Suppose, if possible, that 0 62 co T (x closed and convex, by the strict separation theorem (see p.50, [15]), there exists a nonzero vector d in R n such that Ad ! 0 for all A 2 co T (x From the H-differentiability of f , for the sequence fx k dg, there exist a subsequence fx dg and Ad: Since f(x) f(x ) for all x near x , we see that Ad 0 reaching a contradiction. Hence Remarks When f is differentiable at x with T (x )g, the above optimality condition reduces to the familiar condition rf(x locally Lipschitzian at x, the above result reduces to Proposition 2.3.2 in [2] that see also, Theorem 7 in [17]. The above theorem motivates us to define a stationary point of the problem min f(x) as a point x such that is an H-differential of f at x . By weakening this condition, we may call a point x a quasi-stationary point (semi- stationary point) of the problem min f(x) if While local/global minimizers of min f(x) are stationary points, it is not clear how to get or describe semi- and quasi- stationary points. However, as we shall see in Sections 6, 7, and 8, they are used in formulating conditions for a point x to be a solution of a nonlinear complementarity problem. We now describe a necessary optimality condition for inequality constrained optimization problems. APPLICATIONS OF H-DIFFERENTIABILITY 7 Theorem 4 Suppose that f and g i are real valued functions defined on R n and x is a local optimal solution of the problem minimize f(x) subject to g i (x) 0 for Suppose that f and g i are H-differentiable at x with H-differentials respectively, by T f and I(x Proof. We see that x is a local optimal solution of the problem minimize f(x) subject to g(x) 0: (10) From Theorem 2, we see that g is H-differentiable with T g (x as an H-differential. We have to show that statement is false. Then by the strict separation theorem (see p.50, [15]), there exists a nonzero vector d in R n such that Ad ! 0 for all A 2 T f From the H-differentiability of f and g, for the sequence fx k dg, there exist a subsequence fx dg, matrices Ad and Bd: From we see that f(x We reach a contradiction since x is assumed to be locally optimal to the given problem. Thus we have the stated conclusion. 4. H-differentials of some NCP functions associated with H-differentiable functions In this section, we describe the H-differentials of some well known NCP functions. Example has an H-differential T (x) at x 2 R n . Consider the associated Fischer-Burmeister function [7] where all the operations are performed componentwise. Let Consider the set \Gamma of all quadruples (A; V; W; d) with A 2 T (x), are diagonal matrices satisfying the conditions and when i 62 J(x) when arbitrary when i 2 J(x) and d 2 when i 62 J(x) when arbitrary when i 2 J(x) and d 2 We now claim that \Phi F (or \Phi for simplicity) has an H-differential at x given by To see this claim, let x 1: By the H- differentiability of f , there exist a subsequence ft k j g of ft k g, d k j ! d, and A 2 T (x) such that f(x Let, for ease of notation, d k j . With A and d, define V and W satisfying (11) and (12); let We claim that \Phi(y j To see this, we fix an index i and show that \Phi i (y loss of generality, let 1. We consider two cases: Case In this case we have T is the first row of the identity matrix and APPLICATIONS OF H-DIFFERENTIABILITY 9 Case Subcase In this case, and an easy calculation shows In this case d These arguments prove that \Phi i (y holds for all i. Thus we have the H-differentiability of \Phi with S(x) as an H-differential. Remarks We observe that in the above example, if T (x) consists of P-matrices then S(x) consists of P-matrices. To see this, suppose that every A 2 T (x) is a P-matrix and consider any A is a P-matrix, there exists an index j with x j 6= 0 such that x j in (12) are nonnegative and their sum is positive, x j [Bx] It follows that B is a P-matrix. This observation together with Theorem 1 says that if T (x) consists of P-matrices then the function \Phi F is a P-function. (In fact, \Phi F is a P-function whenever f is a continuous P-function, see [23].) We note that S(x) may not consist of P-matrices if f is merely a P-function on R n . This can be seen by the following example. Let P-function and \Phi F . By a simple calculation, we see that the f2; 0g is an H-differential of \Phi F at zero and that it contains a singular object. Example 6 In the previous example, we described the H-differential of Fischer- Burmeister function. A similar analysis can be carried out for the NCP function [13] where is a fixed parameter in (0; 4). We note that when reduces to the Fischer-Burmeister function, while as ! 0; \Phi(x) becomes Let An H-differential of \Phi in (15) is given by is the set of all quadruples (A; V; W; d) with A 2 T (x), are diagonal matrices satisfying the conditions and when i 62 J(x) when arbitrary when i 2 J(x) and (d when i 62 J(x) when arbitrary when i 2 J(x) and (d Example 7 The following NCP function is called the penalized Fischer-Burmeister function [1] is a fixed parameter. Let For \Phi in (18), a straightforward calculation shows that an H-differential is given by is the set of all quadruples (A; V; W; d) with A 2 T (x), are diagonal matrices with when when arbitrary when i 2 J(x) and d 2 APPLICATIONS OF H-DIFFERENTIABILITY 11 when when arbitrary when i 2 J(x) and d 2 The above calculation relies on the observation that the following is an H- differential of the one variable function t 7! t + at any t: \Delta( Example 8 For an H-differentiable function f consider the NCP function We claim that the H-differential of \Phi is given by To see this claim, let x k ! x: By the H-differentiability of f , there exist a sub-sequence of fx k g, which we continue to write as fx k g for simplicity, and a matrix By considering a suitable subsequence, if necessary, we may write ng as a disjoint union of sets ff and fi where Put We show that \Phi(x k To see this, we fix an index j and show that \Phi j simplicity). We have two cases: Case \Theta \Phi(x k \Theta f(x k Case It is easy to verify that \Phi 1 This proves the above claim. 5. The H-differentiability of the merit function In this section, we consider an NCP function \Phi corresponding to NCP(f) and let Theorem 5 Suppose \Phi is H-differentiable at x with S(x) as an H-differential. Then \Psi := 1jj\Phijj 2 is H-differentiable at x with an H-differential given by Proof. Consider a sequence fx there exist d We have ff \Gamma2 h\Phi(x); \Phi(x)i ff \Gamma2 This gives us lim This completes the proof. 6. Minimizing the merit function under P 0 -conditions For a given function f consider the associated NCP function \Phi and the corresponding merit function It should be recalled that x solves NCP(f): One very popular method of finding zeros of \Phi is to find the local/global minimum points or 'stationary' points of \Psi. Various researchers have shown, under certain that when f is continuously differentiable or more generally locally Lipschitzian, 'stationary' points of \Psi are the zeros of \Psi. In what follows, starting with an H-differentiable function f , we show that under appropriate conditions, a vector x is a solution of the NCP(f) if and only if zero belongs to one of the sets Theorem 6 Suppose f : R n ! R n is H-differentiable at x with an H-differential is an NCP function of f: Assume that \Psi := 1jj\Phijj 2 is H-differentiable at x with an H-differential given by APPLICATIONS OF H-DIFFERENTIABILITY 13 Further suppose that T (x) consists of P 0 -matrices. Then Proof. Clearly, implies that T \Psi by the description of T \Psi (x): Conversely, suppose that 0 2 T \Psi (x), so that for some \Phi(x) T [V A +W yielding A T y Note that for any index which case y i 0, contradicting the P 0 -property of A. We conclude that In the next two successive theorems, we replace the condition conditions relaxations come at the expense of imposing either stronger or different conditions on the H-differential of f . First we recall a definition from [26]. Definition 3. Consider a nonempty set C in R n\Thetan . We say that a matrix A is a row representative of C if for each index row of A is the ith row of some matrix C 2 C. We say that C has the row-P 0 -property (row-P- property) if every row representative of C is a P 0 -matrix (P-matrix). We say that C has the column-P 0 -property (column-P-property) if C Cg has the We have the following result from [26]. Proposition 1 A set C has the row-P 0 -property (row-P-property) if and only if for each nonzero x in R n there is an index i such that x i for all C 2 C. A simple consequence of this proposition is the following. Corollary 1 The following statements hold: (i) Suppose the set of matrices fA has the row-P 0 -property. Then for any collection fV of nonnegative diagonal matrices, the sum A is a P 0 -matrix. In particular, any convex combination of the A i s is a P 0 -matrix. (ii) Suppose the set of matrices fA has the row-P-property. Then for any collection fY of nonnegative diagonal matrices with A is a P-matrix. Proof. (i) Let x 6= 0 in R n . By the above proposition, there exists an index i such that x i 6= 0 and x i This proves the P 0 -property of A . By specializing we get the additional statement. (ii) Let x 6= 0: By Proposition 1, there exists an index i such that x i 6= 0 and Now we have x i the terms of the above sum are nonnegative. If (Z which means that we see that x i is a P-matrix. Remark We note that the implications in the above corollary can be reversed: if every A in (i) ((ii)) is a P 0 -matrix (respectively, P-matrix), then fA has the row-P 0 -property (respectively, row-P-property). Peng [19] proves results similar to Corollary 1 under additional/different hypotheses. Theorem 7 Suppose is H-differentiable at x with an H-differential T (x). Suppose that \Psi is H-differentiable at x with an H-differential given by Further suppose that T (x) has the row-P 0 -property. Then Proof. Suppose and we have co T \Psi versely, suppose 0 2 co T \Psi (x). Then by Carath'eodory's theorem [15], there exist that where Lg. We rewrite (22) as APPLICATIONS OF H-DIFFERENTIABILITY 15 reduces to where diagonal matrix and jZ i the equality unchanged if we replace Y i by jY i j and Z i by jZ i j, we may assume that Y i and Z i are nonnegative for all i. Now suppose, if possible, that By the above corollary, the matrices M and M T are P 0 -matrices. Therefore, there exists an index i such that 0. From \Phi(x) i 6= 0, we see that (W j and so (Z 0: But (\GammaZ u) i is clearly a contradiction since u i This proves that Remarks We note that Theorems 6 and 7 are applicable to the Fischer-Burmeister function This is because, the set T \Psi (x) described in Theorems 6 and 7 is a superset of the H-differential T \Psi described in Example 5. (Note that [\Phi F (x)] i J(x) and hence from (12), v Similarly, we see that Theorems 6 and 7 are applicable to the following NCP functions: (Clarification Example (Clarification Example 7) We state the next result for the Fischer-Burmeister function \Phi. However, as in Theorems 6 and 7, it is possible to state a very general result for any NCP function \Phi. For simplicity, we avoid dealing in such a generality. Theorem is H-differentiable at x with an H-differential T (x) which is compact and having the row-P 0 -property. Let \Phi be the Fischer- Burmeister function as in Example 5 and \Psi := 1jj\Phijj 2 . Let S(x) and T \Psi (x) be as in Example 5 and Theorem 5. Then the following are equivalent: (a) x is a local minimizer of \Psi. (c) x solves NCP(f). Proof. The implication (a) ) (b) follows from Theorem 3. The implication (c) ) (a) is obvious. We now prove that (b) ) (c). Suppose 0 2 co T \Psi (x) and assume that there exists a sequence fC k g of matrices in co S(x) such that Now each C k is a convex combination of at most matrices of the form V A+W 2 S(x) where A 2 T (x), V and W satisfy (11) and (12). Since T (x) is compact and the entries of V and W vary over bounded sets in R, we may assume that C k ! C where C is a convex combination of at most matrices of the form V A +W where A 2 T (x), V and W are nonnegative diagonal matrices satisfying a condition like (11) with and when g. From an equation similar to (22) but now with V i , A i , and W i in place of V i , A i , and respectively. By repeating the argument given in the proof of the previous theorem, we arrive at a contradiction. Hence proving (b) ) (c) . We now state two consequences of the above theorems for the Fischer-Burmeister function (for the sake of simplicity). differentiable and \Phi(x) be the Fischer-Burmeister function and x is a local minimizer to \Psi if and only if x solves NCP(f). This corollary is seen from the above theorem by taking T frf(x)g. If we assume the continuous differentiability of f in the above corollary, we get a result of Facchinei and Soares [5]: For a continuously differentiable P 0 -function f , every stationary point of \Psi solves NCP(f ). (This is because, when f is C 1 , \Psi becomes continuously differentiable, see Prop. 3.4 in [5].) See [9] for the monotone case. locally Lipschitzian. Let \Phi be the Fischer- Burmeister function and the equivalence holds under each of the following conditions. (a) @f(x) consists of P 0 -matrices; (b) @B f(x) has the row-P 0 -property. Proof. The stated equivalence under (a) has already been established by Fischer [8]. In fact, by applying Theorem 6 with T f using his result that @\Psi(x) ' T \Psi (x) for all x, we get the equivalence in (a). Now to see the equivalence under (b), assume (b) holds. Then by Corollary 1, every matrix in Now we have condition (a) and hence the stated equivalence. Remark The condition (b) in the above corollary might be especially useful when the function f is piecewise smooth in which case @B f(x) consists of a finite number of matrices. APPLICATIONS OF H-DIFFERENTIABILITY 17 7. Minimizing the merit function under P -conditions The following theorem is similar to Theorem 6. Theorem 9 Suppose f : R n is H-differentiable at x with an H-differential is an NCP function of f: Assume that \Psi := 1jj\Phijj 2 is H-differentiable at x with an H-differential given by Further suppose that T (x) consists of P-matrices. Then Proof. Suppose 0: Then by description of T \Psi (x); we have T \Psi Conversely, suppose that 0 2 T \Psi (x), so that for some \Phi(x) T [V A +W We claim that 0: Suppose, if possible, that \Phi(x) 6= 0: If which leads to a contradiction since for some Hence y 6= 0 and contradicting the P-property of A. Hence Theorem is H-differentiable at x with an H-differential T (x). Suppose that \Psi is H-differentiable at x with an H-differential given by Further suppose that T (x) has the row-P-property. Then Proof. The proof is similar to that of Theorem 7. To show that 0 2 co T \Psi (x) ) proceed as in the proof of Theorem 7. We have statements (22) and (23) in our new setting where we may assume (as before) that Y i and Z i are nonnegative for all i. Since taking we see that the matrix in (23) is nonsingular. It follows that Remark We note that Theorems 9 and 10 are applicable to the min-function \Phi of Example 8. 8. Minimizing the merit function under regularity (strict regularity) conditions We now generalize the concept of a regular (strictly regular) point [14] in order to weaken the hypotheses in the Theorems 6 and 7. For a given H-differentiable function f and we define the following subsets of I = ng. Definition 4. Consider f , x, and the index sets as above. Let T (x) be an H- differential of f at x. Then the vector x 2 R n is called a regular (strictly regular) point of f with respect to T (x) if for every nonzero vector z 2 R n such that z there exists a vector s 2 R n such that Theorem is H-differentiable at x with an H-differential \Phi be an NCP function satisfying the following conditions: Suppose \Psi is H-differentiable with an H-differential given by x is a regular point if and only if x solves NCP(f). Proof. Suppose that 0 2 T \Psi (x) and x is a regular point. Then for some APPLICATIONS OF H-DIFFERENTIABILITY 19 We claim that 0: Assume the contrary that x is not a solution of NCP(f ). x is a regular point, and y and z have the same sign, by taking a vector s 2 R n satisfying (25) and (26), we have and contradict (30). Hence x is a solution to NCP(f ). The 'if' part of the theorem follows easily from the definitions. Remark Theorem 11 is applicable to the NCP functions of Examples 5, 6 and 7. A slight modification of the above theorem leads to the following result. Theorem is H-differentiable at x with an H-differential \Phi be an NCP function satisfying the following conditions: Suppose \Psi is H-differentiable with an H-differential given by x is a strictly regular point if and only if x solves NCP(f). Proof. The proof is similar to that of Theorem 11. Concluding Remarks In this paper, we considered two applications of H-differentiability. The first application dealt with the necessary optimality condition in H-differentiable op- timization. In the second application, for a nonlinear complementarity problem corresponding to an H-differentiable function, with an associated NCP function \Phi and a merit function described conditions under which every global/local minimum or a stationary point of \Psi is a solution of NCP(f ). We would like to note here that similar methodologies can be carried out for other merit functions. For example, we can consider the Implicit Lagrangian function of Mangasarian and Solodov [16]: \Theta fixed parameter and x y is the Hadamard (=componentwise) product of vectors x and y. (In [16], it is shown that By defining the merit function and formulating the concept of strictly regular point, we can extend the results of [4] for H-differentiable functions. Our results recover/extend various well known results stated for continuously differentiable (locally Lipschitzian, semismooth, C-differentiable) functions. Acknowledgements We thank the referees for their constructive comments. --R "A Penalized Fischer-Burmeister NCP-Function: Theoretical Investigation and Numerical Results," Optimization and Nonsmooth Analysis The Linear Complementarity Problem "On Unconstrained and Constrained Stationary Points of the Implicit Lagrangian," "A New Merit Function for Nonlinear Complementarity Problems and Related Algorithm," "Regularity Properties of a Semismooth Reformulation of Variational Inequalities," "A Special Newton-Type Optimization Method," "Solution of Monotone Complementarity Problems with Locally Lipschitzian Functions," "On the Resolution of Monotone Complementarity Problems," "Algebraic Univalence Theorems for Nonsmooth Functions," "A New Nonsmooth Equations Approach to Nonlinear Complementarity Problems," "Unconstrained Minimization Approaches to Nonlinear Complementarity Prob- lems," "A New Class of Semismooth Newton-Type Methods for Nonlinear Complementarity Problems," "A Semismooth Equation Approach to the Solution of Nonlinear Complementarity Problems," "Nonlinear Complementarity as Unconstrained and Constrained Minimization," "Semismooth and Semiconvex Functions in Constrained Optimization," "On P- and S- Functions and Related Classes of N-Dimensional Nonlinear Mappings," "A Smoothing Function and its Applications," "Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations," "C-differentiability, C-differential Operators and Generalized Newton Methods," "Regularization of P0 -functions in Box Variational Inequality Problems," Variational Analysis "On Characterizations of P- and P 0 - Properties in Nonsmooth Functions," "On Some Properties of P-matrix Sets," --TR --CTR M. A. Tawhid, On the local uniqueness of solutions of variational inequalities under H-differentiability, Journal of Optimization Theory and Applications, v.113 n.1, p.149-164, April 2002 Jong-Shi Pang , Defeng Sun , Jie Sun, Semismooth homeomorphisms and strong stability of semidefinite and Lorentz complementarity problems, Mathematics of Operations Research, v.28 n.1, p.39-63, February

merit function;nonlinear complementarity problem;locally Lipschitzian function;NCP function;h-differentiability;generalized Jacobian

606884

Robust Optimal Service Analysis of Single-Server Re-Entrant Queues.

We generalize the analysis of J.A. Ball, M.V. Day, and P. Kachroo (Mathematics of Control, Signals, and Systems, vol. 12, pp. 307345, 1999) to a fluid model of a single server re-entrant queue. The approach is to solve the Hamilton-Jacobi-Isaacs equation associated with optimal robust control of the system. The method of staged characteristics is generalized from Ball et al. (1999) to construct the solution explicitly. Formulas are developed allowing explicit calculations for the Skorokhod problem involved in the system equations. Such formulas are particularly important for numerical verification of conditions on the boundary of the nonnegative orthant. The optimal control (server) strategy is shown to be of linear-index type. Dai-type stability properties are discussed. A modification of the model in which new customers are allowed only at a specified entry queue is considered in 2 dimensions. The same optimal strategy is found in that case as well.

Figure 1. Re-entrant Server There is much current interest in developing optimal service strategies for queueing systems. The volume by Kelly and Williams [9]includes several articles addressing this. Although queueing models are generally integer-valued and stochastic, Dai [4]and others have developed connections between the stability of Date: November 30, 2001. Research supported by the ASPIRES program of Research and Graduate Studies, and the Millennium program of the College of Artsand Sciences, both of Virginia Tech. stochastic queuing systems and their deterministic fluid limits. Thus optimal strategies for fluid models are recognized as significant for stochastic models. Fluid models for a large class of queueing systems can be described by equations of the general form (5) introduced in Section 2 below. We pursue the same robust control approach as in [2]for such models. Much of what we present here is a further development of ideas from that paper. In particular, Section 2.1 gives explicit representations of the velocity projection map (x, v) of the Skorokhod reflection mechanism which comes into play when one or more queues are empty. Section 3 develops the construction of the value function for our control problem. Here, as in [2], the construction proceeds without regard to the Skorokhod dynamics on the boundary of the nonnegative orthant. (In more general multiple server examples the Skorokhod dynamics will play a more decisive role.) Even so, the solution we construct must be shown to satisfy various inequalities associated with optimality with respect to the Skorokhod dynamics on the boundary. We do not provide a deductive proof of these inequalities, but rely instead on a system of numerical confirmation for individual test cases in Section 4. The explicit representations of (x, v) are important for this, and for the optimality argument of Section 5. The version of that argument given here improves on the one in [2]in that it applies to all admissible strategies, not just those of state feedback form. Our model allows new arrivals and unserved departures in the form of an exogenous load qi(t)foreach xi; see (1) below. In some queueing applications this feature would be inappropriate. For instance in typical re-entrant lines, new arrivals only occur at a specified entry queue and departures only as service is completed at a designated final queue. In Section 6 we will look at the the 2-dimensional case of our model under the more restrictive assumption that exogenous arrivals are only allowed in the entry queue x1. This requires a number changes in our calculations. But we find that this change to the model does not eect the resulting optimal service policy. 2. The Model and Approach of Optimal Control We describe in this section the general model formulation and performance criteria that we will use. Fluid models for a large class of queueing systems can be described by equations of the nominal form (1) x(t)=q(t) Gu(t). The state variable is n-dimensional: x =(x1,.,xn) Rn. For queueing models x(t) must remain in the nonnegative orthant, K in (4) below. For that purpose we will couple (1) with Skorokhod problem dynamics, resulting in (5) below. The term q(t)istheload on the system due to new arrivals (or unserved departures if qi(t) < 0). The service allocation is specified by the control function u(t) whose values are taken from a finite set U0 Rm of possible service control settings. For purposes of an adequate existence theory for solutions to (5) we relax this to allow u(t) to be taken from the convex hull (2) U =convU0. For our single-server examples we willsimply take the standard unit vectors in Rn.Thus 1. The matrix G converts u(t) to the appropriate vector of contributions to x. For the case to be considered here (Figure 1) G will be the lower triangular matrix . . sn1 0 The si > 0 are parameters which specify the service rates for the respective queues. Thus when u(t)=ek (k<n), the eect of Gu(t) in (1) is to drain queue xk at rate sk with the served customers entering xk+1 at the same rate: Multiple server examples are easily modeled by (1) and (5) as well. Consider for example the 2-server re-entrant line in Figure 2. It would be natural to use The first two columns in G correspond to the service allocation at server A and the second two at server B. The u U0 correspond to the 4 dierent combinations in which server A chooses between x1 or x3 and server B chooses between x2 or x4. x3 Figure 2. 2-Server re-entrant line 2.1. Skorokhod Problem Dynamics. We denote by K the nonnegative orthant of Rn: The faces of K are The interior normal to iK is the standard unit vector We will use to denote the set of all coordinate indices. For x K, I(x)={i will denote the set of indices with zero coordinate values. An essential feature of queueing models is that x(t) remains in K for all t. One could simply impose this as a constraint the control functions u(t) and loads q(t) which are considered admissible. Although some constraints on the load are reasonable, we find it much more natural in general to couple (1) with the dynamics of a Skorokhod problem. On each face iK we specify a constraint vector di. If the solution of (1) attempts to exit K through iK, then the idea is to add some positive multiple of di to the right side of (1) to prevent the exit. A precise formulation is the following: given x(0) K and q(t),u(t) (which we will assume to be locally integrable), let Problem is to find a continuous function x(t) K, a measurable function r(t) Rn and a nondecreasing function (t) 0 which satisfy the following for t 0: (0,t] for each t, r(t)= iI(x(t)) idi for some i 0; (t)= (0,t] 1x(s)K d(s). By imposing a normalization there will be a unique solution, provided K and di satisfy certain conditions. Dupuis and Ishii [5]and Dupuis and Ramanan [7]provide a substantial body of theory of Skorokhod problems in general. In particular they show, using a velocity projection map (x, v), that a Skorokhod problem can be expressed as a dierential system. The velocity projection map is of the form for an appropriate choice of i 0. (See (6) below.) The result of coupling our (1) with the appropriate Skorokhod problem is expressed as holding almost surely. The appropriate constraint vectors di are determined by the structure of the system in Figure 1. If x iK and server i is active (ui > 0) but the applied service rate siui exceeds the inflow qi to xi, then according to (1) x would exit K through iK. In an actual network the system could not really use the full service capacity siui allocated to xi. Instead, service would take place at a lower level which exactly balances the inflow and outflow of queue xi. Mathematically this is achieved by adding a positive multiple of the column Gei of G to the right side of the system (1), bringing xi to 0 and producing the correct reduction of the throughput xi xi+1 to the next queue. So we take di = s1 Gei (the normalization being so that 1). The same prescription is appropriate for the example of Figure 2: take di to be the unique column of G having a positive entry in row i, normalized so that ni di =1. At this point we wish to highlight the fact that no restrictions on u(t)andq(t) are needed to keep x(t) in the nonnegative orthant; (5) will determine a state trajectory with x(t) K regardless. Thus we always instruct the server to work at full capacity ( and the Skorokhod dynamics can be viewed as automatically reducing the service rates to the levels that can actually be implemented. The model allows a separate load term qi(t) for each queue. For re-entrant queues, one typically would only want to allow for the entry queue in each re-entrant sequence. In Figure 1 for instance it would be natural to assume nice feature of single servers with respect to the L2 performance criteria of Section 2.4 is that the the optimal strategy is the same for all loads, regardless of which coordinates might be zero. In queueing applications one also naturally assumes that qi 0. However it was argued in [2]that, for purposes of vehicular trac for instance, it is reasonable to consider qi < 0. This would correspond to customers that leave the system without waiting to receive service all the way through. This is a reasonable consideration in some applications. However it is hard to conceive of a realistic interpretation for qi < 0 when xi =0. Even so, (5) will still yield a mathematical solution. The eect of the additional +idi terms in (x, q Gu) might then be seen not as reductions to the service rates but as a transference of the reducing influence of qi < 0 from the empty xi to the queues xj further along in sequence; fluid at xj would be drawn backwards through the system to satisfy to external demand due to qi < 0atpreviousxi. With di defined, we face the important technical issue of existence and regularity properties of the Skorokhod problem. This issue is treated in detail in [5]and [7.] Those treatments consider a more general convex polyhedron in place of our K. Our particular choice of the nonnegative orthant falls within the scope of the earlier work [12]. Let D =[di]be the matrix with the constraint vectors as columns. In our case, I Q where Q is the subdiagonal matrix with entries Assuming that Q has nonnegative entries and spectral radius less than 1, both clearly satisfied for us, [12] provided a direct construction of the solution of the Skorokhod problem. In [7]it is shown that these conditions from [12]fall within the scope of a more general set of sucient conditions for existence and Lipschitz continuity of the Skorokhod map y() x(). Drawing on the ideas of [12], we can give a direct construction of the (x, v) appearing in the dierential formulation (5) of the Skorokhod problem. For any x K and v Rn, we will show that be characterized using a linear complementarity problem: subject to the following constraints for each i I(x): For i/I(x) there is no constraint on wi, and we consider to be implicit. Let =[i]n .Using I Q, and rewriting (6) as it is easy to see that the complementarity problem is equivalent to saying is a fixed point =x()of the defined coordinate-wise by The notation y+ refers to the usual positive part: fixed point representation is a particular case of the general fixed point representation of variational inequalities in Chapter 1 of [10].) We first observe the existence of a unique fixed point. The argument of [12]is to observe that (after a linear change of variables) is a contraction, under the nonnegativity and spectral radius assumption mentioned above. However this is even simpler for our particular Q; =x() reduces to which determine the i sequentially. Iteration of x from any initial will converge to the fixed point after at most n steps. This makes it particularly simple to see that v and thus v (x, v) are continuous, and to evaluate (x, v) numerically. Indeed (x, v) is Lipschitz in v for a fixed x, and is jointly continuous in (x, v)ifx is restricted so that I(x) is constant. We can easily check that (x, v) as defined by the above complementarity problem is indeed the velocity projection map as identified in [5]. First, following an observation of [3], we can check that (y)=(0,y)is the discrete projection map of Assumption 3.1 of [5]. Indeed, if y K then clearly the complementarity problem: (0,y)=y.Fory/K, y (0,y)=y which is of the form for some 0and d(w), where d(x) is the set of reflection directions as defined in [5, 3]. Next, suppose x K, v Rn, and let w,solve the complementarity problem for above. We claim that suciently small. This will imply that which is the characterization of (x, v)in[5,5.3]. Since w,solve (6), it follows that di.We want to see that satify the complementarity conditions (7) - associated with First consider i I(x). Since xi =0,wehave For the prodiuct (9) we have Next consider i/I(x). Provided h>0 is suciently small we have which also confirms the product condition. This verifies (11), as desired. For purposes of our calculations, the characterizations of (x, v) in Lemma 1 below will be useful. If J N we will use NJ =[nj]jJ and DJ =[dj]jJ to denote the matrices whose columns are the normal vectors nj and constraint directions dj for the j J. Given v, and the corresponding i as described above, let From the complementarity problem we know F0 L I(x). Using any F0 F L the values of i, i F are determined by setting wi =0,i F in (6). In other words we can solve for F =[i]iF directly in and consequently where RF is the reflection matrix For simply take More precisely, The fact that wi 0fori/F is equivalent to Suppose that we dont know F0 or L at the outset, but just take an arbitrary F I(x), calculate v. By construction wi =0fori F and i =0fori/F. So item 3 of the complementarity problem is satisfied. If item 1 is satisfied of i F, which is to say and item 2 holds for i I(x) \ F, which is to say then we can say that is in fact (x, v). This discussion proves the following lemma. Lemma 1. Given x K, v Rn,andF I(x), the following are equivalent: 1. (x, v)=RF v; 2. both of the following hold: (a) BF v 0 when 3. for some L with F L I(x) all of the following hold: (a) BF v>0 when (b) NL\F RF v =0when (c) NIT(x)\LRF v>0 when Notice that the strict inequality in 3 (a) simply identifies F as 2.2. The Optimal Control Policy. Our goal is to design a feedback control strategy (x), prescribing a value in the extended control set U for each x K, so that using u(t)=(x(t)) produces optimal performance of the system. The criteria used to determine optimality is based on where >0 is a parameter. Roughly speaking, the control should keep the integrated cost (14) small, so that x(t) remains small compared to the load q(t) in a time-averaged sense. We will give this a more precise formulation in Section 2.4 below. The running cost 1 x2 2 q2 of (14) has it roots in classical H control, and is attractive for its broad familiarity and success in a wide range of control applications. Other choices might be more appropriate for particular queueing applications, such as those associated with optimal draining and time-to-empty criteria; see [18]and [1.] There are however considerations that favor L2 in the trac setting. For a given total customer population, the L2 norm favors balanced queue lengths over a situation in which some queues are empty and others are full. When each customer is a person who has to wait in a queue, a cost structure that can be minimized by using excessive waits for some small class of customers would be considered unacceptable. The optimal policy itself is easy to identify by naive considerations at this point. In order to minimize for a given q(t) one would minimize 1 x(t)2, for which one would naturally try to choose u(t)tominimize d 1 In the interior of K, where (1) applies, this suggests that the optimal u is that which maximizes x Gu over u U.OnK x is given by (5), which makes finding the u to minimize potentially more dicult. However if we assume that all qi 0 then the Skorokhod dynamics do not aect the minimizing u. To see this consider x K with suppose u Umaximizes xGu. It is easy to see that sup x Gu > 0. Observe that for any i I(x), since o-diagonal entries of G are nonpositive, x Gei 0. Thus the i-coordinate of u must be 0, from which we can conclude that ni Gu 0. Since qi 0 by hypothesis, we see that ni (q Gu) 0, for all i I(x). This means that (x, q Gu)=q Gu. Also, for any i I(x)wehavex di 0 because the i coordinate of di is positive. Therefore, for any u Uwe can say that Thus the policy U is an obvious candidate for the optimal service policy. We will see below that it is indeed optimal in the sense to be made precise in Section 2.4. Several comments should be made at this point. First observe that (x) is set-valued. There are inherent discontinuities in (x), as the optimum u jumps among the extreme points of U. When we replace u(t) by (x(t)) in (5), we will want the resulting feedback system to have good existence properties. This is addressed using the Filippov theory of dierential inclusions. For that it is important that (x) have closed graph and be convex set-valued. (The notion of closed graph is often called lower semi-continuity for set-valued functions.) It is easy to see that (15) has these properties. Another important point to make is that the naive reasoning which suggests (x(t)) to us does not actually imply that it achieves the smallest possible value of 1 x(T)2 for a given q() and target time T.Itis conceivable that it might be better to forgo pointwise minimization of d 1 x(t)2 in order to drive x(t) into a dierent region (or section of the boundary) where larger reductions of x(t) could be achieved. Some sort of dynamic programming argument, such as the Hamilton-Jacobi equation developed in Section 3, is needed to adequately address such global optimality issues. Although it may not have much practical import, one might ask whether allowing qi < 0 when xi =0 might aect the choice of u Uwhich minimizes x(x, q Gu). Indeed it can. If qi < 0 is large enough its eect through the Skorokhod dynamics can produce cases in which a u/(x) minimizes x (x, q Gu). This consideration would lead to an enhanced optimal policy which agrees with (x) on the interior of K, but depends on both q and x when x K. Although no longer state-feedback, this enhanced control would (we expect) produce lower values of (14), but only for those negative loads q(t) which, as we described above, have the eect of drawing customers backwards through the system. Even so, this enhanced control would not improve the performance of the system in the worst case sense of the dierential game formulated in Section 2.4, as Theorem 1 below will assert. 2.3. Minimum Performance Criteria. Our strategy is expressed in state feedback form. Given a load q(), the associated control function u(t) would be what results from solving the system This system is a combination of a dierential inclusion, in the sense of Filippov, and a Skorokhod problem as described above. The discussion in [2, Section 1.4]outlined how the arguments of [6]can be adapted to establish the existence of a solution. A proof of uniqueness is more elusive. The usual Filippov uniqueness condition would be that for some L (xa xb) [(q Gua) (q Gub)]= ( xa xb) (Gub Gua) Lxa xb2. This is immediate (using since by definition of (), xa Gub xb Gua and xb Gua xb Gub for all ua (xa), ub (xb). However, as noted in [2], when coupled with Skorokhod dynamics (16) we are unable to conclude uniqueness based on existing results in the literature. Until that issue can be addressed, we must allow the possibility of multiple solutions to (16). The uniqueness question is not essential to our main result Theorem 1, however. We simply need to formulate its statement in such a way that strategies are allowed to produce more than one control function u(t) for a given load q(t). In general a service strategy () maps a pair x(0),q() to one or more control functions u(). We will write u(t)=[x(0),q()](t), although this notation is not quite proper if there are actually more than one u() associated with x(0),q()by. Rather than formulating a cumbersome notation to accommodate this, we will simply use phrases like for any u(t)=[x(0),q()](t) to refer to all possible u(t). A strategy should produce one or more control functions for any x(0) K and load function q() which is locally square-integrable. We insist that a strategy be nonanticipating, in the sense that if q(s)=q(s) for all s t, then for any u(t)=[x(0),q()](s) there is a u(t)=[x(0), q()](s) with u(s)=u(s) for all s t.Given any such x(0), q() and a resulting u(t), the general existence and uniqueness properties of the Skorokhod problem (e.g. [5]) provide a unique state trajectory x(t) K. We will call a strategy () non-idling if for any nonnegative load qi(t) 0 for all i and all t 0, any any u(t)=[x(0),q()](t), the resulting state trajectory x(t) has the property that ui(t) > 0 and occur simultaneously for some i only if In other words, all service eort is allocated to nonempty queues, unless all queues are empty. In particular, our strategy (15) is non-idling, because if x K and is the index of the largest nonzero coordinate of x. One of the features of single servers as in Figure 1 is that for nonnegative loads, a non-idling strategy will never invoke the Skorokhod dynamics on K, until it reaches Indeed if x(t) K \{0} but the non-idling property means that 0, from which the structure of G implies that ni Gu(t) 0. Since qi(t) 0, we conclude that ni (q(t) Gu(t)) 0. Thus unless Multiple servers do not have this property. In the case of Figure 2 for instance, if both then the service eort at B is wasted and the Skorokohd dynamics will definitely come into play, regardless of x1 and x3. The Skorokhod dynamics will thus have a stronger influence on the design of optimal strategies for multiple server models. When considering those fluid models that arise as limits of discrete/stochastic queueing systems, the stability criterion of Dai [4]is important for purposes of positive recurrence of the stochastic model. In that setting the load q(t) is typically constant, with 1/qi equal to the mean time between new arrivals in queue xi. The stability property of [4]is simply that for any x(0) K, the state x(t) reaches at a finite T 0. For our single server model, all non-idling strategies are equivalent in this respect. To see why, consider the vector Observe that T G =(1, 1,. ,1), so that for all u Uwe have 1. For any nonnegative load q(t) and the u(t) resulting form any non-idling strategy, we have (on any interval prior to the first time T when d x(t)= (x(t),q(t) Gu(t)) dt Said another way, W(x)= x is a sort of universal Lyapunov function for all non-idling controls. Thus, the first time T for which does not depend on the choice of non-idling control; it only depends on the load q(t). For constant nonnegative loads q(t) q, the Dai stability property simply boils down to Moreover if q =(q1, 0,. ,0)T then this reduces to the familiar load condition of [4, (1.9)]: q1 < 1. Figure 3 illustrates this stability property. We have taken the optimal strategy (x) for our model with subjected the system to the constant disturbance q(t) (.4, 0). For these parameters we find =(2, 1) and q so that the load condition (18) is indeed satisfied. The figure illustrates the resulting trajectories of (16). When x(t) reaches the ray from the origin in the direction of , the solution of (16) in the Fillipov sense uses the averaged control value which takes x(t) to the origin in finite time directly along the ray. One may check that this ray consists of those x for which (x)=U is multiple-valued. Theorem 1 below considers the optimality of with respect to all strategies that satisfy the following minimum performance criterion: given x(0) K with x(0) < 1, there exists <1 so that whenever q() is a nonnegative load satisfying q(t) 1 for all t, and any u(t)=[x(0),q()](t) the resulting state trajectory satisfies x(t) <for all t 0. It is clear from our discussion above that every non-idling control satisfies the minimum performance criterion; simply take 2.4. The Robust Control Problem. We now want to define more carefully the sense in which our service strategy (x) is optimal. We follow the general approach of Soravia [14]to formulate a dierential game based on (14). The focus is on a value function of the form dt. Here x K, the outer infimum is over strategies (), the inner supremum is over locally square integrable loads q(), all u(t)=[x, q()](t) and bounded time intervals [0,T], with x(t) the resulting solution of (1) for x1 Figure 3. Controlled Trajectories for q(t) (.4, 0). The gain parameter >0 is customary in robust control formulations. However for the structure of our problems scales out of the game (19) in a natural way. To see this, consider a particular load q(t), control u(t), and solution x(t) of (5). Make the change of variables Then x(t)= d x(s), and because K is a cone, (x, q Gu)=(x, q Gu). Thus x(s) solves (5) on the new ds time scale. With ds. If V ()=V1() is the value (19) for then the above implies that V(x)=3V (1x). From this point forward we simply take instead of V. We can only expect V (x) < to hold in a bounded region. To see why, imagine a load q(t) which is large on some initial interval 0 t s so as to drive the state out to a large value X, and then q(t)ischosenfor t>sso as to maintain x(t)=X for t>s: q(t)=Gu(t). If X > supuU Gu, the integral in (19) grows without bound as T , producing infinite value. We must exclude such scenarios from the definition of the game. It turns out that the region in which V (x) will be finite is described using the vector of (17) above: We restrict the T in (19) to those for which x(t) remains in for all 0 t T. This qualification on the state in turn requires us to place some limitations on the strategies () considered as well. We need to exclude controls that cheat by encouraging x(t) to run quickly to the outer boundary of to force an early truncation of the integral in (19). Such controls could achieve an artificially low value by having actually destabilized the system. To exclude such policies we insist that all control strategies () satisfy the minimum performance criterion stated at the end of Section 2.2. With these qualifications, we can now state precisely the optimality properties of the feedback strategy (x). Theorem 1. Let and (x) be as defined above and suppose the boundary verifications of Section 4 have been successfully completed. Using the control (x),forx ,define where the supremum is over all loads q(t), all resulting control functions u(t)=[x, q()](t), and those <T< such that the controlled state from other control strategy () satisfying the minimum performance criterion, with the same qualifications on the supremum. The proof of these assertions will be discussed in Section 5 below. The qualification regarding the boundary verifications of Section 4 will be explained in the last paragraph before Section 3.1. 3. Construction of the Value Function by Staged Characteristics The proof in Section 5 of Theorem 1 is based on showing that the function V (x) of (21) solves the Hamilton-Jocobi-Issacs equation associated with the game (19): The Hamiltonian function is complicated by the special reflection eects on K: The essential property of our strategy (x) for the proof is that, given x point for the supq infU defining H(x, DV (x)) in (24) is given by any u (x). To be specific, the minimum value of over u Uis 0, achieved at and the maximum value of over q Rn is 0, achieved at q. Together these imply (23). Our primary task is to produce V (x)and establish this property of . In general (23) must be considered in the viscosity sense. Lions [13]considers the viscosity-sense formulation of a general class of problems involving Skorokhod dynamics on K. Instead of working with H as in (24), the viscosity sense solutions are described using only the interior form of the Hamiltonian (27), together with special viscosity sense boundary conditions on K. In our case it will turn out that the solution V is actually a classical one. We find the direct formulation in terms of H more natural for our development. We will construct the desired solution V (x) by working in the interior K, where the complicating eects of (x, v) are not present: (x, v)=v so =infHu(x, p). uU Here Hu refers to the the individual Hamiltonian for u U: The supremum is achieved for Also observe that for u Uto achieve the infimum in (27) means simply that u maximizes p Gu.Soforx K, (23) and the saddle point conditions (25) and (26) simply reduce to the statement that for any u (x), uU We turn now to the construction of V (x) of by a generalized method of characteristics. We cover with a family of paths x(t) as described below. The idea is that at a point the gradient of DV (x(t)) should be given by the costate trajectory p(t) that accompanies x(t). Thus a simple covering of by a family of such paths will determine the values of DV (x) in . Knowing V in the region. We itemize the essential features of this family of x(t),p(t) in (30)-(33), and then explain their relation to the Hamilton-Jacobi-Isaacs equation and saddle point property above. To begin, the paths must solve the system of ODEs for some piecewise constant u(t) U. The value of u(t) may change from one time interval to another, but at each time t we require the optimality condition uU Given an initial condition (depending on x(0)) at which both x and p reach the origin: Lastly 0 t<T we require Observe that (30) is the Hamiltonian system p). This is intimately connected with the propertyt that p(t)=DV (x(t)) for a solution of Hu (x, DV We will return to this issue, near the end of Section 3.2 explaining why the manifold of (x, p) formed by our solution family is truly the graph of a gradient also that for 0 t T we have The formula (28) for Hu shows that (34) is indeed satisfied at (32). It is a general property of Hamiltonian systems as in (30) that the value of Hu (x(t),p(t)) is constant with respect to t. Property (31) implies that the jumps in u(t) do not produce discontinuities with respect to t in (34). Therefore (34) follows as a consequence of (30) - (32). Thus (34) and (31) give us (29) for u(t) in particular. The construction of x(t),p(t) in Section 3.2 will show that u(t) (x(t)) and that extends to all u (x(t)). This will provide the saddle point conditions (29) on the interior. The equation H(x, DV has many solutions, if it has any at all. One property of the particular solution we want is that V be associated with the stable manifold of (30), in accord with the general approach of van der Schaft [15, 16, 17]to robust nonlinear control. We see this in the convergence to the origin of above. Another important property is that To this end, notice that the formula (28) for an individual Hamiltonian, together with (34), impliess that So for p(t)=DV (x(t)), (33) is the same as saying d dt If we stipulate that V which is (35). One may wonder why we have insisted on (32). Observe that (33) (in the limit as t 0) implies that necessary if A family of x(t),p(t) as described above will give us a function V (x) which has the desired saddle point properties at interior points. However for x K both (25) and (26) are complicated by the nontrivial structure of (x, v). We claim the V (x) so constructed does in fact satisfy the saddle point conditions (25) and (26) at x K as well. We do not give a mathematical proof of this. Instead we have developed a scheme of numerical confirmation that can be applied to test this claim for any specification of si. This is described in Section 4. We also note the requirement in (32) above that x(t) for all 0 t<T, given . This follows if we can verify that whenever x(t) iK then ni (p Gu) 0. We rely on numerical tests for this fact as well. (See the discussion of (52) in Section 4.) Based on the success of these tests for numerous examples, we conjecture that (25) and (26) are true in general. The reference to the boundary verifications of Section 4 in Theorem 1 indicates that the validity of that result depends on the success of those tests. 3.1. Identification and Properties of the Invariant Control Vectors. We will construct the family x(t),p(t) as above by generalizing the development of [2]. The key is to look for solutions that approach the origin as in (32) using a constant control u. The solution of (30) with constant conditions x(T)=0=p(T)is Observe that for 0 t T /2 the values of both sin(t T) and 1 cos(t T) will be positive. Now consider what (31) requires of (38): uU There are only a finite number of such they provide the key to the explicit representation of the family of solutions x(t),p(t) that we desire. We will call any GU satisfying (39) an invariant control vector. To simplify our discussion here, let denote the columns of G. (In more general models, gi would be the extreme points of GU.) To say GU means that is a convex combination of the gi: 1. For an as in consider the set of indicies It follows from (39) that every j J achieves the maximum value of gi over i N. Therefore Our construction of V (x) depends on the fact that there is a unique such associated with every nonempty subset J N. The existence of J depends on properties of our particular set of gi, but the uniqueness does not. So we present the uniqueness argument separately as the following lemma. Lemma 2. Suppose gi, i =1,. ,m are nonzero vectors in Rn and J {1,. ,m} is nonempty. If there exists a vector J as described in (40) then it is unique. Suppose J and J exist for both J J.Then Proof. We establish (41) first. Without assuming uniqueness, suppose exists as in (40). It follows that Now suppose both J and J exist for J J. Then the same reasoning implies that which is (41). Regarding uniqueness, suppose for the same J. In that case (41) implies But then (42) implies gj J. This means is orthogonal to the span of {gj, But since it is also in the span, we are forced to conclude that =. It is not dicult to determine whether or not J exists for a given J. If it does, the values must be a nonnegative solution of the linear system From such a nonnegative solution we can recover j from check (40). With this observation we can prove that J exists for all J N in our single-server model. Theorem 2. Assume the specific G and U0 of our model (see Section 2). For every nonempty J N there exists a unique invariant control vector J . Moreover the j, j J in (40) are strictly positive. Proof. Let GJ =[gj]jJ be the matrix whose columns are the gj for just those j J. Observe that (43) simply says J =[j]jJ must solve GTJ GJ J =1J . For the existence of nonnegative j in (43) it is enough to show that GTJ GJ is invertible and that all entries of its inverse are nonnegative. Consider the diagonal matrix and let Note that MJ is nothing but GJ for the particular case of all it is enough to show that (MJT MJ )1 exists and has nonnegative entries. Now observe that MJT MJ is block diagonal . Ak1 0 where the A and B are tridiagonal of the form 121 . 121 . One may check by explicit calculation that denoting the size of A, c +1 Since all entries are positive in both cases, it follows that all entries of (MJT MJ )1 and hence (GTJ GJ )1 are nonnegative, as desired. Since no rows are identically 0, all i are positive in (43) and therefore the respective j > 0 can always be found. Next, we need to show that gj J >gi J for i/J, j J. First observe that gj constant over j J. Also note that for We know that j > 0. So if i/J, Therefore, gi J 0 for every i/J,andgj J >gi J . Lemma 2 gives the uniqueness. Observe that of (17) is a scalar multiple of N . Indeed, in the notation of the above proof, It follows that = N /N N . In particular the of (20) is alternately described as Fundamental to our construction is the existence and uniqueness of the following representation of x using a nested sequence of invariant control vectors. Theorem 3. Assume the specific G and U0 of our model. Every nonzero x has a unique representation of the form for some 0 <aj, aj < 1 and J1 . Jk N. Moreover, Proof. Consider any nonzero x . We first solve G. The reader can check that G1 has all nonnegative entries, which implies that all i 0. Therefore every x K can be written as Next, let and consider the invariant control vector Now consider (j aj)gj Our choice of a implies for all j J. However for one or more j J,j aj =0. By induction on the number of positive coecients in (47) it is possible to write J. Simply taking am = a and completes the induction argument. Next notice that since Ji 2. From the hypothesis that x we conclude that aj < 1. Now consider J1 in (45) and any j, j J1. Then j, j Ji for all i, which from (40) tells us that However if j J1 but j / J1 then while for i>1, gj Ji gj Ji (depending on whether j Ji or not). We conclude that gj x>gj x. This proves that J1 is the set of j for which gj x takes its largest possible value, as claimed. Regarding uniqueness, since G is nonsingular the in are uniquely determined, and then Jk from the last term of (45) is necessarily the J above. Since Jk1 Jk, k1 (j akj)gj still must have nonnegative coecients j akj. But only those for j Jk1 can be positive. Hence J. This implies that ak = a as well. Thus Jk and ak are uniquely determined. Repeating the argument on k1 ajJjgives the uniqueness of the other aj,Jj . The following lemma records two other facts that will be used below. Lemma 3. Assuming the G and U0 of our model, a) All coordinates of N are positive. b) If x K is as in Theorem 3, and if xi =0, then i/J1. Proof. We have already observed that is as in (17). This proves a). Next suppose It follows that x gi 0. On the other hand, there does exist j with x gj > 0 (take j to be largest with xj > 0 for instance). Thus the set J1 of those j with does not include i. It is important to realize that the single server re-entrant queue being studied here is special in that a unique J is defined for every J N. This is not the case for more multiserver systems. For example, consider the re-entrant line with two servers in Figure 2. The gi are Gui, ui U0 as in (3). Simple test calculations reveal many J for which no J exists. Moreover it turns out that the gi are linearly dependent so the points representable as in (45) can account for at most a 3-dimensional subset of K. 3.2. Construction of the Characteristic Family. We can now exhibit the desired family of solutions to (30). Consider a nested sequence J1 J2 Jk, and parameters coecient functions ai(t)andi(t)acordingtotheformulas (48) . . . (Here again we use y+ to denote the positive part: k. For all 0 t T we have so that all ai(t)andi(t) are nonnegative. We claim that provide a solution of of (30)-(33). The derivation of (48) is based on the calculations in [2]. Here we will simply present as direct a calculation as possible. To that end, consider the partial sums appearing on the left in (48): Consider t in one of the intervals 1 <t<. Then for i<we have so that For Taking pairwise dierences we see that Using this in (49), we find that for 1 <t< Thus (30) is satisfied for 1 <t< using To confirm property (31) on that interval, observe that since j(t)=0forj<, we know from (46) that p(t) Gu is maximized over u Uat any u for which only the j J coordinates are positive, in particular for Implicit in this construction is a function defined by means of (49). Starting with x , express x as in (45). Then determine p(x)using iJiwhere the partial sums of the i are determined from those of the ai according to for 0 <i /2. This is the gradient map p(x)=DV (x) of our solution to (23). There are several facts to record about p(x) before proceeding. Theorem 4. The map p(x) described above is locally Lipschitz continuous in and satisfies the strict inequality for all x , x =0. To see this, first consider what we will call a maximal sequence J1 J2 . Jn, i.e. and each Ji has precisely i elements, with N. Consider the x representable as in (45) for this particular maximal sequence. The maps x ai ai and i i p are linear. The maps ai i are simply These are Lipschitz so long as ai remains bounded below 1. Since ai n in , we see that p(x) is indeed Lipschitz in any compact subset of , constrained to those x associated with a fixed maximal sequence of Ji. If in (45) we relax the positivity assumption to ai 0, then we can include additional Ji so that every x is associated with one or more maximal sequence of Ji. The Lipschitz continuity argument extends to the x associated with any given maximal sequence in this way. To finish the continuity assertions of the theorem we need to consider x on the boundary between the regions associated with distinct maximal sequences: The uniqueness assertion of Theorem 3 means that the nonzero terms of both representations agree, which implies that the corresponding terms of the expressions for p also agree: Thus p(x) is continuous across the boundaries of the regions associated with dierent maximal sequences. From this it follows that the (local) Lipschitz continuity assertion of the theorem is valid in all of . The argument that p(x)2 < x2 is the same as [2, pg. 334, 335]. The strict inequality comes from the fact that Equality occurs only for n 1). Notice that for x , x sin(n) < 1 implies that p(x) The argument given in [2]that p(x) is indeed the gradient of a function V (x), x also generalizes to the present context. In brief, the standard reasoning from the method of characteristics can be applied to each of the individual Hamiltonians Hu where to see that DV (x)=p(x) in the region associated with a given maximal sequence Ji. Continuity across the boundaries between such regions allows us to conclude that there is indeed a C1 function V in with DV (x)=p(x). Taking V implies that V (x) > by virtue of the discussion of (35) above. Finally, we return to the connection of (31) with (x) and (29). Since U is convex, and so (x) consists precisely of those u in the convex hull of ej, j J1, J1 being as in (45) for x. Because in p(x)= iJi the i are positive when the corresponding ai are positive, we see that (x) has the alternate description U In particular, the u(t) of (31) belongs to (x(t)). Moreover p(x)Gu has the same value for all u (x), so as desired in (29). 4. Verification of Conditions on the Boundary We have completed the construction of V (x) satisfying (29) on the interior of . We now consider the assertion of that for x K the resulting V remains a solution when the H of (27) is replaced by H as in (27), and that any u (x) is a saddle point, as in (25) and (26). Specifically, we want to confirm that for a given x K, its associated any u (x), the following hold: Since we know Hu (x, imply (25), and (52) and (54) imply (26). Together these imply (23). Our validation of (52) - (54) consists of extensive numerical testing, as opposed to a deductive proof. We will describe computational procedures below. Test calculations have been performed on numerous examples (see Section 4.4), confirming (52) - (54) to within machine precision in each case. This gives us confidence in the theoretical validity of (52) - (54), but until deductive arguments can be presented, their theoretical validity must be considered conjectural. 4.1. Inactive Projection. Given x , the corresponding any u (x), (52) is equivalent to the statement that ni (p Gu) 0 for all i I(x). This would be easy to check by direct calculation at a given x. However the second part of the following lemma provides an equivalent condition which is even easier to check. Lemma 4. The following are equivalent 1. ni (p(x) Gu) 0 for all x K with x =0,alu (x),andi I(x); 2. p(x)i 0 for all x K and i I(x); 3. p(x)i > 0 for all i and all x =0in the interior of . Proof. Clearly (2) follows from (3) by continuity of p(x). To see that (2) implies (1), recall from our discussion in Section 2.2 of the fact that is a nonidling policy that that ni Gu 0 for any u (x). Therefore (2) implies ni (p Gu) pi 0. Finally, observe that (1) implies that the characteristic curves (49) do not exit K in forward time. From any x(0) in the interior of , x(t) remains in K up to the time T at which it follows that pi(x) > 0 for all i in x is in the interior of . 4.2. Control Optimality. Now we consider an approach to checking (53) at a given x K with its associated any u (x). We want to check that u is the minimizer of over u U. Observe that by virtue of (52) Since pGu has the same value for all u (x) it suces to consider any single u (x) and to show that it gives the minimum of p(x, pGu)overu U. Since this is a continuous function of u and U is compact, we know that there does exist a minimizing u. Moreover for some F I(x), p(s, pGu)=pRF (pGu), according to (13). So given F we can identify u as a maximizer of p RF Gu subject to the constraints of Lemma 1 part 2). If (53) were false then an exception u would occur as a solution of such a constrained minimization problem, for some F I(x). There is no exception to (53) for because in that case solves a standard linear programming problem: subject to u U BF (p Gu) 0, and If u is an exception to (53) then so is any feasible maximizer uF to (55): To verify (53) computationally we invoke a standard linear programming algorithm for (55) for each nonempty subset F of I(x), and for each feasible maximizer so found, check that We note that when I(x)={i} is a singleton we only need to check itself. In this case it is sucient to check that directly for each of j =1,. ,n. To see why, first observe that the last constraint in (55) is satisfied vacuously. Since BF (p Gu) > 0, the same must hold for all u Usuciently close to u. It follows that u gives a local maximum of p RF Gu over U. Since U is convex, it must be a global maximum. Therefore u must be a convex combination of those ej for which p RF observe that since the constraint BF (p Gu)=ni (p Gu) > 0 is a scalar constraint. It must therefore be satisfied by one of the ej for which p RF This means that this ej also solves (55). Hence when I(x) is a singleton it suces to check just the ej as candidates for u, rather than invoking the linear programming algorithm. 4.3. Load Optimality. Once (52) is confirmed we know that for any u (x), (x, q Gu)=q Gu and that with respect to those q for which (x, q Gu)=q Gu, and that the maximal value is 0. To verify (54) we need to be sure that there are not some other u (x) and q with (x, q Gu) =q Gu and for which Since (x, v) is continuous and piecewise linear, and since (x) is a compact set, it follows that there does exist a u (x) and q which maximizes (56) over q Rn and u (x). We derive necessary conditions contingent on the specification of the subset F I(x) for which (x, q Gu)=RF (q Gu). Using part 3 of Lemma 1 we know that u =uand q =qsatisfy Consider the ane set of all q satisfying (58). Since the inequalities are strict in (57), all q near q and satisfying (58) must also have (x, q Gu)=RF (q Gu). Thus q =qis a local maximum of(59) p RF (q Gu) q2, subject to the constraint NLT\F RF (q Gu)=0.A simple calculation shows that this implies where PL\F is the orthogonal projection onto the kernel of NLT\F the constraint (58) is vacuous and we take Substituting this back into (59) and considering the result as a function of u,it follows that u =uis a local (and hence global by convexity) solution of the quadratic programming problem: subject to u (x), BF PL\F (RFT p Gu) 0, and To verify (54) computationally, we consider all pairs of subsets F L I(x). For each, we invoke a standard quadratic programming algorithm to find a feasible maximizer u, if any exists. If such a u is found, we take and then check by direct calculation whether this is an exception to (54), as in (56). If we consider all F L I(x) but find no such exceptions, then (54) is confirmed for this x, p. Again we note that the quadratic programming calculation can be skipped in some cases. If then (x, q Gu)=q Gu and we know there are no such exceptions to (54). Thus only need be considered. Secondly, suppose I(x)={i} is a singleton. Then the only case to check is In that case if there is an exception to (54), q must maximize and satisfy It follows that But for the latter inequality simplifies to ni (p Gu) <di p. Moreover since di = ni ni+1 (with this is equivalent to But i I(x) means i/J1,soni Gu 0 for all u (x). So (61) would imply pj < 0 for some j.If we have already checked that p 0 in accord with Lemma 4 and our confirmation of (52), then we can be sure no exceptions to (54) occur when I(x) is a singleton. Thus we only need to appeal to the quadratic programming calculations when two or more xi are zero. 4.4. Test Cases. We begin our test of (30)-(33) for a specific choice of parameters s1,. ,sn by calculating all the invariant control vectors J . Then on each face iK a rectangular grid of points x iK with x N N N is constructed. For each grid point x we then compute the representation (45) and then the associated according to (51). We then check that all pi 0 in accord with Lemma 4 and carry out the constrained optimization calculations described above for all possible F L I(x). Obviously, the amount of computation involved will be prohibitive if the number of dimensions n is significant. However, for modest n the calculations can be completed in a reasonable amount of time. We have carried out these computations for numerous examples, including the following: (s1,. ,sn)=(1, 1, 1) No exceptions to (52)-(54) were found. 5. Proof of Optimality: Theorem 1 We turn now to the proof of the optimality assertions of Theorem 1. By hypothesis V (x) is as constructed in Section 3.2, the saddle point conditions (25) and (26) have been confirmed, as well as the equivalent conditions of Lemma 4. We know that () satisfies the minimum performance criterion of Section 2.4. As explained above, V (x) > 0 for all x with load q(). The argument of [2, Theorem 2.1]shows that with respect to (), on any interval [0,T]on which x(t) remains in , we have dt. For a given x(0) , let x(t),p(t) be the particular path constructed according to (30), with x(T)=0. We know that u(t) (x(t)) so x(t) is the controlled path produced by in response to the load q(t). Along it we have from (36) that and therefore dt. This establishes (21). Next we consider an arbitrary strategy satisfying the minimum performance criterion. We would like to produce a load q(t) which is related to the resulting state trajectory x(t)byq(t)=DV (x(t)). In [2, Theorem 2.3]this was accomplished by limiting to state-feedback strategies and appealing to an existence result for Filippov solutions of the dierential inclusion [2, (2.23)]. Here we only approximate such a load. By taking advantage of the properties of (x, ), our argument will not be limited to state-feedback strategies, and will not need the Filippov existence result. Given x we will show that for any =>0 there exists a load q(t) satisfying qi(t) 0andq(t) 1 for all t>0, and such that (for some u(t)=[x, q()](t)) holds for all T. The dicult question of existence for the closed loop system for an arbitrary strategy is easily resolved by introducing a small time lag: The system can now be solved incrementally on a sequence of time intervals [tn1,tn]where For t [tn1,tn]the values of q(t) are determined by x(t) on the previous interval [tn2,tn1], so the basic existence properties of the system under subject to a prescribed q(t) insure the existence of x(t)andq(t) as above. Let u(t)=[x, q()](t) be the associated control function. Since q(t) is always a value of DV (x) at some x , we know qi(t) 0andq(t) 1, and the minimum performance hypothesis insures that x(t) remains in a compact subset of : x(t) , <1. We must explain how the time lag leads to the += term in (62). Observe that because (x, v)=RF v for one of only a finite number of possible matrices RF , and because there is a uniform upper bound on x: Consequently, x(t) x(t =et)B=et. Next, on the subset of x with x , DV (x) is Lipschitz; see Theorem 4. It follows that for some constant C1 (independent of =) such that We observed previously that (x, v) in Lipschitz in v. It follows that for some constant C2 and all =, t > 0 We know that it follows that integrating both sides over [0,T]and replacing = by =/C2 yields (62). With this q(t) in hand the remainder of the argument proceeds as in [2]: if there exists a sequence with x(Tn) 0, then V (x(Tn)) 0 in (62) which implies dt. Suppose no such sequence exists. Then in addition to x(t) <1 we know x(t) does not approach 0; it must remain in a compact subset M of \{0}. From (63), dt. WealsoknowfromTheorem4that1 x2 1 DV (x(t))2 has a positive lower bound. Therefore the right side in (64) is infinite. Thus (64) holds in either case. Since =>0 was arbitrary, (22) follows. 6. An Example with Restricted Entry In this section we reconsider our model in modified so that the exogenous load only applied to queue x1. This is illustrated in Figure 4. The system equations are now where q(t) is a scalar and ,while the control matrix the control values u Uand the constraint directions di all remain as before. We carry out the same general approach to constructing V (x) as outlined at the beginning of Section 3. The details of the analysis are dierent in several regards. This is significant because it shows that our general approach is not exclusive to all the structural features of Sections 3.1 and 3.2. We will find that the optimal policy is the same (x)as given in (15) above. In higher dimensions (n>2) it is interesting to speculate whether the optimal policy would likewise remain unchanged if we removed the exogenous loads qi(t), i>1 . However, at present this has only been explored in 2 dimensions. Figure 4. Re-entrant Loop with Single Input Queue The presence of M in (65) changes the individual Hamiltonian: p1 being the first coordinate of p =(p1,p2). The supremum is achieved for p1. The corresponding Hamiltonian system, for a given u U,is We calculate the invariant control vector as described in Section 3.1. Other than there are no additional J to consider. To simplify notation we will drop the subscript N: The first place we find a significant dierence from our previous analysis is in the calculation of a solution to the Hamiltonian system associated with , analogous to (38). Previously, we did this using being as determined by the construction of because of the missing p2 term in the x2 equation of (66), we must use a Gu(t) which is both dierent from and time dependent. We seek a solution x(t)=a(t),p(t)=(t) (both a(t)and(t) nonnegative) to for some function 0 1. The overbar on x, p distinguishes this special solution from the others encountered below. In light of the p equation and the terminal conditions (32), the solution we seek must be of the form for some function (t) 0. is a scalar multiple of , the right side of the x equation in must also be a scalar multiple of . Since implies a relationship between (t)and(t), which works out to be Using this we can reduce (67) to a single second order dierential equation for (t): (t)+A(t)=1, where A is the constant The solution (for initial conditions It will be convenient for the rest of this discussion to fix so that (One consequence of fixing is that for a given x the t<0 for which x(t)=x depends on x.) For t 0 we confirm that 0 (t) 1, (t) 0anda(t)= (t) 0, as we wished. We now have the desired solution: This special solution provides the final stage of our family of paths x(t),p(t) as in (30)-(33) of Section 3, but with some adjustment. In contrast to Section 3, our u(t)=[(t), 1(t)]T varies continuously, instead of being piecewise constant. This means we have to pay closer attention to (34). Once again it is satisfied at by virtue of the terminal conditions. When we calculate dt Hu(t)(x(t), p(t)), one term does not automatically drop out: d dt Since p is a scalar multiple of and we know g2, we do indeed find that Hu(t)(x(t), p(t)) 0. The analogue of (33) for this example is This is because q2 x2 =(p1(t))2 < x(t)2 when q(t)=p(t). Also note that taking advantage of the fact that Hu (x, to verify (33) along x, p in particular, simply observe that since both and are positive (excepting In the following xi(t), pi(t) will refer to the individual coordinates of this particular solution. Our special solution x(t), p(t) provides the final stage (t1 <t 0) for each of the solutions in the family described at the beginning of Section 3. The initial stage (t<t1 < 0) will be a solution of (66), with either e1 or e2, which joins x, p at some t1 < 0: x(t1)=x(t1), p(t1)=p(t1). In other words we solve (66) backwards from x(t1), p(t1),for the appropriate choice of u. It turns out that using produces that part of the family which covers a region below the line x() in the first quadrant, and using the x(t) which cover a region above x(). This is illustrated in Figure 5, for parameter values s1 =4,s2 =1. Note that the region covered by this family, and hence the domain of V (x), is no longer the simple polygon of (20).1.410.60.20 Figure 5. Characteristics for Restricted Entry Loop We will need to verify that the resulting family indeed satisfies all the conditions outlined in Section 3. These verifications are discussed below. Once confirmed, this implies that the optimal control (x) produces e1 if x is below the line x(t), e2 if x is above the line, and any u Uif x is on the line. So although we will not produce as explicit a construction for x p as we did in Section 3, we still find the same optimal control uU 6.1. Interior Verifications. We have already discussed properties (30)-(33) of Section 3 for the final stage of our family of solutions: x(t)=x(t), p(t)=p(t)fort1 t 0. However we still need to verify (31) and (68) for the initial segment t<t1. In Section 3.2 this followed from properties of the J and the rather explicit formulae for x(t)andp(t) in terms of them. Here we have not developed such an elaborate general structure. Instead we resort to direct evaluation of the needed inequalities. By solving (66) for x(t1)=x(t1), p(t1)=p(t1) we obtain the formulas for the lower half of our family: for t<t1 < 0, For any <t1 < 0, the above will be valid for t<t1 down to the first time at which x(1)(t) either reaches the horizontal axis, or reaches the outer boundary of , curves appearing in the figure, b(t1) <1(t1).) A formula for 1() is easily obtained from the expressions in (69). The value can be identified as the point at which the determinant of the Jacobian of x(1) with respect to vanishes. An explicit formula is possible for b() as well. (For brevity we omit both formulas.) Thus is valid for The points on the horizontal boundary 2K are x(1)(1(t1)) for those t1 with b(t1) 1(t1). The analogous formulas for the upper half of our family are obtained by solving (66) for x(t1)=x(t1), p(t1)=p(t1) to obtain the following expression for t<t1 < 0: This time, for a given <t1 < 0, the valid range of t<t1 is slightly dierent. It turns out that x(2)(t) always reaches the outer boundary of , at a time prior to contacting the vertical boundary 1K. (Once again, an explicit formula for b() is obtained by setting the Jacobian of (70) equal to 0.) Thus given t1 < 0, (70) is valid for The vertical boundary itself is traced out by the solution for t1 =0: x (t)= valid for b(0) <t 0. The availability of these formulas makes it possible to check the inequalities we need for (31) and (33). For (31) we want to verify For (73), it turns out that ),which certainly is positive for t<t1. We resort to numerical calculation to confirm (72). We have already noted that (33) should be replaced by (68): x(i)(t)2 (p(i)(t))2 > 0,for both i =1, 2. It is a straightforward task to prepare a short computer program that, given values for s1,s2, evaluates (72) and (68) for a large number of t<t1 pairs extending through the full range of possibilities. In this way we have confirmed the above inequalities numerically. 6.2. The Horizontal Boundary. Finally we must consider the influence of the projection dynamics at points x K, confirming as we did in Section 4 that our remains a saddle point when (x, v) is taken into account. This entails checking the same three facts, (52), (54), and (53) as before. We consider the two faces of K separately. The reflection matrix for 2K is and (x, v)= . Now observe that which is independent of q. Since it follows that (x, Mq Gu)= Mq Gu, so that (52) reduces to (72). Moreover this independence of q also implies (54) since we know is the saddle point in the absence of projection dynamics. Next consider (53). The u Uare just Note that corresponds to = 1. So for (53) we want to show that the minimum of occurs at little algebra shows that for 0 s1s+2s2 we have so that For s1s+2s2 1wehaven2 (Mp1 Gu) 0, so that Thus the function of in (75) is piecewise linear, in two segments. The slope of the right segment ( s2 1) is which we already know to be negative, by virtue of our work in checking (72). Thus to establish (53) we only need to check that the value for greater than that for =0: which is equivalent to p g1 0. This we have confirmed numerically, by evaluating for various choices of s1,s2 and t1 throughout its range. 6.3. The Vertical Boundary. Recall that along 1K we have and that from (71) we know Thus (x, Mq Gu)=(Mq Gu), confirming (52). The reflection matrix on 1K is We already know that over those q for which (x, Mq g2)=Mq g2. We need to consider the possibility of a global maximum among those q with (x, Mq g2)=R{1}(Mq g2), namely q with n1 (Mq g2)=q 0. However, is maximized at its maximum over q 0 must occur at confirms (54). Finally, we turn to (53). Since any u as in (74) we have Therefore, after a little algebra, which is minimized at corresponds to verifies (53). --R An Introduction to Variational Inequalities and their Applications Queueing Systems: Theory and Applications Reflected Brownian motion on an orthant Neumann type boundary conditions for Hamilton-Jacobi equations Nonlinear state space H1 control theory On optimal draining of re-entrant fluid lines Department of Mathematics --TR

skorokhod problem;queueing;robust control

606888

On an Augmented Lagrangian SQP Method for a Class of Optimal Control Problems in Banach Spaces.

An augmented Lagrangian SQP method is discussed for a class of nonlinear optimal control problems in Banach spaces with constraints on the control. The convergence of the method is investigated by its equivalence with the generalized Newton method for the optimality system of the augmented optimal control problem. The method is shown to be quadratically convergent, if the optimality system of the standard non-augmented SQP method is strongly regular in the sense of Robinson. This result is applied to a test problem for the heat equation with Stefan-Boltzmann boundary condition. The numerical tests confirm the theoretical results.

Introduction We consider an Augmented Lagrangian SQP method (ALSQP method) for the following class of optimal control problems, which includes some meaningful applications to control problems for semilinear partial dierential equations: Minimize subject to y In this setting Y and U are real Banach spaces, are dierentiable mappings, and U ad is a nonempty, closed, convex and bounded subset of U . The operator is a continuous linear operator from Y to U . In general, (P) is a non-convex problem. We will refer to u as the control, and to y as the state. In the past years, the application of ALSQP methods to optimal control or identication problems for partial dierential equations has made considerable progress. The list of contributions to this eld has already become rather extensive so that we shall mention only the papers by Bergounioux and Kunish [6], Ito and Kunisch [13], [14], Kaumann [15], Kunisch and Volkwein [16], and Volkwein [25], [26]. Supported by SFB 393 "Numerical Simulation on Massive Parallel Computers". 2 Parts of this work were done when the third author was visiting professor at the Universite Paul Sabatier in Toulouse. In this paper, we extend the analysis of the ALSQP method to a Banach space setting. This generalization is needed, if, for instance, the nonlinearities of the problem cannot be well dened in Hilbert spaces. In our application, this will concern the nonlinear mapping . A natural consequence of this extension is that, in contrast to the literature about the ALSQP method, we have to deal with the well known two-norm discrepancy. Another novelty in our approach is the presence of the control constraints u 2 U ad in (P) , which complicates the discussion of the method. To resolve the associated diculties, we rely on known results on the convergence of the generalized Newton method for generalized equations. One of the main goals of this paper is to reduce the convergence analysis to one main assumption, which has to be checked for the particular applications { the strong regularity of the optimality system. In this way, we hope to have shown a general way to perform the convergence analysis of the ALSQP method. For (P) we concentrate on a particular type of augmentation, applied only to the nonlinearity of the state equation. Splitting up the state equation into y and z augment only the second equation. This type of augmentation is useful for our application to parabolic boundary control problems. The convergence analysis is conrmed by numerical tests, which are compared with those performed for the (non-augmented) SQP method. We obtain the following main results: If the optimality system of rst order necessary optimality conditions for (P) is strongly regular in the sense of Robinson, then the ALSQP method will be locally quadratic convergent under natural assumptions. This result is applied to a boundary control problem for a semilinear parabolic equation. In [23], the convergence of the (non-augmented) SQP method was shown for this particular problem by verifying this strong regularity assumption. In this way, our result is immediately applicable to obtain the convergence of the augmented method in our application. The paper is organized as follows: In Section 2 we x the general assumptions and formulate rst order necessary and second order sucient optimality conditions. Section 3 contains our example, a semilinear parabolic control problem. The ALSQP method is presented in Section 4, where we show that its iterates are well dened in the associated Banach spaces. The convergence analysis is developed in Section 5 on the basis of the Newton method for generalized equations. The last part of our paper reports on our numerical tests with the ALSQP method. General assumptions and optimality conditions We rst x the assumptions on the spaces and mappings. The Banach spaces Y and U mentioned in the introduction stand for the ones where the following holds: f is a mapping of class C 2 from Y U into R, is a mapping of class C 2 from Y into U . For several reasons, among them, the formulation of the SQP method and the sucient second order optimality conditions, we have to introduce real Hilbert spaces Y 2 and U 2 such that Y (respectively U) is continuously and densely imbedded in Y 2 (respectively U 2 ). Moreover, we identify U 2 with its dual U . Therefore, denoting by U the dual space of U , we have the continuous imbeddings Let us introduce the product space endowed with the norm jjvjj jjujj U , and the space endowed with the norm jjvjj Notations: We shall denote the rst and second order derivatives of f and by Partial derivatives are indicated by associated subscripts such as f y (v), f yu (v), etc. Notice that, by their very denition, f 0 (v) 2 V , U)). The open ball in V centered at v, with radius r is denoted by B V (v; r). The same notation is used in other Banach spaces. We will denote the duality pairing between U and U (resp. Y and Y ) by reserved in this paper for the scalar product of U 2 . Below we list our main assumptions: (A1) is a linear, continuous, and bijective operator from Y 2 to U 2 . Moreover, its restriction to Y , still denoted by , is continuous and bijective from Y to U . In addition, we assume that U ad is closed in U 2 . (Extension properties) For all r > 0 there is a constant c(r) > 0 such that, for all for all v 2 V; (2.1) for all From (2.1) it follows that f 0 (v) can be considered as a continuous linear operator from V 2 to R, and 0 (y) can be considered as a continuous linear operator from Y 2 to U 2 . Since 00 (y belongs to U , and U U , the term k 00 (y ful. Moreover, f 00 (v) (respectively 00 (y)) can be considered as a continuous bilinear operator from V 2 (respectively U ). In the second order derivatives we shall write [v; there is a c(r) > 0 such that for all z 2: (Remainder terms) Let r F denote the i-th order remainder term for the Taylor expansion of a mapping F at the point x in the direction h. Following Ioe [11] and Maurer [18] we assume kr For all y 2 Y , the operator is bijective from Y 2 to U 2 . Its restriction to Y , still denoted by is bijective from Y to U . For all belongs to b Y , where b Y is a Banach space continuously imbedded in Y . For all belongs to U . The restriction of ( Y is continuous from b Y to U . The rst assumption concerns the linearized state equation. The second and third assumptions are needed to get optimal regularity for the adjoint equation. Indeed, the adjoint state corresponding to dened by To study the convergence of the SQP method we need that p belongs to U . Since by denition f u (v) belongs to U , the condition f u (v) 2 U is a regularity condition on f u (v). In the analysis of the Generalized Newton Method, we need the following additional regularity conditions. (A6) For every y 2 Y , 0 (y) belongs to L(U; ^ Y ). The mapping y 7! 0 (y) is locally of class C 1;1 from Y into L(U; b Y ). For every y belongs to L(U; b The mapping (y is locally of class C 1;1 from Y Y into L(U; b (A7) The mapping v 7! f 0 (v) is locally of class C 1;1 from V into ^ Y U . 3 Example - Control of a semilinear parabolic equa- tion Let us consider the following particular case of (P) : Z a u u Z a y y subject to in Here, n is a bounded domain with boundary of class C 2 , T > 0; > 0, y T 2 2 and u a < u b are given xed. The function ' : R ! R is nondecreasing, and locally of class C 2;1 . (The choice ts into this setting.) Let us verify that problem (E) satises all our assumptions. This problem is related to (P) as follows: fy fy where W (0; T ) is the Hilbert space dened by dt The space Y (respectively Y 2 ) is endowed with the norm kyk us check the assumptions. The operator is obviously continuous from Y 2 to U 2 , and is bijective from Y 2 to U 2 (see [17]). It is also a bijection from Y to U . (see [8], [20].) Thus (A1) is satised. continuous imbedding ([8], [20]), we can verify that is a mapping of class C 2 from Y into U , and that f is a mapping of class C 2 from Y U into R. Moreover, for all v f y (y Z (y Z a y (x; t)y(x; t) dSdt f u (y Z Thus, the derivative f y (v can be identied with the triplet (0; y ()). The assumptions (2.1) and (2.3) can be easily satised. To verify assumption (A5), let us introduce the space b This space can be identied with the subspace of Y of all elements having the form y 7! Z Z y Z where (^y y Y . >From the above calculations, it is clear that f y (v belongs to b Y . Let y (d;a;u) be the solution to the equation The operator (d; a; u) 7! y (d;a;u) is continuous and bijective from U 2 into Y 2 ([17]), and from U into Y ([8], [20]). The rst part of (A5) is satised. To prove the second part, let us consider the adjoint equation y For all (d; a; u) 2 U , and all ^ y Y , by using a Green formula, we obtain Z Z Z Z Z y Z Therefore nothing else than (; (0); j ). With this identity, we can easily verify the second part of assumption (A5). Let us nally discuss properties of some second order derivatives. The second derivative For We can interprete 00 (y as an element of L 1 () L 1 () , and (2.2) can be checked. The other assumptions on the second order derivatives, precisely (2.4) and (A4), are also 4 Optimality conditions This section is devoted to the discussion of the rst and second order optimality conditions. Let u) be a local solution of (P) . This means that holds for all v, which belong to a suciently small ball B V (v; ") and satisfy all constraints of (P) . Theorem 1 Let u) be a local solution of (P ) and suppose that the assumptions (A1), (A2), and (A5) are satised. Then there exists a unique Lagrange multiplier such that hf u (y; Proof. Since f is Frechet-dierentiable at u), is of class C 1 from Y to U , and is surjective from Y to U , there exists a unique such that (4.4) and (4.5) be satised (see [12], and also Theorem 2.1 in [1]). The variational equation (4.4) admits a unique solution dened by (v). Due to assumptions (A5), it follows that p belongs to U . 2 We next introduce the Lagrange function The system (4.4)-(4.5) is equivalent to For shortening, we shall write the adjoint equation (4.4) in the form f y (v)+p(+ 0 Thus the rst order optimality system for (P) is hf In what follows, the derivatives in L 0 and L 00 refer only to the variable v, but not to the Lagrange multiplier p. Let us assume that also satises the following: (SSC) Second order sucient optimality condition There is > 0 such that holds for all that satisfy the linearized equation Remark 1 The condition (SSC) is a quite strong assumption, and does not consider active control constraints, which might occur in U ad . For instance, this can be useful for constraints of the type U In concrete applications, the use of an associated second order assumption is possible (see for example [23]). However, we intend to shed light on the main steps, which are needed for a convergence analysis of the augmented Lagrangian SQP method, rather than to present the dicult technical details connected with weakening (SSC) . We shall adress this issue again in section 6. Let us complete this section by some simple results, which follow from the second order sucient condition. Lemma 1 Suppose that the assumptions (A1)-(A5) are satised. Suppose in addition that v satises the second order sucient condition (SSC). Then there exists > 0 such that, for every p) given in B V U ((y; u; for all that satisfy the perturbed linearized equation Proof. We brie y explain the main ideas of this quite standard result, to show where the dierent assumptions are needed. If p) is suciently close to (y; p), then the quadratic form L 00 p) is arbitrarily close to L 00 (y; u; p). By (SSC), (A2), and (A3) we derive that provided that y+ 0 (^y) y analogous estimate has to be shown for the solutions of the perturbed equation (4.11), where 0 is taken at ^ y. Write for short B := L 00 and dene z as the unique solution of z use the rst part of (A5)). Then The assumptions (A1), (A3), and (A5) ensure the estimate ck^y (4. (here and below c stands for a generic constant). Therefore, 7=8 k(z; u)k 2 follows by (4.12), (4.14) and Young inequality, where " > 0 can be taken arbitrarily small. Now we re-substitute z by y arrive by similar estimates at provided that is suciently small. Thus (4.10) is proven. 2 Although we shall not directly apply the next result, we state it to show why the dierent assumptions are needed. Some of them have been assumed to deal with the well known two-norm discrepancy. the optimality system (4:7) of (P ) and the second order sucient condition (SSC). Suppose that the assumptions (A1)-(A5) are fullled. Then there are constants " > 0 and > 0 such that the quadratic growth condition holds for all admissible Proof. The rst order optimality system implies Subtracting the state equations for y and y, analogously to (4.13) we nd that 1 . Then v h := (y solves the linearized equation (4:9), and the coercivity estimate of (SSC) can be applied to v h . Moreover, (A5) yields khk Y2 c kr We insert v h in (4.16), write for short B := L 00 (v; p) and proceed similarly to the estimation of Bv 2 in the last proof: ckv jr L g: In these estimates, the assumptions (A2) and (A3) were used. We have kv v v h k V 2 , and the estimate of h by the rst order remainder term r 1 can be inserted. Let the quadratic growth estimate follows from classical arguments. 2 This Lemma shows that the second order condition (SSC) is sucient for local optimality of (y; u) in the sense of V , whenever (y; u) solves the rst order optimality system. Notice that we cannot show local optimality in the sense of V 2 . 5 Augmented Lagrangian method 5.1 Augmented Lagrangian SQP method In this section we introduce the Augmented Lagrangian SQP method (ALSQP) with some special type of augmentation. For this, we rst represent (P) in the equivalent form Minimize subject to z The augmentation takes into account only the nonlinear equation z ALSQP method is obtained by applying the classical SQP method to the problem Minimize f (y; subject to z where > 0 is given. We dene the Lagrange functional L for ( ~ P ), and the corresponding augmented functional L on Y U 4 as follows: Once again, the derivatives L 0 and L 00 will stand for derivatives with respect to (y; u; z) and do not refer to the Lagrange multipliers (p; ). The same remark concerns L . Let the current iterate of the ALSQP method, and consider the linear-quadratic problem (QP Minimize f 0 subject to z (y n The new iterate (y obtained by taking the solution (y of (QP exists), and the multipliers (p n+1 ; n+1 ) associated with the constraints respectively. For we recover the classical SQP method. Let us also introduce the following problem: QP subject to y The problems (QP QP are equivalent in the sense precised below. Theorem 2 Let (y a solution of (QP associated Lagrange multipliers must solve the problem ( d QP n+1 ), and the multiplier p n+1 is the solution to the equation Moreover, z n+1 and n+1 must satisfy z Conversely, if (y n+1 ; u n+1 ) is a solution of ( d QP are dened by { (5:3), then (y n+1 ; u n+1 ; z n+1 ) is a solution to (QP associated Lagrange multipliers (p n+1 ; n+1 ). Proof. Let us rst assume that (y n+1 ; u To show that (y n+1 ; u n+1 ) solves ( d QP n+1 ) and that the relations (5.1){(5.3) are satised, we investigate the following: Explicit form of (QP We expand all derivatives occuring in the problem (QP . Write for short k and introduce for convenience the functional g(y; Having this, the objective to minimize in (QP n+1 ) is given by The minimization is subject to the constraints z (y n Reduction to ( d QP To reduce the dimension of the problem, we exploit the second one of the equations (5.4): We insert the expression z z n 0 (y n )(y y in the functional J . Then the second and fourth items in the denition of J are constant with respect to (y; z; u). They depend only on the current iterate and can be neglected during the minimization of J . The associated functional to be minimized is ~ Moreover, we can delete the second equation of (5.4) by inserting the expression for z in the rst one. This explains why (y n+1 ; u n+1 ) is a solution of ( d QP Necessary optimality conditions. To derive the necessary conditions for the triplet with the Lagrange functional ~ The conditions are ~ L u (u u n+1 ) 0, for all u 2 U ad . An evaluation yields for We mention for later use, that the equations (5.4) belong to the optimality system of (QP too. The update formulas for p n+1 and n+1 follow from (5.5), (5.6). We have shown one direction of the statement. The converse direction can be proved in a completely analogous manner. If (y QP n+1 ), then we substitute z for in the corresponding positions. Then it is easy to verify that (y n+1 ; u subject to (5.4), and that n+1 is the multiplier associated to the equation z (y n Remark 2 The update rules (5:2) { (5:3) imply that the Lagrange multiplier coincides with p during the iteration, while this is not necessarily true for the initial values of n and p n . Therefore, with possible exception of the rst step, up to a constant, the objective functional of ( d QP n+1 ) is ~ This easily follows by calculating L 00 (y from the formula (4:6). Moreover, we are justifed to replace n by p n in the variational equation (5:1). Theorem 2 shows that the iterates of the ALSQP method can be obtained by solving the reduced problem ( d QP solutions of (QP exist. This question of existence, can be answered by considering ( d QP Theorem 3 Let (y; p) satisfy the assumptions of Lemma 1 and let suciently small, then ( d QP n+1 ) has a unique solution (y Moreover, (y being dened by (5:3)) is the unique solution of (QP Proof. Assume that k(y; us prove the existence for ( d QP In view of the remark above, the functional ~ J can be taken instead of J for the minimization in ( d QP Its quadratic part is where ~ tends to in U , since z n (y z yields that the objective functional of ( d QP n+1 ) is coercive on the set ~ hence it is strictly convex there. The set U ad is non-empty, bounded, convex, and closed in U , and in U 2 as well. We have assumed in (A5) that ( is continuous from U 2 to Y 2 at all y 2 Y , in particular at Therefore, ~ C is non-empty, convex, closed, and bounded in Y 2 U 2 . Now existence and uniqueness of a solution (y QP n+1 ) are standard conclusions. Moreover, U ad U , hence u n+1 2 U , and the regularity properties of ( guarantee that y n+1 2 Y . Further, z n+1 2 U follows from (5.3). Existence and uniqueness for (QP are obtained from Theorem 2. 2 The update rules of Theorem 2 show that (p n+1 ; n+1 ) is uniquely determined in U 2 U 2 . We get even better regularity: Corollary 1 If the initial element (y is taken from Y U 4 , then the iterates generated by the ALSQP method are uniquely determined and belong to Y U 4 . Proof. Existence and uniqueness follows from the last theorem and the update rules (5.2){ (5.3). We also know that (y n+1 ; u . The only new result we have to derive is that (p n+1 ; n+1 ) remains in U U as well. Since we have to verify p n+1 2 U . This, however, follows instantly from the equation (5.1): We know that belong to b Y (assumptions (A5), (A6), (A7)). Moreover, the same holds for ( 00 (y n )(y n+1 y n Therefore, (A5) ensures the solution p n+1 of (5.1) to be in U . 2 5.2 Newton method for the optimality system of (P ) The augmented SQP method can be considered as a computational algorithm to solve the rst order optimality system of (P ) by the generalized Newton method. This equivalence will be our tool in the convergence analysis. The optimality system for (P ) consists of the equations (L (w)) z for the unknown variable The optimality system (5.8) of (P ) is equivalent to a generalized equation. To see this, let us rst introduce the following set-valued mappings: Y f0 U g N(u) f0 U g f0 U and consider F Y U 4 dened by f u (y; u) p z (y)C C C C C C C C C C A Notice that N(u) has a closed graph in U U . It is the restriction to U of the normal cone at U ad in the point u. (For the denition of the normal cone, we refer to [5].) In the rst component of F , due to (A6), we identify with the element which belongs to ^ Y . With (A5) and (A6), we can easily verify that F takes values in ^ Y U 4 . Lemma 3 The optimality system (5:8) of (P ) is equivalent to the generalized equation Proof. By calculating the derivatives of L in (5.8), we easily verify that: (L (w)) y (L (w)) z (L (w)) u z (y)C C C C C C A Therefore, by the denition of F , (5.10) is equivalent to The third relation can be rewritten as: This is just the variational inequality of (5.8), and the equivalence of (5.8) and (5.10) is veried. 2 Next we recall some facts about generalized equations and related convergence results for the Generalized Newton Method (GNM). Let W and E be Banach spaces, and let O be an open subset of W. Let F be a dierentiable mapping from O into E , and T be a set-valued mapping from O into P(E) with closed graph. Consider the generalized equation The generalized Newton method for (5.11) consists in the following algorithm: Choose a starting point For the solution to the generalized equation: The generalized Newton method is locally convergent under some assumptions stated below Equation (5.11) admits at least one solution !. There exist constants ~ r(!) and ~ c(!) such that BW (!; ~ r(!)). Denition 1 The generalized equation is said to be strongly regular at ! 2 O, if there exist constants r(! ) and c(! ), such that, for all )), the perturbed generalized equation has a unique solution S(! for all The theorem below is a variant of Robinson's implicit function theorem ([21], Theorem 2.1). Theorem 4 ([4], Theorem 2:5) Assume that (5.11) is strongly regular at some ! 2 O, and that (C1) and (C2) are fullled. Then there exist (!) > 0, k(!) > 0, and a mapping S 0 from BW (!; (!)) O into BW (!; (!)) such that, for every ! 2 BW (!; (!)), S 0 (! ) is the unique solution to (5:13), and The following theorem is an extension to the generalized equation (5.11) of the well known Newton-Kantorovitch theorem. It is a direct consequence of Theorem 4. Theorem 5 ([4], Theorem 2:6) Assume that the hypotheses of Theorem 4 are fullled. Then there exists ~ (!) > 0 such that, for any starting point (!)), the generalized Newton method generates a unique sequence (! k ) k convergent to !, and satisfying W for all k 1: We apply these results to set up the generalized Newton method for the generalized equation (5.10), which is the abstract formulation of the optimality system of (P ). Lemma 4 The generalized Newton method for solving the optimality system of (P ), dened by (5:12), proceeds as follows: Let w be the current iterate. Then the next iterate w is the solution of the following generalized equation for Proof. This iteration scheme is a conclusion of the iteration rule (5.12) applied to the concrete choice of (5.9) for F . The computations are straightforward. We should only mention the following equivalent transformation, which nally leads to (5.14), (5.15): Due to the concrete expression for F given in (5.12), the rst two relations in are Inserting (5.18) in (5.19), (5.20) we obtain (5.14), (5.15). 2 To apply Theorem 5 to the concrete generalized equation (5.10), we need that (5.10) be strongly regular at and that conditions (C1) and (C2) be satised. The assumption of strong regularity at w must be assumed here. It has to be checked for each particular application. In general, the verication of strong regularity requires a detailed analysis. In the case of the optimal control of parabolic partial dierential equations, we refer to the discussion of the SQP method in Troltzsch [23]. The strong regularity of an associated generalized equation was proved there by means of a result on L 1 -Lipschitz stability from [22]. The associated semilinear elliptic case was studied by Unger [24]. The conditions (C1) and (C2) can be veried with assumptions (A6) and (A7). Lemma 5 The mapping w 7! F (w) is of class C 1;1 from Y U 4 into b Y U 4 . Proof. This statement is an immediate consequence of (A6) and (A7). 2 Theorem 6 Let (y; u) be a local solution of (P ), and let p be the associated adjoint state. Assume that the generalized equation: Find (y; u; p) 2 Y U 2 such that be strongly regular at (y; p). Then the generalized equation Find is strongly regular at p. Proof. Let z ) be a perturbation in b Y U 4 . The linearized generalized equation for (5.22) at the point associated with the perturbation e, is f yy (y (z e f uy (y where f yy stands for f yy (y; u), and the same notations is used for the other mappings. To obtain the two rst equations of (5.23), we refer to the system (5.19), (5.20), where we insert w and replace the left hand side by the perturbation. Since z by straightforward calculations, we can easily prove that the system (5.23) is equivalent to f yy (y e f uy (y Now we observe that the rst, third, and fourth relation of (5.24) form a subsystem for which does not depend on (z; ). Once (y; u; p) is given from this subsystem, (z; ) is uniquely determined by the remaining two equations. Let us set ~ e with ~ The subsystem of (5.24) can be rewritten in the form of the generalized equation f uy (y The generalized equation (5.26) is the linearization of the generalized equation (5.21) at p), associated with the perturbation ~ e. Since (5.21) was assumed to be strongly regular at (y; p), there exist ~ r r(y; p) > 0, and a mapping S from U , such that S(~e) is the unique solution to (5.26) for all ~ e r), and U . Now, we show that (5.22) is strongly regular at w. For any e, let ~ e be given by (5.25). Then and there exists r > 0 such that ~ e belongs to (0; r). Dene a mapping S from B b (0; r) into b where ce z S 3 (~e): Then S(e) is clearly the unique solution to (5.23). We can easily nd c > 0 such that . The proof is complete. 2 Theorem 6 shows that once the convergence analysis for the standard non augmented Lagrange-Newton-SQP method has been done by proving strong regularity of the associated generalized equation, this analysis does not have to be repeated for analyzing convergence of the augmented method. Up to now, we have discussed the Augmented SQP method and the Generalized Newton method separately. Now we shall show that both methods are equivalent. This equivalence is used to obtain a convergence theorem for the augmented SQP method. Theorem 7 Let (y; u) a local solution of (P ), which satises together with the associated Lagrange multiplier p the second order sucient optimality condition (SSC). Dene suppose that the generalized equation (5:21) is strongly regular at w. Then there exists w) > 0 such that, for any starting point in the neighbourhood BW ( w; r), the ALSQP method dened according to Theorem 2 and the generalized Newton method dened in Lemma 4 generate the same sequence of iterates (w n . Moreover, there is a constant c q ( w) such that the estimate is satised for all Proof. First we should mention the simple but decisive fact that w satises the optimality system of (P ), since (y; p) has to satisfy the optimality system for (P). There- fore, it makes sense to determine w by the generalized Newton method. Let w be an arbitrary current iterate, which is identical for the ALSQP method and the generalized Newton method. In the GNM, w n+1 2 W is found as the unique solution of (5.14){(5.18). As concerns the ALSQP method, (y obtained as the unique solution of ( d QP are determined by (5.2). Therefore, (y satises the associated optimality system (5.4), (5.5)-(5.7) which is obviously identical with (5.14)-(5.18). It is clear that both the methods deliver the same new iterate w n+1 2 W . All remaining statements of the theorem follow from the convergence Theorem 5. 2 6 Numerical results 6.1 Test example We apply the augmented SQP method to the following one-dimensional nonlinear parabolic control problem with Stefan-Boltzmann boundary condition: Z( a y (t) subject to u a u(t) This example is a particular case of problem (E) considered in Section 3, where we take (0; ') and make an associated modication of the boundary condition. In an early phase of this work, we studied the numerical behaviour of the SQP method without augmentation. Here, we compare both methods. We performed our numerical tests for the following particular data: a y Lemma 6 The pair (y; u) dened by e 2=3 e 1=3 is a locally optimal solution for (6:27) in C([0; '][0; T ])L 1 (0; T ). The associated adjoint state (Lagrange multiplier) is given by cos(x). The triplet (y; the second order sucient optimality condition (SSC). Proof. The proof is split into four steps. Step 1. State equation. It is easy to see that Now regard the boundary condition at ': The left hand side is The same holds for the right hand side, since Step 2. Adjoint equation. Again, the equations are easy to check. It remains to verify the boundary condition at It is obvious that '. The right hand side of the boundary condition has the same value, since a y (t) Step 3. Variational inequality. We must verify that { which is trivial { and that a It is well known that this holds if and only if (a e 2=3 e 1=3 where P [0;1] denotes projection onto [0; 1]. This is obviously veried. Step 4. Second order sucient condition. The Lagrange function is given by R R l R Ty x (0; t)p(0; R T(y x ('; R T Therefore, Since p is negative, L 00 (y; p) is coercive on the whole space Y U , hence (SSC) is Theorem 8 The pair (y; u) is a global solution of (E). Proof. Let (y; u) be any other admissible pair for (E). Due to the rst order necessary condition, we have p)(y u)2 Z Z >From the positivity of p and of ' 00 (y (independently of s and y), it follows that f(y; u) f(y; u). 2 Next we discuss the strong regularity of the optimality system at (y; p). Theorem 9 The optimality system of (E) is strongly regular at (y; p). Proof. The triplet (y; p) satises (SSC). Moreover, (E) ts into a more general class of optimal control problems for semilinear parabolic equations, which was considered in [23]. It follows from Theorem 5.2 in [22], and Theorem 5.3 in [23] that (SSC) ensures the strong regularity of the generalized equation being the abstract formulation of the associated optimality system. We only have to apply this result to problem (6.27). 2 Remark 3 A study of [23] reveals that convergence of the standard SQP method can be proved for arbitrary dimension of assuming a weaker form of (SSC). It requires coercivity of L 00 only on a smaller subspace that considers strongly active control constraints. This weaker assumption should be helpful for proving the convergence of the augmented SQP method as well. We shall not discuss this, since the technical eort will increase considerably. Now we obtain from Theorem 7 the following result: Corollary 2 The Augmented Lagrangian SQP method for (E) is locally quadratically convergent towards (y; p). 6.2 Algorithm For the convenience of the reader, let us consider the problem ( d QP corresponding to our test example. After simplifying we get Minimize2 subject to (6. with One specic diculty for solving problem (6:27)-(6:29) is partially related to the control constraints. But the main diculty appears also in the unconstrained case where a (large) linear system has to be solved. Let us consider for a moment the unconstrained case. If solution of problem (6:27)-(6:28), then the optimal triplet satises (6:28), the adjoint equation and (a In practice, we solve ( d QP discretization of its optimality system. The result is taken to solve ( d QP ). The discretized version of equation (6.31) corresponds to a large-scale linear system. To solve this system, we need the solutions corresponding to the discretization of two coupled parabolic equations (the state and the adjoint equations). It is clear that the accuracy of the Augmented Lagrangian SQP-method depends on the one for solving the linear system, and consequently on the numerical methods for the partial dierential equations. In our example, the state and adjoint equations are solved by using a second-order nite dierence scheme (Cranck-Nicholson scheme) appropriately modied at the boundary to maintain second order approximation. The linear system is solved by using the CGM (conjugate gradient method), with a step length given by the Polak-Ribiere formula. Let us now take into account the constraints (6.29). The optimality condition (6.31) is replaced by (a (a The management of these restrictions is based on (6.32) and on an projection method by Bertsekas [7]. (See also [9] and [10] where this method is successfully applied.) More precisely, we have the following algorithm: be the vector representing the iterate corresponding to xed grid. Let " and be xed positive numbers, and let I = f1; ; mg be the index set associated to w n . (m is the dimension of the vector w n and depends on the discretization of u n ) and denote by d the vector representing the iterate corresponding to the solution of (6.30). the sets of strongly active inequalities I I where A is the vector representing a u . n for all j 2 I a [ I b . Solve the unconstrained problem (6.27)-(6.28) for w j a [ I remaining components are xed due to 4.) Denote by v n the vector representation of the solution. denotes the projection onto [u a ; and go to 2. Otherwise stop the iteration. 6.3 Numerical tests In the numerical tests, we focused our interest on the aspects concerning the convergence for dierent values of initial data and penalty parameters , and on the rate of convergence. The programs were written in MATLAB. Let us rst summarize some general observations. In our example, the augmented Lagrangian algorithm performed well. In particular, the graphs of the exact solution and that of the numerical solution are (almost) identical. When compared with the SQP method (corresponding to = 0), the augmented Lagrangian SQP has the advantage of a more global behavior. Moreover, it is less sensitive to the start-up values, and is signicantly faster than the SQP method for some points. Graphical correction of the computed controls and precisionof optimal value (up to ve digits) are obtained by taking the discretization parameters with respect to the time and the space equal to 200. For xed data, the number of iterations for the CGM and the Augmented SQP turned out to be independent of the mesh size. In all the sequel, we set ku ku where (u; z) and are the vectors respectiveely corresponding to the exact solution of (E), the numerical solution of (E), and the solution of ( d QP Moreover, we denote by n t and n x the discretization parameters with respect to the time and the space. Optimal controls were determined for the following pairs (n x (200,200), (400,400). Run 1. (SQP method.) The rst test corresponds to The rates for e n , n u , n p , and n z are given in Table 1. Table 1: 100 1.7782e-06 2.5347e-06 1.6610e-06 0.2372 0.3653 1.0575 1.3886 200 1.3725e-06 2.9337e-06 1.0724e-06 0.2472 0.3663 1.0585 0.9980 The SQP method shows a good convergence for this initial point. 4 iterations were needed to get the result. Run 2. (ALSQP method.) The second test corresponds to the point (0:5; 0:5; 0:5), with z Table 2: 100 1.5391e-06 2.6824e-06 1.5013e-06 1.4459e-06 0.0150 1.0004 2.2378 200 1.2318e-06 3.8783e-07 5.2251e-07 5.5521e-07 0.0149 0.9256 1.0015 The ALSQP method has a very good convergence for this choice. Convergence could always be achieved by xing using other values of z 0 and . However, the number of iterations and the speed of the method depend on these choices. As shown in Table 3, three iterations for the ALSQP method were needed, instead of four for the SQP method. The number of iterations for the CGM, the SQP and the ALSQP methods is independent of the mesh-size. The exact value for the cost functional is In Table 3, we give the values of the cost functional corresponding to the dierent steps for Table 3: SQP method Iter f n CGM iter ALSQP method Iter f n CGM iter u2 u1 u3, u4 u2, u3 Figure 1: Controls for Run 1 and Run 2 Figure 2: States y('; t) for Run 1 and Run 2 p3, p4 p2, p3 Figure 3: Adjoint states p('; t) for Run 1 and Run 2 In Figures 1, 2, and 3, we compare the behavior of the control, the state, and the adjoint state obtained by taking 1. It is clear that in the case of the ALSQP method, the second iteration gives a good approximation to the optimal control, the optimal state, and the optimal adjoint state. Run 3. The last test corresponds to the initial point given by Table 4: 100 9.9421e-06 2.7989e-06 4.0979e-06 3.0853e-06 200 1.1864e-05 4.7523e-06 5.0999e-06 2.3842e-06 400 1.2167e-05 5.2817e-06 5.3307e-06 2.2146e-06 200 0.0404 0.3975 1.3537 1.1471 For this initial point, the SQP method (corresponding to does not converge, while the ALSQP method converges for many choices of z 0 . In our tests, the point which gives the best result is given by z 1. For this choice, 4 iterations are needed with 2, 5, 6 and 9 CG steps. The dierents rates are given in Table 4, and the behavior of the solution is shown is Figure 4. Remark 4 The numerical results stated in Table 1, 2, and reft3 were obtained for a xed mesh-size (xed grid). However, we also implemented the ALSQP method with adaptative mesh size, i.e. we started with a coarse grid and used the obtained results as startup values u2 u3, u4 p3, Figure 4: Controls, states, and adjoint states for Run 3 for the next ner grid. This method is signicantly faster, and delivers essentially the same results. --R A Lagrange multiplier theorem for control problems with state constraints The Lagrange-Newton method for in nite-dimensional optimization prob- lems The Lagrange Newton method for in Discretization and mesh independence of Newton's method for generalized equations. Analysis and Control of Nonlinear In Augmented Lagrangian techniques for elliptic state constrained optimal control problems Projected Newton methods for optimization problems with simple constraints Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations Numerical solution of a constrained control problem for a phase Augmented Lagrangian-SQP methods for nonlinear optimal control problems of tracking type Augmented Lagrangian-SQP methods in Hilbert spaces and application to control in the coecients problems Augmented Lagrangian-SQP techniques and their approxi- mations "Linear and quasilinear equations of parabolic type" First and second order su Hamiltonian Pontryagin's principles for control problems governed by semilinear parabolic equations Strongly regular generalized equations. Hinreichende Optimalit Distributed control problems for the Burgers equation. --TR Multiplier methods for nonlinear optimal control Second-order sufficient optimality conditions for a class of nonlinear parabolic boundary control problems Augmented Lagrangian--SQP Methods for Nonlinear OptimalControl Problems of Tracking Type Augemented Lagrangian Techniques for Elliptic State Constrained Optimal Control Problems Pontryagin''s Principle for State-Constrained Boundary Control Problems of Semilinear Parabolic Equations On the Lagrange--Newton--SQP Method for the Optimal Control of Semilinear Parabolic Equations Mesh-Independence for an Augmented Lagrangian-SQP Method in Hilbert Spaces Distributed Control Problems for the Burgers Equation --CTR Hans D. Mittelmann, Verification of Second-Order Sufficient Optimality Conditions for Semilinear Elliptic and Parabolic Control Problems, Computational Optimization and Applications, v.20 n.1, p.93-110, October 2001

optimal control;two-norm discrepancy;control constraints;generalized equation;semilinear parabolic equation;augmented Lagrangian SQP method in Banach spaces;generalized Newton method

606895

Large-Scale Active-Set Box-Constrained Optimization Method with Spectral Projected Gradients.

A new active-set method for smooth box-constrained minimization is introduced. The algorithm combines an unconstrained method, including a new line-search which aims to add many constraints to the working set at a single iteration, with a recently introduced technique (spectral projected gradient) for dropping constraints from the working set. Global convergence is proved. A computer implementation is fully described and a numerical comparison assesses the reliability of the new algorithm.

Introduction The problem considered in this paper consists in the minimization of a smooth with bounds on the variables. The feasi- Department of Computer Science IME-USP, University of S~ao Paulo, Rua do Mat~ao 1010, Cidade Universitaria, 05508-090, S~ao Paulo SP, Brazil. This author was supported by PRONEX-Optimization 76.79.1008-00, FAPESP (Grants 99/08029-9 and 01/04597-4) and CNPq (Grant 300151/00-4). e-mail: egbirgin@ime.usp.br y Department of Applied Mathematics IMECC-UNICAMP, University of Campinas, This author was supported by PRONEX- Optimization 76.79.1008-00, FAPESP (Grant 01/04597-4), CNPq and FAEP-UNICAMP. e-mail: martinez@ime.unicamp.br ble set is dened by Box-constrained minimization algorithms are used as subalgorithms for solving the subproblems that appear in many augmented Lagrangian and penalty methods for general constrained optimization. See [11, 12, 16, 17, 18, 19, 20, 21, 26, 28, 31]. A very promising novel application is the reformulation of equilibrium problems. See [1] and references therein. The methods introduced in [11] and [26] are of trust-region type. For each iterate x k 2 a quadratic approximation of f is minimized in a trust-region box. If the objective function value at the trial point is su-ciently smaller than f(x k ), the trial point is accepted. Otherwise, the trust region is reduced. The dierence between [11] and [26] is that, in [11], the trial point is in the face dened by a \Cauchy point", whereas in [26] the trial point is obtained by means of a specic box-constrained quadratic solver, called QUACAN. See [2, 15, 23, 25, 31] and [13] (p. 459). Other trust-region methods for box-constrained optimization have been introduced in [3, 29]. QUACAN is an active-set method that uses conjugate gradients within the faces, approximate internal-face minimizations, projections to add constraints to the active set and an \orthogonal-to-the-face" direction to leave the current face when an approximate minimizer in the face is met. In [17] a clever physical interpretation for this direction was given. Numerical experiments in [16] suggested that the e-ciency of the algorithm [26] relies, not in the trust-region strategy, but in the strategy of QUACAN for dealing with constraints. This motivated us to adapt the strategy of QUACAN to general box-constrained problems. Such adaptation involves two main decisions. On one hand, one needs to choose an unconstrained minimization algorithm to deal with the objective function within the faces. On the other hand, it is necessary to dene robust and e-cient strategies to leave faces and to add active constraints. Attempts for the rst decision have been made in [6] and [10]. In [10] a secant multipoint minimization algorithm is used and in [6] the authors use the second-order minimization algorithm of Zhang and Xu [36]. In this paper we adopt the leaving-face criterion of [6], that employs the spectral projected gradients dened in [7, 8]. See, also, [4, 5, 32, 33, 34]. For the internal minimization in the faces we introduce a new general algorithm with a line search that combines backtracking and extrapolation. The compromise in every line-search algorithm is between accuracy in the localization of the one-dimensional minimizer and economy in terms of functional evaluations. Backtracking-like line-search algorithms are cheap but, sometimes, tend to generate excessively small steps. For this reason, back-tracking is complemented with a simple extrapolation procedure here. The direction chosen at each step is arbitrary, provided that an angle condition is satised. In the implementation described in this paper, we suggest to choose the direction using the truncated-Newton approach. This means that the search vector is an approximate minimizer of the quadratic approximation of the function in the current face. We use conjugate gradients to nd this direction, so the rst iterate is obviously a descent direction, and this property is easily monitorized through successive conjugate gradient steps. The present research is organized as follows. In Section 2 we describe an \unconstrained" minimization algorithm that deals with the minimization of a function on a box. The algorithm uses the new line-search technique. Due to this technique it is possible to prove that either the method nishes at a point on the boundary where, perhaps, many constraints are added, or it converges to a point in the box where the gradient vanishes. The box-constrained algorithm is described in Section 3. Essentially, we use the algorithm of Section 2 to work within the \current face" and spectral projected gradients [7] to leave constraints. The spectral projected gradient technique also allows one to leave many bounds and to add many others to the working set at a single iteration. This feature can be very important for large-scale calculations. In this section we prove the global convergence of the box-constrained algorithm. The computational description of the code (GENCAN) is given in Section 4. In Section 5 we show numerical experiments using the CUTE collection. In Section 6 we report experiments using some very large problems (up to 10 7 variables). Finally, in Section 6 we make nal comments and suggest some lines for future research. In this section we assume that f : IR ug: The set B will represent each of the closed faces of in Section 3. The dimension n in this section is the dimension of the reduced subspace of the Section 3 and the gradient of this section is composed by the derivatives with respect to free variables in Section 3. We hope that using the notation rf in this section will not lead to confusion. From now on, we denote Our objective is to dene a general iterative algorithm that starts in the interior of B and, either converges to an unconstrained stationary point, or nishes in the boundary of B having decreased the functional value. This will be the algorithm used \within the faces" in the box-constrained method. Algorithm 2.1 is based on line searches with Armijo-like conditions and extrapolation. Given the current point x k and a descent direction d k , we nish the line search if x k +d k satises a su-cient descent criterion and if the directional derivative is su-ciently larger than hg(x k ); d k i. If the su-cient descent criterion does not hold, we do backtracking. If we obtained su-cient descent but the increase of the directional derivative is not enough, we try extrapolation. Let us explain why we think that this philosophy is adequate for large-scale box-constrained optimization. 1. Pure backtracking is enough for proving global convergence of many optimization algorithms. However, to accept the rst trial point when it satises an Armijo condition can lead to very small steps in critical situations. Therefore, steps larger than the unity must be tried when some indicator says that this is worthwhile. 2. If the directional derivative su-ciently larger than we consider that there is not much to decrease increasing the steplength in the direction of d k and, so, we accept the unit steplength provided it satises the Armijo condition. This is reasonable since, usually, the search direction contains some amount of second-order information that makes the unitary steplength desirable from the point of view of preserving a satisfactory order of convergence. 3. If the unitary steplength does not satisfy the Armijo condition, we do backtracking. In this case we judge that it is not worthwhile to compute gradients of the new trial points, which would be discarded if the point is not accepted. 4. Extrapolation is especially useful in large-scale problems, where it is important to try to add as many constraints as possible to the working set. So, we extrapolate in a rather greedy way, multiplying the steplength by a xed factor while the function value decreases. 5. We think that the algorithm presented here is the most simple way in which extrapolation devices can be introduced with a reasonable balance between cost and e-ciency. It is important to stress that this line search can be coupled with virtually any minimization procedure that computes descent directions. For all z 2 IR n , the Euclidean projection of z onto a convex set S will be denoted P S (z). In this section, we denote P (y). The symbol k k represents the Euclidean norm throughout the paper. Algorithm 2.1: Line-search based algorithm The algorithm starts with x 0 2 Int(B). The non-dimensional parameters are given. We also use the small tolerances abs ; rel > 0. Initially, we set k 0. Step 1. Computing the search direction Step 1.1 If kg k Step 1.2 Compute such that Step 2. Line-search decisions Step 2.1 Compute set minf then go to Step 2.2 else go to Step 2.3. Step 2.2 (At this point we have x k If take and go to Step 5 else go to Step 3 (Extrapolation) else go to Step 4 (Backtracking). Step 2.3 (At this point we have x k If take k max and x such that f(x k+1 ) and go to Step 5 (In practice, such a point is obtained performing Step 3 of this algorithm (Extrapolation).) else go to Step 4 (Backtracking). Step 3. Extrapolation Step 3.1 If ( < max and N > max ) then set trial max else set trial N. Step 3.2 If ( max and kP take the execution of Algorithm 2.1. Step 3.3 If (f(P take to Step 5 else set trial and go to Step 3.1. Step 4. Backtracking Step 4.1 Compute new . Step 4.2 If (f(x k take and go to Step 5 else go to Step 4.1. Step 5. If k max terminate the execution of Algorithm 2.1 else set to Step 1. Remarks. Let us explain here the main steps of Algorithm 2.1 and their motivations. The algorithm perform line-searches along directions that satisfy the angle-cosine condition (2). In general, this line search will be used with directions that possess some second-order information, so that the \nat- ural" step must be initially tested and accepted if su-cient-descent and directional-derivative conditions ((3) and (4)) are satised. The rst test, at Step 2.1, asks whether x k is interior to the box. If this is not the case, but f(x k we try to obtain smaller functional values multiplying the step by a xed factor and projecting onto the box. This procedure is called \Extrapolation". If x k is not interior and backtracking. is interior but the Armijo condition (3) does not hold, we also do backtracking. Backtracking stops when the Armijo condition (6) is fullled. If (3) holds, we test the directional derivative condition (4). As we mentioned above, if (4) is satised too, we accept x k new point. However, if (3) holds and (4) does not, we judge that, very likely, taking larger steps along the direction d k will produce further decrease of the objective function. So, in this case we also do Extrapolation. In the Extrapolation procedure we try successive projections of x k +d k onto the box, with increasing values of . If the entry point interior but x k +Nd k is not, we make sure that the point x will be tested rst. The extrapolation nishes when decrease of the function is not obtained anymore or when the distance between two consecutive projected trial points is negligible. The iteration of Algorithm 2.1 nishes at Step 5. If the corresponding iterate x k+1 is on the boundary of B, the algorithm stops, having encountered a boundary point where the functional value decreased with respect to all previous ones. If x k+1 is in the interior of B the execution continues increasing the iteration number. The ux-diagrams in Figures 1 and 2 help to understand the structure of the line-search procedure. x+d -Int aa-amax Line Search bb-condition xnew-x+d End Armijo Backtracking Extrapolation Figure 1: Line Search procedure. Extrapolation a - aamax and dist < ee aa-aatrial aatrial-NNaa a < aamax and NNa > aamax f(P(x+aatrial d)) xnew-P(x+aad) End Figure 2: Extrapolation strategy. In the following theorem we prove that any sequence generated by Algorithm 2.1, either stops at an unconstrained stationary point, or stops in the boundary of B, or generates, in the limit, unconstrained stationary points. Theorem 2.1. Algorithm 2.1 is well dened and generates points with strictly decreasing functional values. If fx k g is a sequence generated by Algorithm 2.1, one of the following possibilities holds. (i) The sequence stops at x k , with g(x k (ii) The sequence stops at x (iii) The sequence is innite, it has at least one limit point, and every limit point x satises g(x Proof. Let us prove rst that the algorithm is well dened and that it generates a sequence with strictly decreasing function values. To see that it is well dened we prove that the loops of Steps 3 and 4 necessarily nish in nite time. In fact, at Step 3 we multiply the nonnull direction d k by a number greater than one, or we take the maximum allowable feasible step. Therefore, eventually, the boundary is reached or the increase condition (5) is met. The loop of Step 4 is a classical backtracking loop and nishes because of well-known directional derivative arguments. See [14]. On exit, the algorithm always requires that f(x k so the sequence strictly decreasing. It remains to prove that, if neither (i) nor (ii) hold, then any cluster point x of the generated sequence satises g(x be an innite subset of IN such that lim Suppose rst that ks k k is bounded away from zero for k 2 K 1 . Therefore, there exists > 0 such that ks k k for all k 2 K 1 . By (3) or (6) we have that for all k 2 K 1 . Therefore, by (2), By the continuity of f this implies that lim k2K 1 Suppose now that ks k k is not bounded away from zero for k 2 K 1 . So, there exists K 2 , an innite subset of K 1 , such that lim k2K 2 ks be the set of indices such that k is computed at Step 2.2 for Analogously, let K 4 K 2 be the set of indices such that k is computed at Step 3 for all k 2 K 4 and let K 5 K 2 be the set of indices such that k is computed at Step 4 for all k 2 K 5 . We consider three possibilities: (i) K 3 is innite. (ii) K 4 is innite. (iii) K 5 is innite. Consider, rst, the case (i). By (4) we have that and ks ks for all k 2 K 3 . Since K 3 is innite, taking an convergent subsequence taking limits in (7) and using continuity, we obtain that Since 2 (0; 1), this implies that hg(x ); di 0. But, by (2) and continuity, Consider, now, Case (iii). In this case, K 5 is innite. For all k 2 K 5 there exists s 0 k such that and ks 0 ks k By (10), lim ks 0 by (9), we have for all k 2 K 5 , So, by the Mean-Value theorem, there exists k 2 [0; 1] such that for all k 2 K 5 . Dividing by ks 0 k k, taking limits for a convergent subsequence d) we obtain that This inequality is similar to (8). So, g(x follows from the same arguments. Consider, now, Case (ii). Since we are considering cases where an innite sequence is generated it turns out that, in (5), P Moreover, by Step 3.1, trial N and P Therefore, for all k 2 K 4 , writing 0 we have that 0 and Therefore, by the Mean-Value theorem, for all k 2 K 4 there exists k 2 k ] such that Thus, for all k 2 K 4 , since 0 we have that dividing by kd k k and taking a convergent subsequence of d k =kd k k, we obtain: hg(x ); di 0: But, by (2), taking limits we get hg(x ); di kg(x )k. This implies that This completes the proof. 2 3 The box-constrained algorithm The problem considered in this section is Minimize f(x) subject to x where is given by (1). As in [23], let us divide the feasible set into disjoint open faces, as follows. For all I We I the smallest a-ne subspace that contains F I and S I the parallel linear subspace to V I . The (continuous) projected gradient at xis dened as For all x 2 F I , we dene I [g P (x)]: The main algorithm considered in this paper is described below. Algorithm 3.1: GENCAN Assume that x 0is an arbitrary initial point, 2 (0; 1) and 0 < min I be the face that contains the current iterate x k . Assume that g P (otherwise the algorithm terminates). At the main iteration of the algorithm we perform the test kg I If (13) takes place, we judge that it is convenient that the new iterate belongs to F I (the closure of F I ) and, so, we compute x k+1 doing one iteration of Algorithm 2.1, with the set of variables restricted to the free variables in F I . So, the set B of the previous section corresponds to F I here. If (13) does not hold, we decide that some constraints should be abandoned and, so, the new iterate x k+1 is computed doing one iteration of the SPG method described by Algorithm 3.2. In this case, before the computation of x k+1 we compute the spectral gradient coe-cient k in the following way. Otherwise, dene and Algorithm 3.2 is the algorithm used when it is necessary to leave the current face, according to the test (13). Algorithm 3.2: SPG Compute as the next iterate of a monotone SPG iteration [7, 8] with the spectral step k . Namely, we dene the search direction d k as and we compute x in such a way that trying rst perhaps, reducing this coe-cient by means of a safeguarded quadratic interpolation procedure. Remark. Observe that x F I if x k 2 F I and x k+1 is computed by Algorithm 3.2. In this case, (13) does not hold, so kg P the components corresponding to the free variables of g I are the same, this means that g P components corresponding to xed variables. Therefore, F I for all > 0. So, F I for all > 0. But, according to the SPG iteration, for some > 0, 0 > 0. This implies that x F I . We nish this section giving some theoretical results. Roughly speaking, we prove that the algorithm is well dened and that a Karush-Kuhn-Tucker point is computed up to an arbitrary precision. Moreover, under dual- nondegeneracy, the (innite) algorithm identies the face to which the limit belongs in a nite number of iterations. Theorem 3.1. Algorithm 3.1 is well dened. Proof. This is a trivial consequence of the fact that Algorithm 2.1 and Algorithm 3.2 (the SPG algorithm [7]) are well dened. 2 Theorem 3.2. Assume that fx k g is generated by Algorithm 3.1. Suppose that there exists k 2 f0; I for all k k. Then, every limit point of fx k g is rst-order stationary. Proof. In this case, x k+1 is computed by Algorithm 2.1 for all k k. Thus, by Theorem 2.1, the gradient with respect to the free variables tends to zero. By a straightforward projection argument, it follows that kg I Since (13) holds, this implies that kg P every limit point is rst-order stationary. 2 Theorem 3.3. Suppose that for all k 2 f0; I , there exists such that x k I . Then, there exists a limit point of fx k g that is rst-order stationary. Proof. See Theorem 3.3 of [6]. 2 Theorem 3.4. Suppose that all the stationary points of (12) are nondegen- erate. ( @f the hypothesis of Theorem 3.2 (and, hence, its thesis) must hold. Proof. See Theorem 3.4 of [6]. 2 Theorem 3.5. Suppose that fx k g is a sequence generated by Algorithm 3.1 and let " be an arbitrary positive number. Then, there exists k 2 f0; such that kg P Proof. This result is a direct consequence of Theorems 3.2 and 3.3. 2 Implementation At iteration k of Algorithm 2.1 the current iterate is x k and we are looking for a direction d k satisfying condition (2). We use a truncated-Newton approach to compute this direction. To solve the Newtonian system we call Algorithm 4.1 (described below) with A r 2 f(x k The following algorithm applies to the problem s: The initial approximation to the solution of (14) is s The algorithm nds a point s which is a solution or satises q(s ) < q(s 0 ). Perhaps, the - nal point is on the boundary of the region dened by ksk and l s u. Algorithm 4.1: Conjugate gradients The parameters << 1 and k max 2 IN are given. The algorithm starts with Step 1. Test stopping criteria set s Step 2. Compute conjugate gradient direction Step 2.1 If else compute Step 2.2 If (p T Step 3. Compute step Step 3.1 Compute ug. Step 3.2 Compute Step 3.3 If ( If ( If ( Step 4. Compute new iterate Step 4.1 Compute Step 4.2 If (b T s k+1 > kbkks k+1 set s Step 4.3 If ( set s Step 5. Compute set to Step 1. This algorithm is a modication of the one presented in [27] (p. 529) for symmetric positive denite matrices A and without constraints. The modications are the following: At Step 2.2 we test if p k is a descent direction at s k , i.e., if hp k ; rq(s k )i < To force this condition we multiply p k by 1 if necessary. If the matrix-vector products are computed exactly, this safeguard is not necessary. However, in many cases the matrix-vector product Ap k is replaced by a nite-dierence approximation. For this reason, we perform the test in order to guarantee that the quadratic decreases along the direction p k . At Step 3.3 we test if p T inequality holds, the step k in the direction p k is computed as the minimum among the conjugate-gradient step and the maximum positive step preserving feasibility. If and we are at the rst iteration of CG, we set k max . In this way CG will stop with s the angle condition of Step 4.2. If we are not at iteration zero of CG, we keep the current approximation to the solution of (14) obtained so far. At Step 4.2 we test whether the angle condition (2) is satised by the new iterate or not. If this condition is not fullled, we stop the algorithm with the previous iterate. We also stop the algorithm if the boundary of the feasible set is achieved (Step 4.3). The convergence criterion for the conjugate-gradient algorithm is dy- namic. It varies linearly with the logarithm of the norm of the continuous projected gradient, beginning with the value i and nishing with f . We dene where a log log and is used in the stopping criterion kg P (x)k 1 < of Algorithm 3.1. The parameter k max is the maximum number of CG-iterations for each call of the conjugate-gradient algorithm. It also varies dynamically in such a way that more iterations are allowed at the end of the process than at the beginning. The reason is that we want to invest a larger eort in solving quadratic subproblems when we are close to the solution than when we are far from it. In fact, where log In the Incremental-quotient version of GENCAN, r 2 f(x k ) is not computed and the matrix-vector products r 2 f(x k )y are approximated by with In fact, only the components correspondent to free variables are computed and the existence of xed variables is conveniently exploited in (15). 5 Numerical experiments with the CUTE collec- tion In order to assess the reliability GENCAN, we tested this method against some well-known alternative algorithms using all the non-quadratic bound- constrained problems with more than 50 variables from the CUTE [12] col- lection. The algorithms that we used for comparing GENCAN are BOX- QUACAN [26] (see, also, [28]), LANCELOT [11, 12] and the Spectral Projected Gradient method (SPG) (described as SPG2 in [7]; see also [8]). All the methods used the convergence criterion kg P stopping criteria were inhibited. In GENCAN we used Algorithm 3.1), and Algorithm 4.1). In all algorithms we used . The parameters of (the line search of) Algorithm 3.2 were the default parameters mentioned in [7] and the same used in (the line search of) Algorithm 2.1, i.e., In LANCELOT we used exact second derivatives and we did not use preconditioning in the conjugate-gradient method. The reason for this is that, in GENCAN, the conjugate gradient method for computing directions is also used without preconditioning. The other options of LANCELOT were the default ones. A small number of modications were made in BOX-QUACAN to provide a fair comparison. These modications were: (i) the initial trust-region radius of GENCAN was adopted; (ii) the maximum number of conjugate-gradient iterations was xed in the accuracy for solving the quadratic subproblems was dynamic in BOX- QUACAN varying from 0:1 to 10 5 , as done in GENCAN, (iv) the minimum trust-region radius min was xed in 10 3 to be equal to the corresponding parameter in GENCAN. The codes are written in Fortran 77. The tests were done using an ultra-SPARC from SUN, with 4 processors at 167 MHz, 1280 mega bytes of main memory, and Solaris 2.5.1 operating system. The compiler was WorkShop Compilers 4.2 Oct 1996 FORTRAN 77 4.2. Finally we have used the ag -O4 to optimize the code. In the rst four tables we report the full performance of LANCELOT, SPG, BOX-QUACAN, GENCAN (true Hessian) and GENCAN (Incremental- quotients). The usual denition of iteration in LANCELOT involves only one function evaluation. However, in order to unify the comparison we call \iteration" to the whole process that computes a new iterate with lower functional value, starting from the current one. Therefore, a single LANCELOT- iteration involves one gradient evaluation but, perhaps, several functional evaluations. At each iteration several trust-region problems are solved approximately and each of them uses a number of CG-iterations. Problems HADAMALS and SCON1LS have bounds where the lower limit is equal to the upper limit. BOX-QUACAN does not run under this circumstances, so the performance of this method in that situation is not reported in the corresponding table. In these tables, we report, for each method: IT: Number of iterations; FE: Functional evaluations; GE: Gradient evaluations; CG: Conjugate gradient iterations, except in the case of SPG, where CG iterations are not computed; Time: CPU time in seconds; nal functional value obtained; of the projected gradient at the nal point. The next 3 tables repeat the information of the rst ones in a more compact and readable form. In Table 5 we report the nal functional value obtained for each method, in the cases where there was at least one dierence between them, when computed with four signicant digits. In Table 7 we report, for each method, the numbers FE and (GE+CG). The idea is that a CG iteration is sometimes as costly as a gradient-evaluation. The cost is certainly the same when we use the incremental-quotient version of GENCAN. Roughly speaking, GE+CG represents the amount of work used for solving subproblems and FE represents the work done on the true problem trying to reduce the objective function. Table 8 reports the computer times for the problems where at least one of the methods used more than 1 second. The computer time used by LANCELOT must be considered under the warning made in [9] page 136, \LANCELOT [.] does not require an interface using the CUTE tools. It is worth noting that LANCELOT exploits much more structure than that Problem n IT FE GE CG Time f(x) kg P (x)k1 6.467D 06 QRTQUAD 120 144 178 145 570 1.39 3.625D+06 3.505D 06 CHEBYQAD 50 22 28 23 463 2.22 5.386D 7.229D 06 LINVERSE 1999 22 28 23 2049 47.22 6.810D+02 3.003D 06 Table 1: Performance of LANCELOT. provided by the interface tools". As a consequence, although GENCAN used less iterations, less functional evaluations, less gradient evaluations and less conjugate-gradient iterations than LANCELOT in SCON1LS, its computer time is greater than the one spent by LANCELOT. In some problems, like QR3DLS and CHEBYQAD, the way in which LANCELOT takes advantage of the SIF structure is also impressive. Now we include an additional table that was motivated by the observation of Table 7. It can be observed that the number of functional evaluations per iteration is larger in GENCAN than in LANCELOT and BOX- QUACAN. The possible reasons are three: Many SPG-iterations with, perhaps, many functional evaluations per iteration. Many TN-iterations with backtracking. Many TN-iterations with extrapolations. We classify the iterations with extrapolation in successful and unsuccessful ones. A successful extrapolation is an iteration where the extrapolation produced a functional value smaller than the one corresponding to the rst Problem n IT FE GE Time f(x) kg P (x)k1 7.896D 06 QRTQUAD 120 598 1025 599 0.20 3.624D+06 8.049D 06 HADAMALS 1024 CHEBYQAD 50 841 1340 842 33.75 5.386D 9.549D 06 NONSCOMP 10000 43 44 44 2.81 3.419D 10 7.191D 06 Table 2: Performance of SPG. trial point. An unsuccessful extrapolation corresponds to a failure in the rst attempt to \double" the steplength. Therefore, in an unsuccessful ex- trapolation, an additional \unnecessary" functional evaluation is done and the \next iterate" corresponds to the rst trial point. According to this, we report, in Table 9, the following features of GENCAN (incremental-quotient SPG-IT: SPG iterations, used for leaving the current face. SPG-FE: functional evaluations in SPG-iterations. TN-IT: TN iterations. TN-FE: functional evaluations in TN-iterations. TN-(Step 1)-IT: TN-iterations where the unitary step was accepted. TN-(Step 1)-FE: functional evaluations in TN-iterations where the unitary step was accepted. This is necessarily equal to TN-(Step 1)- IT. TN-(Backtracking)-IT: TN-iterations where backtracking was necessary Problem n IT FE GE CG Time f(x) kg P (x)k1 5.742D 06 EXPQUAD 120 28 5.23 9.133D+03 6.388D 07 QRTQUAD 120 22 28 23 214 0.10 3.625D+06 5.706D 07 CHEBYQAD 50 52 66 53 960 45.70 5.387D 9.535D 06 6.559D 06 Table 3: Performance of BOX-QUACAN. TN-(Backtracking)-FE: functional evaluations at iterations with back-tracking TN-(Extrap(+))-IT: successful iterations with extrapolation. TN-(Extrap(+))-FE: functional evaluations at successful iterations with extrapolation. TN-(Extrap( ))-IT: unsuccessful iterations with extrapolation. TN-(Extrap( ))-FE: functional evaluations at unsuccessful iterations with extrapolation. This number is necessarily equal to twice the corresponding number of iterations. Problem n IT FE GE CG Time f(x) kg P (x)k1 EXPQUAD 120 3.813D 06 NONSCOMP 10000 17 43 19 9.053D 06 Table 4: Performance of GENCAN (true-hessian version). Problem n IT FE GE CG Time f(x) kg P (x)k1 EXPLIN 120 17 43 19 EXPQUAD 120 21 51 23 53 0.03 3.626D+06 2.236D 06 CHEBYQAD 50 31 43 2.929D 06 NONSCOMP 10000 Table 5: Performance of GENCAN (incremental-quotient version). BDEXP 1.969D 2.744D 1.967D QRTQUAD 3.625D+06 3.624D+06 3.625D+06 3.625D+06 3.625D+06 CHEBYQAD 5.386D 5.386D 5.387D 5.386D 5.386D DECONVB 6.393D 09 4.826D 08 5.664D 6.043D QR3DLS 2.245D 08 1.973D 05 1.450D SCON1LS 5.981D 04 1.224D+00 | 1.269D 4.549D 04 Table Final functional values. Problem LANCELOT SPG BOX-QUACAN GENCAN-QUOT FE GE+CG FE GE FE GE+CG FE GE+CG 43 58 43 EXPQUAD QRTQUAD 178 715 1025 599 28 236 75 101 CHEBYQAD 28 486 1340 842 66 1012 43 918 28 2072 1853 1023 19 415 34 87 SCON1LS 9340 5750468 7673022 5000002 | | 8565 4995260 Table 7: Functional and equivalent-gradient (GE+CG) evaluations. MCCORMCK 4.24 2.27 5.23 4.57 3.56 HADAMALS 4.40 1.63 | 1.80 1.19 CHEBYQAD 2.22 33.75 45.70 13.86 22.26 QR3DLS 439.31 2203.97 2286.09 976.13 523.50 Table 8: Computer time. Type of GENCAN iterations Details of Truncated Newton iterations SPG Iteration TN iterations Step=1 Backtracking Extrap(+) Extrap( ) Problem IT FE IT FE IT FE IT FE IT FE IT FE Table 9: GENCAN features. Observing Table 9 we realise that: 1. The number of SPG-iterations is surprisingly small. Therefore, only in few iterations the mechanism to \leave the face" is activated. So, in most iterations, the number of active constraints remains the same or is increased. Clearly, SPG-iterations are not responsible for the relatively high number of functional evaluations. 2. The number of iterations where backtracking was necessary is, also, surprisingly small. Therefore, extrapolations are responsible for the functional-evaluations phenomenon. Since an unsuccessful extrapolation uses only one additional (unnecessary) functional evaluations, its contribution to increasing FE is also moderate. In fact, unsuccessful extrapolations are responsible for 116 functional evaluations considering all the problems, this means less than 8 evaluations per problem. It turns out that many functional evaluations are used in successful extrapolations. Considering the overall performance of the method this seems to be a really good feature. An extreme case is BDEXP, where only one TN-iteration was performed, giving a successful extrapolation that used 11 functional evaluations and gave the solution of the problem. Further remarks Convergence was obtained for all the problems with all the methods tested, with the exception of SPG that did not solve SCON1LS after more than thirty hours of computer time. The method that, in most cases, obtained the lowest functional values was GENCAN-QUOT, but the dier- ences do not seem to be large enough to reveal a clear tendency. As was already mentioned in [7], the behavior of SPG is surprisingly good. Although it is the only method that fails to solve a problem in reasonable time, its behavior in the problems where it works is quite e-cient. This indicates the existence of families of problems where SPG is, probably, the best possible alternative. This observation has already been made in [8]. BOX-QUACAN has been the less successful method in this set of ex- periments. This is not surprising, since the authors of [16] had observed that this method outperformed LANCELOT in quadratic problems but is not so good when the function is far from being quadratic. In fact, it was this observation what motivated the present work. Nevertheless, there is still a large scope for improvements of BOX-QUACAN, as far as we take into account that improvements in the solution of quadratic subproblems are possible and that sophisticated strategies for updating trust-region radius can be incorporated. 6 Experiments with very large problems We wish to place q circles of radius r in the rectangle [0; d 1 in such a way that for all qg, the intersection between the circle i and the circle j is at most one point. Therefore, given I i the goal is to determine c solving the problem: Minimize subject to r c i r c i The points c are the centers of the desired circles. If the objective function value at the solution of this minimization problem is zero, then the original problem is solved. When I the problem above is known as the Cylinder Packing problem [22]. The present generalization is directed to Sociometry applications. Table describes the main features of some medium and large scale problems of this type. In the problems 9-15 the sets I i were randomly generated with the Schrage's random number generator [35] and seed = 1. In all cases we used 0:5. Observe that n, the number of variables, is equal to 2 q. Tables 11 and 12 show the performances of GENCAN and LANCELOT. Internal limitations of the big-problems installation of CUTE forbid the solution of larger instances of this problems using SIF. We show the CPU Problem n #I i box Table 10: Medium- and large-scale classical and modied cylinder packing problems. times of GENCAN both using SIF (SIF-Time) and Fortran subroutines (FS- for computing function and gradient. We used a random initial point (generated inside the box with the Schrage's algorithm and seed equal to 1). Both methods found a global solution in all the cases. In Table 12 we also report the number of free variables at the solution so far found by GENCAN. GENCAN using Fortran subroutines using SIF LANCELOT IT FE GE CG Time IT FE GE CG Time IT FE GE CG Time Table 11: GENCAN and LANCELOT with cylinder packing problems. 9 28 10649.79 9914682 Table 12: GENCAN with very large problems. 7 Final remarks Numerical algorithms must be analyzed not only from the point of view of its present state but also from considerations related to their possibility of im- provement. The chances of improvement of active-set methods like the one presented in this paper come from the development of new unconstrained algorithms and from the adaptation of known unconstrained algorithms to the specic characteristics of our problem. In our algorithm, the computation of the search direction is open to many possibilities. As we mentioned in the introduction, a secant multipoint scheme (with a dierent procedure for leaving the faces) was considered in [10] and a negative-curvature Newtonian direction for small problems was used in [6], where leaving faces is also associated to SPG. A particularly interesting alternative is the preconditioned spectral projected gradient method introduced in [30]. The extension of the technique of this paper to general linearly constrained optimization is another interesting subject of possible research. From the theoretical point of view, the extension is straightforward, and the convergence proofs do not oer technical di-culties. The only real di-culty is that we need to project onto the feasible set, both in the extrapolation steps and in the SPG iterations. In theory, extrapolation can be avoided without aecting global convergence, but projections are essential in SPG iterations. Sometimes, the feasible polytope is such that projections are easy to compute. See [8]. In those cases, the extension of GENCAN would probably be quite e-cient. Acknowledgements . The authors are very grateful to Nick Gould, who helped them in the use of the SIF language. We are also indebted to two anonymous referees whose comments helped us a lot to improve the nal version of the paper. --R On the resolution of the generalized nonlinear complementarity problem. A limited memory algorithm for bound constrained minimization. Restricted opti- mization: a clue to a fast and accurate implementation of the Common Re ection Surface stack method CUTE: constrained and unconstrained testing environment. Global convergence of a class of trust region algorithms for optimization with simple bounds. A globally convergent augmented Lagrangean algorithm for optimization with general constraints and simple bounds Numerical methods for unconstrained optimization and nonlinear equations. Comparing the numerical performance of two trust-region algorithms for large-scale bound-constrained minimization Optimising the palletisation of cylinders in cases Matrix Computations. Preconditioned spectral gradient method for unconstrained optimization problems On the Barzilai and Borwein choice of steplength for the gradient method The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem. A more portable Fortran random number generator. A class of inde --TR Global convergence of a class of trust region algorithms for optimization with simple bounds A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds A limited memory algorithm for bound constrained optimization Matrix computations (3rd ed.) Gradient Method with Retards and Generalizations Estimation of the optical constants and the thickness of thin films using unconstrained optimization A More Portable Fortran Random Number Generator Trust-region methods Validation of an Augmented Lagrangian Algorithm with a Gauss-Newton Hessian Approximation Using a Set of Hard-Spheres Problems Duality-based domain decomposition with natural coarse-space for variational inequalities0 Algorithm 813 On the Resolution of the Generalized Nonlinear Complementarity Problem A Class of Indefinite Dogleg Path Methods for Unconstrained Minimization The Barzilai and Borwein Gradient Method for the Large Scale Unconstrained Minimization Problem Nonmonotone Spectral Projected Gradient Methods on Convex Sets Newton''s Method for Large Bound-Constrained Optimization Problems Constrained Quadratic Programming with Proportioning and Projections Augmented Lagrangians with Adaptive Precision Control for Quadratic Programming with Equality Constraints --CTR G. Birgin , J. M. Martnez, Structured minimal-memory inexact quasi-Newton method and secant preconditioners for augmented Lagrangian optimization, Computational Optimization and Applications, v.39 n.1, p.1-16, January 2008 G. Birgin , R. A. Castillo , J. M. Martnez, Numerical Comparison of Augmented Lagrangian Algorithms for Nonconvex Problems, Computational Optimization and Applications, v.31 n.1, p.31-55, May 2005 G. Birgin , J. M. Martnez , F. H. Nishihara , D. P. Ronconi, Orthogonal packing of rectangular items within arbitrary convex regions by nonlinear optimization, Computers and Operations Research, v.33 n.12, p.3535-3548, December 2006

active-set strategies;numerical methods;box-constrained minimization;Spectral Projected Gradient

606904

Smoothing Methods for Linear Programs with a More Flexible Update of the Smoothing Parameter.

We consider a smoothing-type method for the solution of linear programs. Its main idea is to reformulate the corresponding central path conditions as a nonlinear system of equations, to which a variant of Newton's method is applied. The method is shown to be globally and locally quadratically convergent under suitable assumptions. In contrast to a number of recently proposed smoothing-type methods, the current work allows a more flexible updating of the smoothing parameter. Furthermore, compared with previous smoothing-type methods, the current implementation of the new method gives significantly better numerical results on the netlib test suite.

Introduction Consider the linear program x s.t. are the given data and A is assumed to be of full rank, m. The classical method for the solution of this minimization problem is Dantzig's simplex algorithm, see, e.g., [11, 1]. During the last two decades, however, interior-point methods have become quite popular and are now viewed as being serious alternatives to the simplex method, especially for large-scale problems. More recently, so-called smoothing-type methods have also been investigated for the solution of linear programs. These smoothing-type methods join some of the properties of interior-point methods. To explain this in more detail, consider the optimality conditions (2) of the linear program (1), and recall that (1) has a solution if and only if (2) has a solution. The most successful interior-point methods try to solve the optimality conditions (2) by solving (inexactly) a sequence of perturbed problems (also called the central path conditions) where > 0 denotes a suitable parameter. Typically, interior-point methods apply some kind of Newton method to the equations within these perturbed optimality conditions and guarantee the positivity of the primal and dual variables by an appropriate line search. Many smoothing-type methods follow a similar pattern: They also try to solve (inexactly) a sequence of perturbed problems (3). To this end, however, they rst reformulate the system (3) as a nonlinear system of equations and then apply Newton's method to this reformulated system. In this way, smoothing-type methods avoid the explicit inequality constraints, and therefore the iterates generated by these methods do not necessarily belong to the positive orthant. More details on smoothing methods are given in Section 2. The algorithm to be presented in this manuscript belongs to the class of smoothing-type methods. It is closely related to some methods recently proposed by Burke and Xu [2, 3] and further investigated by the authors in [13, 14]. In contrast to these methods, however, we allow a more exible choice for the parameter . Since the precise way this parameter is updated within the algorithm has an enormous in uence on the entire behaviour of the algorithm, we feel that this is a highly important topic. The second motivation for writing this paper is the fact that our current code gives signicantly better numerical results than previous implementations of smoothing-type methods. For some further background on smoothing-type methods, the interested reader is referred to [4, 6, 7, 8, 16, 17, 20, 22, 23] and references therein. The paper is organized as follows: We develop our algorithm in Section 2, give a detailed statement and show that it is well-dened. Section 3 then discusses the global and local convergence properties of our algorithm. In particular, it will be shown that the method has the same nice global convergence properties as the method suggested by Burke and Xu [3]. Section 4 indicates that the method works quite well on the whole netlib test suite. We then close this paper with some nal remarks in Section 5. A few words about our notation: R n denotes the n-dimensional real vector space. For we use the subscript x i in order to indicate the ith component of x, whereas a superscript like in x k is used to indicate that this is the kth iterate of a sequence fx k g R n . Quite often, we will consider a triple of the form and s 2 R n ; of course, w is then a vector in R n+m+n . In order to simplify our notation, however, we will usually write instead of using the mathematically more correct is a vector whose components are all nonnegative, we simply write x 0; an expression like x 0 has a similar meaning. Finally, the symbol k k is used for the Euclidean vector norm. 2 Description of Algorithm In this section, we want to derive our predictor-corrector smoothing method for the solution of the optimality conditions (2). Furthermore, we will see that the method is well-dened. Since the main idea of our method is based on a suitable reformulation of the optimality conditions (2), we begin with a very simple way to reformulate this system. To this end, let denote the so-called minimum function and let dened by Since ' has the property that a 0; b 0; ab it follows that can be used in order to get a characterization of the complementarity conditions: Consequently, a vector w a solution of the optimality conditions (2) if and only if it satises the nonlinear system of equations The main disadvantage of the mapping is that it is not dierentiable everywhere. In order to overcome this nonsmoothness, several researchers (see, e.g., [7, 5, 18, 21]) have proposed to approximate the minimum function ' by a continuously dierentiable mapping with the help of a so-called smoothing parameter . In particular, the function has become quite popular and is typically called the Chen-Harker-Kanzow-Smale smoothing function in the literature [5, 18, 21]. Based on this function, we may dene the mappings and (w) := Obviously, is a smooth approximation of for every > 0, and coincides with in the limiting case Furthermore, it was observed in [18] that a vector w solves the nonlinear system of equations if and only if this vector is a solution of the central path conditions (3). Solving the system (4) by, say, Newton's method, is therefore closely related to several primal-dual path-following methods which have become quite popular during the last 15 years, cf. [24]. However, due to our numerical experience [13, 14] and motivated by some stronger theoretical results obtained by Burke and Xu [3], we prefer to view as an independent variable (rather than a parameter). To make this clear in our notation, we write and, similarly, from now on. Since the nonlinear system (4) contains only equations and we add one more equation and dene a mapping cf. [3]. We also need the following generalization of the function : here, 2 (0; 1] denotes a suitable centering parameter, and : [0; 1) ! R is a function having the following properties: (P.1) is continuously dierentiable with For each 0 > 0, there is a constant (possibly depending on 0 ) such that The following functions satisfy all these properties: In fact, it is quite easy to see that all three examples satisfy properties (P.1), (P.2), and (P.3). Furthermore, the mapping being independent of 0 . Also the mapping () := being independent of 0 . On the other hand, a simple calculation shows that the third example does satisfy with depends on 0 . Note that the choice corresponds to the one used in [2, 3], whereas here we aim to generalize the approach from [2, 3] in order to allow a more exible procedure to decrease . Since the precise reduction of has a signicant in uence on the overall performance of our smoothing-type method, we feel that such a generalization is very important from a computational point of view. Before we give a precise statement of our algorithm, let us add some further comments on the properties of the function : (P.1) is obviously needed since we want to apply a Newton-type method to the system of equations hence has to be su-ciently smooth. The second property (P.2) implies that is strictly monotonically increasing. Together with 0 from property (P.1), this means that the nonlinear system of equations is equivalent to the optimality conditions (2) themselves (and not to the central path conditions (3)) since the last row immediately gives The third property (P.3) will be used in order to show that the algorithm to be presented below is well-dened, cf. the proof of Lemma 2.2 (c). Furthermore, properties (P.3) and (P.4) together will guarantee that the sequence is monotonically decreasing and converges to zero, see the proof of Theorem 3.3. We now return to the description of the algorithm. The method to be presented below is a predictor-corrector algorithm with the predictor step being responsible for the local fast rate of convergence, and with the corrector step guaranteeing global convergence. More precisely, the predictor step consists of one Newton iteration applied to the system (x; ; s; followed by a suitable update of which tries to reduce as much as possible. The corrector step then applies one Newton iteration to the system but with the usual right-hand side being replaced by centering parameter 1). This Newton step is followed by an Armijo-type line search. The computation of all iterates is carried out in such a way that they belong to the neighbourhood of the central path, where > 0 denotes a suitable constant. In addition, we will see later that all iterates automatically satisfy the inequality (x; s; ) 0, which will be important in order to establish a result regarding the boundedness of the iterates, cf. Lemma 3.1 and Proposition 3.2 below. The precise statement of our algorithm is as follows (recall that and ; denote the mappings from (5) and (6), respectively). Algorithm 2.1 (Predictor-Corrector Smoothing Method) Choose w 0 := and select k(x 0, and set k := 0. (Termination Criterion) If Compute a solution (w of the linear system then set else compute is the nonnegative integer such that (Corrector Step) Choose R n R m R n R of the linear system such that and go to Step (S.1). Algorithm 2.1 is closely related to some other methods recently investigated by dierent authors. For example, if we take then the above algorithm is almost identical with a method proposed by Burke and Xu [3]. It is not completely identical since we use a dierent update for ^ w k in the predictor step, namely for the case ' This is necessary in order to prove our global convergence results, Theorem 3.3 and Corollary 3.4 below. On the other hand, Algorithm 2.1 is similar to a method used by the authors in [14]; in fact, taking once again almost have the method from [14]. The only dierence that remains is that we use a dierent right-hand side in the predictor step, namely (w k ; k ), whereas [14] uses (w k ; 0). The latter choice seems to give slightly better local properties, however, the current version allows to prove better global convergence properties. From now on, we always assume that the termination parameter " in Algorithm 2.1 is equal to zero and that Algorithm 2.1 generates an innite sequence we assume that the stopping criteria in Steps (S.1) and (S.2) are never satised. This is not at all restrictive since otherwise w k or w k would be a solution of the optimality conditions (2). We rst note that Algorithm 2.1 is well-dened. Lemma 2.2 The following statements hold for any k 2 N: (a) The linear systems (7) and (8) have a unique solution. (b) There is a unique k satisfying the conditions in Step (S.2). (c) The stepsize ^ t k in (S.3) is uniquely dened. Consequently, Algorithm 2.1 is well-dened. Proof. Taking into account the structure of the Jacobians 0 (w; ) and 0 using the fact that 0 () > 0 by property (P.2), part (a) is an immediate consequence of, e.g., [12, Proposition 3.1]. The second statement follows from [13, Proposition 3.2] and is essentially due to Burke and Xu [3]. In order to verify the third statement, assume there is an iteration index k such that for all ' 2 N . Since k(^x we obtain from property (P.3) that Taking this inequality into account, the proof can now be completed by using a standard argument for the Armijo line search rule. 2 We next state some simple properties of Algorithm 2.1 to which we will refer a couple of times in our subsequent analysis. Lemma 2.3 The sequences fw k generated by Algorithm 2.1 have the following properties: (a) A T (b) k 0 (1 denotes the constant from property (P.4). (c) Proof. Part (a) holds for our choice of the starting point Hence it holds for all k 2 N since Newton's method solves linear systems exactly. In order to verify statement (b), we rst note that we get from the fourth block row of the linear equation (8). Since it therefore follows from property (P.4) and the updating rules in steps (S.2) and (S.3) of Algorithm 2.1 that Using a simple induction argument, we see that (b) holds. Finally, statement (c) is a direct consequence of the updating rules in Algorithm 2.1. 2 3 Convergence Properties In this section, we analyze the global and local convergence properties of Algorithm 2.1. Since the analysis for the local rate of convergence is essentially the same as in [3] (recall that our predictor step is identically to the one from [3]), we focus on the global properties. In particular, we will show that all iterates remain bounded under a strict feasibility assumption. This was noted by Burke and Xu [3] for a particular member of our class of methods (namely for the choice () := ), but is not true for many other smoothing-type methods like those from [5, 6, 7, 8, 13, 14, 22, 23]. The central observation which allows us to prove the boundedness of the iterates is that they automatically satisfy the inequality for all k 2 N provided this inequality holds for This is precisely the statement of our rst result. Lemma 3.1 The sequences fw k generated by Algorithm 2.1 have the following properties: (a) (^x (b) Proof. We rst derive some useful inequalities, and then verify the two statements simultaneously by induction on k. We begin with some preliminary discussions regarding statement (a). To this end, let be xed for the moment, and assume that we take ^ in Step (S.2) of Algorithm 2.1. Since each component of the function is concave, we then obtain From the third block row of (7), we have Hence we get from (11): We claim that the right-hand side of (12) is nonpositive. To prove this statement, we rst note that with @ @ 0: Hence it remains to show that However, this is obvious since the last row of the linear system (7) implies We next derive some useful inequalities regarding statement (b). To this end, we still assume that k 2 N is xed. Using once again the fact that is a concave function in each component, we obtain from (8) and this completes our preliminary discussions. We now verify statements (a) and (b) by induction on k. For 0 by our choice of the starting point w and the initial smoothing parameter in Step (S.0) of Algorithm 2.1. Therefore, if we set ^ in Step (S.2) of Algorithm 2.1, we also have ^ On the other hand, if we in Step (S.2), the argument used in the beginning of this proof shows that the inequality (^x holds in this case. Suppose that we have immediately implies that we have Consequently, if we have in Step (S.2) of Algorithm 2.1, we obviously have (^x erwise, i.e., if we set ^ in Step (S.2), the argument used in the beginning part of this proof shows that the same inequality holds. This completes the formal proof by induction. 2 We next show that the sequence fw k g generated by Algorithm 2.1 remains bounded provided that there is a strictly feasible point for the optimality conditions (2), i.e., a vector ^ x Proposition 3.2 Assume that there is a strictly feasible point for the optimality conditions (2). Then the sequence fw k generated by Algorithm 2.1 is bounded. Proof. The statement is essentially due to Burke and Xu [3], and we include a proof here only for the sake of completeness. Assume that the sequence fw k generated by Algorithm 2.1 is un- bounded. Since f k g is monotonically decreasing by Lemma 2.3 (b), it follows from Lemma 2.3 (c) that for all k 2 N . The denition of the (smoothed) minimum function therefore implies that there is no index ng such that x k i !1 on a subsequence, since otherwise we would have '(x k in turn, would imply k(x on a subsequence in contrast to (14). Therefore, all components of the two sequences fx k g and are bounded from below, i.e., and s k where R denotes a suitable (possibly negative) constant. On the other hand, the sequence fw k unbounded by assumption. This implies that there is at least one component ng such that x k on a subsequence since otherwise the two sequences fx k g and fs k g would be bounded which, in turn, would imply the boundedness of the sequence f k g as well because we have A T 2.3 (a)) and because A is assumed to have full rank. be a strictly feasible point for (2) whose existence is guaranteed by our assumption. Then, in particular, we have Since we also have for all k 2 N by Lemma 2.3 (a), we get A by subtracting these equations. Premultiplying the rst equation in (16) with (^x x k ) T and taking into account the second equation in (16) gives Reordering this equation, we obtain for all k 2 N . Using (15) as well as ^ in view of the strict feasibility of the it follows from (17) and the fact that x k on a subsequence for at least one index ng that Hence there exists a component ng (independent of k) such that on a suitable subsequence. using Lemma 3.1 (b), we have for all k 2 N . Taking into account the denition of and looking at the j-th component, this implies for all k 2 N . Using (18) and (15), we see that we necessarily have x k those k belonging to the subsequence for which (18) holds. Therefore, taking the square in (19), we obtain after some simplications. However, since the right-hand side of this expression is bounded by 4 2 0 , this gives a contradiction to (18). 2 We next prove a global convergence result for Algorithm 2.1. Note that this result is dierent from the one provided by Burke and Xu [3] and is more in the spirit of those from [22, 13, 14]. (Burke and Xu [3] use a stronger assumption in order to prove a global linear rate of convergence for the sequence f k g.) Theorem 3.3 Assume that the sequence fw k generated by Algorithm 2.1 has at least one accumulation point. Then f k g converges to zero. Proof. Since the sequence f k g is monotonically decreasing (by Lemma 2.3 (b)) and bounded from below by zero, it converges to a number 0. If 0, we are done. So assume that > 0. Then the updating rules in Step (S.2) of Algorithm 2.1 immediately give for all k 2 N su-ciently large. Subsequencing if necessary, we assume without loss of generality that (20) holds for all k 2 N . Then Lemma 2.3 (b) and ^ Y Y by assumption, it follows from (21) that lim Therefore, the stepsize does not satisfy the line search criterion (9) for all k 2 N large enough. Hence we have for all these k 2 N . Now let w be an accumulation point of the sequence fw k g, and let fw k gK be a subsequence converging to w . Since , we can assume without loss of generality that the subsequence f^ k g K converges to some number ^ Furthermore, since > 0, it follows from (20) and Lemma 2.2 (a) that the corresponding subsequence converges to a vector is the unique solution of the linear equation cf. (8). Using f^ k g K ! 0 and taking the limit k !1 on the subset K, we then obtain from (20) and (22) that On the other hand, we get from (22), (10), property (P.3), (20), Lemma 2.3 (c), and that for all k 2 N su-ciently large. Using (20), this implies is a continuously dierentiable function at due to (24), taking the limit k !1 for k 2 K then gives ^x ^s denotes the solution of the linear system (23). Using (23) then gives a contradiction to (24). Hence we cannot have > 0. 2 Due to Proposition 3.2, the assumed existence of an accumulation point in Theorem 3.3 is automatically satised if there is a strictly feasible point for the optimality conditions (2). An immediate consequence of Theorem 3.3 is the following result. Corollary 3.4 Every accumulation point of a sequence fw k generated by Algorithm 2.1 is a solution of the optimality conditions (2). Proof. The short proof is essentially the same as in [14], for example, and we include it here for the sake of completeness. | Let w be an accumulation point of the sequence fw k K denote a subsequence converging to w . Then we have k ! 0 in view of Theorem 3.3. Hence Lemma 2.3 (c) implies i.e., we have x 0; s 0 and x due to the denition of . Lemma 2.3 (a) also shows that we have A T we see that indeed a solution of the optimality conditions (2). 2 We nally state our local rate of convergence result. Since our predictor step coincides with the one by Burke and Xu [3], the proof of this result is essentially the same as in [3], and we therefore omit the details here. Theorem 3.5 Let the parameter satisfy the inequality > 2 n, assume that the optimality conditions (2) have a unique solution w suppose that the sequence generated by Algorithm 2.1 converges to w . Then f k g converges globally linearly and locally quadratically to zero. The central observation in order to prove Theorem 3.5 is that the sequence of Jacobian matrices 0 (w k ; k ) converges to a nonsingular matrix under the assumption of Theorem 3.5. In fact, as noted in [3, 12], the convergence of this sequence to a nonsingular Jacobian matrix is equivalent to the unique solvability of the optimality conditions (2). We implemented Algorithm 2.1 in C. In order to simplify the work, we took the PCx code from [10, 9] and modied it in an appropriate way. PCx is a predictor-corrector interior-point solver for linear programs, written in C and calling a FORTRAN subroutine in order to solve certain linear systems using the sparse Cholesky method by Ng and Peyton [19]. Since the linear systems occuring in Algorithm 2.1 have essentially the same structure as those arising in interior-point methods, it was possible to use the numerical linear algebra part from PCx for our implementation of Algorithm 2.1. We also apply the preprocessor from PCx before starting our method. The initial point w is the same as the one used for our numerical experiments in [14] and was constructed in the following way: (a) Solve AA T using a sparse Cholesky code in order to compute y 0 (b) (c) Solve AA T using a sparse Cholesky code to compute 0 Note that this starting point is feasible in the sense that it satises the linear equations b. Furthermore, the initial smoothing parameter was set to i.e., 0 is equal to the initial residual of the optimality conditions (2) (recall that the starting vector satises the linear equations in (2) exactly, at least up to numerical inaccuracies). In order to guarantee that however, we sometimes have to enlarge the value of 0 so that it satises the inequalities ng with x 0 Note that the same was done in [14]. We also took the stopping criterion from [14], i.e., we terminate the iteration if one of the following conditions hold: (a) Finally, the centering parameter ^ k was chosen as follows: We let ^ 0:1, start with ^ if the predictor step was successful (i.e., if we were allowed to take ^ otherwise. This strategy guarantees that all centering parameters belong to the interval According to our experience, a larger value of usually gives faster convergence, but the entire behaviour of our method becomes more unstable, whereas a smaller value of the centering parameter gives a more stable behaviour, while the overall number of iterations increases. The dynamic choice of ^ above tries to combine these observations in a suitable way. The remaining parameters from Step (S.0) of Algorithm 2.1 were chosen as follows: We rst consider the function () := (this, more or less, corresponds to the method from All test runs were done on a SUN Enterprise 450 with 480 MHz. Table 1 contains the corresponding results, with the columns of Table 1 having the following meanings: problem: name of the test problem in the netlib collection, m: number of equality constraints (after preprocessing), n: number of variables (after preprocessing), k: number of iterations until termination, P: number of accepted predictor steps, value of k at the nal iterate, value of k(w k )k 1 at the nal iterate, primal objective: value of the primal objective function at nal iterate. Moreover, we give the number of iterations needed by the related method from [14] in parantheses after the number of iterations used by our new method. Table 1: Numerical results for Algorithm 2.1 problem objective 1.758e 04 5.50184589e+03 adlittle aro agg 390 477 22 (23) 17 3.8e 02 6.257e 04 3.59917673e+07 agg2 514 750 22 (25) agg3 514 750 21 (30) beaconfd 86 171 21 (18) 5.156e 04 3.35924858e+04 blend 3.652e 04 3.35213568e+02 4.166e 06 3.15018729e+02 bore3d 81 138 14 (28) 11 5.9e 3.980e brandy 133 238 3.469e 04 1.51850990e+03 9.161e 04 2.69000997e+03 cycle 1420 2773 5.207e d2q06c 2132 5728 48 (57) d6cube 403 5443 Table results for Algorithm 2.1 problem objective degen2 2.901e degen3 d 001 | | | (|) | | | | f800 322 826 28 (36) 17 1.2e 5.876e 04 5.55679564e+05 nnis 438 935 20 (31) 17 2.0e 7.843e 04 1.72791066e+05 8.491e t2d 7.494e 04 6.84642932e+04 9.397e forplan 121 447 26 (28) 17 2.2e 4.722e 04 6.64218959e+02 ganges 1113 1510 20 (25) 19 2.4e 1.218e 04 1.09585736e+05 greenbea | | | (25) | | | | greenbeb 1932 4154 43 (35) 13 1.7e 9.559e 04 4.30226026e+06 israel 174 316 17 (27) 15 1.0e 02 4.732e 04 8.96644822e+05 kb2 43 68 1.653e 06 1.74990013e+03 lot 133 346 23 (35) 12 3.2e 7.087e 04 2.52647043e+01 maros 655 1437 22 (37) 14 2.4e 1.738e 04 5.80637437e+04 8.053e 04 1.49718517e+06 3.330e 04 3.20619729e+02 nesm 654 2922 46 (52) 9 4.7e 04 4.718e 04 1.40760365e+07 perold 593 1374 26 (33) 12 2.1e 6.564e 04 9.38075527e+03 pilot 1368 4543 71 (81) 9 9.0e pilot.ja 810 1804 pilot.we 701 2814 36 9.981e 04 2.72010753e+06 6.888e 04 2.58113924e+03 4.059e 04 4.49727619e+03 recipe 4.205e 2.793e sc50a 8.546e sc50b 48 76 7.714e 06 7.00000047e+01 1.049e 04 1.47534331e+07 4.563e 04 2.33138982e+06 6.230e 04 1.84167590e+04 1.834e 04 3.66602616e+04 9.098e 04 5.49012545e+04 scorpion 340 412 19 (21) 14 2.4e 04 1.815e 05 1.87812482e+03 2.169e 04 9.04293215e+02 Table results for Algorithm 2.1 problem objective 7.203e 06 8.66666364e+00 3.131e 8.910e 06 1.41224999e+03 8.233e 1.051e seba 448 901 19 (23) 12 2.5e 1.550e 06 1.57116000e+04 share1b 112 248 29 (43) 14 2.2e 3.762e 04 7.65893186e+04 share2b 96 162 8.099e shell 487 1451 19 (22) ship04l 292 1905 22 (20) 7.616e 04 1.79332454e+06 ship04s 216 1281 1.561e 04 1.79871470e+06 ship08l 470 3121 25 (21) 15 2.1e 7.592e 04 1.90905521e+06 ship08s 276 1604 15 (20) 13 3.0e 02 7.416e 04 1.92009821e+06 ship12l 610 4171 21 (21) 13 7.0e 2.670e 04 1.47018792e+06 ship12s 340 1943 7.548e 2.548e stair 356 532 standata 314 796 standgub 314 796 standmps 422 1192 14 (18) 12 9.6e 4.418e stocfor2 1980 2868 14 stocfor3 15362 22228 23 (63) 19 2.8e 04 5.514e 05 3.99767839e+04 stocfor3old 15362 22228 23 (70) 19 2.8e 04 5.514e 05 3.99767839e+04 truss 1000 8806 3.621e 04 4.58815785e+05 vtp.base Table clearly indicates that our current implementation works much better than our previous code from [14]. In fact, for almost all examples we were able to reduce the number of iterations considerably. We nally state some results for the function () := giving another complete list, however, we illustrate the typical behaviour of this method by presenting the corresponding results for those test examples why lie between kb2 and scagr7 (this list includes the di-cult pilot* problems) in Table 2. Table 2: Numerical results with quadratic function problem objective kb2 43 68 15 9 2.0e 2.458e lot 133 346 22 9 3.0e 6.715e 04 2.52647449e+01 maros 655 1437 20 11 3.0e 3.805e 04 5.80637438e+04 9.450e 04 1.49718510e+06 modszk1 665 1599 26 11 2.5e 3.087e nesm 654 2922 perold 593 1374 55 11 5.7e 05 2.585e 04 9.38075528e+03 pilot 1368 4543 53 7 1.4e 04 2.953e 04 5.57310815e+02 pilot.we 701 2814 43 4 9.8e 04 9.283e 04 2.72010754e+06 5.672e 04 2.58113925e+03 3.573e 04 4.49727619e+03 recipe 1.928e 1.193e 04 5.22020686e+01 sc50a 3.224e sc50b 48 76 11 9 4.1e 4.955e 9.326e Concluding Remarks We have presented a class of smoothing-type methods for the solution of linear programs. This class of methods has similar convergence properties as the one by Burke and Xu [3], for example, but allows a more exible choice for the updating of the smoothing parameter . The numerical results presented for our implementation of this smoothing-type method are very encouraging and, in particular, signicantly better than for all previous implementa- tions. The results also indicate that the precise updating of the smoothing parameter plays a very important role for the overall behaviour of the methods. However, this subject certainly needs to be investigated further. --R Introduction to Linear Programming. A global and local superlinear continuation-smoothing method for P 0 and R 0 NCP or monotone NCP A global linear and local quadratic noninterior continuation method for nonlinear complementarity problems based on Chen-Mangasarian smoothing functions A class of smoothing functions for nonlinear and mixed complementarity problems. Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities. PCx: An interior-point code for linear programming PCx User Guide. Linear Programming and Extensions. On the solution of linear programs by Jacobian smoothing methods. Improved smoothing-type methods for the solution of linear programs A special Newton-type optimization method A complexity analysis of a smoothing method using CHKS-functions for monotone linear complementarity problems Global convergence of a class of non-interior point algorithms using Chen-Harker-Kanzow-Smale functions for nonlinear complementarity problems Some noninterior continuation methods for linear complementarity prob- lems Block sparse Cholesky algorithm on advanced uniprocessor computers. A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities. Algorithms for solving equations. Analysis of a non-interior continuation method based on Chen-Mangasarian smoothing functions for complementarity problems bounds and superlinear convergence analysis of some Newton-type methods in optimization --TR Block sparse Cholesky algorithms on advanced uniprocessor computers A non-interior-point continuation method for linear complementarity problems A class of smoothing functions for nonlinear and mixed complementarity problems Some Noninterior Continuation Methods for LinearComplementarity Problems Primal-dual interior-point methods Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities A Global Linear and Local Quadratic Noninterior Continuation Method for Nonlinear Complementarity Problems Based on Chen--Mangasarian Smoothing Functions A Global and Local Superlinear Continuation-Smoothing Method for <i>P</i>_{<FONT SIZE="-1">0}</FONT> and <i>R</i>_{<FONT SIZE="-1">0}</FONT> NCP or Monotone NCP A Complexity Analysis of a Smoothing Method Using CHKS-functions for Monotone Linear Complementarity Problems A Complexity Bound of a Predictor-Corrector Smoothing Method Using CHKS-Functions for Monotone LCP

global convergence;linear programs;central path;quadratic convergence;smoothing method

606910

Streams and strings in formal proofs.

Streams are acyclic directed subgraphs of the logical flow graph of a proof representing bundles of paths with the same origin and the same end. The notion of stream is used to describe the evolution of proofs during cut-elimination in purely algebraic terms. The algebraic and combinatorial properties of flow graphs emerging from our analysis serve to elucidate logical phenomena. However, the full logical significance of the combinatorics, e.g. the absence of certain patterns within flow graphs, remains unclear.

Introduction The analytical method which divides proofs into blocks, analyses them separately and puts them together again, proved its failure: by "cutting up" it destroys what it seeks to understand, that is the dynamics within proofs [CS97]. This important point has been understood and emphazised by J-Y. Girard who, in 1987, introduced proof nets to study proofs as global entities and to study the way that formulas interact in a proof through logical connectives. In 1991, another notion of graph associated to proofs has been introduced by S. Buss [Bus91] for different purposes (namely, as a tool to show the undecidability of k-provability) and has been employed in [Car97, Car96, Car97a] to study dynamics in proofs. This graph, called logical flow graph, traces the flow of occurrences of formulas in a proof. The combinatorics and the complexity of the evolution of logical flow graphs of proofs under cut-elimination are particularly complicated and intruiging. An overview can be found in [CS97] and a combinatorial analysis is developed in [Car97b]. These difficulties constitute the main reason for looking at simpler but well-defined subgraphs of logical flow graph and try to study their properties and behavior in proofs. We shall concentrate on streams (defined in Section 3). A stream represents a bundle of paths traversing occurrences of the same atomic formula in a proof and having the same origin and the same target. A proof is usually constituted by several streams. They interact with each other because of logical rules and share common paths because of contractions. There are cases where a bundle of paths needs to be exponentially large in size like in the propositional cut-free proofs of the pigeon-hole principle for instance (this is a consequence of [Hak85] and a formal argument is found in [Car97c]), and the study of streams becomes relevant for the study of complexity of sequent calculus proofs. Our interest lies on the topological properties of streams. We shall be concerned only with a rough description of logical paths in a stream. This description will be based on axioms, cuts and contraction rules occurring in the proof. Rules introducing logical connectives will not play any role. This simplified treatment of logical paths allows for a description of proofs as strings (Section 6), and for a natural algebraic manipulation of proofs (Sections 7 and 8). When logical flow graphs contain cycles, this description leads to precise relations between proofs and finitely presented groups [Car98]. Here, we will only look at proofs whose logical flow graph is acyclic and we shall develop a theory which relates algebraic strings to streams. We prove that any stream can be described by an algebraic string, and that for any string there is always a proof with a logical flow graph which is a stream described by the string (Section 6). Usually, several strings can describe the same stream. We shall characterise the most compact and the most explicit ones (Sections 6). In Section 8 we show that the transformation of streams during the procedure of cut-elimination (Section 2.2) can be simulated by a finite set of rewriting rules (Theorem 26). The notion of stream, though simple, is shown to be very powerful at the computational level: Example 27 illustrates how, at times, a purely algebraic manipulation of streams can completely describe the proof transformation. Theorem 28 pinpoints how weak formulas in a proof influence the complexity of streams during cut-elimination. Theorem 29 says that a growth in complexity is either already "explicit" in a proof (i.e. a proof contains a stream with large arithmetical value; this notion is defined in Section 7) or it is due to purely global effects induced by local rules of transformation. To conclude, let us mention that our algebraic analysis of proofs seems adequate to approach the problem of the introduction of cuts in proofs, an important topic in proof theory and automated deduction. It seems plausible that a theory of the flow of information in a proof might lead to develop methods for the introduction of cuts in proofs. Basic notions and notation In this section we briefly recall known concepts. For limitations of space, in most cases, we shall refer the reader to the literature. Good sources are [Gir87a, Tak87], and also [CS97]. 2.1 Formal proofs Formal proofs are described in the sequent calculus LK. This system is constituted by axioms which are sequents of the form any formula and \Gamma; \Delta are any collections of formulas, by logical rules for the introduction of logical connectives and by two structural rules Cut Contraction We shall extend LK with the rule F -rule where F is a unary predicate and is a binary function. The F -rule is added to LK because it allows to speak more directly about computations. It was considered already in [Car97b, CS96a, Car96, CS97a]. In our notation, a rule is always denoted by a bar. The sequent(s) above the bar is called antecedent of the rule and the sequent below the bar is called consequent. In the following we will frequently use the notion of occurrence of a formula in a proof as compared to the formula itself which may occur many times. Notions as positive and negative occurrence of a formula in a sequent are defined in In an axiom the two formulas A are called distinguished formulas and the formulas in \Gamma; \Delta are called side formulas. A formula A in \Pi which has been introduced as a side formula in some axiom is called A formal proof is a binary tree of sequents, where each occurrence of a sequent in a proof can be used at most once as premise of a rule. The root of the tree is labelled by the theorem, its leaves are labelled by axioms and its internal nodes are labelled by sequents derived from one or two sequents (which label the antecedents of the node in the tree) through the rules of LK and the F -rule. The height of a rule R in a proof \Pi is the distance between the consequent of R and the root of the proof-tree describing \Pi. At times we shall consider proofs \Pi which are reduced in the sense of [Car97b], i.e. there are no superfluous redundancies in the proof which have been built with the help of weak occurrences. More formally, no binary rule or contraction rule is applied to a weak formula, no unary logical rule is applied to two weak formulas and no occurrence in cut-formulas is weak. In [Car97b] it is shown that given any proof, we can always find a reduced proof of the same end-sequent which has a number of lines and symbols bounded by the ones of the original proof. 2.2 Cut-elimination In 1934 Gentzen proved the following result Any proof in LK can be effectively transformed into a proof which never uses the cut-rule. This works for both propositional and predicate logic. The statement holds for the extension of LK with the F -rule as well. This is a fundamental result in proof theory and in [CS97] the reader can find a presentation of its motivations and consequences. The computational aspects of the theorem have been largely investigated but we are still far from an understanding of the dynamical process which can occur within proofs [Car97, Car96, Car97a]. After the elimination of cuts, the resulting proof may have to be much larger than the proof with cuts. For propositional proofs, this expansion might be exponential and for proofs with quantifiers, it can be super- exponential, i.e. an exponential tower of 2's [Tse68, Ore79, Sta73, Sta78, Sta79]. We will not enter into the details of the steps of transformation of the procedure of cut-elimination. The reader which is unfamiliar with them can refer to [Gir87a] or [Tak87] (and also to [CS97] or to [Car97b]). 2.3 Logical flow graphs As described in [Car97], one can associate to a given proof a logical flow graph by tracing the flow of atomic occurrences in it. (The notion of logical flow graph was first introduced by Buss in [Bus91] and a similar notion is due to Girard and appeared in [Gir87]. Here we restrict Buss' notion to atomic formulas.) We will not give the formal definition but we will illustrate the idea with an example. Consider the two formal proofs below formalized in the language of propositional logic and the sequences of edges that one can trace through them cut contraction contraction Each step of deduction manipulates formulas following a logically justified rule, and precise links between the formulas involved in the logical step are traced (the arrows indicated in the figures above represent some of these links). Formulas in a proof correspond to nodes in the graph and logical links induced by rules and axioms correspond to edges. As a side effect different occurrences of a formula in a proof might be logically linked even if their position in the proof is apparently very far apart. Between any two logically linked occurrences there is a path. The graph that we obtain is in general disconnected and each connected component corresponds to a different atomic formula in the proof. The structure of the proof on the left is interesting because shows that paths in a proof can get together through contraction of formulas, and the structure on the right shows that cyclic paths might be formed. The orientation on the edges of a logical flow graph is induced by natural considerations on the validity of the rules of inference which we shall not discuss here (see [Car97]). In the following we will not really exploit the direction of the paths. We will use directions only to establish that a path starts and ends somewhere. We might speak of a path going up or down, and of an edge being horizontal, in case the edge appears in an axiom or between cut-formulas. These latter will be called axiom-edges and cut-edges, respectively. In the sequel, we call bridge any maximal oriented path that starts from a negative occurrence, ends in a positive occurrence and does not traverse cut- edges. The maximality condition implies that both the starting and ending occurrences of a bridge should lie either in a cut-formula or in the end-sequent of the proof. A node of a logical flow graph is called a branching point if it has exactly three edges attached to it. In a proof, branching points correspond to formulas obtained by contraction or by a F :rule. We say that a node is a focussing branching point if there are two edges oriented towards it. A node is called defocussing branching point if the two edges are oriented away from it. A node is called input vertex if there are no edges in the graph which are oriented towards it. A vertex is called output vertex if there are no edges in the graph which are oriented away from it. Input and output nodes are called extremal points. In a proof, extremal points correspond to weak occurrences of formulas and to occurrences of formulas in the end-sequent. By a focal pair we mean an ordered pair (u; w) of vertices in the logical flow graph for which there is a pair of distinct paths from u to w. We also require that these paths arrive at w along different edges flowing into w. Two logical flow graphs have the same topological structure if they can both be reduced to the same graph by collapsing each edge between pairs of points of degree at most 2 to a vertex. The notions of bridge, focal pairs, topological structure, focussing and defo- cussing point, input and output node have been introduced in [CS97a, Car97b] where the reader can find properties and intuition. A stream is an acyclic directed graph with one input vertex v and one output vertex w. The pair (v; w) is called base of the stream. All other vertices in the stream are not extremal. If G is a directed graph, then a stream in G is a subgraph of G which is a stream. A full stream in G based on (v; w) is a stream in G such that all directed paths lying in G between v and w belong to the stream. A substream of a stream is a subgraph of the stream that is based on a pair (v; w), where v; w are nodes of the stream. A substream might have the same base of the stream, but this is not required. A stream of a proof \Pi is a stream in the logical flow graph of \Pi such that the input and output vertices occur in the end-sequent of \Pi. A stream of a proof \Pi is based on the pair of formulas (A; B), where A is a positive occurrence and B is a negative occurrence. In the simplest case, a stream of a proof is a bridge, but usually, the stream of a cut-free proof will look (after stretching it) roughly as follows input vertex output vertex where the bifurcation points correspond to the presence of contractions or F - rules in a proof, the circles correspond to axiom-edges through which each path has to pass, and the paths are oriented from left to right. In this way all bifurcation points on the left hand side of the axiom-edges will correspond to contractions on negative occurrences and the bifurcation points on the right will correspond to contractions on positive occurrences or applications of F -rules. If the proof contains cuts, a stream might be much more complicated. For instance it might contain arbitrarily long chains of focal pairs as illustrated by the following figure output vertex input vertex where each horizontal edge (linking a focussing point to a defocussing point) corresponds to a cut in the proof [Car97b]. It might also contain cyclic paths, but we will not consider this situation here. The reader interested in cycles in proofs can refer to [Car97c, Car97b, Car97a, Car96] for their combinatorics and complexity. Usually, a stream lying in an acyclic logical flow graph associated to a proof and containing cuts, looks roughly as follows where we see a chain of shapes similar to the one associated to cut-free proofs but where single bridges are now allowed to take themselves this shape. As usual, axioms lie along paths between the point where defocussing points end and focussing points start to appear. The following property of streams illustrates their regularity Proposition 1 The number of focussing points in a stream is the same as the number of its defocussing points. Proof. The claim follows from a simple fact. Let P be an acyclic directed graph which is connected, contains one input vertex, n focussing points and m defocussing ones. Then, P has nodes. This is easily proved by induction on the values of n; m by noticing that a defocussing node induces the number of distinct output nodes to increase by 1, and a focussing node induces the number of distinct output nodes to decrease by 1. From this we conclude that if P is a stream, then exactly one input vertex and one output vertex.4 Interaction of streams in proofs A logical flow graph of a proof is a union of connected components. There might be many of them, and typically, each component corresponds to a distinguished atomic formula in the proof. Each component is a directed graph which has input nodes as well as output nodes. When no cycles appear in a proof, this is easy to check. To show the assertion in the general case, one needs to show that cyclic logical flow graphs (e.g. graphs with several nested cycles) must have incoming edges and outcoming ones. This follows from Theorems 25 and 54 of [Car97b]. Each component is usually constituted by several streams as illustrated by the graph on the left x x x which contains two streams, the first has base (x; y 1 ) and the second (x; y 2 ) (as illustrated on the right). Connected components and streams define two different types of interaction in a proof: 1. distinct streams can share subgraphs (as in the example above) and can influence one another. This interaction is analysed in [Car97b] through a combinatorial study of cut-elimination. 2. different connected components belong to the same logical flow graph because of logical connectives. The interaction between distinguished connected components has been studied through the notion of proof-net by Girard, Lafont, Danos, Regnier and many others. Girard's seminal paper [Gir87] introduces the reader to the area. We shall skip here the numerous references. In this paper we shall not address any question concerning interaction with the exception of Section 8. 5 Strings and stream structures In this section we introduce a language to represent a stream as a string formalized in a language of three symbols b; ; +, where b is a constant, is the binary operation of concatenation, and + is the binary operation of bifurcation. We will also need two extra symbols (; ) to be used as separators. The language will be called S and the words in S are simply called strings. string is a word in the language Ssatisfying one of the following conditions i. b is a string ii. if w are strings then w 1 w 2 is a string iii. if w are strings then is a string. Example 3 The string b corresponds to a stream which looks like a sequence of edges. The diagrams below illustrate the behavior of the operations of concatenation and of bifurcation + on streams. Obviously the pair of streams that we consider in the diagrams below can be substituted by any other pair. The topology of streams changes with the application of both operations since new branching points are generated. Through concatenation we create sequences of focal pairs and through bifurcation we create new focal pairs. We define a first order theory whose axioms are universally quantified equations Definition 4 A stream structure is an algebra of terms satisfying the following axioms A3 Axioms us that there is no topological change that can be achieved by concatenating a bridge to a stream. Axioms A2; A2 0 can be illustrated in terms of streams as follows By the point of view of the input vertex, the structure of the two graphs is identical. Namely, the number of paths going from the input vertex to the output vertex remains unchanged for both graphs. Axioms A3 and A4 guarantee that the topological structure of the stream is preserved by commutativity and associativity of +. In fact, we shall be interested in streams up to isomorphism. Definition 5 A stream structure is called associative when it satisfies (associativity of ) and it is called commutative when it satisfies One can think of A1-A6 as universally quantified axioms over variables w, stream structure is constituted by an abelian additive semi-group and by a multiplicative part. The distributivity law holds. Notice also that the operation of concatenation is not commutative in general and that in an associative stream structure we do not distinguish substrings of the form (w 1 Since the operation of bifurcation is associative, in the following we will drop parenthesis when not necessary. For instance, the word (w 1 will be We will also use the shorthand notation w n instead of times and we will call n the multiplicity of w. Proposition 6 The following properties are satisfied in any stream structure 1. w wn 2. wn wn 3. permutation - Proof. To check the three properties it is a routine. Note that they correspond to axioms A2; A3. They are derived from their corresponding axiom with the help of A4.In a stream structure there are infinitely many non-equivalent strings. Namely, for any two positive integers n; m, we have b n Therefore a stream structure contains at least a countable number of non-equivalent terms. Proposition 7 (Normalization) Let w be a string. For every stream struc- ture, there is a unique integer k - 1 such that w is equivalent to b k in the stream structure. Proof. Let the height h(w) of a string w be defined as follows: By induction on the height of the string w we show that there is a k such that w is equivalent to b k . If w is b then If w is of the form w 1 w 2 then by induction hypothesis, w 1 is equivalent to is equivalent to b l . Therefore w 1 w 2 is equivalent to b m b l . On the other hand, by property 2 in Proposition 6 and axiom A1 one derives times times If w is of the form w 1 +w 2 then by induction hypothesis w are equivalent to is equivalent to b m l and hence to b m+l by axiom A6. To show uniqueness, let us interpret the concatenation as the operation of multiplication on natural numbers, the bifurcation + as addition and the constant b as the number 1. In this way, the model of natural numbers becomes a natural model for the stream structure. In particular, given a string w there is exactly one value k such that w is equivalent to b k . If this was not the case, then the natural numbers could not be a model for the stream structure.Proposition 8 (Cancellation modulo torsion) The following properties are satisfied in any stream structure 1. b w 2. w 1 3. w w 4. w 1 Proof. Properties are derived using axioms A1; To derive property 3 we apply Proposition 7 to the string w and observe that there is a positive integer n such that w is equivalent to b n . Hence w w . By right distributivity (i.e. axiom (where the right hand side contains n strings of the form b w 1 ) and analogously 2 . Similarly, one shows property 4. In this case axioms A1 0 and should be used instead.6 Strings and streams of proofs To associate a string to a stream in a proof, we think of the logical flow graph of the proof as being embedded in the plane, we read a bridge as the constant b, we read a cut-edge as performing the operation of concatenation , and we express the nesting of bridges through the operation of bifurcation +. Definition 9 A decomposition of an acyclic directed graph P is a set of streams lying on P such that 1. each directed path in P belongs to exactly one of the streams P j , 2. the input node of P j is an input node of P , 3. the output node of P j is an output node of P . If P is a graph lying in the logical flow graph of a proof \Pi, then the symbol denotes the restriction of P to the logical flow graph of a subproof \Pi 0 of \Pi. \Pi be a proof whose last rule is applied to the subproofs . (If the rule is unary, then consider \Pi 1 only.) A stream P is called an extension of P is a stream of \Pi and fP is a decomposition of In Definition 10, if the last rule of \Pi is a cut, then the number of streams n can be arbitrarily large. In fact, a stream P might pass through the cut- formulas, back and forth, several times. Definition 11 Let P be a stream of a proof \Pi. A string associated to a stream P is built by induction on the height of the subproofs \Pi 0 of \Pi in such a way that the following conditions are satisfied: 1. if \Pi 0 does not contain cuts, let fP be a decomposition of P - \Pi 0 . Each string associated to P i is b m , where m is the number of distinguished directed paths lying in P i , 2. if the last rule of \Pi 0 is not a cut and it is applied to \Pi 1 ; \Pi 2 (if the rule is unary, then consider \Pi be the decomposition of . Then the decomposition of P - \Pi 0 is fP 0 l g where (a) i is an extension of P j and the string associated to P 0 w j is the string associated to P j , or (b) i is a stream obtained by the union of streams P 0 which are extensions of P and are based on the same pair (v string associated to P 0 where w jr is the string associated to P jr , for 3. if the last rule of \Pi 0 is a cut applied to subproofs \Pi is a decomposition of P - \Pi 1 and P - \Pi 2 , then the decomposition of P - \Pi 0 is fP 0 l g, where the P 0 i 's are all possible extensions of the streams in such that (a) i is an extension of P and the string associated to P 0 i is (possibly is the string associated to (b) i is a stream obtained by the union of streams P 0 where each jr is an extension of P jr and all the jr are based on the same pair (v The string associated to P 0 is w 0 jr is w jr ;1 ;s is the string associated to P jr ;s for Let us now give a couple of examples to illustrate how streams of proofs can be read as strings. Example 12 Consider the stream of a proof c c where a path starts on the left hand side of the picture, goes up until it reaches a branching point. Two distinguished paths depart from this branching point, they pass through two axioms in the proof and they rejoin into another branching point to pass through a cut-edge, form a new bridge, pass through a second cut-edge, go up into a new subproof, split once more, join once more and end-up into the right hand side of the picture. The structure of this proof can be described with different strings. For instance, the strings (b 2 b) b 2 , (b b) (b b) b are descriptions of the above structure. One can easily check that the three strings are equivalent in a stream structure. In fact, the first and the second string are equivalent by A2 0 and the second is equivalent to the third by A2. Example 13 Consider two streams of proofs having the following form where the height of a cut in the proof is reflected by the position of horizontal edges in the graph. We read the stream on the left as (b b) b and the stream on the right as b (b b). The parenthesis denote the height of a cut in a proof. In this way the string (b b) b represents a cut between a subproof containing a stream b b and a second subproof containing a bridge. The representation of b (b b) is symmetric. string associated to a stream in a proof \Pi is compact if it is determined as described in Definition 11, where we require that the streams P i lying in decompositions fP relative to subproofs \Pi 0 of \Pi have distinct bases (v Given a stream of a proof, there is a unique compact string associated to it (up to commutativity of +). This is because, for all subproofs \Pi 0 of \Pi, there is only one stream P 0 (up to commutativity of +) which is defined as P - \Pi 0 on a In Example 12, the string (b 2 b) b 2 is compact. Compact strings are a succinct way to represents streams. All other representations have larger complexity, that is a larger number of symbols. Consider, for instance, a string of the form w 1 describing a stream P of a proof based on (v; w). If each w i describes a simple path in P based on (v; w), then w i is of the form b. In this case we say that the string w 1 the stream P explicitly: all paths are described one by one. This description is the most expensive in terms of the number of symbols and we refer to it as explicit representation. Proposition 15 Let \Pi be a cut-free proof. All the streams of \Pi are described by strings of the form b n , where n is bounded by the number of axiom-edges of \Pi. Proof. The claim follows from Definition 11 (assertion 1) and the following observation. Each path belonging to a stream passes through an axiom-edge by definition. Suppose that more than one path belonging to the same stream passes through the same axiom. These paths will pass through the pair of distinguished formulas of the axiom but through distinct atomic occurrences. Therefore there should be a moment along the proof, where the occurrences need to identify (since a stream has one input vertex and one output vertex). But the identification is impossible because of the subformula property which holds for cut-free proofs.Remark 16 The cut-free proof of F (2) which can be constructed from axioms of the form F (t) ! F (t), the F -rule, and contractions on the left, is an example of proof where the number n in Proposition 15 corresponds exactly to the number of axioms in the proof. The logical flow graph of this proof is a stream based on )). Since distinguished formulas in axioms are atomic and all of them are linked to the end-sequent, then the number of axioms in the proof must coincide with the number of paths of the stream. Remark 17 In proofs we can only compose and bifurcate bridges having the same orientation. This justifies the fact that stream structures are not defined to have a group on their additive part but simply an additive semi-group. 6.1 From strings to streams We show that any string is associated to some stream of a proof. Theorem For each string there is a proof with a stream described by the string. Proof. Let w be a string. We shall build a proof \Pi w whose end-sequent is whose logical flow graph is a stream associated to w. The construction is done by induction on the complexity of the substrings. If w is b then \Pi w is an axiom of the form F If w is w 1 w 2 then by induction we know \Pi w1 and \Pi w2 . The end-sequents of \Pi w1 and \Pi w2 are F and g(x). By substituting the occurrences of the variable x in \Pi w2 with the term f(x) we obtain a proof \Pi 0 w2 with end-sequent F (f(x)) ! F (g(f(x))), the same logical structure as \Pi w2 and the same logical graph. (This is straightforward to check.) Then, we combine with a cut on the formula F (f(x)) the proofs \Pi w1 and \Pi 0 w2 and obtain a proof of the sequent F (x) ! F (g(f(x))) whose associated string is w 1 w 2 (by Definition 11). If w is then by induction we know \Pi w1 and \Pi w2 . Their end- sequents are of the form F f(x) and g(x). We apply the F -rule to \Pi w1 and \Pi w2 to obtain a proof of applying a contraction to the occurrences F (x) on the left, we obtain the sequent F (x) ! F (f(x) g(x)) and a proof with associated string (by Definition 11).Remark 19 The proof \Pi constructed in Theorem is formalized in the extension of the propositional sequent calculus with F -rules. Notice that there are no occurrences in \Pi and that cuts are only on atomic formulas. In particular, the proof \Pi is reduced. One might be unhappy with the presence of F -rules in \Pi and might like to look for proofs in pure propositional logic. To find proofs containing a required stream is not difficult once we allow an arbitrary use of occurrences in \Pi. To find a propositional proof \Pi which is reduced is, on the other hand, a very difficult task and it is not at all clear whether there is a uniform algorithm that given a stream, returns a proof which contains it. Remark 20 Theorem associates a stream to a given string. The proof containing such a stream is not unique. Take for instance the following transformation of streams due to the procedure of cut-elimination (the existence of such transformations is proved in [Car97b]) where the contraction (on the proof on the left) lying above w is applied much after the cut rule. The streams, before and after cut-elimination, are described by the same string b b also that this is the only possible string describing the above structures. (The fact that the height of the contraction above w is smaller than the height of the cut rule, plays a crucial role here.) 6.2 Strings and topology of streams Usually, proofs having streams with the same topology (i.e. the same display of branching points), might have different strings associated to them. Take for instance the following transformation of streams due to the procedure of cut-elimination [Car97b] where the proof on the left can be described by the strings b b 2 and (b b) 2 , but the proof on the right can only be described by (b b) 2 . Also, consider the following pair of streams where the proof on the left is described by the strings b b 2 and (b b) 2 , and the one on the right by b (b We say that a stream P in \Pi is minimal if for any subproof \Pi 0 of \Pi whose end-sequent is combined with some cut-rule, the graph P - \Pi 0 does not contain simple bridges as connected components. Proposition be two minimal streams. If G 1 and G 2 have the same topological structure then they are described by the same explicit strings. Proof. Let G 1 be a stream for (v; w) and G 2 be a stream of (x; y). By definition an explicit string for a stream is a bifurcation of strings which are concatenations of b's and describe simple paths in the stream. Since G 1 have the same topological structure, the number of paths between v; w and y must be the same, say n. In particular, the two streams are reduced by hypothesis and therefore they have the same number of cut-edges lying along each path. This is enough to conclude that if w is an explicit string of G 1 then wn must also be a string of G 2 . Moreover, this string is unique up to permutation of the components of the bifurcation operator.7 Arithmetical value of strings and complexity If a proof contains cuts, then the compact descriptions for its streams might be much shorter than the explicit ones. Let us illustrate this point with a concrete example where the presence of a chain of focal pairs in a stream is described by a compact string of size n, and by an explicit string of size 2 n . Example 22 We look at a proof of F (2) (This example is taken from [Car97b].) There is no use of quantifiers and the formalization takes place on the propositional part of predicate logic. Our basic building block is given by which can be proved for each j in only a few steps. (One starts with two copies of the axiom F combines them with the F -rule to get Then one applies a contraction to the two occurrences of F ) on the left and derives the sequent.) We can then combine a sequence of these proofs together using cuts to get a proof of F (2) in O(n) steps. The logical flow graph for the proof of F (2) looks roughly as where the notation \Pi refers to the proofs of F (2) ). The logical flow graph of each \Pi j contains two branches, one for the contraction of two occurrences of F ) on the left, and another for the use of the F -rule on the right. Along the graph we notice a chain of n pairs of branches which gives rise to an exponential number of paths starting at F (2) and ending in F (2 2 n There are no cycles in the proof and the logical flow graph of this proof is a stream. The compact string associated to it is b 2 b 2 where each of the corresponds to a focal pair in the graph. The explicit string is b 2 n Can we detect a chain of focal pairs lying in a stream by reading its associated string? To answer let us introduce some more notation. A string can always be seen as a concatenation of strings either of the form b or w wn , wn are strings and n ? 1. For instance, take the string w of the form (b (b 2 +b b 3 We say that are concatenated to each other and we call them factors of w. A factor of the form w 1 is called non-trivial. The number of non-trivial factors of w is the index of w. Proposition 23 Let P be a stream based on (v; w) and lying in the logical flow graph of a proof \Pi. Let w be the string representing P . If w contains a substring of index n then there is a chain of n focal pairs in the logical flow graph of \Pi. Proof. Let P be a stream and w be the string associated to it. By Definition 11, any substring w 0 of w describes a stream lying in some subproof \Pi 0 of \Pi. If w 0 has index n then w 0 is of the form w 1 are non-trivial factors, for the substrings w correspond to streams lying in subproofs \Pi i linked through cuts, and based on pairs (in \Pi 0 the occurrences A are linked by a cut-edge). In particular, the substrings are of the form By Definition 11, w are strings associated to streams based on the same pair (A; B). Therefore, there is at least a focal pair lying in the subproof (because ? 1). This means that in \Pi 0 we have a chain of focal pairs which is defined by the cut-edges connecting the subproofs \Pi and the focal pairs in the \Pi i j 's.As illustrated in Example 22, a chain of n focal pairs lying in a stream gives rise to at least 2 n distinct paths. In Proposition 24, we show that the number of paths in a stream can be computed precisely by means of an arithmetical interpretation of strings. We say that the arithmetical value t(w) associated to a string w is defined as follows: t(b) is 1, t(w 1 is Proposition 24 Let w be a string associated to a stream P . Then the number of directed paths from the input vertex to the output vertex of P is t(w). Proof. This follows in a straightforward way from the interpretation between streams and strings described in Example 3.Proposition 25 Let w be a string and w 0 be a substring of w. Then any substitution of w 0 with a string w 00 where t(w string w such that Proof. The arithmetical term t(w) (once considered in its syntactical form) contains the arithmetical subterm t(w 0 ). If we substitute the occurrence of t(w 0 ) with t(w 00 ) we shall obtain the arithmetical term t(w ) whose value is t(w) since cut-elimination How does a stream of a proof evolve through the procedure of cut elimination? A more general version of this question was addressed in [Car97, Car97b] where the combinatorial operation of "duplication" on directed graphs was introduced and used to analyse the combinatorics of the transformations induced by cut- elimination. Here we would like to show that the evolution of streams can be analysed through simple algebraic manipulations. We give a number of rewriting rules and show that these rules describe the transformation. The set of rewriting rules that we want to consider contains the computational rules wn wn wn wn w wn \Gamma! w permutation - which follow from the axioms A1-A3 and A5. Axiom A4 does not have a counter-part here because from now on we shall consider only compact strings associated to proofs. It also contains the local structural rule which represents the possibility to duplicate the same substrings, and the global structural rules which cancel some of the substrings. It is clear that local and global structural rules allow a string to grow and shrink. Theorem 26 shows how the process of cut-elimination induces streams to shrink and grow. Notice that if w is a string transformed by R6 into w 0 then t(w) ! t(w 0 ). If w is transformed in w 0 by R7 i , On the other hand, if any of the rules R1-R5 are used then Before stating Theorem 26, we need to introduce some more definitions. A reduction is a sequence of applications of rewriting rules that transforms a string w into a string w 0 . An application of a rewriting rule s ! t to w replaces an occurrence of the substring s in w with the substring t. A reduction of a string is called final if it leads to a string of the form b n , for some n. We say that a path in the logical flow graph of a proof is disrupted by a step of cut-elimination when given two nodes of the path, after the transformation there is no more path between them. A stream is disrupted when one of its paths is disrupted. This notion was introduced in [Car97] where the reader can find examples. Theorem 26 Let \Pi be a proof and let w be the compact string associated to some stream of \Pi. For any process of elimination of cuts which gives a cut-free proof with n axioms, either there is a reduction of w to a string b m (where through the rules R1-R7, or the stream is disrupted by some step of elimination of cuts either on weak occurrences or on contractions. Proof. The proof consists of checking that at each stage of the procedure of cut-elimination, the deformation of streams in the proof is regulated by the set of rules R1-R7. Namely, if w is a compact string associated to a stream P in \Pi, and if \Pi 0 is the proof obtained by transforming \Pi through a step of elimination of cuts, then there is a compact string associated to a stream in \Pi 0 which is obtained from w after rewriting one or several of its substrings with the rules R1-R7. Notice that several substrings of w can be simultaneously affected because several paths belonging to the same stream might be involved in the transformation of the same cut. We shall consider first the behavior of a stream P which passes through the cut-formulas that are simplified by the step of the procedure. Let us start by considering the elimination of a cut when one of the cut- formulas is a distinguished occurrence in an axiom. There might be several paths (of the stream P ) that pass through the distinguished occurrences and each of these paths will be denoted b (in the string w) because of compactness. This is shown by an easy chasing of Definition 11. If the axiom appears on the left, then we use R1 to replace substrings b w 0 with w 0 in w. If the axiom appears on the right, then allows to replace substrings of the form w 0 b with w 0 in w. Clearly the string that we obtain is compact. If a cut is applied to formulas with non-trivial logical complexity, i.e. formulas which are not atomic, then a directed path belonging to the stream might pass through the same cut-formula several times and different portions of the same path might behave differently. In particular, several directed paths belonging to the stream might pass through the same cut-formulas. Their behavior will be captured by a simultaneous applications of rules R1-R7 to substrings of w which describe different portions of the stream involved in the transformation. We shall start to consider the case where a cut is applied to two formulas which are main formulas of two logical rules. Several situations might arise. Let us suppose without loss of generality that the formulas are of the form A - B. First, suppose that there is a substream lying in the stream that passes through both A and B and that it is described by a substring (wA wAB ) wB , where wA describes a substream passing through A in the cut-formula on the right, wAB describes a substream passing through both the A and the B in the cut-formula on the left, and wB describes a substream passing through B in the cut-formula on the right. After cut-elimination, the substream is described either by the string (wA wAB ) wB or by the string wA (wAB wB ), depending on the position of the cuts on A and B. In the first case no rewriting rule is applied and the cut on A preceeds the cut on B; in the second case R5 is applied and the cut on B preceeds the cut on A. Second, suppose that there are paths of the stream that pass through exactly one of the disjuncts. In this case the paths will be simply stretched but no change in their description will take place. These are the only two possible situations that might occur. Of course, a path might pass through A and B several times, or wAB might describe a stream passing through A and B on the cut-formula on the right, or the initial substream might be described by wA (wAB wB ). It is easy to imagine all combinations. The main point is that the treatment specified above adapts easily to all other variants. In particular, rule R5 0 might be used instead of R5. If a cut is applied to a formula A obtained from a contraction on two occurrences then the procedure of cut-elimination yields a duplication of a subproof and this creates quite intriguing situations. We start to handle the simplest case. (This case is illustrated in Remark 20.) Suppose that there is a substream lying in the stream that passes through the cut-formula which lies on the proof that will be duplicated by the procedure. Suppose also that the extremes of the substream do not both lie in the cut- formula. Then, the stream has to pass through the sequent resulting from the application of the cut rule because its extremes lie in the end-sequent. Let w 3 be the substring describing the substream. Let w be the substreams passing through the contraction formulas A 1 ; A 2 such that w 3 (w 1 +w 2 ) is a substring describing the topology of this portion of the stream. After duplication of the subproof the substring will be transformed into the substring (w 3 w 1 )+(w 3 w 2 ) and this is done by applying rules R2; Suppose now that the substream above (i.e. the substream lying in the stream that passes through the cut-formula which lies on the proof that will be duplicated by the procedure) is such that both its extreme points belong to the cut-formula. This case is the most intriguing. Let w 3 be the substring describing the substream. After passing through the cut-edges, the stream will go up to the contraction formulas A 1 ; A 2 and it will depart into four paths: two coming from its input vertex, say w and two going towards its output vertex, say It might be that not all the four paths belong to the stream and because of this we shall handle different cases. We illustrate the transformation of the portion of the stream in the following picture input vertex output vertex input vertex output vertex where we think of the streams as being stretched. If all four paths belong to the stream, the transformation is described by rule R7 1 , where the paths w 1 w 3 w 2 and w 4 w 3 w 5 are lost. If w belong to the stream, the transformation is described by R7 2 or R7 3 . The cases where any three other paths are considered, is handled similarly. If w or belong to the stream, then the substring is unaltered. If the paths belong to the stream, then the stream will be disrupted and the statement holds. This concludes the treatement of the contraction rule. (To be precise, since the operation of bifurcation + is commutative, we might need to use R3 to rearrange the order of the substrings of the form w handling properly the contraction case.) If two cuts are permuted, suppose that the stream passes through both pairs of cut-formulas, say C be subproofs of \Pi to which a cut on C applied and let \Pi 2 be the subproof whose last rule of inference is a cut on D 1 ; D 2 . By Definition 11 a stream P 2 in \Pi 2 passing through might be described either by a compact string of the form w 1 wn , or by a compact string of the form (w 1;1 w 0 be a substring associated to a stream P 1 lying in \Pi 1 , passing through C 1 and connected by a cut-edge to P 2 . In the first case, w will contain a substring of the form w wn ); then, we apply (w 2 wn ). In the second case, w will contain a substring of the form apply R5 and obtain (w If are contained in \Pi 1 , then rule has to be applied instead of R2. If two cuts are permuted but the stream passes through only one of the cut-formulas then no rule is applied. If a cut is pushed upwards then it might happen that a contraction is pushed below the height of the cut. This might imply that a branching point of the stream (maybe several of them) will be pushed below a cut-edge and therefore that the compact description of the stream might change. The new compact description is obtained with the application of rules In all cases treated above, the stream P passed through the cut-formulas simplified by the step of the procedure. We shall consider now the case where the stream does not pass through the pair of cut-formulas. This is the case when one of the cut-formulas is weak. (Remember that the extremes of a stream can only lie on the end-sequent.) Then the procedure of cut-elimination will induce a disruption of the structure of the proof due to the removal of a subproof. In case the stream passes through the subproof which is removed, then the stream will be disrupted and the statement holds. In case a stream passes through the side formulas of the antecedents of a cut but not through their cut-formulas, its paths will be stretched and no modification of the substrings is needed. The only exception is the contraction case, where a substream passing through the side formulas of the subproof to duplicate, might be duplicated while the subproof is duplicated. In this case, rule R6 is used. Let us notice that all along the proof one needs to verify that the string associated to the proof \Pi 0 is compact. This is a straightforward verification and we leave it to the reader. To conclude, if the stream has not been disrupted then from Proposition 15 it follows that w has been reduced to a string b m where m is smaller than the number of axioms in \Pi.Example 27 Consider the proof of the sequent F (2) given in Example 22 and described by the compact string b 2 b 2 where the terms b 2 are exactly n. We can calculate the exponential expansion of this proof after cut-elimination through a purely algebraic manipulation of strings as we shall show for namely for the string b 2 b 2 b 2 b 2 . For an arbitrary n the approach is similar. By applying R4 to the substrings b 2 b 2 we get (b apply R3 to all substrings b 2 b to get (b b (b b and by R1 we or in short b 4 b 4 . By applying again R4; R3 and R1 we get (b 4 (b 4 b (b b (b b (b b (b b Note that b 16 is the minimum expected value for a cut-free proof computing ) from F (2). In fact the minimal tree of computation of F branching points corresponding to 2 times F (2). Theorem 28 Let \Pi be a proof of F atomic cuts and no weak formulas. Let w i be the full compact string associated to procedures of cut-elimination transforming \Pi into \Pi 0 and w i into w 0 are simulated by R1-R5 and w 0 Proof. The proof \Pi has a very simple structure. Here are some properties: 1. the proof \Pi contains no logical rules and all formulas appearing in \Pi are atomic. This is because cuts are defined on atomic formulas and formulas in the end-sequent are atomic; 2. for any sequent in \Pi, exactly one formula lies on the right hand side of the sequent. This follows from 1; 3. there are no contractions on the right in \Pi. This is because there are no weak formulas and no logical rule can be applied on negative formulas in \Pi. Properties 1-3 imply some properties of the flow graph of \Pi: a. no path passes twice through the same cut-formula, since cut-formulas are b. no path passes twice through the side formulas of a sequent used in a cut- rule. This is because at any stage of the procedure, the sequents have the and contractions are on the left only. Therefore, if the cut-formula in the sequent is a positive occurrence of the form are negative occurrences and no path can start and end in them; if the cut-formula is a negative occurrence F (s j ), for some there might be several paths passing through the side formulas (in fact, all of them should pass through the formula on the right of the sequent) but the proof where F occurs cannot be duplicated because contractions can be applied only to negative formulas. Properties a and b ensure that the rewriting rules R6 and R7 are not used by the simulation. In particular R1-R5 are rewriting rules of the form p ! q This implies that the final string w 0 i are of the form b n i where \Pi be a proof of the sequent S such that the number of symbols in S is n. If any cut-free proof \Pi 0 of S has 2 O(n) lines then 1. either there is a stream w in \Pi such that t(w) is 2 O(n) , 2. or any process of cut-elimination from \Pi to \Pi 0 is simulated by R6. Proof. If \Pi 0 has 2 O(n) lines then there is a stream b k in it where k is 2 O(n) . This means that b k has been obtained from a string w in \Pi with or without the help of R6. If R6 has not been used then the arithmetical value t(w) is at least k since rules R1-R5 and R7 cannot augment it.Remark Rule R6 has a global effect. In fact, it does not concern the cut- formulas involved in the step of elimination of cuts, but the structure of the proof itself. It corresponds to the existence of a path in the proof which passes twice through the side formulas of a subproof that is duplicated by the procedure of cut-elimination. Even if a proof might be such that no path passes twice through the side formulas of a sequent applied to a cut-rule, during cut-elimination this property might be lost. It is easy to check that permutation of cuts, contraction and resolution of cut-formulas which are main formulas of logical rules, might produce a proof which falsifies this property. Once the property is violated, rule R6 might play a role in the transformation. Remark 31 The expansion of \Pi 0 can be exponential with respect to the number of sequents in \Pi, as in Example 22. This example shows that cuts on atomic formulas can induce exponential complexity. Problem To decide whether w 1 and w 2 can be reduced to the same string k , for some k, by using rules R1-R5, can be done in polynomial time. In fact can be polynomially reduced to some b k1 ; b k2 for some fixed k and it is sufficient to check whether the values k 1 and k 2 are the same or not. If we allow the rules R1-R7, does the question become NP -complete? --R The undecidability of k-provability In Annals of Pure and Applied Logic Some combinatorics behind proofs. Cycling in proofs and feasibility. Turning cycles into spirals. Duplication of directed graphs and exponential blow up of proofs. The cost of a cycle is a square. Asymptotic cyclic expansion and bridge groups of formal proofs. Looking from the inside and the outside. Propositional proofs via combinatorial geometry and the search for symmetry. Making proofs without modus ponens: An introduction to the combinatorics and complexity of cut elimination. A graphic apology for symmetry and implicitness. The relative efficiency of propositional proof systems. Linear Logic. Proof Theory and Logical Complexity. The intractability of resolution. Lower bounds for increasing complexity of derivations after cut elimination. The lengths of proofs. Structural Complexity of Proofs. Bounds for proof-search and speed-up in predicate calculus Lower bounds on Herbrand's Theorem. Proof Theory. Complexity of a derivation in the propositional calculus. --TR Linear logic Graphic Apology for Symmetry and Implicitness

cut elimination;structure of proofs;proof complexity;duplication;cycles in proofs;directed graphs;logical flow graphs

606912

On the complexity of data disjunctions.

We study the complexity of data disjunctions in disjunctive deductive databases (DDDBs). A data disjunction is a disjunctive ground clause R(c-1pt1)...R(ck),K 2, which is derived from the database such that all atoms in the clause involve the same predicate R. We consider the complexity of deciding existence and uniqueness of a minimal data disjunction, as well as actually computing one, both for propositional (data) and nonground (program) complexity of the database. Our results extend and complement previous results on the complexity of disjunctive databases, and provide newly developed tools for the analysis of the complexity of function computation.

Introduction During the past decades, a lot of research has been spent to overcome the limitations of conventional relational database systems. The field of deductive databases, which has emerged from logic programming [29], uses logic as a tool for representing and querying information from databases. Numerous logical query languages, which extend Horn clause programming for dealing with various aspects such as incomplete or indefinite information, have been proposed to date, cf. [1, 33]. In particular, the use of disjunction in rule heads for expressing indefinite information was proposed in Minker's seminal paper [32], which started interest in disjunctive logic programming [30, 10]. For example, the rule lives in(x; US) - lives in(x; canada) - lives in(x; mexico) / lives in(x; n america) (1) informally states that a person living in north America lives in one of the three countries there. The semantical and computational aspects of disjunctive logic programming, and in particular, disjunctive deductive databases, have been investigated in many papers (see [33] for an overview). The results of this paper have been presented at the international workshop "Colloquium Logicum: Complexity," Vienna, October 9-10, 1998. This work was partially supported by the Austrian Science Fund Project N Z29-INF, and by the British Council-Austria ARC Programme for collaborative research (Oracle Computations within Descriptive Complexity Theory). In this paper, we are interested in a restricted type of disjunction, which has been previously considered e.g. in [6, 5, 12, 19, 15]. A data disjunction [19] is a ground clause R(-c 1 which all atoms are different and involve the same predicate R. For example, the head of the rule (1) for Joe, is a data disjunction, as well as the disjunctive fact loves(bill; monica) - loves(bill; hillary): A data disjunction expresses indefinite information about the truth of a predicate on a set of arguments; in database terminology, it expresses a null value on this predicate, whose range is given by the arguments of its atoms. In the context of deductive databases, null values of this form in the extensional database and their complexity have been considered e.g. in [20], and in many other papers. If, in the above example, the fact lives in(joe; n america) is known, then the data disjunction lives in(joe; US) - lives in(joe; canada) - lives in(joe; mexico) can be derived from rule (1). If a clause C is entailed from a database, then also any clause C 0 subsumed by C is entailed. For example, the clause C - lives in(joe; usbekistan) is entailed by virtue of C as well. We thus adopt the natural condition that a data disjunction C must be minimal, i.e., no proper subclause of C is entailed. The question we address here is the complexity of data disjunctions in a disjunctive deductive database (DDDB). Table 1 summarizes the problems studied in this paper (see Section 3 for precise definitions), and the main complexity results obtained. They complement previous results on reasoning from DDDBs. Deciding whether an arbitrary disjunction, rather than a data disjunction, follows from a DDDB has \Pi Pdata and propositional complexity, and exponentially higher expression and combined complexity [14]; various syntactic restrictions lower the complexity to coNP or even polynomial time [9]. On the other hand, evaluating a conjunctive query over a disjunctive extensional database is coNP-complete [20], and hence deciding entailment of a single ground atom a has coNP data and propositional complexity. Thus, data disjunctions have intermediate complexity between arbitrary clauses and single atoms. Observe that Table 1 contains also results on actually computing a data disjunction (assuming at most one exists). While all the results in this table could be derived in the standard way, i.e., by proving membership in class C and reducing a chosen C-hard problem to the problem in question, we pursue here an "engineering" perspective of complexity analysis in databases, proposed e.g. in [18], which utilizes tools from descriptive and succinct complexity theory and exploits properties of the deductive database semantics. By means of these tools, hardness results can be derived at an abstracted level of consideration, without the need for choosing a fixed C-hard problem. Such tools (in particular, complexity upgrading) have been developed for decision problems, but are not available for function problems. We overcome this by generalizing the tools for propositional problems in a suitable way. Thus, the main contributions of this paper can be summarized as follows. Firstly, we determine the complexity of data disjunctions. We obtain natural and simple logical inference problems complete for the class \Theta P 2 of the refined polynomial hierarchy [45], and, in their computational variants, complete problems for the function classes FP NP k and FL NP log [log] and their exponential analogs. Secondly, we provide upgrading techniques for determining the complexity of function computations. They generalize available tools for decisional problems and may be fruitfully applied in other contexts as well. The rest of this paper is organized as follows. Section 2 states preliminaries, and Section 3 formalizes the problems. In Section 4, the decision problems are considered, while Section 5 is devoted to computing Data Disjunction Input: A disjunctive deductive database a collection of (possibly disjunctive) ground facts, and - are the inference rules, plus a distinguished relation symbol R. propositional complexity data complexity expression complexity combined complexity 9DD: does DB have a data disjunction on R? \Theta P PSpace NP PSpace NP 9!DD: does DB have a unique data disjunction on R? \Theta P PSpace NP PSpace NP -DD: Computation of the unique data disjunction on R. FPSpace NP FPSpace NP k -DD: Computation of the unique data disjunction on R, if it has at most k disjuncts (k constant). log [log] FL NP log [log] FPSpace NP [pol] FPSpace NP [pol] Table 1: Complexity of Data Disjunctions. data disjunctions. Logical characterizations of function computations are given through a generalization of the Stewart Normal Form (SNF) [38, 39, 17], which has been first used to characterize the class \Theta P . For deriving the expression and combined complexity of function computations, upgrading results are developed in Section 6. The final Section 7 applies the results to the area of closed-world reasoning and gives some conclusions. Preliminaries 2.1 Deductive databases For a background on disjunctive deductive databases, we refer to [30]. Syntax. A finite relational language is a tuple where the R i are relation symbols (also called predicate symbols) with associated arities a , and the c i are constant sym- bols. An atom is a formula of the form R i (-v), where - v is a tuple of first order variables and constant symbols. A disjunctive datalog rule is a clause of the form over a finite relational language, where the a i 's are atoms forming the head of the clause, and the b j 's are atoms or inequalities of the form u (where u and v are variables or constants) forming the body of the clause. A disjunctive deductive program (short program) is a finite collection of disjunctive datalog rules; it is ground, if no variables occur in the rules. If a predicate symbol occurs only in rule bodies, it is called an input predicate, otherwise it is called a derived predicate. A disjunctive deductive program with input negation is a program where input predicates are allowed to appear negated. A ground fact is a clause of the form a 1 where a 1 is a variable-free atom; a disjunctive ground fact is a clause of the form a 1 - a n where the a i 's are variable-free atoms. A disjunctive deductive database (DDDB) is a tuple E) where - is a program, and E is a finite set of disjunctive ground facts. Here, E represents the input database, also called the extensional part, and - are inference rules, called the intensional part of the database DB. Remark: Note that -; E, and - [ E are all disjunctive deductive programs, i.e., ground facts can be included into the programs, and in fact we shall do this for defining the semantics. However, for methodological and complexity issues, it is important to distinguish the input data from the inference rules. For example, the complexity of evaluating DB is exponentially lower when - is fixed. In section 3, we shall define data and query complexity to give a formal meaning to this intuition. Semantics. The semantics of DDDBs has been defined in terms of their minimal models [32, 30]. For a E), we denote by HU DB its Herbrand universe, i.e., the set of all constants occurring in DB. The Herbrand base HB DB (resp., disjunctive Herbrand base DHB DB ) is the set of all ground atoms disjunctive ground facts) of predicates in DB over HU DB . The ground instantiation of a program - over a set of constants C is denoted by ground(-; C); the ground instance of DB, denoted ground(DB), is ground(-; HB DB An (Herbrand) interpretation of DB is a subset H ' HB DB . An interpretation H of DB is a model of DB, if it satisfies each rule in ground(DB) in the standard sense. A model H of DB is minimal, if it does not contain any other model of DB properly; by MM(DB) we denote the set of all minimal models of DB. We write DB is true in every M 2 MM(DB), and say that ' is entailed from DB. Example 2.1 Let is the rule q(x) / p(x) and E contains the single disjunctive fact are among the models of DB; M is while M is not. The minimal models of DB are fq(b)gg. Remark. It is easy to see [32] that for each positive clause C , DB only iff DB where is satisfaction in all models of DB. We will repeatedly use this fact. The set of minimal models of DB has been characterized in terms of a unique least model-state MS (see [30]), i.e., a subset of DHB DB , which can be computed by least fixpoint iteration of an operator T S generalizing the standard T P operator of logic programming [29]. In general, the computation of MS takes exponential space and time, even if the program - of DB is fixed. 2.1.1 Negation Introducing negation in disjunctive deductive databases is not straightforward, and gave rise to different semantics, cf. [33]. We restrict here to input negation, i.e., the use of negated atoms :R( - t) in rule bodies where R is an extensional predicate, and adopt a closed-world assumption (CWA) on models imposing the following condition: any accepted model M of E), restricted to the extensional part, must be a minimal model of E. Unless stated otherwise, a model of a DDDB must satisfy this kind of closed-world assumption. Observe that this condition is satisfied by each M 2 MM(DB) if - is negation-free; furthermore, if E contains no disjunctive facts, then :R(-c) is true in every M 2 MM(DB) iff As for complexity, it is easy to see that checking whether the restriction of M to its extensional part is a minimal model of E is possible in polynomial time. Hence, the complexity of model checking and of deciding DB does not increase through the CWA on models. Furthermore, if E is restricted to disjunction-free ground facts, input negation can be eliminated in computation as follows. Definition 2.1 Let - be a finite relational language, and let - the class of all finite - 0 structures A where for all relations R in - , R 0 A is the complement of R A . Proposition 2.1 Extending a given -structure to its corresponding NEG - 0 structure and replacing literals in a program - by R 0 possible in LOGSPACE. In the derivation of hardness results, we shall consider DDDBs E) using input negation but where E is disjunction-free. Hence, all hardness results in this paper hold for DDDBs without negation and non-disjunctive (i.e., relational) facts as well. 2.2 Complexity In this section, we introduce some of the more specific complexity classes and notions employed in the paper; we assume however some familiarity with basic notions of complexity theory such as oracle computations, NP, PSpace, L etc. The class \Theta P 2 contains the languages which are polynomial-time truth-table reducible to sets in NP. It has a wide range of different characterizations [45, 21]. In particular, the following classes coincide with \Theta P polynomial time computation with k rounds of parallel queries to an NP oracle [27]. log : polynomial time computation where the number of queries to an NP oracle is at most logarithmic in the input size [25]. log : logarithmic space computation where the number of queries to an NP oracle is at most logarithmic in the input size [26]. 1 1 Observe that the space for the oracle tape is not bounded. Unbounded oracle space is also assumed for all other classes using an oracle in this paper. III FL NP log log IV II log [log] log [log] QUERY I Figure 1: Function classes corresponding to \Theta P . For an overview of different characterizations and their history, consult [45, 21]. It is shown in [21, 4, 37, 40] that this picture changes when we turn to function computation. The above mentioned list gives rise to at most three presumably different complexity classes FP NP log , and FL NP log , which are shown in Figure 1. Here, for any function class FC, we denote by FC[log] the restriction of FC to functions with logarithmic output size. Moreover, k[k] denotes k rounds of parallel queries, where k is a constant. The relationships between the complexity classes in Figure 1 have been attracting quite some research efforts, which led to a number of interesting results. ffl II=III is equivalent to ffl I=II is equivalent to the property that SAT is O(log n) approximable. This was shown in [2], after I)II was proved in [8]. (Here f -approximability of a set A means that there is a function g such that for all x holds that g(x ffl Furthermore, if I=II, then (1SAT,SAT), i.e., promise SAT, is in P [4, 40], FewP=P, NP=R [37], n), and NP ' DTIME(2 n O(1= log log n) To compare the complexity of functions, and to obtain a notion of completeness in function classes, we use Krentel's notion of metric reducibility [25]: Definition 2.2 A function f is metric reducible (- mr -reducible) to a function g (in symbols, f - mr g), if there is a pair of polynomial-time computable functions h 1 and h 2 such that for every x, Proviso 1. Let C be a complexity class. Unless stated otherwise, we use the following convention: C-completeness is defined with respect to LOGSPACE reductions, if C is a class of decision problems, and with respect to metric reductions, if C is a class of function problems. Some complete problems for function classes are shown in Figure 1. The canonical FP NP -complete problem is QUERY, i.e., computing the string -(I 1 SUPREMUM is computing, given a Boolean formula F string there is a satisfying assignment to the variables of F such that x SIZE is computing the size of a maximum clique in a given graph. Note that this problem is also complete for FP NP log . All these problems, turned into proper decision problems, are \Theta P 2 -complete. In particular, deciding whether the maximum clique size in a graph is even and deciding whether the answer string to QUERY contains an even number of 1's are \Theta P -complete, cf. [45]. 2.3 Queries and descriptive complexity Definition 2.3 Let - be a finite relational language, and let fRg be a language containing a single relational symbol R. A query Q is a function which maps -structures to ffi-structures over the same domain, s.t. Q(A) and Q(B) are isomorphic, if A and B are isomorphic. If R is nullary, then Q is a Boolean query. A Boolean query Q is regarded as a mapping from -structures to f0; 1g s.t. for isomorphic A; B, Remarks. (1) If we disregard nonelementary queries, we can identify queries with higher order definable relations. (2) Note that "query" is also used for oracle calls. (3) Since queries are functions, we shall also write them as sets of pairs (A; Q(A)). Definition 2.4 Let - be a finite relational language with a distinguished binary relation succ, and two constant symbols min; max. Then SUCC - is the set of all finite structures A with at least two distinct elements where succ A is a successor relation on jAj, and min A ; max A are the first and last element wrt the successor relation, respectively. Note that queries are not defined over SUCC - , but over arbitrary -structures; this is called "order independence" of queries. Many query languages however seem to require a built-in order for capturing complexity classes, i.e., capturing requires that the -structures are extended by a contingent ordering to structures from SUCC - . Thus, when we write on ordered structures/databases, or on SUCC - , we mean that the queries are computed on -structures which are extended to SUCC - structures. The following theorems provide examples of this phenomenon. Definition 2.5 A SNF formula (Stewart Normal Form) is a second-order formula of the form where ff and fi are \Pi 1 1 second order formulas with equality having the free variables - y. An SNF sentence is a SNF formula without free variables. The Skolem functions for the variables - x are called SNF witnesses. Lemma 2.2 ([38, 39, 17]) Every \Theta P 2 -computable property on SUCC - is expressible as where oe is a SNF sentence. This result, in equivalent terms of first-order logic with NP-computable generalized quantifiers, is contained for particular cases of generalized quantifiers in [38, 39], and was given for broad classes of generalized quantifiers in [17]. On a structure A, a formula '(-x) with free variables - x defines the relation ' A given by f-c j A '(-c)g. A program - defines a relation R on A, if (-; on A. In particular, if R is nullary, ' (resp., -) defines a property on A. Lemma 2.3 (immediate from [13, 14]) Every \Pi 1 1 definable property ' on SUCC - is expressible by a disjunctive datalog program - ' using input negation. Remark: Note that Lemma 2.3 does not require inequalities in rule bodies, since inequality is definable in the presence of order, cf. [14]. 3 Data Disjunctions Definition 3.1 Given a DDDB DB, a disjunctive ground fact R-c 1 - R-c n , n - 2, is called a data disjunction, if 1. DB 2. for all S ae In this case, we say that DB has a data disjunction on R. A data disjunction can be seen as a kind of null value in a data base. Example 3.1 The DDDB E) has a data disjunction Pa - P b - P c. Definition 3.2 Given a DDDB DB, the maximal disjunction on R (in symbols, md(DB;R)) is the disjunctive ground fact R-c: Lemma 3.1 DB has a data disjunction on R if and only if DB Proof. If DB has a data disjunction ffi on R, then no atom R-c of ffi is implied by DB. Therefore, ffi is a subclause of md(DB;R), and thus DB md(DB;R). On the other hand, if DB clearly md(DB;R) is not empty. Either md(DB;R) is a data disjunction itself, or atoms of md(DB;R) can be removed until a minimal disjunction ffi is reached such that DB by definition no atomic subformula of md(DB;R) is implied by DB, ffi must contain at least two different atoms. J In measuring the complexity of data disjunctions, we distinguish several cases following Vardi's [41] distinction between data complexity, expression complexity (alias program complexity), and combined complexity. Definition 3.3 The problems 9DD, 9!DD, -DD, and k -DD are defined as follows: Instance: A DDDB E), and a relation symbol R. Query: 9DD: Does DB have a data disjunction on R ? 9!DD: Does DB have a unique data disjunction on R ? -DD: Compute the unique data disjunction on R if it exists, and # otherwise. k -DD: Compute the unique data disjunction on R, if it exists and has at most k disjuncts, and # otherwise. Observe that 9DD, called ignorance test in [5], has been used in [5, 6] to discriminate the expressive power of different query languages based on nonmonotonic logics over sets of disjunctive ground facts. Problem 9!DD corresponds to the unique satisfiability problem. The uniqueness variant of a problem has often different complexity. Definition 3.4 Let \Pi be one of 9DD, 9!DD, -DD, or k -DD. ffl The data complexity of \Pi is the complexity of \Pi with parameter - fixed. ffl The expression complexity of \Pi is the complexity of \Pi with parameter E fixed. ffl The propositional complexity of \Pi is the complexity of \Pi where - is ground. ffl The (unconstrained) complexity of \Pi is also called the combined complexity of \Pi. Problem \Pi has combined (or propositional) complexity C , if \Pi is C-complete with respect to combined complexity. \Pi has data (or expression) complexity C , if \Pi is in C with respect to data (resp. expression) complexity for all choices of the parameter, and \Pi is C-complete with respect to data (resp. expression) complexity for a particular choice of the parameter. 4 Existence of Data Disjunctions Theorem 4.1 Let Q be a fixed Boolean query. Over ordered databases, the following are equivalent: 1. Q is \Theta P -computable. 2. Q is definable by a SNF sentence. 3. There exist a program - and a relation symbol R s.t. (-; A) has a data disjunction over R iff A 4. Q is equivalent to a SNF sentence whose SNF witnesses are uniquely defined. 5. There exist a program - and a relation symbol R s.t. (-; A) has a unique data disjunction over R iff A Proof. The equivalence of 1: and 2: is stated in [17]. A close inspection of the proof in [17] shows that in fact 1: is also equivalent to 4: 3: ! 1:: By Lemma 3.1, the following algorithm determines if DB has a data disjunction on R: Algorithm DDExistence(DB; R) 1: M := ;; 2: for all R-c 2 HBDB 3: if not (DB 4: ' := 5: if DB return true else return false; Note that in line 4, ' equals md(DB;R). The algorithm DDExistence works in polynomial time and makes two rounds of parallel queries to an NP oracle, and thus the problem is in P NP . 5: ! 1:: The following algorithm DDUniqueness is an extension of DDExistence. Algorithm DDUniqueness(DB; R) 02: for all R-c 2 HBDB 03: if not (DB 05: if not (DB 07: for all R-c 2 M 08: if not (DB 10: if DB Note that lines 1 to 4 coincide with DDExistence. On line 5, the algorithm terminates if no data disjunction exists. Otherwise, all possible data disjunctions are subclauses of to 9 construct a subclause /; it contains all those literals R-c of md(DB;R) in N which cannot be removed from md(DB;R) without destroying the data disjunction, i.e., it contains those literals which necessarily appear in every data disjunction. Thus, if ' is a data disjunction, it is the unique one. On the other hand, if a unique data disjunction exists, it is by construction equal to '. Like the algorithm DDExistence, this algorithm also works in polynomial time making a constant number of rounds of parallel queries to an NP oracle. Hence, the problem is in \Theta P . 2: ! 3:: Let ' be a formula of the form By Lemma 2.3 there exist programs -A and -B containing predicate symbols A and B such that for all Let - be the program -A [ -B with the additional rules Observe that P does not occur in -A [ -B , and thus, by well-known modularity properties [14, Section 5], the minimal models of - on A are obtained by extending the minimal models of -A [ -B on A. It is easy to see that has a data disjunction on P if and only if there exists a tuple - c on A such that A indeed computes property ' on SUCC - . From the equivalence of 2: and 4:, it follows that the data disjunction of program - in the proof of 2: ! 3: is unique. J Corollary 4.2 The propositional and data complexity of 9DD and 9!DD are in \Theta P . The expression and combined complexity of 9DD and 9!DD are in PSpace NP . Proof. It remains to consider expression and combined complexity. When the program is not fixed, the size of HB DB is single exponential in the input, and thus the algorithm DDExistence takes exponentially more steps. Thus, the problem is in EXPTIME NP k , which coincides with PSpace NP [18]. J Since \Theta P 2 has complete problems, we obtain from Theorem 4.1 the following. Corollary 4.3 There is a program - for which 9DD and 9!DD are \Theta P Hence, we obtain the announced result. Theorem 4.4 The data complexity and propositional complexity of 9DD and 9!DD is \Theta P . Note that the propositional complexity of 9DD has been stated in [12]. The hardness proof there, given by a standard reduction, is far more involved; this indicates the elegance of using the descriptional complexity approach. Since the data complexity of a query language is uniquely determined by its expressive power, two languages with the same expressive power will always have the same data complexity. Hence, data complexity is a property of semantics. Expression and combined complexity, however, depend on the syntax of the language. Therefore, it is in general not possible to determine the expression complexity of a query language L from its expressive power. Indeed, both the syntax and the semantics of L impact on its expression complexity. In spite of these principal obstacles, the typical behavior of expression complexity was often found to respect the following pattern: If L captures C , then the expression complexity of L is hard for a complexity class exponentially harder than C . The main result of [18] shows that all query languages satisfying simple closure properties indeed match the above observation. Suppose that in a database, domain elements are replaced by tuples of domain elements. This operation is natural when a database is redesigned; for instance, entries like "John Smith" in a database A can be replaced by tuples ("John","Smith") in a database B. It is natural to expect that a query QA over A can be easily rewritten into an equivalent query Q B over B. We call Q B a vectorized variant of QA . This is the essence of the first closure property: Vector Closure: A query language is uniformly vector closed, if the vectorized variants of query expressions can be computed in LOGSPACE. The second closure property is similar. Suppose again that a database A is replaced by a database B in such a way that all relations of A can be defined by views which use only unions and intersections of relations in B. Then, it is again natural to expect that a query QA over A can be translated into an equivalent query Q B over B. In this case, we call Q B an interpretational variant of QA . Interpretation Closure: A query language is uniformly interpretation closed, if the interpretational variants of queries can be computed from the database schemata in LOGSPACE. In conclusion, we have the following closure condition (see [18] for a formal definition). Definition 4.1 A query language is uniformly closed, if it is uniformly vector closed and uniformly interpretation closed. Lemma 4.5 ([18]) The language of DDDBs is uniformly closed. Combining Corollary 4.2 and the following Proposition 4.6, we obtain the expression and combined complexity of 9DD and 9!DD. Proposition 4.6 ([18]) If a language is uniformly closed, and expresses all \Theta P 2 properties of SUCC - , then its expression complexity and combined complexity are at least PSpace NP . Theorem 4.7 The expression complexity and the combined complexity of 9DD and 9!DD is PSpace NP . 5 Computation of Data Disjunctions 5.1 Data complexity and propositional complexity Theorem 5.1 Let Q be a fixed query. Then, over ordered databases the following are equivalent. 1. Q is FP NP computable. 2. Q is definable by a SNF formula. 3. There exist a program - and a relation symbol R s.t. Q(A) is polynomial time computable from the unique data disjunction of (A; -) on R. Proof. 1: ! 2:: The problem of deciding whether a given tuple - c on A fulfills - easily seen to be in \Theta P 2 . Thus, Lemma 2.2 implies there is a SNF sentence oe such that A; - c (provide - c through a designated singleton relation R - c , and use 9-yR - c (-y) to access - c). Hence, there is an 2: ! 3:: Similarly as in the proof of Theorem 4.1, let ' be the SNF formula having the free variables - y. By Lemma 2.3 there exist programs -A and -B containing predicate symbols A and B such that for all A(-c; - d) 2 HB -A and B(-c; - d) 2 HB -B d) iff A d) and (A; -B ) d) iff A We have to construct a program - whose unique data disjunction on input A over relation R contains the information about all tuples in ' A . To this end, consider the program in Figure 2. Let a be the arity of A and B there. Then the new relation symbols T and S also have arity a, and R has arity a+2. The program Figure 2: DDDB program for FP NP queries. requires that a successor relation over tuples is available. The lexicographical successor relation can be easily defined using datalog rules. Lines 1 to 3 enforce that S(-c) holds for at least one - c. Consequently, the unique data disjunction on - c is the positive clause containing all possible S(-c) on A. Consider lines 1 to 4 now. If the program contained only these rules, then line 4 would enforce that the unique data disjunction on R would be the clause d d; max; max): Lines 5 and 6, however, remove certain literals from this clause. In particular, it holds that A d) iff there is a - c on A such that A d) iff there is a - c on A such that the (unique) data disjunction of (-; A) on R contains the clause R(-c; - d; min; min)-R(-c; - d; min; max) but not R(-c; - d; max; max). Therefore, ' A is polynomial time computable from the unique data disjunction on R. 3: ! 1:: The algorithm DDUniqueness in the proof of Theorem 4.1 computes the unique data disjunction in its variable /, provided it exists. It is easily modified to output # in the other case. J The data and propositional complexity of -DD is an easy corollary to this result. Corollary 5.2 Problem -DD has data complexity and propositional complexity FP NP Definition 5.1 A domain element query (DEQ) is a query whose answer relation is a singleton, i.e., a query Q s.t. for all A it holds that Theorem 5.3 Let Q be a fixed DEQ. Then, over ordered databases the following are equivalent: 1. Q is FP NP 2. Q(A) is a tuple of SNF witnesses to a SNF sentence. 3. There exist a program - and a relation symbol R s.t. Q(A) is definable as a projection of the unique data disjunction over R, where the data disjunction contains at most two atoms. 4. There exist a program - and a relation symbol R s.t. Q(A) is polynomial time computable from the unique data disjunction over R where the data disjunction contains at most two atoms. Proof. 1: ! 2:: For constant - c, the problem of deciding whether - c 2 -(A) is easily seen to be in \Theta P . Thus, there is a SNF formula '(-x) having the free variables - x for the constants s.t. A; - c -(A). By Theorem 4.1 we may w.l.o.g. suppose that the SNF witnesses in ' are unique. 2: ! 3:: Let ' be a SNF sentence which has unique SNF witnesses. Then, the program for ' in part 2: ! 3: of the proof of Theorem 4.1 has a unique data disjunction of the form P (-c; - d; max). From this data disjunction, the projection on the - d tuple yields the desired result. 3: M be the polynomial time Turing Machine which computes the answer from the unique data disjunction. Then, we use the algorithm DDUniqueness in the proof of Theorem 4.1 which computes the unique data disjunction in its variable /. It remains to check if the data disjunction is small, and to simulate M . By assumption, the result has logarithmic size. The data and propositional complexity of k -DD is an immediate corollary to this result. Corollary 5.4 The data complexity and propositional complexity of k -DD are FL NP log [log]. At this point, the question arises whether we could not have surpassed the reduction in the proof of 2: ! 3: in Theorem 5.1, by exploiting the completeness result on k -DD. The next result tells us that this is (presumably) not possible, and that disjunctions not bounded by a constant are needed in general to have hardness for FP NP Proposition 5.5 Problem -DD is metric reducible to k -DD with respect to data complexity, for some only if FP NP log . Proof. !: Suppose -DD is metric reducible to k -DD. Then, Corollary 5.2 implies that k -DD is metric complete for FP NP k . Since FL NP log [log] ' FP NP log , this implies that FP NP log ) where denotes the closure of C under metric reductions. Clearly, - mr log log , and thus log holds. Combined with FP NP log ' FP NP (cf. Figure 1), it follows that FP NP log . /: Suppose that FP NP . Let f be any function complete for FP NP log (such an f exists). Then, by hypothesis. We use the following fact: Every function f in FP NP log is - mr -reducible to some function g in FL NP log [log]. Indeed, Krentel showed that his class OptP[O(log n)] satisfies FP NP log ' - mr (OptP[O(log n)]), and that CLIQUE SIZE (cf. Section 2) is OptP[O(log n)]-complete [25]. Since CLIQUE SIZE is clearly in FP NP log [log], the claimed fact follows by transitivity of - mr . This fact and Corollary 5.2, together with the hypothesis FP NP k imply that -DD - mr f - mr g - mr -DD. By transitivity of - mr , we obtain -DD - mr k -DD. J 5.2 Expression and combined complexity Finally, we determine the expression complexity of computing the unique data disjunction. Theorem 5.6 The expression and combined complexity of -DD is FPSpace NP , and the expression and combined complexity of k -DD is FPSpace NP [pol]. The proof of the theorem uses succinct upgrade techniques for function problems whose inputs are given in succinct circuit description. These techniques are described in detail in the following section. 6 Problems with Succinct Inputs 6.1 Previous work and methodology A problem is succinct, if its input is not given by a string as usual, but by a Boolean circuit which computes the bits of this string. For example, a graph can be represented by a circuit with 2n input gates, such that on input of two binary numbers v; w of length n, the circuit outputs if there is an edge from vertex v to vertex w. In this way, a circuit of size O(n) can represent a graph with 2 n vertices. Suppose that a graph algorithm runs in time polynomial in the number of vertices. Then the natural algorithm on the succinctly represented graph runs in exponential time. Similarly, upper bounds for other time and space measures can be obtained. The question of lower bounds for succinct problems has been studied in a series of papers about circuits [35, 22, 31, 24, 3, 7, 44], and also about other forms of succinctness such as representation by Boolean formulas or OBDDs [42, 43]. The first crucial step in these results is a so-called conversion lemma. It states that reductions between ordinary problems can be lifted to reductions between succinct problems: Here, s(A) denotes the succinct version of A, X and Y denote suitable notions of reducibility, where - Y is transitive. For the second step, an operator 'long' is introduced which is antagonistic to s in the sense that it reduces the complexity of its arguments. For a binary language A, long(A) can be taken as the set of strings w whose size jwj written as a binary string is in A. Contrary to s, long contains instances which are exponentially larger than the input to A. For a complexity class C , long(C) is the set of languages long(A) for all A 2 C . It remains to show a second lemma: Compensation Lemma A - Y s(long(A)). Then the following theorem can be derived: Theorem Let C be complexity classes such that long(C 1 let A be C 2 -hard under reductions. Suppose that the Conversion Lemma and the Compensation Lemma holds. Then s(A) is reductions. Proof. To show C 1 -hardness, let B be an arbitrary problem in C 1 . By assumption, long(B) 2 C 2 , and therefore, long(B) - X A. By the Compensation Lemma, B - Y s(long(B)), and by the Conversion Lemma, we obtain s(long(B)) - Y s(A). Since - Y is transitive, s(A) is C 1 -hard. J 6.2 Queries on succinct inputs For any -structure A, let enc(A) denote the encoding of A by a binary string. The standard way to encode A is to fix an order on the domain elements, and to concatenate the characteristic sequences of all relations in A. 2 All Turing machine based algorithms (and in particular, all reductions) in fact work on A. Therefore, we shall usually identify A and enc(A) without further notice. We use the further notation: ffl enc(-) denotes the binary language of all encodings of finite -structures. ffl char(A) is the value of the binary number obtained by concatenating a leading 1 with enc(A). Given a binary circuit C with k input gates, gen(C) denotes the binary string of size 2 k obtained by evaluating the circuit for all possible assignments in lexicographical order. The idea of succinct representation is to represent enc(A) in the form gen(C). To overcome the mismatch between the fact that the size of enc(A) can be almost arbitrary, while the size of gen(C) has always the form 2 k , we use self-delimiting encodings: Definition 6.1 Let . The self-delimiting encoding of w is defined as 1). For a number n, denotes the binary representation of the number n. Thus, from a string sd(w)v, the string w can be easily retrieved by looking for the first 1 at an even position in the string. Definition 6.2 ([42]) For a binary language L, let sd(L) denote the language 2 The characteristic sequence of a relation is the binary string which for all tuples in lexicographical enumeration describes membership in the relation by 1, and non-membership by 0; for graphs, this means writing down the adjacency matrix line by line. Thus, sd(L) is the language obtained from L by adding the length descriptor and then some dummy string that pads its size to a power of 2. Definition 6.3 An FPLT function f is computed by two polylogarithmic time bounded deterministic Turing Machines N and M , such that on input x, N computes the size of the output jf(x)j, and on input x and i, M computes the i-th bit of f(x). A PLT reduction is a reduction computed by an FPLT function. Modulo PLT reductions, self-delimiting encoding is equivalent to standard encoding: Lemma 6.1 ([42]) For a nonempty binary language L, L j PLT sd(L). In particular, this means that there exists an FPLT function extract, which extracts a word from its self-delimiting encoding. Definition 6.4 Let F be a query on -structures. The succinct version s(F ) of F is given by If denote the corresponding -structure, otherwise gen - (C) denotes some default -structure. Using gen - , we can rephrase the definition of s(F ) as follows: The weak reducibility needed for the antecedent of the conversion lemma is given by so-called forgetful metric reductions; they differ from metric reductions in that the complexity of the inner function is restricted to FPLT, and that the outer ("forgetful") function may not access the original input. Definition 6.5 A function f is forgetfully metric reducible to a function g (in symbols, f - mr f g), if there is an FPLT function h 1 and a polynomial time computable function h 2 such that for every x, It is not hard to see that - mr f is transitive. The crucial observation needed to generalize the results about succinct decision problems to succinct function problems is that the succinct representation affects only the inner computation in the metric reduction (i.e., h 1 ), because the result of the succinct function s(F ) is not succinct. Thus, if we are able to lift the inner reductions from ordinary instances to succinct instances, then we can leave the outer computation (i.e., h 2 ) unchanged. This lifting is achieved by the following lemma: Lemma 6.2 (immediate from [44]) Let f be a FPLT function which maps -structures to oe-structures. Then there exists an FPLT function F s.t. for all circuits C With this background, the conversion lemma is easy to show: Lemma 6.3 (Conversion Lemma) Let F be a query over -structures, and G be a query over oe-structures. f s(G). Proof. By assumption we have F We have to show that there exist an FPLT function H 1 and a polynomial time function H 2 such that By Lemma 6.2 there is an H 1 s.t. h 1 (gen - Then we can set H and the lemma is proven. J It remains to define a suitable long operator. Recall that it has to simplify the complexity of its argu- ment. Following [42], we obtain the following definition for long on queries: Definition 6.6 Let (R 1 ) be a signature with a single unary relation symbol, and let Q be a convex query over signature - . Then the query long(Q) is defined as follows: where char(A) is the value of the binary number obtained by concatenating a leading 1 with the characteristic sequence of the tuples in A in lexicographical order. Lemma 6.4 (Compensation Lemma) Let F be a query. Then F - mr Proof. As in Lemma 6.3, it is sufficient to show that every input T of F can be translated into a This was shown (using somewhat different terminology) in [42, Lemma 6]. J Theorem 6.5 Let F 1 be two classes of functions, such that long(F 1 . If a query F is hard for f -reductions, then s(F ) is hard for F 1 under - mr f -reductions. 6.3 Succinctness and expression complexity Succinct problems and expression complexity are related by the following methods, which was used in [23, 14] and generalized in [18]: Suppose that a language L can express a C 2 -complete property A. Then its data complexity is trivially . If the language is rich enough to simulate a Boolean circuit by a program of roughly the same size, then it is possible to combine a program for A with a program for circuit simulation, thus obtaining a program for s(A). Consequently, there is a reduction from s(A) to the expression complexity of L. In [14], it was shown how a negation-free DDDB can simulate a Boolean circuit: Let be a boolean circuit that decides a k-ary predicate R over f0; 1g, i.e., for any tuple supplied to C as input, a designated output gate of C , which we assume is g t , has value 1 iff We describe a program -C that simulates C using the universe f0; 1g. For each gate g i , -C uses a k-ary predicate G i , where G i (-x) informally states that on input of tuple - x to C , the circuit computation sets the output of g i to 1. Moreover, it uses a propositional letter False , which is true in those models in which the G i do not have the intended interpretation; none of these models will be minimal. The clauses of -C are the following ones. For each gate of C , it contains the clause Depending on the type a i , -C contains for additional clauses: The clauses (00) ensure that if a model of ground(-C ; f0; 1g) contains False , it is the maximal interpretation (which is trivially a model of -C ). In fact, this is the only model of -C that contains False. Let MC denote the interpretation given by takes value 1 on input t to C g. Lemma 6.6 ([14]) For any Boolean circuit C , MC is the unique minimal model of -C . Theorem 6.7 Problem -DD is complete for FPSpace NP . Proof. Define as usual a problem whose input contains a uniform circuit with constant input gates which generates the instance to be solved. Then it is easy to see that FP NP k contains a query Q which is complete under - mr f -reductions. (For example, QUERY is such a problem: the function h 1 of any metric reduction g - mr QUERY can be shifted inside the oracle queries in polylog-time, and the bits of the input string x provided through dummy oracle queries to h 2 .) It is not hard to see that long(FLinSpace NP , and therefore by Theorem 6.5, s(Q) is complete for FLinSpace NP . By standard padding arguments, completeness for FLinSpace NP implies completeness for FPSpace NP . Thus, it remains to reduce s(Q) to -DD. By Lemma 6.6, a circuit C with k input gates can be converted into a disjunctive program -C whose k-ary output relation R describes the string gen(C). Consider the query Q(extract(A)), where A is an ordered input structure which describes a string by a unary relation. Since Q(extract(A)) is easily seen to be in FP NP k , Theorem 5.1 implies that there is a program - whose data disjunction describes the result of Q(extract(A)). Since DDDBs are uniformly closed, - can be rewritten into a program - 0 whose input relation R has arity k. As in the proof of Theorem 5.1 we can assume that there is a lexicographical successor relation on k-tuples. By well-known modularity properties [14, Section 5], the program - 0 [-C indeed computes Q on the succinctly specified input. J It is not hard to show that also FL NP log has - mr f -complete queries (e.g., a variant of CLIQUE SIZE in which circuits computing the functions h of a metric reduction to CLIQUE SIZE are part of the problem instance). The following theorem is then shown analogously: Theorem 6.8 Problem k -DD is complete for FPSpace NP [pol]. 7 Further Results and Conclusion 7.1 Closed world reasoning The results on data disjunctions that we have derived above have an immediate application to related problems in the area of closed-world reasoning. Reiter [36] has introduced the closed-world assumption (CWA) as a principle for inferring negative information from a logical database. Formally, For example, CWA(fP (a); (a)g. It follows from results in [11] that computing CWA(DB) has propositional complexity FP NP Observe that CWA(DB) may not be classically consistent (under Herbrand interpretations); for exam- ple, CWA(fP - which has no model. As shown in [11], deciding whether CWA(DB) is consistent is in \Theta P 2 and coNP-hard in the propositional case; the precise complexity of this problem is open. In a refined notion of partial CWA (cf. [16]), which is in the spirit of protected circumscription [34], only atoms A involving a particular predicate P or, more general, a predicate P from a list of predicates P may be negatively concluded from DB: (b)g. Definition 7.1 (P-minimal model) Let P be a list of predicates. The preorder -P on the models of a DDDB DB is defined as follows: M -P M 0 , for every P (-c) 2 HB DB such that P 2 P it holds that . A model M is P-minimal for DB, if there exists no model M 0 such that We remark that a P-minimal model is a special case of the notion of model [28], given by an empty list of fixed predicates in a circumscription. Proposition 7.1 Let DB be a DDDB and P a predicate. Then, the following statements are equivalent: 1. DB has a data disjunction on P . 2. PCWA(DB;P ) is not consistent (with respect to classical Herbrand models). 3. DB does not have a global P -minimal model M , i.e., M - P M 0 for all models M 0 of DB. Proof. 1: ! 2:: Suppose 2, is a data disjunction of DB. Then, implies that which means PCWA(DB;P ) is not consistent. Suppose DB has a global P -minimal model M . Then, for each atom P (-c) 2 HB DB it holds that DB 6j= P (-c) iff M 6j= P (-c), since M has the unique smallest P -part over all models of DB. Hence, M is model of PCWA(DB;P ), and thus PCWA(DB;P ) is consistent. 3: ! 1:: Suppose DB has no global P -minimal model. Let M be the collection of all P - minimal models M of DB, where w.l.o.g. M 1 6- P M 2 and M 2 6- P M 1 . Let X be the set of all atoms in arbitrary atoms. Then, holds for every which means DB contains a data disjunction on predicate P (which contains P As for P-minimality, a list P of predicates can, by simple coding, be replaced with a single predicate where the first argument in P codes the predicate. This coding is compatible with P-minimality, i.e., P-minimal and P -minimal models correspond as obvious. From Proposition 7.1 and the results of the previous sections, we thus obtain the following result. Theorem 7.2 Deciding consistency of PCWA(DB;P) and existence of some global P-minimal model of DB have both \Theta P propositional and data complexity, and PSpace NP program and expression complexity. By the same coding technique, the result in Theorem 7.2 holds even if the language has only two predicates and P contains a single predicate. On the other hand, if the language has a only one predicate , then the existence of a data disjunction on p is equivalent to the consistency of CWA(DB), whose precise complexity is open. 7.2 Restricted data disjunctions In [15], a stronger notion of data disjunction R-c is considered, which requests in addition that all disjuncts R-c i are identical up to one argument of the list of constants - c i ; we call such data disjunctions restricted. Note that all data disjunction considered in Section 1 are restricted. For the problems reformulated to restricted data disjunctions, Table 1 in Section 1 is the same except that the expression and combined complexity of -DD is FPSpace NP [pol]. Indeed, a restricted data disjunction C has at most m disjuncts where is the number of constants, and thus -DD has log n) many output bits in the combined complexity case, where n is the size of DB. The number of maximal disjunctions md(DB;R), adapted to restricted data disjunctions, is polynomial in the data size, and thus the same upper bounds can be easily derived as for unrestricted data disjunctions. All hardness results are immediate from the proofs except for propositional and data complexity of -DD; here, mapping of elements to newly introduced (polynomially many) domain elements is a suitable technique for adapting the construction in Figure 2 in the proof of Theorem 5.1. Finally, we remark that Lemma 2.3 remains true even if all disjunctions in the program - ' describe restricted data disjunctions. Thus, by a slight adaptation of the programs in proofs and exploiting the fact that disjunction-free datalog with input negation is sufficient for upgrading purposes [14], the complexity results for restricted data disjunction remain true even if all disjunctions in DB must be restricted data disjunctions. 7.3 Conclusion In this paper, we have considered the complexity of some problems concerning data disjunctions in deductive databases. To this aim, we have taken an "engineering perspective" on deriving complexity results using tools from the domain of descriptive complexity theory, and combined them with results for upgrading complexity results on normal to succinct representations of the problem input. In particular, we have also investigated the complexity of actually computing data disjunctions as a function, rather than only the associated decision problem. This led us to generalize upgrading techniques developed for decision problems to computations of functions. These upgrading results, in particular Theorem 6.5, may be conveniently used in other contexts. The tools as used and provided in this paper allow for a high-level analysis of the complexity of prob- lems, in the sense that establishing certain properties and schematic reductions are sufficient in order to derive intricate complexity results as eg. for the case of data disjunctions in a clean and transparent way, without the need to deal with particular problems in reductions. While this relieves us from spelling out detailed technical constructions, the understanding of what makes the problem computationally hard may be blurred. In particular, syntactical restrictions under which the complexity remains the same or is lowered can not be immediately inferred. We leave such considerations for further work. Another interesting issue for future work is the consideration of computing data disjunctions viewed as a multi-valued function, which we have not done here. Acknowledgment We are grateful to Iain Stewart and Georg Gottlob for discussions and remarks. --R Foundations of Databases. Sparse sets The complexity of algorithmic problems on succinct instances. Autoepistemic logics as a unifying framework for the semantics of logic programs. Querying disjunctive databases through nonmonotonic logics. Succinct circuit representations and leaf languages are basically the same concept. Six hypotheses in search of a theorem. The complexity of propositional closed world reasoning and circumscription. Semantics of logic programs: Their intuitions and formal properties. Propositional circumscription and extended closed world reasoning are The complexity class Normal forms for second-order logic over finite structures Disjunctive datalog. A tractable class of disjunctive deductive databases. Logical Foundations of Artificial Intelligence. Relativized logspace and generalized quantifiers over ordered finite structures. Succinctness as a source of complexity in logical formalisms. Complexity of query processing in databases with or-objects Computing functions with parallel queries to NP. The computational complexity of graph problems with succinct multigraph representation. Why not negation by fixpoint Vector language: Simple descriptions of hard instances. The complexity of optimization problems. Relativization questions about logspace computability. Comparison of polynomial-time reducibilities Computing circumscription. Foundations of Logic Programming. Foundations of Disjunctive Logic Programming. The complexity of graph problems for succinctly represented graphs. On indefinite data bases and the closed world assumption. Logic and databases: A 20 year retrospective. Computing protected circumscription. A Note on succinct representations of graphs. On closed-world databases A taxonomy of complexity classes of functions. Logical characterizations of bounded query classes I: Logspace oracle machines. Logical characterizations of bounded query classes II: Polynomial-time oracle machines On polynomial-time truth-reducibilities of intractable sets to p-selective sets The complexity of relational query languages. Languages represented by Boolean formulas. How to encode a logical structure by an obdd. Succinct representation Bounded query classes. --TR A note on succinct representations of graphs Logical foundations of artificial intelligence Foundations of logic programming; (2nd extended ed.) The complexity of optimization problems Complexity of query processing in databases with OR-objects The complexity of graph problems for succinctly represented graphs Vector language: simple description of hard instances On truth-table reducibility to SAT Bounded query classes Why not negation by fixpoint? Foundations of disjunctive logic programming Propositional circumscription and extended closed-world reasoning are MYAMPERSANDPgr;<supscrpt>p</supscrpt><subscrpt>2</subscrpt>-complete The complexity of algorithmic problems on succinct instances A taxonomy of complexity classes of functions Computing functions with parallel queries to NP Querying disjunctive databases through nonmonotonic logics Succinct circuit representations and leaf language classes are basically the same concept Disjunctive datalog Languages represented by Boolean formulas Succinct representation, leaf languages, and projection reductions Foundations of Databases The Complexity Class Theta2p Logic and Databases On Indefinite Databases and the Closed World Assumption Six Hypotheses in Search of a Theorem How to Encode a Logical Structure by an OBDD The complexity of relational query languages (Extended Abstract)

deductive databases;conversion lemma;complexity upgrading;computational complexity;data disjunction

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Guarded fixed point logics and the monadic theory of countable trees.

Different variants of guarded logics (a powerful generalization of modal logics) are surveyed and an elementary proof for the decidability of guarded fixed point logics is presented. In a joint paper with Igor Walukiewicz, we proved that the satisfiability problems for guarded fixed point logics are decidable and complete for deterministic double exponential time (E. Grdel and I. Walulkiewicz, Proc. 14th IEEE Symp. on Logic in Computer Science, 1999, pp. 45-54). That proof relies on alternating automata on trees and on a forgetful determinacy theorem for games on graphs with unbounded branching. The exposition given here emphasizes the tree model property of guarded logics: every satisfiable sentence has a model of bounded tree width. Based on the tree model property, we show that the satisfiability problem for guarded fixed point formulae can be reduced to the monadic theory of countable trees (SS), or to the -calculus with backwards modalities.

Introduction Guarded logics are dened by restricting quantication in rst-order logic, second-order logic, xed point logics or innitary logics in such a way that, semantically speaking, each subformula can 'speak' only about elements that are 'very close together' or `guarded'. Email address: graedel@informatik.rwth-aachen.de (Erich Gradel). URL: www-mgi.informatik.rwth-aachen.de (Erich Gradel). Preprint submitted to Elsevier Preprint 5 October 2000 Syntactically this means that all rst-order quantiers must be relativised by certain 'guard formulae' that tie together all the free variables in the scope of the quantier. Quantication is of the form where quantiers may range over a tuple y of variables, but are 'guarded' by a formula that must contain all the free variables of the formula that is quantied over. The guard formulae are of a simple syntactic form (in the basic version, they are just atoms). Depending on the conditions imposed on guard formulae, one has logics with dierent levels of 'closeness' or `guardedness'. Again, there is a syntactic and a semantic view of such guard conditions. Let us start with the logic GF, the guarded fragment of rst-order logic, as it was introduced by Andreka, van Benthem, and Nemeti [1]. Denition 1.1. GF is dened inductively as follows: (1) Every relational atomic formula Rx i 1 or x belongs to GF. (2) GF is closed under boolean operations. (3) If x; y are tuples of variables, (x; y) is a positive atomic formula, and (x; y) is a formula in GF such that free( ) y, then also the formulae belong to GF. Here free( ) means the set of free variables of . An atom (x; y) that rel- ativizes a quantier as in rule (3) is the guard of the quantier. Hence in GF, guards must be atoms. But the really crucial property of guards (also for the more powerful guarded logics that we will consider below) is that it must contain all free variables of the formula that is quantied over. The main motivation for introducing the guarded fragment was to explain and generalize the good algorithmic and model-theoretic properties of propositional modal logics (see [1,26]). Recall that the basic (poly)modal logic ML (also called propositional logic by the possibility to construct formulae hai and [a] (for any a from a given set A of 'actions' or `modalities') with the meaning that holds at some, respectively each, a-successor of the current state. (We refer to [4] or [22] for background on modal logic). Although ML is formally a propositional logic we really view it as a fragment of rst-order logic. Kripke structures, which provide the semantics for modal logics, are just relational structures with only unary and binary relations. There is a standard translation taking every formula 2 ML to a rst-order formula (x) with one free variable, such that for every Kripke structure K with a distinguished node w we have that K; w if and only if K (w). This translation takes an atomic proposition P to the atom Px, it commutes with the Boolean connectives, and it translates the modal operators by quantiers as follows: where E a is the transition relation associated with the modality a. The modal fragment of rst-order logic is the image of propositional modal logic under this translation. Clearly the translation of modal logic into rst-order logic uses only guarded quantication, so we see immediately that the modal fragment is contained in GF. The guarded fragment generalizes the modal fragment by dropping the restrictions to use only two variables and only monadic and binary predicates, and retains only the restriction that quantiers must be guarded. The following properties of GF have been demonstrated [1,10]: (1) The satisability problem for GF is decidable. (2) GF has the nite model property, i.e., every satisable formula in the guarded fragment has a nite model. (3) GF has (a generalized variant of) the tree model property. Many important model theoretic properties which hold for rst-order logic and modal logic, but not, say, for the bounded-variable fragments do hold also for the guarded fragment. (5) The notion of equivalence under guarded formulae can be characterized by a straightforward generalization of bisimulation. Further work on the guarded fragment can be found in [7{9,17]. Based on this kind of results Andreka, van Benthem, and Nemeti put forward the 'thesis' that it is the guarded nature of quantication that is the main reason for the good model-theoretic and algorithmic properties of modal logics. Let us discuss to what extent this explanation is adequate. One way to address this question is to look at the complexity of GF. We have shown in [10] that the satisability problem for GF is complete for 2Exptime, the class of problems solvable by a deterministic algorithm in time 2 2 p(n) , for some polynomial p(n). This seems very bad, in particular if we compare it to the well-known fact that the satisability problem for propositional modal logic is in Pspace [19]. But dismissing the explanation of Andreka, van Benthem, and Nemeti on these grounds would be too supercial. Indeed, the reason for the double exponential time complexity of GF is just the fact that predicates may have unbounded arity (wheras ML only expresses properties of graphs). Given that even a single predicate of arity n over a domain of just two element leads to 2 2 n possible types already on the atomic level, the double exponential lower complexity bound is hardly a surprise. Further, in most of the potential applications of guarded logics the arity of the relation symbols is bounded. But for GF-sentences of bounded arity, the satisability problem can be decided in Exptime [10], which is a complexity level that is reached already for rather weak extensions of ML (e.g. by a universal modality) [25]. Thus, the complexity analysis does not really provide a decisive answer to our question. To approach the question from a dierent angle, let us look at extensions of ML. Indeed ML is a very weak logic and the really interesting modal logics extend ML by features like path quantication, temporal operators, least and greatest xed points etc. which are of crucial importance for most computer science applications. It has turned out that many of these extended modal logics are algorithmically still manageable and actually of considerable practical importance. The most important of these extensions is the modal -calculus , which extends ML by least and greatest xed points and subsumes most of the modal logics used for automatic verication including CTL, LTL, CTL , PDL, and also many description logics. The satisability problem for L is known to be decidable and complete for Exptime [5]. Therefore, a good test for the explanation put forward by Andreka, van Benthem, and Nemeti is the following problem: If we extend GF by least and greatest xed points, do we still get a decidable logic? If yes, what is its complexity? To put it dierently, what is the penalty, in terms of complexity, that we pay for adding xed points to the guarded In [13] we were able to give a positive answer to this question. The model-theoretic and algorithmic methods that are available for the -calculus on one side, and the guarded fragments of rst-order logic on the other side, can indeed be combined and generalized to provide positive results for guarded xed point logics. (Precise denitions for these logics will be given in the next section.) In fact we could establish precise complexity bounds. Theorem 1.2 (Gradel, Walukiewicz). The satisability problems for guarded xed point logics are decidable and 2Exptime-complete. For guarded xed point sentences of bounded width the satisability problem is Exptime- complete. By the width of a formula , we mean the maximal number of free variables in the subformulae of . For sentences that are guarded in the sense of GF, the width is bounded by the maximal arity of the relation symbols, but there are other variants of guarded logics where the width may be larger. Note that for guarded xed point sentences of bounded width the complexity level is the same as for -calculus and for GF without xed points. The proof that we give in [13] relies on alternating two-way tree automata (on trees of unbounded branching), on a forgetful determinacy theorem for parity games, and on a notion of tableaux for guarded xed point sentences, which can be viewed as tree representations of structures. We associate with every guarded xed point sentence an alternating tree automaton A that accepts precisely the tableaux that represent models for . This reduces the satisability problem for guarded xed point logic to the emptiness problem for alternating two-way tree automata. In this paper we discuss other variants of guarded logics, with more liberal notions of guarded quantication and explain alternative possibilities to design decision procedures for guarded xed point logics. Already in [3], van Benthem had proposed loosely guarded quantication (also called guarded quantication) as a more general way of restricting quantiers, and proved that also LGF, the loosely guarded fragment of rst-order logic, remains de- cidable. Here we motivate and introduce clique guarded quantication, which is even more liberal than loosely guarded quantication, but retains the same decidability properties. The techniques for establishing decidability results for guarded xed point logics that we explain in this paper exploit a crucial property of such logics, namely the (generalized variant of the) tree model property, saying that every satisable sentence of width k has a model of tree width at most k 1. The tree width of a structure is a notion coming from graph theory which measures how closely the structure resembles a tree. Informally a structure has tree width k, if it can be covered by (possibly overlapping) substructures of size at most are arranged in a tree-like manner. The tree model property for guarded logics is a consequence of their invariance under a suitable notion of bisimulation, called guarded bisimulation. Guarded bisimulations play a fundamental role for characterizing the expressive power of guarded logics, in the same way as usual bisimulations are crucial for understanding modal logics. For instance, the characterization theorem by van Benthem [2], saying that a property is denable in propositional modal logic if and only if it is rst-order denable and invariant under bisimulation. has a natural analogue for the guarded fragment: GF can dene precisely the model classes that are rst-order denable and invariant under guarded bisimulation [1]. We will explain and prove a similar result for the clique-guarded fragment in Section 3. There is a similar and highly non-trivial characterisation theorem for the modal -calculus, due to Janin and Walukiewicz [18], saying that the properties denable in the modal -calculus are precisely the properties that are denable in monadic second-order logic and invariant under bisimulation. And, as shown recently by Gradel, Hirsch, and Otto [12], this result also carries over to the guarded world. Indeed, there is a natural fragment of second-order logic, called GSO, which is between monadic second-order logic and full second-order logic, such that guarded xed point logic is precisely the bisimulation-invariant portion of GSO. Outline of the paper. In Sect. 2 we discuss dierent variants of guarded logics. In particular we introduce the notion of clique-guarded quantication and present the precise denitions and elementary properties of guarded xed point logics. In Sect. 3 we explain the notions of guarded bisimulations, of tree width and of the unraveling of a structure. We prove a characterization theorem for the clique-guarded fragment and discuss the tree model property of guarded logics. Based on the tree model property we will present in Sect. 4.2 a simpler decidability proof for guarded xed point logic that replaces the automata theoretic machinery used in [13] by an interpretation argument into the monadic second-order theory of countable trees (S!S) which by Rabin's famous result [23] is known to be decidable. We then show in Sect. 4.3 that instead of using S!S, one can also reduce guarded xed point logic to the -calculus with backwards modalities which has recently been proved to be decidable (in Exptime, actually) by Vardi [27]. We remark that this paper is to a considerable extent expository. The new results, mostly concerning clique-guarded logics, can also be derived using the automata theoretic techniques of [13]. However, it is worthwhile to make the role of guarded bisimulations explicit and to show how one can establish decidability results for guarded xed point logics via reductions to well-known formalisms such as S!S or the -calculus. Even if the automata theoretic method gives more ecient algorithms, the reduction technique provides a simple and high-level method for proving decidability, avoiding any explicit use of automata-theoretic machinery (the use of automata is hidden in the decision algorithms for S!S or the -calculus). For the convenience of the reader, we have included explicit proofs of some facts that are known | or straightforward variations of known results | but where the proofs are hard to nd. Guarded logics There are several ways to dene more general guarded logics than GF. On one side, we can consider other notions of guardedness, and on the other side we can look at guarded fragments of more powerful logics than rst-order logic. We rst consider other guardedness conditions. Loosely guarded quantication. The direct translation of temporal formulae say over the temporal frame (N ; <), into rst-order logic is which is not guarded in the sense of Denition 1.1. However, the quantier 8z in this formula is guarded in a weaker sense, which lead van Benthem [3] to the following generalization of GF. Denition 2.1. The loosely guarded fragment LGF is dened in the same way as GF, but the quantier-rule is relaxed as follows: is in LGF, and (x; is a conjunction of atoms, then belong to LGF, provided that free( ) any two variables z 2 y, z 0 2 x[ y there is at least one atom j that contains both z and z 0 . In the translation of ( until ') described above, the quantier 8z is loosely guarded by (x z^z < y) since z coexists with both x and y in some conjunct of the guard. On the other side, the transitivity axiom Exz) is not in LGF. The conjunction Exy ^ Eyz is not a proper guard of 8xyz since x and z do not coexist in any conjunct. Indeed, it has been shown in [10] that there is no way to express transitivity in LGF. Clique-guarded quantication. In this paper we introduce a new, even more liberal, variant of guarded quantication, which leads to what we may call clique-guarded logics. To motivate this notion, let us look at the semantic meaning of guardedness. Denition 2.2. Let B be a structure with universe B and vocabulary . A set guarded in B if there exists an atomic formula Note that every singleton set guarded (by the atom is guarded some guarded set X. Clearly, sentences of GF can refer only to guarded tuples. Consider now the LGF-sentence While the rst quantier is guarded even in the sense of GF, the second one is only loosely guarded: the quantied variables u; v coexist in an atom of the guard (in fact in both of them) and they also coexist with each of the other z in one of the atoms. The subformula ' can hence talk about quadruples (y; z; u; v) in a structure, that are not guarded in the sense of the denition just given, but in weaker sense. The corresponding semantic denition of a loosely guarded set in a structure B is inductive. Denition 2.3. A set X is loosely guarded in the structure B if it either is a guarded set, or if there exists a loosely guarded set X 0 such that for every there is a guarded set Y with fa; bg Y X. For instance in the structure and a ternary relation consisting of the triples (a; b; c); (b; d; e); (c; d; e), the set fb; c; d; eg is loosely guarded. Note that the elements of a loosely guarded set need not coexist in a single atom of the structure, but they are all 'adjacent' in the sense of the locality graph or Gaifman graph of a structure. Denition 2.4. The Gaifman graph of a relational structure B (with universe B) is the undirected graph there exists a guarded set X B with a; a 0 2 Xg: set X of elements of a structure B is clique-guarded in B if it induces a clique in G(B). A tuple (b clique-guarded if its components form a clique-guarded set. Lemma 2.5. Every loosely guarded set is also clique-guarded. Proof. We proceed by induction on the denition of a loosely guarded set. If X is guarded, then it is obviously also clique-guarded. Otherwise there exists a loosely guarded set X 0 and, for every a 2 X X 0 , b 2 X a guarded set containing both a and b. Hence all such a and b are connected in G(B). It remains to consider elements b that are both contained in X [ X 0 . In that case, a and b are connected in G(B) because, by induction hypothesis, X 0 induces a clique in G(B). The converse is not true, as the following example shows. Consider a structure and one ternary relation R containing the triangles (a 1 ; a is neither guarded nor loosely guarded, but induces a clique in G(A). Note that for each nite vocabulary and each k 2 N , there is a positive, existential rst-order formula clique(x B and all b induce a clique in G(B): Denition 2.6. The clique-guarded fragment CGF of rst-order logic is dened in the same way as GF and LGF, but with the clique-formulae as guards. Hence, the quantication rule for CGF is (3) 00 If (x; y) is a formula CGF, then belong to CGF, provided that free( ) y. Note that quantiers over tuples are in principle no longer needed in CGF (contrary to GF and LGF), since they can be written as sequences of clique- guarded quantiers over single variables. Alternative denitions of CGF. In practice, one will of course not write down the clique-formulae explicitly. One possibility is not to write them down at all, i.e., to take the usual (unguarded) rst-order syntax and to change the semantics of quantiers so that only clique-guarded tuples are considered. More precisely: let '(x; y) be a rst-order formula, B a structure and a be a clique-guarded tuple of elements of B. Then B there exists an element b such that (a; b) is clique-guarded and B for universal quantiers. It is easy to see that, for nite vocabularies, this semantic denition of CGF is equivalent to the one given above. An alternative possibility is to permit as guards any existential positive formul (x) that implies clique(x). This is what Maarten Marx [20,16] uses in his denition of the packed fragment PF. The dierences between the clique- guarded fragment and the packed fragment are purely syntactical. PF and CGF have the same expressive power. The work of Maarten Marx and ours have been done independently. Every LGF-sentence is equivalent to a CGF-sentence. (The analogous statement for formulae is only true if we impose that the free variables must be interpreted by loosely guarded tuples. However, in this paper, we restrict attention to sentences.) We observe that CGF has strictly more expressive power than LGF. Proposition 2.7. The CGF-sentence 8xyz(clique(x; Rxyz) is not equivalent to any sentence in LGF. The good algorithmic and model-theoretic properties of GF go through also for LGF and CGF. In most cases, in particular for decidability, for the characterization via an appropriate notion of guarded bisimulation and for the tree model property, the proofs for GF extend without major diculties. An exception is the nite model property which, for GF, has been established in [10], and where the extension to LGF and CGF, recently established by Ian Hodkinson [15], requires considerable eort. Notation. We use the notation i.e., we write guarded formulae in the form When this notation is used, then it is always understood that is indeed a proper guard as specied by condition (3), (3) 0 , or (3) 00 . Guarded xed point logics. We now dene guarded xed point logics, which can be seen as the natural common extensions of GF, LGF and CGF on one side, and the -calculus on the other side. Denition 2.8. The guarded xed point logics GF; LGF, and CGF are obtained by adding to GF; LGF, and CGF, respectively, the following rules for constructing xed point formulae: Let W be a k-ary relation a k-tuple of distinct vari- ables, and (W; x) be a guarded formula that contains only positive occurrences of W , no free rst-order variables other than x is not used in guards. Then we can build the formulae The semantics of the xed point formulae is the usual one: Given a structure A providing interpretations for all free second-order variables in , except W , the formula (W; x) denes an operator on k-ary relations W A k , namely Since W occurs only positively in , this operator is monotone (i.e., W W 0 implies A (W ) A (W 0 )) and therefore has a least xed point LFP( A ) and a greatest xed point GFP( A ). Now, the semantics of least xed point formulae is dened by A and similarly for the greatest xed points. Least and greatest xed point can be dened inductively. For a formula (W; x) with k-ary relation variable W a structure A, and ordinals set ~ W for limit ordinals The relations W (resp. ~ are called the stages of the LFP-induction (resp. GFP-induction) of (W; x) on A. Since the operator A is monotone, we have ~ W +1 , and there exist ordinals ; 0 such that W ~ These are called the closure ordinals of the LFP-induction resp. GFP-induction of (W; x) on A. Finite and countable models. Contrary to GF, LGF, CGF, and also to the modal -calculus, guarded xed point logics do not have the nite model property [13]. Proposition 2.9. Guarded xed point logic GF (even with only two vari- ables, without nested xed points and without equality) contains innity axioms Proof. Consider the conjunction of the formulae 9xyFxy The rst two formulae say that a model should contain an innite F -path and the third formula says that F is well-founded, thus, in particular, acyclic. Therefore every model of the three formulae is innite. On the other side, the are clearly satisable, for instance by (!; <). While the nite model property fails for guarded xed point logics we recall, for future use, that the Lowenheim-Skolem property is true even for the (un- guarded) least xed point logic every satisable xed point sentence has a countable model. This result is part of the folklore on xed point logic, but it is hard to nd a published proof. Our exposition follows the one in [6]. Theorem 2.10. Every satisable sentence in hence every satisable sentence in GF, LGF, or CGF, has a model of countable cardinality. Proof. Consider a xed point formula of form (x) := [LFP Rx : '(R; x)](x), with rst-order ' such that A j= (a) for some innite model A. For any ordinal , let R be stage of the least xed point induction of ' on A. Expand A by a monadic relation U , a binary relation <, and a m relation (where m is the arity of R) such that (1) (U; <) is a well-ordering of length 1, and < is empty outside U . describes the stages of ' A in the following way ; u is the -th element of (U; <), and b 2 R g: In the expanded structure A := (A; U; <; S) the stages of the operator ' A are dened by the sentence := 8u8x Here '[Ry=9z(z < is the formula obtained form '(R; x) by replacing all occurrences of subformula Ry by 9z(z < be a countable elementary substructure of A , containing the tuple a. Since A the stages of ' B . Since also B j= 9uSua, it follows that a is contained in the least xed point of ' B , i.e., B (a). A straightforward iteration of this argument gives the desired result for arbitrary nestings of xed point operators, and hence for the entire xed point Guarded innitary logics. It is well known that xed point logics have a close relationship to innitary logics (with bounded number of variables). Denition 2.11. GF are the innitary variants of the guarded fragments GF, LGF, and CGF, respectively. For instance GF 1 extends GF by the following rule for building new formulae: If GF 1 is any set of formulae, then also W and V are formulae of GF 1 . The denitions for LGF 1 and CGF 1 are anologous. In the sequel we explicitly talk about the clique-guarded case only, i.e., about CGF and CGF 1 , but all results apply to the guarded and loosely guarded case as well. The following simple observation relates CGF with CGF 1 . Proposition 2.12. For each 2 CGF of width k and each cardinal , there is a 0 2 CGF 1 , also of width k, which is equivalent to on all structures up to cardinality Proof. Consider a xed point formula [LFP Rx : '(R; x)](x). For every ordinal , there is a formula ' (x) 2 CGF 1 that denes the stage of the induction of '. Indeed, let ' 0 (x) := false, let ' +1 (x) := '[Ry=' (y)](x), that is, the formula that one obtains from '(R; x) if one replaces each atom Ry (for any y) by the formula ' (y), and for limit ordinals , let ' (x) := On structures of bounded cardinality, also the closure ordinal of any xed-point formula is bounded. Hence for every cardinal there exists an ordinal such that [LFP Rx : '(R; x)](x) is equivalent to ' (x) on structures of cardinality at most Remark. Without the restriction on the cardinality of the structures, this result fails. Indeed there are very simple xed point formulae, even in the modal -calculus (such as well-foundedness axioms), that are not equivalent to any formulae of the full innitary logic L1! . Guarded bisimulation and the tree model property Tree width is an important notion in graph theory. Many dicult or undecidable computational problems on graphs become easy on graphs of bounded tree width. The tree width of a structure measures how closely it resembles a tree. Informally, a structure has tree width k, if it can be covered by (pos- sibly overlapping) substructures of size at most k + 1 which are arranged in a tree-like manner. For instance, trees and forests have tree width 1, cycles have tree width 2, and the n n-grid has tree width n. Here we need the notion of tree width for arbitrary relational structures. For readers who are familiar with the notion of tree width in graph theory we can simply say that the tree width of a structure is the tree width of its Gaifman graph. Here is a more detailed denition. Denition 3.1. A structure B (with universe B) has tree width k if k is the minimal natural number satisfying the following condition. There exists a directed tree E) and a function assigning to every node v of T a set F (v) of at most k +1 elements of B, such that the following two conditions hold. (i) For every guarded set X in B there exists a node v of T with X F (v). (ii) For every element b of B, the set of nodes fv connected (and hence induces a subtree of T ). For each node v of T , F (v) induces a substructure F(v) B of cardinality at most k + 1. (Since F (v) may be empty, we also permit empty substructures.) is called a tree decomposition of width k of B. Remark. A more concise, but equivalent, formulation of clause (i) would be that v2T F(v). By denition, every guarded set X B is contained in some F (v). A simple graph theoretic argument shows that the same is true for loosely guarded and clique-guarded sets. Lemma 3.2. Let hT; (F(v) v2T )i be a tree decomposition of B and X B be a clique-guarded set in B. Then there exists a node v of T such that X F (v). Proof. For each b 2 X, let V b be the set of nodes v such that b 2 F (v). By the denition of a tree decomposition, each V b induces a subtree of T . For all is non-empty, since b and b 0 are adjacent in G(B) and must therefore coexist in some atomic fact that is true in B. It is known that any collection of pairwise overlapping subtrees of a tree has a common node (see e.g. [24, p. 94]). Hence there is a node v of the T such that F (v) contains all elements of X. Guarded bisimulations. The notion of bisimulation from modal logic generalises in a straightforward way to various notions of guarded bisimulation that describe indistinguishability in guarded logics. We focus here on clique- bisimulations, the appropriate notion for clique-guarded formulae. The notions of guarded or loosely guarded bisimulations can be dened analogously. Denition 3.3. A clique-k-bisimulation, between two -structures A and B is a non-empty set I of nite partial isomorphisms Y from A to B, where X A and Y B are clique-guarded sets of size at most k, such that the following back and forth conditions are satised. For every I, for every clique-guarded set X 0 A of size at most k there exists a partial isomorphism in I such that f and g agree on X \ X 0 . back: for every clique-guarded set Y 0 B of size at most k there exists a partial isomorphism in I such that f 1 and g 1 agree on Y \Y 0 . Clique-bisimulations are dened in the same way, without restriction on the size of X; Y; X 0 and Y 0 . Two -structures A and B are clique-(k-)bisimilar if there exists a cliqe-(k-)bisimulation between them. Obviously, two structures are clique-bisimilar if and only if they are clique-k-bisimilar for all k. Remark. One can describe clique-k-bisimilarity also via a guarded variant of the innitary Ehrenfeucht-Frasse game with k pebbles. One just has to impose that after every move, the set of all pebbled elements induces a clique in the Gaifman graph of each of the two structures. Then A and B are clique- k-bisimilar if and only if Player II has a winning strategy for this guarded game. Adapting basic and well-known model-theoretic techniques to the present sit- uation, one obtains the following result. Theorem 3.4. Let A and B be two -structures. The following are equivalent: (i) A and B are clique-k-bisimilar. (ii) For all sentences 2 CGF 1 of width at most k, A Proof. (i) =) (ii): Let I be a clique-k-bisimulation between A and B, let formula in CGF 1 with width at most k such that A We show, by induction on , that there is no partial isomorphism f 2 I with . By setting the claim follows. If is atomic this is obvious, and the induction steps for are immediate. Hence the only interesting case concerns formulae of the form Since A j= (a), there exists a tuple a 0 in A such that A j= clique(a; a '(a; a 0 ). Suppose, towards a contradiction, that some f 2 I takes a to b. Since the set a[a 0 is clique-guarded there exists a partial isomorphism g 2 I, taking a to b and a 0 to some tuple b 0 in B. But then the tuple b[b 0 is clique-guarded and B contradicts the induction hypothesis. (ii) =) (i): Let I be the set of all partial isomorphisms f : a 7! b, taking a clique-guarded tuple a in A to a clique-guarded tuple b in B such that for all of width at most k, A and B cannot be distinguished by sentences of width k in CGF 1 , I contains the empty map and is therefore non-empty. It remains to show that I satises the back and forth properties. For the forth property, take any partial isomorphism I and any clique-guarded set X 0 in A of size at most k. Let X g. We have to show that there exists a g 2 I, dened on X 0 that coincides with f on X \ X 0 . Suppose that no such exists. Let a = a and let T be the set of all tuples b s such that b[b 0 is clique-guarded in B. Since there is no appropriate g 2 I there exists for every tuple b (a; a 0 But then we can construct the formula Clearly, A j= (a) but B which is impossible, given that f 2 I maps a to b. The proof for the back property is analogous. In particular, this shows that clique-(k-)bisimilar structures cannot be separated by CGF-sentences (of width k). Characterizing CGF via clique-guarded bisimulations. We show next that the characterisations of propositional modal logic and GF as bisimulation- invariant fragments of rst-order logic [1,2] have their counterpart for CGF and clique-guarded bisimulation. The proof is a straightforward adaptation of van Benthems proof for modal logic, but for the convenience of the reader, we present it in full. However, we assume that the reader is familiar with the notions of elementary extensions and !-saturated structures (see any textbook on model theory, such as [14,21]). We recall that every structure has an !- saturated elementary extension. Theorem 3.5. A rst-order sentence is invariant under clique-guarded bisimulation if and only if it is equivalent to a CGF-sentence. Proof. We have already established that CGF-sentences (in fact even sentences from CGF 1 ) are invariant under clique-guarded bisimulations. For the converse, suppose that is a satisable rst-order sentence that is invariant under clique-guarded bisimulations. Let be the set of sentences ' 2 CGF such that It suces to show that j= . Indeed, by the compactness theorem, already a nite conjunction of sentences from will then imply, and hence be equivalent to, . Since was assumed to be satisable, so is . Take any model A j= . We have to prove that A j= . Let TCGF (A) be the CGF-theory of A, i.e., the set of all CGF-sentences that hold in A. is satisable. Otherwise there were sentences such that CGF-sentence implied by and is therefore contained in . But then A which is impossible since This proves the claim. Take any model B be !-saturated elementary extensions of A and B, respectively. are clique-bisimilar. Let I be the set of partial isomorphisms Y from clique-guarded subsets of A + to clique-guarded subsets of B + such that, for all formulae '(x) in CGF and all tuples a from X, we have that A + '(fa). The fact that A are !-saturated implies that the back and forth conditions for clique-guarded bisimulations are satised by I. Indeed, let f 2 X, and let X 0 be any clique-guarded set in A + , with X 0 \ g. Let be the set of all formulae of form For every formula '(fa; y) 2 , we have A therefore Hence is a consistent type of which is, by !-saturation, realized in B + by some xed tuple b such that (fa; b) is clique-guarded. Hence the function g taking a to fa and a 0 to b is a partial isomorphism with domain X 0 that coincides with f on . The back property is proved in the same way, exploiting that A + is !-saturated. We can now complete the proof of the theorem. Since B an elementary extension of B, we have that By assumption, is invariant under clique-guarded bisimulations, so A + j= and therefore also A An analogous result applies to clique-k-bisimulations and CGF-sentences of width k, for any k 2 N . Unravelings of structures. The k-unraveling B (k) of a structure B is dened inductively. We build a tree T , together with functions F and G such that for each node v of T , F (v) induces a clique-guarded substructure F(v) B, and G(v) induces a substructure G(v) B (k) that is isomorphic to F(v). Further, hT; (G(v)) v2T i will be a tree decomposition of B (k) . The root of T is , with F Given a node v of T with F r g we create for every clique-guarded set in B with s k a successor node w of v such that F and G(w) is a set fb s g which is dened as follows. For those i, such that b j so that G(w) has the same overlap with G(v) as F (w) has with F (v). The other b in G(w) are fresh elements. G(w) be the bijection taking b i to b F(w) being the substructure of B induced by F (w), dene G(w) so that f w is an isomorphism from F(w) to G(w). Finally B (k) is the structure with tree decomposition hT; (G(v) v2T )i. Note that the k-unraveling of a structure has tree width at most k 1. Proposition 3.6. B and B (k) are k-bisimilar. Proof. Let I be the set of functions f nodes v of T . It follows that no sentence of width k in CGF 1 , and hence no sentence of width k in CGF distinguishes between a structure and its k-unraveling. Since every satisable sentence in CGF has a model of at most countable cardinality, and since the k-unraveling of a countable model is again countable we obtain the following tree model property for guarded xed point logic. Theorem 3.7 (Tree model property). Every satisable sentence in CGF with width k has a countable model of tree width at most k 1. Remark. In fact the decision algorithms for guarded xed point logics imply a stronger version of the tree model property, where the underlying tree has branching bounded by O(j 4 Decision procedures Once the tree model property is established, there are several ways to design decision algorithms for guarded logics. We focus here on guarded xed point logics (in fact on CGF which contains GF and LGF). 4.1 Tree representations of structures Let hT; be a tree decomposition of width k 1 of a -structure D with universe D. We want to describe D by a tree with a nite set of labels. To this end, we x a set K of 2k constants and choose a function f : D ! K assigning to each element d of D a constant a d 2 K such that the following condition is satised. If v; w are adjacent nodes of T , then distinct elements of F(v) [ F(w) are always mapped to distinct constants of K. For each constant a 2 K, let O a be the set of those nodes v 2 T at which the constant a occurs, i.e., for which there exists an element d 2 F(v) such that a. Further, we introduce for each m-ary relation R of D a tuple R := (R a of monadic relations on T with R a := fv 2 T : there exist d The tree E) together with the monadic relations O a and R a (for called the tree structure T (D) associated with D (and, strictly speaking, with its tree decomposition and with K and f ). Lemma 4.1. Two occurrences of a constant a 2 K at nodes u; v of T represent the same element of D if and only if a occurs in the label of all nodes on the link between u and v. (The link between two nodes u; v in a tree T is the smallest connected subgraph of T containing both u and v.) An arbitrary tree E) with monadic relations O a and R does dene a tree decomposition of width k 1 of some structure D, provided that the following axioms are satised. (1) At each node v, at most k of the predicates O a are true. (2) Neighbouring nodes agree on their common elements. For all m-ary relation we have the axiom 8x8y a2a (O a x ^ O a y) (R a x $ R a y) These are rst-order axioms over the vocabulary := fEg [ fO a : a 2 Kg. Given a tree structure T with underlying tree E) and monadic predicates O a and R a satisfying (1) and (2), we obtain a structure D such that T as follows. For every constant a 2 K, we call two nodes u; w of T a-equivalent if T j= O a v for all nodes v on the link between u and w. Clearly this is an equivalence relation on O T a . We write [v] a for the a-equivalence class of the node v. The universe of D is the set of all a-equivalence classes of T for a 2 K, i.e., O a vg: For every m-ary relation symbol R in , we dene (and hence all) v 2 4.2 Reduction to S!S We now describe a translation from CGF into monadic second-order logic on countable trees. Given a formula '(x we construct a monadic second-order formula ' a (z) with describe in the associated tree structure T (D) the same properties of clique-guarded tuples as '(x) does in D. (We will make this statement more precise below). On a directed tree E) we can express that U contains all nodes on the link between x and y by the formula For any set a K we can then construct a monadic second-order formula link a a2a O a z) saying that the tuple a occurs at all nodes on the link between x and y. The translation is now dened by induction as follows: (1) If '(x) is an atom Sx i 1 (3) If '(x) := clique(x), let clique a a;a2a 9y(link a;a 0 _ _ (z). ' a (z) := 9y link a _ O ' a (z) := 8S Here S is a tuple (S b ) b2K m of monadic predicates where m is the arity of S. Theorem 4.2. Let '(x) be a formula in CGF and D be a structure with tree decomposition hT; For an appropriate set of constants K and a (D) be the associated tree structure. Then, for every node v of T and every clique-guarded tuple d F(v) with Proof. We proceed by induction on '. The non-trivial cases are the clique- guards, existential quantication and least xed points. For the clique-guards, note that the translated formula clique a (v) says that for any pair a; a 0 of components of a, there is a node w, such that a; a 0 occur at all nodes on the link from v to w and hence represent the same elements at w as they do at v. some predicate R and some tuple b that contains both a and a 0 . By induction hypothesis, this means that d; d 0 are components of some tuple d 0 such that D Hence T (D) clique a (v) if and only if the tuple d induces a clique in the Gaifman graph G(D). Suppose now that and that D there exists a tuple d 0 such that D j= clique(d; d By Lemma 3.2 there exists a node w of T such that all components of d [ d 0 are contained in F(w). induction hypothesis it follows that O Let U be the set of nodes on the link between v and w. Then the tuple d occurs in F(u) for all nodes u 2 U . It follows that T (D) link a (v; w). Hence Conversely, if T (D) (v) then there exists a node w such that the constants a occur at all nodes on the link between v and w (and hence correspond to the same tuple d) and such that T (D) clique ab (w) for some tuple b. By induction hypothesis this implies that D j= clique(d; d some tuple d 0 , hence D Finally, let and only if d is contained in every xed point of the operator D , i.e., is in every relation S such that c)g. We rst observe that, for guarded tuples d, this is equivalent to the seemingly weaker condition that d is contained in every S such that c 2 S i D for all guarded tuples c. Indeed, this is obvious since (S; x) is a Boolean combination of quantier-free formulae not involving x, of positive atoms of the form Su where u is a recombination of the variables appearing in x and of formulae starting with a guarded existential quantier. Therefore the truth values of Sc for unguarded tuples c never matters for the question whether a given guarded tuple is in ' D (S). Recall that the formula associated with '(x) and a is ' a (z) := (8S) Consider any tuple of monadic relations on T (D) that satisies the consistency axiom such that This tuple S denes a relation S on D such that for all nodes w of T and all tuples c in F(w) with Conversely, each relation S on D denes such a tuple S of monadic relations on T (D) which describes the truth values of S on all guarded tuples of D. Since T (D) induction hypothesis that D Further d 2 S if and only if v 2 S a . Hence the formula ' a (v) is true in T (D) if and only if d is contained in all relations S over D such that for all guarded tuples c, c 2 S i c 2 D (S). By the remarks above, this is equivalent to saying that d is in the least xed point of D . Theorem 4.3. The satisability problem for CGF is decidable. Proof. Let be a sentence in CGF of vocabulary and width k. We translate into a monadic second-order sentence such that is satisable if and only if there exists a countable tree E) with T Fix a set K of 2k constants and let O be the tuple of monadic relations O a for a 2 K. Further, for each m-relation symbol R 2 , let R be the tuple of monadic relation R a where a 2 K m . The desired monadic second-order sentence has the form Here is the rst-order axiom expressing that the tree T expanded by the relations O and R does describe a tree structure T (D) associated to some -structure D. We have shown above that this can be done in rst-order logic. The formula ; (x) is the translation of (and the empty tuple of constants) into monadic second-order logic, as described by Theorem 4.2. If is satisable, then by Theorem 3.7, has a countable model D of tree width k 1. By Theorem 4.2, the associated tree structure T (D) satises there exists a tree T such that T j= . Conversely, if there exists an expansion which satises and hence describes the tree decomposition of a -structure D. Since T it follows by Theorem 4.2 that D j= . The decidability of CGF now follows by the decidability of S!S, the monadic second-order theory of countable trees, a famous result that has been established by Rabin [23]. Note that while this reduction argument to S!S gives a somewhat more elementary decidability proof (modulo Rabin's result, of course), it does not give good complexity bounds. Indeed, even the rst-order theory of countable trees is non-elementary, i.e. its time complexity exceeds every bounded number of iterations of the exponential function. 4.3 Reduction to the -calculus with backwards modalities Instead of reducing the satisability problem for CGF to the monadic second-order theory of trees, we can dene a similar reduction to the -calculus with backward modalities and then invoke Vardi's decidability result for this logic [27]. For a set of actions A, the -calculus with backwards modalities L , permits, for each action a 2 A, besides the common modal operators hai and [a] also the backwards operators ha i and [a ] corresponding to the backwards transitions a := f(w; g. Hence ha i' is true at state w in a Kripke structure K if and only if there exists a state v, such that K; v reachable from v via action a. Here we will need L on trees (V; E) with only one transition relation. We can write h+i, [+] for the forward modal operators, and hi, [ ] for the backwards operators, and then use the abbreviations Hence 3 and 2 are the usual modal operators on symmetric Kripke structures. Finally, it is convenient for our reduction argument to permit the use of simultaneous least and greatest xed points in L . Let a sequence of propositional variables, and a sequence of L -formulae in which all occurrences of X are posi- tive. Then, for each i r, the expressions [X : '(X)] i and [X : '(X)] i are formulae in L On every Kripke structure K with universe V , the sequence '(X) denes an operator ' K that maps any tuple to a new tuple r (S)) where ' K Since the variables in X occur only positive in ', the operator ' K has a least xed point r ). Now, the semantics of simulteneous least xed point formulae is given by The meaning of a simultaneous greatest xed point similarly. It is well-known that simultaneous xed points can be rewritten as nestings of simple xed points, so the use of simultaneous xed points does not change the expressive power of L . Theorem 4.4 (Vardi). Every satisable formula in L has a tree model. Further, the satisability problem for L is decidable and Exptime-complete. On connected Kripke structures (in particular on trees), the universal modality is denable in L . For every formula ', we write 8' to abbreviate the formula It is easy to see that 8' is satised at some state of a connected Kripke structure K if and only if ' is satised at all states of K. Let D be a structure of bounded tree width, and let T (D) be its tree representation as described in Sect. 4.1. We view T (D) as a Kripke structure, with atomic propositions O a and R a . Having available an universal modality, the axioms for tree representations T (D) given in the previos subsection, can easily be expressed by modal formulae. For instance, the consistency axioms can be written a2a O a _ a2a :O a _ R a Theorem 4.5. Let D be a structure with tree decomposition hT; For an appropriate set of constants K and a function f : D ! K, let T (D) be the associated tree structure. For every formula '(x every tuple a 2 K m we can construct a formula ' a 2 L such that, for every node v of T and every clique-guarded tuple d F(v) with Proof. The translation is very similar to the translation into monadic second-order logic that was given in the previous section. (1) If '(x) is an atom Sx i 1 then ' a true if a (3) For the guard formulae clique(x), let clique a _ _ 3(O a ^ O a 0 ' a _ a2a O a ' a Here S is a tuple of xed point variables S b and (S) is the tuple of the (S) for all b 2 K m (where m is the arity of S). The proof that the translation is correct is analogous to the proof of Theorem 4.2. We now get another proof for the decidability of guarded xed point logic. Given a sentence 2 CGF, we translate it into the L according to Theorem 4.5 and take the conjunction with the consistency axioms in L for tree representations T (D). Then use Vardi's decidability result for L . By the tree model property of L , the tree model property of CGF and Theorem 4.5 this gives a decision procedure for CGF. However, it is not clear whether this argument can be modied to provide the optimal complexity bounds for guarded xed point logic. --R Modal Correspondence Theory Dynamic bits and pieces Modal Logic The complexity of tree automata and logics of programs On the (in A superposition decision procedure for the guarded fragment with equality The two-variable guarded fragment with transitive relations Model Theory Loosely guarded fragment of Interpolation in guarded fragments. Beth De On the expressive completeness of the propositional mu-calculus with respect to monadic second order logic The computational complexity of provability in systems of propositional modal logic Tolerance logic Cours de th First Steps in Modal Logic Decidability of second-order theories and automata on in nite trees Tree width and tangles: A new connectivity measure and some applications Complexity of modal logics Why is modal logic so robustly decidable? --TR First steps in modal logic Beth Definability for the Guarded Fragment Reasoning about The Past with Two-Way Automata On the Expressive Completeness of the Propositional mu-Calculus with Respect to Monadic Second Order Logic Guarded Fixed Point Logic The Two-Variable Guarded Fragment with Transitive Relations A Superposition Decision Procedure for the Guarded Fragment with Equality Back and Forth between Guarded and Modal Logics --CTR Antje Nowack, A Guarded Fragment for Abstract State Machines, Journal of Logic, Language and Information, v.14 n.3, p.345-368, June 2005 Dirk Leinders , Maarten Marx , Jerzy Tyszkiewicz , Jan Bussche, The Semijoin Algebra and the Guarded Fragment, Journal of Logic, Language and Information, v.14 n.3, p.331-343, June 2005 Maarten Marx, Queries determined by views: pack your views, Proceedings of the twenty-sixth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems, June 11-13, 2007, Beijing, China Erich Grdel , Wolfgang Thomas , Thomas Wilke, Literature, Automata logics, and infinite games: a guide to current research, Springer-Verlag New York, Inc., New York, NY, 2002

decidability;guarded logics;fixed point logics

606915

On an optimal propositional proof system and the structure of easy subsets of TAUT.

In this paper we develop a connection between optimal propositional proof systems and structural complexity theory--specifically, there exists an optimal propositional proof system if and only if there is a suitable recursive presentation of the class of all easy (polynomial time recognizable) subsets of TAUT. As a corollary we obtain the result that if there does not exist an optimal propositional proof system, then for every theory T there exists an easy subset of TAUT which is not T-provably easy.

Introduction The first classification of propositional proof systems by their relative efficiency was done by S. Cook and R. Reckhow [4] in 1979. The key tool for comparing the relative strength of proof systems is p- simulation. Intuitively a proof system h p-simulates a second one g if there is a polynomial time computable function translating proofs in into proofs in h. A propositional proof system is called p-optimal if it p-simulates any propositional proof system. The question of the existence of a p-optimal propositional proof system and its nondeterministic counterpart, an optimal propositional proof system, posed by J. Kraj'i-cek and P. Pudl'ak [9], is still open. It is not known whether many-one complete languages for exist. For these and other promise classes no recursively enumerable representation of appropriate sets of Turing machines is known. Moreover, J. Hartmanis, L. Hemachandra in [5] and W. Kowalczyk in [7] pointed out that NP " co-NP and UP possess complete languages if and only if there are recursive enumerations of polynomial time clocked Turing machines covering languages from these classes. In this paper we show that the question of the existence of optimal (p-optimal) propositional proof systems can be characterized in a similar manner. The main result of our paper shows that optimal proof systems for TAUT (the set of all propositional tautologies) exist if and only if there is a recursive enumeration of polynomial time clocked Turing machines covering all easy (recognizable in polynomial time) subsets of TAUT . This means that the problem of the existence of complete languages for promise classes and the problem of the existence of optimal proof systems for TAUT , although distant at first sight, are structurally similar. Since complete languages for promise classes have been unsuccesfully searched for in the past our equivalence gives some evidence of the fact that optimal propositional proof systems might not exist. Our result can be related to the already existing line of research in computational complexity. After the revelation of the connection between the existence of optimal proof systems and the existence of many-one complete languages for promise classes in [12] and [15], this subject has been intensively investigated. J. K-obler and J. Messner in [8] formalized this relationship introducing the concept of test set, and showed that the existence of p-optimal proof systems for TAUT and for SAT (the set of all satisfiable boolean formulas) suffices to obtain a complete language for NP " co-NP. J. Messner and J. Tor'an showed in [12] that a complete language for UP exists in case there is a p-optimal proof system for TAUT . We believe that our results make the next step towards deeper understanding of this link between optimal proof systems and complete languages for promise classes. The paper is organized as follows. In Section 2 we set down notation that will be used throughout the paper. Background information about propositional proof systems is presented in Section 3. The problems of the existence of complete languages for the classes " co-NP and UP and their characterization in terms of polynomial time clocked machines covering languages from these classes are presented in Section 4. In Section 5 we give a precise definition of a family of propositional formulas which will be used in the proofs of our main results. In Section 6 our main results are stated and proved. In the last section we discuss corollaries arising from the main results of the paper. Preliminaries We assume some familiarity with basic complexity theory, see [1]. The symbol \Sigma denotes, throughout the paper, a certain fixed finite alphabet. The set of all strings over \Sigma is denoted by \Sigma ? . For a string x, jxj denotes the length of x. For a language A ae \Sigma ? the complement of A is the set of all strings that are not in A. We use Turing machines (acceptors and transducers) as our basic computational model. We will not distinguish between a machine and its code. For a deterministic Turing machine M and an input w, denotes the computing time of M on w. When M is a nondeterministic Turing machine TIME(M , w) is defined only for w's accepted by M and denotes the number of steps in the shortest accepting computation of M on w. For a Turing machine M the denotes the language accepted by M . The output of a Turing transducer M on input w 2 \Sigma ? is denoted by M(w). We consider deterministic and nondeterministic polynomial time clocked Turing machines with uniformly attached standard clocks which stop their computations in polynomial time (see [1]). We impose some restrictions on our encoding of these machines. From the code of any polynomial time clocked Turing machines we can detect easily (in polynomial time) the natural k such that n k +k is its polynomial time bound. Let D 1 , D 2 , D 3 , . and N 1 , N 2 , N 3 , . be, respectively, standard enumerations of all deterministic and nondeterministic polynomial time clocked Turing machines. Recall that the classes P, NP, co-NP are, respectively, the class of all languages recognized by deterministic Turing machines working in polynomial time, the class of all languages accepted by nondeterministic Turing machines working in polynomial time and the class of complements of all languages from NP. The symbol TAUT denotes the set (of encodings) of all propositional tautologies over a fixed adequate set of connectives, SAT denotes the set of all satisfiable boolean formulas. Finally, standard polynomial time computable tupling function. 3 Propositional proof systems The abstract notion of a propositional proof system was introduced by S. Cook and R. Reckhow [4] in the following way: Definition 1. A propositional proof system is a function f \Gamma! TAUT computable by a deterministic Turing machine in time bounded by a polynomial in the length of the input. A string w such that we call a proof of a formula ff. A propositional proof system that allows short proofs to all tautologies is called a polynomially bounded propositional proof system. Definition 2. (Cook, Reckhow) A propositional proof system is polynomially bounded if and only if there exists a polynomial p(n) such that every tautology ff has a proof of length no more than p(jffj) in this system. The existence of a polynomially bounded propositional proof system is equivalent to one of the most fundamental problems in complexity theory. Fact 1. (Cook, Reckhow) NP=co-NP if and only if there exists a polynomially bounded propositional proof system. S. Cook and R. Reckhow were the first to propose a program of research aimed at attacking the NP versus co-NP problem by classifying propositional proof systems by their relative efficiency and then systematically studying more and more powerful concrete proof systems (see [2]). A natural way for such a classification is to introduce a partial order reflecting the relative strength of propositional proof systems. It was done in two different manners. Definition 3. (Cook, Reckhow) Propositional proof system P polynomially simulates (p-simulates) propositional proof system Q if there exists a polynomial time computable function f that for every w, if w is a proof of ff in Q, then f(w) is a proof of ff in P . Definition 4. (Kraj'i-cek, Pudl'ak) Propositional proof system P simulates propositional proof system Q if there exists a polynomial p such that for every tautology ff, if ff has a proof of length n in Q, then ff has a proof of length - p(n) in P . Obviously p-simulation is a stronger notion than simulation. We would like to pay attention to the fact that the simulation between proof systems may be treated as a counterpart of the complexity-theoretic notion of reducibility between problems. Analogously the notion of a complete problem (a complete language) would correspond to the notion of an optimal proof system. The notion of an optimal propositional proof system was introduced by J. Kraj'i-cek and P. Pudl'ak in [9] in two different versions. Definition 5. A propositional proof system is optimal if it simulates every other propositional proof system. A propositional proof system is p-optimal if it p-simulates every other propositional proof system. The following open problem, posed by J. Kraj'i-cek and P. Pudl'ak will be studied in our paper. Open Problem: (1) Does there exist an optimal propositional proof system? (2) Does there exist a p-optimal propositional proof system? The importance of these questions and their connection with the NP versus co-NP problem is described by the following fact. Fact 2. If an optimal (p-optimal) propositional proof system exists, then NP=co-NP if and only if this system is polynomially bounded. Complete languages for NP " co-NP and for UP The classes NP " co-NP and UP are called promise classes because they are defined using nondeterministic polynomial time clocked Turing machines which obey special conditions (promises). The problem whether a given nondeterministic polynomial time clocked Turing machine indeed defines a language in any of these classes is undecidable and because of this complete languages for these classes are not known. Since there exist relativizations for which these two classes have complete languages as well as relativizations for which they do not the problems of the existence of complete languages for seem to be very difficult. It turns out that the existence of complete languages for these classes depends on a certain structural condition on the set of machines defining languages from these classes. Since this condition is the chief motivation for our main theorems we survey known results in this direction. The class NP " co-NP is most often defined using complementary pairs of nondeterministic Turing machines. We will use strong nondeterministic Turing machines to define this class. A strong non-deterministic Turing machine is one that has three possible out- comes: "yes", "no" and "maybe". We say that such a machine accepts a language L if the following is true: if x 2 L, then all computations end up with "yes" or "maybe" and at least one with "yes", if x 62 L, then all computations end up with "no" or "maybe" and at least one with "no". . is a standard enumeration of all nondeterministic polynomial time clocked Turing machines then strong nondeterministicg. The following theorem links the question of the existence of a complete language for NP " co-NP with the existence of a recursively enumerable list of machines covering languages from NP " co-NP. In [7] this list of machines is called a "nice" presentation of Theorem 1. (Kowalczyk) There exists a complete language for NP " co-NP if and only if there exists a recursively enumerable list of strong nondeterministic polynomial time clocked Turing machines N i 1 ::: such that This theorem can be exploited to obtain the following independence result. Let T be any formal theory whose language contains the language of arithmetic, i. e. the language f0,1, -, =, +, \Delta g. We will not specify T in detail but only assume that T is sound (that is, in T we can prove only true theorems) and the set of all theorems of T is recursively enumerable. Theorem 2. (Kowalczyk) If NP " co-NP has no complete languages, then for any theory T there exists L 2 NP " co-NP such that for no nondeterministic polynomial time clocked N i with L(N i can it be proven in T that N i is strong nondeterministic. The class UP is closely related to a one-way function, the notion central to public-key cryptography (see [13]). This class can be defined using categorical (unambiguous) Turing machines. We call a nondeterministic Turing machine categorical or unambiguous if it has the following property: for any input x there is at most one accepting computation. We define UP=fL(N i g. As we can see from the following theorems the problem of the existence of a complete language for UP is similar to its NP " co-NP counterpart. Theorem 3. (Hartmanis, Hemachandra) There exists a complete language for UP if and only if there exists a recursively enumerable list of categorical nondeterministic polynomial time clocked Turing machines N i 1 ::: such that fL(N i k Theorem 4. (Hartmanis, Hemachandra) If UP has no complete languages, then for any theory T there exists such that for no nondeterministic polynomial time clocked can it be proven in T that N i is categorical. In Sections 6 and 7 we will show that the similarity between the problems of the existence of complete languages for NP " co-NP and for UP is also shared by the problem of the existence of an optimal propositional proof system. 5 Formulas expressing the soundness of Turing machines In this section we construct boolean formulas which will be used to verify for a given deterministic polynomial time clocked transducer M and integer n that M on any input of length n produces propositional tautologies. We use these formulas in the proofs of Theorems 5 and 6. For any transducer N we will denote by fN the function computed by N (f Definition 6. A Turing transducer N is called sound if fN maps To any polynomial time clocked transducer M we will assign the set AM =f Sound 1 M ,.g of propositional formulas such M is a propositional tautology if and only if for every input of length n , the machine M outputs a propositional tautology. So , for any polynomial time clocked transducer M , it holds: M is sound if and only if AM ae TAUT . Let N be a fixed nondeterministic Turing machine working in polynomial time which accepts a string w if and only if w is not a propositional tautology. For any fixed polynomial time clocked transducer M , let us consider the set BM=fhM; 0 n i: There exists a string x of length n such that M(x) 62 TAUT g. Using the machines M and N we construct the nondeterministic Turing machine M 0 which guesses a string x of length n, runs M on input x and then runs N on output produced by M . works in polynomial time and accepts BM . Let FM;n be Cook's Theorem formula (see [3]) for the machine M 0 and the input hM; 0 n i. We define Sound n M as :FM;n and then the formula M is a tautology if and only if for every input of length n, M outputs a tautology. From the structure of Cook's reduction (as FM;n clearly displays M and n) it follows that for any fixed M , the set AM is in P. Moreover, the formulas describing the soundness of Turing machines possess the following properties: (1) Global uniformity property There exists a polynomial time computable function f such that for any polynomial time clocked transducer M with time bound (2) Local uniformity property Let M be any fixed polynomial time clocked transducer. There exists a polynomial time computable function fM such that for any w 2 \Sigma ? 6 Main results A class of sets is recursively presentable if there exists an effective enumeration of devices for recognizing all and only members of this class ([10]). In this paper we use the notions of recursive P- presentation and recursive NP-presentation which are mutations of the notion of recursive presentability. Definition 7. By an easy subset of TAUT we mean a set A such that A ae TAUT and A 2 P ( A is polynomial time recognizable). Definition 8. An optimal nondeterministic algorithm for TAUT is a nondeterministic Turing machine M which accepts TAUT and such that for every nondeterministic Turing machine M 0 which accepts TAUT there exists a polynomial p such that for every tautology ff Let A be any easy subset of TAUT . We say that nondeterministic polynomial time clocked Turing machine M names the set A if A. Obviously A may possess many names. The following theorem states that an optimal propositional proof system exists if and only if there exists a recursively enumerable list of names for all easy subsets of TAUT . We would like to pay attention to the similarity between the next theorem and Theorems 1 and 3 from Section 4. Theorem 5. Statements (i) - (iii) are equivalent. (i) There exists an optimal propositional proof system. (ii) There exists an optimal nondeterministic algorithm for TAUT . (iii) The class of all easy subsets of TAUT possesses a recursive NP- presentation. By the statement (iii) we mean: there exists a recursively enumerable list of nondeterministic polynomial time clocked Turing machines ::: such that (2) For every A ae TAUT such that A 2 P there exists j such that Proof. (i) With every propositional proof system we can associate a non-deterministic "guess and verify" algorithm for TAUT . On an input ff this algorithm guesses a string w and then checks in polynomial time whether w is a proof of ff. If successful, the algorithm halts in an accepting state. Symmetrically any nondeterministic algorithm for TAUT can be transformed to a propositional proof system. The proof of a formula ff in this system is a computation of M accepting ff. Let Opt denote an optimal propositional proof system and let M denote a nondeterministic Turing machine associated with Opt (a "guess and verify" algorithm associated with Opt). It can be easily checked that M accepts TAUT and for any nondeterministic Turing machine M 0 accepting TAUT there exists a polynomial p such that for every tautology ff it holds: Let M be an optimal nondeterministic algorithm for TAUT. A recursive NP-presentation of all easy subsets of TAUT we will define in two steps. In the first step we define a recursively enumerable list of nondeterministic Turing machines F 1 , F 2 , F 3 ,. The machine F k is obtained by attaching the shut-off clock n k +k to the machine M . On any input w, the machine F k accepts w if and only if M accepts w in no more than jwj. The sequence F 1 , F 2 , F 3 , F 4 , . of nondeterministic Turing machines possesses the properties (1) and (2): (1) For every i it holds (2) For every A which is an easy subset of TAUT there exists j such that A ae L(F j ) In the second step we define the new recursively enumerable list of nondeterministic polynomial time clocked Turing machines K 1 , as the machine which simulates steps of F i and steps of N j (see Section 2 for definition of N j ) and accepts w if and only if both F i and N j accept w. Let A be any fixed easy subset of TAUT . There exist k and m such that A = L(N k ) and A ae L(Fm ). It follows from the definition of the sequence K 1 , K 2 , K 3 , . that A is accepted by the machine provides a recursive NP-presentation of all easy subsets of TAUT . Let G be the machine generating the codes of the machines from the sequence N i 1 ,. forming a recursive NP-presentation of all easy subsets of TAUT . We say that a string v 2 \Sigma ? is in good form if G; Comp \Gamma Sound jwj where: M is a polynomial time clocked Turing transducer with time bound, G is a computation of the machine G. This computation produces a code of a certain machine N i j M is a computation of the machine N i j accepting the formula Sound jwj is the sequence of zeros (padding). We call a Turing transducer n-sound if and only if on any input of length n it produces a propositional tautology. Let us notice, that if v is in good form then Sound jwj M as a formula accepted by a certain machine from NP-presentation is a propositional tautology. This clearly forces M to be n-sound, where so M on input w produces a propositional tautology. Let ff 0 be a certain fixed propositional tautology. We define in the following way: is in good G; Comp \Gamma Sound jwj and ff is a propositional tautology produced by M on input w, otherwise \Gamma! TAUT . In order to prove that Opt is polynomial time computable it is sufficient to notice that using global uniformity property we can check in polynomial time whether v is in good form. Hence Opt is a propositional proof system. It remains to prove that Opt simulates any propositional proof system. Let h be a propositional proof system computed by the polynomial time clocked transducer K with time bound n l the set AK= fSound 1 K , .g is an easy subset of TAUT , there exists the machine N i j from the NP-presentation such that Let ff be any propositional tautology and let x be its proof in h. Then ff possesses a proof in Opt of the form: G; Comp \Gamma Sound jxj The word Comp \Gamma G is the computation of G producing the code of K is a computation of N i j accepting Sound jxj K . Let us notice that jComp \Gamma is a constant. Because is polynomial time clocked there exists a polynomial p such that p(jxj). The constants c 1 , l and the polynomial depend only on N i j which is fixed and connected with K. This proves that Opt simulates h. The following definition is a nondeterministic counterpart of Definition 7. Definition 9. By an NP-easy subset of TAUT we mean a set A such that A ae TAUT and A 2 NP. A slight change in the previous proof shows that also the second version of Theorem 5 is valid. In this version condition (iii) is replaced by the following one: (iv) The class of all NP-easy subsets of TAUT possesses a recursive NP-presentation. Now we will translate the previous result to the deterministic case. Definition 10. An almost optimal deterministic algorithm for TAUT is a deterministic Turing machine M which accepts TAUT and such that for every deterministic Turing machine M 0 which accepts TAUT there exists a polynomial p such, that for every tautology ff We name such an algorithm as an almost optimal deterministic algorithm for TAUT because the optimality property is stated for any input string x which belongs to TAUT and nothing is claimed for other x's (compare the definition of an optimal acceptor for TAUT in [11]). The equivalence (i) $ (ii) in the next theorem is restated from [9] in order to emphasize the symmetry between Theorem 5 and Theorem 6. Theorem 6. Statements (i) - (iii) are equivalent. (i) There exists a p-optimal propositional proof system. (ii) There exists an almost optimal deterministic algorithm for TAUT . (iii) The class of all easy subsets of TAUT possesses a recursive P- presentation. By the statement (iii) we mean: there exists a recursively enumerable list of deterministic polynomial time clocked Turing machines ::: such that (2) For every A ae TAUT such that A 2 P there exists j such that Proof. (i) This follows by the same arguments as in the proof of (ii) ! (iii) from Theorem 5. The only change is the use of deterministic Turing machines instead of the nondeterministic ones. A string v 2 \Sigma ? is in good form if G; Comp \Gamma Sound jwj where the appropriate symbols mean the same as before. We define analogously as in the proof of Theorem 5: is in good form G; Comp \Gamma Sound jwj and ff is a propositional tautology produced by M on input w, otherwise is a certain fixed propositional tautology. It remains to prove that Opt p-simulates any propositional proof system. Let h be a propositional proof system computed by a polynomial time clocked transducer K with time bound n l the set AK= fSound 1 K , .g is an easy subset of TAUT , there exists the machine D i j from the P-presentation such that ). The function G; Comp \Gamma Sound jxj translates proofs in h into proofs in Opt. The word Comp \Gamma G in the definition of t is the computation of G producing the code of D i j K is a computation of D i j accepting Sound jxj K . From the fact that D i j is deterministic and works in polynomial time and from local uniformity property (see Section that Comp \Gamma Sound jxj K can be constructed in polynomial time. This proves that t is polynomial time computable. Definition 11. A Turing machine acceptor M is called sound if ae TAUT . The question, whether the set of all sound deterministic (non- deterministic) polynomial time clocked Turing machines yields the desired P-presentation (NP-presentation) (that is, whether this set is recursively enumerable) occurs naturally in connection with Theorems 5 and 6. The negative answer to this question is provided by the next theorem. Theorem 7. The set of all sound deterministic (nondeterministic) polynomial time clocked Turing acceptors is not recursively enumerable This follows immediately from Rice's Theorem (see [14]). 7 Independence results Let T be any formal theory satisfying the assumptions from Section 4. The notation T ' fi means that a first order formula fi is provable in T . Let M be a Turing machine. By "L(M) ae TAUT " we denote the first order formula which expresses the soundness of M , i.e. 8w2L(M) [w is a propositional tautology] Definition 12. A deterministic (nondeterministic) Turing machine Definition 13. A set A ae TAUT is T -provably NP-easy if there exists a nondeterministic polynomial time clocked Turing machine fulfilling (1) - (2) (1) M is As in the case of the classes NP " co-NP and for UP we can obtain the following independence result. Theorem 8. If there does not exist an optimal propositional proof system, then for every theory T there exists an easy subset of TAUT which is not T -provably NP-easy. Proof. Suppose, on the contrary, that there exists a theory T such that all easy subsets of TAUT are T -provably NP-easy. Then the following recursively enumerable set of machines\Omega is a nondeterministic polynomial time clocked Turing machine which is creates a recursive NP-presentation of the class of all easy subsets of TAUT . By Theorem 5, this implies that there exists an optimal propositional proof system, giving a contradiction. The following result can be obtained from the second version of Theorem 5. Theorem 9. If there does not exist an optimal propositional proof system, then for every theory T there exists an NP-easy subset of TAUT which is not T -provably NP-easy. The translation of this result to the deterministic case goes along the following lines. Definition 14. A set A ae TAUT is T -provably easy if there exists a deterministic polynomial time clocked Turing machine M fulfilling (1) M is Theorem 10. If there does not exist a p-optimal propositional proof system, then for every theory T there exists an easy subset of TAUT which is not T -provably easy. 8 Conclusion In this paper we related the question of the existence of an optimal propositional proof system to the recursive presentability of the family of all easy subsets of TAUT by means of polynomial time clocked Turing machines . The problems of the existence of complete languages for the classes NP " co-NP and for UP have a similar characterization. From this characterization a variety of interesting results about the promise classes NP " co-NP and UP were derived by recursion-theoretic techniques (see [7], [5]). Although recursion-theoretic methods seem unable to solve the problem of the existence of an optimal propositional proof system we believe that our main results from Section 6 allow the application of these methods (as it was in case of promise classes, see [5], [6]) to further study of this problem. --R Structural Complexity I (Springer-Verlag Lectures on Proof Theory. The complexity of theorem proving procedures The relative efficiency of propositional proof systems Complexity classes without machines: On complete languages for UP On complete problems for NP Some connections between presentability of complexity classes and the power of formal systems of reasoning Complete Problems for Promise Classes by Optimal Proof Systems for Test Sets Propositional proof systems On the structure of sets in NP and other complexity classes On optimal algoritms and optimal proof systems Optimal proof systems for Propositional Logic and complete sets Computational Complexity Classes of recursively enumerable sets and their decision problems On an optimal quantified propositional proof system and a complete language for NP --TR Complexity classes without machines: on complete languages for UP Some Connections between Representability of Complexity Classes and the Power of Formal Systems of Reasoning On Complete Problems for NP$\cap$CoNP Optimal Proof Systems for Propositional Logic and Complete Sets On an Optimal Quantified Propositional Proof System and a Complete Language for NP cap co-NP Complete Problems for Promise Classes by Optimal Proof Systems for Test Sets The complexity of theorem-proving procedures --CTR Christian Glaer , Alan L. Selman , Samik Sengupta, Reductions between disjoint NP-pairs, Information and Computation, v.200 n.2, p.247-267, 1 August 2005

complexity of computation;complexity classes;complexity of proofs;classical propositional logic

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Concepts and realization of a diagram editor generator based on hypergraph transformation.

Diagram editors which are tailored to a specific diagram language typically support either syntax-directed editing or free-hand editing, i.e., the user is either restricted to a collection of predefined editing operations, or he is not restricted at all, but misses the convenience of such complex editing operations. This paper describes DIAGEN, a rapid prototyping tool for creating diagram editors which support both modes in order to get their combined advantages. Created editors use hypergraphs as an internal diagram model and hypergraph parsers for syntactic analysis whereas syntax-directed editing is realized by programmed hypergraph transformation of these internal hypergraphs. This approach has proven to be powerful and general in the sense that it supports quick prototyping of diagram editors and does not restrict the class of diagram languages which it can be applied to.

Introduction Diagram editors are graphical editors which are tailored to a specific diagram language; they can be distinguished from pure drawing tools by their capability of "understanding" edited diagrams to some extent. Furthermore, diagram editors do not allow to create arbitrary drawings, but are restricted to visual components which occur in the diagram language. For instance, an editor for UML class diagrams typically does not allow to draw a transistor symbol Preprint submitted to Elsevier Science 27 March 2001 which would be possible in a circuit diagram editor. Current diagram editors support either syntax-directed editing or free-hand editing. Syntax-directed editors provide a set of editing operations. Each of these operations is geared to modify the meaning of the diagram. This editing mode requires an internal diagram model that is primarily modified by the opera- tions; diagrams are then updated according to their modified model. These models are most commonly described by some kind of graph; editing operations are then represented by graph transformations (e.g. [1,2]). Diagram editors providing free-hand editing are low-level graphics editors which allow the user to directly manipulate the diagram. The graphics editor becomes a diagram editor by o#ering only pictorial objects which are used by the visual language and by combining it with a parser. A parser is necessary for checking the correctness of diagrams and analyzing the syntactic structure of the diagram. There are grammar formalisms and parsers that do not require an internal diagram model as an intermediate diagram representation, but operate directly on the diagram (e.g., constraint multiset grammars [3]). Other approaches use an internal model which is analyzed by the parser (e.g., VisPro [4]). Again, graphs are the most common means for describing such a model. The advantage of free-hand editing over syntax-directed editing is that a diagram language can be defined by a concise (graph) grammar editing operations can be omitted. The editor does not force the user to edit diagrams in a certain way since there is no restriction to predefined editing operations. However, this may turn out to be a disadvantage since editors permit to create any diagram; they do not o#er explicit guidance to the user. Furthermore, free-hand editing requires a parser and is thus restricted to diagrams and (graph) grammars which o#er e#cient parsers. editors either support syntax-directed editing or free-hand editing. An editor that supports both editing modes at the same time would combine the positive aspects of both editing modes and reduce their negative ones. Despite this observation, there is only one such proposal which has not yet been realized known to us: Rekers and Schurr propose to use two kinds of graphs as internal representations of diagrams [5]: the spatial relationship graph (SRG) abstracts from the physical diagram layout and represents higher level spatial relations. Additionally, an abstract syntax graph (ASG) that represents the logical structure of the diagram is kept up-to-date with the SRG. Context-sensitive graph grammars are used to define the syntax of both graphs. Free-hand editing of diagrams is planned to modify the first graph, syntax-directed editing is going to modify the second. In each case, the other graph is modified accordingly. Therefore, a kind of diagram semantics is available by the ASG. However, this approach requires almost a one-to-one relationship between SRG and ASG. This is not required in the approach of this paper. We will come back to this approach in the conclusions (cf. Section 6). This paper describes DiaGen, a rapid-prototyping tool for creating diagram editors that support both editing modes at the same time. DiaGen (Diagram editor Generator) supports free-hand editing based on an internal hypergraph model which is parsed according to some hypergraph grammar. Attribute evaluation which is directed by the syntactic structure of the diagram is then used for creating a user-specified semantic representation of the diagram. This free-hand editing mode is seamlessly extended by a syntax-directed editing mode, which also requires an automatic layout mechanism for diagrams. Support for automatic diagram layout which is used for both syntax-directed editing and free-hand editing is briefly outlined, too. The next section gives an overview of the DiaGen tool and the common architecture of editors being created with DiaGen. Section 3 then explains the free-hand editing mode of these editors and the diagram analysis steps which are necessary for translating freely edited diagrams into some semantic rep- resentation. The integration of additional syntax-directed editing operations into such editors is explained in Section 4. An automatic layout mechanism, which is required by syntax-directed editing, is outlined in Section 5. Section 6 concludes the paper. DiaGen provides an environment for rapidly developing diagram editors. This section first outlines this environment and how it is used for creating a diagram editor that is tailored to a specific diagram language. Each of such DiaGen editors is based on the same editor architecture which is adjusted to the specific diagram language. This architecture is described afterwards. 2.1 The DiaGen environment DiaGen is completely implemented in Java and consists of an editor frame-work and a program generator. DiaGen is free software and can be down-loaded from the DiaGen web site [6]. Fig. 1 shows the structure of DiaGen and the process of using it as a rapid-prototyping tool for developing diagram editors. The framework, as a collection of Java classes, provides the generic editor functionality which is necessary for editing and analyzing diagrams. In order to create an editor for a specific DiaGen editor framework generator Program Specification code Generated program code program specific Editor IAGEN Editor developer Diagram editor Fig. 1. Generating diagram editors with DiaGen. diagram language, the editor developer primarily has to supply a specifica- tion, which textually describes syntax and semantics of the diagram language. Additional program code which is written "manually" can be supplied, too. Manual programming is necessary for the visual representation of diagram components on the screen and for processing specific data structures of the problem domain, e.g., for semantic processing when using the editor as a component of another software system. The specification is then translated into Java classes by the program generator. The generated classes, together with the editor framework and the manually written code, implement an editor for the specified diagram language. This editor can run as a stand-alone program. But it can also be used as a software component since the editor framework as well as the generated program code is conformable with the JavaBeans standard, the software component model for Java. Common integrated development environments (IDEs, e.g., JBuilder by Imprise/Borland, VisualCafe by Symantec or Visual Age for Java by IBM.) can be used to visually plug in generated editors into other software systems without much programming e#ort. Diagram editors which have been developed using DiaGen (such editors are called "DiaGen editors" in the following) provide the following features: . DiaGen editors always support free-hand editing. The editor framework contains a generic drawing tool which is adjusted to the specified diagram language by the program generator. The visual representation of diagram components which are used by the drawing tool has to be supplied by the editor developer. The editor framework provides an extensive class library for that purpose. Diagrams that are drawn using the drawing tool are internally modeled by hypergraphs which are analyzed primarily by a hypergraph parser (cf. Section 3). The hypergraph grammar which is used by the hyper-graph parser is the core of the diagram language specification. The analysis results are used to provide user feedback on diagram parts which are not correct with respect to the diagram language. . Diagrams which are created using a DiaGen editor are translated into a semantic representation. This process is driven by the syntactic analysis and makes use of program code and data structures which are provided as "ed- itor specific program code" in Fig. 1. The reverse translation, i.e., creating diagrams from external representations, is also supported by a mechanism that is similar the one of syntax-directed editing operations. . DiaGen editors optionally support syntax-directed editing, too, if the editor developer has specified syntax-directed editing operations. These operations are primarily hypergraph transformations which modify the internal hypergraph model of edited diagrams (cf. Section 4). DiaGen editors can be specified and developed in a rapid prototyping fashion without any syntax-directed editing operation. Any diagram of the diagram language can be created by free-hand editing only. Desirable editing operations can be added later. . Automatic layout is an optional DiaGen editor feature, too, but which is obligatory when specifying syntax-directed operations. The automatic lay-out mechanism adjusts the diagram layout after applying syntax-directed editing operations which have modified the internal diagram model. Automatic layout also assists free-hand editing: After each layout modification by the user, the layout mechanism changes the diagram such that the structure of the diagram remains unchanged. DiaGen o#ers constraints for specifying the layout mechanism in a declarative way (cf. Section 5), or a programming interface for plugging in other layout mechanisms. DiaGen comes with some general layouting mechanisms like a force-driven layout and simple constraint propagation methods which can be parameterized by the editor developer. The rest of this paper presents the concepts and realization of these features by means of a formal specification based on hypergraph transformation and generating the editor using such a specification. Each of these editors has the same architecture which is considered next. 2.2 The DiaGen editor architecture Fig. 2 shows the structure which is common to all DiaGen editors and which is described in the following paragraphs. Ovals are data structures, and rectangles represent functional components. Gray rectangles are parts of the editor framework which have been adjusted by the DiaGen program generator based on the specification of the specific diagram language. Flow of information is represented by arrows. If not labeled, information flow means reading resp. creating the corresponding data structures. The editor supports free-hand editing by means of the included drawing tool which is part of the editor framework, but which has been adjusted by the operations selects selects operations adds/removes components modifies reads reads reads modifies reads marks syntactically correct subdiagrams modifies gathers gathers gathers Layouter Diagram information Derivation structure Reduced hypergraph model Drawing tool Hypergraph transformer Modeler Reducer Parser semantic representation Attribute evaluation Hypergraph model Fig. 2. Architecture of a diagram editor based on DiaGen. program generator. With this drawing tool, the editor user can create, arrange and modify diagram components which are specific to the diagram language. Editor specific program code which has been supplied by the editor developer is responsible for the visual representation of these language specific compo- nents. Examples are rectangular text boxes or diamond-shaped conditions in flowcharts. Fig. 3 shows a screenshot of such an editor whose visual appearance is characterized by its drawing tool. When components are selected, so-called handles - like in conventional drawing tools - show up which allow to move or modify single or grouped diagram components like with common o#-the-shelf drawing tools (cf. Fig. 9a). The drawing tool creates the data structure of the diagram as a set of diagram components together with their attributes (position, size, etc. The sequence of processing steps which starts with the modeler and ends with attribute evaluation (cf. Fig. 2) realizes diagram analysis which is necessary for free-hand editing: The modeler first transforms the diagram into an internal model, the hypergraph model. The task of analyzing this hypergraph model is quite similar to familiar compiler techniques: The reducer - which corresponds to the scanner of a compiler - performs some kind of lexical analysis and creates a reduced hypergraph model which is then syntactically analyzed by the hypergraph parser. This processing step identifies maximal parts of diagram which are (syntactically) correct and provides visual feedback to the user by coloring each subdiagram with a di#erent color. A correct diagram is thus entirely colored with just a single color, and errors are indicated by missing colors. Driven by the syntactic structure of each subdiagram and similar to the semantic analysis step of compilers, attribute evaluation is then used to create a semantic representation for each of these subdiagrams. Fig. 3. Screenshot of a diagram editor for flowcharts. The layouter modifies attributes of diagram components and thus the diagram layout by using information which has been gathered by the reducer and the parser or by attribute evaluation (cf. Section 5). The layouter is necessary for realizing syntax-directed editing: Syntax-directed editing operations modify the hypergraph model by means of the hypergraph transformer and add or remove components to resp. from the diagram. The visual representation of the diagram and its layout is then computed by the layouter. These processing steps, which have been outlined referring to Fig. 2, are described in more detail in the following sections. 3 Free-Hand Editing This section describes the processing steps of a DiaGen editor which are used for free-hand editing and which are shown in Fig. 2. DiaGen has been used for creating editors for many diagram languages (e.g., UML diagrams, ladder di- agrams, Petri nets). As a sample diagram language, this paper uses flowcharts although it is an admittedly simple language. However, other languages are less suited for presentation in a paper. 3.1 The hypergraph model Each diagram consists of a finite set of diagram components, each of which is determined by its attributes. for flowcharts, there are rectangular text boxes and diamond-shaped conditions whose positions are defined by their x and y coordinates and their size by a width and a height attribute. Vertical as well as horizontal lines and arrows have x and y coordinates of their starting and end points on the canvas. However, attributes describe an arrangement of diagram components only in terms of numbers. The meaning of a diagram is determined by the diagram components and their spatial arrangement. The specific arrangement of flowchart components is made up of boxes and diamonds which are connected by arrows and lines in a very specific way. Arrangements can always be described by spatial relationships between diagram components. for that purpose, each diagram component typically has several distinct attachment areas at which it can be connected to other diagram components. A flowchart diamond, e.g., has its top vertex as well as its left and right one where it can be connected to lines and arrows, whereas lines and arrows have their end points as well as their line (please note that arrows can be connected to the middle of another arrow as shown in Fig. 3) as attachment areas. Connections can be established by spatially related (e.g., overlapping) attachment areas as with flowcharts where an arrow has to end at an exact position in order to be connected to a diamond. DiaGen uses hypergraphs to describe a diagram as a set of diagram components and the relationships between attachment areas of "connected" com- ponents. Hypergraphs consist of two finite sets of nodes and hyperedges (or simply edges for short). Each hyperedge carries a type and is connected to an ordered sequence of nodes. The sequence has a certain length which is called arity of the hyperedge and which is determined by the type of the edge. Each node of this sequence is called "visited" by the hyperedge. Familiar directed edge-labeled graphs are special hypergraphs where each hyperedge has arity 2. Hypergraphs are an obvious means for modeling diagrams: Each diagram component is modeled by a hyperedge. The kind of diagram component is the hyperedge type, the number of attachment areas is its arity. Attachment areas are modeled by nodes which are visited by the hyperedge. The sequence of visited nodes determines which attachment area is modeled by which node. The set of diagram components is thus represented by a set of nodes and a set of hyperedges where each node is visited by exactly one hyperedge. Relationships between attachment areas are modeled by hyperedges of arity 2. They carry a type which describes the kind of relationship between related attachment areas. Fig. 4 shows the hypergraph model of a subdiagram of the one shown in Fig. 3. Nodes are depicted by black dots. Component edges which represent diagram components are shown as gray rectangles that are connected to visited nodes by thin lines. Line numbers represent the sequence of visited nodes. Relation edges which represent relationships between attachment areas are depicted as arrows between connected nodes. The arrow direction indicates the node sequence. Fig. 4 shows the hypergraph in a similar way as the represented sub- diagram. Rectangular boxes and diamond-shaped conditions are represented by box edges resp. cond edges with arity 2 resp. 3. Vertical and horizontal arrows resp. lines are shown as vArrow, hArrow, vLine, and hLine edges, resp. cond hLine hLine flowIn flowOut flowOut vArrow hArrow join box flowIn flowOut vArrow Fig. 4. A part of the flowchart which is shown in Fig. 3 and its corresponding hypergraph model. Relationship edge types are flowIn, flowOut, and join. The relationship of a vertical arrow which ends at the upper attachment area of a box or a diamond is represented by a flowIn relation between the "end node" of the arrow and the "upper node" of the corresponding vArrow and box edges. A flowOut relationship is used in a similar way for leaving arrows. A join relation connects an arrow end with lines or arrows. Hypergraph models are created by the modeler of DiaGen editors: The modeler first creates component edges for each diagram component and nodes for each of their attachment areas. Afterwards, the modeler checks for each pair of attachment areas whether they are related as defined in the specification. 2 The language specification describes such relationships in terms of relations on attribute values of corresponding attachment areas. E.g., in the flowchart ex- ample, the end attachment area of a vertical arrow and the upper attachment area of a rectangular box are flowIn-related if both attachment areas overlap, i.e., have close positions on the canvas. for each relationship which is detected, the modeler adds a corresponding relation edge between corresponding nodes. 3.2 The reduced hypergraph model Hypergraph models tend to be quite large even for small diagrams. for in- stance, Fig. 4 shows only a small portion of the hypergraph model of the really small flowchart of Fig. 3. The hypergraph model represents each diagram component and each relationship between them directly. The structure and meaning of a diagram, however, is generally represented in terms of larger groups of components and their relationship. for flowcharts, e.g., the crucial 2 for e#ciency reasons, only pairs of attachment areas with overlapping bounding boxes are actually considered. a=b=c a=b=c a=b a a a vArrow c 3 flowIn vArrow cconn conn statement2 box vArrow continue connect flowOut2a beginaFig. 5. Some reduction rules for flowcharts. information is contained by the set of boxes and conditions which are inter-connected by lines and arrows. The specific path of lines and arrows between connected boxes is irrelevant. DiaGen editors therefore do not analyze the hypergraph model directly, but first identify such groups of components and relationships. Similar to common compiler techniques where lexical analysis is used to group input stream characters to tokens (e.g., identifiers and key- words) and leaving other characters unconsidered (e.g., comments), the reducer searches for all matches of specified patterns and creates a reduced hypergraph model which then represents the diagram structure directly. Similar to compiler generators which require a specification of lexical analy- sis, the reducer has to be specified for a specific diagram language. DiaGen provides reduction rules to this end: Each rule consists of a pair (P, R) of hypergraphs and additional application conditions. P is the pattern whose occurrences are searched in the hypergraph model. The hypergraph R ("result") describes a modification to the reduced hypergraph for each match of P which also satisfies the application conditions. Fig. 5 shows five reduction rules for flowcharts in the form P # R. The pattern of the rightmost rule actually consists of the vArrow edge with its three visited nodes only. The gray, crossed out sub-hypergraphs are negative application conditions: A match for the vArrow edge is used for rule application if and only if none of the three crossed out sub-hypergraphs can be matched as well, i.e., the match is valid if there is no additional flowIn, continue, or connect edge which is connected to the start node of the vArrow edge (continue edges are not further considered here). The hypergraph R of each rule shows the hypergraph which is added to the reduced hypergraph model for each valid match of the P -hypergraph. Same node labels indicate corresponding nodes of the hypergraph model and the reduced one. Hypergraph model nodes which lie in di#erent pattern occurrences (not necessarily of di#erent patterns) always correspond to the same node of the reduced model. Three special cases have to be mentioned here: conn conn conn conn conn conn conn conn conn conn conn conn conn conn conn conn conn conn conn statement f d c e a statement statement statement condition condition statement statement begin Fig. 6. The reduced hypergraph model of the flowchart of Fig. 3. . Nodes which are matched by no P -hypergraph of any rule do not have corresponding nodes in the reduced model. . If there are nodes which lie in di#erent pattern occurrences where none of these pattern nodes has a corresponding node in its R graph, these nodes do not have corresponding nodes in the reduced model. . Two or more P -nodes may correspond to a single R-node (e.g., a the second and fourth rule). All the nodes of the hypergraph model which match these "identified" P -nodes correspond to a single node of the reduced hypergraph model. Fig. 6 shows the reduced hypergraph model of the flowchart of Fig. 3 and which is created by these reduction rules. The structure of this model is similar to the structure of the hypergraph model. Because of the reduction rules which identify nodes, a much cleaner hypergraph model is created. The conn edges are grayed out since they are actually not needed for the following syntactic analysis; the corresponding reduction rules could be omitted for pure free-hand editing editors. Section 4 however shows why they are needed in the context of syntax-directed editing operations. The concept of reduction rules is similar to hypergraph transformation rules L ::= R (or L # R) with L (left-hand side, LHS) and R (right-hand side, being hypergraphs [7,8]. A transformation rule L ::= R is applied to a hypergraph H by finding L as a subgraph of H and replacing this match by R obtaining hypergraph H # . We say, H # is derived from H in one (deriva- step. A derivation sequence is a sequence of derivation steps where the resulting hypergraph of each step is immediately derived in the next step. The following observations show that specifying the reducer and the reducing process for a specific diagram language would be rather di#cult if the reducer had been defined in terms of such derivation sequences from hypergraph models to reduced ones. Instead, the reducer applies all reduction rules to all occurrences of their left-hand sides in some kind of parallel fashion: . Patterns frequently overlap. This is so since the meaning of a group of diagram components and relationships - and this "meaning" is tried to be represented by the edges of the reduced hypergraph model - often depends on context which is part of another group. E.g., the last rule of Fig. 5 uses a flowOut edge as (negative) context which also occurs in the pattern of the third rule. Applying one rule would change the context of the other one if regular hypergraph transformations were used. It would be a di#cult task to specify the desired reducing semantics. . There are in general many di#erent derivation sequences starting at a specific hypergraph which would produce di#erent reduced hypergraphs because of these overlapping patterns. The editor developer had to take measures to avoid this nondeterminism. However, it is a nontrivial task to set up such confluent sets of transformations [9]. Instead, reduction rules are applied as follows: All possible matches of all rule patterns are searched first without changing the hypergraph. But only those matches are selected which satisfy the corresponding application conditions. In a second step, the corresponding result hypergraphs are instantiated in parallel for each valid match of the corresponding pattern. All these hypergraphs are connected by common nodes according to the correspondence between nodes of the hypergraph model and the reduced one. 3 The reduced hypergraph model now directly represents the structure of the diagram which is syntactically analyzed by the parser. 3.3 Parsing The syntactic structure of a diagram is described in terms of its reduced hyper-graph model, i.e., a diagram language corresponds to a class of hypergraphs. In the literature, there exist two main approaches for specifying graph or hypergraph classes. The first one uses a graph schema which is a kind of Entity-Relationship diagram that describes how edges and nodes of certain types may interconnect (e.g., EER [11]). The other one uses some kind of graph or hyper- 3 In a formal treatment, each reduction rule (P, R) represents a hypergraph morphism P#R is the union of the pattern with the result hypergraph. Corresponding nodes of P and R as well as identified nodes of R are identified in P#R. The reduced hypergraph model is computed by first creating the colimes of all match morphisms of the di#erent patterns into the hypergraph model together with all these morphisms P # P # R, and then removing the edges and "unnecessary" nodes of the hypergraph model from the colimes hypergraph [10]. graph grammar (e.g. [12]) which generalizes the idea of Chomsky grammars for strings which are also used by standard compiler generators [13]. Because of the similarity of diagram analysis with program analysis being performed by compilers and the availability of derivation trees and directed acyclic graphs (DAGs, see below) which easily allow to represent the syntactic structure of a diagram, DiaGen uses a hypergraph grammar approach for specifying the class of reduced hypergraph models of the diagram language. As already mentioned, hypergraph grammars are similar to string grammars. Each hypergraph grammar consists of two finite sets of terminal and nonterminal hyperedge labels and a starting hypergraph which contains nonterminally labeled hyperedges only. Syntax is described by a set of hypergraph transformation rules which are called productions in this context. The hypergraph class or language of the grammar is defined by the set of terminally labeled hypergraphs which can be derived from the starting hypergraph in a finite derivation sequence. There are di#erent types of hypergraph grammars which impose restrictions on the LHS and RHS of each production as well as the allowed sequence of derivation steps. Context-free hypergraph grammars are the simplest ones: each LHS has to consist of a single nonterminally labeled hyperedge together with the appropriate number of nodes. Application of such a production removes the LHS hyperedge and replaces it by the RHS. Matching node labels of LHS and RHS determine how the RHS has to fit in after removing the LHS hyperedge. The productions of Fig. 7 are context-free ones. Productions L . with the same LHS are drawn as L ::= R 1 |R 2 | . Actually, Fig. 7 shows the productions of a hypergraph grammar whose language is just the set of all reduced hypergraph models of structured flowcharts, i.e., flowcharts whose blocks have a single entry and a single exit only. The types statement, condition, and conn are terminal hyperedge labels being used in reduced hypergraph models. The set of nonterminal labels consists of Flowchart, BlockSeq, Block, and Conn. Flowchart edges do not connect to any node (arity 0). The starting hypergraph consists of just a single Flowchart edge. Again, conn edges, and now Conn edges, too, are grayed out since they are actually not required for free-hand editing, but for syntax-directed editing (cf. Section 4). Context-free hypergraph grammars can describe only very limited hypergraph languages [12,14] and, therefore, are not suited for specifying the syntax of many diagram languages. 4 Context-free hypergraph grammars with embeddings are more expressive than context-free ones. They additionally allow 4 Actually, the only diagram languages that we know about and which can be described by context-free grammars are Nassi-Shneiderman diagrams [15], syntax diagrams [16] and flowcharts as used in this paper. a a a a a a a a a a a a BlockSeq Block Conn Block Flowchart begin Block condition Conn Conn statement condition Conn condition Conn condition Conn condition Conn condition Conn condition Conn Conn conn Conn conn Fig. 7. Productions of a grammar for the reduced hypergraph models of flowcharts. embedding productions L ::= R where the RHS R extends the LHS L # R by some edges and nodes, which are "embedded" into the context provided by the LHS when applying such a production. This very limited treatment of context has been chosen since it has proven su#cient for all diagram languages which have been treated with DiaGen, but still allows for e#cient parsing; context-free hypergraph grammars with embeddings even appear to be suitable for all possible kinds of diagram languages. 5 Parsing algorithms and a more detailed description of both grammar types can be found in [19,17,10]. The most prominent feature of the parsing algorithms being used in DiaGen editors is their capability of dealing with diagram errors: Erroneous diagrams resp. their reduced hypergraph models are not just rejected. Instead, maximal subdiagrams resp. sub-hypergraphs are identified which are correct with respect to the hypergraph grammar. Feedback about these correct subdiagrams is provided to the user by drawing all diagram components with the same color whose representing edges belong to the same correct sub-hypergraph. The result of this step of diagram analysis is the derivation structure of the reduced hypergraph which describes the syntactic structure of the diagram. The derivation structure - similar to context-free string grammars - is a derivation tree if a context-free hypergraph grammar is used (for context-free hypergraph grammars with embeddings, it is a directed acyclic graph, the derivation DAG [17,10]). The tree root represents the nonterminal edge of the starting hypergraph, and the (terminal) edges of the reduced hypergraph 5 Plain context-free grammars with embeddings may be too restricted for some diagram languages, e.g., UML class diagrams [17]. However, DiaGen allows to restrict productions by application conditions. With this feature, DiaGen can be applied to real-world languages like Statecharts and UML class diagrams [18,6]. begin(a) BlockSeq(a,i) statement(a,b) Block(b,c) statement(b,c) BlockSeq(h,c) Block(h,c) statement(h,c) statement(e,i) Flowchart Fig. 8. Derivation tree of the reduced hypergraph model of Fig. 6 according to the grammar of Fig. 7 when omitting any conn edge. model are represented as leaves of the tree. Fig. 8 shows the derivation tree of the reduced hypergraph model of Fig. 6. Any conn edge, however, has been omitted for simplicity. Edges are written as their edge labels together with the labels of their visited nodes in parentheses. 3.4 Attribute evaluation The task of the final step of diagram analysis is translating the diagram into some data structure which is specific for the application domain where the diagram editor is used. If, e.g., the flowchart editor is used as part of a programming tool, it should probably create some textual representation of the flowchart. for that purpose, DiaGen uses a common syntax-directed translation mechanism based on attribute evaluation similar to those of attribute string grammars [13]: Each hyperedge carries some attributes. Number and types of these attributes which have to be specified by the editor developer depend on the hyperedge label. Productions of the hypergraph grammar may be augmented by attribute evaluation rules which compute values of some attributes that are accessible through those edges which are referred to by the production. After parsing, attribute evaluation works as follows: Each hyperedge which occurs in the derivation tree (or DAG in general) has a distinct number of attributes; grammar productions which have been used for creating the tree impose rules how attribute values are computed as soon the value of others are known. Some (or even all) attribute values of terminal edges are already known; they have been derived from attributes of the diagram components during the reducing step (This feature has been omitted in Section 3.2). The attribute evaluation mechanism of the editor then computes a valid evaluation order. Please note that DiaGen does not require a specific form of attributed definition like S- or L-attributed definitions [13]. At least when dealing with derivation DAGs these forms would fail. The editor developer, therefore, is allowed to define evaluation rules rather freely for each grammar production, and the evaluation mechanism has to determine an evaluation order for each diagram analysis run anew. Of course, the developer has to be careful in order not to introduce inconsistencies or cyclic attribute dependencies. Syntax-directed translation in the context of flowcharts is rather simple. An obvious data structure representing a flowchart is textual program, e.g., in Pascal-like notation which is possible since syntactically flowcharts are well structured (at least when using the hypergraph grammar as shown in Fig. 7). for that purpose, each hyperedge needs a single attribute of type String : the terminal hyperedges contain the text of their corresponding diagram components whereas the nonterminal hyperedges contain the program text of their sub-diagram. Attribute evaluation rules are straight-forward. Attribute evaluation is the last step of diagram analysis when editing diagrams by free-hand editing. The following section shows that syntax-directed editing is seamlessly integrated into DiaGen which means that editors make use of the diagram analysis as it has been described above even when editing diagrams in a syntax-directed way. Syntax-directed Editing As discussed in the introduction, syntax-directed editing has several important benefits. Other approaches for free-hand editing which do not make use of abstract internal models (e.g., the Penguins system being based on constraint multiset grammars [3,20]) cannot extend free-hand editing by syntax-directed editing, which requires such an abstract model. But since the DiaGen approach is based on such a model (the hypergraph model), it is quite obvious to o#er syntax-directed editing, too. However, free-hand editing using a parser requires that the hypergraph grammar remains the only syntax description of the reduced hypergraph model and thus the diagram language. Syntax-directed editing operations must not change the syntax of the diagram language; they can only o#er some additional support to the user. This requirement has two immediate consequences: . It is possible to specify editing rules that deliberately transform a correct diagram into an incorrect one with respect to the hypergraph grammar. This might appear to be an undesired feature; but consider the process of creating a complex diagram: the intermediate "drawings" need not, and generally do not make up a correct diagram, only the final "drawing". In order to support those intermediate incorrect results, syntax-directed editing operations have to allow for such "disimprovements", too. . Editing operations are quite similar to macros in o#-the-shelf text and graphics editors; they combine several actions, which can also be performed by free-hand editing, into one complex editing operation. However, syntax-directed editing rules are actually much more powerful than such macros which o#er only recording of editing operations and their playback as a complex operation: syntax-directed editing operations also take care of providing a valid diagram layout where this is possible (incorrect diagrams in general have no valid layout.) Furthermore, editing operations can take into account context information, and they may have rather complex application conditions. This makes use of graph transformation an obvious choice for adding syntax-directed editing to the free-hand editing mode: Editing operations are specified by hypergraph transformations on the hypergraph model as shown in Fig. 2. In the following it is explained why hypergraph transformations may have to use information from the reduced hypergraph model and the derivation structure, too. Whenever the hypergraph model has been changed by some transformation, it has to be parsed again. The results of the parser are then used to indicate correct subdiagrams and to create a valid layout for them (cf. Section 5). Please note that the hypergraph model is directly modified by the transformation rules; the modeling step, which is necessary for free-hand editing, does not take place. In the following, two examples of editing operations for a flowchart editor are used for describing specification and realization of syntax-directed editing operations. The first example demonstrates the use of simple hypergraph transformation rules whereas the second one shows why additional information from the reduced hypergraph model as well as the derivation DAG may be necessary. 4.1 Example 1: Simple hypergraph transformation rules Fig. 9 shows an example of a syntax-directed editing operation which adds a new statement below an existing one in a flowchart editor. The situation just before applying the editing operation is depicted in Fig. 9a. The topmost statement has been selected which is indicated by a thick border and gray handles; the editing operation whose hypergraph transformation rule is shown in Fig. 9b adds a new statement just below this selected one. The result is shown in Fig. 9c. The hypergraph transformation rule in Fig. 9b is depicted as before: LHS and RHS are separated by "#", corresponding edges and nodes of LHS and RHS carry the same labels. Host nodes and edges which match the LHS without a) b)2 x a flowOut box flowOut flowOut flowIn vArrow box x a a b22 a e c) Fig. 9. A syntax-directed editing operation which inserts a new statement below a selected one. an identically labeled counterpart in the RHS are removed when applying the rule. The marked box hyperedge of the LHS indicates that this edge has to match the hypergraph model edge of the diagram component which has been selected by the editor user. When applied, this rule removes the flowOut relation edge which connects the selected statement box with an outgoing line or arrow (which is not specified here); a new vertical arrow and a new statement box together with some relation edges are added. After applying the rule, the resulting hypergraph is reduced and parsed (cf. Fig. 2). The layouter can then properly layout the resulting diagram which now contains a new statement box (this box carries the default text "Action" in Fig. 9c.) Fig. 10 shows the concrete specification of this simple editing operation together with its transformation rule. In DiaGen, syntax-directed editing operations are specified in terms of simple rules and complex operations quite similar to rules and transformation units in GRACE [21] as shown in the following. A rule (add rule in Fig. 10) is specified as its LHS (as a list of edges) and how its RHS "di#ers" from its LHS, i.e., which edges are removed (indicated by -) and which ones are added (indicated by +) by the rule. Each hyperedge is again written as its edge type together with its visited nodes in parentheses. The node hyperedges are special: they are actually pseudo edges which allow to refer to nodes with the same notation as edges. The LHS in Fig. 10 consists of a box edge, a flowOut edge, some nodes and a node pseudo edge which is used to rule add_rule: box(_,a) f:flowOut(a,b) n:node(a) do -f { OperationSupport.createVArrow(n) } { OperationSupport.createBox(n) } operation add_stmt_after_stmt "Add statement" : specify box b "select statement" do add_rule(b); Fig. 10. DiaGen specification of adding a statement below another statement. refer to node a. Applying the rule removes the flowOut edge (indicated by -f where f is the edge reference introduced in the LHS). Furthermore, a vArrow instance etc. are added to the hypergraph model. The Java methods in curly braces are responsible for creating the corresponding diagram components, i.e., a vertical arrow and a statement box. Each syntax-directed editing operation is specified by a complex operation defined in terms of rules; a control program describes how the operation is defined by a sequence of rules or more complex control structures. Control programs in DiaGen have been inspired by [21] and [22], but their semantics is much simpler because backtracking is not performed [10]. Fig. 10 shows the operation add stmt after stmt which uses the trivial control program that simply calls a single rule. The operation of Fig. 10 requires a statement box as parameter b (indicated by specify box b. ) and which simply calls the add rule rule that has been described above. The parameter b that is passed to this rule simply defines a partial match when applying this rule. The corresponding formal parameters are the first edges which are specified in the LHS of the "invoked" rule. An important issue of syntax-directed editing is the question how to select those parts of the diagram that are a#ected by the application of an editing operation. In DiaGen, this has been solved by adding parameters to complex operations (indicated by specify box b. in Fig. 10). When the user selects an editing operation for application, the editor requests the user to specify a single diagram component for each of the parameters of the operation. The hyperedges that internally represent these components specify a partial match which is then used to select where the operation and its rules have to be ap- plied. DiaGen simplifies this user interaction process: When a diagram component is selected, the editor o#ers those editing operations to the user which require a diagram component of the selected type as a first parameter. When the user selects one of those operations, the editor asks for the missing param- eters. However, many operations, e.g., the add stmt after stmt operation, require just a single parameter, i.e., no further user interaction is necessary after selecting the operation. 4.2 Example 2: Utilizing additional information The former example has been rather simple in the sense that its operation can be described with just a single transformation rule. Furthermore, it uses only information which is readily available in the hypergraph model. This subsection outlines that editing operations are in general more complicated and have to use additional information beyond the plain hypergraph model. Fig. 11 shows such an operation in action with screenshots just before and after a) b) Fig. 11. A syntax-directed editing operation which removes a conditional block. applying it. 6 Its task is removing a conditional block which the user has chosen by selecting its condition diamond. Unlike the former example, the number of edges which have to be removed is unknown when the operation is being specified. It is, moreover, di#cult to decide whether a diagram component and its hyperedge belong to the conditional block when solely considering the hypergraph model. However, since this is a problem of diagram syntax, it is quite an easy task when also using syntactic information from the last parsing step: The operation has to remove all leaves of the Block(d, h)-subtree of the derivation tree in Fig. 8. The crucial task of the editing operation is thus to find the Block(d, h)-node of the derivation tree and - from there - all terminal hyperedges which can be reached by paths from this tree node. Finally, their corresponding component edges as well as diagram components have to be identified. Apparently, editing operations have to take into account information which has been collected during diagram analysis, i.e., information from the reduced hypergraph model and from the derivation structure (cf. Fig. 2). DiaGen editors make this information available by so-called cross-model links which connect corresponding nodes and edges of hypergraph model, reduced hypergraph model, and derivation DAG. Path expressions allow to specify how to navigate in and between models using these cross-model links. for our sample operation, this is shown in Fig. 12 which, because of lack of space, does neither show these path expressions nor the hypergraph model, but only the diagram, its reduced hypergraph model, and its (simplified) derivation tree (cf. Figures 11a, 6, and 8). Thick arrows indicate how models are used to find, starting from the selected condition diamond, those terminal statement hyperedges which belong to the conditional block. Dashed edges show how they correspond to the diagram components (resp. their component hyperedges which are omitted here) which have to be removed from the diagram. Please note that not only statement boxes and condition diamonds have to be removed by this operation, but also lines and arrows. In order to also match them by path expressions, these components must have been represented in the reduced hypergraph model as well as in the derivation tree. This was the 6 Actually, Fig. 11a shows the same diagram as Fig. 3, but with a condition selected. conn conn conn conn conn conn conn conn conn conn conn conn conn conn conn conn conn conn conn statement d c e a begin(a) BlockSeq(a,i) statement(a,b) Block(b,c) statement(b,c) BlockSeq(h,c) Block(h,c) statement(h,c) statement(e,i) Flowchart statement statement statement condition condition statement statement begin Fig. 12. Using cross-model information for editing operations reason for using the conn and Conn edges which, for clarity, have been omitted in Section 3 and also in the derivation tree of Fig. 12. 5 Automatic Layout As it has become clear in the previous section, transformations on the hypergraph model modify the structure of the internal model, but they do not describe their e#ects on the position or the size of the diagram components; an automatic layout mechanism which considers the diagram syntax is needed. DiaGen o#ers two kinds of automatic layout support: Tailored layout modules can be programmed by hand. Such a layout is connected to diagram analysis by a generic Java interface to attribute evaluation (cf. Fig. 2). Information about the syntactic structure of the diagram has to be prepared by syntax-directed attribute evaluation first. The layout module then uses this information to compute a diagram layout. Some generic layout modules have been realized already, e.g., a force-directed layout algorithm (cf. [23]) which is used in a Statechart as well a UML class diagram editor [18,6]. Programming such a layout module by hand is quite complicated. for reducing this e#ort, DiaGen o#ers constraint-based specification of diagram layout and computing diagram layout by a constraint solver as in earlier work of ours [24]: The main idea is to describe a diagram layout in terms of values which are assigned to the attributes of the diagram components (e.g., their position). A valid diagram layout is specified by a set of constraints on these attributes; the constraint set is determined by the syntactic structure of the diagram similar to the syntax-directed translation by attribute evaluation: Hyperedges of the hypergraph model and terminal as well as nonterminal hyperedges of the reduced hypergraph model carry additional layout attributes, and reduction step rules as well as grammar productions are augmented by constraints on their accessible attributes. These constraints are added to the set of constraints which specify a diagram layout whenever the corresponding rule or production is instantiated during the reduction step or parsing process. It is important to define layout constraints not only in the hypergraph grammar which is used during the parsing step, but also in the rule set which specifies the reduction step (cf. Fig. 2). This is so because the reduction step may "reduce away" the explicit representation of some specific diagram components (e.g., lines in our flowchart example). If we had restricted specification of layout constraints to the hypergraph grammar, we would not be able to describe the layout of those diagram components. for flowcharts, e.g., constraints have to require a minimum length of lines and arrows. Automatic layout is not restricted to syntax-directed editing. The same information is also available during free-hand editing. Editors being specified and generated by DiaGen therefore o#er an intelligent diagram mode where diagram components may be modified arbitrarily, but the other components, especially their position, may be a#ected by these modifications, too. The layouter takes care of modifying the overall appearance of the diagram such that its syntax is preserved and the layout beautified. This work on intelligent diagrams is similar to the approach by Chok, Marriott, and Paton [20]. 6 Conclusions This paper has presented DiaGen, a rapid-prototyping tool based on hypergraph transformation for creating diagram editors that support free-hand editing as well as syntax-directed editing. By supporting both editing modes in one editor, it combines the positive aspects of both modes, i.e., unrestricted editing capabilities and convenient syntax-directed editing. The approach has proven to be powerful and general in the sense that it supports quick prototyping of diagram editors and does not restrict the class of diagram languages which it can be applied to. This has been demonstrated by several diagram languages for which diagram editors have already been generated, e.g., flowcharts, Nassi-Shneiderman diagrams [19], syntax diagrams [16], a visual #-calculus [25], ladder diagrams [26], MSC [17], UML class diagrams, signal interpreted Petri nets and SFC diagrams [27]. The approach which has been presented in this paper appears to be quite similar to the approach of Rekers and Schurr [5] which has already been outlined in Section 1. Both approaches make use of two hypergraphs resp. graphs. The spatial relationship graph (SRG) in Reker's and Schurr's approach is quite similar to the hypergraph model of DiaGen. But their abstract syntax graph (ASG), which represents the abstract meaning of the diagram, has been introduced for a di#erent reason than the reduced hypergraph model of Dia- Gen: hypergraph models (and also SRGs) are generally quite complicated such that there is no hypergraph parser which can analyze an hypergraph model. Therefore, DiaGen reduces the hypergraph model and parses the much simpler reduced hypergraph model instead of the hypergraph model. As we have demonstrated, parsing of the reduced hypergraph model can be performed efficiently [10]. However in Reker's and Schurr's approach, SRG and ASG are always strongly coupled since they use triple graph grammars for defining the syntax of the SRG and the ASG with one formalism; the ASG has not been introduced for reducing complexity. Instead, a graph grammar parser has to analyze the SRG directly; the ASG (i.e., the abstract meaning of the diagram) is not parsed, it is created as a "side-e#ect" of the parsing of the SRG during free-hand editing. The requirement for a graph parser for the SRG imposes a strong restriction on this approach. The concepts of this paper have been implemented with constraint-based automatic layout based on the constraint solver QOCA by Chok and Marriott [20]. Their Penguins system also allows to generate free-hand editors, however they do not generate an internal model, but use constraint multiset grammars (CMGs) [3]. The hypergraph grammar approach of DiaGen appears to be better suited to the problem since they report a performance that is about two orders of magnitude worse than the performance of DiaGen editors on comparable computers. Furthermore, their system cannot support syntax-directed editing since they do not use an intermediate internal model. As the examples of syntax-directed editing operations suggest, it appears to be unsatisfactory to some extent to specify syntax-directed editing operations on the less abstract hypergraph model instead of the reduced one which appears to be better suited for syntax-directed editing (cf. Reker's and Schurr's approach [5]). However, since the mapping from the hypergraph model to the reduced one is non-injective, the approach which has been presented in this paper does not leave much choice if expressiveness should not be sacrificed. However, future work will investigate where specifying syntax-directed editing operations on the more abstract hypergraph model is su#cient. --R GenGEd: A generic graphical Graph grammars and diagram editing Automatic construction of user interfaces from constraint multiset grammars VisPro: A visual language generation toolset A graph based framework for the implementation of visual environments DiaGen web site http://www2. Algebraic approaches to graph transformation - part I: Basic concepts and double pushout approach Computing by graph rewriting Specifying and generating diagram Graph based modeling and implementation with EER/GRAL Grammars and Languages Hyperedge replacement graph grammars Flowchart techniques for structured programming Pascal User Manual and Report Application of graph transformation to visual languages Diagram editing with hypergraph parser support Programmed graph replacement systems An experimental comparison of force-directed and randomized graph drawing algorithms Specification of diagram Automatically generating environments for dynamic diagram languages Creating semantic representations of diagrams International Standard 61131 A: Programmable Logic Controllers Handbook of Graph Grammars and Computing by Graph Transformation --TR Compilers: principles, techniques, and tools Handbook of graph grammars and computing by graph transformation Hyperedge replacement graph grammars Algebraic approaches to graph transformation. Part I Algebraic approaches to graph transformation. Part II Programmed graph replacement systems Graph transformation for specification and programming Application of graph transformation to visual languages Hyperedge Replacement Pascal-User Manual and Report Creating Semantic Representations of Diagrams Graph Based Modeling and Implementation with EER / GRAL An Experimental Comparison of Force-Directed and Randomized Graph Drawing Algorithms Graph grammars and diagram editing Automatic construction of user interfaces from constraint multiset grammars A graph based framework for the implementation of visual environments Diagram Editing with Hypergraph Parser Support VisPro Automatically Generating Environments for Dynamic Diagram Languages GenGEd - A Generic Graphical Editor for Visual Languages based on Algebraic Graph Grammars Constraint-Based Diagram Beautification Flowchart techniques for structured programming --CTR Ewa Grabska , Andrzej achwa , Grazyna Slusarczyk , Katarzyna Grzesiak-Kopec , Jacek Lembas, Hierarchical layout hypergraph operations and diagrammatic reasoning, Machine Graphics & Vision International Journal, v.16 n.1, p.23-38, January 2007 O. G. Sharov , A. N. Afanas'Ev, Syntax-Directed Implementation of Visual Languages Based on Automaton Graphical Grammars, Programming and Computing Software, v.31 n.6, p.332-339, November 2005 Mark Minas, Syntax analysis for diagram editors: a constraint satisfaction problem, Proceedings of the working conference on Advanced visual interfaces, May 23-26, 2006, Venezia, Italy Hans Vangheluwe , Juan de Lara, Foundations of multi-paradigm modeling and simulation: computer automated multi-paradigm modelling: meta-modelling and graph transformation, Proceedings of the 35th conference on Winter simulation: driving innovation, December 07-10, 2003, New Orleans, Louisiana Gennaro Costagliola , Vincenzo Deufemia , Giuseppe Polese, Visual language implementation through standard compiler-compiler techniques, Journal of Visual Languages and Computing, v.18 n.2, p.165-226, April, 2007 Frank Drewes , Berthold Hoffmann , Mark Minas, Context-exploiting shapes for diagram transformation, Machine Graphics & Vision International Journal, v.12 n.1, p.117-132, January Frank Drewes , Berthold Hoffmann , Detlef Plump, Hierarchical graph transformation, Journal of Computer and System Sciences, v.64 n.2, p.249-283, March 2002 Berthold Hoffmann, Abstraction and control for shapely nested graph transformation, Fundamenta Informaticae, v.58 n.1, p.39-65, November Berthold Hoffmann, Abstraction and Control for Shapely Nested Graph Transformation, Fundamenta Informaticae, v.58 n.1, p.39-65, January Jun Kong , Kang Zhang , Xiaoqin Zeng, Spatial graph grammars for graphical user interfaces, ACM Transactions on Computer-Human Interaction (TOCHI), v.13 n.2, p.268-307, June 2006 Gennaro Costagliola , Vincenzo Deufemia , Giuseppe Polese, A framework for modeling and implementing visual notations with applications to software engineering, ACM Transactions on Software Engineering and Methodology (TOSEM), v.13 n.4, p.431-487, October 2004

diagram editors;hypergraph grammar;hypergraph transformation;rapid prototyping

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A Hybrid Index Technique for Power Efficient Data Broadcast.

The intention of power conservative indexing techniques for wireless data broadcast is to reduce mobile client tune-in time while maintaining an acceptable data access time. In this paper, we investigate indexing techniques based on index trees and signatures for data disseminated on a broadcast channel. Moreover, a hybrid indexing method combining strengths of the signature and the index tree techniques is proposed. Different from previous studies, our research takes into consideration of two important data organization factors, namely, clustering and scheduling. Cost models for the three indexing methods are derived for various data organization accommodating these two factors. Based on our analytical comparisons, the signature and the hybrid indexing techniques are the best choices for power conservative indexing of various data organization on wireless broadcast channels.

Introduction Due to resource limitations in a mobile environment, it is important to efficiently utilize wireless bandwidth and battery power in mobile applications. Wireless broadcasting is an attractive approach for data dissemination in a mobile environment since it tackles both bandwidth efficiency and power conservation problems [BI94, IVB96, SRB97, HLL98c]. On one hand, data disseminated through broadcast channels allows simultaneous access by an arbitrary number of mobile users and thus allows efficient usage of scarce bandwidth. On the other hand, the mobile computers consume less battery power when passively monitoring broadcast channels than actively interacting with the server by point-to-point communication. Three criteria are used in this paper to evaluate the data access efficiency of broadcast channels: ffl Access Time: the average time elapsed from the moment a client 1 issues a query to the moment when all the requested data frames are received by the client, The author is now with Department of Computer Science, University of Waterloo, Waterloo, Ontario, Canada 1 In this paper, we use 'client' or 'mobile client' to refer to a user with a mobile computer. ffl Tune-in Time: the period of time spent by a mobile computer staying active in order to obtain the requested data. Indexing Efficiency: The tune-in time saved per unit of access time overhead for indexing 2 . While access time measures the efficiency of access methods and data organization for broadcast channels, tune-in time is frequently used to estimate the power consumption by a mobile computer. Indexing efficiency, which correlates the access time and tune-in time, is used to evaluate the efficiency of indexing techniques in terms of minimizing the tune-in time while maintaining an acceptable access time overhead. In other words, a power conservative indexing technique has to balance out the index overhead (in terms of access time increased) and the time saved in order to maximize the indexing efficiency. To facilitate efficient data delivery on broadcast channels, scheduling and clustering are frequently used to select and organize data for broadcast. Broadcast scheduling policies determine the content and organization of data broadcasting based on aggregate user data access patterns. Broadcast disk [AAFZ95] is one of the well-known broadcast scheduling methods. In contrast, flat broadcast refers to the broadcast scheduling where each data frame is broadcast once in every cycle [HLL98c]. When all frames with the same attribute value are broadcast consecutively, the data broadcast are called clustered on that attribute 3 . In contrast, the data broadcast are non-clustered on an attribute when all frames with the same value of that attribute are not broadcast consecutively. Clustering allows continuous reception of data with a specific attribute value. Several indexing techniques for broadcast channels have been discussed in the literature[IVB96, LL96b, CYW97, SV96]. The basic idea behind these techniques is that, by including information about the arriving schedule of data frames in the broadcast channels, mobile computers are able to predict the arrival time of the requested data frames and thus, selective tuning in can be realized. Signature and index tree techniques [IVB96, LL96b] represent two different classes of indexing methods for broadcast channels. According to [IVB96, LL96b], the index tree method is based on clustered data organization, while the signature methods don't presume a clustered data organization 4 . Moreover, indexing techniques used in these papers only took flat broadcast into consideration. In [IVB96] index frames were treated in the same way as a data frame, although a better approach is to separate index frames from data frames. [LL96b] did not consider the clustered data. Although the authors demonstrated that the signature size played an important role in terms of data filtering efficiency and access latency, the optimal signature size was not given. In this paper, we extend the existing works further. For the index tree method, since the size of an index frame is normally not equal to that of a data frame, to accurately estimate the access time and the tune-in time, we separate the index frames from data frames. For the signature method, we derive formulae to estimate the optimal signature size. In addition, scheduling and clustering are considered together with the index methods. The tune-in time and the access time cost formulae are developed to cover the cases: i) flat scheduling and clustered; ii) flat scheduling and non-clustered; and iii) broadcast disks scheduling and clustered. To our knowledge, there is no systematical study and comparison of the power conservative indexing techniques in the literature which takes both clustering and scheduling issues into account. In addition, we propose a new indexing method, called hybrid index, which takes strengths of both index tree and signature methods. Those three index methods are evaluated based on our criteria, namely, access time, tune-in time and indexing efficiency. Our results show that clustering and scheduling have major impacts on data organization of wireless data broadcast. We also conclude that the hybrid and signature methods give superior performance to the index tree method for various broadcast data organization accommodating the clustering and scheduling factors. Every indexing technique usually introduces non-zero access time overhead. 3 In this paper, we only consider the case of single attribute indexing and clustering. Issues involving multiple attribute indexing and clustering are addressed in [HLL98b]. 4 Even so, the integrated signature and multi-level signature schemes can benefit from a clustered data organization for broadcast channels. The rest of the paper is organized as follows. Section 2 gives an informal introduction of the broadcast channels, indexing techniques and system parameters used in performance evaluation and comparisons. In Section 3, indexing techniques based on index tree and signature methods are re-examined by taking the clustering and scheduling factors into consideration. In Section 4, a hybrid index scheme and corresponding cost models for access time and tune-in time are developed. Section 5 evaluates the indexing techniques in terms of tune-in time, access time, and indexing efficiency. Section 6 is a review of related work. Finally, Section 7 concludes the paper. 2 The Data Organization for Wireless Broadcast In this section, we briefly introduce the concept of broadcast channels and some of the terminologies used. We assume that a base station is serving the role of an information server which maintains various kinds of multimedia data, including texts, images, audio/video and other system data. The server periodically broadcasts, on a specific channel, data of popular demands as a series of data frames to a large client population. These data frames vary in size and each frame consists of packets which are the physical units of broadcast. The header of a frame contains signals for synchronization and meta-information such as the type and the length of the frame. Logically the data frames are classified into two types: record frames and index frames, where record frames contain data items and index frames contain indexing information such as index tree nodes or signatures for a set of data items. Those two types of frames are interleaved together in a cycle. The clients retrieve the frames of their interest off the air by monitoring the broadcast channel. Since a set of data frames is periodically broadcast, a complete broadcast of the set of data frames is called a broadcast cycle. The organization of data frames in a broadcast cycle is called a broadcast schedule. Data organization on the broadcast channels have great impacts on data access efficiency and power consump- tion. Data frames can be clustered based on attributes. Based on the results in [IVB96], index tree techniques result in more efficient access for clustered information than non-clustered one. In this paper, we take both clustered and non-clustered data organization into consideration. Generally speaking, a non-clustered data organization can be divided into a number of segments, called meta segments [IVB96], with non-decreasing or non-increasing values of a specific attribute. These meta segments can be considered as clustered and thus the indexing techniques for clustered data can be applied to them. To facilitate our study, we use the scattering factor M , the number of meta segments in the data organization, to model the non-clusterness of a data organization 5 . Since access latency is directly proportional to the volume of data being broadcast, the volume of data in a broadcast cycle should be limited in such a way that only the most frequently accessed data frames are broadcast on the channel while the remaining data frames can be requested on demand through point-to-point connections [HLL98c, AFZ97, SRB97]. The server must determine the set of data frames to be broadcast by collecting statistics about data access patterns. Due to some timely events, the client access pattern sometimes shows skewed distributions, which may be captured by Zipf or Gaussian distribution functions. In this case, scheduling data frames in broadcast disks (refer to Appendix and [AAFZ95] for detail) can achieve a better performance in terms of the access time and is a very important technique. As indicated in [AAFZ95], in addition to performance benefits, constructing a broadcast schedule on multi-disks can give clients the estimated time when a particular data frame is to appear on the broadcast channel. This is particularly important for selected tune-in, data prefetching [AFZ96b], hybrid pull and push technique [SRB97, AFZ97], and updates [AFZ96a]. Therefore, the application of the index methods to broadcast disks is studied in this paper. The broadcast disks method has better access time when the data frames with the same attribute values are clustered in one of the minor cycles. By receiving the cluster of data frames together, the mobile computer can answer the query without continuing to monitor the rest of broadcast cycle. This can be achieved by placing all of the data frames with the same indexed attribute value as a cluster on the same broadcast disk. The whole 5 To simplify our discussion, we neglect the variance of the meta segment size. cluster of data frames are brought to the broadcast channel as a unit. Depending on speed of broadcast disks where this cluster is located, these data frames may appear several times in minor cycles. Thus, the resulting broadcast cycle is different from the completely clustered broadcast cycle. For broadcast scheduling adopting broadcast disks without using clustering, we simply consider the resulting broadcast cycle as non-clustered. In that case, broadcast disks lose their advantages over flat broadcast. Thus, when we consider index techniques for broadcast disks in the later sections of this paper, we only consider the clustered case. We assume only one broadcast channel since a channel with large bandwidth is logically the same as multiple channels with combined bandwidth of the same capacity. Moreover, it incurs smaller overheads of administrating than multiple channels. With the same token, we assume that index information is disseminated on the same broadcast channel. Finally, we assume that updates are only reflected between cycles. In other words, a broadcast schedule is fixed before a cycle begins. D Number of information frames (excluding index frames) in the broadcast cycle F Number of distinct information frames in the broadcast cycle Average number of packets in an information frame S Selectivity of query: average number of frames containing the same attribute value Scattering factor of an attribute, which is the number of meta segments of the attribute Table 1. System Parameter Setting Table 1 gives the parameters which describe the characteristics of a broadcast cycle. The cost models for the various index methods discussed later in this paper are derived based on these parameters. Although the sizes of data frames may vary, we assume frames to be a multiple of the packet size. Both access time and tune-in time are measured in terms of number of packets. Before we develop the cost models for various index methods in broadcast disks, we derived a theorem 6 for the optimal broadcast scheduling based on multiple broadcast disks (please refer to Appendix for proof). The broadcast schedules derived from the theorem is used in our analysis later. Given the number of data frames to be broadcast, D, the number of disks, N , the size of disk and the broadcast frequency 7 of disk i, broadcasting a data frame d on disk i, 8i fixed inter-arrival time can achieve the optimal access time. In that case, to retrieve data frame d from disk i, client needs to scan, on an average, D frames. 3 Basic Indexing Techniques In this section we discuss the basic ideas behind the index tree and the signature methods. We describe the distributed indexing and integrated signature techniques because they are the best methods of their class for single attribute indexing. The analytical cost models for the access time and the tune-in time for clustered and non-clustered data broadcast are presented. Moreover, the application of these index techniques to broadcast disks is also considered. Due to space limitations, we don't give all the derivations of these cost models. Interested readers can refer to [IVB96] and [LL96b] for more details. the leaves of an index tree. The access method for retrieving data frames with an index tree technique involves the following steps: ffl Initial probe: The client tunes into the broadcast channel and determines when the next index tree is broadcast The client follows a list of pointers to find out the arrival time of the desired data frames. The number of pointers retrieved is equal to the height of the index tree. ffl Retrieve: The client tunes into the channel and downloads all the required data frames. Figure 1 depicts an example of an index tree for a broadcast cycle which consists of 81 data frames [IVB96]. The lowest level consists of square boxes which represent a collection of three data frames. The index tree is shown above the data frames. Each index node has three pointers 8 . a2 a3 Non Replicated Replicated Part I Figure 1. A Full Index Tree h Height of the whole index tree t Number of upper levels in the index tree that are replicated T Number of packets in an index tree node n Number of search keys plus pointers that a node can hold Table 2. Parameter Setting for Index Tree Schemes Table 2 gives the parameter setting for the index tree cost model. To reduce access time while maintaining a similar tune-in time for the client, the index tree can be replicated and interleaved with the information. Distributed indexing is actually one index replication and interleaving method. The index tree is broadcast every 1 d of the file during a broadcast cycle. However instead of interleaving the entire index tree d times, only the part of the index tree which indexes the data block immediately following it is broadcast. The whole index tree is divided into two parts: replicated and non-replicated parts. The replicated part constitutes the upper t levels of the index tree and each node in that part is replicated a number of times equal the number of children it has, while the non-replicated part consists of the lower and each node in this part appears only once in a given broadcast cycle. Since the lower levels of an index tree take up much more space than the upper part (i.e., the replicated part of the index tree), the index overheads can be greatly reduced if the lower levels of the index tree are not replicated. In this way, access time can be improved significantly without much deterioration in tune-in time. To support distributed indexing, every frame has an offset to the root of the next index tree. The first node of each distributed index tree contains a tuple, with the first field as the primary key of the record that was broadcast last and the second field as the offset to the beginning of the next broadcast cycle. This is to guide the clients that 8 For simplicity, the three pointers of each index node in the lower most index tree level is represented by just one arrow. have missed the required record in the current cycle to tune to the next broadcast cycle. There is a control index at the beginning of every replicated index to direct the client to a proper branch in the index tree. This additional index information for navigation together with the sparse index tree provides the same function as the complete index tree. We assume that each node of the index tree takes up T packets and X[h] and X[t], respectively, are the total number of nodes of the full index tree and the replicated part of the index tree. The number of nodes in the i-th level of the index tree is denoted as L[i]. In the distributed index tree, each node, p, in the replicated part is repeated as many times as the number of children that p has. Thus, the root is broadcast L[2] times and nodes at level 2 are broadcast L[3] times etc. For the index tree in Figure 1, since each node has three children, the root and nodes at level 2, i.e., a 1 , a 2 , and a 3 , are broadcast 3 and 9 times, respectively. Therefore, in a broadcast cycle, the total number of nodes in the replicated part is 1. Additionally, the number of index nodes that are located below the t-th level of the index (i.e., the non-replicated) is, Hence, the total number of index nodes in a cycle is, which equals to X[h] As a result, the index overhead is In the above discussion, we assumed that the file is clustered. For a non-clustered broadcast cycle, we can still apply index tree techniques to each meta segment. Instead of using one index tree for the entire broadcast cycle, an index tree is created for each meta segment. However, each index tree indexes all the values of the non-clustered attribute rather than indexing just the attribute values that appear in the current meta segment. For attribute values that do not appear in the current meta segment, a pointer in the index tree points to the next occurrence of the data frame with the desired attribute values. Thus, there are M distinct index trees for a broadcast cycle consists of M meta segments, The total overhead for putting index trees in a broadcast cycle is T packets. To simplify the cost models, we average the index tree overhead to each data frame so that the size of a frame is considered to consist of a data part and an index overhead part. Of course, the actual index tree overhead for each data frame is different, but from the statistics point of view we can assume that all data frames have the same average index tree overhead. The average overhead for each data frame is The replicated index tree part is broadcast every 1=L[t fraction of each meta segment. Therefore, a broadcast cycle is divided into M \Delta L[t data blocks with replicated index nodes at the beginning of each block. Let P be the average number of packets for a data frame, the length of each block is Flat Broadcast: Let us derive the access time and the tune-in time estimates for flat broadcast first. Since each frame is broadcast once in a cycle, the number of data frames in the broadcast, D, is equal to the number of distinct frames F . The initial probe period is the time to reach the index frame at the beginning of the next data block and can be estimated as: For a clustered broadcast, the scattering factor, and the expected number of data frames before the arrival of the desired frames is Hence, the access time is: initial probe time +waiting time before first desired frame arrives waiting time for retrieving all desired frames in the broadcast For a non-clustered broadcast cycle (M ? 1), the access time is: waiting time for retrieving all desired frames in the broadcast The tune-in time for both clustered and non-clustered broadcast cycle depends on the initial probe, the scanning of index tree, the extra scanning of index tree in subsequent meta segments, and the retrieval of S data frames. Thus, the tune-in time of the index technique is upper bounded by: For a fully balanced indexing tree, the height of the tree, the number of nodes at i-th level, and the number of nodes in the upper t levels of the index tree are: F According to [IVB96], the optimal height of the replicated part of the index tree for a broadcast, denoted as - t, can be estimated as: (log F while for a non-clustered broadcast cycle, the optimal number of replicated levels - t within a meta segment is: (log Broadcast Disks: For broadcast disks, as discussed in Section 2, data frames with the same attribute values are clustered in one minor cycle. In this case, we can treat each minor cycle of the broadcast disks as a meta segment 9 . An index tree can be built for each minor cycle. Similar to flat broadcast, the initial probe period, the time to reach the index frame at the beginning of the next data block, can be estimated as: where the number of data frames in the broadcast is and the scattering factor, M , is equal to the number of minor cycles in the broadcast (i.e., the LCM of the relative frequency of the disks). Hence, the access time for a clustered broadcast is: initial probe time +waiting time before first desired frame arrives waiting time for retrieving all desired frames in the broadcast Note that in the above equation, based on Theorem 1, the expected number of data frames before the arrival of the desired frames is and the optimal number of replicated levels within a minor cycle can be derived from Equation 2. 9 Note that it is different from the meta segments for a non-clustered flat broadcast, where frames with the same attribute value may be scattered in several meta segments. Since all the desired data frames are clustered in one minor cycle, the tune-in time is the same as in flat broadcast for a clustered broadcast cycle, i.e., 3.2 The Signature Technique Signature methods have been widely used for information retrieval. A signature of a data frame is basically a bit vector generated by first hashing the values in the data frame into bit strings and then superimposing them together. The signature technique interleaves signatures with their associated data frames in data broadcasting [LL96b]. To answer a query, a query signature is generated in a similar way as a data frame signature based on the query specified by the user. The client simply retrieves information signatures from the broadcast channel and then matches the signatures with the query signature by performing a bitwise AND operation. When the result is not the same as the query signature, the corresponding data frame can be ignored. Otherwise, there are two possible cases. First for every bit set in the query signature, the corresponding bit in the data frame signature is also set. This case is called true match. Second the data frame in fact does not match the search criteria. This case is called false drop. Obviously the data frames still need to be checked against the query to distinguish a true match from a false drop. The primary issue with different signature methods is the size and the number of levels of the signatures. The access method for a signature scheme involves the following steps: ffl Initial probe: The client tunes into the broadcast channel for the first received signature. ffl Filtering: The client accesses the successive signatures and data frames to find the required data. On an average, it takes half of a broadcast cycle for the client to get the first frame with the required attribute. ffl Retrieve: The client tunes in to get the successive desired data frames from the channel. k number of information frames indexed by an integrated signature p number of bits in a packet R the size (number of packets) of an integrated signature Table 3. Parameter Setting for Signature Scheme The number and the size of the signatures and the average false drop probability of the signatures 10 affect tune-in time and access time. The average false drop probability may be controlled by the size of the signatures. The initial probe time is related to the number of signatures interleaved with the data frames. Table 3 defines the parameters for signature cost models. Estimation of the average false drop probability is given in the following Lemma [LL96b]: Given the size of a signature, R, the number of bit strings superimposed into the signature, s, the average false drop probability for the signature is, Each data frame may have different false drop probabilities. To simplify the cost model, we use average false drop probability to estimate the access time and the tune-in time when a large number of queries are sampled (i.e., many data frames are retrieved). In [LL96b], three signature algorithms, namely simple signature, integrated signature, and multi-level signa- ture, were proposed and their cost models for access time and tune-in time were given. For simple signatures, the signature frame is broadcast before the corresponding data frame. Therefore, the number of signatures is equal to the number of data frames in a cycle. An integrated signature is constructed for a group of consecutive frames, called a frame group. The multi-level signature is a combination of the simple signature and the integrated signature methods, in which the upper level signatures are integrated signatures and the lowest level signatures are simple signatures. Since the three signature algorithms have been extensively compared in the literature [LL96b, HLL98a], we don't repeat the comparisons here. In the context of this study, simple signature is not very efficient since it will be generated from only one attribute. Thus, we select the integrated signature method to compare with the index tree method and the new index methods proposed later in this paper. A Frame Group Info Frame Info Frame Info Frame Info Info Frame A Broadcast Cycle Integrated Signature Info Frame Frame Figure 2. An Example of the Integrated Signature Technique Figure 2 illustrates an integrated signature scheme. An integrated signature indexes all of the data frames between itself and the next integrated signature. The integrated signature method is general enough to accommodate both clustered and non-clustered data broadcast. For clustered data broadcast, a lot of data frames can be indexed by one integrated signature. According to Lemma 1, the smaller the number of bit strings s superimposed into an integrated signature, the lower the false drop probability. The integrated signature generated for a clustered broadcast cycle has the effect of reducing the number of bit strings superimposed. To maintain a similar false drop probability for a non-clustered broadcast cycle, the number of data frames indexed by an integrated signature may be reduced. Determining the number of data frames for signature generation requires further study. To simplify our discussion, we assume that frames with the same attribute value for an attribute a are evenly distributed in each meta segment. Consequently, the number of frames with the same attribute value in each meta segment is dS=Me, where the attribute a has a selectivity S and a scattering factor M . Let k be the number of data frames indexed by an integrated signature. The number of distinct attribute values used for signature generation, s, can be estimated as dk=dS=Mee. For frames in a meta segment, the average number of qualified frames corresponding to a matched integrated signature, called locality of true matches l (1 - l - k), can be estimated as, l = k=dk=dS=Meee, for frames which are randomly distributed over the file, l is equal to 1. Flat Broadcast: Next, we derive the access time and the tune-in time for clustered and non-clustered broadcast cycles. Let SIG be the average signature overhead for each data frame. Then, Once again, we assume that the expected number of data frames before the arrival of the desired frames is C . For clustered data broadcast, the access time can be derived as follows: initial probe time +waiting time before first desired frame arrives waiting time for retrieving all the desired frames in the broadcast (R and the tune-in time is: true match frames in the broadcast integrated signatures before the first desired frame false drop frames before the first desired frame For a non-clustered broadcast cycle, the access time is: waiting time for the first desired frame to arrive waiting time for retrieving all the desired frames in the broadcast (R and the tune-in time is: true match frames in the broadcast integrated signatures for the retrieval of all the desired frames false drop frames for the retrieval of all the desired frame According to Lemma 1, we have, P We differentiate Equation (4) or (5) with respect to R and let @TUNE=@R equal zero. Then the optimal signature size (number of packets), - R, can be computed as: s s Broadcast Disks: For broadcast disks, the access time and the tune-in time can also be obtained by Equations (3) and (4) respectively. Compared with flat broadcast, the difference is in the parameter C , i.e., for flat broadcast broadcast disks and 4 The Hybrid Index Approach Both the signature and the index tree techniques have advantages and disadvantages in one aspect or the other. For example, the index tree method is good for random data access, while the signature method is good for sequentially structured media such as broadcast channels. The index tree technique is very efficient for a clustered broadcast cycle, but the signature method is not affected much by clustering factor. While the signature method is particularly good for multi-attribute retrieval, the index tree provides a more accurate and complete global view of the data frames based on its indexed value. Since the clients can quickly search in the index tree to find out the arrival time of the desired data, the tune-in time is normally very short. Since a signature does not contain global information about the data frames, it can only help the clients to make a quick decision on whether the current frame (or a group of frames) is relevant to the query or not. The filtering efficiency heavily depends on the false drop probability of the signature. As a result, the tune-in time is normally high and is proportional to the length of the broadcast cycle. Data Block a3 Sparse Index Tree Signature of the following frame group I Data Block Data Block Info Frame Info Info Info Info Frame Frame Frame Frame Frame Frame Info Info Info Frame A Broadcast Cycle Figure 3. The Hybrid of Index Tree and Signature In this section, we develop a new index method, called hybrid index, which builds on top of signatures a sparse index tree to provided global view for data frames and their corresponding signatures. A key-search pointer node in the sparse index tree points to a data block of consecutive frames and their corresponding signatures (refer to Figure 3). The index tree is called sparse tree because only upper t levels of the whole index tree are constructed. Ob- viously, the sparse index tree overhead depends on t. The larger the t, the more precise location information the sparse tree provides, and the higher the access time overhead. One extreme case is t equals h the number of the whole index tree levels. The hybrid index evolves to the index tree method. On the other hand, if t equals zero, the hybrid index method becomes the signature method. To retrieve information, the client can search the sparse index tree to obtain the approximate location information about the desired data frames. Since the size of the upper t levels of an index tree is usually small the overhead for this additional index is very small. Since the hybrid index technique is built on top of signature method, it retains all of the advantages that a signature method has. However, the global information provided by the sparse index tree improves tune-in time considerably. The general access method for retrieving data with this technique now becomes: ffl Initial probe: The client tunes into the broadcast channel and determines when the next index tree arrives. ffl Upon receipt of the index tree, the client accesses a list of pointers in the index tree to find out when to tune into the broadcast channel to get to the nearest location where the required data frames can be found. ffl Filtering: At the nearest location, a successive signature filtering is carried out until the desired data frames are found. ffl Retrieval: The client tunes into the channel and downloads all the required data frames. 4.1 Cost Model Analysis Based on the above definition of the hybrid indexing method, we derive an estimates of the access time and the tune-in time. The sparse index tree is the same as the replicated part of the index tree method. The average waiting time for retrieving one data frame from the broadcast cycle with M meta segments can be expressed as: where TREE and SIG are the index overheads of the index tree parts and the signature parts of a frame. The average number of data frames in one data block D[B] can be calculated in a similar way as in the index tree method, which is D=(M 1]). Thus, the total index tree and signature overheads in a data block are respectively. Hence, the average initial probe time for the index tree is half of the data block: Flat Broadcast: For the clustered broadcast cycle with flat broadcast scheduling, the expected access time for hybrid indexing method is: and the expected number of data frames before the arrival of the desired frames C is F=2. If the broadcast cycle is non-clustered, then there is one sparse index tree for each meta segment. Index tree technique is applied in each meta segment. Hence, the expected access time is: For both clustered and non-clustered broadcast cycle with flat broadcast scheduling, the tune-in time primarily depends on the initial probe of the client to determine the next occurrence of the control index, the access time for the index tree part which equals to the number of levels t of the sparse index tree, the tune-in time for the data block B, the selectivity of a query S, and the successive access to M meta segments. Therefore, it is upper bounded by: where TUNEB is defined as the tune-in time for filtering data block B with the signature technique. It can be estimated as follows: every signature in half the length of the data block data frames in half the length of the data block B Broadcast Disks: For broadcast disks, the access time for hybrid indexing method can be obtained by Equation (7) (i.e., for broadcast disks and The tune-in time of broadcast disks is the same as that of flat broadcast for a clustered cycle (i.e., let Equation (8)). Note: according to Equation (8), the tune-in time is proportional to M . Hence, the hybrid method is efficient only for the broadcast cycle with small M . Actually, the sparse index tree introduces overhead for the non-clustered broadcast cycle with large M . In this case, retrieval based on signatures can result in better tune-in time. However, the hybrid method supports multi-attribute indexing very well [HLL98b]. For an attribute with small scattering factor, a sparse index tree can be built to reduce the tune-in time. For an attribute with high scattering factor, there is no need to build the sparse index tree and the client simply filters out the requested data frames sequentially and ignores the sparse index tree. We extend the hybrid index with control information, which includes the size of the sparse index tree and the size of the data block. When a query is specified on a non-clustered attribute, this control information is used to direct the client to the beginning of the next data block. Starting there, the client matches the signatures one by one for each data frame in that data block. Hence, the access time for non-clustered information is the same as Equation (7). In order to skip each of these index trees, we assume that the client needs to retrieve an index node to get information such as the size of the sparse index tree and the size of the data block. Therefore, the tune-in time is: where TUNE Sig is defined to be the tune-in time for the corresponding signature scheme used (i.e., integrated signature in this paper). 5 Evaluation of Index Methods In this section, we compare the access time, the tune-in time, and the indexing efficiency of the index tree, the integrated signature, and the hybrid techniques. We also include the case where no index is used (denoted as non-index) as a baseline for comparisons. Our comparisons are based on the cost models developed previously. Orthogonal to the index method, frames can be broadcast based on broadcast disks or flat broadcast. Thus, there are various combinations to be considered. For flat broadcast, each data frame appears once in a given broadcast cycle. Therefore, the number of data frames in the broadcast D equals the number of distinct frames F . For a clustered broadcast cycle (i.e., on average, half of a broadcast cycle needs to be scanned before the desired frames arrive (i.e., For broadcast disks (M=number of minor cycles), D is greater than F due to frame duplication in the broadcast cycle. The access time and the tune-in time on different disks i may be different. We denote the average access probability, the access time, and the tune-in time for frames on disks i as P i , Access i , and Tune i , respectively. For disk i with frequency f i , the expected number of frames scanned before the arrival of the desired frames, C , is given by Theorem 1. Therefore, the estimates for the average access time and tune-in time are: The study for a non-clustered broadcast cycle is especially important in multi-attribute indexing where cycle can be clustered on at most one attribute, while query requests on other attributes get a reply via indexes built on the non-clustered broadcast cycle. For a non-clustered broadcast cycle, M is greater than 1 and the client needs to scan the entire broadcast to retrieve all the desired frames. to 200 Table 4. Parameters of the cost models Table 4 lists the parameter values used in the comparisons. Both access time and tune-in time are measured in number of packets and are compared with respect to the number of distinct frames in a broadcast cycle which is varied from 10 3 to 10 6 . We made the following assumption in the comparisons: a frame has capacity packets and a tree node takes up packets which can contain search keys and pointers, the size of a packet is are grouped together in an integrated signature, the index tree is balanced (all leaves are on the same level) and each node has the same number of children. In order to make comparison, the sparse tree levels t of the hybrid method is set to the same as the replicated tree levels in the index tree method, which can be obtained via Equation (1). A broadcast cycle with selectivity S ? 1 is logically equal to a broadcast cycle with selectivity and the data frame size S times of the original broadcast cycle. Thus, in this paper, we only explore the case where the query selectivity S is 1. For broadcast disks, we assume that three disks are adopted (i.e., 3). The sizes of fast, medium, and slow disk are, respectively, 1=10, 1=2:5, and 1=2 of the total number of frames and the relative spin speeds are 3, 2, and 1. The aggregate client access probability for each disk is the same (i.e., P each disk, all data frames have equal average access probability. Therefore, the average access probability for each data frame is inversely proportional to the size of the disk where the data frame is located. For a non-clustered broadcast cycle, we vary the scattering factor M (i.e., from 1 to 200) to examine its impact on the performance of the index methods. In what follows, we will first evaluate the access time, the tune-in time, and the indexing efficiency of index methods for clustered broadcast cycle and then for non-clustered broadcast cycle. For the clustered broadcast cycle, we consider both of the broadcast disks and flat broadcast as broadcast scheduling policies while for the non-clustered broadcast cycle, we only consider the flat broadcast scheduling. 5.1 The Clustered Broadcast Cycle In this section we study the access time and the tune-in time of the index methods for a clustered broadcast cycle. Figures 4 and 5 depict the access time and the tune-in time comparisons, where the y coordinate is in logarithmic scale and the access time is the overhead with respect to non-index for broadcast disks scheduling. First we consider the access time in Figure 4. The curves representing the access time overhead of the hybrid, the signature, and the non-index methods (denoted as hybrid, sig, and non, respectively) overlap each other for 1.0 2.0 4.0 6.0 8.0 10.0 Access Time Number of Frames in Cycle (x 1e+05 Frames) tree tree BD sig sig BD hybrid hybrid BD non Figure 4. Access Time Overhead Comparisons for Clustered Cycle flat broadcast. Generally, amongst all broadcast scheduling and indexing methods, the non-index method with broadcast disks gives the shortest access time which is proportional to the size of a broadcast cycle. For any particular indexing methods, the access time for broadcast disks (denoted with BD in the figures) is always better than that for flat broadcast because of the skewed client access pattern. When we consider flat broadcast only, the access time for the signature and the hybrid methods is similar to the non-index method as indicated by the overlapping curves in Figure 4 while the access time for the index tree method gives an obviously worse access time. Compared with the non-index method, the index overhead for the index methods (especially the signature and the hybrid methods) does not deteriorate the access time much for a clustered broadcast cycle. In the broadcast disks method, the broadcast cycle is longer than that scheduled in flat broadcast. Since the longer the broadcast cycle, the higher the index overhead, all three index methods give a much worse access time than the non-index BD. The signature method performs better than the hybrid and the tree methods. Since the index tree is replicated in every minor cycle, its index overhead for broadcast disks is the highest. Thus, the difference between the index tree method and the other two index methods for broadcast disks is much larger than that for flat broadcast. Next, we consider the tune-in time of the index methods. Figure 5 shows that the curves representing the index tree method (denoted as tree) and the hybrid method are overlapping for both broadcast disks and flat broadcast. The non-index methods give much worse results than the index methods. This suggests that indexing can improve client tune-in time considerably. If we focus on the index methods only, the index tree method gives the best tune-in time and the signature method has the worst tune-in time. Broadcast disks can also improve the tune-in time of the index methods. As shown in Figure 5, the broadcast disks improve the tune-in time of the index methods and such improvement for the non-index and the signature methods is more than for others. In order to investigate the relationship between the tune-in time and the access time, we demonstrate in Figure 6 the indexing efficiency of indexing methods for various sizes of a broadcast cycle. The tune-in time saved and the access time overhead is calculated with respect to the non-index method for broadcast disks and flat broadcast. Intuitively, the larger the amount of tune-in time saved per unit access time overhead, the better the index methods. We can observe that the amount of tune-in time saved per unit of access time overhead increases as the number of frames in a cycle increases. The figure tells us that the signature method can give the largest amount of tune-in time saved per unit of access time overhead and the index tree method gives the least amount of saving which is much less than the other two methods. Indexing broadcast disks results in less amount of tune-in time saved than 1.0 2.0 4.0 6.0 8.0 10.0 Tune-in Time Number of Frames in Cycle (x 1e+05 Frames) tree tree BD sig sig BD hybrid hybrid BD non non BD Figure 5. Tune-in Time Comparisons for Clustered Cycle1010001.0 2.0 4.0 6.0 8.0 10.0 Indexing Efficiency Number of Frames in Cycle (x 1e+05 Frames) tree tree BD sig sig BD hybrid hybrid BD Figure 6. Indexing Efficiency for Clustered Cycle the indexing flat broadcast. In conclusion, when a broadcast cycle is clustered by attributes, the hybrid scheme is the best when the access time, the tune-in time, and indexing efficiency are considered. If only the tune-in time is considered, then the index tree scheme shows the best performance. If we consider the indexing efficiency, then the signature is the most efficient index method. Broadcast disks approach can improve both the access time and the tune-in time when the client access patterns are skewed, although the improvement in the tune-in time is not as significant as that in the access time. 5.2 The Non-Clustered Broadcast Cycle In this section, we investigate the index methods for a non-clustered broadcast cycle (i.e., M ? 1). To examine the influence of M on the system performance, we fix the number of frames in a cycle to 10 5 and vary M from 1 to 200. The access time overhead is obtained with respect to the non-index method. Figures 7 and 8 illustrate the results. As expected, the scattering factor has great impact on the access time of the index tree method. Since there is an index tree corresponding to every meta segment, as M is increased, the index tree overhead increases Access Time Scattering Factor in Broadcast Cycle with 1e+05 Frames index tree signature hybrid Figure 7. Access Time Overhead vs Scattering Factor48121620 Tune-in Time Scattering Factor in Broadcast Cycle with 1e+05 Frames tree sig hybrid Figure 8. Tune-in Time vs Scattering Factor rapidly. For the hybrid method, although there is a sparse index tree for each meta segment, the sparse index tree overhead is very small and as M increases, the initial probe time for index tree node decreases. Therefore, M has little influence on the access time in the hybrid method. As shown in Figure 8, the tune-in time of the index tree and the hybrid methods goes up quickly as M is increased, while the tune-in time of the signature index method remains the same. Since both the index tree and the hybrid methods need to probe each meta segment for the possible arrival of the desired frames, the major advantage of the index tree and the hybrid methods, namely, short tune-in time, disappears when M is greater than 33. However, there is no impact on the signature method for both the access time and tune-in time when the scattering factor changes. This suggests that the index tree and the hybrid methods are not applicable to a broadcast cycle with a large scattering factor. Similar to the previous section, Figure 9 depicts the indexing efficiency with respect to different scattering factors in a broadcast cycle. The tune-in time saved for the index tree is very low while the tune-in time saved for the signature method is the highest. Finally, we use the same parameter settings as in the clustered broadcast cycle case, but we assume that the broadcast cycle is non-clustered with a scattering factor set to 100 (refer to Figures 10 and 11). The access time Indexing Efficiency Scattering Factor in Broadcast Cycle with 1e+05 Frames tree sig hybrid Figure 9. Indexing Efficiency vs Scattering Factor1000001e+071e+091.0 2.0 4.0 6.0 8.0 10.0 Access Time Number of Frames in Cycle (x 1e+05 Frames) tree sig hybrid Figure 10. Access Time Comparisons for Non-Clustered Cycle overhead is obtained with respect to that of the non-index. Similar to the clustered broadcast cycle, the access time of the index tree method is much worse than that of the other two index methods. The signature method has the closest access time to the non-index method. Since we assume that M is fixed at 100 for any broadcast length and there is an index overhead for each meta segment, unlike the clustered cycle, the tune-in time of the index tree and hybrid methods is not always better than that of the signature method. That is, for small broadcast cycle (i.e., less than methods have the best tune-in time among the three methods. When the length of a cycle increases, the tune-in time of the signature method increases quickly due to false drops and becomes worse than the other methods again. As in the case of clustered cycle, the tune-in time of the hybrid method is always a little bit worse than that of the index tree method. In Figure 12, we illustrate the indexing efficiency for various cycle lengths. All index methods display similar interrelation to that in the case of clustered cycle. For both clustered and non-clustered broadcast cycle, we observe that the tune-in time of the signature schemes is proportional to the length of the broadcast cycle, while the other two methods have the tune-in time independent of the length of the broadcast cycle. The reason is that the size of the index tree can be adjusted automatically according to the length of the broadcast cycle F and the height of the index tree h increases very slowly (n h - F ) 2.0 4.0 6.0 8.0 10.0 Tune-in Time Number of Frames in Cycle (x 1e+05 Frames) tree sig hybrid Figure 11. Tune-in Time Comparisons for Non-Clustered Cycle110010000 1.0 2.0 4.0 6.0 8.0 10.0 Indexing Efficiency Number of Frames in Cycle (x 1e+05 Frames) tree sig hybrid Figure 12. Indexing Efficiency for Non-Clustered Cycle and only h affects the tune-in time of the clients. 6 Related Works The basic idea of constructing index on broadcast data was investigated by a number of projects [IVB94b, IVB96, LL96b]. To reduce the power consumption of clients, [IVB94a, IVB96] proposed two methods, (1; d) indexing and distributed indexing. In (1; d) indexing method, the index tree is broadcast d times during one broadcast cycle. The full index tree is broadcast following every 1 d fraction of the file. All frames have an offset to the beginning of the next index segment. The first frames of each index segment has a tuple, with the first field as the attribute value of the record that was broadcast last and the second field as the offset to the beginning of the next cycle. This is to guide the clients that have missed the required frames in the current cycle and have to tune to the next cycle. Notice that there is no need to replicate the entire index between successive data segments, the distributed indexing techniques was developed, it interleaves and replicates index tree with data, in the sence that most frequently access index part (the upper level of the index tree) is replicated the number of times equal to the number of children. The project in [IVB94b] discussed the hashing schemes and a flexible indexing method for organizing broadcast cycle. In the hashing schemes, instead of broadcasting a separate directory with the information frames, the hashing parameters are included in the frames. Each frame has two parts: the data part and the control part. The control part is the "investment" which helps guide searches to minimize the access time and the tune-in time. It consists of a hash function and a shift function. The shift function is necessary since most often the hash function is not perfect. In such a case there can be collisions and the colliding frames are stored immediately following the frame assigned to them by the hashing function. The flexible indexing method first sorts the data in ascending (or descending) order and then divides the cycle into p segments numbered 1 through p. The first frame in each of the data segments contains a control part consisting of the control index. The control index is a binary index which , for a given key K , helps to locate the frame which contains that key. In this way, we can reduce the tune-in time. The parameter p makes the indexing method flexible since depending on its value we can either get a very good tune-in time or a very good access time. [LL96b] investigated the signature techniques for flat data broadcasting. Three signature methods, simple signature, integrated signature, and multi-level signature, were proposed and their cost models for the access time and the tune-in time were given. Based on the models, they made comparisons for the performance of different signature methods. Work in [LL96a] explored the influencies of caching signatures in the client side to the system performance. Four caching strategies were developed and the tune-in time and the access time were compared. With reasonable access time delay, all the caching strategies help in reducing the tune-in time for the two-level signature scheme. All those index methods can reduce the power consumption to some extent with a certain amount of access overhead. However, the index techniques developed previously didn't consider the characteristics of skewed access patterns. Recent work in [CYW97] developed an imbalanced index tree on broadcast data. The index tree is constructed in accordance with data access frequencies in such a way that the expected cost of index probes for data access is minimized. In contrast to [IVB96], the variant fanouts for index nodes was also exploited. Since the cost of index probes takes up small part of the overall cost, such imbalanced index tree gives only limited improvement. To reduce the overall access time, as mentioned in the introduction section, Broadcast disks [ZFA94, AAFZ95] is an efficient technique which can improve the overall access time for skewed data access patterns. In their later work, [AFZ96b] studied the opportunistic prefetching from broadcast disks by the client, [AFZ96a] considered the case when update presents in broadcast disks, and [AFZ97] studied the performance of a hybrid data delivery in broadcast disks environments, where clients can retrieve their desired data items either by monitoring broadcast channel (push-based) or by issuing explicit pull request to server (pull-based). These studies indicate that data prefetching and hybrid data delivery with caching can significantly improve performance over pure pull-based caching and pure push-based caching. While updates have no great influence on the system performance. [HV97] further developed an O(log(n)) time-complexity scheduling algorithm which can determines the broadcast frequency of each data item according to data access patterns for both single and multiple broadcast channels. In their models, the length of data items is not necessarily of the same. However, no study explores index on broadcast disks. 7 Conclusion and Future Work In a mobile environment, power conservation of the mobile clients is a critical issue to be addressed. An efficient power conservative indexing method should introduce low access time overhead, consume low tune-in time, and produce high indexing efficiency. Moreover, an ideal index method should perform well under both clustered and non-clustered broadcast cycle, with different broadcast scheduling policies, such as flat broadcast and broadcast disks. In this paper, we evaluate the performance of power conservative indexing methods based on index tree and signature techniques. Combining strengths of the signature and the index tree techniques, a hybrid indexing method is developed in this paper. This method has the advantages of both the index tree method and the signature method and has a better performance than the index tree method. A variant of the hybrid indexing method has been demonstrated to be the best choice for multiple attributes indexing organization in wireless broadcast environments [HLL98b]. Our evaluation of the indexing methods takes into consideration the clustering and scheduling factors which may be employeed in wireless data broadcast. Access time, tune-in time, and indexing efficiency are the evaluation criteria for our comparisons. We develop cost models for access time and tune-in time of the three indexing methods and produce numerical comparisons under various broadcast organization based on the formulae. Through our comparisons for both clustered and non-clustered data broadcast cycles, we find that the index tree method has low tune-in time only for the clustered broadcast cycles or the non-clustered broadcast cycles with low scattering factor. The index tree always produces high access time overhead. For a broadcast cycle with high scattering factor, the signature method is the best choice. Since the signature method needs further filtering to determine whether a data item really satisfies a query, the tune-in time for signature methods may be high. However, the variations of data organization for the broadcast channels have very limited impact on performance of the signature method. Moreover, the access time overhead is low. The hybrid method has the advantages of both the index tree method and the signature method. It performs well for clustered broadcast cycles or non-clustered cycles with low scattering factor (i.e., low tune-in time similar to the index tree method and low access time overhead similar to the signature method). If we only consider the indexing efficiency, the signature method has the best performance for various broadcast organization. Finally, through our comparisons for flat broadcast and broadcast disks, we observe that broadcast disks can reduce the access time for any index methods and the tune-in time for the signature and non-index methods. As a related study [HLL98b], we have studied the performance of multi-attribute index methods for wireless broadcast channels. Since the access time and the tune-in time of the index methods may be different for queries based on different attributes, We have estimated the average access time and tune-in time of the client according to the queries arrival rate for each attribute. In the future, we plan to incorporate the index schemes with data caching algorithms to achieve an improved system performance and obtain a better understanding of the wireless broadcast systems. Appendix Broadcast disks were proposed to improve data access efficiency [AAFZ95]. The idea is to divide data frames to be broadcast into broadcast disks based on their access frequency and then interleave data frames on these disks into an information stream for broadcast. This imitates multiple disks each spinning at a different speed. The relative speeds of disks are differentiated by the number of broadcast units 11 on the disks. Data located on a disk with less broadcast units is scheduled for broadcast more frequently than a disk with many broadcast units. The relative speeds and broadcast frequency of broadcast disks inversely proportional to the number of broadcast units on those disks. Thus, data frames with higher demands usually are placed on a higher speed broadcast disk. The broadcast units on broadcast disks, called chunks, have equal size 12 . The broadcast schedule is generated by broadcasting a chunk from each disk and cycling through all the chunks sequentially over all the disks. A minor cycle is defined as a sub-cycle consisting of one chunk from each disk. Consequently, chunks in a minor cycle are repeated only once and the number of minor cycles in the broadcast equals the least common multiple (LCM) of the relative frequency. Unlike traditional disks where the number and the capacity of the disks are fixed by hardware, The broadcast disks has flexibility in deciding the number, the size, the relative spinning speed, and the data frame placement of broadcast unit may consist of one or many data frames, e.g., a cluster of data frames with the same attribute value. In real implementation, chunks can be replaced by variable-sized data frames or a group of data frames. each disk. Broadcast schedules can be programmed once the data frames, relative speed of each disk, the number of data frames placed on the disks, and the size of each disk are determined. c Chunks Data Set Fast Disks Slow COLD a a a b f A Broadcast Cycle a g Minor Cycle b d a c e a b f a c Figure 13. An Example of a Seven-page, Three-disk Broadcast Program Figure 13 illustrates an example where seven chunks are divided into three ranges of similar average access probabilities [AAFZ95]. Each of which will be assigned to a separate disk in the broadcast. In the figure, chunk, refers to the j th chunk of disk i. Chunks in the first disk are to be broadcast twice as frequently as chunks in the second one and four times as often as those of the slowest disk. However, the reason that multi-disk broadcast can achieve better performance than a random broadcast schedule and the expected access time for retrieving data frames from the broadcast disks were not given in [AAFZ95]. In the following, we prove Theorem 1 used in the paper. PROOF: The data frame d is scheduled to broadcast f i times in a cycle, the length of the broadcast (in number of frames) is D: Assuming the inter-arrival time for each broadcast of d is D is the number of frames between two consecutive copies of d. k is the number of frames between the frame where clients begin monitoring channels and the next copy of d. Therefore, the expected access time for d can be estimated as:D \Delta (D will have a minimum value. Note that D is the number of frames from one broadcast of d to the next broadcast of d. If all the f i broadcasts of d are equally spaced, then we have As a result, the minimum expected number of data frames retrieved before the desired one arrives for data on disk with frequency f i , denoted as C , can be expressed as follows. --R Broadcast disks: Data management for asymmetric communications environments. Dissemination updates on broadcast disks. Prefetching from a broadcast disk. Balancing push and pull for data broadcast. Sleepers and workaholics: Caching strategies for mobile environments. Indexed sequential data broadcasting in wireless mobile comput- ing A comparison of indexing methods for data broadcast on the air. Optimal channel allocation for data dissemination in mobile computing environments. Efficient algorithms for a scheduling single and multiple channel data broadcast. Energy efficiency indexing on air. Power efficiency filtering of data on air. Data on the air - organization and access On signature caching of wireless broadcast and filtering services. Using signature techniques for information filtering in wireless and mobile environments. Adaptive data broadcast in hybrid networks. Scheduling data broadcast in asymmetric communication environ- ments Are disks in the air' just pie in the sky? --TR Power efficient filtering of data on air Sleepers and workaholics Energy efficient indexing on air Broadcast disks Balancing push and pull for data broadcast Efficient indexing for broadcast based wireless systems Signature caching techniques for information filtering in mobile enviroments A study on channel allocation for data dissemination in mobile computing environments Data on Air Prefetching from Broadcast Disks Disseminating Updates on Broadcast Disks Adaptive Data Broadcast in Hybrid Networks A Comparision of Indexing Methods for Data Broadcast on the Air Indexed Sequential Data Broadcasting in Wireless Mobile Computing Optimal Channel Allocation for Data Dissemination in Mobile Computing Environments Efficient Algorithms for Scheduling Single and Multiple Channel Data Broadcast Scheduling Data Broadcast in Asymmetric Communication Environments --CTR Quinglong Hu , Wang-Chien Lee , Dik Lun Lee, Indexing techniques for wireless data broadcast under data clustering and scheduling, Proceedings of the eighth international conference on Information and knowledge management, p.351-358, November 02-06, 1999, Kansas City, Missouri, United States Qingzhao Tan , Wang-Chien Lee , Baihua Zheng , Peng Liu , Dik Lun Lee, Balancing performance and confidentiality in air index, Proceedings of the 14th ACM international conference on Information and knowledge management, October 31-November 05, 2005, Bremen, Germany Jianting Zhang , Le Gruenwald, Prioritized sequencing for efficient query on broadcast geographical information in mobile-computing, Proceedings of the 10th ACM international symposium on Advances in geographic information systems, November 08-09, 2002, McLean, Virginia, USA Qinglong Hu , Dik Lun Lee , Wang-Chien Lee, Performance evaluation of a wireless hierarchical data dissemination system, Proceedings of the 5th annual ACM/IEEE international conference on Mobile computing and networking, p.163-173, August 15-19, 1999, Seattle, Washington, United States Jianliang Xu , Wang-Chien Lee , Xueyan Tang, Exponential index: a parameterized distributed indexing scheme for data on air, Proceedings of the 2nd international conference on Mobile systems, applications, and services, June 06-09, 2004, Boston, MA, USA KwangJin Park , MoonBae Song , Chong-Sun Hwang, Adaptive data dissemination schemes for location-aware mobile services, Journal of Systems and Software, v.79 n.5, p.674-688, May 2006 Jianting Zhang , Le Gruenwald, Efficient placement of geographical data over broadcast channel for spatial range query under quadratic cost model, Proceedings of the 3rd ACM international workshop on Data engineering for wireless and mobile access, September 19-19, 2003, San Diego, CA, USA Kwang-Jin Park , Moon-Bae Song , Chong-Sun Hwang, Broadcast-based spatial queries, Journal of Computer Science and Technology, v.20 n.6, p.811-821, November 2005 Wen-Chih Peng , Ming-Syan Chen, Efficient channel allocation tree generation for data broadcasting in a mobile computing environment, Wireless Networks, v.9 n.2, p.117-129, March Jianliang Xu , Dik-Lun Lee , Qinglong Hu , Wang-Chien Lee, Data broadcast, Handbook of wireless networks and mobile computing, John Wiley & Sons, Inc., New York, NY, 2002 Sunil Prabhakar , Yuni Xia , Dmitri V. Kalashnikov , Walid G. Aref , Susanne E. Hambrusch, Query Indexing and Velocity Constrained Indexing: Scalable Techniques for Continuous Queries on Moving Objects, IEEE Transactions on Computers, v.51 n.10, p.1124-1140, October 2002 Chi-Yin Chow , Hong Leong , Alvin T. S. Chan, Distributed group-based cooperative caching in a mobile broadcast environment, Proceedings of the 6th international conference on Mobile data management, May 09-13, 2005, Ayia Napa, Cyprus

index technique;wireless data broadcast;power conservation;signature method

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Parallel Mining of Outliers in Large Database.

Data mining is a new, important and fast growing database application. Outlier (exception) detection is one kind of data mining, which can be applied in a variety of areas like monitoring of credit card fraud and criminal activities in electronic commerce. With the ever-increasing size and attributes (dimensions) of database, previously proposed detection methods for two dimensions are no longer applicable. The time complexity of the Nested-Loop (NL) algorithm (Knorr and Ng, in Proc. 24th VLDB, 1998) is linear to the dimensionality but quadratic to the dataset size, inducing an unacceptable cost for large dataset.A more efficient version (ENL) and its parallel version (PENL) are introduced. In theory, the improvement of performance in PENL is linear to the number of processors, as shown in a performance comparison between ENL and PENL using Bulk Synchronization Parallel (BSP) model. The great improvement is further verified by experiments on a parallel computer system IBM 9076 SP2. The results show that it is a very good choice to mine outliers in a cluster of workstations with a low-cost interconnected by a commodity communication network.

Introduction Data mining or knowledge discovery tasks can be classified into four general categories: (a) dependency detection (e.g. association rules [1]) (b) class identification (e.g. classification, data clustering [6, 14, 17]) (c) class description (e.g. concept generalization [7, 11]), and (d) excep- tion/outlier detection [12, 13]. Most research has concentrated on the first three categories while most of the existing work on outliers detection has lied in the field of statistics [2, 8]. Although The author is currently in the Department of Computer Science at the University of Maryland, College Park, but the work in this paper was done when he was at the University of Hong Kong. Hung and Cheung outliers have also been considered in some existing algorithms, they are not the main target and the algorithms only try to remove or tolerate them [6, 14, 17]. In fact, the identification of outliers can be applied in the areas of electronic commerce, credit card fraud detection, analysis of performance statistic of professional athletes [10] and even exploration of satellite or medical images [12]. For example, in a database of transactions containing sales information, most transactions would involve a small amount of money and items. Thus a typical fault detection can discover exceptions in the amount of money spent, type of items purchased, time and location. As a second example, satellite nowadays can be used to take images on the earth using visible lights as well as electromagnetic waves to detect targets such as potential oil fields or suspicious military bases. Detection of exceptional high energy or temperature or reflection of certain electromagnetic waves can be used to locate possible targets. A simple algorithm called Nested-Loop algorithm (NL) has been proposed in [13], which, however, has a complexity of O(kN 2 ), (k is the number of dimensions and N is the number of data objects), and the number of passes over the dataset is linear to N. By real implementation and performance studies, we find that the major cost is from the calculation of distances between objects. Though NL is a good choice when the dataset has high dimensionality, the large number of calculations makes it unfavorable. A cell-based algorithm has been proposed in [13]. It needs only at most three dataset passes. However, it is not suitable to high dimensions because its time complexity is exponential to the number of dimensions. NL always outperforms the cell-based algorithm when there are more than four dimensions [13]. In this paper, we will improve NL for high dimensional dataset, which is very common in data warehouse. One approach to improve the NL algorithm is to parallelize it. In this paper, the definition of outliers and the original NL is described in the next few subsections in this introduction. NL is improved to reduce the number of calculations. The resulted algorithm, ENL, is given in Section 2. In Section 3, ENL is parallelized to further reduce the execution time in a shared-nothing system. In Section 4, the performance and improvement are analyzed by using Bulk Synchronization Parallel (BSP) model. After that, performance studies are given in Section 5. Finally, we give a discussion and related works in Section 6. This paper mainly focuses on identification of distance-based outliers, although our parallel algorithm can also be modified to perform the most expensive step of finding density-based outliers [4]. Related works on density-based outliers are described in Section 6. 1.1 Distance-Based Outliers As given in [13], the definition of an outlier is as the following: Given parameter p and D, an object O in a dataset T is a DB(p; D)-outlier if at least fraction p of the objects in T lies greater than distance D from O. From the definition, the maximum number of objects within distance D of an outlier O is is the number of objects. Let F be the underlying distance function that gives the distance between any pair of objects in T . Then, for an object O, the D- Parallel Mining of Outliers in Large Database 3 neighbourhood of O contains the set of objects Q 2 T that are within distance D of O (i.e. This notion of outliers is suitable for any situation in which the observed distribution does not fit any standard distribution. Readers are referred to [12] for the generalization of the notions of distance-based outliers supported by statistical tests for standard distributions. Works on density-based outliers are described in Section 6. 1.2 Assumption and Notation in NL Algorithm Algorithm NL [13] is a block-oriented, nested-loop design. Here a block can involve one or more disk I/O, i.e. it may be necessary to take one or more disk I/O to read in a block. In this paper, a page needs one disk I/O access only. Thus, the total number of pages in a particular dataset is constant, but the total number of blocks in that dataset will change with the size of a block. NL algorithm is designed for a uniprocessor system with one local memory and one local disk. The effect of cache is insignificant here. We make some assumptions here that are generally acceptable in real systems: there is no additional disk buffer from the operating system beside the buffer we will use in the algorithm; the disk access is sequential. Let n be the number of blocks in the dataset T , k be the dimensionality, N be the number of objects in the dataset, P be the number of pages contained in a block, so the number of objects in a page is let the time of accessing a page from the disk be t I=O , and the time of computing the distance between 2 objects be t comp , which is linear to the dimensionality. 1.3 Original NL The original NL algorithm from [13] with some clarification is described here. Assume that the buffer size (for storing dataset) is B% of the dataset size. Then the buffer is first divided into 2 equal halves, called the first and second arrays. The dataset is read into the first or second array in a predefined order. For each object t in the first array, distance between t and all the other objects in the arrays are directly computed. A count of objects in its D-neighbourhood is maintained. Algorithm NL 1. fill the first array (of size B% of dataset) with a block of objects from T 2. for each object t in the first array, do: (a) count the number of objects in first array which are close to t (distance - D); if the count as a non-outlier, where 3. repeat until all blocks are compared to first array, do: (a) fill the second array with another block (but save a block which has never served as the first array, for last) 4 Hung and Cheung (b) for each unmarked object t in the first array, do: i. increase its count by the number of objects in second array close to t (distance - D); if the count ? M , mark t as a non-outlier 4. report unmarked objects in the first array as outliers 5. if second array has served as the first array before, stop; otherwise, swap the names of first and second arrays and repeat the above from step 2. The time complexity is stated as O(kN 2 ) in [13], where k is the dimensionality and N is the number of objects in the dataset. The disk I/O time is considered briefly only in [13]. In fact, both the CPU time and I/O time should be considered. The detailed analysis of the computation time and disk I/O time is given below. For computation time, the total time for calculation of distance between pairs of objects has an upper bound of N 2 t comp , i.e. the number of calculations of distance is quadratic to the number of objects in the dataset. The actual number of calculations depends on the distribution of data, the location of data in the blocks, the distance D, and the number M , which in turn depends on the fraction p. Since in usual case, the number of outliers should be small, so M is small. No more calculation for a particular object t will be done if its count exceeds M . As a result, the actual number of calculations is much less than N 2 , usually, within half, as shown in Section 5. For disk I/O time, since the dataset is divided into blocks, the total number of block reads is n+(n\Gamma2)(n\Gamma1), and the number of passes over the dataset is n . The total number of page reads is nP 1)P . It is noted that P is directly proportional to the buffer size. With a fixed buffer size, n is directly proportional to N . So the , which has a complexity of O(N 2 ). Example 1 The following is an example. Consider 50% buffering and 4 blocks of the dataset denoted as A, B, C, D, i.e. each block contains4 of the dataset. The order of filling the arrays and comparing is as the following: 1. A with A, then with B, C, D, for a total of 4 blocks reads; 2. D with D (no read required), then with A (no read), B, C, for a total of 2 blocks reads; 3. C with C, then with D, A, B, for a total of 2 blocks reads; 4. B with B, then with C, A, D, for a total of 2 blocks reads. The table below shows the order of the blocks loaded, and the blocks staying in the two arrays of the buffer. Each row shows a snapshot after step 1 or step 3 (a). Parallel Mining of Outliers in Large Database 5 Block Buffer disk I/O order A B C D Array 1 Array 2 6 L * D C 9 buffer (unchanged) The total number of disk I/O is n blocks. The total number of dataset passes is4 2:5. Enhanced NL In NL, there are redundant block reading and comparison. In this section, a new order is proposed which results in reduced computation time and disk I/O time. The arrangement is that: in each turn, the blocks are read into the second array in a predefined order until the end of the series of ready blocks is reached, then the block in the first array is marked as done, the names of the two arrays are swapped and the order is reversed. The above is repeated until all blocks are done. The resultant is Enhanced NL (ENL) Algorithm, which is described below. For each object in the dataset, there is a count. Algorithm ENL 1. label all blocks as "ready" (the block is either in a "ready" or a "done" state) 2. fill the first array (of size B% of dataset) with a block of objects from T 3. for each object t in the first array, do: (a) increase the count by the number of objects in first array which are close to t (distance - D); if the count ? M , mark t as a non-outlier 4. set the block-reading order as "forward" 5. repeat until all "ready" blocks (without marked as done) are compared to first array in the specified block-reading order, do: (a) fill the second array with next block (b) for each object t i in the first array, do: i. for each object t j in the second array, if object t i or t j is unmarked, then if 6 Hung and Cheung A. increase count i and count j by 1; if count i ? M , mark t i as a non-outlier, proceed to next t as a non-outlier 6. report unmarked objects in the first array as outliers 7. if second array is marked as done, stop; otherwise, mark the block in the first array as done, reverse the block-reading order, swap the names of first and second arrays and repeat the above from step 3. Although the time complexity of ENL is still O(kN 2 ), where k is the dimensionality and N is the number of objects in the dataset, the cost of computation and disk I/O are both reduced compared with those of NL. For computation time, the upper bound of the total time for calculation of distance between pairs of objects which is still linear to the dimensionality. In original NL algorithm, only the count of the object in first array is updated. However, in ENL, the counts of both objects are updated for each comparison. Thus, the upper bound of the number of calculations of distances is reduced to almost half (compared with NL). For the actual reduction of the number of calculations, once again, it depends on the distribution of data, the locations of data in the blocks, the distance D, and the number M , which in turn depends on the fraction p. The comparions of performance of NL and ENL is shown in Section 5. For simplicity, the upper bound of number of distance calculations between the objects in the same block was said to be (NP P ) 2 but in fact it only needs (NP P )(NP times. For disk I/O time, if the dataset is divided into blocks, then the total number of block reads is and the number of passes over the dataset is n. The total number of page reads is With a fixed buffer size, n is directly proportional to N . So the disk I/O time , which has a complexity of nearly half of that of NL. Example 2 Example 1 is extended here to illustrate ENL. Consider 50% buffering and 4 blocks of the dataset denoted as A, B, C, D, i.e. each block contains4 of the dataset. The order of filling the arrays and comparing is as the following: 1. A with A, then with B, C, D, for a total of 4 blocks reads; 2. D with D (no read required), then with C, B for a total of 2 blocks reads; Parallel Mining of Outliers in Large Database 7 3. B with B, then with C, for a total of 1 blocks reads; 4. C with C, for a total of 0 blocks reads. The table below shows the order of the blocks loaded, and the blocks staying in the two arrays of the buffer. Each row shows a snapshot after step 2 or step 5 (a). Block Buffer disk I/O order A B C D Array 1 Array 2 in buffer (unchanged) The total number of disk I/O is 1 blocks. The total number of dataset passes is4 1:75. 3 Parallel ENL PENL (Parallel ENL) is a parallel version of ENL, running in a shared-nothing system. Actually, when running in a processor, PENL is almost reduced to ENL. Most of block reading in ENL is replaced by transfer of blocks among processors through the communication network. Its major advantage is distributing the costly computations nearly evenly among all processors. 3.1 Assumptions and Notation To extend the ENL to PENL, the assumptions and notations are extended. For simplicity, it is assumed that in the shared-nothing system, each node has only one processor. Each node has its own memory and local disk. The dataset is distributed equally in size to the local disk of each node without overlapping. Communication is done by message passing. The network architecture is designed such that each node can send message and receive message at the same time. (For simplicity of analysis in later section, we make the previous assumption. This requirement is not strict. At least it is required that internode communication is possible among nodes.) Besides, the nodes are arranged in a logical ring so that each node has two neighbour nodes. Logical arrangement means that physically the network can be in other architecture, e.g. bus, which does not affect the effectiveness of our algorithm, but only the performance. Let n be the number of blocks in the dataset, nP be the total number of pages of the dataset, k be the dimensionality, N be the number of objects in the dataset, P be the number of pages contained in a block, so the number of objects in a page is , and the number 8 Hung and Cheung of objects in a block is N . Let the time of computing the distance between 2 objects be t comp , the time of internode communication between 2 nodes to transfer a page of data be t comm , and the time of accessing a page from the local disk be t I=O . Let p be the number of processors or nodes, m be the size of local memory used for disk buffer. To simplify the algorithm and analysis, it is assumed that the local memory buffers are all the same size for all nodes, and so are the data in local disk; and the number of data blocks in a local disk is an integer, i.e. number of pages in dataset (nP ) is a multiple of the product of the number of processors and the number of pages contained in a block (pP ). 3.2 Algorithm Each node has part of the dataset in its local disk. The number of pages of local data is nP The number of objects in local data is N . Each node has a local memory of size m, which is divided into 3 arrays. Each array can contain a block with the size of P pages. The first and second arrays function similarly to those in ENL, while the third array is used as a temp buffer to store data received from a neighboring node. Besides, a count of objects in D-neighbors for each object is maintained. PENL is modified from ENL. The basic principle is: in a node, each time after a block is read from the node's local disk and the distance calculations are all done, then the block is transferred to the node's neighbor node and distance calculations are done using the block received from the node's another neighbour node. This is repeated until the block has passed all neighbors, then the node reads another block from the local disk and repeats the above again. Most of disk I/O operations are replaced by relatively fast internode communication. The huge number of calculations are now distributed by all nodes, which greatly reduces the execution time, i.e. the response time. Algorithm PENL(node id x) 1. label all blocks as "ready" (the block is either in a "ready" or a "done" state) 2. fill the first array with a block of objects 3. set the block-reading order as "forward" 4. set counter b to 0, set counter s to 0 5. repeat until all "ready" blocks are compared to first array in the specified block-reading order, do: (a) set counter c to 0 for each object t in the first array, do: i. increase the count by the number of objects in first array which are close to t (distance - D); if the count ? M , mark t as a non-outlier (c) if set b to 1 and go to step (f) Parallel Mining of Outliers in Large Database 9 (d) if b 6= 0, fill the second array with the next block for each object t i in the first array, do: i. for each object t j in the second array, if object t i or t j is unmarked, then if dist(t A. if increase count i by 1, otherwise, increase count i and count j by 1; if count as a non-outlier reverse the order of execution of steps (g) and (h) send the data in the first array to the neighbor node send the data in the second array to the neighbor node receive the data from the neighbor node store it in the temp buffer (third array) (i) increment counter c by 1, swap the names of the second array and third array; if counter continue the iteration in step 5, otherwise go to step 5(e) 6. if second array is marked as done, report unmarked objects in the first array as outliers; otherwise, mark the block in the first array as done, reverse the block-reading order, swap the names of first and second arrays and repeat the above from step 4. The time analysis of PENL is given below briefly. Later a detailed analysis will be given using the Bulk Synchronous Parallel (BSP) model. For each node, the operations are similar to ENL, except that each block is transferred to other nodes after computations are done on it. Thus, in brief, the upper bound of computation time in each node is@ . Each node has n blocks of local data. The local disk I/O time is the same as that of ENL executing with n blocks of data, so the disk I/O time for each node is For the internode communication time in a node, it is It is obvious that the upper bound of the computation time is linear to the reciprocal of the number of processors, the internode communication time decreases with increasing number of Hung and Cheung processors, the disk I/O time is quadratic to the reciprocal of the number of processors. Please note that P changes with the size of buffer. If the total size of all local memory is fixed, i.e. pP is a constant, then the internode communication time in each node only varies with the dataset size, but not the number of processors, while the computation time and local disk I/O time in each node is linear to the reciprocal of the number of processors. Although only the upper bound of the computation time is shown, actually the calculations are distributed quite evenly among all the nodes, and so it is nearly linear to the reciprocal of the number of processors, as shown in the performance studies in Section 5. 3.3 Example The following gives an example of execution of PENL on a dataset of 16 blocks using four nodes. Each node has four local blocks because the 16 blocks are evenly distributed in the four nodes. The four local blocks of node x are denoted as Ax ; Bx ; Cx ; Dx . The order of filling the arrays by disk I/O or internode communication and comparing in node 0 is as the following: 1. A0 with A0 (I/O), (3 communications), for a total of 4 blocks reads and 12 communications; 2. D0 with D0 (no read required), nications), B0 (I/O), B1 ; B2 ; B3 (3 communications), for a total of 2 blocks reads and 9 communications 3. B0 with B0 (no read required), B1 nications) for a total of 1 blocks reads and 6 communications; 4. C0 with C0 (no read required), C1 ; C2 ; C3 (3 communications) for a total of 0 blocks reads and 3 communications. For each node, the number of disk I/O is 1 blocks. The number of dataset passes is4 1:75; the internode communication is 3(4 times. The table in Appendix A shows the details: the order of the blocks loaded in node 0 , the blocks transferred to and from node 0, and the blocks staying in the two arrays of the buffer of node 0. If we run ENL using a single node with the same amount of memory as that in a node in PENL, then the total number of disk I/O will be blocks. The total number of dataset passes is 121 7:5625. The ratio of disk I/O in ENL to that in PENL using 4 nodes = 7:5625 4:32. The improvement is very significant. However, if we give the ENL the amount of memory same as the total sum of memory of all nodes, then we do not gain any benefit in the disk I/O because the size of a block is larger now, and so the total number of blocks will be much less than 16. Nevertheless, we still have a significant improvement in performance because the computation time, which is the major cost, is nearly evenly distributed by other nodes in PENL, as shown in Section 5. Parallel Mining of Outliers in Large Database 11 3.4 Optimization We can do the following optimization on PENL. For the outer iterations in step 5 (5(a) to 5(i)) with 1, we can only do approximately the first half of all inner iterations (5(e) to 5(i)) to reduce redundancy of block transmission and computation. For the example in Section 3.3, the inner iterations for the computations of blocks A 0 and A 3 , D 0 and D 3 and C 3 are skipped. Then the upper bound of the cost of computation of that outer iteration can be reduced from p(NP P '- Therefore the upper bound of the total computation cost is reduced from Therefore the computation-cost-reduction-ratio (the ratio of reduction of upper bound of computation cost by the optimization of PENL (p\Gamma1)k2 !n is the number of blocks in each node. For the example in Section 3.3, so the reduction is 0:1. In practice, we have a large number of local blocks, so the reduction will not be significant. In this paper, we will refer to the original PENL algorithm, unless specified, for the simplicity of implementation and analysis of the parallel algorithm Theoretical Analysis using BSP Model Before studying the performance of real execution of the algorithms, the theoretical analysis is given here. The BSP (Bulk Synchronous Parallel) model [3] is used to analyze the PENL algorithm because the hardware and software characteristics of the model match with PENL's platform requirement and working principle. A BSP computer consists of a set of processor-memory pairs, a communication network that delivers messages in a point-to-point manner, and a mechanism for the efficient barrier synchronization of the processors. The BSP computer is a two-level memory model, i.e. each processor has its own physically local memory module and all other memory is non-local and accessible in a uniformly efficient way. PENL requires each node to have a local memory buffer. The accesses of other blocks in the buffers of other nodes are done by synchronous communication. Block transfers are done in a node-to-node manner. The BSP computer operates in the following way. A computation consists of a sequence of parallel supersteps, where each superstep is a sequence of steps, followed by a barrier synchronization at which point any non-local memory accesses take effect. PENL requires a barrier synchronization for block transfers. 12 Hung and Cheung During a superstep, each processor has to carry out a set of programs or threads, and it can do the following: (i) perform a number of computation steps, from its set of threads, on values held locally at the start of the superstep; (ii) send and receive a number of messages corresponding to non-local read and write requests. During each superstep, PENL performs computations of items in one or two blocks, accesses disk for loading a new block, and executes block transmissions. As a simple model that bridges hardware and software, BSP model provides portability across diverse platforms with predictable efficiency. It can be seen that the model is very suitable for PENL because PENL has coarse granularity and each superstep consists of a lot of distance calculations followed by message passing. 4.1 Cost Analysis Define the following variables: L: barrier, synchronization cost d: ratio of the time (cost) of local disk I/O accessing an object i:e: time of local disk I=O accessing a page to the time of computation on a distance between two objects g: ratio of the time of internode communication of transferring an object i:e: time of internode communication of transferring a page to the time of computation on a distance between two objects Costs and barriers are added in the PENL algorithm, as shown below: Algorithm PENL(node id x) 1. label all blocks as "ready" (the block is either in a "ready" or a "done" state) 2. fill the first array with a block of objects pages in a block, NP objects in a page) 3. set the block-reading order as "forward" 4. set counter b to 0 5. repeat until all "ready" blocks are compared to first array in the specified block-reading order, do: (a) set counter c to 0, set counter s to 0 for each object t in the first array, do: i. increase the count by the number of objects in first array which are close to t (distance - D); if the count ? M , mark t as a non-outlier Parallel Mining of Outliers in Large Database 13 (c) if set b to 1 and go to step (f) (d) if b 6= 0, fill the second array with the next block, for each object t i in the first array, do: i. for each object t j in the second array, if object t i or t j is unmarked, then if dist(t A. if increase count i by 1, otherwise, increase count i and count j by 1; if count as a non-outlier (f) set a barrier, if reverse the order of execution of steps (g) and (h) send the data in the first array to the neighbor node send the data in the second array to the neighbor node receive the data from the neighbor node store it in the temp buffer (third array) (i) set a barrier, increment counter c by 1, swap the names of the second array and third continue the iteration in step 5, otherwise go to step 6. if second array is marked as done, report unmarked objects in the first array as outliers; otherwise, mark the block in the first array as done, reverse the block-reading order, swap the names of first and second arrays and repeat the above from step 4. Therefore the total costs of the algorithm is The derivation can be found in Appendix B. The first term is computation, the second one is disk I/O, the third one is communication, the last one is synchronization. Please notice that the time of computation is only a upper bound. Therefore this theoretical analysis does not give a reliable value of the actual execution time, but it still acts as a good reference for the comparison with ENL later. When the block size, the page size and object size are constant, then it can be found that, if the dataset size is large (N AE pNP , i.e. local block number is large), ffl the computation cost is quadratic to the dataset size and linear to the reciprocal of the number of processors; ffl the disk I/O cost is quadratic to the dataset size and quadratic to the reciprocal of the number of processors; 14 Hung and Cheung ffl the communication cost is quadratic to the dataset size and linear to the reciprocal of the number of processors; ffl the synchronization cost is quadratic to the dataset size and linear to the reciprocal of the number of processors. Please note that P changes with the size of buffer. If the total size of all local memories is fixed, i.e. pP is a constant, then all costs are still quadratic to the dataset size if the dataset size is large (N AE pNP , i.e. local block number is large); besides, ffl the computation cost is still linear to the reciprocal of the number of processors; ffl the disk I/O cost is now linear to the reciprocal of the number of processors; ffl the synchronization cost is now linear to the number of processors. The above analysis tells us that when the total memory is fixed, then it is still beneficial to increase the number of processors as it is shown that the major cost, the computation, is linear to the reciprocal of the number of processors. On the other hand, if the number of processors is kept unchanged, but the buffer size in each node (i.e. block size P ) varies, then the computation cost is linear proportional to the number of pages in a block (P ). Thus, with smaller block size, fewer computations are necessary. Besides, the local block number increases, which makes the computation-cost-reduction-ratio of the optimization of the algorithm (in Section 3.4) become smaller. However, it is not recommended to use a too small buffer because when P is too small, then N AE pNP P and the effect of reduction of computation cost by small P will be very small. Besides, smaller block size also increases the cost of disk I/O, communication and synchronization. 4.2 Comparision of PENL with ENL What about comparing PENL with ENL when both are given the same amount of memory? For ENL, the corresponding cost is: ii is the number of pages in a block in ENL. In ENL, the buffer is divided into two arrays, so the total amount of memory is 2P 1 . In PENL, each local buffer is divided into three arrays, so the total amount of memory is 3pP . Giving the same amount of memory to PENL and ENL, . Better implementation can be made so that it is sufficient to divide each local buffer into two arrays. However, here and in later sections, we will still choose three arrays in order to give advantage to the sequential algorithms for comparisons but we can still show that our parallel algorithm outperforms them. Parallel Mining of Outliers in Large Database 15 Assume that, in a worse case, the ratio of number of calculations actually done in ENL to the total sum of number of calculations actually done in all nodes in PENL is 2. Let f be the fraction of number of calculations actually done in ENL, so the fraction of that of PENL is 2f . In ENL, the computation cost is )In PENL, the computation cost is CPENL;comp is linear to the number of processors. Thus it is always a better choice to use PENL even if the total buffer size is fixed. Performance Studies 5.1 Experimental Setup and Implementation In our experiments, our base dataset is an 248-object dataset consisting of trade index numbers of HKSAR from 1992 to 1999 March [5]. Each object is from one of the four categories: imports, domestic exports, re-exports, and total exports. The four categories have equal number of objects. Each object has six attributes: index value, year-on-change percentage change of index value, index unit value, year-on-change percentage change of index unit value, index quantum, year-on-change percentage change of index quantum. Since this real-life dataset is quite small, and we want to test our algorithms on a large, disk-resident dataset, we generate a large number of objects simulating the distribution of the orginal dataset. In our testing, the distance D is defined so that the number of outliers is restricted to within a few percents of all objects to simulate the real situation. The programs were run in an IBM 9076 SP2 system installed in the University of Hong Kong. The system consists of three frames. Each frame consists of 16 160 MHz IBM P2SC RISC processors. Individual node has its own local RAM (128 MB or 256 MB) and local disk storage (2 Gb, for system files and local scratch spaces). Each node in a frame is interconnected by high performance switches and the three frames are also linked up by an inter-frame high performance switch. The theroretical peak performance for each processor is 640 MFLOPS. In our tests, the sequential programs and parallel programs were run in dedicated mode using loadleveler (the batch job scheduler). In Section 5.2, NL and ENL were run in another system because the SP2 system has a time limit of 10 hours on running. It is a Sun Enterprise Ultra 450 with 4 UltraSPARC-II CPU running at 250MHz, with 1GB RAM and four 4.1GB hard disks. In SP2, in order to make the comparison fair, we will fix the total amount of memory of all nodes for PENL to be the same as that for NL and ENL, which is able to hold 75000 objects. The number of objects is chosen so that the number of blocks in our test can be more reasonable. Hung and Cheung As a result, the number of objects in a block (NP P ) for NL, PENL with 2, 4, 8, processors is 37500, 12500, 6250, 3125, 1563 respectively. The number of objects is 50000, 100000, 200000, 400000, 800000, which implies that the number of blocks is 2, 4, 8, 16, 32 for PENL and 2, 3, 6, 11, 22 for NL. We have implemented the three algorithms, NL, ENL and PENL using C. MPI (Message- Passing Interface) library was used in PENL for message passing among multiple processors [15]. For PENL, better implementation can be made so that it is sufficient to divide each local buffer into two arrays, rather than three arrays. However, we still chose three arrays for simplicity in order to give advantage to the sequential algorithms for comparisons but we can still show that our parallel algorithm PENL outperforms them. It should be noted that for PENL, in each node, a part of memory is needed to act as counts for all objects. The decrease in memory to act as counts will decrease P (the number of pages in a block) and increase the total cost. However, the addition of the cost is small compared with the improvement from NL. It is because an object in a database usually contains tens or even hundreds of attributes, which may be integers, floating points or even strings. The size of a count is very small compared with the size of an object, so P will only decrease a bit. When the total number of objects are very huge, it is undesirable to hold all counts in the memory, then the counts can be stored in local disks. The total size of the counts is very small compared with the size of datasets. Thus, the extra disk I/O time accessing the counts affects the performance a bit only. In our implementation of ENL and PENL, we have the counts resident in disk and we load them only if they are required. This method is good, but it induces extra disk I/O In our experiments, we decide to define an object to have six dimensions (long integer data type), in order to make the effect of reading and writing the counts more significant. However, our results show that the effect is very minor, compared with the reduction of computation cost. Besides, it needs extra communication to transfer the counts of objects to the node containing the objects in its disk, so that the outliers can be reported as soon as possible. The better way is that, in the end, the counts are gathered to a node and then the outliers are reported from combining the counts. The extra communication cost is little compared with the computation cost. We chose the second method in our implementation. The final point to note is the computer architecture. Each processor has its own cache, so more the processors, the larger is the total cache capacity. Thus the hit ratio can be larger and the performance is further enhanced. Besides, with existing workstations, a cluster can be formed on them with low cost to perform PENL, rather than installing a new advanced but costly supercomputer. Although our experiments were conducted in a supercomputer, our results show that the commuication cost is very minor. Thus, communication network of low cost is sufficient. 5.2 NL vs ENL In this section we will compare the performance of NL and ENL. Parallel Mining of Outliers in Large Database 17 object number 2000 8000 execution time computation time disk I/O time calculation number 974467 618697 23095637 17763925 ENL execution time NL execution time 0.7459 0.8487 Figure 1: Table of comparisons of NL and ENL object number 32000 128000 execution time computation time disk I/O time calculation number ENL execution time NL execution time 0.8636 0.8743 Figure 2: Table of comparisons of NL and ENL (cont.) From Figure 3 we can see that ENL is better than NL, although the improvement is not very great. Moreover, from the tables in Figure 1 and 2, it is found that the major cost is from the computation of distance, which can be greatly reduced in PENL as we will show later. Besides, we can see that the increase in execution time is approximately quadratic to the increase of objects, i.e. the execution time complexity is O(N 2 ), indicating that it is very unlikely to use NL or ENL to deal with large number of objects. However, PENL can help to reduce the time. 5.3 Sequential vs Parallel Here we will compare the performance of sequential program NL and parallel program PENL. Figure 4 and 5 show that PENL with various number of processors outperforms NL, whatever the number of objects is. Even the number of processor is two only, the performance is improved by more than 100 percents. As we have said, the total amount of memory given to PENL is the same as that to NL, so it is very clear that PENL is always a better choice when a multiprocessor system or a cluster of workstations is available. The result of NL with 400000 and 800000 objects and that of PENL with 2 processors and 800000 objects are not available because the execution time exceeds the time limit of a job in the SP2 system. Hung and Cheung0.750.852000 8000 32000 128000 Object number Figure 3: Comparison of execution time of NL and ENL processor number number of objects 50000 100000 200000 400000 800000 Figure 4: Table of comparisons of execution time of NL and PENL in seconds100100001(NL) 2 4 8 Processor number Execution time (sec) 50000 objects 333100000 objects 200000 objects 222400000 objects \Theta \Theta \Theta \Theta \Theta 800000 objects 44 Figure 5: Execution time against number of processors Parallel Mining of Outliers in Large Database 19 processor number execution CPU I/O communication synchronization Figure Table of comparisons of different costs in NL and PENL with 100000 objects in seconds 5.4 Variation of processor number In this section we will see how the performance of PENL is related to the number of processors. From Figure 4 and 5, we can see that the nearly straight lines are dropping steadily almost in parallel, indicating that the scalability is stable. In all cases, the execution time is almost halved when the number of processors is doubled, which is near to what our theoretical analysis predicts, i.e. the execution time is approximately linear to the reciprocal of the number of processors. Again, the increase in execution time is approximately quadratic to the number of objects, i.e. the execution time complexity is O(N 2 ). But it should be noted that the execution time is linear to the dimensionality, thus it is still preferable for those database of high dimensionality. 5.5 Comparison of computation, disk I/O, communication time, synchronization time In this section we will look more clearly into the contribution of execution time from the com- putation, disk I/O, communication and synchronization time in PENL. Figure 6 shows that over 99 percents of the execution cost comes from computation time. Since PENL distributes the computation operations among all processors nearly evenly, so the execution time can be reduced greatly. Further improvement can be considered to focus on how to reduce the computation operations. On the other hand, the disk I/O, communication and synchronization time are much more minor. Their trends are not the same as our theoretical analysis. The disk I/O time increases very slowly with the number of processors because there are more reading and writing of counts as the block size is smaller and the number of blocks loaded and transfered is larger (when a new block comes, the counts of the old block are written and the counts of the new block are read). The sum of the number of pages accessed by all processors should be close no matter how many processors are being used, but the total number of pages accessed for the counts increases with the number of processors as the counts are read and written more in times. Thus the disk time increases a bit. The communication time and synchronization time depend much on the system at the moment of execution, e.g. the bandwidth and condition of the communication network. 20 Hung and Cheung 6 Discussion and Related Works NL algorithm is a straight forward method to mine outliers in a database. ENL proposed reduces both of the computation and disk I/O costs. Furthermore, the algorithm PENL is proposed to parallelize ENL. The analysis shows that if the total buffer size in the system is fixed, then the computation cost is linear to the reciprocal of the number of processors, which is verified by our performance studies. The great improvement is caused by the nearly even distribution of computation operations among all processors. Our performance studies further indicate that over 99 percents of the execution time comes from the computation, so the execution time is also linear to the reciprocal of the number of processors. These results show that PENL is very efficient comparing with NL and ENL, and further improvement can be focused on how to reduce the computation operations. Since other costs like communication time is very minor, so a low-cost cluster of workstations with commodity processors, interconnected by a low-cost communication network can be chosen as the platform of running PENL, rather than much more expensive supercomputer. A cluster is also much cheaper and easier to build, maintain and upgrade to achieve the similar performance that NL has in a single high performance processor system. Breunig, et al. introduced a definition of a new kind of outliers (density-based outliers) and investigates its applicability [4]. Their heuristic can identify meaningful local outliers that the notion of distance-based outliers cannot find. The first step of computation of LOF (local outlier factor) is the materialization of the MinPtsUB-nearest neighborhoods (pages 102-103 of [4]). Modification can be made in our parallel algorithm to perform that step, which is also the most expensive step in computation of LOF. Instead of updating the count of objects in D-neighborhood of each object, now each node stores the temporary MinPtsUB-nearest neighborhood of each object. The final MinPtsUB-nearest neighborhood of each object is obtained by combining all the temporary MinPtsUB-nearest neighborhoods of that object calculated by all nodes. We can choose to parallelize NL algorithm instead of using PENL algorithm for simplifying the implementation and reducing the disk storage space for temporary MinPtsUB-nearest neighborhoods. In that case, each node stores the MinPtsUB-nearest neighborhoods of objects in the block that stays in the first array only. The only difference is the reading order of blocks and the increase of number of blocks I/O and computation (by amost doubling). Similarity search in high-dimensional vector space using the VA-File method outperforms other methods known [16]. Detection of outliers based on the VA-File is an approach different from the approaches of nested-loop or cell-structure. We will take that (using VA-File) into consideration in our future works. --R "Mining association rules between sets of items in large databases," Outliers in Statistical Data "Scientific Computing on Bulk Synchronous Parallel Architetures," "LOF: Identifying Density-Based Local Outliers," "Trade Index Numbers" "A density-based algorithm for discovering clusters in large spatial databases with noise," "Knowledge discovery in databases: An attribute-oriented approach," of Outliers "Parallel Algorithm for Mining Outliers in Large Database," On digital money and card technologies "Finding aggregate proximity relationships and commonalities in spatial data mining," "A unified notion of outliers: Properties and computation," "Algorithms for Mining Distance-Based Outliers in Large Datasets," "Efficient and effective clustering methods for spatial data mining," "A Quantitative Analysis and Performance Study for Similarity-Search Methods in Hifh-Dimensional Spaces," "BIRCH: An efficient data clustering method for very large databases," --TR Mining association rules between sets of items in large databases Finding Aggregate Proximity Relationships and Commonalities in Spatial Data Mining A Quantitative Analysis and Performance Study for Similarity-Search Methods in High-Dimensional Spaces Algorithms for Mining Distance-Based Outliers in Large Datasets Knowledge Discovery in Databases Efficient and Effective Clustering Methods for Spatial Data Mining A unified approach for mining outliers On Digital Money and Card Technologies

outlier detection;data mining;parallel algorithm

607278

Simple 8-Designs with Small Parameters.

We show the existence of simple 8-(31,10,93) and 8-(31,10,100) designs. For each value of we show 3 designs in full detail. The designs are constructed with a prescribed group of automorphisms using the method of Kramer and Mesner KramerMesner76. They are the first 8-designs with small parameters which are known explicitly. We do not yet know if PSL(3,5) is the full group of automorphisms of the given designs. There are altogether 138 designs with designs with PSL(3,5) as a group of automorphisms. We prove that they are all pairwise non-isomorphic. For this purpose, a brief account on the intersection numbers of these designs is given. The proof is done in two different ways. At first, a quite general group theoretic observation shows that there are no isomorphisms. In a second approach we use the block intersection types as invariants, they classify the designs completely.

Introduction In this paper, t-designs with prescribed automorphism group are constructed. The method was introduced by Kramer and Mesner in [8]. We choose as group PSL(3; 5) and construct 8-(31; 10; ) designs with two different values of . We get 1658 designs with designs with questions immediately arise: 1. Are the designs all distinct, i.e. pairwise non-isomorphic, or, if not, which of them form a transversal of the isomorphism classes? 2. What is the full group of automorphisms of each of the designs? 3. Are there more designs for other values of ? 4. Are there more designs with a possibly smaller group of automorphisms? In the following sections, we will answer question 1 twice and question 3 partly. Problem 2 would be easily solved if it were known that PSL(3; 5) is a maximal subgroup of S 31 . Note that this fact would imply that 1 is true. Indeed, we will show in Section 7 that designs with the same automorphism group are isomorphic if and only if they are isomorphic under the normalizer of this group. The plan of this paper is the following: In Sections 2 and 3, the method of Kramer and Mesner is briefly sketched. We will give a list of all orbits of the group on 10- subsets which is needed to describe the designs. In Section 4, we recall basic facts about parameters of designs and about intersection numbers. We introduce the equations of Mendelsohn and Kohler. Moreover, we define intersection numbers of higher order and list the relevant generalizations of the parameter equations due to Tran van Trung, Qiu-rong Wu and Dale M. Mesner. We also define global intersection numbers and use the generalized equations to provide means for checking them. The following two Sections 5 and 6 are devoted to the 8-(31; 10; 100) and 8- respectively. For each of these cases, the parameter equations are shown. As the numbers involved tend to become quite large in some cases, this can be of great help avoiding tedious hand-calculations. In fact, all these calculations were done by a computer using a long-integer arithmetic. For each value of designs are listed in full detail. They should serve as examples. The interested reader may reconstruct the full set of designs using our program which is freely available on the internet. The numbering of designs is imposed by the order in which the solutions are computed by the equation solver of (this program is deterministic, so that the order is always the same). See [17] for a more detailed treatment on solving large equation systems with integral coefficients. Finally, in Section 7 the two announced proofs of Problem 1 are given. The first applies group theoretic methods together with some (small) computer calculations. The second is a more combinatorial one. It uses intersection numbers as invariants to show that no two designs are isomorphic. Problem 4 is beyond the scope of this paper. 2. The Group and its Orbits We denote the elements of the field GF (5) by 0; 4. The elements of the projective geometry PG 2 (5) can be identified with the one-dimensional subspaces of GF (5) 3 . We number them in the following way using representatives (a; b; c) t for the one-dimensional subspace h(a; b; c) t i generated by scalar multiples: 26 " 28 " The group PSL(3; 5), represented as a permutation group on PG 2 (5) is generated by the following permutations: The group is of order We are now going to construct t-(v; k; ) designs on the set vertices and with contained in their automorphism groups. Thus, the parameter v is 31 and we want to construct 8-designs, i.e. 8. Moreover we leave open, in fact our method of construction shows that and are fine. Consider a putative design which has A as a subgroup of its automorphism group. In this case, the set B of blocks decomposes into a collection of full orbits of A on k-sets. In order to describe the design, we only need to know which orbits (among the total set of orbits of A on the set \Delta of all the k-subsets of the set V of points) belong to the design. Therefore, we label the A-orbits and refer to these numbers later on. In our case, we compute the orbits of PSL(3; 5) on i-subsets for all i which are less than or equal to 10. Table 1 shows the number of orbits. Table 1. Number of orbits of PSL(3; 5) on i-subsets of PG 2 (5) # i-orbits of A 1 The following table shows all 10-orbits of A on V . The stabilizer order is indicated by a subscript. The orbit length is the index of the stabilizer in A. We give the lexicographically minimal representative within each orbit. This list of representatives is not lexicographically ordered, due to the fact that we do not generate orbits via orderly generation. Instead of orderly generation, we use an algorithm Leiterspiel [12] (snakes and ladders) to provide orbit representatives and further knowledge needed for the evaluation of Kramer-Mesner matrices (see below). As the representatives all start with the sequence 1; 2; 3; consecutive numbers, only the last of these numbers is shown. The first part is replaced by the symbol ': : :'. So, the set displayed as 28g. 10-orbits: 2: 3: 5: 7: 8: 3. Orbit Selection The designs are constructed using the Kramer-Mesner matrix M A consists in our case of 42 rows and 174 columns (recall that 1). The entry m ij is the number of k-subsets in the j-th orbit of A on k-subsets containing the representative of the i-th orbit on t-subsets. Hence, the f0; 1g-solutions of the Diophantine system of equations (2) are exactly the possible ways of choosing suitable block orbits (the chosen columns) which fulfil all the conditions of a t-(v; k; ) design admitting the prescribed automorphism group A. Namely, such a solution is a collection of group orbits on k-sets such that each representative T of a t-orbit is contained in exactly k-sets from all the chosen k-orbits. This system was completely solved by the LLL-based algorithm as described in [17]. There are exactly 138 solutions for solutions for solutions exist for other values of 126 for this system of equations. The enumeration of all solutions is a backtracking-algorithm over the integral linear combinations of the LLL-reduced basis-vectors of the corresponding Kramer- Mesner system. In order to speed up the search one can parallelize the algorithm as described in [6]. Nevertheless, the designs presented here were found with the sequential version of the program within a few hours. 4. Intersection Numbers of Designs In this section, we recall some basic facts about parameters and intersection numbers of designs. We will make use of intersection numbers in Section 7 when proving the fact that the 8-designs with PSL(3; 5) are pairwise non-isomorphic. Intersection numbers have a long history in design theory, early results were obtained by Mendelsohn [10] and Stanton and Sprott [14]. They can be generalized to higher t, we will show them soon. The equations of Kohler [7] support the evaluation of these formulae. We will also speak about generalized intersection numbers, which already appeared in [10]. Recent progress was made by Tran van Trung, Qiu-rong Wu and Dale M. Mesner [16]. be a simple t-(v; k; ) design on the set of points V with g be the blocks with Fix disjoint subsets I and J of V with t. Define r the number of blocks which contain a given point and Ray-Chaudhuri and Wilson proved in [11] that these numbers i;j are independent of the choice of the sets I and J (depending only on their cardinalities i and j). They can be computed by the following formula The following recursion holds for This is the same recursion as in the well-known Pascal-triangle of binomial coeffi- cients. Here, one also speaks of the intersection triangle of the design. For sake of simplicity, put i := i;0 . For an arbitrary fixed m-subset M of V we define for 6 BETTEN, KERBER, LAUE, WASSERMANN the i-th intersection number of M with D. The reference to the set M will sometimes be omitted. It should then be clear from the context which set M we are referring to. Again, let M be an arbitrary m-subset of V . Fix an integer i with 0 i t. Counting the set in two different ways, one arrives at the equations of Mendelsohn [10]: (for all Writing down the system of equations we obtain a rectangularly shaped matrix with integeral coefficients fomed by the binomial numbers. In its first min(t+1; m) columns, this matrix is upper triangular with all diagonal entries equal to 1. In the case m ? t we have some additional columns corresponding to intersection numbers ff t+1 (M Of particular importance for our applications are the block intersection numbers. Here, M is chosen to be a block B of the design itself (and thus In this case, ff k (B 0 ) is always equal to 1 since we allow only simple designs. The equations of Mendelsohn read as t. We remark the following fact for the case m ? t. Assume we know the intersection numbers ff t+1 (M late intersection numbers). Then, since the coefficient matrix is upper triangular and has ones on the main diagonal one can easily compute the remaining numbers ff 0 (M numbers). Kohler gives explicit equations for the early intersection numbers. In [7], he proves that for t. The terms early and late intersection numbers should not be mixed up with intersection numbers of higher order which will be introduced in the sequel. For any B 0 2 B, the vector (ff 0 (B called the block intersection type of B 0 (in the design D). The equations of Kohler show that only the essential block intersection numbers are needed, that is (ff t+1 (B call this vector the essential block intersection type. Clearly, block intersection types are constant on orbits of the automorphism group. So, when computing designs as orbits of some automorphism (sub-) group, we need only specify block intersection types for each of our orbit representatives. We will do so later when we specify the 8-(31; 10; ) designs as sets of orbits. Let now be the representing sets for the A-orbits of blocks in the design (not to be mixed up with all A orbits as in Section 2). Let K A h be the corresponding orbit under the group A define the global intersection number as By adding up the intersection types of all blocks of the design one gets the following formula - we count all intersections twice, therefore the factor 1: The vector (ff is the global intersection type of pairs of blocks of the design. Clearly, but we will find more equations for global block intersection types in the follow- ing. In order to achieve this, let us introduce intersection numbers of higher order (already introduced by Mendelsohn [10]). For an arbitrary fixed m-subset M of V and b s 1 define for ff s the i-th intersection number of order s of M with D. In the case when reduces to ordinary intersection numbers. If s is at least two and if m is at least k, ff as we have excluded designs with repeated blocks. It can be shown (see Tran van Trung, Qiu-rong Wu, Dale M. Mesner [16]) that the equations of Mendelsohn can be generalized in the following way (for an arbitrary m-subset M of V , b s 1 and 0 i t): ff '' i;0 s The following generalization of Kohler's equations has also been proved in [16]. Again, let M be an (arbitrary) m-subset of V and let 0 i t. Then, for each ff s For reduce to (9), i.e. the equations for ordinary intersection numbers. Again, we see that only the essential block intersection numbers (of higher need to be specified (for B a block of the design). Global intersection numbers of order s of the design D can be defined in the following way: ff s Clearly, in the case back the values ff i (D) which we already know. Again, global intersection numbers can be computed by cumulating intersections over all block orbits: ff s s These numbers can be checked in the following way: Choose apply (14). This gives for ff '' i;0 s To see this, one verifies that ff by definition. We stopped the summation on the left after the k-th coefficient since clearly, ff 1). In the case when get back Equation (12) - recall that ff (2) Applying are able to compute ff (ff (D)). The latter vector is called the essential global block intersection type of order s of the design. For s ? 1, ff k (D) vanishes. 5. 8-(31; 10; 100) Designs 5.1. Parameters and Intersection Equations The intersection triangle of i;j for The following values are helpful for the verification of some of the intersection numbers: The System (7) of Mendelsohn for arbitrary M ' V of size The Equations (9) of Kohler are: These equations are important in particular if M is a block B 0 of the design. In this case ff 10 (B and the essential block intersection type consists of just one namely ff 9 (B 0 ). If we consider generalized Mendelsohn Systems (14), only the right hand side differs from the case 1. For the 8-(31,10,100) designs, we get the following vectors for 30; 140215; 072989; 385000 266; 928655; 539000 Next, we evaluate the generalized Kohler Equations (15). Choosing using the equalities ff ff (2) 9 (D) ff (2) 9 (D) ff (2) 9 (D) ff (2) 9 (D) ff (2) 9 (D) ff (2) 9 (D) ff (2) 9 (D) ff (2) 9 (D) ff (2) 9 (D) For ff (3) 9 (D) ff (3) 9 (D) ff (3) 9 (D) ff (3) 9 (D) ff (3) 9 (D) ff (3) 9 (D) ff (3) 9 (D) ff (3) 9 (D) ff (3) 9 (D) 5.2. The Designs We display 3 out of the 1658 designs for 100. The designs are collections of full orbits from the list of 10-orbits of the group. Here, we list only the orbit numbers using the labelling of orbits of Section 2. 66, 67, 68, 70, 72, 76, 77, 78, 79, 84, 87, 88, 91, 92, 94, 96, 100, 102, 105, 106, 108, 113, 114, 117, 118, 120, 121, 124, 126, 136, 139, 141, 143, 145, 147, 148, 149, 151, 153, 156, 157, 160, 164, 165, 171, 173. Block intersection types: ff 9 (B h ff 9 (B h The following table shows the global intersection numbers of all 2-sets of blocks (all 3-sets of blocks). The column sums are respectively. The values of these tables contain a lot of redundancy. According to (15), only ff (2) 9 (D) and ff (3) 9 (D) really matters. The other values follow. In fact, all the numbers in these tables have been computed from the orbit data. So, verification of (15) via (24) and (25) really is a good test for the correctness of our algorithms to compute intersection numbers. 6 4,353419,605500 13800,712024,071000 9 699,336750 1699,063500 The number ff (2) 9 (D) can be computed according to (11) as the following sum. The pairs of numbers of the form 'a \Theta b' give the multiplicity (a) together with the intersection number ff 9 (b). The greatest common divisor of all the multiplicities is taken out of the sum. \Theta 73+30 \Theta 75+78 \Theta 76+72 \Theta 77+128 \Theta 78+ 112, 113, 120, 126, 130, 131, 132, 134, 141, 142, 144, 145, 146, 148, 149, 151, 153, 156, 157, 160, 164, 170, 171, 173. Block intersection types: ff 9 (B h ff 9 (B h ff 9 (B h Global intersections: 6 4,353200,869500 13800,710711,655000 9 701,940750 1714,687500 74+24 \Theta 75+36 \Theta 76+132 \Theta 77+116 \Theta 78+84 \Theta 79+ 216 \Theta 80+112 \Theta 81+138 \Theta 82+123 \Theta 83+68 \Theta 84+12 \Theta 85+4 \Theta 87+12 \Theta 88+12 \Theta 89) 117, 119, 120, 125, 128, 129, 130, 132, 133, 134, 140, 141, 143, 144, 146, 148, 149, 153, 157, 160, 163, 164, 170, 171, 173. Block intersection types: ff 9 (B h 68, 99, 106, 111, 144, 148, 163g, ff 9 (B h for f164g. Global intersections: 6 4,353521,161500 13800,712695,903000 9 698,127750 1691,065500 73+12 \Theta 74+60 \Theta 75+72 \Theta 76+114 \Theta 77+106 \Theta 78+ 129 \Theta 79+252 \Theta 80+164 \Theta 81+48 \Theta 82+24 \Theta 83+48 \Theta 84+42 \Theta 85+42 \Theta 86+6 \Theta 87) Designs 6.1. Parameters and Intersection Equations Again, we list 3 of the designs, now with We have 435240 Some useful values are: The system of Mendelsohn for The equations of Kohler are: The generalized Mendelsohn Systems (14) have the following right hand side (for 94716; 711180 The generalized Kohler equations applied to are (the ff (D)-terms with are left out): ff (2) 9 (D) ff (2) 9 (D) ff (2) 9 (D) ff (2) 9 (D) ff (2) 9 (D) ff (2) 9 (D) ff (2) 9 (D) ff (2) 9 (D) ff (2) 9 (D) ff (3) 9 (D) ff (3) 9 (D) ff (3) 9 (D) ff (3) 9 (D) ff (3) 9 (D) ff (3) 9 (D) ff (3) 9 (D) ff (3) 9 (D) ff (3) 9 (D) 6.2. The Designs 57, 60, 64, 65, 72, 75, 76, 81, 83, 84, 85, 91, 92, 94, 96, 98, 103, 105, 107, 109, 113, 114, 116, 120, 124, 125, 126, 128, 131, 132, 136, 138, 139, 141, 147, 148, 149, 150, 152, 159, Block intersection types: ff 9 (B h for for f2g. Global intersections: 6 3,765061,999125 11100,433974,021000 9 603,084075 1347,012000 1200 \Theta 70+11400 \Theta 71+22800 \Theta 72+4800 \Theta 73+17400 \Theta 74+ 15800 \Theta 75+ 7200 \Theta 57, 60, 63, 69, 70, 72, 75, 78, 80, 84, 85, 90, 94, 95, 98, 100, 101, 103, 104, 105, 110, 116, 117, 121, 122, 125, 128, 130, 134, 135, 137, 138, 139, 143, 147, 148, 149, 152, 156, 159, 163, 167, 169, 170, 172. Block intersection types: ff 9 (B h for ff 9 (B h 78, 101, 105, 110g, ff 9 (B h 57, 69, 134, 137g, ff 9 (B h ff 9 (B h f13g. Global intersections: 6 3,764718,271125 11100,431120,037000 9 607,176075 1380,988000 3000 \Theta 71+14400 \Theta 72+13200 \Theta 73+22800 \Theta 74+12000 \Theta 75+ 7200 \Theta 76+ 4800 \Theta 55, 58, 60, 62, 64, 66, 71, 75, 76, 77, 78, 81, 85, 88, 90, 94, 97, 98, 105, 109, 111, 113, 116, 118, 119, 120, 125, 126, 127, 131, 132, 133, 136, 138, 140, 145, 146, 149, 151, 152, 159, 167, 169, 172. Block intersection types: ff 9 (B h ff 9 (B h ff 9 (B h ff 9 (B h 105, 113, 169g, ff 9 (B h for h 2 f2g. Global intersections: 6 3,765140,119125 11100,433336,041000 9 602,154075 1354,607000 (3 \Theta 10+100 \Theta 48+400 \Theta 60+200 \Theta 66+600 \Theta 67+2400 \Theta 21000 \Theta 7. Isomorphism Problems This section addresses Problem 1 of Section 1. We answer the question posed there by showing that all designs are non-isomorphic. This claim is proved in two different ways. 7.1. First Proof General group theoretic tools quite often suffice to solve the isomorphism problem for designs constructed by the Kramer-Mesner method. This approach was already partly used in [12], [13] and we first briefly report the basic idea from that papers. be the full symmetric group on the underlying point set V: The following lemma is useful when constructing objects with a prescribed automorphism group. be designs with a group A as their (full) group of au- tomorphisms. Assume that g 2 S V maps D 1 onto D 2 . Then g belongs to the normalizer of A in SV . If the prescribed group of automorphisms A is a maximal subgroup of S V different from the alternating group then all designs found are pairwise non-isomorphic. If A is not a maximalsubgroup one can apply a Moebius inversion on the subgroup lattice to single out those designs having A as their full automorphism group and then form the NSV (A) orbits on the set of these designs. These orbits, all of length jN SV (A)=Aj; are just the different isomorphism types. A severe drawback of this approach is that it relies on the knowledge of the set of groups containing A in SV : Often the information on overgroups can be obtained in some way from the classification of the finite simple groups. We want to show here that in important cases we can avoid this laborious task by a localization technique. We regard A as a guess for the automorphism group of the designs constructed. A good guess might at least find a correct Sylow subgroup of the automorphism group. Then the following holds. Lemma 2 Let a finite group G act on a 2\Omega be fixed by a p-subgroup P of G and g 2 G such that ! g be a Sylow subgroup of there is some x 2 NG (! 2 ) such that P by the Sylow Theorem. Then If the prescribed subgroup A of the automorphism group of the objects that are searched for contains a Sylow subgroup P of all the automorphism groups of the objects then only elements from NG (P ) have to be applied to the objects as possible isomorphisms. In our applications to t-designs it is often possible to show that no design exists if the proposed subgroup P is extended to a larger p-group. Then the assumptions of the lemma are of course fulfilled. There is a problem if A is not normalized by NG (P usually, the set of fixed points of A is not closed under NG (P just form the orbits of in order to solve the isomorphism problem. Lemma 3 Let G be a finite group acting on a set\Omega and A G: Let A contain a Sylow subgroup P of all designs admitting A as a group of automorphisms. If acts on the set of fixed points Fix\Omega (A) of A in\Omega . If Fix\Omega (A) and g 2 H with NH (A)g Figure 1. Subgroups of Lemma 3 @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ A G NH (A)NG (P Suppose that D 1 and D 2 are two designs admitting A as an automorphism group. If A is a large subgroup of the full symmetric group Sn then it happens very often that If this situation appears for each then the orbits of NH (A) on Fix\Omega (A) are the different isomorphism types appearing in Fix\Omega (A). If in addition designs admitting A as an automorphism group are pairwise non-isomorphic. Algorithmically, only representatives from the cosets NH (A)g in H have to be considered in forming hA; A g i. A remarkable feature of this approach is that the individual designs are not touched upon. So, the isomorphism problem may be solved without knowing details like orbit representatives etc. of the designs. To solve the isomorphism problem for the 8-designs of this paper we use the fact that contains a Sylow-31 subgroup P of S 31 (cf. Figure 2). Figure 2. Special Situation for PSL(3; 5) @ @ @ @ @ @ We choose the group P generated by The normalizer of P in the full symmetric group is the holomorph of i.e. the semidirect product of P with its automorphism group. This normalizer is not contained in A but jNS31 (P This intersection has 10 right cosets in NS31 (P Representatives of these cosets are given by the powers of the element 28 21)(3 22 1230 6 For i and in each case obtain A a design which has A 31 as a group of automorphisms must be the complete design. Thus, by the above theory, all designs obtained as solutions of the Kramer-Mesner system for ! are pairwise non-isomorphic. 7.2. Second Proof The second proof of the fact that all designs are non-isomorphic is done using the intersection numbers of Section 4. In a first step, the global intersection type ff (2) 9 (D) is used in order to distinguish between the designs. Clearly, two designs which have different intersection numbers are non-isomorphic. Coming back to Sections 5 and 6, we find ff (2) 9 (D 1 ff (2) 9 (D 2 ff (2) 9 (D 3 for the designs with ff (2) 9 (D 1 ff (2) 9 (D 2 ff (2) 9 (D 3 for those with These numbers are all distinct (which is fine for our pur- poses!) but for the whole set of designs, there are coincidences. For the 138 designs with different values of ff (2) 9 (D) in the range from 591,366075 to 611,268075. The following table shows the classes of designs with sorted according to the value of ff (2) 9 (D). For each value, the indices i of the designs D i are given. 593; 226075 for f110g 594; 342075 for f95g 595; 830075 for f111g 596; 853075 for f87g 597; 039075 for f102g 597; 225075 for f107g 597; 318075 for f23; 128g 597; 504075 for f5; 35g 597; 597075 for f15; 46g 597; 969075 for f8g 598; 248075 for f126g 598; 341075 for f14g 598; 434075 for f118; 132g 598; 527075 for f96g 598; 806075 for f79g 598; 899075 for f30; 70; 112g 598; 992075 for f97; 100g 599; 085075 for f48g 599; 643075 for f49g 599; 829075 for f44; 119g 599; 922075 for f64g 602; 433075 for f13; 109g 602; 712075 for f34g SIMPLE 8-DESIGNS 21 608; 199075 for f114g 610; 152075 for f78g 610; 803075 for f58g Let us make some statistics first: The class sizes are distributed in the following way: # of classes of size i 48 The average class size is = 1:643, we get for the variance V ar = 0:86 and the standard deviation A better choice for an invariant is the multiset of all block intersection types of a design. So, one starts with Equation (11) and collects equal terms of ff 9 (B h ) (in the case of the 8-designs). This leads to an additive decomposition of ff (2) 9 (D) which is a much finer invariant. For example, the class of designs with ff (2) 9 (D has the following different types of block intersections (sorted lexicographically by the coefficients of the terms): 7200 \Theta 76 (3 \Theta 10+400 \Theta 63+1200 \Theta 64+600 \Theta 67+2400 \Theta 68+ 3600 \Theta 69+6000 \Theta 70+6000 \Theta 71+12950 \Theta 72+8400 \Theta 73+14400 \Theta 74+11400 \Theta 75+ 17100 \Theta 76+12000 \Theta 77+5600 \Theta 78+1200 \Theta 79+600 \Theta 80+700 \Theta 84+600 \Theta 85+30\Theta90) (3 \Theta 10+200 \Theta 57+1800 \Theta 64+600 \Theta 67+6000 \Theta 68+ 400 \Theta 69+4200 \Theta 70+4800 \Theta 71+15000 \Theta 72+10800 \Theta 73+16200 \Theta 74+13200 \Theta 75+ 11400 \Theta 77 22 BETTEN, KERBER, LAUE, WASSERMANN 30+100 \Theta 48+600 \Theta 64+1800 \Theta 65+800 \Theta 66+ 900 \Theta 68+3600 \Theta 69+6000 \Theta 70+3600 \Theta 71+13600 \Theta 72+10800 \Theta 73+13200 \Theta 74+ 17000 \Theta 15000 \Theta As a matter of fact, all designs (for can be distinguished using this invariant. The major drawback of using the ff (3) 9 (D) for classification purposes is simple: these numbers are quite hard to compute because lots of intersections are involved. For sake of completeness, we list the orbit indices of the remaining 5 designs (D 1 has already been shown in Section 6): 51, 55, 60, 61, 64, 66, 68, 72, 75, 79, 84, 85, 86, 90, 96, 98, 100, 105, 107, 108, 109, 111, 114, 117, 120, 127, 128, 130, 131, 133, 134, 135, 137, 138, 141, 142, 144, 149, 151, 152, 154, 159, 167, 169, 170, 172. 55, 56, 58, 60, 64, 66, 71, 72, 75, 76, 77, 81, 90, 92, 94, 95, 98, 100, 101, 102, 103, 52, 55, 56, 58, 59, 64, 67, 68, 72, 78, 80, 82, 83, 86, 91, 93, 96, 100, 101, 107, 112, 113, 114, 115, 116, 117, 119, 120, 124, 125, 126, 132, 135, 136, 147, 149, 152, 153, 156, 159, 162, 165, 167, 170, 171, 172. 55, 58, 60, 61, 62, 64, 67, 70, 72, 78, 79, 81, 83, 85, 86, 88, 92, 97, 98, 100, 103, 107, 111, 115, 117, 118, 119, 120, 121, 127, 131, 133, 134, 137, 139, 143, 149, 150, 151, 152, 155, 159, 162, 165, 167, 170, 172. 58, 60, 64, 66, 67, 71, 72, 73, 76, 77, 80, 82, 83, 87, 88, 100, 101, 104, 105, 109, 112, 117, 120, 121, 122, 127, 128, 130, 133, 137, 140, 141, 144, 147, 148, 149, 150, 151, 152, 154, 155, 156, 159, 165, 167, 169, 172. In the case of the 1658 designs of type 8-(31; 10; 100) we get 219 different values ff (2) 9 (D) in the range from 688; 455750 to 716; 169750. The distribution of class sizes is the following: # of classes of size i 38 20 15 15 14 13 6 8 11 7 # of classes of size # of classes of size 20 The average class size is = 7:57. We have The largest class of designs is of size 25. Namely, one gets ff (2) 9 (D 1127, 1208, 1288, 1299, 1426, 1459, 1507, 1545, 1585g. As remarked above, in all cases the use of block intersection numbers allows to distinguish between the designs. 8. Acknowledgement The first author likes to express his thanks to the Deutsche Forschungsgemeinschaft which supported him under the grant Ke 201 / 17-1. --R Design theory. The discovery of simple 7-designs with automorphism group P \GammaL(2 Simple 6 and 7-designs on 19 to 33 points Some simple 7-designs Allgemeine Schnittzahlen in t-designs Intersection numbers of t-designs Block intersections in balanced incomplete block designs. Combinatorial configurations: designs High order intersection numbers of t-designs Finding simple t-designs with enumeration techniques --TR Design theory Combinatorial configurations, designs, codes, graphs The Discovery of Simple 7-Designs with Automorphism Group PTL (2, 32) MYAMPERSAND#123;0, 1MYAMPERSAND#125;-Solutions of Integer Linear Equation Systems --CTR Anton Betten , Reinhard Laue , Alfred Wassermann, A Steiner 5-Design on 36 Points, Designs, Codes and Cryptography, v.17 n.1-3, p.181-186, Sept. 1999 Reinhard Laue, Solving isomorphism problems for t-designs, DESIGNS 2002: Further computational and constructive design theory, Kluwer Academic Publishers, Norwell, MA, Johannes Grabmeier , Erich Kaltofen , Volker Weispfenning, Cited References, Computer algebra handbook, Springer-Verlag New York, Inc., New York, NY, Johannes Grabmeier , Erich Kaltofen , Volker Weispfenning, Cited References, Computer algebra handbook, Springer-Verlag New York, Inc., New York, NY,

kramer-mesner method;isomorphism problem;group action;t-design;intersection number

607292

Jacobi Polynomials, Type II Codes, and Designs.

Jacobi polynomials were introduced by Ozeki in analogy with Jacobi forms of lattices. They are useful to compute coset weight enumerators, and weight enumerators of children. We determine them in most interesting cases in length at most 32, and in some cases in length 72. We use them to construct group divisible designs, packing designs, covering designs, and (t,r)-designs in the sense of Calderbank-Delsarte. A major tool is invariant theory of finite groups, in particular simultaneous invariants in the sense of Schur, polarization, and bivariate Molien series. A combinatorial interpretation of the Aronhold polarization operator is given. New rank parameters for spaces of coset weight distributions and Jacobi polynomials are introduced and studied here.

Introduction While the use of invariants of finite groups to study weight enumerators of self-dual codes has a long and distinguished history [22, Chap.19], they were introduced only recently in the study of the weight distribution of cosets of self-dual codes [2, 19]. The polynomial invariant of the code on which a finite group acts by linear substitutions of the variables is the Jacobi polynomial. Roughly speaking, Jacobi polynomials are to binary codes what Jacobi forms [13] are to lattices. If the code is self-dual, they are invariant under the same group that fixes the weight enumerator of the code and contain more information than the coset weight distribution or outer distribution matrix in Delsarte sense. They are, however, strongly related to the t-distribution matrix of [7] or outer distribution matrix in a Johnson scheme. New rank parameters for spaces of Jacobi polynomials (b t ) and the outer distribution of a code (a t ) are introduced in this paper. Introduced by Ozeki [19] Jacobi polynomials were studied by Bannai and Ozeki [2] by polarization techniques. A very simple combinatorial interpretation of polarization is given here for codes whose codewords hold t\Gammadesigns. Since the motivation of these authors was to build large spaces of modular forms they used a group of order 96: For combinatorial purposes it seems more natural to use the larger group of order 192: Our motivation is to construct designs of various kinds and in particular group divisible designs when the Assmus-Mattson theorem cannot yield classical designs which do not exist anyways. The paper is organized as follows. Section 2 collects definitions and some basic results. Subsections 2.3 and 2.5 are required reading for understanding the rest of the paper. Section 3 derives ( in a different way than [19]) a MacWilliams formula for Jacobi polynomials thus solving a covering open problem of long standing. Section 4 is devoted to bivariate Molien series. Section 5 studies invariants and how polarization produces them. Section 6 studies examples of Jacobi polynomials in lengths 8,16,24,32, and 72. Notations and definitions 2.1 Codes All codes here are binary linear of length n. By weight w(x) we mean the number of 1's in x and by Hamming composition the ordered pair (n \Gamma w(x); w(x)): A self-dual code is said to be Type II if its weights are multiple of 4 and Type I otherwise. Let A i stands for the number of codewords of weight i; and A i (x) for the weight distribution of the coset x C: The weight of a coset x + C is the smallest i ? 0 such that A i (x) ? 0: The Covering Radius R is the largest weight of a coset. The weight enumerator WC (x; y) is the generating series The joint weight enumerator J A;B of two codes A and B is i;j;k;l where A i;j;k;l is the number of pairs (a; b) in A \Theta B containing i patterns 10, j patterns 11, k patterns 00, l patterns 01. 2.2 Designs A design with parameters N )) is a collection of k\Gammasets called blocks of a v-set (the varieties) and a partition of the set of all t\Gammauples into N groups such that every t\Gammaset within a group of a i such t-sets is contained in exactly - i blocks of size k: A t-design with no further precision will mean a design with A packing (resp. covering) design with parameters t \Gamma (v; k; -) is a design with 9]. The minimum (resp. maximum) size of a covering (resp. packing) design is denoted by C - (v; k; t) (resp. D - (v; k; t)). A group-divisible block design (GDD for short) [5] noted GD(k; with blocks of size k and N groups of size g such that pairs in the same group occur in - 1 blocks and pairs with varieties belonging to two different groups occur in - 2 blocks. GDD incidence matrices produce distance regular graphs of diameter 3 [5, x1.10]. Observe that a GDD is a design with a 1 =number of pairs in the same block and a 2 =number of pairs in two different blocks. We shall need the following consequence of the Assmus-Mattson theorem. Theorem 1 If C is an extremal Type II code of length n congruent to 0 (respectively 8; 16) modulo 24 then the vectors of any given weight in C hold a 5\Gammadesign (respectively a 3\Gammadesign, a 1\Gammadesign). Define the Assmus-Mattson index am(n) of an integer n divisible by 8 as congruent to s modulo 3 and 2: The above theorem says that an extremal Type II code C of length n is an am(n) design. 2.3 Enumerators By the Jacobi polynomial attached to a set T of coordinate places of a code C of length n over F 2 we shall mean the polynomial in four variables w m0 (c) z m1 (c) x n0 (c) y n1 (c) where T ' [n]; and where m i (c) is the Hammingcomposition of c on T and n i (c) is the Hammingcomposition on The basic observation is that if T supports a coset leader x T of x is the weight enumerator of x is constant for all T such that jT then the codewords whose monomials have hold a t-design possibly with several groups (one for each possible J C;T ). Relations with two other polynomial invariants of a code are to be noted. 2.4 Joint weight enumerators First, the Jacobi polynomial is, up to variables permutation, the joint weight enumerator of C with the singleton code reduced to the characteristic vector x T of T: Let J A;B denote the joint weight enumerator of two linear codes A and B. Denote by ! T ? the monodimensional code spanned by x T : The following is immediate from [22, p.147]. 2.5 (t; r)-designs Second, consider the set of codewords of weight j of C as a collection B j of j \Gammasubsets of an n\Gammaset. Let the coefficient of w i z t\Gammai x n\Gammaj y j in J C;T . Then the matrix with generic element is the t-distribution matrix D t of [7]. A t\Gammaform is a row vector which is sent to a constant vector by left multiplication by D t : A (t; r) design in the sense of [7] is a collection B j such that the space of t\Gammaforms has dimension t is shown in [7] that the rank of D is either r of r + 1). It is shown in [7] that a t\Gammadesign with design and conversely. (C) denote the dimension of the real vector space of Jacobi polynomials J C;T with T ranging over t-sets. Any upper bound on b t (C) yields an upper bound on r uniform in j: For instance that all codewords of given weight hold a t\Gammadesign entails b (the converse is false as can be seen for e 8 and that all codewords of given weight hold a (t; 1)\Gammadesign. In the case of C of type II a trivial upper bound on b t (C) is the coefficient of u n\Gammat v t in the Molien series defined below. A trivial lower bound is the rank a t (C) say, of the bound is not applicable (NA in the tables below) for t ? R: A well-known result due to Delsarte is that R a t where s 0 is the number of nonzero dual weights. Recently there was some work on J C;T with jT [8, 6, 7, 18] but little on jT 1: In this article we will show how to construct GDD in some of these two cases, but, as examples of C for will show, there is little hope of a general upper bound on b am(n)+1 even for the restricted class of extremal codes . 3 A MacWilliams formula for Jacobi Polynomials We give an independent derivation of the MacWilliams relation for Jacobi polynomials of [19]. Theorem 2 (Ozeki) Let C be a binary linear code. Proof:Follows by Lemma 1 and the MacWilliams relation for the ordinary and joint [22, p.148] weight enumerator. 2 Observe that this essentially solves Open Problem 15 of [10, xIX]. 4 Molien Series It has been known since Gleason's address to the International Congress of Mathematics of Nice 1970 [14] that the weight enumerator of a type II binary code is left invariant by a group G 2 of order 192 generated by two matrices of size two by two defined as and corresponding respectively to the MacWilliams Transform, and mod 4 congruence condition. It is a simple exercise, using Theorem 1, to show that, for any T , the Jacobi polynomial is invariant under the same group acting in the same way on each pair of variables. This is therefore a simultaneous invariant in the sense of Issai Schur [24] for the diagonal action of G There is a bivariate Molien series that enumerates invariant polynomials by their homogeneous degrees in w; z and x; y. It can be shown [25, equ. (13)] that where det(h) stands for the determinant of a matrix For instance in the case of the group G 2 a Magma computation yields an expression for f(u; v) whose denominator factors as d(u)d(v) with or, more suggestively A Taylor decomposition yields after reordering terms of degree 40 6 terms of degree 5 terms of degree 24, 6 5 terms of degree 16, terms of degree 8 and the constant term Assume an Hironaka decomposition [24] of the algebra of invariants I of the type where the j i s are so-called secondary invariants and where P is the algebra of primary invariants (or homogeneous system of parameters) of the type Then the bivariate Molien series can be written as P a In that expression dm (:) (resp. dn (:)) denote the degree in the first (resp. second) set of two variables. In the case at hand we have primary invariants and a = 192 secondary invariants. One may take as h.s.o.p. the system (1) (2) 5 Invariants In principle some simultaneous invariants for a group G can be computed from usual invariants of the Aronhold polarization operator A [24] which we now describe. Let P 0 y ) denote the partial derivative with respect to variable x (resp. y). Let P (x; y) denote a polynomial in two variables. Define the polarization operator A as x The following basic lemma can be found in a much more general form in [24]. is an invariant for G 2 then A:P is a simultaneous invariant for G 2 . The map P 7! AP is injective, the inverse map being given by Proof:For every complex scalar i the quantity is a simultaneous invariant of G 2 . A Taylor expansion yields Each coefficient of this expansion in powers of i is a simultaneous invariant. Setting y in the preceding expansion and identifying coefficients of i on both sides yields the inversion formula. 2 There is a simple combinatorial interpretation of the polarization of the weight enumerator of a code. The code obtained from C by puncturing (resp. shortening) at coordinate place i will be denoted by We shall denote by C the coset of C=i into C \Gamma i. We shall say that a code is homogeneous if the codewords of every given weight hold a 1-design, and more generally t-homogeneous if the codewords of every given weight hold a t-design. In that situation, the Jacobi polynomial J C;T does not depend on T for t; and for convenience, this common value is denoted J Theorem 3 For every binary code C and every coordinate place i we get If C contains no word of weight 1 we have If C is 1\Gammahomogeneous then Proof:The first and second assertion are restatements of the definition of the Jacobi polynomial and of the polarization operator. The third assertion follows by noticing that, in the second assertion, for an homogeneous code the polynomials WC=i and WC+i are independent of i: It also follows from [22, Pb. (37) p.233] which is a re-statement of Prange Theorem [21, Th. 80] in terms of generating series. 2 The generalization for all t is a bit more cumbersome to write down but no more difficult. We leave the proof to the reader. Theorem 4 If C is t\Gammahomogeneous and contains no word of weight - t then A useful corollary concerns children C of a self-dual code D obtained by subtraction, i.e. taking all codewords whose value on two given coordinates is 00 or 11 and puncturing at those places. Corollary 1 If D is a self-dual t\Gammahomogeneous binary code with t - 2 then the weight enumerator of C is As an application we recover the weight enumerators of both the extremal [70; 35; 14] [11, 18] extremal Type I code and the elusive shadow extremal [68; 34; 12] code obtained [11] by subtraction from the putative extremal [72; 36; 16] Type II code, whose Jacobi polynomials are calculated in x6.5. They are, respectively, and 5.1 When polarization fails The question arises: how many invariants in w; z; x; y are polarizations of invariants in x; y? The answer is given in terms of generating series. Let D denote the bivariate generating series enumerating by bidegree the invariants in four variables that cannot be obtained by successive polarizations of two-variable invariants. Proposition denote the Molien series of G Proof:By the injectivity of the polarization operation every term m j u j in the Taylor series of M (u) yields geometrically as m j The Taylor series up to degree 40 can be written in decresing degree order as: In this case we use the Reynolds operator which is defined as in [22, p.609]. Let M 2 G 2 . Denote its action on P (w; z; x; y) as M ffi P: With these notations we define the Reynolds operator as The following result is a special case of [22, Th.4,p.609]. Proposition 2 The polynomial R ffi P is a simultaneous invariant of G An example is given in x5.1. It was shown by EmmyNoether in 1916 that all invariants can be constructed in that way [26, Th. 2.1.4]. 6 Examples Let w 8 and g 24 denote the weight enumerator of the extended Hamming and Golay codes of length 8 and 24: These codes hold respectively 3\Gamma and 5\Gamma designs. Let J 8;t and J 24;m denote for t - 3 and m - 5 their Jacobi we corresponding to coset leaders of weight t and m. Similarly for the code E 16 of [20] let J 16;i be the Jacobi polynomial of index i for 2: We denote by f [a] the homogeneous part of degree a of the Molien Series f: 6.1 Length 8 This corresponds to J 8;s with 3: Alternatively this corresponds to A s w 8 with 3: As the coefficient of w s y 8\Gammas in J 8;s should be 1; and the coefficient of u n\Gammas v s in the Molien series f(u; v) is 1 we see that for This also follows from Theorem 1. Specifically we get: To actually compute a basis of the invariant space, we use the Reynolds operator acting on (wzxy) 2 which gives, up to scalar multiple r := (w 4 Let T be of size 4: Then we know that J C;T is a combination of r and AJ 8;3 . Two cases occur: 1. If T supports a codeword, then it can be seen combinatorially (minimum distance 4 and so on) that 2. If T does not support a codeword then J r)=4 as the term in x 4 z 4 must vanish 6.2 Length This counts The calculation of J 16;1 is consistent with [1, p.361, Table I]. Taking the linear combination 8;1 )=3 we obtain which yields after the substitution x the coset weight enumerator Similarly the linear combination 8;1 )=21 yields the Jacobi polynomial which corresponds to the coset weight distribution Combining these two equations (accounting for all the 2-designs with parameters This gives packing and covering designs In fact these are indeed GDD's with parameters because the 8 pairs in the same coset are disjoint, the minimum distance being 4: The space of Jacobi polynomials J d16 ;T with jT may be generated by the two polynomials J 1631 =42 A 3 w which gives the packing and covering designs The space of Jacobi polynomials J d16 ;T with jT may be generated by the following three polynomials z which gives the packing and covering designs d d 6.3 Length 24 This counts respectively Observe that This is consistent with the Pascal triangle for the Witt design on 24 points [22, p.68]. In this paragraph, we call B 6 the basis of invariants for jT The space of Jacobi polynomials JG24;T with jT generated by two polynomials. If T is not contained in an octad, then it can be seen combinatorially (using that J C;T has the following decomposition relatively to B 6 If we call J 2461 this polynomial, we have Note that the coefficient of wz 5 x 15 y 3 is 6, which corresponds to the number of possible 5\Gammasets in T . If T is contained in an octad, then J with the following decomposition yielding z 6 y Note that this polynomial contains the monomial z 6 x which corresponds to an octad of the Golay. From these polynomials, we obtain the packing and covering designs Golay Golay Golay Golay Golay 5 NA 1 5 Golay 6 NA 2 6 6.4 Length We focus our study on The Reed-Muller code of length 32. Observe that From Table I in [9] it seems that the second order Reed Muller code in length would yield a 4-design with 2 classes. Furthermore since the second type of coset contains exactly 8 vectors of weight 4 any such coset define a partition of [32] into disjoint (by Hamming distance quadruples. Consider the following two polynomials belonging to the invariant space z 4 x and which correspond respectively to the two cosets of weight 4 and 14336 y 14 x We obtain by this way a 4-design with parameters and four other designs with parameters From the packing and covering point of view Note that the first covering design is the record owner in [15]. Here we know that b 4 is indeed 2. A basis of the invariant space is given by J The denotations of the codes and the information on a t are from [9]. CP means an extremal Type II code of length 32. 6.5 Length 72 It is still an open problem to know if there exist a [72,36,16] binary type II code. However, its weight enumerator can be computed by using invariant theory [17]. By theorem 1, the vectors of any given weight in the code hold a 5-design. Theorem 4 then gives the Jacobi polynomials for jT We have: 4397342400 x 44 y 28 4397342400 x 28 y 44 9223731055 zx 28 y 43 2119532800 wzx 43 y 43719104 w 3 zx 21 y 43719104 wz 3 x 30888000 z 5 x 44 y z 5 y 28 y Acknowledgements P. Sol'e thanks Christine Bachoc and Eiichi Bannai for helpful discussions, and Michio Ozeki for sending him [19]. A. Bonnecaze and P. Sol'e thank Allan Steel for programming help in Magma [3, 4]. --R On the covering radius of extremal self-dual codes Construction of Jacobi forms from certain combinatorial polynomials The Magma algebra system I: The user language. Magma: A new computer algebra system. of Discr. Extending the t A strengthening of the Assmus-Mattson theorem Cosets weight enumerators of the extremal self-dual binary codes of length New Extremal Self-dual codes of Length 68 Contemporary Design Theory: a collection of surveys Wiley The theory of Jacobi forms Actes Congr'es International de Math'ematiques Nice New Constructions for Covering Designs A coding theoretic approach to extending designs An upper bound for self-dual codes On self-dual doubly even extremal codes On the notion of Jacobi polynomials for codes A classification of self-orthogonal codes over GF (2) Introduction to the theory of error correcting codes The theory of error-correcting codes On the classification and enumeration of self-dual codes Invariants of Finite Groups and their Applications to Combinatorics Algorithms in Invariant Theory --TR --CTR Christine Bachoc, On Harmonic Weight Enumerators of Binary Codes, Designs, Codes and Cryptography, v.18 n.1-3, p.11-28, December 1999 Y. Choie , P. Sol, A Gleason formula for Ozeki polynomials, Journal of Combinatorial Theory Series A, v.98 n.1, p.60-73, April 2002

group divisible designs;type II codes;jacobi polynomials;invariant theory;packing and covering designs

607301

On Harmonic Weight Enumerators of Binary Codes.

We define some new polynomials associated to a linear binary code and a harmonic function of degree k. The case k=0 is the usual weight enumerator of the code. When divided by (xy)^k, they satisfy a MacWilliams type equality. When applied to certain harmonic functions constructed from Hahn polynomials, they can compute some information on the intersection numbers of the code. As an application, we classify the extremal even formally self-dual codes of length 12.

Introduction In the theory of lattices, some modular forms play a special role, the so-called theta series with spherical coe-cients. They are generalizations of the theta series of the lattice which counts the number of vectors of given norm; they are a powerful tool for the study of the spherical codes supported by the vectors of an even unimodular lattice, as shown in [24], and also provide some knowledge on the values of the scalar product of the vectors of the lattice with a given vector of the Euclidean space (i.e. on the so-called Jacobi theta series of the lattice). For example, they have allowed B. Venkov to settle \a priori" the list of the possible root systems of an even unimodular 24-dimensional lattice [7, Chapter 18]. See [1] for a generalization of these methods to non-unimodular lattices. Inspired by the analogy pointed out in [8], [9] between the theory of combinatorial and euclidean designs and their connection in both cases with harmonic spaces, we dene here analogues of these for linear binary codes. More precisely, we associate to a binary code C and a harmonic function f of degree k in the sense of [8], a polynomial W C;f (x; y), which, when divided by (xy) k , behaves, up to a sign, like the usual weight enumerator WC (x; y) under the MacWilliams transform. In particular, when C is a doubly even self-dual code, we get a whole set of polynomials which are relative invariants under the usual group G 1 of 2 2-matrices of order 192 generated byp1 1 and ( 1 0 In the case of an even formally self-dual code, the group to be considered is the subgroup G 2 generated by 1 and 1 0 , and the polynomials (xy) k (W C;f W C ? ;f ) are relative invariants for G 2 . In both cases, these results can be used to derive some information on the way a given t-set T meets the codewords. In particular, we give another proof of the fact that the words of xed weight in an extremal code (resp. and its dual in the case formally self-dual) support \t 1 "-designs, as shown in [6], [16]. More generally, we can derive some \invariant linear forms" in the sense of [5] on the so-called intersection numbers: (1) not only in the case when jT and one has t-designs, but also for all value of through the explicit description of the space of relative invariant polynomials in which (xy) k (W C;f W C ? ;f ) falls. Therefore, we specialize to certain harmonic functions associated to T , which have the property that H k;T (u) only depends on t, juj, and ju \T j; they are constructed from Hahn polynomials. As an example and application, we derive a classication of the extremal even formally self-dual codes of length 12. This classication has been extended in [11], [12], where intersection numbers play an important role. Another method is used in [14] to derive analogous results. It involves some other kinds of polynomials, the so-called overlap and covering polyno- mials, which are closely connected to Ozeki's Jacobi polynomials([18]). The author is grateful to one referee for pointing out this reference. This paper is organized in the following way: Section 1 contains the needed denitions and properties of harmonic functions and binary linear codes. Section 2 contains the denition of harmonic weight enumerators and the proof of the MacWilliams-type formula (Theorem 2.1). The consequences on the invariance properties of these polynomials in the cases of doubly even self-dual codes and of even formally self-dual codes are stated in Corollaries 2.1 and 2.2. Section 3 gathers the needed results of invariant theory. In Section 4, we reprove Assmus-Mattson theorem and Calderbank- Delsarte strengthening of it for doubly even self-dual codes. Section 5 explains the method based on Hahn polynomials used to compute the intersection numbers, and Section 6 contains the classication of the extremal even formally self-dual codes of length 12 (Theorem 6.1). We now recall some denitions and properties of discrete harmonic func- tions, which are developed in [8]. Let ng be a nite set (which will be the set of coordinates of the code C) and let X be the set of its subsets, while, for all k is the set of its k-subsets. We denote by RX, RX k the free real vector spaces spanned by respectively the elements of X, X k . An element of RX k is denoted by f(z)z (2) and is identied with the real-valued function on X k given by z ! f(z). The complementary set of z is denoted by z. Such an element f 2 RX k can be extended to an element ~ setting, for all u 2 X, ~ (In the notations of [8], the restriction of ~ f to RX n is dened to be (f ).) We may later on denote again ~ f by f . If an element g 2 RX is equal to some ~ f , for f 2 RX k , we say that g has degree k. The dierentiation is the operator dened by linearity from for all z 2 X k and for all is the kernel of Harm k := Ker( Concerning codes, we take the following notations: we freely identify words of F n 2 and subsets of the weight of an element u 2 F n 2 is also the cardinality of its support and is denoted by wt(u) or juj. We recall some basic notions of coding theory, for which we refer to [17], [22]; we only consider linear codes. The weight enumerator WC (x; y) of a binary code C is where A i is the number of codewords of weight i and satises the MacWilliams A code C is said to be formally self-dual if It is even if doubly even if wt(u) 0 mod 4 for all codes are even and formally self-dual, while the converse is not true; see [16], [22] for examples. If a formally self-dual code is in addition doubly even, then it is necessarily self-dual. From the facts that the polynomial WC is invariant under the group G 1 in the self-dual doubly even case (resp. under G 2 in the even formally self-dual case), one deduces the inequalities for the minimal weight d(C) of C: d(C) 4([n=24] (respectively meeting these bounds is said to be extremal; its weight enumerator is then uniquely determined. Harmonic weight enumerators In this section, we dene the harmonic weight enumerators associated to a binary linear code C and prove a MacWilliams type equality. Denition 2.1 Let C be a binary code of length n and let f 2 Harm k . The harmonic weight enumerator associated to C and f is ~ Theorem 2.1 Let WC;f (x; y) be the harmonic weight enumerator associated to the code C and the harmonic function f of degree k. Then where Z C;f is a homogeneous polynomial of degree n 2k, and satises x y Proof. Like in the classical case of MacWilliams formula for weight enu- merators, the proof relies on Poisson summation formula, which we recall here: Theorem 2.2 (Poisson summation formula) Let : F n R be a function taking its values into a ring R, and let ^ be its Fourier transform, dened by (v) := Then, for all linear code C F n (v) (12) We shall apply Poisson formula to each term of Z C;f , namely to: Therefore, we compute the Fourier transform of , rst in the case 2.2), and in the general case but for harmonic functions in Lemma 2.3. In order to prove that the ZC;f are actually polynomials, we start with a technical lemma on harmonic functions. Lemma 2.1 Let f 2 Harm k and v 2 F n . Let f (i) (v) := Then, for all 0 i k, f (i) ~ f(v). Proof. For all 0 i k 1, which means (from (4)) that tz The evaluation at v 2 F n is: tz The proof then follows by induction on k i, since clearly f and the previous equality implies ~ We can notice now that, for all u such that wt(u) < k, from denition (3) of ~ f , ~ 0, and from Lemma 2.1, ~ is a polynomial. We now compute the Fourier transform of (see (13)). Lemma 2.2 Let Proof. (v) := We can write runs through F n k and are then reduced to the usual formula for the Fourier transform of x n k wt(v\z) y wt(v\z) . We now consider the case of a harmonic function of degree k and prove Lemma 2.3 Let f 2 Harm k . Then Proof. Since f(z)z, and from Lemma 2.2, To conclude for Lemma 2.3, we need another last lemma: Lemma 2.4 Let f 2 Harm k . Then, for all v 2 F n Proof. Let B s be the coe-cient of x k s y s in this polynomial. We must show that B We sum over with the notations of Lemma 2.1, it is equal to j+l=s l j+l=s l i;j;l j+l=s j;l;t;r j+l=s where the last equality is the computation of the coe-cient of x s in the specialization of Theorem 2.1 now follows from Lemma 2.3 and the Poisson summation formula (12). In the special case of doubly even self-dual codes, an immediate consequence of Theorem 2.1 is that the polynomials Z C;f are relative invariants for the group G 2 . This result is stated in Corollary 2.1, and an analogous result for even formally self-dual codes is stated in Corollary 2.2. We take the following notations: We consider the group G 1 =< together with the characters k dened by: and the group G 2 =< together with the characters dened by Corollary 2.1 If C is a self-dual, doubly even code of length n, for all , the polynomial Z C;f (x; y) satises Z C;f (A(x; for all matrix A 2 G 1 . Corollary 2.2 If C is an even formally self-dual code of length n, for all , the polynomials Z C;f Z C ? ;f satisfy for all matrix A 2 G 2 . 3 Some invariant theory We gather here some well-known results of invariant theory that will be of further use. We denote by C [x the polynomial algebra in n variables, together with the left action of the algebra M n (C ) of nn complex matrices given by (M:P the transposition). If G is a subgroup of M n (C ), we denote by IG the algebra of invariants of G, namely If is a character of G, the space of relative invariants with respect to is I G; = fP It is clearly a module over IG . In view of our situation, we need to compute I , for the characters k dened in (15). It is well-known to be, in the case k 0 mod 4, the polynomial algebra C [P . The other cases are probably also very classical, but we recall the result: Lemma 3.1 I Proof. The dimension a ;d of (I G; ) d , the homogeneous component of degree d of (I G; ) is computed by Molien's series: a ;d X In the case of the group G 1 , and for the characters k given by (15), we nd respectively 1=((1 X 8 )(1 X 24 )), X It is easy to verify that the polynomials announced in the lemma do belong to the spaces I G1 ; k ; the result then follows from the equality of the dimensions. The case of the group G 2 goes the same; we have I and the I G 2 ; for the characters (16), (17), are principal ideals. Clearly these characters only depend on k mod 2. Lemma 3.2 I R 4 I G2 if 4 New proofs of some classical results In this section, we recover the classical results on t-designs supported by words of binary linear codes, using the harmonic weight enumerators previously dened, and the characterization of designs in terms of the harmonic spaces given in [8]: a set B of blocks is a t-design if and only if ~ for all f 2 Harm k , 1 k t. Hence, the set of words of xed weight in a code C form a t-design if and only if W C;f (x; t. We start with Assmus-Mattson theorem: Theorem 4.1 (Assmus-Mattson) Let C be a binary code of length n and distance d, and let C ? be its dual, of distance e. If t d is such that the number of non zero weights of C ? which are lower or equal to n t, is at most d t, then the set of codewords of C (respectively C ? ) of xed weight w form a t-design, for d w n (respectively e w n t). Proof. Let f 2 Harm k , 1 k t. Write A i;f := ~ f(u) and ~ f(u). We want to prove that, for all i (rep. i n Theorem 2.1 translates, in terms of these, into: kin k for all j, k j n k, where the P (n 2k) are the Krawtchouck polynomials ([17, Chap 5]). Since C has distance d, we have A which leads to d k independent equations in the B i;f , k i n k. By hypothesis, there are at most d k unknowns and hence the only solution is trivial . Hence B t. Now the n 2k equations in the A i;f , d i n d, using equations (18) applied to C since k d and the equations are independent, the only solution is trivial. In the case of extremal doubly even self-dual codes, we can prove the result directly from the description of the relative invariants of the group G 1 , avoiding the use of Krawtchouck polynomials; moreover, the extra property that the t-designs are \t 1"-designs (which was shown rst by B. Venkov by means of spherical theta series, then in [6] in a combinatorial setting) follows easily, and is very similar to the initial proof of B.Venkov [24] concerning the spherical designs in extremal even unimodular lattices. We recall the slightly more general denition of the notion of a T -design, for a subset T of ng: a set B of blocks is called a T -design if and only if ~ 0 for all f 2 Harm k and for all k 2 T . Hence a t-design is a design. Theorem 4.2 ([5]) Let C be an extremal self-dual doubly even code of length n. If n 0 mod 24, the codewords of xed weight in C form a If n 8 mod 24, the codewords of xed weight in C form a If n 16 mod 24, the codewords of xed weight in C form a Proof. Let the extremality of C means that We prove that WC;f (x; the other cases being similar. From Theorem 2.1 and Lemma 3.1, for all f 2 Harm k , W C;f (x; Since the valuation at y of Q, (i.e. the least power of y in Q) is 4(m+1) k 3, We compute the degree of this polynomial is non zero, it has degree n 2k Notice that, if the polynomial Q 0 is determined up to a scalar: it is proportional to 1 if respectively to P 8 if 8. Remark 4.1 With the same method, we can recover the results of [16] on the designs supported by codewords of xed weight in C [C ? , when C is an extremal even formally self-dual code. We omit the proof. 5 Harmonic weight enumerators and the computation of Jacobi polynomials In this section, we show how harmonic weight enumerators can be used to compute Jacobi polynomials. We rst recall the denition of these: Let C be a binary code of length n and T ng. JC;T (v; z; x; y) := where, for is the number of coordinates of u\T (respectively of u\T ) equal to i. They have been introduced by Ozeki [18] in analogy with Jacobi forms of lattices, and studied by A. Bonnecaze, P. Sole et al. [2], [3], [4] in the case of type II binary and Z 4 -codes. In particular they point out the following characterization of codes supporting designs: the set of codewords of a code C form a t-design for every xed weight, if and only if the Jacobi polynomial JC;T for a t-set T is independent of T . Since we can also characterize this property of a code C by the set of conditions: a natural question is: how can one compute JC;T for a t-set T given in (19) from the set of conditions (20)? The answer lies in the fact that one can attach to every t-set T some harmonic functions H k;T of degree k, 1 k t; the values H k;T (u) are expressed in terms of Hahn polynomials, and only depend on juj and ju \ T j. They are described in [8] as the orthogonal projection of T 2 RX t over Harm k . In view of our applications, we need to generalize [8, Theorem 5] to the case of subsets of non equal cardinality. For the denition and properties of Hahn polynomials, we refer to [15]. Proposition 5.1 [8, Theorem 5] Let T be a t-subset of ng. For all for all t-set u, where ([15]) Q t are orthogonal Hahn polynomials. Then H k;T 2 Harm k . Proof. In the notations of [8], H k;T Proposition 5.2 With the same hypothesis, as an element of RX, the H k;T (u) for all subsets u of ng only depend on We set H k;T I where Proof. From [8, Theorem 3] applied to have H k;T zx jx \ uj which leads to the announced formula by setting Remark: The same argumentation as in [8] applied to H k;T for juj > t show that they are also linked to Hahn polynomials but for the parameters (with the notations of [15]): Q k (x; juj n From (1) and (19), the numbers nw;i (T ) are the coe-cients of JC;T : On the other hand, the harmonic weight enumerators WC;H k;T have the following WC;H k;T f Hence the set of equations (20) for leads for every w to the following t linear equations in the t We denote by Cw the set of codewords of weight w. Its cardinality is is equal to the coe-cient Aw of the weight enumerator of the code C, dened by (6). Then, another equation, corresponding to the degree 0 case, is: For all w, the nw;i (T ) are the solutions of the system of equations (29), (30). Remark: In the cases when the polynomials ZC;H k;T are invariant polyno- mials, i.e. in the cases of doubly even self-dual codes or of even formally self-dual codes, we can more generally get some information on the nw;i (T ), not only when the codewords support t-designs, in the following way: a condition of the type Z C;f 2 I G; , joined with the knowledge of d(C), says that Z C;f sits in a nite-dimensional vector space, which is explicitly described. Hence this information can be turned into linear equations in the nw;i (T ). Of course, the smaller this dimension is, the more equations we get, and the case when the codewords support designs is the 0-dimensional case. The higher d(C) is, the smaller are these dimensions, the most interesting cases being the extremal codes. An example of this method is treated in next section. 6 A classication result In this section, we classify, with the help of harmonic weight enumerators, the extremal even formally self-dual codes of length 12. These codes have weight 4 and their weight enumerator is There is a unique code which is self-dual; it is the code B 12 with component d 12 of [19], [20]; we nd two other codes which are isodual, one of them is described in [22, Chap.3]. They both appear in [13] as double circulant codes. First step in this classication result is the computation of the w;i (T )g 0it (see (1)). We rst show that, if T is a word of C of weight 4 or 6, there are only two solutions for fnw;i (T ); n therefore, we use the results of the previous section to derive some equations satised by these numbers. Lemma 6.1 Let C be an extremal even formally self-dual code of length 12 and let T 2 C, of weight 4 or 6. There are only two possibilities for w;i (T )g 0it , which are given in the following tables: If wt(T If wt(T Proof. We rst make some easy remarks: since odd. Moreover, since the all-one word 1 belongs to C \C ? , From (31), we have: and 4. Some of the entries are easily computed from the hypothesis on the code C: n 4;3 (T otherwise the sum with T would be a weight 2 word in C, and clearly n 4;4 (T Taking into account the equations (32) and (33), we are reduced to the set of six unknowns: )g. We now consider the harmonic weight enumerators WC;H k;T dened in the previous section. From Corollary 2.2 and Lemma 3.2, ZC;H 1;T but, since C and C ? have weight 4, ZC;H 1;T must be a multiple of (xy) 3 , and hence of Q 8 P 0 8 . This last polynomial is of degree while ZC;H 1;T is of degree 10; hence it is zero. A similar discussion shows that ZC;H 1;T We derive the following equations: which, in terms of our six unknowns, the coe-cients h k;4 (w; i) being computed from equation (), lead to: Since we look for positive integral solutions, we see from the rst two equations of (35) that the only possibilities are n the two announced solutions. Clearly, n depending whether T belongs to C ? or not. arguments lead to the result. Theorem 6.1 There are exactly three extremal formally self-dual codes with even weights of length 12; one is the unique self-dual code B 12 and the two others are given by the following generator matrices: Permutation group of order 384 C (2) C (2) Permutation group of order 120 Proof. Let C be such a code. Let T 6 2 C be a word of weight 6, not belonging to C ? . From Lemma 6.1 we know that n is a unique word u 4 of weight 4 in C ? whose support is contained in T 6 . belong to C because 2. On the other hand, since n we see that each such u 4 is associated to exactly two weight 6 words of C (we have reversed the roles of C and C ? in Lemma 6.1). Hence the number of weight 6 words in C but not in C ? is at most there is at least one pair of words of weight 6 belonging to C \ C ? . Each of the words of weight 4 in C intersects T in two positions, which are never the same, otherwise the sum of two such words with 1+T would be a weight 2 word in C. Hence there is a one-to-one correspondence between the 15 elements in C 4 and the 2-subsets of T (respectively of 1+T ). We denote them by u Let u be a xed weight 4 word in C. Up to permutation, we can assume that T , u are in the following position: We assume rst that u 2 C \ C ? . From Lemma 6.1, there are 8 words meeting u in two positions; since t(u 0 are four possibilities for u \ Assume one of them appears at least three times, say the rst one and for Again because t and t are bijections, there is up to permutation only one possibility: generates the self-dual code B 12 ([19],[20]). We can next assume that the eight u 0 reach exactly twice the four possibilities for Again for the same argument, there is up to permutation only one possibility: and now generates the code C (1) 12 . The last case to consider is the case when no weight 4 word in C belongs to C ? . Hence C \ C ? is the 2-dimensional code generated by T and 1. From Lemma 6.1, we know that eight words of weight 4 in C meet u in one position. Then, at least one position is reached at least twice, say by must share another position outside u. Up to permutation, they are in the following positions: If a third word u 4 meets u again in the same position as u 2 and u 3 , this is also true for the other pairs but then either or u 2 +u 3 +u 4 +T +1 has weight 2, which is not possible. Hence each position in u corresponds to a pair of weight 4 words in C intersecting at that position. From the previous discussion, the sum is a weight 4 word which is disjoint from u; there are exactly two such words since n 4;0 and they are necessarily disjoint (if w is one of them, the other is w corresponds to and let be such that We have two choices up to permutation for the common position of u; u can be (on u) either 1000 or 0010. But it is easy to see that the rst one is not possible under the condition that t, t are bijective and that the second one leads to only one possibility: In that case, f1; generate the code C (2) 12 . Since we nd up to permutation two codes, which are distinguished by the dimension of C\C ? , and since the dual of an extremal even formally self-dual code is again an extremal formally self-dual code with even weights, these codes are necessarily equivalent to their duals. The automorphism groups have been computed with Magma. Remark 6.1 By \construction A", these codes construct non-isometric lattices which are 4-modular and extremal in the sense of H.-G. Quebbemann [21]. --R lattices and spherical designs preprint On error-correcting codes and invariant linear forms SIAM J A strengthening of the Assmus-Mattson theorem IEEE Trans Spherical codes and designs Geom. Overlap and covering polynomials with applications to designs and self-dual codes The On designs and formally self-dual codes De- signs On the notion of Jacobi polynomial for codes Math. A classi On the classi A shadow identity and an application to isoduality Abh. Handbook of Coding Theory Even unimodular extremal lattices --TR --CTR Koichi Betsumiya , Masaaki Harada, Classification of Formally Self-Dual Even Codes of Lengths up to 16, Designs, Codes and Cryptography, v.23 n.3, p.325-332, August 2001 Koichi Betsumiya , Masaaki Harada, Binary Optimal Odd Formally Self-Dual Codes, Designs, Codes and Cryptography, v.23 n.1, p.11-22, May 2001 J. E. Fields , P. Gaborit , W. C. Huffman , V. Pless, On the Classification of Extremal Even Formally Self-DualCodes, Designs, Codes and Cryptography, v.18 n.1-3, p.125-148, December 1999 Christine Bachoc , Philippe Gaborit, Designs and self-dual codes with long shadows, Journal of Combinatorial Theory Series A, v.105 n.1, p.15-34, January 2004 Olgica Milenkovic, Support Weight Enumerators and Coset Weight Distributions of Isodual Codes, Designs, Codes and Cryptography, v.35 n.1, p.81-109, April 2005 Kenichiro Tanabe, A Criterion for Designs in {\tf="P101461" Z}_4 David Masson, Designs and Representation of the Symmetric Group, Designs, Codes and Cryptography, v.28 n.3, p.283-302, April

codes;formally self-dual codes;harmonic functions;weight enumerator

607562

Factored Edge-Valued Binary Decision Diagrams.

Factored Edge-Valued Binary Decision Diagrams form an extension to Edge-Valued Binary Decision Diagrams. By associating both an additive and a multiplicative weight with the edges, FEVBDDs can be used to represent a wider range of functions concisely. As a result, the computational complexity for certain operations can be significantly reduced compared to EVBDDs. Additionally, the introduction of multiplicative edge weights allows us to directly represent the so-called complement edges which are used in OBDDs, thus providing a one to one mapping of all OBDDs to FEVBDDs. Applications such as integer linear programming and logic verification that have been proposed for EVBDDs also benefit from the extension. We also present a complete matrix package based on FEVBDDs and apply the package to the problem of solving the Chapman-Kolmogorov equations.

Introduction Over the past decade a drastic increase in the integration of VLSI chips has taken place. Conse- quently, the complexity of the circuit designs has risen dramatically so that today's circuit designers rely more and more on sophisticated computer-aided design (CAD) tools. The goal of CAD tools is to automatically transform a description in the algorithmic or behavioral domains to one in the physical domain, i.e. down to a layout mask for chip production. We divide this process into four different levels: system, behavioral, logic and layout. At the logic level, the behavior of the circuit is described by boolean functions. The efficiency of the algorithms applied in this level depends largely on the chosen data structure. Originally, representations such as the sum of products form or factored form representations were predominant. Today, the most popular data structure for boolean functions is the Ordered Binary Decision Diagram (OBDD) which provides a compact and canonical representation. In the wake of the successful introduction of the concept of function graphs by OBDDs, various other function graphs have been proposed which are not constrained to boolean functions but can be used to denote arithmetic functions. These function graphs have been used for state reduction in finite state machines and logic verification of higher-level specifications. Additionally, they have been applied to problems outside CAD, such as integer linear programming and matrix representation. Since the introduction of OBDDs by R. E. Bryant [5], several different forms of function graphs have been proposed. Functional Decision Diagrams (FDD) have been presented as an alternative to OBDDs for representing boolean functions [3]. Ordered Kronecker Functional Decision Diagrams (OKFDD) have been introduced in [10] as a generalization of OBDDs and FDDs. Multi-Terminal Binary Decision Diagrams (MTBDD) [9] have been proposed to represent integer valued functions and extended to functions on finite sets [2]. Edge-Valued Binary Decision Diagrams (EVBDD) [12][13][14] provide a more compact means of representing such functions. Recently Binary Moment Diagrams (BMD and *BMD) [7] were introduced which permit efficient word-level verification of arithmetic functions (including multipliers of up to 62-bit word size). This paper presents Factored Edge-Valued Binary Decision Diagrams (FEVBDD) as an extension to EVBDDs. By associating both an additive and a multiplicative weight with the edges, FEVBDDs can be used to represent a wider range of functions concisely. As a result, the computational complexity for certain operations can be significantly reduced compared to EVBDDs. Additionally, the introduction of multiplicative edge weights allows us to directly represent the complement edges which are used in OBDDs. This paper also describes uses of FEVBDDs in applications such as integer linear programming, logic verification and matrix representation and manipulation. 2 Review of Edge-Valued Binary Decision Diagrams Edge-Valued Binary Decision Diagrams, which were proposed by Lai, et al. [12][13][14] offer a direct extension to the concept of OBDDs. By associating a so-called edge value ev to every then-edge of the OBDD they are capable of representing pseudo-boolean functions such as integer valued functions. Their application has proven successful in such areas as formal verification and integer linear programming, spectral transformation, and function decomposition. Definition 2.1 An EVBDD is a tuple hc; fi where c is a constant value and f is a rooted, directed acyclic graph E) consisting of two types of vertices. ffl A nonterminal vertex f 2 V is represented by a quadruple child t (f); child e (f); evi, where variable(f) 2 fx is a binary variable ffl The single terminal vertex f 2 T with value 0 is denoted by 0. There is no nonterminal vertex f such that child t child e (f) and ev = 0, and there are no two nonterminal vertices f and g such that g. Furthermore, there exists an index function such that the following holds for every nonterminal vertex. If child t (f) is also nonterminal, then we must have child e (f) is nonterminal, then we must have Definition 2.2 An EVBDD hc; fi denotes the arithmetic function c f is the function f denoted by evi. The terminal node 0 represents the constant denotes the arithmetic function Definitions (2.1), (2.2) provide a graphical representation of pseudo-boolean functions. As a consequence integer variables have to be encoded in binary as in is a n-bit integer variable. It has been shown that EVBDDs form a canonical representation of functions. Definition 2.3 Given an EVBDD hc; fi representing f(x function F that for each variable x assigns a value F(x) equal to either 0 or 1, the function EVBDDeval is defined as c f is the terminal node 0 child e (f)i; F) boolean arithmetic Table 1: Arithmetic equivalents of boolean functions Boolean functions can be represented in EVBDDs by using the integers 0 and 1 to denote the boolean values true and false. Boolean operations are implemented through arithmetic operations as shown in Table 1. A method has been described by Lai, et al. that converts any OBDD representation of a boolean function to its corresponding EVBDD representation. It can be proven that both function graphs OBDD v and EVBDD denoting the same function f share the same topology except that the terminal node 1 is absent from the EVBDD and the edges connected to it are redirected to the single terminal node 0. Additionally, it was shown that boolean operations executed on EVBDDs have the same time complexity O(jf j \Delta jgj) as boolean operations on OBDDs. The concept of complement edges can not be realized in EVBDDs. As has been done for OBDDs, a generic operation apply can be defined that implements arbitrary arithmetic operations on the EVBDD representations of two arithmetic functions f and g. In general, the time complexity of such an operation on two EVBDDs and the flattened EVBDDs of respectively. A flattened EVBDD is defined in exactly the same manner as an MTBDD. For operations such as addition, subtraction, scalar-multiplication, etc. the time complexity of apply can be drastically reduced by exploiting certain properties. A scalar multiplication c \Delta f) can be done with time complexity O(jhc f ; f ij) by simply multiplying all edge values by c. All operations op, such as addition, that fulfill the additive property have the reduced time complexity O(jhc f Based on EVBDDs, the concept of structured EVBDDs (SEVBDDs) has been developed in [14]. SEVBDDs allow the modeling of conditional expressions and vectors. Their main use lies in the field of formal verification. 3 Factored Edge-Valued Binary Decision Diagrams Factored Edge-Valued Binary Decision Diagrams (FEVBDD) are an extension to EVBDDs. By associating both an additive and a multiplicative weight with the true-edges 1 FEVBDDs offer a more compact representation of linear functions, since common subfunctions differing only by an affine transformation can now be expressed by a single subgraph. Additionally, they allow the notion of complement edges to be transferred from OBDDs to FEVBDDs. Definition 3.1 An FEVBDD is a tuple hc; w; f; rulei where c and w are constant values, f is a rooted, directed acyclic graph E) consisting of two types of vertices, and rule is the set of weight normalizing rules applied to the graph. ffl A nonterminal vertex f 2 V is represented by a 6-tuple 2 child t (f); child e (f); ev; w is a binary variable. ffl The single terminal vertex f 2 T with value 0 is denoted by 0. By definition all branches leading to 0 have an associated weight There is no nonterminal vertex f such that child t there are no two nonterminal vertices f and g such that g. Furthermore, there exists an index function such that the following holds for every nonterminal vertex. If child t (f) is also nonterminal, then we must have If child e (f) is nonterminal, then we must have Definition 3.2 A FEVBDD denotes the arithmetic function c f is the function f denoted by i. The terminal node 0 represents the constant denotes the arithmetic function Definition 3.3 Given a FEVBDD representing that for each variable x assigns a value F(x) equal to either 0 or 1, the function FEVBDDeval is defined as: 1 The GCD rule requires also a multiplicative weight to be associated with the else-edges. 2 If we use the rational rule it holds that w nodes. Thus we can represent a nonterminal vertex by a 5-tuple hvariable(f); child t (f); child e (f); ev; w t c f f is the terminal node 0 Figure As an example, we construct the various function graphs based on the different decompositions of function f given in its tabular form in Figure 1. (2) 9 +3 (y(3 +2 z) Equation (2) is in a form that directly corresponds to the function decomposition for MTBDDs or ADDs and the tabular form. Equations (3) and (4) reflect the structure of the decomposition rules for EVBDDs and FEVBDDs, respectively. The different function graphs are shown in Figure 1. Figure goes here. Figure 3 goes here. Figure 4 goes here. Representations of signed integers based on FEVBDDs are presented in Figure 2 and representations of word-level sum and product are given in Figures 3 and 4. Lemma 3.1 Given two FEVBDDs which have been generated using the same weight normalizing rule and with f and g being non-isomorphic, it holds that there exists an assignment F 2 f0; 1g n such that c f +w f \Delta f 6= c g +w g \Delta g for this assignment. Proof: Case 1: if c f 6= c g then let . Case 2: c by the definition of non-isomorphism it holds that 9F such that Consequently, we have that for this assignment F. Case 3: c we assume that it holds that f and g are non-isomorphic and that for all assignments F. This implies that w f g. Consequently, f and g are isomorphic which contradicts the original assumption. Thus, it holds that 9F such that c f g. 2 Theorem 3.1 Two FEVBDDs that have been generated using the same weight normalizing rule, i.e. rule , denote the same function, i.e. only if c , and f and g are isomorphic. Proof: Sufficiency: If c and f and g are isomorphic, then 8F, directly from the definitions of isomorphism and FEVBDDeval. Necessity: If c f 6= c g then let holds that FEVBDDeval(hc f then let F be an arbitrary assignment such that FEVBDDeval(h0; it holds that FEVBDDeval(hc f and g are isomorphic then it holds by the definition of isomorphism and FEVBDDeval that It follows that c f +w f \Delta val 6= c g +w g \Delta val. If f and g are non-isomorphic lemma 3.1 holds. Nowwe have to prove the lemma for the last condition f being isomorphic to g. We need to show that if f and g are not isomorphic, then 9F 2 f0; 1g n such that FEVBDDeval(h0; Without loss of generality, we assume index(variable(f)) - index(variable(g)). Let prove the lemma by induction on k. Base: If and g are terminal nodes. Furthermore, and g are isomorphic. Induction hypothesis: Assume the above holds for Induction: We show that the hypothesis holds for i. Case 1: If ev f 6= ev g then let F(x n\Gammak it holds that ev g and w t f then let F be an arbitrary assignment such that F(x n\Gammak holds that FEVBDDeval(h0; \DeltaFEVBDDeval(h0; and g t are isomorphic it holds that FEVBDDeval(h0; val. Thus we have that ev f +w t f and g t are nonisomorphic then lemma 3.1 is applicable. Almost the identical prove can be given for ev , and w e f and , or f e and g e are nonisomorphic. Subcase 1: If f t and g t are nonisomorphic, then from and the induction hypothesis, we see that there exists some F such that FEVBDDeval(h0; let F 0 be defined as F 0 Subcase 2: Otherwise f e and g e are nonisomorphic, then by similar arguments, letting By definition of a reduced FEVBDD, we cannot have ev being isomorphic to f e . If ev f 6= 0, let F(x n\Gammak then FEVBDDeval(h0; g is independent of the first n \Gamma k bits. If ev then let F be an assignment such that F(x n\Gammak index(variable(g)))). Furthermore, let F be such that FEVBDDeval(h0; val f 6= 0 and FEVBDDeval(h0; If the corresponding subgraph of f with top-variable x n\Gammak and g are isomorphic then it holds that val val g . If the graphs are non-isomorphic we can apply the same reasoning as we did in the proof of lemma 3.1. Oth- erwise, f t and f e are non-isomorphic and at least one of them is not isomorphic to g. If f t and are non-isomorphic, then by induction hypothesis, there exists an assignment F such that 1 and F 0 It holds that FEVBDDeval(h0; As shown above, FEVBDDs form a canonical representation of a function only for specific weight normalizing rules that uniquely determine how the node weight of a new node is computed based on its both descendants. We propose two basic rules that can be used to guarantee canonicity for FEVBDDs. Given two FEVBDDs rule the node weight w of hc; w; f ; rulei is computed as follows 1. GCD rule: 2. RATIONAL rule: make new node(x i ,hc f /* compute the new weights */ /* guarantee uniqueness */ return Table 2: Make New Node These weight normalizing rules (cf. Table are applied whenever a new node is generated using the make new node routine. (cf. Table 2). This routine enforces both the canonicity of the function graph as well as its uniqueness. The routine find or add preserves the uniqueness of all nodes. Before a new node is actually created a quick hash table lookup is performed and, if the node is already a member of the table, the stored node with its unique ID is returned. Otherwise, a new node entry in the hash table is created and the new node with its unique ID is returned. Thus it is guaranteed that every node is stored only once in the hash table. Although the GCD rule requires a multiplicative weight to be associated with both the true- and the else-edges, there are some cases where it might be the rule of choice. If the function range is purely integer the GCD rule avoids dealing with fractions. This is particularly valuable, since all arithmetic operations on fractions are significantly more time consuming than the built in hardware routines for integers. Furthermore, the restriction to integers by use of the GCD rule brings a clear advantage in memory efficiency. Even though we need to store an additional weight, the memory consumption per node is less than when using the rational rule which requires the use of fractions. This is because every fraction is internally represented as one integer for the numerator and one norm weight(ev; w f case 'GCD': else if(w T 6= else return(sign case 'RATIONAL': return else if(w T 6= return else return ev; break; Table 3: Norm Weight for the denominator. Of course, as soon as the application requires the use of fractions the rational rule should be preferred. Nevertheless, the GCD rule is still applicable since we define: gcd( u 3.1 Operations As has been done for OBDDs [5] and EVBDDs [14], we provide a generic algorithm apply that implements arbitrary arithmetic operations on two FEVBDDs (cf. Table 4). Apply takes two FEVBDDs rule g i, as well as an operation op as its arguments. Both FEVBDDs have to be based on the same weight normalizing rule. The algorithm recursively branches at the top variable, i.e. the variable with the least index in f or g until it reaches a terminal case. Terminal cases depend on the operation op; as an example, for op='+' we have the terminal case The computational efficiency of this algorithm can be improved significantly by taking advantage of a computation cache. Before the recursive process is started, a quick lookup in the computation cache is performed and if successful, then the result of op is returned immediately without further computation. The entries of the cache are uniquely identified by a key consisting of the operands and the operation op. Whenever a new result is computed it is stored in the computation cache. In general the complexity of operations performed by apply is O(khc ik). As mentioned before we can further improve the computational complexity of apply by making use of properties of specific operations. We adapt the concept of an additive property proposed for EVBDDs by Lai, et al., [14] and extend it to the so-called affine property for FEVBDDs. Definition 3.4 An operator op applied to is said to satisfy the affine property if The factor w is defined as can be of arbitrary value. 3 3 Similar to the rational rule we can alternatively define the affine property as follows: All the benefits of the affine property remain the same. /* check for a terminal case */ if(terminal return /* is the result of op already available in the computation cache */ if(comp table lookup(hc rule g i,op,hc ans ; w ans ; ans; rule ans i)) return (hc ans ; w ans ; ans; rule ans i); /* perform the recursive computation of op*/ child t (g); rulei; else f child t (f); rulei; child e (f); rulei; else f ge /* store the result in the computation cache */ comp table insert(hc return Table 4: Apply Operations that satisfy the affine property are addition, subtraction, scalar multiplication and logical bit shifting. The main advantage of the affine property lies in reducing the computational complexity of apply. Since we can separately compute the parts of the result generated by the constants c f and c g and by the two subgraphs h0; w rulei, the hit ratio of the computation cache can be drastically increased by separating the influence of the constants and always storing only the results for This concept is applied to every recursion step so that the constant value is never passed down to the next recursion level. Unfortunately, we still have to pass the multiplicative weights w f and w g since they cannot be separated from the functions f and g. To achieve a further improvement in the hit ratio, we extract the common divisor w from w f and w g and promote only w 0 f and w 0 g . This is an advantage in such cases as reducing the problem of performing to the already computed problem (0 quantify the influence of the GCD extraction the worst case computational complexity for operations satisfying the affine property is given as O(jhc f the EVBDDs corresponding to the FEVBDDs respectively. Scalar multiplication and logical-bit shifting offer a better computational complexity since they can be computed in time independent of the size of the function graph. Scalar multiplication only requires the weights of the root node to be multiplied. In the case of EVBDDs we have to multiply every edge weight with the scalar; a task of complexity O(jf j). Since multiplication does not satisfy the affine property we are basically required to use the original version of apply. For the multiplication of two functions that both have a high percentage of reconverging branches, the following approach tends to improve the cache efficiency: We now have only O(jhc calls to multiply but every call requires three calls to apply for adding the separate terms. The first addition is not costly since the first term is always a constant, however, the second and third addition are potentially costly. In addition to the additive property, two further properties - the bounding property and the domain-reducing property - have been introduced by Lai, et al. [14] [12]. As has been done for the additive property, these properties can be easily adapted to FEVBDDs. 3.2 Representation of Boolean Functions Boolean Functions are represented in FEVBDDs by encoding the boolean values true and false as integers 1 and 0, respectively. All the basic boolean operations can be easily represented using only arithmetic operations. Thus we can easily represent any boolean function using FEVBDDs. Although we could implement the boolean operations based on their corresponding arithmetic functions, it is by far better in terms of computational complexity to directly use apply for boolean operations. All we need to do is to provide the necessary terminal cases for apply(hc boolean op). In the case of the boolean conjunction operation for example the terminal cases are: 1. 2. 3. if(hc To convert a boolean function from its OBDD to its FEVBDD representation we can adapt the algorithm suggested by Lai in [14]. Additionally, the concept of multiplicative weights allows us to directly represent the so called complement edges, so that we need to take care of this case in the algorithm: 1. convert the terminal node 0 to h0; 0; 0; rulei and 1 to h1; 0; 0; rulei. 2. for each nonterminal node hx i ; t; ei in the OBDD such that t and e have already been converted to FEVBDDs as the following conversion rules are applied: 3. if the branch leading fromnode hx i ; t; ei to t or e is a complement edge we have to perform the complementation by computing e, respectively. This is achieved by multiplying both weights c t (c e ) and w t (w e ) by \Gamma1 and later adding 1 to c t (c e ). The four basic conversion rules are listed below: The above conversion rules are not complete in the case of FEVBDDs since we can now also have variations in the multiplicative weights which can either be +1 or \Gamma1. These cases however are handled exactly according to the norm weighting rule that has been presented before, so that we do not explicitly list them here. As it has been done for EVBDDs [14], it can be shown that the following theorems hold. Theorem 3.2 Given an OBDD representation v of a boolean function with complement edges being allowed and an FEVBDD have the same topology except that the terminal node 1 is absent from the FEVBDD v 0 and the edges connected to it are redirected to the terminal node 0. Theorem 3.3 Given two OBDDs f and g with complement edges being allowed and the corresponding FEVBDDs time complexity of boolean operations on FEVBDDs (using apply) is O(jfj \Delta An example of a FEVBDD representing a boolean function with complement edges is given in Figure 5. This FEVBDD represents the four output functions of a 3-bit adder. It has the same topology (except for the terminal edges) as the corresponding OBDD depicted in the same figure. As it is shown in this example, FEVBDDs successfully extend the use of EVBDDs to represent boolean functions as they inherently offer a way to represent complement edges. Furthermore, the boolean operation 'not' can now be performed in constant time since it only requires manipulation of the weights of the root node. Figure 5 goes here. 3.3 Logic Verification The purpose of logic verification is to formally prove that the actual implementation satisfies the conditions defined by the specification. This is done by formally showing the equivalence between the combinational circuit, i.e. the description of the design and the specification of the intended behaviour. In general, the implementation is represented by an array of boolean functions f b and the specification is given by a word-level function fw . In order to transform the bit-level representation to the word-level we can use any encoding function to encode the binary input signals to the circuit. The set of input signals is partitioned into several subsets of binary signals x every array x i is then encoded using an encoding function encode i that provides a word-level interpretation of the binary input signals. Common encoding functions are signed-integer, one's-complement and two's-complement. The corresponding FEVBDDs are shown in Figure 2. Thus, the implementation can be described by an array of boolean functions f b The specification is given as a word-level function fw (X Verification is then done by proving the equivalence between an encoding of the binary output signals of the circuit, i.e. the array of boolean functions, and the word-level function of the encoded input signals: encode out (f b This strategy for logic verification was first proposed by Lai, et al., using EVBDDs [12][14]. Since FEVBDDs can describe both bit-level and word-level functions, they can be successfully applied to logic verification. Although all word-level operations can be represented by FEVBDDs, the space complexity of certain operations becomes exponential so that their application is limited to small word-length. Both EVBDD and FEVBDD representations of word-level multiplication are exponential; FEVBDDs however offer significant savings in memory consumption over EVBDDs. As can be seen in Figure 4 for word-level multiplication of two three-bit integers, the EVBDD contains 28 internal nodes whereas the FEVBDD representation requires only 10 nodes. In general, the EVBDD denoting the multiplication of two n-bit integers has (n nodes. The corresponding FEVBDD contains only n+ nodes and the ratio of EVBDD nodes to FEVBDD nodes is n+1 . As can be seen from this ratio, the savings in the number of nodes in the FEVBDD representation are of order n. As an example, a 16-bit multiplier requires 1,114,095 EVBDD nodes but only 65,551 FEVBDD nodes. Even if we take into account that a FEVBDD node requires 20 bytes versus only 12 bytes per EVBDD node, the savings remain significant (EVBDD:13.3 Mbyte, FEVBDD: 1.3 Mbyte). As has been done for EVBDDs [14], FEVBDDs can also be extended to structured FEVBDDs which allow the modeling of conditional expressions and vectors. 3.4 Integer Linear Programming An algorithm FGILP for solving Integer Linear Programming (ILP) problems based on EVBDDs has been proposed by Lai, et al. in [15]. FGILP realizes an ILP solver based on function graphs, which uses a mixed branch-and-bound/implicit-enumeration strategy. It has been shown that this approach can successfully compete with other branch-and-bound strategies that require the solution of the corresponding Linear Programming problems. The latter strategy is the one most widely applied in commercial programs. An ILP problem can be formulated as follows: minimize subject to with x i integer Since both EVBDDs and FEVBDDs allow only binary decision variables, the encodings shown in Figure 2 have to be applied. A 32-bit integer, for example, can be represented by an EVBDD or FEVBDD with nodes. Since FEVBDDs form an extension of EVBDDs we can also apply FEVBDDs to solve ILP problems. We expect a reduction in the memory requirement for FGILP when using FEVBDDs. This is due to the fact that different multiples of the integer variables x i appear in equations (5) and (6). If we use EVBDDs to represent these multiples of x i , we have to build an EVBDD for every different coefficient a ij since scalar multiplication on EVBDDs is performed by multiplying all edge weights with the factor. If we use FEVBDDs, however, we only have to store the FEVBDD representing x i once. Multiples of x i can be easily realized by associating the corresponding multiplicative edge weights with dangling incoming edges leading to x i . As an example, storing 6x, 7x and 5x requires 96 nodes if we use EVBDDs but only nodes if we apply FEVBDDs. 3.5 Implementation of Arbitrary Precision Arithmetic The introduction of multiplicative weights in combination with the RATIONAL rule for weight normalizing makes it necessary to extend the value range of the edge weights from the integer domain to the rational domain. This is done in a way such that any future expansion to other domains such as the complex domain can be easily achieved. All operations on edge weights are accessed through a standardized interface that invokes the specified function and then executes the requested operation depending on the current mode. Thus, the FEVBDD code remains fully independent of the selected domain. By changing to another mode we can easily switch from the integer domain to the rational domain, for example. This means we can still use the fast routines for single precision integers when necessary. Multiple precision integers are realized as arrays of integers and the arithmetic operations are implemented based on the algorithms for multiple precision arithmetic given by Knuth in [11]. Multiple precision fractions are implemented as arrays of two multiple precision integers where one integer represents the numerator and the other one the denominator. It is enforced by the package that the numerator and denominator remain relative prime and only the numerator can be signed. This is achieved by computing the greatest common divisor (GCD) of numerator and denominator and dividing both the numerator and denominator by the GCD. This operation is performed whenever an input is given. Internally the data is guaranteed to remain in the normalized form as this form is strictly enforced by all operations. Thus, a rational value is always uniquely represented by the numerator and denominator. The GCD can be computed very fast by Euclid's algorithm or the binary GCD algorithm [11]. For multi-precision fractions we use the binary GCD algorithm since it works very fast for integers of multiple word length. It only relies on subtraction and right shifting and does not require division operations. For single word precision fractions we employ the classical version of Euclid's algorithm since division can be executed very efficiently for single word integers. The basic arithmetic operations for fractions are realized as follows: ffl multiplication: ffl division: U ffl addition: 3.5.1 Symbolic Operations and Finite Fields FEVBDDs are not constrained to integer valued functions. As one can already see in the use of a rational rule, we can easily represent functions with rational function values. Complex values are also feasible; additionally, we can use symbolic computation. Even though the value ranges can be extended by using rational or complex edge weights, the decision variables still have to be binary. Thus, if we want to represent linear functions containing variables from the above value ranges, we have to encode them binarily such as it has been done for integers. Generally this approach leads to a means to represent any function on finite fields by FEVBDDs as it has been proposed for ADDs [2]. In this case the FEVBDD generally represents the function where \Phi and fi denote operations on the finite field. The ITE operator acts as a switch that either selects the subfunction denoted by the true- or else-edge. Contrary to the ADD approach we can exploit relationships between the subgraphs. 4 Matrix Representation and Manipulation Matrices have been successfully represented using MTBDDs [8] [9] and ADDs [2] and implementations of the basic matrix operations such as addition and multiplication have been given. A popular class of matrices that can be efficiently represented by MTBDDs and EVBDDs is the class of Walsh matrices which can be generated by a recursive rule. 4.1 Representation of Matrices The basic idea in using function graphs to represent matrices is to encode both the row and column position of the matrix elements using binary variables. An 8 \Theta 8 matrix, for example, requires 3 binary variables for the rows and another 3 for the columns. Basically, we can view the problem of representing a m \Theta n matrix as representing a function from the finite set of all element positions to the finite set R of its elements. The binary variables giving the row position are called row designators x 2 fx g, the ones denoting the column position are called column designators y 2 fy g. For the imposed variable ordering row and column designators are mixed together such that the order is g. Because of this chosen variable ordering subtrees in the function graph directly correspond to submatrices in the given matrix, as can be seen in Figure 6. Based on this correspondence the pseudo-boolean function denoting the matrix M can be given easily: fM xy Figure 6 goes here. Furthermore, this ordering allows matrices to be be represented compactly if they have submatrices that are identical (MTBDDs) or can be transformed into each other by an affine transformation 4 (FEVBDDs). Since the concept of square matrices, i.e. vertical size to keep many algorithms efficient and simple we will from now on only consider square matrices with max(m;n). To make non-square matrices square we can easily pad them with rows or columns filled with zeros. This does not significantly increase our memory consumption for storing the matrix since the padded blocks are uniform and can therefore be represented by only a few nodes. As it has already been mentioned, MTBDDs only offer a compact and memory efficient representation of matrices that feature identical subblocks. They require a different terminal node for each distinct matrix element. FEVBDDs can do far better than that. The concept of FEVBDDs allows two subblocks to be represented by the same subgraph if they differ only by an affine transformation of their elements. We will now introduce a special class of matrices that can always be represented by a FEVBDD of linear size. For this class of matrices the sizes of the corresponding MTBDD, EVBDD and *BMD are likely to be exponential. Definition 4.1 A recursively-affine matrix is recursively generated using the following rules: 1. we begin with a 1 \Theta 1 matrix M is a integer or rational constant value 2. in every recursion step a new matrix M n+1 is created based on the previous result M n such with being arbitrary integer or rational numbers. 4 An affine transformation is a transformation of the form y ! a Figure 7 shows the general structure of the FEVBDD that corresponds to a recursion step in building up a recursively affine matrix. In every recursion step a structure as shown in Figure 7 is added to the already constructed FEVBDD. Figure 7 goes here. As can be seen from Figure 7 we only need 3 nodes to represent a recursively-affine matrix of size n x n. As an example of a recursively affine matrix we build in Figure 8 the FEVBDD for the matrix M given below: 9 5 26 22 64 Figure 8 goes here. An important class of matrices that belongs to the family of recursively-affine matrices is the set of Walsh matrices in the Hadamard ordering [17]. These matrices can be used to compute spectral transforms of boolean functions. They are recursively defined as follows: Figure 9 shows both the FEVBDD and EVBDD representations of the Walsh matrix H h 3 . As can be seen in Figure 9, the size of the FEVBDD representation is 2 \Delta n where n denotes the order of the Walsh matrix. The size of the EVBDD representation is 4 Figure 9 goes here. Generally speaking, employing function graphs such as MTBDDs or FEVBDDs to represent sparse matrices offers the following advantages: 1. In comparison with normal sparse data structures, function graphs do provide a uniform log 2 (N) access time, where N is the number of real elements being stored in the function graph (for example, all non-zero elements of a sparse matrix) 2. Function graphs may not be able to beat sparse-matrix data structures in terms of worst space complexity. However, recombination of isomorphic subgraphs may give a considerable practical advantage to function graphs over other data structures. This is particularly valid for FEVBDDs since the same subgraph can represent all the matrices that can be generated by an affine transformation of the matrix represented by the subgraph. 4.2 Operations Operations on matrices can be divided into two major groups. The first group comprises termwise operations such as scalar multiplication, addition, etc. The second group is formed by matrix multiplication, matrix transpose and matrix inversion. Termwise operations are easily implemented based on function graphs. We can simply use apply to compute all termwise operations on matrices. This is obviously possible since apply(op) performs the operation op on every single function value, i.e. it works in a termwise manner. Matrix specific operations such as transposition require their own tailored algorithms. Matrix multiplication is clearly a non-termwise operation since it requires computing the scalar vector product of a row of the left matrix with a column of the right matrix to get the value of a single matrix element of the product matrix. Therefore, we will present two different recursive procedures to perform matrix multiplication on function graphs. The first method was proposed by McGeer [9]. This algorithm has the most direct link to the common conventional method for matrix multiplication. In every recursion step the problem is divided into four subproblems until a terminal case has been reached. In these steps operands are expanded with regard to a pair of row and column designators. This expansion even takes place if the function graphs are actually not dependent on the current pair of internal variables. By doing so there is no need for a scaling step as is necessary in the second method. Let matrix multiplication be denoted by ? and matrix addition by +. This method can be formally stated as: or written in terms of matrices:4 h xy h xy f xz f xz5 ?4 g zy g zy zy g zy5 The computations performed in every recursion step are: Obviously, this method requires eight calls to matrix multiply and four calls to matrix add in every recursion step, i.e. for every internal variable pair. The second method was proposed by Bahar [2]. Unlike the previous method it only expands the top variable of the two operands ffx g. In the process of matrix multiplication, the following variable order is imposed to decide whether the top variable of f or g has to be selected as the top variable for expansion. Depending on the character of the expansion variable var one of the following computations is being made in every recursion step. This approach only expands internal variables that are actually encountered in the function graphs f and g. It requires to keep track of missing z variables in f and g since every z expansion step corresponds to performing matrix addition. If p gives the number of omitted z expansions between two recursion steps we have to scale the result by 2 p before returning it. When using a cache we always store the unscaled results and scale the entry accordingly when reading the cache. Another method was proposed by Clarke [9]. Its basic idea is to take all the products first and then compute all the sums. For our matrix package we have implemented the second method which appears to be superior to the other two [2]. We implemented two different versions of this method. Version 1 passes the value of the edge weights down with every recursion step of matrix multiply and is of O(kfk \Delta kgk) complexity. As we have done for multiplication of two FEVBDDs we suggest a second version for function graphs with a high ratio of reconverging branches (e.g. for recursively-affine matrices) as follows. [f The operations rowadd and coladd which generate matrices such that a i a 0i a i a in only have complexity O(jf j). This second version requires only O(jf j \Delta jgj) calls to matrix multiply but every recursive call to matrix multiply also requires three calls to matrix add. It improves the cache efficiency of matrix multiplication considerably, if both operands are represented by FEVBDDs with a high ratio of reconverging branches. This outweighs the added overhead of three calls to matrix add. If this is not the case, it is better to use the original approach since it does not require the additional overhead. Matrix transposition is performed by exchanging the roles of column and row designators belonging to the same expansion level. To maintain the imposed variable ordering the nodes in the function graph have to be exchanged and it is not sufficient to just interpret row as column designators and vice versa. Transposition can be done in O(jf Matrix inversion is done by performing Gaussian elimination on the original matrix and the identity matrix at the same time. In other words we solve the system of linear equations A ? with the use of pivoting and row transformations. The steps required by Gaussian elimination consist of [19]: ffl selecting a partial pivot in every step j such that ja pj ffl normalizing the selected row by multiplying the row by the inverse of the pivot 1 ffl swapping rows j and p according to above pivot selection subtracting multiples of the pivot row j from all rows i ? j such that a All of the above operations except for row swapping can be implemented efficiently in time O(jAj) or O(jAj \Delta jRj) where R denotes the FEVBDD representing the pivot row. Row swapping is performed by matrix multiplication of matrix A with a permutation matrix P and therefore is of complexity Permutation matrices can be obtained by denotes a permutation matrix swapping rows i and j, I represents the identity matrix and M ij designates a matrix rs rs In general, partial pivoting is done in order to improve the numerical accuracy of Gaussian elimi- nation. Since our implementation relies on fractions of arbitrary precision we always use the exact values and numerical stability is not an issue. In order to avoid unnecessary row swapping we only perform the partial pivoting if it holds in step j that a In addition to the basic matrix operations, fast search operations for specific matrix elements have been implemented. Algorithms for searching both the value and position of the minimal, maximal or absolute maximal element in a given matrix were developed. This approach makes use of the min and max fields that can be associated with every node. The computational complexity for finding both the value and position of the minimal or maximal element in a n \Theta n matrix is O(log 2 (dne)). We will now explain the basic idea behind the algorithms in the case of searching for the maximal element. Given a FEVBDD node f and its two successors f t and f e we can easily determine which edge leads to the maximal element. Based on the values of the max and min fields of f t and f e we simply recompute the max field of f and select the successor that originally generated the max field of f. If only the value of the maximal or minimal element is of interest, it can be computed directly from the min and max field of the top node f without any further computation. value EVBDD FEVBDD range GCD RATIONAL integer 12 bytes 20 bytes 24 bytes fractions Table 5: Memory requirement per node 4.3 Experimental Results We have applied our FEVBDD based matrix package to the problem of solving the Chapman-Kolmogorov equations [18] that arise when computing the global state probabilities of FSMs. Though the memory consumption of our inversion routine is relatively low (8M for inverting a 64x64 matrix), the run time is very high. This is due to several factors. First, the algorithm for Gaussian elimination is purely sequential whereas FEVBDDs are recursively defined. Consequently, computation caching for matrix inversion does not exist. A recursive algorithm for matrix inversion will perform much better on FEVBDDs. Secondly, when using fractions of arbitrary length all operations need substantially more time than is necessary for ordinary integers. We therefore use the obtained inverses primarily as examples of real life non-sparse matrices that can be represented compactly using FEVBDDs and compare them with their EVBDD representations. As can be seen from the table below using FEVBDDs gives savings of up to 50% compared to EVBDDs in the number of nodes required to represent the non-sparse inverse. Of course, one has to consider that the storage requirement per node is higher for FEVBDDs than for EVBDDs. An overview of the memory usage per node in the various modes available for EVBDDs and FEVB- DDs is given in table 5. We assume that every EVBDD node consists of an integer or fractional edge value and two pointers to the children. Every FEVBDD node consists of two fractional edge weights and two pointers in the RATIONAL mode or three integer edge weights and two pointers in the GCD mode. The total memory consumption for storing the matrices using EVBDDs and FEVBDDs is shown in tables 6 and 7, respectively. The given memory usage is based on EVBDDs and FEVBDDs using fractions. The FEVBDDs have been generated using the RATIONAL rule. In the case of CK- Equations we have to use fractions for the edge weights since the matrix elements are fractions. As can be seen from the tables, FEVBDDs do better for the inverses but lose for the original matrices in terms of total memory consumption. This is due to the fact that the original matrices are sparse whereas the inverses are non-sparse. In the case of sparse matrices the additional properties of FEVBDDs are not exploited so that EVBDDs and FEVBDDs perform similarly in the number of nodes. FEVBDDs, however, lose in terms of memory requirement because of the higher cost per FEVBDD node. Since EVBDDs do at least as good as MTBDDs this also gives an idea of the performance of FEVBDDs compared to MTBDDs. 5 Conclusion We showed that by associating both an additive and a multiplicative weight with the edges of an Edge-Valued Binary Decision Diagram, EVBDDs could successfully be extended to Factored Edge-Valued Binary Decision Diagrams. The new data structure preserves the canonical property of the EVBDD and allows efficient caching of operational results. All properties that have been defined for EVBDDs could be adapted to FEVBDDs. The additive property was extended to the affine property. It was shown that FEVBDDs provide a more compact representation of arithmetic functions than EVBDDs. Additionally, the complexity of certain operations could be reduced significantly. We showed that FEVBDDs representing boolean functions allow us to incorporate the concept of complement edges that has originally been proposed for OBDDs. Furthermore, we showed that the EVBDD based Integer Linear Programming solver FGILP benefits from using FEVBDDs instead of EVBDDs. In combination with the FEVBDD package we also implemented an arithmetic package which supplies arithmetic operations on both integers and fractions of arbitrary precision. A complete matrix package based on FEVBDDs was introduced. We applied the package to solving the Chapman-Kolmogorov equations. The experimental results show that in the majority of cases FEVBDDs win over the corresponding EVBDD representation of the matrices in terms of number of nodes and memory consumption. Acknowledgement The authors like to thank Y.-T. Lai for supplying them with the EVBDD package and many helpful discussions. --R "Binary decision diagrams," "Al- gebraic Decision Diagrams and their Applications" "On the relation between BDDs and FDDs" "Efficient Implementation of a BDD Package," "Graph-Based Algorithms for Boolean Function Manipulation," "Symbolic Boolean Manipulation with Ordered Binary-Decision Diagrams," "Verification of Arithmetic Functions with Binary Moment Diagrams," "Spectral transforms for large Boolean functions with application to technology mapping" "Multi-terminal binary decision diagrams: an efficient data structure for matrix representation," "Efficient Representation and Manipulation of Switching Functions Based on Ordered Kronecker Functional Decisiond Diagrams" "The Art of Computer Programming Volume 2: Seminumerical Algorithms" "Edge-valued binary decision diagrams for multi-level hierarchical verification" Vrudhula, "EVBDD-based algorithms for integer linear programming, spectral transformation and function decomposition" "Edge-valued binary decision diagrams" Vrudhula, "FGILP: An integer linear program solver based on function graphs" "Representation of switching circuits by binary-decision-programs" "Fast Transforms. Algorithms, Analyses, Applications" "A First Course in Probability" "Introduction to Numerical Analysis" "Factored Edge-Valued Binary Decision Diagrams and their Application to Matrix Representation and Manipulation" --TR Graph-based algorithms for Boolean function manipulation Efficient implementation of a BDD package Symbolic Boolean manipulation with ordered binary-decision diagrams Edge-valued binary decision diagrams for multi-level hierarchical verification Spectral transforms for large boolean functions with applications to technology mapping Algebraic decision diagrams and their applications The art of computer programming, volume 2 (3rd ed.) Fast Transforms Formal Verification Using Edge-Valued Binary Decision Diagrams Verification of Arithmetic Functions with Binary Moment Diagrams --CTR Rolf Drechsler , Bernd Becker , Stefan Ruppertz, K*BMDs: A New Data Structure for Verification, Proceedings of the 1996 European conference on Design and Test, p.2, March 11-14, 1996 Rolf Drechsler , Wolfgang Gnther , Stefan Hreth, Minimization of word-level decision diagrams, Integration, the VLSI Journal, v.33 n.1, p.39-70, December 2002

logic verification;Ordered Binary Decision Diagrams;integer linear programming;pseudo-boolean functions;matrix operations;affine property

607573

Hierarchical Reachability Graph Generation for Petri Nets.

Reachability analysis is the most general approach to the analysis of Petri nets. Due to the well-known problem of state-space explosion, generation of the reachability set and reachability graph with the known approaches often becomes intractable even for moderately sized nets. This paper presents a new method to generate and represent the reachability set and reachability graph of large Petri nets in a compositional and hierarchical way. The representation is related to previously known Kronecker-based representations, and contains the complete information about reachable markings and possible transitions. Consequently, all properties that it is possible for the reachability graph to decide can be decided using the Kronecker representation. The central idea of the new technique is a divide and conquer approach. Based on net-level results, nets are decomposed, and reachability graphs for parts are generated and combined. The whole approach can be realized in a completely automated way and has been integrated in a Petri net-based analysis tool.

Introduction Petri Nets (PNs) are an established formalism to describe and analyze dynamic systems. Among the large number of available analysis techniques, the generation of the set of all reachable markings and all possible transitions is the most general approach, which is theoretically applicable for every bounded net. The resulting graph is denoted as the reachability graph (RG) or occurrence graph. The set of reachable markings is denoted as the reachability set (RS). Reachable markings of the PN build the vertices of the graph and transitions describe the edges. Edges may be labeled with the corresponding transition identifier from the PN description. The RG contains the full information about the dynamic behavior of the PN and can be easily analyzed to gain results about the functional behavior as required for the verification of system properties. RGs are generated by an algorithm computing all successor markings for discovered markings, starting with the initial marking of the net. This approach is conceptually simple and is integrated in most software tools developed for the analysis of PNs. In practice, unfortunately, the size of RGs often grows exponentially with the size of the PN in terms of places and tokens. Hence RG generation is usable only for relatively small nets, much smaller than most practically relevant examples are. Consequently, a large number of approaches has been published to increase the size of RGs which can be handled. A straightforward idea is to increase the available computing power and memory to increase the size of RGs. This is done by using powerful parallel or distributed computer architectures. Examples for this approach can be found in [1, 9] describing implementations on various parallel architectures and [14, 25], where workstation clusters are used for RG generation. These approaches describe RG exploration for Generalized Stochastic Petri Nets (GSPNs), however, they apply for RG exploration of PNs as well. The general problem of parallel/distributed state space generation is still, that an exponentially growing problem is attacked by increasing the available resources at most linearly. Additionally, the problem of an efficient parallelization of RG generation arises. Efficient realization of the RG generation algorithm in a distributed way is non-trivial since the different distributed tasks are dependent and require synchronization introducing additional overhead. In particular, the speedup that can be reached by a parallel implementation is model dependent which makes the problem of an efficient general purpose realization of parallel RG generation even harder. An alternative to handle large RGs is to reduce their size without loosing relevant information. This idea can be exploited at two different levels. First, the net can be simplified by reducing the number of places and transitions. The corresponding approaches are denoted as reduction rules, published for uncolored PNs in [4] and subsequently for colored PNs (CPNs) in [16]. Reduction rules are defined with respect to the properties of interest. Thus, first properties need to be defined and then reduction rules which preserve these properties can be introduced, which yields a set of predefined rules for a set of predefined properties as in [4, 16]. The main drawback of reduction rules is that their applicability is restricted to relatively specific structures. Consequently, the gain obtained by reduction rules is for most nets relatively small and reduction rules can most times only be used as an a priori step which does not solve the problem of large RGs. The second approach to reduce the size of RGs is to perform the reduction at the level of reachable markings. Such an approach requires a compositional state space generation such that generation and reduction can be interleaved. Different techniques exploiting this idea exist. The usual way is to define the complete PN as a collection of interacting components. Usually component RGs are much smaller than the complete RG. Thus, RGs for the components are generated efficiently and are reduced according to some reduction rules which preserve relevant properties. Subsequently, reduced component state spaces are composed. In most approaches in this context, components interact via synchronized transitions. In [13], an approach for CPNs is introduced where RGs of components are generated in parallel by considering only local transitions. Additionally, a synchronization graph describing synchronized transitions is defined. By interleaving local and synchronized transition firing the complete RG can be generated or properties holding on the complete RG can be proved. Similarly in [29], complete component RGs are generated first, which are finally combined and reduced such that important properties like deadlocks or boundedness are preserved. In [34], a compositional analysis method for place-bordered subnets is presented. It is also based on the interleaving of composition and behavior preserving reduction. In [23], a different approach for components composed via synchronized transitions is proposed. The approach introduces a compact representation of the complete RG and an efficient way to characterize RS. The idea is that the incidence matrix characterizing RG can be composed via Kronecker operations from incidence matrices of component RGs and RS is a subset of the cross product of component reachability sets. Knowing RS and the RGs of the components, RS and RG of the CPN are completely characterized. In [5], an approach for hierarchical RG generation is proposed for hierarchically structured CPNs. Similar to the previous approach the RG is described using component RGs by composing incidence matrices via Kronecker operations. The approach requires that the complete net is structured in an appropriate way. The disadvantage of all these methods for efficient RG generation is that the component structure has to be defined by the modeler and all methods are very sensitive for the component structure. Techniques which reduce the size of RG by behavior preserving reduction depend on the required results. If relatively detailed results are required, most reduction fail or have only a small effect on the size of RG. Other methods for efficient RG computation include the stubborn set method [35], which eliminates unnecessary interleavings from RG during generation, and the exploitation of symmetries to reduce RG [10, 22]. In both methods some additional computation is necessary during RG generation and the CPN has to observe several structural conditions that the methods can be used in an efficient way (i.e., to exploit symmetries, the CPN has to contain symmetric parts, otherwise the reduction has no effect). The idea of symmetry exploitation combined with a compact representation of RG by composing component RGs is described in [17] for quantitative analysis. Techniques based on ordered binary decision diagrams (OBDDs) rely on symmetries as well, in [30] Pastor et. al. describe OBDD-algorithms mainly for 1- bounded PNs. Apart from techniques to characterize the complete or reduced RG in an efficient and compact way, several approaches to derive results without generating RS and RG exist. Usually these approaches yield only partial results in the sense that we can not formally prove results, we can only disprove some by finding failure states. These techniques include simulation and invariant analysis [22]. In this paper, we introduce an approach which is related to the work presented in [23] and [5]. RS and RG are handled in a compositional way, which allows the representation and generation of large RSs/RGs. In contrast to other known methods performing compositional analysis, our approach represents the complete RG. Behavior preserving reduction is not applied. Consequently, arbitrary properties can be be checked on the resulting RG. However, it is also possible to combine the approach with behavior preserving reduction, although this is not considered in this paper. The proposed technique can be completely automated for a large class of PNs including all PNs which are covered by P-invariants. We present the approach here for uncolored PNs to simplify notation. Keeping in mind that every CPN with finite color sets can be unfolded to a uncolored PN [22], it obvious that the approach can be applied for a large class of CPNs too. The structure of the paper is as follows. In Sect. 2, the PN class is defined, reachability and invariant analysis are introduced. Sect 3 describes the definition of regions which divide a PN into subnets. In Sect. 4 an abstraction operator is described which allows us to abstract from details in the net description to reduce the size of RS. Afterwards, Sect. 5, introduces a hierarchical and compositional representation of RS and RG. Then, different analysis approaches are proposed which exploit the hierarchical representation of RS and RG. Sect. 7 contains a non-trivial example to clarify the advantages of the new approach compared to conventional RG generation. Basic Definitions and Known Results We assume that the reader is familiar with PNs and the related basic concepts. For details about these fundamentals we refer to [21, 22, 28]. net is a 5 tuple is a finite and non-empty set of places, is a finite and non-empty set of transitions are the backward and forward incidence functions, and IN is the initial marking. The initial marking is a special case of a marking marking M can be interpreted as an integer (row) vector which includes per place p one element which describes the number of tokens on place p. gives the set of input places for a transition t, and gives the set of output places. Analogously we define The notion can directly be extended to sets. In the sequel we consider connected nets, i.e. each place, transition has at least one incoming and one outgoing arc. Transition t 2 T is enabled in marking M , transition enabled in M can fire, changing the marking of any p 2 P to marking M 0 This will be indicated by M [t?M 0 , M [t? denotes that t is enabled in M and M [? describes the set of enabled transitions in M . Considering firing sequences yields to the definition of the language which is the set of all reachable markings for PN . The reachability graph RG(PN) contains nodes for every M 2 RS(PN) and an arc necessary, arcs can be labeled with the corresponding transition and/or a transition rate as for stochastic Petri nets (SPNs). SPNs [27] extend the above class slightly by the association of exponentially distributed firing times with transitions. We define a function is the set of non-negative numbers. W (t; M) is the rate of an exponential distribution associated with transition t in marking M . We assume that W (t; M) ? 0:0 if t is enabled in M . The RG of a SPN results from the RG of the corresponding PN by adding transition rates to the edges. RS is identical in both cases. SPNs can be used for performance analysis by analyzing the continuous time Markov chain described by the SPN [27]. The incidence matrix C is a matrix which contains for each place p 2 P a row and for each transition a column such that C(p; It can be used to define net-level properties of a net PN . covered by positive -invariants, if for each place A vector y covered by positive T -invariants, if for each transition t 2 T a T -invariant y - 0 with y(t) ? 0 exists. An algorithm for computation of invariants is given in [26], although its time complexity is exponential for a worst case, usually invariant computation is much easier than generation of RS and RG. Incidence matrix and invariants ensure certain properties, however they do not completely characterize RS. The following theorem summarizes some classical results. Theorem 1 For a PN with a set of P -invariants X and a set of T -invariants Y the following results hold. ffl If marking M 0 is reachable from marking M , then an integer vector z exists such that M M+Cz T . This implies that for every M 2 RS(PN) an integer vector z M with exists. ffl If x; x Analogously for Y . ffl For each reachable marking M the relation Mx has to hold for all x 2 X. ffl If PN is covered by positive P -invariants, then it is bounded. ffl If PN is bounded and live, then it is covered by positive T -invariants. Proof: Proofs can be found in standard books on PNs. ffi Although invariants offer some insight in the dynamic behavior of the modeled system, they are most times not sufficient to obtain the required results. Thus, RS and RG have to be generated for a detailed analysis. Usually, first RS is generated and the arcs of RG are computed in a second step. The following algorithm computes RS for a PN , it terminates if RS contains a finite number of markings 1 . generate RS (PN) while (U 6= ;) do remove M from U ; for all t 2 M [? do od od Set U contains markings for which successors have not been generated, whereas RS contains all generated markings. For U a simple data structure like a queue or stack is sufficient since elements only have to be added and removed. For RS a data structure allowing an efficient membership test is necessary. Consequently, RS can be realized using an appropriate hash function or a tree like structure allowing a membership test with an effort logarithmic in the number of elements. The problem with hashing are possible collisions. It is usually very hard to avoid collisions for general PNs. Therefore, most software tools use binary trees for the generation of RS. We briefly analyze the effort required for the generation of RG, when a binary tree is used to store RS. Let n be the number of markings in RS and let n \Delta d the number of arcs in RG. Hence the mean slightly extended version catches infinite RS, see coverability graph construction in PN literature. number of successors per marking is d. The time required for the generation of RS is in the order of d which is approximately d \Delta n \Delta log 2 (n). Additionally, memory limitations have to be taken into account. Even if more sophisticated data structures are used for RS, the number of markings which can be generated on a standard workstation lies between 150; 000 and 1; 500; 000. For PNs including a large number of places, the value can be much smaller. In case of certain symmetries, ordered binary decision diagrams (OBDDs) are able to handle extremely large sizes of RS and RG (see [8] among others). Pastor et al [30] describe how OBDD-techniques can be applied for PNs. However, the use of OBDDs requires the existence of symmetries to yield a compact representation. After RS has been generated, the arcs in RG are generated in a second step. RG can be represented by a n \Theta n incidence matrix Q. If transition identities and rates are not relevant, Q can be stored as a Boolean matrix. Otherwise Q has to include the required information. Autonomous Regions in PNS In this section, we define parts of a PN which will be latter substituted by a less detailed representation in an abstraction operation. These parts, which are denoted as regions, have a place border at the input and a transition output border. This is different from other hierarchical constructs in the PN area [22], where places or transitions are refined. However, the definition is natural from a behaviorally oriented point of view (see also [11]), because a region describes a part acting for its own. Communication is performed by receiving tokens from the environment (place bordered input) and sending tokens to the environment (transition bordered output). of a set of transitions of r r are the corresponding functions of PN restricted to P r and T r , respectively. PN r defines a region iff the input bags of transitions in T r and in T are disjoint, i.e., P r . For a region PN r , the set of output transitions T out consists of all t 2 T r such that I Analogously the set of input transition is T in A region describes an autonomous part of a PN, which will be used to define a hierarchical structure. A region is minimal if it contains no region as a proper subset 2 . The concept is illustrated by the following example, which will serve as a running example to accomplish the line of argumentation. Example 1 We consider a producer/consumer model where a producer A successively fills two buffers B1 and B2. Fig. 1 shows the corresponding PN, where places fp1; p2; p3g describe the state of the producer. Places fp5; p7g are buffer places, whose capacity is limited by places fp4; p6g. Buffer B2 is always filled with two items/tokens at once, while B1 obtains single tokens. The model contains two consumers of equal behavior. A consumer non-deterministically takes tokens from each buffer, but the first buffer is only considered if two consumers are willing to consume. Places fp8; p9; p10g give the state of both consumers. The model is clearly artificial and just intended to illustrate our concepts. Minimal regions in this model are shown in Fig. 1 by shaded polygons. Proposition 1 For a PN with regions N r1 ; N 1. minimal regions are disjoint, i.e. if N r1 , N r2 are minimal and N r1 6= N r2 then P 2. minimal regions define a partition, i.e. for each exactly one minimal region ri with ri ), 3. regions are closed under union, i.e. if N r1 , N r2 are regions then T r1 [T r2 defines a subnet N r which is a region. Minimal regions coincide with the equivalence relation of the conflict relation [33]. p3 p6 Producer A Buffer B1 Consumer C22Buffer B2 Figure 1: Producer/Consumer Model and its partition into minimal regions Proof. Straightforward for nets, where each place has at least one outgoing arc. If this is not the case, one need to define an additional region, which consists of places with empty set of output transitions. Minimal regions can be generated using the simple algorithm shown below. while UT 6= ; do remove t 0 from UT ; while do remove t with fflt " fflT i 6= ; from UT ; od od Once the algorithm terminated, sets T i contains transitions which are used to define regions according to Def. 3 (i.e., (fflT i 4 Generation of Abstract Views Let PN be a Petri net. We want to enhance information associated to a place by the following kind of vector: Definition 4 A p-vector v p for a place p 2 P is a vector v p 2 ZZ n+m with index Entries in v p are referenced by v p (x) for x obtain lower index values. An aggregation function AG : ZZ n+m \ThetaIN n \Gamma! IN for p-vectors and markings is defined as AG(v A linear combination LC : ZZ n+m \Theta ZZ n+m \Theta T \Gamma! ZZ n+m of p-vectors v a , v b is defined for a t 2 T , lcm(jv a (t)j; jv b (t)j)=jv a (t)j and c lcm gives the least common multiple of two integers and gcd of an integer vector is the greatest common divisor of the elements. Note that v c We inductively define extended nets which result from a sequence of net transformations based on linear combinations: be a sequence of transitions of a PN , and ffl denote the empty sequence. An extended net is a tuple (N inductively defined as follows: (N is an extended net where Let for better readability be an abbreviation for the resulting new vectors of a linear combination w.r.t. transition t, and Used = fv a A s g denote the vectors used to generate New for an extended net (N s st st ; A st ) is an extended net if (N s is an extended net, s 2 (Tnftg) and st st st st st s I st I s st Note that c a ; c b - 0 by construction. We additionally distinguish ordinary places P ord those generated in the extension sequence denoted by P agg s . The definition separates available vectors A s from the total set of vectors V s in order to ensure that vectors are used in at most one step of a sequence s. This restriction is made in order to focus on those linear combinations which are relevant in the following. The net transformation basically mimicks the computation of P-invariants according to [26]. (N s contains all transitions t 2 T exactly once describes an extended net where each P -invariant is realized by a place p 2 P agg . For an aggregated place p representing a P -invariant, the marking is constant (i.e., I all We can interpret the marking of an aggregated place as a macro marking which includes an abstract view of the detailed marking. Since the complete net does not exchange tokens with the environment, macro markings representing P-invariants are invariant. However, if sequence s contains only a subset of transitions, then the marking of an aggregated place represents possibly only a macro marking for a subset of places belonging to a P -invariant. In this case, the marking of the aggregated place changes whenever tokens are added to or removed from the partial P-invariant it represents. Since the net transformation follows the computation of P-invariants, the effort is limited to the effort for computing P-invariants. Often the effort is much smaller, since only a subset of transitions is used in s. Example 2 Before we go into further details we come back on our running example: it contains P- invariants (described as formal sums): and a T-invariant (2 Fig. 2 shows the extended net for sequence are hatched and arcs are dotted to indicate the differences from N ffl . The minimal regions which are connected via t1, t6, and t7 have been merged; shaded polygons denote the new and larger regions in N s . p-vectors and corresponding linear combinations are given in the tabular below: p3 p6 Figure 2: Extended net Note that the definition of extended net defines arc weights for new arcs connected to new places as weighted arcs of the original net. This allows to consider bidirectional arcs (self-loops) appropriately, as the arcs connected to places p12 and p13 in Fig. 2 illustrate. A place represents a set of places in P ord , and gives an aggregated marking for the marking of this set. Aggregating information is the crucial point in deducing a hierarchy. Before we describe a way to split an extended net into a high level net and a set of low level nets to obtain the desired hierarchy, we formalize the aggregation and subsequently consider reachability and language invariance of the net extension. Proof. We consider an induction over transition sequences s, initially all trivially fulfill the lemma and P agg ;. For the induction step we consider a st which results from v where v a ; v b 2 New at step s (the case p 2 P s is trivial in N st since for these places M 0 , I \Gamma and I+ remain unchanged). We further consider an induction over firing sequences oe, initially M 0 holds by definition. For the induction step, we consider M where the induction assumption ensures we have to show M 0 after extending oe by a transition -. according to the definition of successor marking and extended net. By induction assumption such that we can replace M(p) in the equation above. Observe that I \Gamma ,I remain invariant for z 2 P ffl , such that we obtain: st st (z; -)) ut The way in which places are added to a net in an extension sequence, ensures that the reachability set and the language remain the same. Lemma 2 For all s 2 T for which an extended net (N s Proof. by induction over sequences s, initially trivially fulfilled. For the induction step we start with the special case directly implies equality. This case can occur e.g. if A or if 6 9v a For the general case we give a proof by contradiction: case st [oe ? M st but oe not possible in PN s . Hence 9t 0 2 oe which is not enabled in PN s . i.e. there are less tokens in a place p 2 P s . Contradiction to definition of extended nets, because M 0 ; I changes only with respect to new places. case but oe not possible in PN st . Hence 9t 0 2 oe which is not enabled in PN st . i.e. there are less tokens in a place st nP s . According to Lemma 1, st z;t and obviously for each z 2 z;t , we obtain a contradiction. In summary equality holds. Equivalence of languages follows by the same line of argumentation. ut A direct consequence of Lemma 2 is that invariants remain valid, T-invariants due to language equivalence and P-invariants due to additivity of invariants, cf. Theorem 1. Furthermore we can decide for places like p13 in our example, where I transitions t, whether a given initial marking M 0 (p13) ensures that a transition is dead due to M 0 or whether the place can safely by omitted since M(p13) - I \Gamma (p13; t) for all M 2 RS. If the former is the case, it is clear that the net is not live. So far we have described a way to add places to a net without changing its reachability set or language. The notion of an extended net is only a formal prop to introduce a hierarchical net; it simplifies argumentation why a hierarchical net indeed includes the reachability set or language of its N ffl . The key issue for a hierarchy is abstraction: at a higher level the state of a subsystem must be represented in less detail than at a lower level. We use aggregated places to obtain an aggregated state representation and the notion of subsystem is build on the concept of region. Let R(N ffl ) denote the set of minimal regions w.r.t. to an extended net When we extend this net for a transition t 2 T out r of a region N r 2 R(N ffl ), then the new places connect N r with regions that contain tffl. Consequently we merge all these regions with N r , which yields a new region N 0 r according to Prop. 1. Since we start from a partition into regions, the resulting set of regions is a partition again, but this partition is less fine. t becomes internal in N 0 r , because fflt [ r , and new places give an aggregated description of the internal behavior of N 0 r w.r.t. transition t. Following this procedure over a sequence s of transitions yields (N s partition into regions, where some regions have internal transitions and aggregated places. In this situation a decomposition of an extended net into a high level net using the aggregated description and a set of low level nets resulting from regions with an internal behavior gives the two-level hierarchy we aim for. More formally, a high level net for a given extended net (N s results from a projection with respect to A s . be an extended net, its corresponding high level net and I are the corresponding projections of I \Gamma , I Example 3 Fig. 3 shows the high level net for the extended net of our running example in Fig. 2.2p4 p6 t5222 Figure 3: High level net for Proof. uses previous lemma about equality for N ffl and N s , and that HN is deduced by omitting places (releases enabling conditions thus increases RS) and by omitting transitions, which are isolated (since all elements in the pre- and postset of such transitions are used in linear combinations and thus not contained in A s anymore). Isolated transitions have no effect on RS. ut P-invariants of HN are linear combinations of P-invariants of N ffl , hence if N ffl is covered by P-invariants, so is HN . Consequently, we can guarantee finiteness of RS(HN) if N ffl is covered by P-invariants. Lemma 3 states that the HN indeed considers a more abstract net, such that the detailed net can only behave in a way which is consistent with this abstraction/aggregation. If a region in the extended net contains all places of the set Used for a transition t, it shows an internal behavior, which allows to define a non-trivial isolated low-level net: Definition 7 A low level net for a region r in an extended net (N s with respect to N ffl ). I I I L+ I H+ (p)(t) if I If LN and the corresponding region r H in HN do not differ in their transitions, LN is trivial and can be neglected. Otherwise LN is non-trivial. Observe that both, HN and LNs do only pick aggregated places from A s , elements in V s nA s are neglected, because their information is sufficiently represented by the linear combinations they contributed to. A LN for a region r and its HN share places transitions T out r . These common net elements form an interface for the HN to communicate with the LN in an asynchronous manner. The HN puts tokens via transitions t 2 T in sends signals to the LN - and experiences output behavior through firing of t 2 T out r . The notion of hierarchy is justified, since the HN abstracts from the details inside the LN, tokens on P describe a so-called macro marking of the LN, and T out r represent the aggregated behaviour of the LN. However an LN can be merged with the HN to observe the detailed behavior. Formally we describe this as an extended LN (EN) as are defined analogously. An EN is an ordinary PN, with RS(EN) and L(EN) defined as before. The relationship between HN and LN is not symmetric, because the HN has an aggregated description of the LN by P r , but not vice versa. As seen from the LN, a HN provides an environment, with whom the LN interacts in an asynchronous manner, but the LN lacks the aggregated description of the HN behavior. Hence the reachability of a LN cannot be seen independently of a HN, such that the reachability set RS(LN) of the LN for a given environment HN needs the notion of EN and results from the projection of RS(EN)jLN . In the following section we will use these concepts to give a hierarchical/compositional representation of RS(N) based on RS(HN) and RS(i)j i for each LN i. Example 4 Figs. 4,5 show the extended low level nets for the two non-trivial low level nets - indicated by shaded polygons - of our running example. They result from the producer part, where transition t1 becomes internal, and from the consumer part, where transitions t6 and t7 become internal. For the consumer region, only p13 becomes part of A t1t6t7 because v12 has been used for its construction. The region with transition t3 has no internal transition. Obviously selection of s has a massive impact on the resulting hierarchical net description. Consideration of "optimal" sequences is subject to further investigations. At this state we can only formulate goals and rules of thumb to follow: 1. it is clear, that a non-trivial LN results from merging adjacent regions, i.e. if for N s with regions in s, then all t should become element of s. p6 t722 Figure 4: Low level net and extended low level net for the consumer region p3 p3 p6 t522 Figure 5: Low level net and extended low level net for the producer region 2. deriving a hierarchy is a divide and conquer strategy, so a sequence should yield a set of non-trivial LNs, such that complexity is equally distributed over this set. This means that only those regions are merged, whose result will not cover a majority of the net. 3. aggregated places introduce overhead, especially building all linear combinations can impose an unacceptable increase of net elements. This is the reason for an exponential worst case time complexity of invariant computation. In our case we have the freedom to select transitions and to consider only a subset of transitions. Hence those transitions t are preferred where w.r.t. N s is relatively small. In our implementation of the proposed approach, we have integrated these heuristic rules to generate appropriate transition sequences. First experiences with several examples (e.g., the example presented in Sect. 7) are very encouraging. The program automatically chooses a sequence of transitions which partions complex nets into non-trivial parts and a non-trivial HPN. Hierarchical Representations of RS and RG Dividing PN into HN and LNs allows us to generate and represent RS(PN) and RG(PN) in a space and time efficient way. For notational convenience we assume that PN is decomposed into one HN and J LNs, which are consecutively numbered 1 through J . Furthermore we assume in the sequel that all reachability sets for the HN and the extended LNs are finite. Consequently, reachability sets are isomorphic to finite sets of consecutive integers. Thus let x corresponds to the x-th marking in RS(HN ), and we use x and M x interchangeably. We can represent RG(HN) by a nH \Theta nH any two markings in RG(HN) at most one transition exists. If more than one transition between M x and M y exists, describes a list of transition indexes. We use for generality the notation t 2 Q H (x; y) for all t that fulfill M x The reachability set RS(j) of a LN j depends on the environment given by HN. Hence, we consider the EN e that corresponds to j and define RS(j) as a projection of RS(e) on the places of j. Since any LN j and the HN share some places as a projection of RS(j) onto places from . Markings from g RS(j) are macro markings and allow to partition RS(j). Macro markings are useful for RS(j) generation since full details of an HN are irrelevant at the LN level, one can redefine transitions of T in , such that their firing is marking dependent with respect to RS(HN)j P j "P h ' g RS(j). So EN is only a formal prop to obtain a clear notion of RS(j), in practice however, computation of RS(j) can be performed more efficiently by using only macro marking dependent transitions of T in since transitions local to HN are ignored. The resulting set RS(j) might contain markings which are not in RS(PN ), but these can be eliminated in a subsequent step, cf. Sect. 6. Let g and denote by RS(j; ~ x) with ~ x 1g the set of markings from which belong to marking ~ x in g RS(j). Markings from a set RS(j; ~ x) are indistinguishable in the HN, i.e., the marking of the places from is the same. Since reachability sets are assumed to be finite, each set RS(j; ~ x) can be represented by a set of integers 1g. A marking M x 2 RS(HN) uniquely determines the macro markings for all LNs. We denote by x j the macro marking of LN j belonging to marking x and obtain M x Markings of PN s can be characterized using J +1-dimensional integer vectors Jg. xH describes a marking from RS(HN) and a marking from RS(j; x j H describes the macro marking of LNj when the marking of HN equals xH . This implies M xH j P H "P . Since the previous relation holds, each integer vector of the previously introduced form determines a marking of the extended net. We define a hierarchically generated reachability set Observe that the number of markings in RS H (PN s ) equals Y Lemma 4 The hierarchically generated reachability set and the reachability set are related as follows. Proof. The previous lemmas imply that RS(PN s )j Jg, and for each M 2 RS(PN s By construction of the hierarchically generated reachability set M 2 RS H (PN s ) follows. The second relation follows since Equation (8) describes a very compact way to represent huge reachability sets by composing a few smaller sets. Observe that a few reachability sets with some hundred of markings are enough to describe sets with several millions or billions of markings. To keep the representation compact, it has to be assured that reachability sets of the LNs are roughly of the same size. Of course, this is hard to assure a priori, but it is possible to generate regions in a way that they include a similar number of places and transitions which is often sufficient to yield reachability sets of a similar size for the different regions. However, the reachability set of the original net is not equal to RS H (PN s ), it is only included in the hierarchically generated reachability set. Before we compute RS(PN s ) as part of RS H (PN s ), the reachability graph is represented in a compact form similarly to the compact representation of the reachability set. First of all, we define the effect of transitions locally for LNs. Two different classes of transitions have to be distinguished with respect to LN j. H the set of local transitions in LN j. j the set of transitions, which describe the communication between LN j and HN. The effect of transitions at marking level is defined using Boolean matrices. As usual we assume that multiplication of Boolean values is defined as Boolean and and summation as Boolean or. Thus let t [~x; ~ y] be a n j (~x) \Theta n j (~y) matrix describing transitions in the reachability graph of LN j due to firing transition t. Q transition t is enabled in marking x 2 RS j [~x] and firing of t yields successor marking y 2 RS j [~y]. All remaining elements in the matrices are 0. Since transitions t 2 LT j do not modify the marking of the HN, Q y and t 2 LT j . Furthermore, we define for y and 0 n j (~x);n j (~y) otherwise. I n is a n \Theta n matrix with 1 at the diagonal and 0 elsewhere. 0 n;m is a n \Theta m matrix with all elements equal to 0. The reason for this definition is that a transition does not modify the marking of LN j and cannot be disabled by LN j. This is exactly described by matrices I and 0. Define for If q j enabled by LN j in marking x. For the HN we define q H In all other cases q H The matrices describe the effect of transitions with respect to the HN or a single LN. The next step is to consider the effect of a transition with respect to the global net. Transition t is enabled in marking Y It is straightforward to prove this enabling condition. Since q j enabling depends only on the marking of parts where the transition belongs to. A transition is enabled if it is enabled in all parts simultaneously. In a similar way we can characterize transitions between markings. Transition t is enabled in marking its firing yields successor marking (y H ; y Y This relation allows us to characterize the reachability graph completely. To do this in a more elegant way, we define Kronecker operations for matrices. Definition 8 The Kronecker product A\Omega B of a nA \Theta mA matrix A and a nB \Theta mB matrix B is defined as a nAnB \Theta mAmB matrix The Kronecker sum A \Phi B is defined for square matrices only as A\Omega I nB \Thetan I nA \Thetan The definition of Kronecker sums/products does not include the data type of the matrix elements. Indeed all kinds of algebraic rings can be used. In particular we consider here Boolean or real values. Since the Kronecker product is associative we can define a generalization for J matrices A j of dimension O Y I l j O O I In the same way the Kronecker sum can be defined for n as I l j O O I Observe that C is a matrix with columns. If we consider the number of non-zero elements in C in terms of the number of non-zero element in A j and denote the number of non-zero elements in a matrix A as nz(A), then we obtain Y Kronecker sums and product are a very compact way to represent huge matrices. Implicitly Kronecker operations realize a linearization of a J dimensional number. Row indices of matrix C or D are computed from the row indices of the matrices A j using the relation Y where x is the row index in C or D, x j is the row index in A j and n j is the number of rows of A j . In the same way column indices are computed from the relation Y y is the column index in C or D, y j is the column index in A j and m j is the number of columns of A j . These representations are denoted as mixed radix number representations. Obviously x (y) determines all x j (y j ) and vice versa. For complementary information about Kronecker operations and mixed radix number schemes we refer to [15], considering an example with is recommended. Mixed radix numbering schemes can as well be applied to number markings in RS H (PN s ). However, we use a two level scheme, where the first number describes the HN marking and the second number is computed from the numbers of LN markings. Thus marking where Y Using this numbering scheme, RG H (PN s ) can be represented using Kronecker products of Boolean matrices. We define Q H t as the incidence matrix of the reachability graph considering only transition t. Using the two level marking number, Q H t has a block structure with n 2 H block matrices. includes all transitions between markings belonging to HN marking x and markings belonging to HN marking y due to transition t in the net. Each submatrix can be represented as a Kronecker product of LN matrices. O This form describes a very compact representation of a huge matrix. Assume that Q t is a Boolean matrix describing transition t at RS(PN ), then after appropriate ordering of markings (i.e., markings from RG(PN) " RG H (PN) are followed by markings from RG H (PN) n RG(PN )). If the initial marking is part of RG(PN) " RG H (PN ), the above representation implies that successors of reachable markings can be computed using matrices Q H t and, consequently, also reachability analysis can be performed using these matrices. The incidence matrix of RG H (PN) can be represented as For a compact representation, the Kronecker representation is definitely preferable. It can be applied in various analysis algorithms as shown in the Sect. 6. Local transitions cause a specific matrix pattern of nonzero elements. Since Q i t [x; x] equals an identity matrix for t 2 LT j , j 6= i and Q j y, I l j ]\Omega I u j (x) if collecting local transitions in one matrix l [x we obtain the following representation for a submatrix of Q H l [x does not distinguish between different local transitions of the same LN. If such a distinction is necessary, transitions which have to be visible can be excluded from the sets LT j . In this way it is possible to keep all relevant information in the representation of RG H (PN s ). If we consider SPNs, transitions are enhanced by a transition rate. Thus Q contains real instead of Boolean values. However, the Kronecker representation of the matrix is very similar. If all transitions have marking independent transition rates - t , matrix Q H is given by In this case the elements of Q t are interpreted as real values 1:0 and 0:0, respectively. Example 5 The running example is rather small so that we can not expect practical gain from representing RS or RG in a compositional way as proposed in this section. However, even for this simple example the representation becomes more compact and the example allows us to clarify the general concepts. The following table summarizes the number of markings in column RS and the number of transitions in the RG in the corresponding column for the various nets considered here. Obviously the HN has a RG which is significantly reduced compared to PN . Marking description Successor markings no ap1 ap2 p2 p4 p5 p6 p7 tnr no tnr no tnr no 22 26 Table 1: Reachable markings and possible transitions of the HN. RS RG transitions PN 254 622 All markings and possible transitions of the HN are shown in Tab. 1. Macro markings with respect to LN 1 are defined by projection of the HN marking on the places ap1; p4; macro markings for LN 2 are defined by projection of the HN marking on ap2; p5; p7. In both cases 9 macro markings are generated. From the extended nets, RS j and the matrices Q j are computed. For LN 1 a macro marking represents in the average 2 markings. This small number is not surprising since LN 1 consists internally of two places connected via single transition, such that macro markings only abstract from the internal place where tokens reside. For the second LN more internal details are hidden by the aggregated description used in the HN. Consequently, a macro marking of LN 2 represents in the average 9 detailed markings. The Kronecker representation requires 195 transitions to represent the complete reachability graph with 622 transitions. Of course, this comparison does not consider overhead to store different matrices in the Kronecker representation. However, the overhead depends on the number of transitions in [ST i and the number of LNs. Both quantities are negligible compared to the number of markings if we consider large nets. The hierarchically generated reachability set RS h includes 270 markings, which means that markings are unreachable. We consider this point in the subsequent section. 6 Hierarchical Analysis Approaches We now introduce analysis approaches which rely on the Kronecker representation of RG(PN ). In particular it is necessary to introduce a method to characterize RS(PN) and not only a superset in form of RS h (PN s ). The central idea of reachability analysis is that the numbering of markings in RS h (PN s ) is a perfect hash function for markings in RS(PN ). This has first been exploited for efficient reachability analysis of SGSPNs, a class of generalized SPNs consisting of components synchronized via transitions, in the work of Kemper [23]. We can use a similar approach here, but do not necessarily rely on it, see e.g., [12] as an alternative. Let the number of markings in RS h when the marking of the HN is M x . Let r[x] be a Boolean vector of length n(x) which is used to store results of the reachability analysis. Thus r[x H ](x L marking after termination that Formally we use here one Boolean vector per HN marking, but it is obviously possible to store all these vectors consecutively in a single Boolean vector of appropriate length. Reachability analysis requires, apart from the vectors r[x] and the different matrices introduced in the previous section, a set U to store unexplored markings, similar to the set U used in generate RS. However, now U only has to store integer pairs instead of complete marking vectors. Let be the number of the initial marking, then r[x 0H ](x 0L ) is initialized with 1, all remaining vector components are zero. Additionally, U is initialized with the following algorithm is used to determine reachable markings. generate structured RS (PN) while (U 6= ;) do remove to J do for all y j with l do // compute successor in subnet j for all y H with Q H do // compute successor in subnet HN for all j with t 2 T j do L with exists then (*) else if y L - 0 then In the step indicated by ( ), the algorithm exploits the fact that firing of transition t always yields a unique successor marking. Therefore each row of a matrix Q i t can include at most one element. The approach can be easily extended for PNs where different successor markings are possible. This situation occurs in nets where probabilistic output bags for transitions are allowed. Since the algorithm computes all successor markings of reachable markings, it is straightforward to prove that generate structured RS generates RS(PN) and terminates when RS(PN) is finite which is the case here, since RS h (PN s ) has assumed to be finite. The remaining point is the comparison of generate structured RS and generate RS. As before we assume that the reachability set contains n markings and in the average d transitions are possible in each marking. The theoretical time complexity of generate RS is O(nd log 2 n) if insert and member functions on RS use log 2 n operations. The complexity of generate structured RS is in O(nd), since the Boolean vectors allow us to test in O(1) whether a marking has been reached before. The reduction by a logarithmic factor seems to be not too much on a first glance. However, the approach is used for large reachability sets such that this implies a reduction by at least an order of magnitude. Additionally the constants behind the asymptotic complexity are much lower for generate structured RS. The reason is that all operations are performed with simple integer operations, while several operations of generate RS are time consuming. For example, if a new marking M is found in generate RS; a data structure to hold M has to be allocated and inserted into the data structure storing the already generated markings. Since this data structure is usually a tree, pointers have to be modified. In generate structured RS the same operation only requires to set a bit in vector r. Thus usually we can expect an improvement of run times which is around two orders of magnitude for large reachability sets. However, to apply generate structured RS, PN has to be decomposed first and the reachability sets and matrices for the subnets have to be generated. The complexity of both problems is for large nets much lower than reachability analysis. This can also be seen in the example presented below. Apart from time complexity, we also have to compare space complexity. Of course, also the difference in memory requirements depends on the concrete example. However, if the net has been decomposed into LNs with roughly identical reachability set sizes and the size of RS h (PN s ) and RN(PN) does not differ too much (i.e., not by several orders of magnitude), then (8) assures that the size of the LN/HN reachability sets and matrices is negligible compared with the size of the complete reachability set and graph. Experiences show that the approach allows us to handle much larger reachability sets. An additional advantage of generate structured RS is that we make use of secondary memory in a very efficient way. Since vector r is structured into subvectors and successor markings are computed consecutively for subvectors, it is possible to preload required subvectors from secondary memory. After the reachability set has been computed by setting the values in vector r, it can be decided in O(1) whether a marking is reachable or not. Furthermore, all successor markings for a marking can be computed from the Kronecker representation for local transitions in a constant time and for others in a time at most linear in the number of subnets. Since the Kronecker representation includes information about transitions yielding to successors, even successors reachable by specific transitions can be computed. Based on these basic steps, standard algorithms for model checking can be applied for the nets. In a similar way the Kronecker representation can be exploited for the quantitative analysis of SPNs. The basic step here is to realize the product of a vector with a sum of Kronecker products of matrices. However, this step is already known in numerical analysis and can be combined with various iterative numerical analysis techniques [3]. Thus the Kronecker representation allows the analysis of large SPN models, which cannot be handled using standard means. For further details we refer to the literature Example 6 The size of the running example is so small that is useless to compare runtimes for reachability graph generation. Instead we briefly consider unreachable markings appearing in the hierarchical representation. As already mentioned RS h contains 270 markings, but only 254 of them are reachable. As an example for unreachable markings we consider markings of the form (0; the vector includes the number of tokens on the places p1 \Gamma p7. For the places p8; p9; p10, we now consider possible markings. Obviously all three places are part of a P-invariant such that the sum of tokens on these places has to equal 2. In the hierarchical generated reachability set all possible distributions of 2 tokens over the places p8; p9; p10 are included. However, reachability analysis shows that only markings are reachable where place p10 is empty. The reason for this restriction can be explained by considering the behavior of the net in some more detail. A token on p10 implies that t5 has fired after t4. But since p6 is empty, t3 fired after t5 and, since p2 is non-empty t1 and t2 fired also after t5. Now after firing t2 a token resides on p5 which has to be transferred to p4 by firing t4. However, this means that t4 fired after t5 and p10 has to be empty. The restriction which assures that p10 has to be empty when the marking of the places p1 \Gamma p7 is as shown above, is a global restriction which depends on the whole net. So it is not visible in an isolated part and the above mentioned markings belong to RS h , but reachability analysis shows that they are not reachable and are not part of RS. Two other optimizations can be used to improve generate structured RS. First optimization As noticed in [23] certain unnecessary interleavings due to internal transitions can be eliminated. The idea is that local transitions in different LNs do not interfere. Thus if t 1 2 LT i are both enabled in some marking, then the sequences t 1 t 2 and t 2 t 1 are both possible and yield an identical successor marking. Consequently it is only necessary to consider one sequence. More general, for a set of local transitions which belong all to different LNs and which are enabled in some marking, only those transition sequences which are described by a subset of T and where transitions occur in the order as described in T need to be considered. This reduces the number of possible sequences from l \Delta l! to l \Delta . In this way the time complexity of reachability analysis can be reduced. Second optimization In [6] an approach is discussed which reduces time and space complexity. The idea is to reduce a priori the marking sets of LN by combining some markings which are always together reachable or not. As a simple example consider two markings x; y 2 RS(i) and a pair of transitions is enabled in x and its firing yields y and t 0 is enabled in y yielding successor marking x, then x is reachable whenever y is reachable and vice versa. We denote this as identical reachability of markings. Obviously identical reachability holds for all markings in an irreducible subset of a matrix l [x; y]. In [6] it is shown that this condition can be further relaxed. However, this extension is beyond the scope of this paper. Markings which are identically reachable can be aggregated a priori. Aggregation in this case means that a set of identically reachable markings is substituted by a single aggregate marking such that all transitions entering or leaving one marking in the subset are substituted by transitions entering/leaving the aggregate marking and transitions between markings in the subset are substituted by transitions starting and ending in the aggregate marking. These transformations are easily performed by adding in the matrices Q i all rows and columns belonging to markings in the subset to be aggregated. The size of RS(i) and RG(i) is reduced by this aggregation which implies that the size of RS h (PN s ) and also the effort for reachability analysis are reduced too. After reachability analysis the reachability of an aggregated marking implies that all markings represented by this aggregated marking are also reachable and vice versa; if the aggregated marking is not reachable, then the detailed markings are also not reachable. Both optimizations depend on the net which is considered. However, for most nets the effort for reachability analysis can be reduced significantly. 7 An Application Example jsj reg, total reg, non-tr P agg RS h RS(HN) max RS(j) percent RS(PN) Table 2: Hierarchical representation for sequence s The running example we considered so far is only useful to illustrate formal concepts, in order to demonstrate applicability of our approach we consider the production cell of [24], which has been subject to modeling and analysis by a variety of tools and which is known to be non-trivial. The production cell model originates from an existing production cell in an industrial setting, which physically consists of six components: a elevating rotary table, a rotable robot with two extendable arms, a traveling crane gen hierarchy gen struct RS jsj CPU user CPU user 111 2.5 3.0 73.3 74.0 112 2.6 3.0 64.9 65.0 Table 3: Computation characteristics for sequence s and two conveyor belts. The production cell performs transportation and processing metal plates in a (cyclic) pipeline. A feeding conveyor belt transports metal plates to the elevating table, the table lifts plates for the robot, the robot inserts plates into the press and takes them after pressing from the press onto the second conveyor belt. Originally plates leave the system by the second belt, but in order to have a closed system, the crane is installed to put plates from the second belt onto the feeding belt, such that the number of plates within the system is constant. Thanks to the work of Heiner et al. [18, 19] a Petri net model exists, which considers processing of 5 plates. Refinement is used to organize a model of this size, however the dynamic behavior of the model is not defined unless all refined subnets are available in full detail. This kind of hierarchy is very common for modeling purposes, but useless in terms of analysis. Hence our analysis starts from a flat Place/Transition net with 231 places and 202 transitions 3 . From [19] it is known that the net is live and 1-bounded. The reachability set contains 1,657,242 markings and the reachability graph 6,746,379 transitions. The algorithm to derive a hierarchy starts from a partition into minimal regions and considers a sequence s, which starts with transitions being internal in minimal regions (which is the case for 74 transitions in our example), subsequently it considers small regions first. Fig. 6 shows how the total number of regions decreases once the internal regions have been considered. On the other hand, the number of non-trivial regions increases in an initial phase since the algorithm prefers small regions and finally decreases when there are no trivial regions left and non-trivial regions are merged. Table 2 indicates the influence of s on the hierarchical representation of RS, it gives the number of regions (non-trivial and in total), the number of aggregated places P agg , cardinalities of the hierarchical reachability set RS h , the reachability set of the high level net RS(HN ), and the maximal number of markings observed among the low level nets. The quality of the whole construction is shown in column "percent RS(PN )", which gives the reachable fraction of RS h . Table 3 gives corresponding computation times in seconds for the computation of the two-level hierarchy and the subsequent computation of the reachability set RS(PN) contained in RS h (PN ), times are given as CPU time and user (wall clock) time. These times have been observed on a SPARCstation 4 with 64 MB main memory, 890 MB virtual memory, and 110 MHz CPU. Obviously computation times are uncritical if the number of aggregated places does not explode. It is worth mentioning that it takes slightly more than a minute to generate the complete reachability set and reachability graph and represent them in a very space efficient way. About 6 Megabyte memory space are necessary to generate and represent RG and RS. These values are excellent compared to conventional RG generation algorithms. In [18] the same model has been analyzed on a similar workstation using different PN analysis tools. RG generation with the tool PROD needs about 14 hours (see [18]). The small runtimes and storage requirements show that much larger systems can be handled with the approach. We have also analyzed an open version of the production cell for which other tools where not able to generate RS (see [18]). For this version our method needs about 3 minutes real time to generate RS with 2,776,936 markings and RG with 13,152,132 arcs. As already noticed in [19], computation of a generating set of semi-positive P-invariants is difficult for this net. Our approach is closely related to invariant computation: if we compute an extended net for a sequence covering all transitions T , we obtain a generating set of P-invariants as well. However this 3 We thank J. Spranger for translating the model into the APNN format [2] used in our implementation. length of sequence s10305070 regions non-trivial regions Figure Total number of regions, number of non-trivial regions length of sequence s50150250350450550 places regions Figure 7: Number of aggregated places, number of non-trivial regions extreme is not suitable and we consider only a subset of transitions, in order to remain some activity in the HN. From a pragmatic point of view the approach allows us to consider those transitions which can be handled with acceptable computational costs and stop the derivation of a hierarchy if it becomes too expensive. Fig. 7 clearly indicates that a careful selection of transitions can avoid high computational costs. However there is a sharp increase after 108 steps, and the hierarchy derivation stops after 113 steps. For a P-invariant computation 202 steps are necessary, hence Fig. 7 also illustrates the difficulties for invariant computation observed in [19]. According to the results in Table 2, the number of regions and a limit for the number of aggregated places give suitable parameters to stop the automatic hierarchy generation where it makes sense. Conclusions We have proposed a new approach for the efficient generation and compact representation of reachability sets and graphs of large PNs. In contrast to other approaches, the technique can be applied to general nets without definition of a hierarchical structure and without inherent symmetries. The structuring of the PN into asynchronously interacting regions is done automatically by an algorithm which uses a basic step related to invariant computation to make a transition internal to a region. The algorithm considers a sequence of distinct transitions which can be arbitrary in principle. For our implementation we use some heuristic rules in order to structure a net into regions of approximately the same size. The algorithm stops once a user given number of regions has been obtained. Usually the number of regions should not be chosen too large, to avoid a too complex HN. For nets covered by P-invariants termination is guaranteed, however we cannot ensure termination for general PNs. The problem is that reachability sets of some part, HN or a LN, can become unbounded, even if the reachability set of the complete net is bounded. This problem can not occur for nets which are covered by P-invariants. The non-trivial example considered in this paper illustrates our experience with the algorithm exercised on a set of examples: the new approach allows the time and space efficient generation and representation of huge reachability sets and graphs. This is, of course, a step towards the analysis of complex PNs. Our current research aims at the integration of the algorithms for model-checking with the Kronecker representation of the reachability graph. First results indicate that this approach allows to analyze much larger nets than conventional means. Additionally, the Kronecker representation can be used for the efficient analysis of SPNs using numerical analysis techniques. For an overview of these techniques we refer to [7]. --R State space construction and steady state solution of GSPNs on a shared-memory multiprocessor Abstract Petri net notation Complexity of Kronecker operations and sparse matrices with applications to the solution of Markov models Transformation and decomposition of nets Hierarchical high level Hierarchical Structuring of Superposed GSPNs Structured Analysis Approaches for Large Markov Chains States and Beyond Parallel state space exploration for GSPN models On well-formed coloured nets and their symbolic reachability graph Distributed simulation of Storage alternatives for large structured state spaces Modular state space analysis of coloured Distributed state-space generation of discrete-state stochastic models IEEE Trans. A. reduction theory for coloured Asynchronous composition of high level Petri net based design and analysis of reactive systems A case study in developing control software on manufacturing systems Reachability trees for high-level Coloured Coloured Reachability analysis based on structured representations Formal development of reactive systems Analysis of large GSPN models: a distributed solution tool A simple and fast algorithm to obtain all invariants of a generalized Petri net Performance analysis using stochastic Hierarchical reachability graph generation of bounded Petri net analysis using Boolean manipulation A comparative study of methods for efficient reachability analysis The numerical solution of stochastic automata networks A class of modular and hierarchical cooperating systems Compositional analysis with place bordered subnets State of the art report: stubborn sets --TR Automatic verification of finite-state concurrent systems using temporal logic specifications Transformations and decompositions of nets A reduction theory for coloured nets The concurrency workbench Using partial orders for the efficient verification of deadlock freedom and safety properties A symbolic reachability graph for coloured Petri nets Colored Petri nets (vol. Automated parallelization of discrete state-space generation On generating a hierarchy for GSPN analysis Structured analysis approaches for large Markov chains Communication and Concurrency Distributed Simulation of Petri Nets Hierarchical Reachability Graph of Bounded Petri Nets for Concurrent-Software Analysis Structured Solution of Asynchronously Communicating Stochastic Modules Hierarchical Structuring of Superposed GSPNs Application and Theory of Petri Nets On Limits and Possibilities of Automated Protocol Analysis An analysis of bistate hashing Saturation Modular State Space Analysis of Coloured Petri Nets Parallel State Space Exploration for GSPN Models A Toolbox for the Analysis of Discrete Event Dynamic Systems Reachability Analysis Based on Structured Representations {SC}*ECS A survey of equivalence notions for net based systems A Simple and Fast Algorithm to Obtain All Invariants of a Generalized Petri Net Reliable Hashing without Collosion Detection Compositional Analysis with Place-Bordered Subnets Hierarchical High Level Petri Nets for Complex System Analysis Superposed Generalized Stochastic Petri Nets Petri Net Analysis Using Boolean Manipulation Storage Alternatives for Large Structured State Spaces State Space Construction and Steady--State Solution of GSPNs on a Shared--Memory Multiprocessor Analysis of large GSPN models --CTR Michael Muskulus , Daniela Besozzi , Robert Brijder , Paolo Cazzaniga , Sanne Houweling , Dario Pescini , Grzegorz Rozenberg, Cycles and communicating classes in membrane systems and molecular dynamics, Theoretical Computer Science, v.372 n.2-3, p.242-266, March, 2007

invariant analysis;hierarchical structure;reachability graph;reachability set;petri nets

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Latent Semantic Kernels.

Kernel methods like support vector machines have successfully been used for text categorization. A standard choice of kernel function has been the inner product between the vector-space representation of two documents, in analogy with classical information retrieval (IR) approaches.Latent semantic indexing (LSI) has been successfully used for IR purposes as a technique for capturing semantic relations between terms and inserting them into the similarity measure between two documents. One of its main drawbacks, in IR, is its computational cost.In this paper we describe how the LSI approach can be implemented in a kernel-defined feature space.We provide experimental results demonstrating that the approach can significantly improve performance, and that it does not impair it.

Introduction Kernel-based learning methods (KMs) are a state-of-the-art class of learning algo- rithms, whose best known example is Support Vector Machines (SVMs) [3]. In this approach, data items are mapped into high-dimensional spaces, where information about their mutual positions (inner products) is used for constructing classification, regression, or clustering rules. They are modular systems, formed by a general purpose learning module (e.g. classification or clustering) and by a data-specific ele- ment, called the kernel, that acts as an interface between the data and the learning machine by defining the mapping into the feature space. Kernel-based algorithms exploit the information encoded in the inner-product between all pairs of data items. Somewhat surprisingly, this information is sufficient to run many standard machine learning algorithms, from the Perceptron Convergence algorithm to Principal Components Analysis (PCA), from Ridge Regression to nearest neighbour. The advantage of adopting this alternative representation is that often there is an efficient method to compute inner products between very complex, in some cases even infinite dimensional, vectors. Since the explicit representation of feature vectors corresponding to data items is not necessary, KMs have the advantage of accessing feature spaces that would otherwise be either too expensive or too complicated to represent. Strong model selection techniques based on Statistical Learning Theory [26] have been developed for such systems in order to avoid overfitting in high dimensional spaces. It is not surprising that one of the areas where such systems work most naturally is text categorization, where the standard representation of documents is as very high-dimensional vectors, and where standard retrieval techniques are based precisely on the inner-products between vectors. The combination of these two methods has been pioneered by Joachims [10], and successively explored by several others [6, 11]. This approach to documents representation is known as the 'bag of words', and is based on mapping documents to large vectors indicating which words occur in the text. The vectors have as many dimensions as terms in the corpus (usually several thousands), and the corresponding entries are zero if a term does not occur in the document at hand, and positive otherwise. Two documents are hence considered similar if they use (approximately) the same terms. Despite the high dimensionality of such spaces (much higher than the training set size), Support Vector Machines have been shown to perform very well [10]. This paper investigates one possible avenue for extending Joachims' work, by incorporating more information in the kernel. When used in Information retrieval (IR) this representation is known to suffer from some drawbacks, in particular the fact that semantic relations between terms are not taken into account. Documents that talk about related topics using different terms are mapped to very distant regions of the feature space. A map that captures some semantic information would be useful, particularly if it could be achieved with a "semantic kernel", that computes the similarity between documents by also considering relations between different terms. Using a kernel that somehow takes this fact into consideration would enable the system to extract much more information from documents. One possible approach is the one adopted by [23], where a semantic network is used to explicitly compute the similarity level between terms. Such information is encoded in the kernel, and defines a new metric in the feature space, or equivalently a further mapping of the documents into another feature space. In this paper we propose to use a technique known in Information Retrieval as Latent Semantic Indexing (LSI) [4]. In this approach, the documents are implicitly mapped into a "semantic space", where documents that do not share any terms can still be close to each other if their terms are semantically related. The semantic similarity between two terms is inferred by an analysis of their co-occurrence pat- terns: terms that co-occur often in the same documents are considered as related. This statistical co-occurrence information is extracted by means of a Singular Value Decomposition of the term by document matrix, in the way described in Section 3. We show how this step can be performed implicitly in any kernel-induced feature space, and how it amounts to a 'kernel adaptation' or `semantic kernel learning' step. Once we have fixed the dimension of the new feature space, its computation is equivalent to solving a convex optimization problem of eigenvalue decomposition, so it has just one global maximum that can be found efficiently. Since eigenvalue decomposition can become expensive for very large datasets we develop an approximation technique based on the Gram-Schmidt orthogonalisation procedure. In practice this method can actually perform better than the LSI method. We provide experimental results with text and non-text data showing that the techniques can deliver significant improvements on some datasets, and certainly never reduce performance. Then we discuss their advantages, limitations, and their relationships with other methods. 2. Kernel Methods for Text Kernel methods are a new approach to solving machine learning problems. By developing algorithms that only make use of inner products between images of different inputs in a feature space, their application becomes possible to very rich feature spaces provided the inner products can be computed. In this way they avoid the need to explicitly compute the feature vector for a given input. One of the key advantages of this approach is its modularity: the decoupling of algorithm design and statistical analysis from the problem of creating appropriate function/feature spaces for a particular application. Furthermore, the design of kernels themselves can be performed in a modular fashion: simple rules exist to combine or adapt basic kernels in order to construct more complex ones, in a way that guarantees that the kernel corresponds to an inner product in some feature space. The main result of this paper can also be regarded as one such kernel adaptation procedure. Though the idea of using a kernel defined feature space is not new [1], it is only recently that its full potential has begun to be realised. The first problem to be considered was classification of labelled examples in the so-called Support Vector Machine [2, 3], with the corresponding statistical learning analysis described in [20]. However, this turned out to be only the beginning of the development of a portfolio of algorithms for clustering [17] using Principal Components Analysis (PCA) in the feature space, regression [24], novelty detection [19], and ordinal learning [7]. At the same time links have been made between this statistical learning approach, the Bayesian approach known as Gaussian Processes [13], and the more classical Krieging known as Ridge Regression [16], hence for the first time providing a direct link between these very distinct paradigms. In view of these developments it is clear that defining an appropriate kernel function allows one to use a range of different algorithms to analyse the data concerned potentially answering many practical prediction problems. For a particular application choosing a kernel corresponds to implicitly choosing a feature space since the kernel function is defined by for the feature map OE. Given a training set g, the information available to kernel based algorithms is contained entirely in the matrix of inner products known as the Gram or kernel matrix. This matrix represents a sort of 'bottleneck' for the information that can be exploited: by operating on the matrix, one can in fact 'virtually' recode the data in a more suitable manner. The solutions sought are linear functions in the feature space for some weight vector w, where 0 denotes the transpose of a vector or matrix. The kernel trick can be applied whenever the weight vector can be expressed as a linear combination of the training points, implying that we can express f as follows Given an explicit feature map OE we can use equation (1) to compute the corresponding kernel. Often, however, methods are sought to provide directly the value of the kernel without explicitly computing OE. We will show how many of the standard information retrieval feature spaces give rise to a particularly natural set of kernels. Perhaps the best known method of this type is referred to as the polynomial kernel. Given a kernel k the polynomial construction creates a kernel - k by applying a polynomial with positive coefficients to k, for example consider for fixed values of D and integer p. Suppose the feature space of k is F , then the feature space of - k is indexed by t-tuples of features from F , for Hence, through a relatively small additional computational cost (each time an inner product is computed one more addition and exponentiation is required) the algorithms are being applied in a feature space of vastly expanded expressive power. As an even more extreme example consider the Gaussian kernel ~ k that transforms the kernel k as follows: whose feature space has infinitely many dimensions. 3. Vector Space Representations Given a document, it is possible to associate with it a bag of terms (or bag of words) by simply considering the number of occurrences of all the terms it contains. Typically words are "stemmed" meaning that the inflection information contained in the last few letters is removed. A bag of words has its natural representation as a vector in the following way. The number of dimensions is the same as the number of different terms in the corpus, each entry of the vector is indexed by a specific term, and the components of the vector are formed by integer numbers representing the frequency of the term in the given document. Typically such a vector is then mapped into some other space, where the word frequency information is merged with other information (e.g. word importance, where uninformative words are given low or no weight). In this way a document is represented by a (column) vector d in which each entry records how many times a particular word stem is used in the document. Typically d can have tens of thousands of entries, often more than the number of documents. Furthermore, for a particular document the representation is typically extremely sparse, having only relatively few non-zero entries. In the basic vector-space model (BVSM), a document is represented by a vertical vector d indexed by all the elements of the dictionary, and a corpus by a matrix D, whose columns are indexed by the documents and whose rows are indexed by the We also call the data matrix D the "term by document" matrix. We define the "document by document" matrix to be and the "term by term" matrix to be If we consider the feature space defined by the basic vector-space model, the corresponding kernel is given by the inner product between the feature vectors In this case the Gram matrix is just the document by document matrix. More gen- erally, we can consider transformations of the document vectors by some mapping OE. The simplest case involves linear transformations of the type P is any appropriately shaped matrix. In this case the kernels have the form We will call all such representations Vector Space Models (VSMs). The Gram matrix is in this case given by D 0 P 0 PD that is by definition symmetric and positive definite. The class of models obtained by varying the matrix P is a very natural one, corresponding as it does to different linear mappings of the standard vector space model, hence giving different scalings and projections. Note that Jiang and Littman [9] use this framework to present a collection of different methods, although without viewing them as kernels. Throughout the rest of the paper we will use P to refer to the matrix defining the VSM. We will describe a number of different models in each case showing how an appropriate choice of P realises it as VSM. Basic Vector Space Model The Basic Vector Space Model (BVSM) was introduced in 1975 by Salton et al. [15] (and used as a kernel by Joachims [10]) and uses the vector representation with no further mapping. In other words the VSM matrix I in this case. The performance of retrieval systems based on such a simple representation is surprisingly good. Since the representation of each document as a vector is very sparse, special techniques can be deployed to facilitate the storage and the computation of dot products between such vectors. A common map P is obtained by considering the importance of each term in a given corpus. The VSM matrix is hence a diagonal, whose entries P (i; i) are the weight of the term i. Several methods have been proposed, and it is known that this has a strong influence on generalization [11]. Often P (i; i) is a function of the inverse document frequency idf , that is the total number of documents in the corpus divided by the number of documents that contain the given term. So if for example a word appears in each document, it would not be regarded as a very informative one. Its distance from the uniform distribution is a good estimation of its importance, but better methods can be obtained by studying the typical term distributions within documents and corpora. The simplest method for doing this is just given by P (i; Other measures can be obtained from information theoretic quantities, or from empirical models of term frequency. Since these measures do not use label information, they could also be estimated from an external, larger unlabelled corpus, that provides the background knowledge to the system. As described in the previous section as soon as we have defined a kernel we can apply the polynomial or Gaussian construction to increase its expressive power. Joachims [10] and Dumais et al. [5] have applied this technique to the basic vector space model for a classification task with impressive results. In particular, the use of polynomial kernels can be seen as including features for each tuple of words up to the degree of the chosen polynomial. One of the problems with this representation is that it treats terms as uncorrelated, assigning them orthogonal directions in the feature space. This means that it can only cluster documents that share many terms. But in reality words are correlated, and sometimes even synonymous, so that documents with very few common terms can potentially be on closely related topics. Such similarities cannot be detected by the BVSM. This raises the question of how to incorporate information about semantics into the feature map, so as to link documents that share related terms? One idea would be to perform a kind of document expansion, adding to the expanded version all synonymous (or closely related) words to the existing terms. Another, somehow similar, method would be to replace terms by concepts. This information could potentially be gleaned from external knowledge about correlations, for example from a semantic network. There are, however, other ways to address this problem. It is also possible to use statistical information about term-term correlations 7derived from the corpus itself, or from an external reference corpus. This approach forms the basis of Latent Semantic Indexing. In the next subsections we will look at two different methods, in each case showing how they can be implemented directly through the kernel matrix, without the need to work explicitly in the feature space. This will allow them to be combined with other kernel techniques such as the polynomial and Gaussian constructions described above. Generalised Vector Space Model An early attempt to overcome the limitations of BVSMs was proposed by Wong et al. [27] under the name of Generalised VSM, or GVSM. A document is characterised by its relation to other documents in the corpus as measured by the BVSM. This method aims at capturing some term-term correlations by looking at co-occurrence information: two terms become semantically related if they co-occur often in the same documents. This has the effect that two documents can be seen as similar even if they do not share any terms. The GVSM technique can provide one such metric, and it is easy to see that it also constitutes a kernel function. Given the term by document data matrix D, the GVSM kernel is given by The matrix DD 0 is the term by term matrix and has a nonzero ij entry if and only if there is a document in the corpus containing both the i-th and the j-th terms. So two terms co-occurring in a document are considered related. The new metric takes this co-occurrence information into account. The documents are mapped to a feature space indexed by the documents in the corpus, as each document is represented by its relation to the other documents in the corpus. For this reason it is also known as a dual space method [22]. In the common case when there are less documents than terms, the method will act as a bottle-neck mapping forcing a dimensionality reduction. For the GVSM the VSM matrix P has been chosen to be D 0 the document by term matrix. Once again the method can be combined with the polynomial and Gaussian kernel construction techniques. For example the degree p polynomial kernel would have features for each (- p)-tuple of documents with a non-zero feature for a document that shares terms with each document in the tuple. To our knowledge this combination has not previously been considered with either the polynomial or the Gaussian construction. Semantic Smoothing for Vector Space Models Perhaps a more natural method of incorporating semantic information is by directly using an external source, like a semantic network. In this section we briefly describe one such approach. Siolas and d'Alch'e-Buc [23] used a semantic network (Word-net [12]) as a way to obtain term-similarity information. Such a network encodes for each word of a dictionary its relation with the other words in a hierarchical fashion (e.g. synonym, hypernym, etc). For example both the word 'husband' and `wife' are special cases of their hypernym 'spouse'. In this way, the distance between two terms in the hierarchical tree provided by Wordnet gives an estimation of their semantic proximity, and can be used to modify the metric of the vector space when the documents are mapped by the bag-of-words approach. Siolas and d'Alch'e-Buc [23] have included this knowledge into the kernel by hand-crafting the entries in the square VSM matrix P . The entries the semantic proximity between the terms i and j. The semantic proximity is defined as the inverse of their topological distance in the graph, that is the length of the shortest path connecting them (but some cases deserve special attention). The modified metric then gives rise to the following kernel or to the following distance Siolas and d'Alch'e-Buc used this distance in order to apply the Gaussian kernel construction described above, though a polynomial construction could equally well be applied to the kernel. Siolas and d'Alch'e-Buc used a term-term similarity matrix to incorporate semantic information resulting in a square matrix P . It would also be possible to use a concept-term relation matrix in which the rows would be indexed by concepts rather than terms. For example one might consider both 'husband' and `wife' examples of the concept 'spouse'. The matrix P would in this case no longer be square symmetric. Notice that GVSMs can be regarded as a special case of this, when the concepts correspond to the documents in the corpus, that is a term belongs to the i-th 'concept' if it occurs in document d i . 4. Latent Semantic Kernels Latent Semantic Indexing (LSI) [4] is a technique to incorporate semantic information in the measure of similarity between two documents. We will use it to construct kernel functions. Conceptually, LSI measures semantic information through co-occurrence analysis in the corpus. The technique used to extract the information relies on a Singular Value Decomposition (SVD) of the term by document matrix. The document feature vectors are projected into the subspace spanned by the first singular vectors of the feature space. Hence, the dimension of the feature space is reduced to k and we can control this dimension by varying k. We can define a kernel for this feature space through a particular choice of the VSM matrix P , and we will see that P can be computed directly from the original kernel matrix without direct computation of the SVD in the feature space. In order to derive a suitable matrix P first consider the term-document matrix D and its SVD decomposition where \Sigma is a diagonal matrix with the same dimensions as D, and U and V are orthogonal (ie U I). The columns of U are the singular vectors of the feature space in order of decreasing singular value. Hence, the projection operator onto the first k dimensions is given by I k is the identity matrix with only the first k diagonal elements nonzero and U k the matrix consisting of the first k columns of U . The new kernel can now be expressed as The motivation for this particular mapping is that it identifies highly correlated dimensions: i.e. terms that co-occur very often in the same documents of the corpus are merged into a single dimension of the new space. This creates a new similarity metric based on context information. In the case of LSI it is also possible to isometrically re-embed the subspace back into the original feature space by defining P as the square symmetric (U k U 0 This gives rise to the same kernel, since We can then view - P as a term-term similarity matrix making LSI a special case of the semantic smoothing described in Solias and d'Alch'e-Buc [23]. While they need to explicitly work out all the entries of the term-by-term similarity matrix with the help of a semantic network, however, we can infer the semantic similarities directly from the corpus, using co-occurrence analysis. What is more interesting for kernel methods is that the same mapping, instead of acting on term-term matrices, can be obtained implicitly by working with the smaller document-document Gram matrix. The original term by document matrix D gives rise to the kernel matrix since the feature vector for document j is the j-th column of D. The SVD decomposition is related to the eigenvalue decomposition of K as follows so that the i-th column of V is the eigenvector of K, with corresponding eigenvalue . The feature space created by choosing the first k singular values in the LSI approach corresponds to mapping a feature vector d to the vector UI k U 0 d and gives rise to the following kernel matrix where k is the matrix with diagonal entries beyond the k-th set to zero. Hence, the new kernel matrix can be obtained directly from K by applying an eigenvalue decomposition of K and remultiplying the component matrices having set all but the first k eigenvalues to zero. Hence, we can obtain the kernel corresponding to the LSI feature space without actually ever computing the features. The relations of this computation to kernel PCA [18] are immediate. By a similar analysis it is possible to verify that we can also evaluate the new kernel on novel inputs again without reference to the explicit feature space. In order to evaluate the learned functions on novel examples, we must show how to evaluate the new kernel - k between a new input d and a training example, - k(d d). The function we wish to evaluate will have the form d) d) i The expression still, however, involves the feature vector d which we would like to avoid evaluating explicitly. Consider the vector of inner products between the new feature vector and the training examples in the original space. These inner products can be evaluated using the original kernel. But now we have I d, showing that we can evaluate f(d) as follows t. Hence to evaluate f on a new example, we first create a vector of the inner products in the original feature space and then take its inner product with the precomputed row vector ff 0 V I k V 0 . None of this computation involves working directly in the feature space. The combination of the LSK technique with the polynomial or Gaussian construction opens up the possibility of performing LSI in very high dimensional feature spaces, for example indexed by tuples of terms. Experiments applying this approach are reported in the experimental section of this paper. If we think of the polynomial mapping as taking conjunctions of terms, we can view the LSK step as a soft disjunction, since the projection links several different conjunctions into a single concept. Hence, the combination of the polynomial mapping followed by an LSK step produces a function with a form reminiscent of a disjunctive normal form. Alternatively one could perform the LSK step before the polynomial mapping (by just applying the polynomial mapping to the entries of the Gram matrix obtained after the LSK step), obtaining a space indexed by tuples of concepts. Here the function obtained will be reminiscent of a conjunctive normal form. We applied this approach to the Ionosphere data but obtained no improvement in performance. We conjecture that the results obtained will depend strongly on the fit of the style of function with the particular data. The main drawback of all such approaches is the computational complexity of performing an eigenvalue decomposition on the kernel matrix. Although the matrix is smaller than the term by document matrix it is usually no longer sparse. This makes it difficult to process training sets much larger than a few thousand examples. We will present in the next section techniques that get round this problem by evaluating an approximation of the LSK approach. 5. Algorithmic Techniques All the experiments were performed using the eigenvalue decomposition routine provided with Numerical Recipes in C [14]. The complete eigen-decomposition of the Kernel matrix is an expensive step, and where possible one should try to avoid it when working with real world data. More efficient methods can be developed to obtain or approximate the LSK solution. We can view the LSK technique as one method of obtaining a low rank approximation of the kernel matrix. Indeed the projection onto the first k eigenvalues is the rank k approximation which minimises the norm of the resulting error matrix. But projection onto the eigensubspaces is just one method of obtaining a low-rank approximation. We have also developed an approximation strategy, based on the Gram-Schmidt decomposition. A similar approach to unsupervised learning is described by Smola et al. [25]. The projection is built up as the span of a subset of (the projections of) a set of k training examples. These are selected by performing a Gram-Schmidt orthogonalisation of the training vectors in the feature space. Hence, once a vector is selected the remaining training points are transformed to become orthogonal to it. The next vector selected is the one with the largest residual norm. The whole transformation is performed in the feature space using the kernel mapping to represent the vectors obtained. We refer to this method as the GK algorithm. Table 1 gives complete pseudo-code for extracting the features in the kernel defined feature space. As with the LSK method it is parametrised by the number of dimensions T selected. Table 1. The GSK Algorithm Given a kernel k, training set d do to T do do return feat[i; j] as the j-th feature of input To classify a new example x: to T do return newfeat[j] as the j-th feature of the example 5.1. Implicit Dimensionality Reduction An interesting solution to the problem of approximating the Latent Semantic solution is possible in the case in which we are not directly interested in the low-rank matrix, unlike in the information retrieval case, but we only plan to use it as a kernel in conjunction with an optimization problem of the type: where H is the Hessian, obtained by pre- and post-multiplying the Gram matrix by the diagonal matrix containing the f+1; \Gamma1g labels, Note that H and K have the same eigenvalues since if It is possible to easily (and cheaply) modify the Gram matrix so as to obtain nearly the same solution that one would obtain by using a (much more expensive) low rank approximation. The minimum of this error function occurs at the point ff which satisfies q+Hff 0. If the matrix H is replaced by H then the minimum moves to a new point e ff which satisfies q us consider the expansion of H in its eigenbasis: and the expansions of ff and e ff in the same basis: Substituting into the above formulae and equating coefficients of the i-th eigenvalue gives ff implying that e ff The fraction in the above equation is a squashing function, approaching zero for values of - i - and approaching 1 for - i AE -. In the first case e in the second case e ff i - ff . The overall effect of this map, if the parameter - is chosen carefully in a region of the spectrum where the eigenvalues decrease rapidly, is to effectively project the solution onto the space spanned by the eigenvectors of the larger eigenvalues. From an algorithmic point of view this is much more efficient than explicitly performing the low-rank approximation by computing the eigenvectors. This derivation not only provides a cheap approximation algorithm for the latent semantic kernel. It also highlights an interesting connection between this algorithm and the 2-norm soft margin algorithm for noise tolerance, that also can be obtained by adding a diagonal to the kernel matrix [21]. But note that there are several approximations in this view since for example the SVM solution is a constrained optimisation, where the ff i 's are constrained to be positive. In this case the effect may be very different if the support vectors are nearly orthogonal to the eigenvectors corresponding to large eigenvalues. The fact that the procedure is distinct from a standard soft margin approach is borne out in the experiments that are described in the next section. 6. Experimental Results We empirically tested the proposed methods both on text and on non-text data, in order to demonstrate the general applicability of the method, and to test its effectiveness under different conditions. The results were generally positive, but in some cases the improvements are not significant or not worth the additional computation. In other cases there is a significant advantage in using the Latent Semantic or Gram-Schmidt kernels, and certainly their use never hurts performance. 6.1. Experiments on Text Data This section describes a series of systematic experiments performed on text data. We selected two text collections, namely Reuters and Medline that are described below. Datasets Reuters21578 We conducted the experiments on a set of documents containing stories from Reuters news agency, namely the Reuters data-set. We used Reuters- 21578, the newer version of the corpus. It was compiled by David Lewis in 1987 and is publicly available at http://www.research.att.com/lewis. To obtain a training set and test set there exists different splits of the corpus. We used the Modified Apte ("ModeApte") split. The "ModeApte" split comprises 9603 training and 3299 test documents. A Reuters category can contain as few as 1 or as many as 2877 documents in the training set. Similarly a test set category can have as few as 1 or as many as 1066 relevant documents. Medline1033 The Medline1033 is the second data-set which was used for experi- ments. This dataset comprises of 1033 medical documents and queries obtained from National library of medicine. We focused on query23 and query20. Each of these two queries contain 39 relevant documents. We selected randomly 90% of the data for training the classifier and 10% for evaluation, while always having 24 relevant documents in the training set and 15 relevant documents in the test set. We performed 100 random splits of this data. Experiments The Reuters documents were preprocessed. We removed the punctuation and the words occurring in the stop list and also applied Porter stemmer to the words. We weighted the terms according to a variant of the tfidf scheme. It is given by here tf represents term frequency, df is used for the document frequency and m is the total number of documents. The documents have unit length in the feature space. We preprocessed the Medline documents by removing stop words and punctuation and weighted the words according to the variant of tfidf described in the preceding paragraph. We normalised the documents so that no bias can occur because of the length of the documents. For evaluation we used the F1 performance measure. It is given by 2pr=(p is the precision and r is the recall. The first set of experiment was conducted on a subset of 3000 documents of Reuters21578 data set. We selected randomly 2000 documents for training and the remaining 1000 documents were used as a test set. We focused on the top 0.97dimension baseline Figure 1. Generalisation performance of SVM with GSK, LSK and linear kernel for earn. 5 Reuters categories (earn, acq, money-fx, grain, crude). We trained a binary classifier for each category and evaluated its performance on new documents. We repeated this process 10 times for each category. We used an SVM with linear kernel for the baseline experiments. The parameter C that controls the trade off between error and maximisation of margin was tuned by conducting preliminary experiments. We chose the optimal value by conducting experiments on ten splits of one category. We ran an SVM not only in the reduced feature space but also in a feature space that has full dimension. The value of C that showed the best results in the full space was selected and used for all further experiments. For the Medline1033 text corpus we selected the value of C by conducting experiments on one split of the data. We ran an SVM in a feature space that has full dimension. The optimal value of C that showed best results was selected. Note that we did not use that split for further experiments. This choice does not seem perfect but on the basis of our experimental observation on Reuters, we conclude that this method gives an optimal value of C. The results of our experiments on Reuters are shown in Figures 1 through 4. Note that these results are averaged over 10 runs of the algorithm. We started with a small dimensional feature space. We increased the dimensionality of the feature space in intervals by extracting more features. These figures demonstrate that the performance of the LSK method is comparable to the baseline method. The generalisation performance of an SVM classifier varies by varying the dimensionality of the semantic space. By increasing the value of k, F1 numbers rise reaching a maximum and then falls to a number equivalent to the baseline method. However this maximum is not substantially different from the baseline method. In other words sometimes we obtain only a modest gain by incorporating more information into a kernel matrix. Figure 6 and Figure 7 illustrate the results of experiments conducted on the two Medline1033 queries. These results are averaged over 100 random runs of the algorithm. For these experiments we start with a small number of dimensions. The dimensionality was increased in intervals by extracting more features. The results dimesnsion baseline Figure 2. Generalisation performance of SVM with GSK, LSK, and linear kernel for acq. baseline Figure 3. Generalisation performance of SVM with GSK, LSK and linear kernel for money-fx. dimension Figure 4. Generalisation performance of SVM with GSK, LSK and linear lkernel for grain. dimension baseline Figure 5. Generalistaion performance of SVM with GSK, LSK and linear kernel for crude. baseline Figure 6. Generalisation performance of SVM with GSK, LSK and linear kernel for query23. baseline Figure 7. Generalisation performance of SVM with GSK, LSK and linear kernel for query20. Table 2. F1 numbers for varying dimensions of feature space for a SVM classifier with LSK and SVM classifier with linear kernel (baseline) for ten Reuters categories category k baseline 100 200 300 money-fx 0.62 0.673 0.635 0.6 grain 0.664 0.661 0.67 0.727 crude 0.431 0.558 0.576 0.575 trade 0.568 0.683 0.66 0.657 interest 0.478 0.497 0.5 0.517 ship 0.422 0.544 0.565 0.565 wheat 0.514 0.51 0.556 0.624 micro-avg 0.786 0.815 0.815 0.819 for query23 are very encouraging showing that the LSK has a potential to show a substantial improvement over the baseline method. Thus the results (Reuters and show that in some cases there can be improvements in performance, while for others there can be no significant improvements. Our results on Reuters and Medline1033 datasets demonstrates that GSK is a very effective approximation strategy for LSK. In most of the cases the results are approximately the same as LSK. However it is worth noting that in some cases such as Figure 6, GSK may show substantial improvement not only over the baseline method but also over LSK. Hence the results demonstrate that GSK is a good approximation strategy for LSK. It can improve the generalisation performance over LSK as is evident from the results on the Medline data. It can extract informative features that can be very useful for classification. GSK can achieve a maximum at a high dimension in some situations. This phenomenon may cause practical limitations for large data sets. We have addressed this issue and developed a generalised GSK algorithm for text classification. Furthermore, we conducted another set of experiments to study the behaviour of and SVM classifier with a semantic kernel and an SVM classifier with a linear kernel in a scenario where a classifier is learnt using a small training set. We selected randomly 5% of the training data (9603 documents). We focused on the top 10 categories (earn, 144), (acq, 85), (money-fx, 29), (grain, 18), (crude, 16), (trade, 28), (interest, 19), (ship, 12), (wheat, 8), (corn, 6). Note that the number of relevant documents are shown with the name of the categories. A binary classifier was learnt for each category and was evaluated on the full test set of (3299) documents. C was tuned on one category. F1 numbers obtained as a results of these experiments are reported in Table 2. Micro-averaged F1 numbers are also given. We set the value of It is to be noted that there is gain for some categories, but that there is loss in performance for others. It is worth noting that an SVM classifier trained with a semantic kernel can perform approximately the same as the baseline method even with 200 dimensions. These results demonstrate that the proposed method is capable of performing reasonably well in environments with very few labelled documents. 6.2. Experiments on Non-text Data Figure 8. Generalization error for polynomial kernels of degrees 2,3, 4 on Ionosphere data (aver- aged over 100 random splits) as a function of the dimension of the feature space. Now we present the experiments conducted on the non-text Ionosphere data set from the UCI repository. Ionosphere contains 34 features and 315 points. We measured the gain of the the LSK by comparing its performance with an SVM with polynomial kernel. The parameter C was set by conducting preliminary experiments on one split of the data keeping the dimensionality of the space full. We tried The optimal value that demonstrated minimum error was chosen. This value was used for all splits and for the reduced feature space. Note that the split of the data used for tuning the parameter C was not used for further experiments. The results are shown in Figure 8. These results are averaged over 100 runs. We begin experiments by setting k to a small value. We increased the dimensionality of the space in intervals. The results show that test error was greatly reduced when the dimension of the feature space was reduced. The curves also demonstrate that the classification error of an SVM classifier with semantic kernel reaches a minimum. It makes some peaks and valleys before showing results equivalent to the baseline method. These results demonstrate that the proposed method is so general that it can be applied to domains other than text. It has a potential to improve the performance of a SVM classifier by reducing the dimension. However in some cases it can show no gain and may not be successful in reducing the dimension. 7. A Generalised Version of GSK algorithm for Text Classification In this section we present a generalised version of the GSK algorithm. This algorithm arose as a result of experiments reported in Section 6. Some other preliminary experiments also contributed to the development of the algorithm. The GSK algorithm presented in the previous section extracts features relative to the documents but irrespective of their relevance to the category. In other words, features are not computed with respect to the label of a document. Generally the category distribution is very skewed for text corpora. This establishes a need to bias the feature computation towards the relevant documents. In other words, if we can introduce some bias in this feature extraction process, the computed features can be more useful and informative for text classification. The main goal of developing the generalised version of the GSK algorithm is to extract few but more informative features, so that when fed to a classifier it can show high effectiveness in a low number of dimensions. To achieve the goal described in the preceding paragraph we propose the algorithm shown in Figure 9. GSK is an iterative procedure that greedily selects a document at each iteration and extracts features. At each iteration the criterion for selecting a document is the maximum residual norm. The generalised version of GSK algorithm focuses on relevant documents by placing more weight on the norm of relevant documents. The algorithm transforms the documents into a new (reduced) feature space by taking a set of documents. As input an underlying kernel function, number T and bias B are also fed to the algorithm. The number T specifies the dimension of the reduced feature space, while B gives the degree to which the feature extraction is biased towards relevant documents. The algorithm starts by measuring the norm of each document. It concentrates on relevant documents by placing more weight on the norm of these documents. As a next step a document with a maximum norm is chosen and features are extracted relative to this document. This process is repeated T times. Finally the documents are transformed into a new T dimensional space. The dimension of the new space is much smaller than the original feature space. Note that when there is enough positive data available for training, equal weights can be given both to relevant and irrelevant documents. The generalised version of the GSK algorithm provides a practical solution of the problem that may occur with the GSK-algorithm. This algorithm may show good Require: A kernel k, training set f(d 1 number T to n do end for to T do to n do if (y i == +1) then else end for to n do end for end for return feat[i; j] as the j-th feature of input To classify a new example d: to T do end for return newfeat[j] as the j-th feature of the example d; Figure 9. A Generalised Version of GSK Algorithm generalisation at high dimension when there is not enough training data. In that scenario the generalised version of the GSK-algorithm shows similar performance at lower dimensions. The complete pseudo-code of the algorithm is given in Figure 9. 8. Experiments with Generalised GSK-algorithm We employed the generalise the GSK algorithm to transform the Reuters documents into a new reduced feature space. We evaluated the proposed method by conducting experiments on the full Reuters data set. We used the ModeApte version and performed experiments on 90 categories that contain at least one relevant document both in the training set and test set. In order to transform documents into a new space, two free parameters T (dimension of reduced space) and B (bias) need to be tuned. We analysed the generalistion performance of an SVM classifier with respect to B by conducting a set of experiments on 3 Reuters categories. The results of these experiments are shown in Table 3. For this set of experiments we set the dimensionality of space (T ) to 500 and varied B. The results demonstrate that the extraction of features in a biased environment can be more informative and useful when there is insufficient training data. On the basis of these experiments we selected an optimal value of B for our next set of experiments. Note that we selected the optimal value of C by conducting preliminary experiments on one Reuters category. We set the value of 1000. The results of this set of experiments are given in Table 4. We have given F1 value for 500, and 1000 dimensional space. Micro-averaged F1 values are also shown in the table. In order to learn a SVM classifier we used SV M light [8] for the experiments described in this section. These results show that the generalised GSK algorithm can be viewed as a substantial dimensionality reduction technique. Our observation is that the proposed method shows results that are comparable to the baseline method at a dimensionality of 500. Note that for the baseline method we employed an SVM with a linear kernel. It is to be noted that after 500 dimensionality there is a slow improvement in generalisation performance of the SVM. The micro-averaged F1 values for an SVM with generalised GSK is 0.822 (at 500 dimensions), whereas the micro-averaged F1 number for an SVM with linear kernel is 0.854. These results show that the performance of the proposed technique is comparable to the baseline method. These results show that the generalised GSK algorithm is a practical approximation of LSK. If the learning algorithm is provided with enough positive training data, there is no need to bias the feature extraction process. However, when the learning algorithm does not have enough positive training data, an SVM may only show good performance at high dimensionality leading to practical limitations. However the introduction of bias towards relevant documents will overcome this problem, hence making it a technique that can be applied to large data sets. Table 3. F1 numbers for acq, money- fx and wheat for different values of B. 1.0 0.922 0.569 0.707 1.1 0.864 0.695 0.855 1.2 0.864 0.756 0.846 2.0 0.864 0.748 0.846 2.2 0.864 0.752 0.855 2.4 0.864 0.756 0.846 2.6 0.864 0.748 0.846 2.8 0.864 0.752 0.846 6.0 0.864 0.752 0.857 10. Table 4. F1 numbers for top-ten Reuters categories Category T baseline 500 1000 acq 0.923 0.934 0.948 money-fx 0.755 0.754 0.775 grain 0.894 0.902 0.93 crude 0.872 0.883 0.880 trade 0.733 0.763 0.761 interest 0.627 0.654 0.691 ship 0.743 0.747 0.797 wheat 0.864 0.851 0.87 corn 0.857 0.869 0.895 micro-avg 9. Conclusion The paper has studied the problem of introducing semantic information into a kernel based learning method. The technique was inspired by an approach known as Latent Semantic Indexing borrowed from Information Retrieval. LSI projects the data into a subspace determined by choosing the first singular vectors of a singular value decomposition. We have shown that we can obtain the same inner products as those derived from this projection by performing an equivalent projection onto the first eigenvectors of the kernel matrix. Hence, it is possible to apply the same technique to any kernel defined feature space whatever its original dimensionality. We refer to the derived kernel as the Latent Semantic Kernel (LSK). We have experimentally demonstrated the efficacy of the approach on both text and non-text data. For some datasets substantial improvements in performance were obtained using the method, while for others little or no effect was observed. As the eigenvalue decomposition of a matrix is relatively expensive to compute, we have also considered an iterative approximation method that is equivalent to projecting onto the first dimension derived from a Gram-Schmidt othogonalisation of the data. Again we can perform this projection efficiently in any kernel defined feature space and experiments show that for some datasets the so-called Gram-Schmidt Kernel (GSK) is more effective than the LSK method. Despite this success, for large imbalanced datasets such as those encountered in text classification tasks the number of dimensions required to obtain good performance grows quite large before relevant features are drawn from the small number of positive documents. This problem is addressed by biasing the GSK feature selection procedure in favour of positive documents hence greatly reducing the number of dimensions required to create an effective feature space. The methods described in the paper all have a similar flavour and have all demonstrated impressive performance on some datasets. The question of what it is about a dataset that makes the different semantic focusing methods effective is not fully understood and remains the subject of ongoing research. Acknowledgements The authors would like to thank Thorsten Joachims and Chris Watkins for useful discussions. Our work was supported by EPSRC grant number GR/N08575 and by the European Commission through the ESPRIT Working Group in Neural and Computational Learning, NeuroCOLT2, Nr. 27150. KerMIT. and the 1st Project 'Kernel methods for images and text', KerMIT, Nr. 1st-2000-25431. --R Theoretical foundations of the potential function method in pattern recognition learning. A training algorithm for optimal margin classifiers. An Introduction to Support Vector Machines. Indexing by latent semantic analysis. Inductive learning algorithms and representations for text categorization. Automatic cross-language retrieval using latent semantic indexing Large margin rank boundaries for ordinal regression. Making large-scale SVM learning practical Approximate dimension equalization in vector-based information retrieval Text categorization with support vector machines. Five papers on wordnet. 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Text Categorization with Suport Vector Machines Ridge Regression Learning Algorithm in Dual Variables Approximate Dimension Equalization in Vector-based Information Retrieval Support Vector Machines Based on a Semantic Kernel for Text Categorization --CTR Qiang Sun , Gerald DeJong, Explanation-Augmented SVM: an approach to incorporating domain knowledge into SVM learning, Proceedings of the 22nd international conference on Machine learning, p.864-871, August 07-11, 2005, Bonn, Germany Yaoyong Li , John Shawe-Taylor, Using KCCA for Japanese---English cross-language information retrieval and document classification, Journal of Intelligent Information Systems, v.27 n.2, p.117-133, September 2006 Yaoyong Li , John Shawe-Taylor, Advanced learning algorithms for cross-language patent retrieval and classification, Information Processing and Management: an International Journal, v.43 n.5, p.1183-1199, September, 2007 Yonghong Tian , Tiejun Huang , Wen Gao, Latent linkage semantic kernels for collective classification of link data, Journal of Intelligent Information Systems, v.26 n.3, p.269-301, May 2006 Mehran Sahami , Timothy D. Heilman, A web-based kernel function for measuring the similarity of short text snippets, Proceedings of the 15th international conference on World Wide Web, May 23-26, 2006, Edinburgh, Scotland Haixian Wang , Zilan Hu , Yu'e Zhao, An efficient algorithm for generalized discriminant analysis using incomplete Cholesky decomposition, Pattern Recognition Letters, v.28 n.2, p.254-259, January, 2007 Kevyn Collins-Thompson , Jamie Callan, Query expansion using random walk models, Proceedings of the 14th ACM international conference on Information and knowledge management, October 31-November 05, 2005, Bremen, Germany Serhiy Kosinov , Stephane Marchand-Maillet , Igor Kozintsev , Carole Dulong , Thierry Pun, Dual diffusion model of spreading activation for content-based image retrieval, Proceedings of the 8th ACM international workshop on Multimedia information retrieval, October 26-27, 2006, Santa Barbara, California, USA Kristen Grauman , Trevor Darrell, The Pyramid Match Kernel: Efficient Learning with Sets of Features, The Journal of Machine Learning Research, 8, p.725-760, 5/1/2007 Vikramjit Mitra , Chia-Jiu Wang , Satarupa Banerjee, Text classification: A least square support vector machine approach, Applied Soft Computing, v.7 n.3, p.908-914, June, 2007 Francis R. Bach , Michael I. Jordan, Kernel independent component analysis, The Journal of Machine Learning Research, 3, p.1-48, 3/1/2003

gram-schmidt kernels;text categorization;latent semantic indexing;kernel methods;latent semantic kernels

607623

Hidden Markov Models for Text Categorization in Multi-Page Documents.

In the traditional setting, text categorization is formulated as a concept learning problem where each instance is a single isolated document. However, this perspective is not appropriate in the case of many digital libraries that offer as contents scanned and optically read books or magazines. In this paper, we propose a more general formulation of text categorization, allowing documents to be organized as sequences of pages. We introduce a novel hybrid system specifically designed for multi-page text documents. The architecture relies on hidden Markov models whose emissions are bag-of-words resulting from a multinomial word event model, as in the generative portion of the Naive Bayes classifier. The rationale behind our proposal is that taking into account contextual information provided by the whole page sequence can help disambiguation and improves single page classification accuracy. Our results on two datasets of scanned journals from the Making of America collection confirm the importance of using whole page sequences. The empirical evaluation indicates that the error rate (as obtained by running the Naive Bayes classifier on isolated pages) can be significantly reduced if contextual information is incorporated.

Figure 1. Bayesian network for the Naive Bayes classifier. 2.2. Hidden Markov models HMMs have been introduced several years ago as a tool for probabilistic sequence modeling. The interest in this area developed particularly in the Seventies, within the speech recognition research community (Rabiner, 1989). During the last years a large number of variants and improvements over the standard HMM have been proposed and applied. Undoubt- edly, Markovian models are now regarded as one of the most significant state-of-the-art approaches for sequence learning. Besides several applications in pattern recognition and molecular biology, HMMs have been also applied to text related tasks, including natural language modeling (Charniak, 1993) and, more recently, information retrieval and extraction (Freitag and McCallum, 2000; McCallum et al., 2000). The recent view of the HMM as a particular case of Bayesian networks (Bengio and Frasconi, 1995; Lucke, 1995; Smyth et al., 1997) has helped their theoretical understanding and the ability to conceive extensions to the standard model in a sound and formally elegant framework. An HMM describes two related discrete-time stochastic processes. The first process pertains to hidden discrete state variables, denoted Xt , forming a first-order Markov chain and taking realizations on a finite set {x1,.,xN }. The second process pertains to observed variables or emissions, denoted Dt . Starting from a given state at time 0 (or given an initial state distribution P(X0)) the model probabilistically transitions to a new state X1 and correspondingly emits observation D1. The process is repeated recursively until an end state is reached. Note that, as this form of computation may suggest, HMMs are closely related to stochastic regular grammars (Charniak, 1993). The Markov property prescribes that Xt+1 is conditionally independent of X1,.,Xt1 given Xt . Furthermore, it is assumed that Dt is independent of the rest given Xt . These two conditional independence assumptions are graphically depicted using the Bayesian network of figure 2. As a result, an HMM is fully specified by the following conditional probability P(Xt | Xt1) (transition distribution) (2) P(Dt | Xt ) (emission distribution) Since the process is stationary, the transition distribution can be represented as a square stochastic matrix whose entries are the transition probabilities abbreviated as P(xi | x j ) in the following. In the classic literature, emissions are restricted HIDDEN MARKOV MODELS 199 Figure 2. Bayesian networks for standard HMMs. to be symbols in a finite alphabet or multivariate continuous variables (Rabiner, 1989). As explained in the next section, our model allows emissions to be bag-of-words. 3. The multi-page classifier We now turn to the description of our classifier for multi-page documents. In this case the categorization task consists of learning from examples a function that maps the whole document sequence d1,.,dT into a corresponding sequence of page categories, c1,.,cT . This section presents the architecture and the asociated algorithms for grammar extraction, training, and classification. 3.1. Architecture The system is based on an HMM whose emissions are associated with entire pages of the document. Thus, the realizations of the observation Dt are bag-of-words representing the text in the t-th page of the document. HMM states are related to pages categories by a deterministic function that maps state realizations into page categories. We assume that is a surjection but not a bijection, i.e. that there are more state realizations than categories. This enriches the expressive power of the model, allowing different transition behaviors for pages of the same class, depending on where the page is actually encountered within the sequence. However, if the page contents depends on the category but not on the context of the category within the sequence,5 multiple states may introduce too many parameters and it may be convenient to assume that P(Dt | xi This constrains emission parameters to be the same for a given page category, a form of parameters sharing that may help to reduce overfitting. The emission distribution is modeled by assuming conditional word independence given the class, like in Eq. (1): |dt | P(dt | ck . (4) Therefore, the architecture can be graphically described as the merging of the Bayesian networks for HMMs and Naive Bayes, as shown in figure 3. We remark that the state (and 200 FRASCONI ET AL. Figure 3. Bayesian network describing the architecture of the sequential classifier. hence the category) at page t depends not only on the contents of that page, but also on the contents of all other pages in the document, summarized into the HMM states. This probabilistic dependency implements the mechanism for taking contextual information into account. The algorithms used in this paper are derived from the literature on Markov models (Rabiner, 1989), inference and learning in Bayesian networks (Pearl, 1988; Heckerman, 1997; Jensen, 1996) and classification with Naive Bayes (Lewis and Gale, 1994; Kalt, 1996). In the following we give details about the integration of all these methods. 3.2. Induction of HMM topology The structure or topology of an HMM is a representation of the allowable transitions between hidden states. More precisely, the topology is described by a directed graph whose vertices are state realizations {x1,.,xN }, and whose edges are the pairs An HMM is said to be ergodic if its transition graph is fully-connected. However, in almost all interesting application domains, less connected structures are better suited for capturing the observed properties of the sequences being modeled, since they convey domain prior knowledge. Thus, starting from the right structure is an important problem in practical Hidden Markov modeling. As an example, consider figure 4, showing a (very simplified) graph that describes transitions between the parts of a hypothetical set of books. Possible state realizations are {start, title, dedication, preface, toc, regular, index, end} (note that in this simplified example is a one-to-one mapping). Figure 4. Example of HMM transition graph. HIDDEN MARKOV MODELS 201 While a structure of this kind could be hand-crafted by a domain expert, it may be more advantageous to learn it automatically from data. We now briefly describe the solution adopted to automatically infer HMM transition graphs from sample multi-page documents. Let us assume that all the pages of the available training documents are labeled with the class they belong to. One can then imagine to take advantage of the observed labels to search for an effective structure in the space of HMMs topologies. Our approach is based on the application of an algorithm for data-driven model induction adapted from previous works on construction of HMMs of text phrases for information extraction (McCallum et al., 2000). The algorithm starts by building a structure that can only explain the available training sequences (a maximally specific model). This initial structure has as many paths (from the initial to the final state) as there are training sequences. Every path is associated with one sequence of pages, i.e. a distinct state is created for every page in the training set. Each state x is labeled by (x), the category of the corresponding page in the document. Note that, unlike the example shown in figure 4, several states are generated for the same category. The algorithm then iteratively applies merging heuristics that collapse states so as to augment generalization capabilities over unseen sequences. The first heuristic, called neighbor-merging, collapses two states x and x if they are neighbors in the graph and (x). The second heuristic, called V-merging, collapses two states x and x if they share a transition from or to a common state, thus reducing the branching factor of the structure. 3.3. Inference and learning Given the HMM topology extracted by the algorithm described above, the learning problem consists of determining transition and emission parameters. One important distinction that needs to be made when training Bayesian networks is whether or not all the variables are observed. Assuming complete data (all variables observed), maximum likelihood estimation of the parameters could be solved using a one-step algorithm that collects sufficient statistics for each parameter (Heckerman, 1997). In our case, data are complete if and only if the following two conditions are met: 1. there is a one-to-one mapping between HMM states and page categories (i.e. and for 2. the category is known for each page in the training documents, i.e. the dataset consists of sequences of pairs ({d1, c1},.,{dT , cT }), being ct the (known) category of page t and being T the number of pages in the document. Under these assumptions, estimation of transition parameters is straightforward and can be accomplished as follows: where N(ci, cj ) is the number of times a page of class ci follows a page of class cj in the trainingset.Similarly,estimationofemissionparametersinthiscasewouldbeaccomplished exactly like in the case of the Naive Bayes classifier (see, e.g. Mitchell (1997)): P(w | where N(w, ck) is the number of occurrences of word w in pages of class ck and |V | is the vocabulary size (1/|V | corresponds to a Dirichlet prior over the parameters (Heckerman, 1997) and plays a regularization role for whose words which are very rare within a class). Conditions 1 and 2 above, however, are normally not satisfied. First, in order to model more accurately different contexts in which a category may occur, it may be convenient to have multiple distinct HMM states for the same page category. This implies that page labels do not determine a unique state path. Second, labeling pages in the training set is a time consuming process that needs to be performed by hand and it may be important to use also unlabeled documents for training (Joachims, 1999; Nigam et al., 2000). This means that label c may be not available for some t. If assumption 2 is satisfied but assumption 1 is not, we can derive the following approximated estimation formula for transition parameters: where N(xi, x j )counts how many times state xi follows x j during the state merge procedure described in Section 3.2. However, in general, the presence of hidden variables requires an iterative maximum likelihood estimation algorithm, such as gradient ascent or expectation-maximization (EM). Our implementation uses the EM algorithm, originally formulated in Dempster et al. (1977) and usable for any Bayesian network with local conditional probability distributions belonging to the exponential family (Heckerman, 1997). Here the EM algorithm essentially reduces to the Baum-Welch form (Rabiner, 1989) with the only modification that some evidence is entered into state variables. Since multiple states are associated with a category and even for labeled documents only the page category is known, state evidence takes the form of findings (Jensen, 1996). State evidence is taken into account in the E-step by changing forward propagation as follows: ct where . is the forward variable in the Baum-Welch algorithm. The emission probability P(dt is obtained from Eq. (4), using ck TheM-stepisperformedinthestandardwayfortransitionparameters,byreplacingcounts in Eq. (5) with their expectations given all the observed variables. Emission probabilities HIDDEN MARKOV MODELS 203 are also estimated using expected word counts. If parameters are shared as indicated in Eq. (3), these counts should be summed over states having the same label. Thus, in the case of incomplete data, Eq. (6) is replaced by P(w | ck) where S is the number of training sequences, N(w, ck) is the number of occurrences of word w in pages of class ck and P(Xt = xi | d1,.,dT ) is the probability of being in state xi at page t given the observed sequence of pages d1 .dT . Readers familiar with HMMs should recognize that the latter quantity can be computed by the Baum-Welch procedure during the E-step. The sum on p extends over training sequences, while the sum on t extends over pages of the p-th document in the training set. The E- and M-steps are iterated until a local maximum of the (incomplete) data likelihood is reached. Note that if page categories are observed, it is convenient to use the estimates computed with Eq. (7) as a starting point, rather than using random initial parameters. Similarly, an initial estimate of the emission parameters can be obtained from Eq. (6). It is interesting to point out a related application of the EM algorithm for learning from labeled and unlabeled documents (Nigam et al., 2000). In that paper, the only concern was to allow the learner to take advantage of unlabeled documents in the training set. As a major difference, the method in Nigam et al. (2000) assumes flat single-page documents and, if applied to multi-page documents, would be equivalent to a zero-order Markov model that cannot take contextual information into account. 3.4. Page classification Given a document of T pages, classification is performed by first computing the sequence of states x1,.,xT that was most likely to have generated the observed sequence of pages, and then mapping each state to the corresponding category (xt ). The most likely state sequence can be obtained by running an adapted version of Viterbi's algorithm, whose more general form is the max-propagation algorithm for Bayesian networks described in Jensen (1996). Briefly, the following quantity is computed using the following recursion: 204 FRASCONI ET AL. The optimal state sequence is then retrieved by backtracking: Finally, categories are obtained as ct = (xt ). By contrast, note that the Naive Bayes clas- sifier would compute the most likely categories as ct Comparing Eqs. (11)-(15) to Eq. (16) we see that both classifiers rely on the same emission model P(dt | cj ) but while Naive Bayes employs the prior class probability to compute its final prediction, the HMM classifier takes advantage of a dynamic term (in square brackets in Eq. (12)) that incorporates grammatical constraints. 4. Experimental results In this section, we describe a set of experiments that give empirical evidence of the effectiveness of the proposed model. The main purpose of our experiments was to make a comparison between our multi-page classification approach and a traditional isolated page classification system, like the well known Naive Bayes text classifier. The evaluation has been conducted over real-world documents that are naturally organized in the form of page sequences. We used two different datasets associated with two journals in the Making of America (MOA) collection. MOA is a joined project between the University of Michigan and Cornell University (see http:/moa.umdl.umich.edu/about.html and Shaw and Blumson (1997)) for collecting and making available digitized information about history and evolution processes of the American society between the XIX and the XX century. 4.1. Datasets The first dataset is a subset of the journal American Missionary, a sociological magazine with strong Christian guidelines. The task consists of correctly classifying pages of previously unseen documents into one of the ten categories described in Table 1. Most of these categories are related to the topic of the articles, but some are related to the parts of the journal (i.e. Contents, Receipts, and Advertisements). The dataset we selected contains 95 issues from 1884 to 1893, for a total of 3222 OCR text pages. Special issues and final report issues (typically November and December issues) have been removed from the dataset as they contain categories not found in the rest. The ten categories are temporally stable over the The second dataset is a subset of Scribners Monthly, a recreational and cultural magazine printed in the second half of the XIX century. Table 2 describes the categories we have selected for this classification task. The filtered dataset contains a total of 6035 OCR text pages, organized into issues ranging from year 1870 to 1875. Although spanning a shorter temporal interval, the number of pages in this second dataset is larger than in the first one because issues are about 3-4 times longer. HIDDEN MARKOV MODELS 205 Table 1. Categories in the American Missionary domain. Name Description Cover and index of surveys Editorial articles Afro-Americans' surveys American Indians' surveys Reports from China missions Articles about female condition Education and childhood Magazine information Lists of founders Contents is mostly graphic, with little text description Table 2. Categories in the Scribners Monthly domain. Name Description 1. Article 2. Books and Authors at Home and Abroad Generic articles Book reviews Broad cultural news Poems or tales Articles on home living Scientific articles Articles on fine arts News reports Category labels for the two datasets were obtained semi-automatically, starting from the MOAXMLfilessuppliedwiththedocumentscollections.Theassignedcategorieswerethen manually checked. In the case of a page containing the end and the beginning of two articles belonging to different categories, the page was assigned the category of the ending article. Each page within a document is represented as a bag-of-words, counting the number of word occurrences within the page. It is worth remarking that in both datasets, instances are text documents output by an OCR system. Imperfections of the recognition algorithm and the presence of images in some pages yields noisy text, containing misspelled or nonexistent words, and trash characters (see Bicknese (1998) for a report of OCR accuracy in the MOA digital library). Although these errors may negatively affect the learning process and subsequent results in the evaluation phase, we made no attempts to correct and filter out misspelledwords,exceptforthefeatureselectionprocessdescribedinSection4.3.However, since OCR extracted documents preserve the text layout found in the original image, it was necessary to rejoin word fragments that had been hyphenated due to line breaking. 206 FRASCONI ET AL. 4.2. Grammar induction In the case of completely labeled documents, it is possible to run the structure learning algorithm presented in Section 3.2. In figure 5 we show an example of induced HMM topology for the journal The American Missionary. This structure was extracted using 10 issues (year 1884) as a training set. Each vertex in the transition graph is associated with one HMM state and is labeled with the corresponding category index (see Table 1). Edges are labeled with the transition probability from source to target state, estimated in this case by counting state transitions during the state merging procedure (see Eq. (7)). These values are also used as initial estimates of P(xi |x j )and subsequently refined by the EM algorithm. The associated stochastic grammar implies that valid sequences must start with the index page (class 1), followed by a page of general communications (class 8). Next state is associated with a page of an editorial article (2). Self transition here has a value of 0.91, meaning that with high probability the next page will belong to the editorial too. With lower probability (0.07) next page is one of The South survey (3) or (probability 0.008) The Indians (4) or Bureau of Women's work (6). In figure 6 we show one example of induced HMM topology for journal Scribners Monthly,obtainedfrom12trainingissues(year1871).AlthoughissuesofScribnersMonthly are longer and the number of categories is comparable to those in the American Missionary, the extracted transition diagram in figure 6 is simpler than the one in figure 5. This reflects less variability in the sequential organization of articles in Scribners Monthly. Note that category 7 (Home and Society) is rare and never occurs in 1871. 4.3. Feature selection Text pages were first preprocessed with common filtering algorithms including stemming and stop words removal. Still, the bag-of-words representation of pages leads to a very high-dimensional feature space that can be responsible of overfitting in conjunction to algorithms based on generative probabilistic models. Feature selection is a technique for limiting overfitting by removing non-informative words from documents. In our experi- ments, we performed feature selection using information gain (Yang and Pedersen, 1997). This criterion is often employed in different machine learning contexts. It measures the average number of bits of information about the category that are gained by including a word in a document. For each dictionary term w, the gain is defined as K K P(ck | w)log P(ck | w) where w denotes the absence of word w. Feature selection is performed by retaining only the words having the highest average mutual information with the class variable. OCR errors, however, can produce very noisy features which may be responsible of poor performance HIDDEN MARKOV MODELS 207 Figure 5. Data induced HMM topology for American Missionary, year 1884. Numbers in each node correspond 208 FRASCONI ET AL. Figure 6. Data induced HMM topology for Scribners Monthly, year 1871. Numbers in each node correspond to HIDDEN MARKOV MODELS 209 even if feature selection is performed. For this reason, it may be convenient to prune from the dictionary (before applying the information gain criterion) all the words occurring less than a given threshold h in the training set. Preliminary experiments showed that best performances are achieved by pruning words having less than occurrences. 4.4. Accuracy comparisons In the following we compare isolated page classification (using standard Naive Bayes) to sequential classification (using the proposed HMM architecture). Although classification accuracy could be estimated by fixing a split of the available data into a training and a test set, here we suggest a method that attempts to incorporate some peculiarities of digital libraries domain. In particular, hand-labeling of documents for the purpose of training is a very expensive activity and working with large training sets is likely to be unrealistic in practical applications. For this reason, in most experiments we deliberately used small fractions of the available data for training. Moreover, there is a problem of temporal stability as the journal organization may change over time. In our test we attempted to address this aspect by assuming that training data is available for a given year and we decided to test generalization over journal issues published in different years. Splitting according to publication year can be an advantage for the training algorithm since it increases the likelihood that different issue organizations are represented in the training set. The resulting method is related to k-fold cross-validation, a common approach for accuracy estimation that partitions the dataset into k subsets and iteratively use one subset for testing and the other k 1 for training. In our experiments we reversed the proportions of data in the training and test sets, using all the journal issues in one year for training, and the remaining issues for testing. We believe that this setting is more realistic in the case of digital libraries. In the following experiments, the HMM classifiers were trained by first extracting the transition structure, then initializing the parameters using Eqs. (6) and (7), and finally tuning the parameters using the EM algorithm. We found that the initial parameter estimates are very close to the final solution found by the EM algorithm. Typically, 2 or 3 iterations are sufficient for EM to converge. 4.4.1. American Missionary dataset. The results of the ten resulting experiments are shown in figure 7. The hybrid HMM classifier (performing sequential classification) consistently outperforms the plain Naive Bayes classifier working on isolated pages. The graph on the top summarizes results obtained without feature selection. Averaging the results over all the ten experiments, NB achieves 61.9% accuracy, while the HMM achieves 80.4%. This corresponds to a 48.4% error rate reduction. The graph on the bottom refers to results obtained by selecting the best 300 words according to the information gain criterion. The average accuracy in this case is 69.8% for NB and 80.6% for the HMM (a 35.7% error rate reduction). In both cases, words occurring less than 10 times in their training sets were pruned. When using feature selection, NB improves while the HMM performance is essentially the same. Moreover, the standard deviation of the accuracy is smaller for NB (2.8%, compared to 4.2% for the HMM). The larger variability in the case of the HMM is due to Figure 7. Isolated vs. sequential page classification on the American Missionary dataset. For each column, classifiers are trained on documents of the corresponding year and tested on all remaining issues. the structure induction algorithm. In facts, the sequential organization of journal issues is temporally less stable than article contents. 4.4.2. Scribners Monthly dataset. Similar experiments have been carried out on the Scrib- ners Monthly journal. Results using no feature selection are shown on the top of figure 8. The average accuracy is 81.0%, for isolated page classification and 89.6% for sequential classi- fication (the error reduction is 42.5%). After feature selection, the average accuracy drops to 75.3% for the isolated page classifier, while it remains similar for the sequential classifier. HIDDEN MARKOV MODELS 211 Figure 8. Isolated vs. sequential page classification on the Scribners Monthly dataset. Noticeably, feature selection has different effects on the two datasets when coupled with the Naive Bayes classifier: it tends to improve accuracy for the American Missionary and tends to worsen for the Scribners Monthly. On the other hand, the HMM is almost insensitive to feature selection, in both datasets. This is apparently counterintuitive since the emission model is almost the same for the two classifiers (except for the EM tuning of emission parameters in the case of the HMM). However, it should be remarked that the Naive Bayes' final prediction is biased by the class prior (Eq. (16)) while the HMM's prediction is biased by extracted grammar (Eqs. (11)-(15)). The latter provides more robust information that effectively compensates for the crude approximation in the emission model, prescribing 212 FRASCONI ET AL. conditional word independence. This robustness also affects positively performance if a suboptimal set of features is selected for representing document pages. 4.5. Learning using ergodic HMMs The following experiments provide a basis for evaluating the effects of the structure learning algorithm presented in Section 3.2. In the present setting, we trained an ergodic HMM with ten states (each state mapped to exactly one class). Emission parameters were initialized using Eq. (6) while transition probabilities were initialized with random values. In this case the EM algorithm takes the full responsibility for extracting sequential structure from data. After training, arcs with associated probability less than 0.001 were pruned away. The evaluation was performed using the American Missionary dataset, training on single years as in the previous set of experiments. As expected (see figure 9), results are worse than those obtained in conjunction with the grammar extraction algorithm. However, the trained HMM outperforms the Naive Bayes classifier also in this case. 4.6. Effects of the training set size To investigate the effects of the size of the training set we propose a set of experiments alternative to those reported in Section 4.4. In these experiments we selected a variable number of sequences (journal issues) n for training (randomly chosen in the dataset) and tested generalization on all the remaining sequences. The accuracy is then reported as a function of n, after averaging over 20 trials (each trial with the same proportion of training and test sequences). All these experiments were performed on the American Missionary dataset. As shown in figure 10, generalization for both the isolated and the sequential classifier tends to saturate after about 15 sequences in the training set. This is slightly more Figure 9. Comparison between the ergodic HMM and the HMM based on the extracted grammar. HIDDEN MARKOV MODELS 213 Figure 10. Learning curve for the sequential and the isolated classifiers. than the average number of issues in a single year. The sequential classifier consistently outperforms the isolated page classifier. 4.7. Learning with partially labeled documents Since labeling is an expensive human activity, we evaluated our system also when only a fraction of the training documents pages are labeled. In particular, we are interested in measuring the loss of accuracy due to missing page labels. Since structure learning is not feasible with partially labeled documents, we used in this case an ergodic (fully connected) HMM with ten states (one per class). We have performed six different experiments on the American Missionary dataset, using different percentages of labeled pages. In all the experiments, all issues of year 1884 form the training set and the remaining issues form the test set. Table 3 shows detailed results of the experiment. Classification accuracy is reported for single classes and for the entire test set. Using 30% of labeled pages the HMM fails to learn a reliable transition structure and the Naive Bayes classifier (trained with EM as in Nigam et al. (2000)) obtains higher accuracy Table 4). However, with higher percentages of known page labels the comparison favors again the sequential classifier. Using only 50% of labeled pages, the HMM outperforms the isolated page classifier that was trained on completely labeled data. With greater percentages of labeled documents, performances begin to saturate reaching a maximum of 80.24% when all the labels are known (this corresponds to the result obtained in Section 4.5). 5. Conclusions We have presented a text categorization system for multi-page documents which is capable of effectively taking into account contextual information to improve accuracy with respect to traditional isolated page classifiers. Our method can smoothly deal with unlabeled pages within a document, although we have found that learning the HMM structure further 214 FRASCONI ET AL. Table 3. Results achieved by the model trained by Expectation-Maximization, varying percentage of labeled documents. Percentage of labeled documents Category Another aspect is the granularity of document structure being exploited. Working at the level of pages is straightforward since page boundaries are readily available. However, actual category boundaries may not coincide with page boundaries. Some pages may contain portions of text belonging to different articles (in this case, the page would belong to multiple categories). Although this is not very critical for single-column journals such as the American Missionary, the case of documents typeset in two or three columns certainly deservesattention.Afurtherdirectionofinvestigationisthereforerelatedtothedevelopment of algorithms capable of performing automatic segmentation of a continuous stream of text, without necessarily relying on page boundaries. Finally, text categorization methods that take document structure into account may be extremely useful for other types of documents natively available in electronic form, including web pages and documents produced with other typesetting systems. In particular, hypertexts (like most documents in the Internet) are organized as directed graphs, a structure that can be seen as a generalization of sequences. However, devising a classifier that can capture context inhypertextsbyextendingthearchitecturedescribedinthispaperisstillanopenproblem:al- though the extension of HMMs from sequences to trees is straightforward (see e.g. Diligenti etal.(2001)),thegeneralcaseofdirectedgraphsisdifficultbecauseofthepresenceofcycles. Preliminary research in this direction (based on simplified models incorporating graphical transition structure) is presented in Diligenti et al. (2000) and Passerini et al. (2001). Acknowledgments We thank the Cornell University Library for providing us data collected within the Making of America project. This research was partially supported by EC grant # IST-1999-20021 under METAe project. Notes 1. A related formulation would consist of assigning a global category to a whole multi-page document, but this formulation is not considered in this paper. 2. After observing the text. 3. A Bayesian network is an annotated graph in which nodes represent random variables and missing edges encode conditional independence statements amongst these variables. Given a particular state of knowledge, the semantics of belief networks determine whether collecting evidence about a set of variables does modify one's belief about some other set of variables (Jensen, 1996; Pearl, 1988). 4. We adopt the standard convention of denoting variables by uppercase letters and realizations by the corresponding lowercase letters. Moreover, we use the table notation for probabilities as in Jensen (1996); for example P(X) is a shorthand for the table denotes the two-dimensional table with entries 5. Of course this does not mean that the category is independent of the context. --R An Input Output HMM Architecture. Bayesian Networks dor Data Mining. An Introduction to Bayesian Networks. Text Categorization with Support Vector Machines: Learning with Many Relevant Fea- tures Transductive Inference for Text Classification using Support Vector Machines. An Experimental Evaluation of OCR Text Representations for Learning Document Classifiers. A New Probabilistic Model of Text Classification and Retrieval. Hierarchically Classifying Documents using Very Few Words. A Sequential Algorithm for Training Text Classifiers. Comparison of Two Learning Algorithms for Text Categorization. Bayesian Belief Networks as a Tool for Stochastic Parsing. Automating the Construction of Internet Portals with Machine Learning. Machine Learning. Feature Selection Text Classification from Labeled and Unlabeled Documents using EM. Evaluation Methods for Focused Crawling. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition. Online Searching and Page Presentation at the University of Michigan. Probabilistic Independence Networks for Hidden Markov Probability Models. Hidden Markov Model Induction by Bayesian Model Merging. An Example-Based Mapping Method for Text Classification and Retrieval A Comparative Study on Feature Selection in Text Categorization. --TR Probabilistic reasoning in intelligent systems: networks of plausible inference An example-based mapping method for text categorization and retrieval A sequential algorithm for training text classifiers Bayesian Belief Networks as a tool for stochastic parsing Probabilistic independence networks for hidden Markov probability models Feature selection, perception learning, and a usability case study for text categorization Text Classification from Labeled and Unlabeled Documents using EM Statistical Language Learning Machine Learning Introduction to Bayesian Networks Bayesian Networks for Data Mining Automating the Construction of Internet Portals with Machine Learning Text Categorization with Suport Vector Machines Hierarchically Classifying Documents Using Very Few Words A Comparative Study on Feature Selection in Text Categorization Transductive Inference for Text Classification using Support Vector Machines Hidden Markov Model} Induction by Bayesian Model Merging Focused Crawling Using Context Graphs Image Document Categorization Using Hidden Tree Markov Models and Structured Representations Information Extraction with HMM Structures Learned by Stochastic Optimization Evaluation Methods for Focused Crawling A New Probabilistic Model of Text Classification and Retrieval TITLE2: --CTR Fabrizio Sebastiani, Machine learning in automated text categorization, ACM Computing Surveys (CSUR), v.34 n.1, p.1-47, March 2002

text categorization;hidden Markov models;multi-page documents;naive bayes;digital libraries

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A Study of Approaches to Hypertext Categorization.

Hypertext poses new research challenges for text classification. Hyperlinks, HTML tags, category labels distributed over linked documents, and meta data extracted from related Web sites all provide rich information for classifying hypertext documents. How to appropriately represent that information and automatically learn statistical patterns for solving hypertext classification problems is an open question. This paper seeks a principled approach to providing the answers. Specifically, we define five hypertext regularities which may (or may not) hold in a particular application domain, and whose presence (or absence) may significantly influence the optimal design of a classifier. Using three hypertext datasets and three well-known learning algorithms (Naive Bayes, Nearest Neighbor, and First Order Inductive Learner), we examine these regularities in different domains, and compare alternative ways to exploit them. Our results show that the identification of hypertext regularities in the data and the selection of appropriate representations for hypertext in particular domains are crucial, but seldom obvious, in real-world problems. We find that adding the words in the linked neighborhood to the page having those links (both inlinks and outlinks) were helpful for all our classifiers on one data set, but more harmful than helpful for two out of the three classifiers on the remaining datasets. We also observed that extracting meta data from related Web sites was extremely useful for improving classification accuracy in some of those domains. Finally, the relative performance of the classifiers being tested provided insights into their strengths and limitations for solving classification problems involving diverse and often noisy Web pages.

Introduction As the size of the Web expands rapidly, the need for good automated hypertext classication techniques is becoming more apparent. The Web contains over two billion pages connected by hyperlinks, making the task of locating specic information on the Web increasingly di-cult. A recent user study [2] showed that users often prefer navigating through directories of pre-classied content, and that providing a category-based view of retrieved documents enables them to nd more relevant information in a shorter time. The common use of category hierarchies for navigation support in Yahoo! and other major Web portals has also demonstrated the practical utility of hypertext categorization. Automated hypertext classication poses new research challenges because of the rich information in a hypertext document and the connectivity among documents. Hyperlinks, HTML tags, category distributions over a linked neighborhood, and meta data extracted from related Web sites all provide rich information for hyper-text classication, which is not normally available in traditional text classication. Researchers have only recently begun to explore the issues of exploiting rich hyper-text information for automated classication. Chakrabarti et al. [1] studied the use of citations in the classication of IBM patents where the citations between documents (patents) were considered as \hy- perlinks", and the categories were dened in a topical hierarchy. Similar experiments on a small set of Web pages (only 900 pages from Yahoo!) with real hyperlinks were also conducted. By using the system-predicted category labels for the linked neighbors of a test document to reinforce the category decision(s) on that docu- ment, they obtained a 31% error reduction, compared to the baseline performance when using the local text in the document alone. They also tested a more naive way of using the linked documents, treating the words in the linked documents as if they were local. This approach increased the error rate of their system by 6% over the baseline performance. Oh et al. [18] reported similar observations on a collection of online Korean encyclopedia articles. By using the system-predicted categories of the linked neighbors of a test document to reinforce the classication decision(s) on that document, they obtained a 13% improvement in F 1 (dened in Section 4.2) over the base-line performance (when using local text only). On the other hand, when treating words in the linked neighborhood of a document as if they were local words in that document, the performance of their classier (Naive Bayes) decreased by 24% in micro-averaged F 1 . Instead of using all the links from a document, they decided to use only a subset of the linked documents based on the cosine similarity between the \bags of words" of pairwise linked documents { the links with low similarity scores were ignored. This ltering process yielded a 7% improvement in F 1 over naively using all the links. Furnkranz [10] used a set of Web pages from the WebKB University corpus (Sec- tion to study the use of anchor text (words on a link) and the words \near" the anchor text in a Web page to predict the class of the target page pointed to by the links. By representing the target page using the anchor words on all the links that point to it, plus the headlines that structurally precede the sections where links occur, the classication accuracy of a rule-learning system (Ripper [5]) improved by 20%, compared to the baseline performance of the same system when using the local words in the target page instead. Slattery and Mitchell [23] also used the WebKB University corpus, but studied alternative learning paradigms, namely, a First Order Inductive Learner which exploits the relational structure among Web pages, and a Hubs & Authorities style algorithm [15] exploiting the hyperlink topology. They found that a combined use of these two algorithms performed better than using each alone. Joachims et al. [14] also reported a study using the WebKB University corpus, focusing on Support Vector Machines (SVMs) with dierent kernel functions. Using one kernel to represent a document based on its local words, and another kernel to represent hyperlinks, they give evidence that combining the two kernels leads to better performance in two out of three classication problems. Their experiments suggest that the kernels can make more use of the training-set category labels in the linked neighborhood of a document compared to the local words in that document. Whereas the work summarized above provides initial insights in exploiting information in hypertext documents for automated classication, many questions still remain unanswered. For example, it is not entirely clear why the use of anchor words improve classication accuracy in Furnkranz's experiments on the WebKB pages, but the inclusion of all linked words decreased performance in Chakrabarti's experiments on the IBM patents. Recall that anchor words are a subset of the linked words (from the in-links). What would happen if Furnkranz expanded the subset to the full set of linked words in the WebKB pages, or if Chakrabarti et al. selected a subset (the words from the anchor elds of the in-link pages only, for example) instead of the full set of words in the IBM patents? How much did the dierence between the data contribute to the reported performance variance? How much did the particular algorithms used in those experiments in uence the obser- vations? Since most of the experimental results are not directly comparable (even true for the results on the WebKB corpus because of dierent subsets of documents and categories used in those experiments), the answers to these questions are not clear. In order to draw general conclusions about hypertext classication, we need more systematic experiments and better analysis about the potential reasons behind the observed performance variances. As a step in that direction, we begin with hypotheses about hypertext regularities (Section 2.1), then report systematic examinations of these hypotheses on the cross product of three data collections (Section 3), three classication algorithms (Section 2.3), and various representations for hypertext data (Section 2.2). We also provide direct data analysis in support of our empirical results with our classiers for the hypertext regularities being tested (Section 4), leading toward generalizable observations and conclusions (Section 5). 2. Methodology The purpose of the experiments presented in this paper is to explore various hypotheses about the structure of hypertext especially as it relates to hypertext clas- sication. While the scope of the experimental results presented is necessarily conned to the three classication problems described in Section 3, we hope that the analysis that follows will help future research into hypertext classication by providing some ideas about various types of regularities that may be present in other Table 1. Denitions of ve possible regularities we can use when classifying documents of class A. Regularity Denition None Documents neighboring class A documents exhibit no pattern. Encyclopedia Documents neighboring class A documents are all of class A. Co-referencing Documents neighboring class A documents all share the same class, but are not of class A. Preclassied A single document points only to all documents of class A. Meta data Relevant text extracted from sources external to the Web document, or internal but not visible on that document. hypertext corpora and how one should construct a classier to take advantage of them. 2.1. Regularities Before we can exploit patterns in a hypertext corpus, we need to understand what kind of regularities to expect. This section presents a list of what we believe are the simplest kinds of hypertext regularities we might consider searching for. Of course we still expect the content the document being classied to be a primary source of information and this list is meant to explore where we might look for more information in a hypertext classication problem. Succinct denitions of these regularities are given in Table 1. 2.1.1. No Hypertext Regularity It is important to be aware that for some hypertext corpora and classication tasks on them, the only useful place to look for information about the class label of a document is the document itself. In cases like this, looking outside the document for information about the label is not going to help and may in some cases hurt classication performance. However, we believe that for many real-world hypertext classication tasks, extra information is available to improve upon the performance of a classier which only uses the content of each document. 2.1.2. Encyclopedia Regularity Perhaps the simplest regularity we might hope to nd is one where documents with a given class label only link to documents with the same class label. We might expect to nd approximately this regularity in a corpus of encyclopedia articles, such as the ETRI-Kyemong encyclopedia corpus used in [18], since encyclopedia articles generally reference other articles which are topically similar. 2.1.3. Co-Referencing Regularity Instead of having the same class, neighboring documents can have some other topic in common with each other. For example, news articles about a particular current event may link to many articles about the background for that event. As another example, previous work [22] found that when learning to classify course home pages, a group of the neighboring pages about homework assignments were found to be useful, even though those homework assignment pages were not part of the learning task and not labelled in the data. It is important to realize that, in general, the topic of these linked documents may not correspond to any class in the classication problem. A variant of this regularity relaxes the requirement that all of the neighboring documents share the same topic. Instead, it may be the case that only some neighboring documents of a class share the same topic. For example, all faculty home pages may contain a link to a page describing research interests. If we can nd this regularity, it can help us with classication. In previous work [12] we described this type of regularity as a partial co-referencing regularity. This type of regularity is particularly di-cult to exploit because it requires searching for subsets of the neighboring documents that have some unseen topic in common. 2.1.4. Preclassied Regularity 1 While the encyclopedia and co-referencing regularities consider the topic of neighboring documents, there can also be regularities in the hyperlink structure itself. One such regularity prevalent on the Web consists of a single document which contains hyperlinks to documents which share the same topic. Finding this "hub" document would help us in classifying all the documents that are linked from it. Categories on the Yahoo! topic hierarchy are a perfect example of this regularity. If a category on the Yahoo! hierarchy happens to corresponds to a class in our classication problem, then we say that the pages linked to that category exhibit a preclassied regularity, since in eect the creator of those hyperlinks has preclassied all the documents of some class for us. 2.1.5. Meta Data Regularity For many classication tasks that are of practical and commercial importance, meta data are often available from external sources on the Web that can be exploited in the form of additional features. Examples of these types of meta data include movie reviews for movie classication, online discussion boards for various other topic classication tasks (such as stock market predictions or competitive analysis). Meta data is also implicitly present in many Web documents in the form of text within tags and within ALT and TITLE tags (which are not visible when viewing the Web page through most browsers). If we can extract rich and predictive features from such sources, we can build classiers that can use them alone or combine them with the hyperlinks and textual information. 2.2. Hypertext Classication Approaches Depending on which of the above regularities holds for the hypertext classication task under consideration, dierent classier designs should be considered. Likewise for given applications, using dierent classier designs, we can search for various hypertext regularities in the dataset. In our experiments for this paper, for simplicity we did not distinguish the links to and from a page. Our examination consists of the following components: 2.2.1. No Hyperlink Regularity With no regularity, we expect no benet from using hyperlinks and would use standard text classiers on the text of the document itself. Using such classiers as performance baselines and comparing them to classiers which take hyperlinks into account will allow us to test for the existence of various hyperlink regularities. It is quite possible that for a substantial fraction of hypertext classication tasks there is no hyperlink regularity that can be exploited and the best performance we can hope for is with a standard text classier. Indeed in such cases, learners looking for hyperlink regularities may be led astray and end up performing worse than a simple text classier. 2.2.2. Encyclopedia Regularity If the encyclopedia regularity holds in a given data set, then augmenting the text of each document with the text of its neighbors should produce better classication results because more topic-related words would be present in the document representation. Chakrabarti et al. applied this approach to a database of patents and found that classication performance suered, suggesting that the patent database is unlikely to have this structure [1]. 2.2.3. Co-Referencing Regularity If the co-referencing regularity holds, then augmenting each document as above except treating the additional words as if they come from a separate vocabulary should help classication. A simple way to do this is to prex the words in the linked documents with a tag. Chakrabarti et al. also tried this approach on the patent database and again found that performance suered, suggesting that the patent database is unlikely to have this structure [1]. If instead we have a partial co-referencing regularity, we need to identify the linked pages which are topically similar to each other (the \research interests" pages from the faculty home page example in Section 2.1.3). This can be achieved by computing the text-based similarity among all the documents linked to documents in the same class and clustering them accordingly. In contrast to the previous approaches which can be characterized as using various \bag-of-words" representations and standard text classication algorithms, this approach requires a more elaborate algorithm. One such algorithm is the Foil algorithm described in the next section. Craven et al. [7] applied Foil to the WebKB University corpus and found that it did improve classication performance, indicating that this corpus does have this kind of regularity. The cosine-similarity based ltering of linked neighbors by Oh et al. (Section 1) is another example of utilizing this regularity. 2.2.4. Preclassied Regularity If the classication scheme of our corpus is already embedded in the hyperlink structure, we have no need to look at the text of any document. We just need to nd those pages within the hypertext \graph" that have this property. We can search for those pages by representing each page with only the names of the pages it links with. If any of these linked pages are correlated with a class label, a reasonable learning algorithm should be able to recognize it as a predictive feature and use it (often together with other features) to make classication decisions. The successful use of the SVM kernel function for hyperlinks by Joachims et al. (Section 1) illustrated one way of exploiting this regularity. In this paper we show alternative approaches and use of this regularity with the kNN, NB and Foil algorithms. 2.2.5. Meta Data Regularity When external sources of information are available that can be used as meta data, we can collect them, possibly using information extraction techniques. In particular, we look for features that relate two or more entities/documents being classied. Following the approaches outlined above for hyperlinks, these extracted features can then be used in a similar fashion by using the identity of the related documents and by using the text of related documents in various ways. Any information source from the Web about the entity being classied can be used as a meta data resource and the availability and quality of such resources will certainly depend on the classication task. Cohen [4] described some experiments where he automatically located and extracted such features for several (non-hypertext) classication tasks. We also look for \meta data" contained within Web pages such as META and TITLE tags. The information contained within these HTML tags in a page is technically not meta data because it is internal rather than external to the page. Nevertheless, these tagged elds can be treated dierently from other parts of Web pages and can a useful source for classication. 2.3. Learning Algorithms Used Our experiments used three existing classiers: Naive Bayes, kNN and Foil. Naive Bayes and kNN have been thoroughly evaluated for text classication on benchmark collections and oer a strong baseline for comparison. Foil is a relational learner which has shown promise for hypertext classication. The following notation is used for the descriptions of Naive Bayes and kNN: { Set of training documents { Set of training documents in class c j n(t) { Number of training documents containing t N(t) { Number of occurrences of t N(t; d) { Number of occurrences of t in document d 2.3.1. Naive Bayes Naive Bayes is a simple but eective text classication algorithm [16, 17]. The parameterization given by Naive Bayes denes an underlying generative model assumed by the classier. In this model, rst a class is selected according to class prior probabilities. Then, the generator creates each word in a document by drawing from a multinomial distribution over words specic to the class. Thus, this model assumes each word in a document is generated independently of the others given the class. Naive Bayes forms maximum a posteriori estimates for the class-conditional probabilities for each term in the vocabulary, V , from labeled training data D. This is done by calculating the frequency of each term t over all the documents in a class supplemented with Laplace smoothing to avoid zero probabilities: N(t; d) N(t; d) We calculate the prior probability of each class (Pr(c j )) from the frequency of each document label in the training set. At classication time we use these estimated parameters by applying Bayes' rule (using a word independence assumption) to calculate the posterior probability of each class label (Pr(c j jd)) for a test document d, and taking the most probable class as the prediction (since all the documents in our datasets used for this study belong to one and only one class; see Section3 for details): Y t2d 2.3.2. K-Nearest Neighbor (kNN) kNN, an instance-based classication method, has been an eective approach to a broad range of pattern recognition and text classication problems [8, 25, 26, 28]. In contrast to \eager learning" algorithms (including Naive Bayes) which have an explicit training phase before seeing any test document, kNN uses the training documents \local" to each test document to make its classication decision on that document. Our kNN uses the conventional vector space model, which represents each document as a vector of term weights, and the similarity between two documents is measured using the cosine value of the angle between the corresponding vectors. We compute the weight vectors for each document using one of the conventional TF-IDF schemes [20], in which the weight of term t in document d is dened as: Given an arbitrary test document d, the kNN classier assigns a relevance score to each candidate category using the following formula: where set R k (d) are the k nearest neighbors (training documents) of document d. By sorting the scores of all candidate categories, we obtain a ranked list of categories for each test document; by further thresholding on the ranks or the scores, we obtain binary decisions, i.e. the categories above the threshold will be assigned to the document. There are advantages and disadvantages of dierent thresholding strategies [26, 27]. In this paper, we use the simplest strategy { assigning the top-ranking category only to each document as a baseline; for more exible trade-o between recall and precision, we further threshold on the scores of the top-ranking candidates, as described in Section 4.4. 2.3.3. FOIL Quinlan's Foil [19] is a greedy covering algorithm for learning Horn clauses. It induces each clause by beginning with an empty tail and using a hill-climbing search to add literals until the clause covers as few negative instances as possible. The evaluation function used to guide the hill-climbing search is an information-theoretic measure. Foil has already been used for text classication to exploit word order [3] and hyperlink information [7]. Here Foil is used as described in [7], using a unary has word(page) relation (where word is a variable) for each word and a link to(page,page) relation for hyperlinks between pages. The former allows Foil to distinguish informative words from non-informative ones, and the latter gives Foil the power to recognize predictive links among pages among all the links. It is a common practice to apply class-driven feature selection to documents before training a classier, for reducing the computational cost and possibly improving the eectiveness. However, for relational hypertext classication, this is less than straightforward (how would we know that words relating to assignments in a linked page would help to classify course home pages without searching for that regularity rst?). The experiments presented here use document frequency feature selection. All the Foil experiments in this paper were done by running the algorithm on binary subproblems, one for each class in the problem. For each test example, the system computed a score for each class by picking the matching rule with highest condence score from each of the learned binary classiers. The condence scores were based on the training-set accuracies of the rules. This process results in a list of scored categories for each test example, allowing further thresholding for classication decisions, as described in the kNN section above. This is perhaps the simplest approach to combining the outputs of several Foil classiers, and more elaborate strategies would almost certainly do better. 3. Datasets To test our proposed approaches to hypertext classication, we needed datasets that would re ect the properties of real-world hypertext classication tasks. We wanted a variety of problems so we could get a general sense of the usefulness of each regularity described in the previous section. We found three hypertext classication problems for this study: two of them are about classication of company Web sites, and the third one is a classication task for university Web pages. 3.1. Hoovers-28 and Hoovers-255 The Hoovers corpora of company Web pages was assembled using the Hoovers On-line Web resource (www.hoovers.com) which contains detailed information about a large number of companies and is a reliable source of corporate information. Ghani et al. [11] obtained a list of the names and home-page URLs for 4285 companies on the Web and used a custom crawler to extract information from company Web sites. This crawler visited 4285 dierent company Web sites and searched up to the rst 50 Web pages on each site (in breadth rst order), examining just over 108,000 Web pages.50150250350 COUNTS Hoovers-28 (a) Hoovers-28103050700 50 100 150 200 250 COUNTS (b) Hoovers-255500150025003500 Other Student Course Faculty Project Staff COUNTS Univ-6 (c) Univ-6101000 50 100 150200250 COUNTS Univ-6 Hoovers-28 (d) All problems, logarithmic scale Figure 1. Category distributions for all three problems Two sets of categories are available from Hoovers Online: a coarse classication scheme of 28 classes (\Hoovers-28", dening industry sectors such as Oil & Gas, Sporting Goods, Computer Software & Services) and a more ne grained classi- cation scheme consisting of 255 classes (\Hoovers-255"). These categories label companies, not particular Web pages. For this reason, we constructed one synthetic page per company by concatenating all the pages (up to crawled for that company and ignoring the inner links between pages of that company. Therefore our task for this dataset is Web site classication rather than Web page classication due to the granularity of the categories in this application. Figures 1(a) and 1(b) show the category distributions for these two problems. Previous work with this dataset [11] extracted meta data about these Web sites from Hoovers Online, which provided information about the company names, and names of their competitors. The authors constructed several kinds of wrappers (from simple string matchers to statistical information extraction techniques) to extract additional information about the relationships between companies from the Web pages in this dataset, such as whether one company name is mentioned by another in its Web page, whether two companies are located in the same state (in U.S.) or the same country (outside of U.S.), and so forth. In the results section, we only report our experiments using the competitor information because of space limitations. The resulting corpora (namely, Hoovers-28 and Hoovers-255) consist of 4,285 synthetic pages with a vocabulary of 256,715 unique words (after removing stop words and stemming), 7,762 links between companies (1.8 links per company) and 6.0 competitors per company. Each Web site is classied into one category only for each classication scheme. 3.2. Univ-6 Dataset The second corpus comes from the WebKB project at CMU [6]. This dataset was assembled for training an intelligent Web crawler which could populate a knowledge base with facts extracted directly from the Web sites of university computer science departments. The dataset consists of 4,165 pages with a vocabulary of 45,979 unique words (after removing stop words and stemming). There are 10,353 links between pages in the corpus (2.5 links per page). Figure 1(c) shows the category distribution and Figure 1(d) shows how the category distributions for all three problems compare in logarithmic scales. The pages were manually labelled into one of 7 classes: student, course, faculty, project, sta, department and other. The department class was ignored in our experiments as it had only 4 instances. The most populous class (\other") is a catch-all class which is assigned to documents (74% of the total) that do not belong in any of the dened classes of interest. 4. Empirical Validation We conducted experiments aimed at testing the performance of each of the algorithms from Section 2.3 using Web page representations based on the discussion from Section 2.2. Note that each problem may or may not contain any of the regularities dened previously. Therefore, if a method does not perform well with a particular representation, it may be interpreted as either that the regularity does not exist in the task, or as evidence that the method is not well suited for making an eective use of the representation. 4.1. Experiments To examine the six possible regularities discussed in Section 2.1, we tested NB, kNN and Foil with the following representations of Web pages for all the three datasets (the hypertext regularity being considered is given in parentheses): Page Only (No Regularity) Use only the words on the pages themselves (used with NB, kNN, Foil) Linked Words (Encyclopedia Regularity) Add words from linked pages (used with NB, kNN; not applicable to Foil) Tagged Words (Co-Referencing Regularity) Add words from linked pages but distinguish them with a prex (used with NB, kNN; not needed for Foil) Tagged Words (Partial Co-Referencing Regularity) Represent Web pages individually and use a binary relation to indicate links (used with Foil; not applicable to NB and kNN) Linked Names (Preclassied Regularity) Represent each Web page by the names (or identiers) of the Web pages it links to and ignore the words on the Web page entirely (used with NB, kNN and Foil) HTML Title (Meta Data Regularity) Use the HTML title of a Web page (used with NB, kNN and Foil) HTML Meta (Meta Data Regularity) Use the text found in META tags on a Web page (used with NB, kNN and Foil). In addition to the above representations, we also explored the use of the following representation for the Hoovers experiments: Competitors (Meta Data Regularity) Use the competitor identiers (aka \com- petitors") of a company to represent that company instead of the original Web page (used with NB, kNN, Foil) All of the results of the experiments are averages of ve runs: each dataset was split into ve subsets, and each subset was used once as test data in a particular run while the remaining subsets were used as training data for that run. The split into training and test sets for each run was the same for all the classiers. Table 2. Micro-averaged F 1 results for each classier on each representation. Best results for each dataset with each representation are shown in bold. Hoovers28 Hoovers255 Univ6 Page Only 55.1 58.1 31.5 32.5 32.0 11.6 69.6 83.0 82.7 Linked Words 40.1 38.6 N/A 18.9 20.4 N/A 74.1 86.2 N/A Tagged Words 49.2 49.0 31.8 24.0 26.9 12.1 76.3 88.0 86.0 HTML Title 40.8 43.3 28.7 17.9 22.6 11.5 78.6 81.5 86.3 HTML Meta 48.6 49.8 29.3 23.1 28.3 13.1 73.3 78.6 81.5 Linked Names 14.8 13.3 12.3 5.0 5.9 4.6 81.7 87.2 86.6 Competitor Names 75.4 74.5 33.8 52.0 53.0 12.0 N/A N/A N/A Table 3. Macro-averaged F1 results for each classier on each representation. Best results for each dataset with each representation are shown in bold. Hoovers28 Hoovers255 Univ6 Page Only 54.3 55.3 31.6 24.6 19.8 8.0 31.4 46.4 51.3 Linked Words 40.3 35.1 N/A 14.8 12.0 N/A 38.3 53.0 N/A Tagged Words 49.0 46.5 31.9 17.9 15.9 8.3 46.1 59.1 52.9 HTML Title 37.1 39.9 27.5 10.2 14.5 9.6 43.2 41.8 50.1 HTML Meta 45.1 47.4 29.8 13.8 18.7 10.6 14.1 40.7 39.3 Linked Names 11.8 12.8 9.5 2.4 5.2 3.6 47.0 44.3 62.9 Competitor Names 75.2 74.2 33.7 40.8 44.5 8.3 N/A N/A N/A 4.2. Overall Results Tables 2, 3 and Figure 2 summarize the main results, where the performance of each classier is measured using the conventional micro-averaged and macro-averaged recall, precision and F 1 values [24, 26]. Recall (r) is the ratio of the number of categories correctly assigned by the system to the test documents to the actual number of relevant document/category pairs in the test set; precision (p) is the ratio of the number of correctly assigned categories to the total number of assigned categories. The F 1 measure is dened to be F recall and precision in a way that gives them equal weight. The recall, precision and F 1 scores can rst be computed for individual categories, and then averaged over categories as a global measure of the average performance over all categories; this way of averaging is called macro-averaging. An alternative way, micro-averaging, is to count the decisions for all the categories in a joint pool and compute the global recall, precision and F 1 values for that global pool. Micro-averaged scores tend to be dominated by the performance of the system on common categories, while macro-averaged scores tend to be dominated by the performance on rare categories if the majority of categories in the task are rare. For skewed category distributions (Figure 1(c) in Section 3) in our tasks, providing both types of evaluation scores gives a clearer picture than considering either type alone. Since our datasets used in this study are single-label-per-document tasks, micro-averaged accuracy, precision, recall and F 1 are all equal. We therefore report the F 1 score in our micro-averaged results, although all of the above measures can be used interchangeably. However, in general, the macro-averaged recall, precision and F 1 values are not the same. Further discussions on this issue can be found in the text categorization literature [26]. All the results reported are for optimal vocabulary sizes for each algorithm. The eect of feature selection on the categorization performance of our classiers is analyzed in Section 4.3. Some general observations can be made from the results of these experiments. The performance of a classier depends on the characteristics of the problem, the information encoded in the document representation and the capability of the clas- sier in identifying regularities in documents. For the Univ-6 problem, all the three classiers performed better when using hyperlink information (including Linked Names, Tagged Words and Linked Words) compared to using Page Only. For the two Hoovers problems, on the other hand, both kNN and NB suered a signi- cant performance decrease when using hyperlink information, except Competitor Names, while Foil's performance was not signicantly aected when given information about hyperlinked documents. As for our specic hypotheses on hypertext regularities, we have the following observations: 4.2.1. Page Only The Page Only results tell us something about the overall di-culty of each task. Unsurprisingly, problems with more classes proved to be more di-cult. On the Hoovers problems, NB and kNN have roughly equal per- formance, while Foil's performance is more competitive on Univ-6 than on the Hoovers datasets. The big contrast between Foil performance on Univ-6 and its performance on the Hoovers datasets is surprising, suggesting that Foil may not be as robust or stable as NB and kNN for conventional text categorization. In particular Foil is known to have a tendency to overt the training data, since it was designed for learning logic programs. However, it is interesting that Foil is the only classier (among these three) with no performance degradation on all the three datasets when using Tagged Words instead of the Page Only setting, while NB and kNN suered on Hoovers datasets due to the highly noisy hyperlinks (Section 4.2.2). 4.2.2. Linked Words The kNN and Naive Bayes results under the Linked Words condition in the Hoovers sets show that performance suers badly when compared to the baseline. It is quite clear that these datasets do not exhibit this regularity. A close look at these datasets revealed that 56.8% of the Hoovers pages have links (1.8 links per page), but, according to the Hoovers-255 labelling, only 6.5% of the linked pairs of pages belong to the same category. This means that at most 3.7% (i.e., 56.8% 6.5%) of the total pages would be possibly helped when using the system-assigned category labels to linked pages to reinforce the classication of those pages. In other words, a \perfect" hyperlink classier (making perfect use of the category labels of linked pages) would show improved performance in classifying at most 3.7% of the total pages. Since our classiers are not perfect, Page Only Linked Words Tagged Words Title Meta Linked Names Competitor Names Score (a) Hoovers-28, micro-avg F 110305070Page Only Linked Words Tagged Words Title Meta Linked Names Competitor Names Score (b) Hoovers-28, macro-avg F 1103050Page Only Linked Words Tagged Words Title Meta Linked Names Competitor Names Score (c) Hoovers-255, micro-avg F1515253545 Page Only Linked Words Tagged Words Title Meta Linked Names Competitor Names Score (d) Hoovers-255, macro-avg F11030507090Page Only Linked Words Tagged Words Title Meta Linked Names Score Only Linked Words Tagged Words Title Meta Linked Names Score (f) Univ-6, macro-avg F1 Figure 2. Performance of classiers on Hoovers-28, Hoovers-255 and Univ-6 dumping the words from linked documents into the document having these links adds a tremendous amount of noise to the representation of that document. It is unsurprising, therefore, that the performance of kNN and NB with Linked Words suered signicantly on the Hoovers datasets. Even Foil, designed for leveraging relational information in data, gains no improvement by using Tagged Words instead of Page Only. In the Univ-6 dataset, on the other hand, 98.9% of pages have links (2.5 links per page on average), and 22.5% of the linked pairs of pages belong to the same category. This means that a perfect hyperlink classier would improve performance on 22.3% of the total pages on Univ-6, which is much higher than the 3.7% of the Hoovers datasets. As a result, all the classiers have improved results on Univ-6 when using Linked Words or Tagged Words instead of Page Only, indicating that hyperlinks in this dataset are informative for those classiers. A potential problem in the algorithm proposed by Oh et al. [18] for exploiting the Encyclopedia regularity is that it calculates the likelihood for each document belonging to a certain class by multiplying the system-estimated class probability (using the words in the Web page) by the fraction of neighbors that are in the same class. Applying their method to our Hoovers datasets would result in very poor performance since 96.3% of the Web pages do not have any linked neighbors in the same class and multiplying such a low probability to the likelihood scores based on page words will result in a zero or near zero probability for a category candidate for many documents. 4.2.3. Tagged Words The Tagged Words experiments examine the impact of the Co-Referencing regularity. Unlike previous results reported by Chakrabarti et al. [1] where tagging the words from the neighbors (treating them as if they're from a separate vocabulary) did not aect results, we nd that this document representation results in signicant performance degradation from the baseline for two (Hoovers-28 and Hoovers-255) of our problems but signicant performance improvement from the baseline on one problem (Univ-6). An interesting observation is that using Tagged Words instead of Linked Words yielded less severe performance degradation for NB and kNN on the Hoovers datasets, and better performance for all the classiers on Univ-6. One possible explanation for this is that tagging linked words allows a feature selection procedure to remove irrelevant linked words leaving the learner (classier) with less noise in the data (evidence supporting this hypothesis is shown in Figure 3(e)), and allows the classier to weight linked words dierently from within-page words when making classication decisions. Note that if our classiers could handle noise \perfectly", then the performance would be at least as good as the baseline which ignores link information. This suggests that the feature selection methods and term weighting schemes we used with NB and kNN may have problems handling noise which could be overcome with more training data. The Foil results under Tagged Words look for partial co-referencing regularities. Here we see a slight improvement over the result of the baseline Foil for Hoovers- 255, and a small degradation for Hoovers-28 (in contrast to the severe performance degradations in NB and kNN), indicating that some small partial \co-referencing" regularity does exist and that Foil was capable of exploiting it with its relational representation. 4.2.4. Linked Names The Linked Names experiments examine the impact of the Pre-classied Regularity. Our results show that this representation works surprisingly well for Univ-6 (which is consistent with the observations by Joachims et al. [14] for SVMs on another version of the same corpus), but was the worst choice (for all three classiers) for the Hoovers datasets. Figure 2 shows the clear contrast that this regularity holds strongly in Univ-6 but not nearly so strongly in Hoovers datasets. The names or identiers of linked Web pages in Univ-6 are at least as informative as words (local, linked or tagged) in those pages for the clas- sication task; however, they do not provide as much information for the Hoovers classication tasks. 4.2.5. Meta Data Regularity The meta data we report results from is the Competitors data. We treat the competitor information in the same way as we do the hyperlinks and use the meta data in two ways: as category labels (Preclassied Reg- ularity) and as links between pages of the same class (Encyclopedia Regularity). In the case of the Preclassied Regularity, we use only the names of the competitors and nd a sharp boost in performance for both Hoovers-28 and Hoovers-255. For the Encyclopedia Regularity, we use the words from the competitors in the same way as we use the words from hyperlinked neighbors. Since the two representations yielded an almost equal performance boost for our classiers, which was reported in a previous paper [12], we only include the results for Competitor Names in Tables 2, 3 and Figure 2. Evidently, the competitor information is more useful than any other representations we examined for classifying the Hoovers Web sites, including the local text in a page or the linkage among pages. A detailed analysis reveals that 70% of the pairs of competitors share the same class label, which is much higher than the 6.5% of the hyperlinked pairs sharing the same class. Another point to note is that using the names of competitors as the hypertext representation had much smaller vocabulary size than using the words in competitor pages, and thus making it much more e-cient to train and test the classier based on competitor names. As for the other types of meta data, including the text in the HTML meta and title elds in Web pages, our results show that they are quite informative for the classication tasks although not as predictive as Competitor Names and Page Only for all the classiers on the Hoovers datasets, and not as good as Linked Names and Tagged Words on the Univ-6 dataset. Nevertheless, using these kinds of meta information in addition to Web pages (and links) can improve the classication performance than using Web pages alone, which has been shown in our experiments as reported in a previous paper [12]. 4.3. Performance Analysis with respect to Feature Selection The results discussed so far are for approximately optimal vocabulary sizes we found in feature selection. For NB we used Information Gain as the feature selection criterion because our NB system supported this functionality. For kNN we used 2 statistics (after removing the words which occur only once or twice in the training set) because we found this worked slightly better than Information Gain for our kNN in a previous study[29]. For Foil we used Document Frequency to rank and select words, for the reason discussed in Section 2.3.3. Since the original vocabulary sizes of the Hoovers datasets are very large for some representations (over 300,000 terms for Tagged Words, for example), it would take too long to test each classier with the full vocabulary sizes for these representations. We then only tested each classier with subsets of features with increasing sizes until the performance curve of that classier approached a stable plateau. Figure 3 shows the curves in micro-averaged F 1 for our classiers with three representations: Page Only, Linked Words and Tagged Words. We omit the macro-averaged curves and the results for other representations because of space limita- tions; we also omit the curves on Hoovers-28 which are similar in shape to those on Hoovers-255. Notice that we do not have the curves for Foil with the Linked Words representation because this method always treats words in linked pages as Tagged Words by denition. We nd that the observed performance variations over the cross product of the datasets, representations and classiers are larger than those reported in feature selection for conventional text categorization[29, 13]. Not only do the performance curves of our classiers peak with very dierent vocabulary sizes, the shapes of those curves also show a larger degree of diversity than those previously reported. Perhaps the highly noisy nature of Web pages makes feature selection more important for performance optimization in hypertext classiers. The inconsistent shapes of the curves for NB suggests a potential di-culty in obtaining stable or optimized performance for this method on highly noisy and heterogeneous hypertext collec- tions. Foil exhibited smaller performance variances with respect to vocabulary- size changes, but larger performance dierences across datasets and when switching from micro-averaged measures to macro-averaged measures. 4.4. Recall-Precision Trade-o In addition to evaluating our classiers at a single point using the F 1 metric, we also examined their potential for making exible trade-os between recall and precision. In practice, it is important to know whether or not a classier can produce either high-precision decisions, or conversely high-recall output, depending on the type of real-world application being addressed. Cross-classier comparison also allows us to get a deeper understanding of the strengths and weaknesses of our methods in hypertext classication. Figure 4 shows the recall-precision trade-o curves for NB, kNN and Foil on Hoovers-28, Hoovers-255 and Univ-6. The micro-averaged and macro-averaged MicroAvg Number of Selected Features FOIL (a) Classiers on Hoovers-255 Page Only0.50.70.90 5000 10000 15000 20000 25000 MicroAvg Number of Selected Features FOIL (b) Classiers on Univ-6 Page Only0.050.150.250.350 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 MicroAvg Number of Selected Features (c) Classiers on Hoovers-255 Linked Words0.50.70.90 2000 4000 6000 8000 10000 MicroAvg Number of Selected Features (d) Classiers on Univ-6 Linked Words0.050.150.250.350 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 MicroAvg Number of Selected Features FOIL (e) Classiers on Hoovers-255 Tagged Words0.50.70.90 2000 4000 6000 8000 10000 MicroAvg Number of Selected Features FOIL (f) Classiers on Univ-6 Tagged Words Figure 3. Performance of classiers with respect to feature selection. curves are compared side-by-side. For each classier and dataset, we present its curve for the representation with which the averaged F 1 score of this classier is optimized. For example, on the Univ-6 dataset, Tagged Words is the choice for Foil and Linked Names is the choice for kNN and NB. For the Hoovers datasets, on the other hand, Competitor Names is the choice for all the three classiers. These curves were generated by thresholding over the system-generated ranks and scores of the candidate categories for test documents, by the following procedure: 1. All the candidate categories are rst ordered by their rank (R for each test document) { the higher the rank, the more relevant they are considered. 2. For the candidate categories with the same rank, their scores are used for further ordering. 3. Move (automatically) the threshold over the candidate categories, compute the recall and precision values for each threshold, and interpolate the resulting plots. A break-even line is shown in each graph for reference, on which the recall and precision are equally valued, and around which the F 1 values are typically optimized. The interesting observations from these graphs are: Both kNN and NB are exible in trading-o recall and precision from the high-precision extreme to the high-recall extreme, and in both micro-averaged and macro-averaged evaluations. Foil's curves exhibit a large performance variance across datasets. Its curves on Univ-6 are competitive with kNN's in both micro-averaged and macro-averaged measures. However, it has di-culty in getting high-recall results for the Hoovers task. Improving the pruning strategy for rules in Foil and inventing richer scoring schemes may be potential solutions for this kind of problem, which requires future investigation. It may also worth mentioning that Univ-6 has the most skewed distribution of categories, and performance on the largest class \Other" (miscellaneous) tends to dominate the overall performance of a classier on Univ-6 if the system is not su-ciently sensitive to rare categories. The dierent performance curves of these classiers suggest an intriguing potential of combining these classiers for a classication system that is more robust than using each method alone. Similar questions have been raised in both conventional text categorization and hypertext mining [28, 9, 21]; how to solve it for better hypertext categorization also requires future research. 4.5. Run-Time Observation Table 4 shows the running times (including training and testing) in CPU minutes for our classiers in some of the experiments. When using the representations of Linked Names or Competitor Names, all the classiers were very fast and we omit MicroAVG Precision MicroAvg Recall kNN (competitor names) NB (competitor names) FOIL (competitor names) break-even line (a) NB, kNN & FOIL on Hoovers-280.20.61 MacroAVG Precision MacroAvg Recall kNN (competitor names) NB (competitor names) FOIL (competitor names) break-even line (b) NB, kNN & FOIL on Hoovers-280.20.61 MicroAVG Precision MicroAvg Recall kNN (competitor names) NB (competitor names) FOIL (competitor names) break-even line (c) NB, kNN & FOIL on Hoovers-2550.20.61 MacroAVG Precision MacroAvg Recall kNN (competitor names) NB (competitor names) FOIL (competitor names) break-even line (d) NB, kNN & FOIL on Hoovers-2550.20.61 MicroAVG Precision MicroAvg Recall kNN (tagged words) (linked names) FOIL (linked names) break-even line MacroAVG Precision MacroAvg Recall kNN (tagged words) (linked names) FOIL (linked names) break-even line (f) NB, kNN & FOIL on Univ-6 Figure 4. Performance in Recall-Precision Trade-o. Table 4. Average running times in CPU minutes for each algorithm with dierent representations. Page Only Linked Words Tagged Words the time for those runs; on the other hand, for the representations of Page Only, Linked Words and Tagged Words, the computations were much more intensive due to the large vocabulary sizes. Table 4 shows the times for the latter. Since we used dierent machines for running those classiers, the computation times are not directly comparable, but rather indicative for a rough estimate. On average, Naive Bayes experiments run faster than kNN while Foil takes considerably longer than both kNN and Naive Bayes. 5. Concluding Remarks The rich information typically available in hypertext corpora makes the classi- cation task signicantly dierent from traditional text classication. In this pa- per, we presented the most comprehensive examinations to date, addressing some open questions of how to eectively use hypertext information by examining the cross product of explicitly dened hypertext regularities (ve), alternative representations multiple established hypertext datasets (three) from dierent domains, and several well-known supervised classication algorithms (three). This systematic approach enabled us to explicitly analyze potential reasons behind the observed performance variance in hypertext classication, leading toward generalizable conclusions. Our major ndings include: The identication of hypertext regularities (Section 2.1) in the data and the selection of appropriate representations for hypertext in particular domains are crucial for the optimal design of a classication system. The most surprising observations are that Linked Names are at least as informative as words (local words, linked words or target words all together) on Univ6, and that Competitor Names are more informative than any other alternative representations we explored on the Hoovers datasets. These observations suggest that Preclassied Regularity strongly in uences the learnability of those problems, although for one problem it appears in hyperlinks, and for the other problem it exhibits in meta data beyond the Web pages themselves. The \right" choice of hypertext representation for a real-world problem is crucial but seldom obvious. Adding linked words (tagged or untagged) to a local page, for example, improved classication accuracy on the Univ-6 dataset for all three classiers, but had the opposite eect on the performance of NB and kNN on the Hoovers datasets (which is consistent with previously reported results by Chakrabarti et al. and Oh et al. on dierent data). Moreover, Linked Names and Tagged Words were almost equally informative for all the three classiers on Univ-6; but this phenomenon was not observed from the experiments on the Hoovers datasets. Our extensive experiments over several domains show that drawing general conclusions for hypertext classication without examinations over multiple datasets can be seriously misleading. Meta data about Web pages or Web sites can be extremely useful for improving the classication performance, as shown by the Hoovers Web site classication tasks. This suggests the importance of examining the availability of meta data in the real world, and exploiting information extraction techniques for automated acquisition of meta data. Recognizing useful HTML elds in hypertext pages and using those elds jointly in making classication decisions can also improve classication performance, which is evident in our experiments on Univ-6. kNN and NB, with extended document representations combining within-page words, linked words and meta data in a naive fashion, show simple and eective ways of exploiting hypertext regularities. Their simplicity as algorithms allows them to scale well for very large feature spaces, and their relatively strong performance (with the \right choice" for hypertext representation) across datasets makes them suitable choices for generating baselines in comparative studies. Algorithms focusing on automated discovery of the relevant parts of the hypertext neighborhood should have an edge over more naive approaches treating all the links and linked pages without distinction. Foil, with the power for discovering relational regularities, gave mixed results in this study, indicating the discovery problem to be non-trivial especially given the noisy nature of links and inviting future investigation for improving this algorithm and on other algorithms of this kind. The use of micro-average and macro-average together is necessary to actually understand the results and relative performance of classiers. This is of special signicance for datasets where the category distribution is extremely skewed. Also, investigating the precision-recall tradeo is important in order to observe the performance of classiers in specic regions of interest. This issue becomes essential for hypertext applications where high precision results are extremely important. While the scope of the experimental results presented is necessarily conned to our datasets, we hope that our analysis will help future research into hypertext classication by dening explicit hypotheses for various types of regularities that may be present in a hypertext corpus, by presenting a systematic approach to the examination of those regularities, and by providing some ideas about how one should construct a classier to take advantage of each. Acknowledgments The authors would like to thank Bryan Kisiel for his signicant help in improving the evaluation tools and applying them to the output of dierent classiers, and Thomas Ault for editorial assistance. This study was funded in part by the National Science Foundation under the grant number KDI-9873009. Notes 1. Thorsten Joachims provided the name and the intuition behind this regularity (personal communication). --R Enhanced hypertext categorization using hyperlinks. 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Legrand, Identifying ontology components from digital archives for the semantic web, Proceedings of the 2nd IASTED international conference on Advances in computer science and technology, p.7-12, January 23-25, 2006, Puerto Vallarta, Mexico Pvel Calado , Marco Cristo , Edleno Moura , Nivio Ziviani , Berthier Ribeiro-Neto , Marcos Andr Gonalves, Combining link-based and content-based methods for web document classification, Proceedings of the twelfth international conference on Information and knowledge management, November 03-08, 2003, New Orleans, LA, USA Xin Li , Hsinchun Chen , Zhu Zhang , Jiexun Li, Automatic patent classification using citation network information: an experimental study in nanotechnology, Proceedings of the 2007 conference on Digital libraries, June 18-23, 2007, Vancouver, BC, Canada Dmitry Davidov , Evgeniy Gabrilovich , Shaul Markovitch, Parameterized generation of labeled datasets for text categorization based on a hierarchical directory, Proceedings of the 27th annual international ACM SIGIR conference on Research and development in information retrieval, July 25-29, 2004, Sheffield, United Kingdom Jan Bakus , Mohamed S. Kamel, Higher order feature selection for text classification, Knowledge and Information Systems, v.9 n.4, p.468-491, April 2006 Baoping Zhang , Yuxin Chen , Weiguo Fan , Edward A. Fox , Marcos Gonalves , Marco Cristo , Pvel Calado, Intelligent GP fusion from multiple sources for text classification, Proceedings of the 14th ACM international conference on Information and knowledge management, October 31-November 05, 2005, Bremen, Germany Yonghong Tian , Tiejun Huang , Wen Gao, Latent linkage semantic kernels for collective classification of link data, Journal of Intelligent Information Systems, v.26 n.3, p.269-301, May 2006 Nayer M. Wanas , Dina A. Said , Nadia H. Hegazy , Nevin M. Darwish, A study of local and global thresholding techniques in text categorization, Proceedings of the fifth Australasian conference on Data mining and analystics, p.91-101, November 29-30, 2006, Sydney, Australia A. Georgakis , H. Li, User behavior modeling and content based speculative web page prefetching, Data & Knowledge Engineering, v.59 n.3, p.770-788, December 2006 Using web structure and summarisation techniques for web content mining, Information Processing and Management: an International Journal, v.41 n.5, p.1225-1242, September 2005 Lise Getoor, Link mining: a new data mining challenge, ACM SIGKDD Explorations Newsletter, v.5 n.1, July Brian D. Davison, Predicting web actions from HTML content, Proceedings of the thirteenth ACM conference on Hypertext and hypermedia, June 11-15, 2002, College Park, Maryland, USA Fabrizio Sebastiani, Machine learning in automated text categorization, ACM Computing Surveys (CSUR), v.34 n.1, p.1-47, March 2002

hypertext classification;web mining;text mining;machine learning

607698

Grain Filters.

Motivated by operators simplifying the topographic map of a function, we study the theoretical properties of two kinds of grain filters. The first category, discovered by L. Vincent, defines grains as connected components of level sets and removes those of small area. This category is composed of two filters, the maxima filter and the minima filter. However, they do not commute. The second kind of filter, introduced by Masnou, works on shapes, which are based on connected components of level sets. This filter has the additional property that it acts in the same manner on upper and lower level sets, that is, it commutes with an inversion of contrast. We discuss the relations of Masnou's filter with other classes of connected operators introduced in the literature. We display some experiments to show the main properties of the filters discussed above and compare them.

Introduction Filters used to simplify an image and satisfying a minimal set of invariance properties are scarce. Actually, only one of them has the maximal set of invariance properties, and it is driven by the parabolic partial dierential equation [1, 26, 18]: @t where curvu(x) is the curvature of the level line of u at the point x, this being restricted to the regular points of u. However, the previous lter is optimal (in terms of invariance) among regular lters, that is, lters driven by a P.D.E. This property of regularity, while desirable in theory, has the drawback of modifying all contours, and, in particular, of destroying T -junctions, which are important clues for occlusion [5, 4]. If we drop this requirement, a bunch of other lters satisfying the same invariance properties are available. Motivated by the study of a family of lters by reconstruction [9, 10, 21, 31, 32], Serra and Salembier [30, 25] introduced the notion of http://pascal.monasse.free.fr c 2001 Kluwer Academic Publishers. Printed in the Netherlands. V. Caselles and P. Monasse connected operators. Such operators simplify the topographic map of the image. These lters have become very popular in image processing because, on an experimental basis, they have been claimed to simplify the image while preserving contours. This property has made them very attractive for a large number of applications, such as noise cancellation or segmentation [17, 33]. More recently, they have become the basis of a morphological approach to image and video compression [23, 24, 22, 6]. Dierent classes of connected operators have been studied by Meyer [15, 16], Serra [29] or Heijmans [8] (see also references therein). In this article, we study the theoretical properties of two kinds of connected operators: the extrema lters and the \shape" lters. Each of them simplies the topographic map of the image, but with dierent senses given to the term topographic map. The maxima lter removes connected components of upper level sets of insu-cient area, while keeping the other ones identical [31, 32]. This ensures that regional maxima of the ltered image have a minimal grain size. Similarly, the minima lter removes too small connected components of lower level sets. For these lters, the \grain" corresponds to a connected component of a level set, and small grains are considered as noise. This can be seen as the pruning of the tree of the connected components of upper, or lower, level sets. In a previous work [3], we introduced the notion of \shapes", designed to deal symmetrically with upper and lower level sets. The shapes are also organized in a tree, driven by inclusion. When applied to images of positive minimal grain size, we showed that the structure of this tree is nite. As shown here, any image resulting from the application of the extrema lters has this property. This new tree also provides the denition of another grain lter, for which the grain is a shape [13, 19, 20]. It removes small shapes while preserving the ones of su-cient area. The essential improvement over the extrema lters is that it deals in the same manner with upper and lower level sets. In the vocabulary of mathematical morphology [14, 27, 28], this lter is selfdual when applied to continuous functions. The present article is organized as follows. Section 2 recalls the foundations of mathematical morphology and underlines the link between morphological lters and set operators. Section 3 introduces the main properties of extrema lters and proves them. In Section 4 we prove the analogous properties for the grain lter and, in particular, that it is a self-dual lter on continuous functions, generalizing a result of [2]. Finally, in Section 5 we illustrate these lters with an experiment. 2. General results from mathematical morphology Throughout this section, we shall consider real functions dened in a subset of IR N . If u is a real function dened on D IR N , we denote by [u ] the set fx 2 D : u(x) g, 2 IR. Similarly, we dene the sets [u > ], [u ], [u < ]. 2.1. Level sets If u is any real function, and X which implies, in particular, that Conversely, if X is a family of sets satisfying then X is the level set at level of the function u dened by namely, X Under the weaker hypothesis of monotonicity of (X ) 2IR , Guichard and Morel show in [7] that X a.e. and for almost every 2 IR. 2.2. Contrast invariance A contrast change, in the restrictive sense, is a strictly increasing continuous IR. It is therefore a homeomorphism of IR onto an open interval of IR. A direct consequence is For an image u, the contrast change g applied to u is g-u. A contrast change g will indierently be considered as a function dened on IR or as an operator acting on functions u. A direct consequence of (1) is which can also be written as 4 V. Caselles and P. Monasse showing that the families of level sets of g - u and of u are the same, only their level changes. A morphological lter ~ T is a map acting on functions u that commutes with any contrast change: g - ~ g. 2.3. Link between set operator and morphological filter If T is an operator acting on sets, a necessary requirement for T to transform the level sets of a function into the level sets of a function is thus The lter ~ T associated to T is dened by or equivalently ~ If we denote by B then we observe ([11, 12]) that ~ Indeed, let ~ be the right hand side term of this equality. If is such that x 2 T ([u ]), by denition of B x we deduce that [u on the other hand, since u(y) we get immediately ~ Tu(x). Conversely, if u(y) n That is, B n [u 1=n] and, therefore, Taking the intersection over all n, we get that x 2 ~ ~ We say that contrast change if g is nondecreasing and upper semicontinuous. For a general contrast change, we have that where We use the convention If T is only dened on closed sets and satises (2) when the sets F n are closed, then ~ T is dened on upper semicontinuous functions. Then it is easy to check that ~ T and g commute when applied to upper semicontinuous functions. We get contrast invariance in a strong sense, since g can have constant stretches, and needs not be continuous. 2.4. Spatial invariance properties If D IR N , f : D ! D is a map and ~ T a morphological lter, whose associated set operator is T , and whose structuring elements are given by the family B x at x, we denote by f ~ T the morphological operator dened by (f ~ and we dene ~ by ( ~ is an image. g. If f(B x is easy to see that f ~ . Then we say that T is invariant with respect to f . For instance, if B T is translation invariant. The lters we shall discuss below satisfy the property that their respective family of structuring elements is globally invariant with respect to any area preserving map, i.e., a special a-ne map. These lters are thus special a-ne invariant. They depend on a parameter ". If x is the set of structuring elements of such a lter at x, they also satisfy that B s" x for any s > 0. In particular, this implies that, for any a-ne map f , we have f ~ 3. Extrema lters 3.1. Definition Extrema lters are constructed in such manner that the connected components of level sets of an image have a minimum area. We call them extrema lters because a connected component of level set contains a regional extremum. This is achieved in two steps: rst, connected components of upper level sets are ltered, then lower level sets. We dene the set operators ensuring such properties. Let us rst x some notation. Let be a set homeomorphic to the closed unit ball of IR N (N 2), and be the 6 V. Caselles and P. Monasse interior of Note that, in particular, is compact, connected and locally connected. Moreover, is unicoherent, i.e., for any A; B closed connected sets such that A\B is connected. For a set X, we denote cc(X) any of its connected components and by cc(X; x) the one containing the point x, provided x 2 X, and by extension and C is connected, cc(X; C) is cc(X; x), with x 2 C. " be a parameter, representing an area threshold, and let X be a subset of . We dene the lters "g. We dene the maxima lter M " and the minima lter M " by sup We will show that they are morphological lters, whose associated set operators are M " and M 0 " , respectively. The denitions are voluntarily not symmetric, so that both can act on (upper) semicontinuous func- tions. To avoid the cases where suppose that " < j. 3.2. Preliminary results LEMMA 1. If (C n ) n2IN is a nonincreasing sequence of compact sets and If (O n ) n2IN is a nondecreasing sequence of open sets and O then Proof. It is clear that cc(C; x) cc(C n ; x) for any n. Conversely, is an intersection of continua, thus, it is a continuum. Since it contains x and is included in C, we get the other inclusion. For any n, cc(O On the other hand, O being open and locally connected, cc(O; x) is an open set. Hence, for any y 2 cc(O; x), there is some continuum K y cc(O; x) containing x and y. Since K y S O n and it is a compact set, we can extract a nite covering of K y , and as the sequence (O n ) is non- decreasing, there is some n such that K y O n . Since K y is connected and contains x, we have that y 2 K y cc(O n ; x). We conclude that PROPOSITION 1. We have the following properties for M " and M 0 " are nondecreasing on subsets of . upper semicontinuous on compact sets: being a non-increasing sequence of compact sets, then M " " is lower semicontinuous on open sets: (O n ) n0 being a nondecreasing sequence of open sets, then M 0 " O n . Proof. Property (i) is a direct consequence of the denitions. " is monotone, we have that M " F Applying Lemma 1, we observe that cc(F; x). In particular, O n . Since M 0 " is monotone, S " O. Now, let " O. Then is such that jU j > ". Let U Lemma 1 proves that ". Hence for n large enough, jU n j > ". We conclude that x 2 U n M 0 " O n . If A and B are two families of sets, we say that A is a basis of B if A B and for any B 2 B, there is some A 2 A such that A B. 2. B " is a basis of fX " is a basis of " Xg. Proof. This is a direct consequence of the denitions. COROLLARY 1. Applied to upper semicontinuous functions, M morphological lter whose associated set operator is M " 8 V. Caselles and P. Monasse precisely, for all , " Proof. Let Xg. A consequence of Proposition 1 we have ( u(y) We now use the fact that B " is a basis of C " , as shown in Lemma 2. As we deduce that u (x) sup there is some u(y) inf " and by taking the supremum over all B, we get sup u A similar proof applies to link M " and M 0 " . PROPOSITION 2. Let u; v IR. Then " (ii) If u v, then M " " v. The proofs are immediate and we will not include the details. COROLLARY 2. Let u IR be such that u n ! u uniformly in . Then M " uniformly in . Proof. Given - > 0, let n 0 be such that u - u n Using Proposition 2, (ii), (iii) and (iv), we obtain for all n n 0 . Hence uniformly in . Similarly, we prove that M + " uniformly in . 3.3. Properties PROPOSITION 3. If u is an upper semicontinuous function, also are " u. If u is continuous, also are M " u. Proof. Let u be an upper semicontinuous function. Then, for any " As [u ] is a closed set, its connected components are closed. Since M " [u ] is a nite union of some of them, it is closed. Thus M " u is upper semicontinuous. In the same manner, [M " u < is an open set, its connected components are also open, and its image by M 0 is thus a union of open sets, which is open. This proves that M " u is upper semicontinuous. Finally, suppose that u is continuous. We just have to prove that " closed for any 2 IR. Using Corollary 1, we write " " We claim that If C). Since O is open and contains the closed set C, we deduce that O n C is open and not empty, and, thus, of positive measure. Hence jOj > jCj ". This proves that be such that jOj > ". Then, thanks to Lemma 1, we have V. Caselles and P. Monasse There is some n such that ". This proves the remaining inclusion in (). The right hand side of this equality being open, we conclude that [M " closed. To prove the same result for M " , we write We claim that the last set coincides with "g. If O n is open and C is closed, we have that " jCj < jO n j. Hence, n ] for all n. Then we have Applying Lemma 1, we get yielding Hence jcc([u ]; This proves that and, thus, the equality of both terms. The right hand side term, being a nite union of closed sets, it is also closed. PROPOSITION 4. When restricted to upper semicontinuous function- s, M " is idempotent. Proof. Let u be an upper semicontinuous function and 2 IR. Clearly, we have . Applying this equality to the set [u ], and using Corollary 1, we get " " Now, thanks to Proposition 3, we have that [M " closed and we can apply again Corollary 1 to the left hand side of this equality to obtain " " Since this equality holds for any 2 IR, we conclude that M + " . The same proof applies to M " . PROPOSITION 5. Let u 2 Proof. (i) Both cases being similar, it will be su-cient to prove the rst part of the assertion. Let " u(x). Observe that v is lower semicontinuous. Assume that M := max 0be such that contains an open set, hence, Letting choosing Mwhich is a contradiction. We conclude that and the proposition is proved. (ii) Both cases being similar, we shall only prove that M + uniformly as " ! 0+. For that, let us write " " Given - > 0, let " 0 > 0 be small enough so that V. Caselles and P. Monasse and " " i.e., Collecting these facts we have that 3.4. Interpretation Let u be an upper semicontinuous function and jCj ", we have thus, C is a connected component of " C is not a connected component of [M " u ]: it does not even meet this set, since [M " so that " Conversely, if " u ]), we have being a connected component of [u ], and since C 6= ;, we have Summing up these remarks, we can see that the connected components of [M " are exactly the connected components of [u ] of measure ". In particular, since the connected components of upper level sets have a structure of tree driven by inclusion, the tree of M " is the tree of u pruned of all nodes of insu-cient measure. The same observations can be made concerning M " and the connected components of [u < ]. The tree of M " is the tree of connected components of lower level sets of u pruned of all nodes of insu-cient measure. Summarizing the above discussion, we have the following result. PROPOSITION 6. If " u ]) 6= ;, then jXj . If jcc([u ])j (resp. " u < ])j In particular, the above result implies that the connected components of [M ". The same thing can be said of connected components of the upper and lower level sets of " u. 3.5. Composition LEMMA 3. Let u be an upper semicontinuous function and 2 IR. Then Proof. Let " u < ]). If the consequence was false, we would have C [u ]. If then we have " since This contradicts the denition of C. If C , C being open, we have C ) C and @C [M " Then x belongs to a connected component D of [u ] such that connected and included in [u ]. As jD[Cj ", we get D[C [M " contradicting the denition of C. This proves our claim that there is some x 2 C such that u(x) < . arguing as above, we have which contradicts the hypothesis. Thus, we may assume that C . Let Since C is closed and D open, then C ( D. Thus there is some x in meaning that jcc([u < ]; x)j > ". This component must be D and, therefore jDj > ". Finally, we observe that D [M " u < ], which is a contradiction. THEOREM 1. The operators M (i) transform upper semicontinuous functions into upper semicontinuous functions, and continuous functions into continuous functions; (ii) are idempotent on upper semicontinuous functions. Proof. (i) is a direct consequence of the equivalent properties we have proved for " and M " in Proposition 3. As a consequence of Proposition 2, we have " any function u. Applying this to M of u, we get 14 V. Caselles and P. Monasse We apply the rst part of Lemma 3 to that We nd a point x 2 C \ [v < ]. Let We know that jDj ". Since M that D [M and D is connected, we have This proves that thus, we have the equality " to each member and using its idempotency, we conclude that With a similar proof, using the second part of Lemma 3, we prove that " is idempotent. 4. Grain lter 4.1. Definitions The problem with the extrema lters presented above, is that we have two operators which act on both upper and lower level sets. In general, they do not commute. Moreover, none of them has the property to deal symmetrically with upper and lower level sets. Actually, they work in two steps: rst upper, then lower level sets are treated (or in the opposite order). That is, these operators are not selfdual. DEFINITION 1. A morphological lter ~ T , associated to the set operator T , is said to be selfdual on continuous functions if the following equivalent properties hold for any continuous function u 1. ~ 2. 8x, sup B2B inf y2B describing the structuring elements of ~ T . 3. 8, T [u To show the equivalence of the rst two properties, it su-ces to write the second one with u instead of u. By taking instead of , we can see that the third property is equivalent to Now, using the contrast invariance of ~ amounts to write which means exactly that ~ Tu. We recall the denition and the essential properties of a saturation operator. DEFINITION 2. Let p 1 be a xed point of . For X , we call holes of X the connected components of X. The external hole of X is X. The other holes are the internal holes of X. The saturation of X, Sat(X), is the union of X and its internal holes. The following results are proved in [3]: PROPOSITION 7. 1. 2. 3. X connected ) Sat(X) connected and connected 4. X open (resp. closed) ) Sat(X) open (resp. closed). 5. closed 6. @Sat(X) is a connected subset of @X. 7. H internal (resp. external) hole of X 8. are nested or disjoint. 9. Let X be open or closed, X) such that x 2 Sat(O) Sat(C). If u is an upper semicontinuous function, we call shapes of u the elements of The tree structure of u is expressed by the properties [3]: . are nested or disjoint. V. Caselles and P. Monasse We restrict the denition of our set operator to closed sets, since later we will dene our lter on upper semicontinuous images. If K is a compact set, we dene f internal hole of C; jC 0 j, we clearly have that G " compact K. Thus we will always suppose that " j. Remark. We have dened the grain lter for compact sets since this will be su-cient for the extension of this lter to upper semicontinuous functions in Obviously, the denition has sense for any subset of 4.2. Preliminary results LEMMA 4. G " is nondecreasing on compact sets. Proof. Let K L be compact sets. Let be such that be the family of internal holes of C such that ". Then we observe that there is some C On the other hand, if H 0 is an internal hole of C 0 such that included in a hole H of C, and, thus, jHj > ". From these observations we deduce that i , are the families of internal holes of C, resp. C 0 , whose measure is > ". We conclude that G " K G " L. LEMMA 5. If K is compact, then Proof. Let us denote by (C i ) i2I the family of connected components of K and let I " I be the set of indices for which jSat(C i )j ". monotone, we have that S the other inclusion is immediate from the denition, we have the identity We also observe that G " C i , if not ;, is the union of C i and some of its internal holes and, thus, it is connected. Let J be a subset of I " whose cardinal is at least 2, and let . We now prove that D is not connected. not connected, there is an open and closed subset E 0 in D 0 dierent from ; and D 0 . Observe that E 0 is an union of C j , since each of those sets is connected. To be precise, let us write We prove that E is open and closed in D, hence, D is not connected. Clearly, we have that ; 6= E ( D. Since E 0 is open in D 0 , there is an open set U 0 such that U 0 \ D the union of the connected components of U 0 that meet some C l , l 2 L. The set U 00 , as a union of open sets, is open, and the set U , as the union of U 00 and some internal holes of C l , which are all open, is also open. If j 2 J nL, we have that E\G " Indeed, if the last equality does not hold, we would have There is a connected component O of U 0 meeting some C l , l 2 L (by denition of U 00 ), and C j . Indeed, if it meets some hole H of C j with jHj ", then H cannot not contain C l . Otherwise, since H n Sat(C l ) is open and not empty, thus of positive measure, we would have jHj > l )j ". Thus O is connected and meets two dierent connected components of and the component containing C l ), hence O \ C j 6= ;. This implies that U 0 \ C j 6= ;, contradicting the identity We conclude that () does not hold, i.e., U 00 \G " and therefore U 00 \ (D n E) = ;, proving that U \ We have shown that the set E is open in D. Applying the same argument to of D we prove that E is also closed in D. LEMMA 6. Let (C n ) n2IN be a nonincreasing sequence of continua and n C n . If H is a hole of C, there exists n 0 2 IN and a non-decreasing sequence being a hole of C n , such that Proof. If x 2 H, then there is some n 0 such that x 62 C n for n n 0 . Thus, if n n 0 , x is in some hole H n of C n . Obviously, H n H, and therefore S H n H. U be a neighborhood of y and V be a connected neighborhood of y such that V U . Since y 2 H n there is some and, by monotonicity of (H n ), we may write nnn H n , we also have that V. Caselles and P. Monasse From () and (), the connectedness of V implies that It follows that V \ C 6= ;, which implies that U \ C 6= ;. This being true for any neighborhood U of y, we get that y 2 @ This implies that H \ @ H n is closed in H. Since it is also open, as a union of open sets, the connectedness of H implies that THEOREM 2. The operator G " is upper semicontinuous, i.e., if (K n ) is a nonincreasing sequence of compact sets, then Proof. The inclusion of the left hand side term into the right hand side one is due to the monotonicity of G " . We just have to show the other inclusion. As of Lemma 5, for any n, there is some . For each n 1, C n is inside some implying, due to Lemma 5, that This shows that the sequence of continua (C n ) is nonincreasing, so that their intersection C is a continuum, thanks to Zoretti's theorem. Since C is contained in some component of T remains to show that x 2 G " C. By taking H proves that H so that . Thus jSatCj ". Moreover, if x belongs to some internal hole H of C, since x 2 G " C n , the associated sequence H n obtained from Lemma 6 is such that jH n j " and thus ". Hence x 4.3. Properties LEMMA 7. If A is a closed set, C a connected component of A and H an internal hole of A, then, for any x in H, G: Proof. The right hand side term of this equality, H 0 , is obviously contained in H. Suppose this inclusion is strict. The set H being connected and H 0 being open (as a union of open sets), H 0 is not closed in H, so that H \ @H 0 6= ;. Let y 2 H \ @H 0 and let U be a connected neighborhood of y. We claim that U\A 6= ;. Otherwise, we would have U A, and as U meets some n A with U [ G 0 being connected, contradicting G A). We conclude that y 2 A. Then there is some G Since G is an open set, it contains some neighborhood of y 2 @H 0 , in particular, it meets some G n A)) such that x 2 G 00 H. Thus G and G 00 are nested, and, as G 6 H 0 , we have that G 00 G. Thus, x 2 G and G H, implying that G H 0 , which contradicts our assumption since y 2 G. We conclude that H LEMMA 8. Let A; B be closed sets. Proof. Suppose that A and B are disjoint. Taking components of A and B instead of A and B, we may assume that A and B are connected. The result being obvious if G " A, or G " B, is empty, we may also assume none of them is empty. Then Sat(A) and Sat(B) are either nested or disjoint. If they are disjoint, since G " A Sat(A) and G " B Sat(B), (5a) is obvious. If they are nested, without loss of generality, we may assume that Sat(A) Sat(B). Then Sat(A) is inside a hole H of B. We get that jHj jSatAj ". Since Sat(A) is closed and H is open, is an open and nonempty set, thus, it has positive measure. This yields jHj > ", and therefore Sat(A) \ (5a). Now, we suppose that A [ Let x 62 G " A. In particular, this implies that Sat(C) for any C = cc(A). For any such set C, n SatC is connected, and also is n SatC, which is a compact set. By Lindelof's theorem, the intersection of a family of continua can be written as the intersection of a sequence of them. We can thus nd a sequence (C n ) of connected components of A such that V. Caselles and P. Monasse Let D Clearly, (D n ) is a nonincreasing sequence of continua, and we may write which is thus a continuum D. We observe that D B. Indeed, if y is in some connected component C of A. Since yn Sat(C), obviously we have that y 2 @Sat(C) @C. Now, we prove that any connected neighborhood U of y meets a point of A. Otherwise, C[U A would be connected, which, in turn, implies that U C. Since n A B, we conclude that y B, a contradiction with our choice of y. This proves that D B. We also observe that , and, thus, jSat(D)j ". one of the following cases happens: (iii) There is some such that x 2 Sat(C) and jSat(C)j ". Suppose that (i) holds. Then x 2 D. Since jSat(D)j ", we have Suppose that (ii) holds. Let D be the set dened above. If x 2 D we have again that x 2 D G " D G " B. If x 62 D, then x is in a hole H of D. Thanks to Lemma 7, we can write G: Thanks to Lindelof's theorem, this union of open sets can be written as the union of a sequence G n of these sets. For each n, Since that G 0 and we obtain that jG n j jSat(C)j < ". We conclude that sup n2IN jG n j ". Thus x 2 H G " D G " B. Finally, we suppose that (iii) holds. Let C: We claim that being a connected component of A. If there is a nite number of sets E in the intersection above, the result is just due to the fact that they are nested. If there is an innite number of such sets E, we can write their intersection as the intersection of a sequence E n of them, thanks to Lindelof's theorem. By taking T n instead of E n , we may also assume that this sequence is nonincreasing. The set T is therefore a continuum and jT connected neighborhood of y. Since U 6 T , so there is some n such that U n E n 6= ;. Since U is connected and meets its complement, U also meets @E n @A A. We obtain that A. This shows that @T A. On the other hand, we observe that the complement of T is connected, being the union of an increasing sequence of connected sets. Thus @T can be written as the intersection of two continua, thus, it is connected, since is unicoherent. Thus, there is some T such that @T T 0 , and this implies that jEj ", we have that T 0 \E 6= ;, and, therefore, sat(T 0 ) E. It follows that Sat(T 0 ) T. We have, thus, the equality Now, observe that x must be in a hole H of T 0 and we must have jHj > ", since otherwise x A. Using Lemma 7, we write G; and, as above, we may write the above union as the union of a non-decreasing sequence of such sets G n . Thus there is some n such that ". We write G n . If x is in a hole H 0 of contained in a connected component K of A and ". In this case, we also have that x n . Since G 0 n is contained in some connected component K 0 of B, we get that x THEOREM 3. Restricted to continuous functions, ~ G " is selfdual. Proof. We shall prove that condition 3 of Denition 1 holds. According to (5a), we may write for any 2 IR, and, since G " [u n ], by taking the intersection over all n, we get Therefore 22 V. Caselles and P. Monasse let x be such that for all n > 0, x Due to (5b), we have that x 2 This proves that and, thus, By taking the complement of each part, we get: which proves that and, actually, we have the equality of both sets. THEOREM 4. ~ semicontinuous functions into upper semicontinuous functions and continuous functions to continuous functions Proof. We have to show that the image of a compact set K is a compact set. Let sequence of points of G " K converging to x. We shall prove that x 2 G " K. As shown by Lemma 5, x n belongs to some G " K n , where K cc(K). If the family fK n ; n 2 INg is nite, we can extract a subsequence of belonging to some G " K n 0 , which is a closed set since its complement is a union of holes of K n 0 , and the holes are open sets. Thus, x . We may now assume that fK n ; n 2 INg is innite, and, maybe after extraction of a subsequence, that Km \ K any m 6= n. We have that Sat(G " K n only a nite number of these saturations are two by two disjoint, since each of them has measure at least ". Thus, after extraction of a subsequence, if necessary, we may assume that they all intersect, so that they form either a decreasing or an increasing sequence. If the sequence (Sat(K n )) n2IN is decreasing, then their intersection is a set Sat(K 0 ), cc(K). This can be shown as in Lemma 8. Then we have that jSat(K 0 (otherwise, we would have K su-ciently large n), and contained in a hole of K n for any n, thus x n 62 Sat(K 0 ) for any n. Since x 2 Sat(K 0 ) we conclude that the desired result. Let us assume that (Sat(K n )) n2IN is increasing. Since the lim inf of the sequence K n is nonempty (it contains x), its lim sup is a continuum C, according to Zoretti's theorem. Since K is compact, it follows that C). We observe that x 2 K 0 . We shall prove that jSat(K 0 )j ", and more precisely that all Sat(K n ), which have measure ", are in the same internal hole of K 0 . This result implies If K n \K 0 6= ;, then K since both are connected components of K. Since we are assuming that the sets K n are two by two disjoint, this cannot happen twice. Thus we may assume that K n \ K for any n. The sequence (Sat(K n being increasing, all K n are in the same hole of K 0 . Suppose that this hole is the external hole H of K 0 . Since H is open, there is a continuum L joining p 1 and an arbitrary point y 0 of K 0 . Since K 0 is in an internal hole of K 1 , there is some . In this manner, we can construct a sequence (y n ) n2N such that y n 2 L\K n for all n. L being compact, some subsequence of (y n ) converges to a point y 2 L\C. It follows that K 0 \L 6= ;, contrary to the assumption that L was in a hole of K 0 . This proves our claim. Applying the preceding result, if u is continuous, ~ " u is upper semi- continuous. Since u is also upper semicontinuous, ~ " u is also upper semicontinuous, hence ~ " u is lower semicontinuous. Thus, ~ " u is continuous. PROPOSITION 8. For " 0 ", ~ Therefore ~ G " is idempotent. Proof. The conclusion that ~ G " is idempotent derives from the previous statement by taking " The result amounts to show that for any , G " 0 [u We distinguish three families among the connected components I V. Caselles and P. Monasse We observe that I . Thanks to Lemma 5, we may write internal hole of C internal hole of C the measure of is For the same reason, if In conclusion, this yields The following properties are an easy consequence of the denition of ~ LEMMA 9. The operator ~ " satises the following properties (i) If u v, then ~ function u and any 2 IR. LEMMA 10. Let u be an upper semicontinuous function and let Proof. Let us x " < -. Since [ ~ it will be su-cient to prove that G " [v almost all 2 IR. Let denote the family of connected components of K and H(K) the family of internal holes of K. We observe that C2C(K);jSat(C)j" internal hole of C; jHj > "g: Since, by Proposition 6, the connected components of K have measure -, any C 2 C(K) satises jSat(C)j ". Similarly, since any internal hole of K contains a connected component of [v < ], it has also measure - > ". Hence, the family of sets H 2 H(K); jHj " is empty. We conclude that G " PROPOSITION 9. Let u 2 0+. Proof. Using the above Lemma, we have ~ for any "; - > 0 such that " < -. By Proposition 5, given > 0, there is some - 0 > 0 such that This implies that ~ u) ~ and, therefore, u) ~ Now, we choose " < - 0 and we obtain that u) ~ 2: The proposition follows. 4.4. Relations with connected operators We want to compare the grain lter described above with the notion of grain operators as dened in [8]. To x ideas, we shall work in with the classical connectivity. Thus, we denote by C the family of connected sets of . A grain criterion is a mapping c : C ! f0; 1g. Given two grain criteria, f for the foreground and b for the background, the associated grain operator f;b is dened by or In [8], Heijmans characterizes when the grain operators are selfdual and when are increasing. Indeed, he proves that f;b is self-dual if and only b. He also proves that f;b is increasing if and only if f and b are increasing and the following condition holds for any X and any x. We shall say that a grain criterion c : C ! f0; 1g is upper semicontinuous on compact sets if c(\ n K n decreasing sequence of continua K n , n 2 IN . 26 V. Caselles and P. Monasse PROPOSITION 10. Let be a self-dual and increasing grain oper- ator in associated to the grain criterion c. Assume that c upper semicontinuous. . Proof. We have that either or for some point x. In the rst case, we deduce that any nonempty subset X of and, therefore, we have that for any ; 6= X Since also we have that any X In the second case, using the upper semicontinuity of c we have that c(B(x; choose observe that c(cc(X [fxg; which contradicts (6). The above proposition says that there are no nontrivial translation invariant, increasing and self-dual grain operators. Other types of connected operators called attenings and levelings were introduced by F. Meyer in [15],[16] and further studied in [29]. In particular, Serra proves that there exist increasing and selfdual attenings and levelings based on markers [29]. Finally, let us prove that the grain lter we have introduced above corresponds to a universal criterion to dene increasing and self-dual lters. Let us recall the denition of connected operator [25], [30], [8]. For that, given a set X , we denote by P (X) the partition of constituted by the cc(X) and cc(X c ). The family of all subsets of X will be denoted by P(DEFINITION 3. An operator : P( P( connected if the partition P ( (X)) is coarser than P (X) for every set X . Given a connected operator P( P( we shall say that (i) is increasing if (X) (Y ) for any X Y (ii) acts additively on connected components if when X i is the family of connected components of X. (iii) is self-dual if (IR N n any open or closed set X (iv) is bounded if p 1 It is not di-cult to see that if is a connected operator which is increasing and self-dual then it induces an increasing and self-dual lter on continuous functions. PROPOSITION 11. Let : P( P( ) be a connected operator. Suppose that is increasing, self-dual, bounded and acts additively on connected components. Let Ker := fX be an open or closed connected set. Then, if Sat(X) 62 Ker , we have where H(X) denotes the family of internal holes of X. Proof. Since is self-dual, without loss of generality, we may assume that p 1 62 X. Since is a connected operator, if Z is simply connected, then (Z) must be one of the sets f;; g. Since is bounded we must have that either In particular, either Sat(X). In the rst case, we have that Sat(X) 2 Ker . In the second case, using the additivity of on connected components and the observation at the beginning of the proof, we have Now, using the self-duality of we have (X) Obviously, if is increasing, then Ker is an ideal of sets, i.e., if Y X and X 2 Ker , then Y 2 Ker . 4.5. Interpretation Similar remarks to those for the extrema lters can be made concerning the shapes of ~ . The shapes of ~ " u are the shapes of u of su-cient measure. ~ " corresponds to a pruning of the tree of shapes of u. 5. Experiment Theoretically, the lters M " and G " are dierent. This is illustrated in Figure 1. The second row shows the ltered images 3 u; they are all dierent, stressing that their respective notions of grains are dierent. The dierence appears in presence of holes. Concerning natural im- ages, the dierence would be the most apparent on certain images of 28 V. Caselles and P. Monasse11202 Figure 1. Top-left: original image u. The three constant regions are supposed of area 2. Bottom left: G3u. Middle column: M 3 u and M 3 u and 3 u. Figure 2. Texture image of a carpet, size 254 173. textures, for which the nestedness of shapes would be important. Figure 2 shows a complex texture, and Figure 3 the image ltered according to the three lters of parameter pixels. Whereas they are actually dierent, they are visually equivalent, and distinguishing them requires some eort. We can explain that by the fact that connected components of level sets having a hole of greater area than themselves are scarce. In other words, the situation illustrated by Figure 1 is not frequent. Figure 3. Three grain lters applied to the image u of Figure 2. Left column: G30u, Acknowledgements We acknowledge partial support by the TMR European project \Vis- cosity solutions and their applications", reference FMRX-CT98-0234 and the CNRS through a PICS project. The rst author acknowledges partial support by the PNPGC project, reference BFM2000-0962-C02- V. Caselles and P. Monasse --R 'Image Iterative Smoothing and P.D.E.'s'. Random Sets and Integral Geometry. Mathematical Morphology and Its Application to Signal and Image Processing Mathematical Morphology and Its Application to Signal and Image Processing Image Analysis and Mathematical Morphology. --TR Introduction to mathematical morphology Watersheds in Digital Spaces Affine invariant scale-space Morphological multiscale segmentation for image coding Self-dual morphological operators and filters From connected operators to levelings The levelings Fundamenta morphologicae mathematicae A Compact and Multiscale Image Model Based on Level Sets --CTR Renato Keshet, Shape-Tree Semilattice, Journal of Mathematical Imaging and Vision, v.22 n.2-3, p.309-331, May 2005 Renato Keshet, Adjacency lattices and shape-tree semilattices, Image and Vision Computing, v.25 n.4, p.436-446, April, 2007

connected sets;extrema filters;connected operators;grain filters;mathematical morphology

607707

Algebraic geometrical methods for hierarchical learning machines.

Hierarchical learning machines such as layered perceptrons, radial basis functions, Gaussian mixtures are non-identifiable learning machines, whose Fisher information matrices are not positive definite. This fact shows that conventional statistical asymptotic theory cannot be applied to neural network learning theory, for example either the Bayesian a posteriori probability distribution does not converge to the Gaussian distribution, or the generalization error is not in proportion to the number of parameters. The purpose of this paper is to overcome this problem and to clarify the relation between the learning curve of a hierarchical learning machine and the algebraic geometrical structure of the parameter space. We establish an algorithm to calculate the Bayesian stochastic complexity based on blowing-up technology in algebraic geometry and prove that the Bayesian generalization error of a hierarchical learning machine is smaller than that of a regular statistical model, even if the true distribution is not contained in the parametric model.

Introduction Learning in artificial neural networks can be understood as statistical estimation of an unknown probability distribution based on empirical samples (White, 1989; Watanabe & Fukumizu, 1995). Let p(y|x, w) be a conditional probability density function which represents a probabilistic inference of an artificial neural network, where x is an input and y is an output. The parameter w, which consists of a lot of weights and biases, is optimized so that the inference p(y|x, w) approximates the true conditional probability density from which training samples are taken. Let us reconsider a basic property of a homogeneous and hierarchical learning machine. If the mapping from a parameter w to the conditional probability density p(y|x, w) is one-to-one, then the model is called identifiable. If otherwise, then it is called non-identifiable. In other words, a model is identifiable if and only if its parameter is uniquely determined from its behavior. The standard asymptotic theory in mathematical statistics requires that a given model should be identifiable. For example, identifiablity is a necessary condition to ensure that both the distribution of the maximum likelihood estimator and the Bayesian a posteriori probability density function converge to the normal distribution if the number of training samples tends to infinity (Cramer, 1949). When we approximate the likelihood function by a quadratic form of the parameter and select the optimal model using information criteria such as AIC, BIC, and MDL, we implicitly assume that the model is identifiable. However, many kinds of artificial neural networks such as layered perceptrons, radial basis functions, Boltzmann machines, and gaussian mixtures are non-identifiable, hence either their statistical property is not yet clarified or conventional statistical design methods can not be applied. In fact, a failure of likelihood asymptotics for normal mixtures was shown from the viewpoint of testing hypothesis in statistics (Hartigan, 1985). In researches of artificial neural networks, it was pointed out that AIC does not correspond to the generalization error by the maximum likelihood method (Hagiwara, 1993), since the Fisher information matrix is degenerate if the parameter represents the smaller model (Fukumizu, 1996). The asymptotic distribution of the maximum likelihood estimator of a non-identifiable model was analyzed based on the theorem that the empirical likelihood function converges to the gaussian process if it satisfies Donsker's condition (Dacunha-Castelle & Gassiat, 1997). It was proven that the generalization error by the Bayesian estimation is far smaller than the number of parameters divided by the number of training samples (Watanabe, 1997; Watanabe, 1998). When the parameter space is conic and sym- metric, the generalization error of the maximum likelihood method is di#erent from that of a regular statistical model (Amari & Ozeki, 2000). If the log likelihood function is analytic for the parameter and if the set of parameters is compact, then the generalization error by the maximum likelihood method is bounded by the constant divided by the number of training samples (Watanabe, 2001b). Let us illustrate the problem caused by non-identifiability of layered learning machines. If p(y|x, w) be a three-layer perceptron with K hidden units and if w 0 is a parameter such that p(y|x, w 0 ) is equal to the machine with K 0 hidden units then the set of true parameters consists of several sub-manifolds in the parameter space. Moreover, the Fisher information matrix, log p(y|x, w) log p(y|x, w)p(y|x, w)q(x)dxdy, where q(x) is the probability density function on the input space, is positive semi-definite but not positive definite, and its rank, rank I(w), depends on the parameter This fact indicates that artificial neural networks have many singular points in the parameter space (Figure 1). A typical example is shown in Example.2 in section 3. By the same reason, almost all homogenous and hierarchical learning machines such as a Boltzmann machine, a gaussian mixture, and a competitive neural network have singularities in their parameter spaces, resulting that we have no mathematical foundation to analyze their learning. In the previous paper (Watanabe, 1999b; Watanabe, 2000; Watanabe, 2001a), in order to overcome such a problem, we proved the basic mathematical relation between the algebraic geometrical structure of singularities in the parameter space and the asymptotic behavior of the learning curve, and constructed a general formula to calculate the asymptotic form of the Bayesian generalization error using resolution of singularities, based on the assumption that the true distribution is contained in the parametric model. In this paper, we consider a three-layer perceptron in the case when the true probability density is not contained in the parametric model, and clarify how singularities in the parameter space a#ect learning in Bayesian estimation. By employing an algebraic geometrical method, we show the following facts. (1) The learning curve is strongly a#ected by singularities, since the statistical estimation error depends on the estimated parameter. (2) The learning e#ciency can be evaluated by using the blowing-up technology in algebraic geometry. (3) The generalization error is made smaller by singularities, if the Bayesian estimation is applied. These results clarify the reason why the Bayesian estimation is useful in practical applications of neural networks, and demonstrate a possibility that algebraic geometry plays an important role in learning theory of hierarchical learning machines, just same as di#erential geometry did in that of regular statistical models (Amari, 1985). This paper consists of 7 sections. In section 2, the general framework of Bayesian estimation is formulated. In section 3, we analyze a parametric case when the true probability density function is contained in the learning model, and derive the asymptotic expansion of the stochastic complexity using resolution of singularities. In section 4, we also study a non-parametric case when the true probability density is not contained, and clarify the e#ect of singularities in the parameter space. In section 5, the problem of the asymptotic expansion of the generalization error is considered. Finally, section 6 and 7 are devoted to discussion and conclusion. Bayesian Framework In this section, we formulate the standard framework of Bayesian estimation and Bayesian stochastic complexity (Schwarz 1974; Akaike, 1980; Levin, Tishby, & Solla, 1990; Mackay, 1992; Amari, Fujita, & Shinomoto, 1992; Amari & Murata, 1993). Let p(y|x, w) be a probability density function of a learning machine, where an input x, an output y, and a parameter w are M , N , and d dimensional vectors, respectively. Let q(y|x)q(x) be a true probability density function on the input and out space, from which training samples {(x i , y i are independently taken. In this paper, we mainly consider the Bayesian framework, hence the estimated probability density # n (w) on the parameter space is defined by exp(-nH n (w))#(w), log where Z n is the normalizing constant, #(w) is an arbitrary fixed probability density function on the parameter space called an a priori distribution, and H n (w) is the empirical Kullback distance. Note that the a posteriori distribution # n (w) does not depend on {q(y i |x i constant function of w. Hence it can be written in the other form, The inference p n (y|x) of the trained machine for a new input x is defined by the average conditional probability density function, The generalization error G(n) is defined by the Kullback distance of p n (y|x) from q(y|x), { # q(y|x) log q(x)dxdy}, (1) represents the expectation value overall sets of training samples. One of the most important purposes in learning theory is to clarify the behavior of the generalization error when the number of training samples are su#ciently large. It is well known (Levin, Tishby, Solla, 1990; Amari, 1993; Amari, Murata, 1993) that the generalization error G(n) is equal to the increase of the stochastic complexity F (n), for an arbitrary natural number n, where F (n) is defined by The stochastic complexity F (n) and its generalized concepts, which are sometimes called the free energy, the Bayesian factor, or the logarithm of the evidence, can be seen in statistics, information theory, learning theory, and mathematical physics (Schwarz, 1974; Akaike, 1980; Rissanen, 1986; Mackay, 1992; Opper & Haussler, 1995; Meir & Merhav, 1995 ; Haussler & Opper, 1997; Yamanishi, 1998). For example, both Bayesian model selection and hyperparatemeter optimization are often carried out by minimization of the stochastic complexity before averaging. They are called BIC and ABIC, which are important in practical applications. The stochastic complexity satisfies two basic inequalities. Firstly, we define H(w) and F (n) respectively by q(x)dxdy, Note that H(w) is called the Kullback information. Then, by applying Jensen's inequality, holds for an arbitrary natural number n (Opper & Haussler, 1995; Watanabe, 2001a). Secondly, we use notations F (#, n) = F (n) and F (#, n) = F (n) which explicitly show the a priori probability density #(w). Then F (#, n) and F (#, n) can be understood as a generalized stochastic complexity for a case when #(w) is a non-negative function. If #(w) and #(w) satisfy then it immediately follows that Therefore, the restriction of the integrated region of the parameter space makes the stochastic complexity not smaller. For example, we define exp(-nH(w))#(w)dw, (7) with su#ciently small # > 0, then These two inequalities eq.(4) and eq.(8) give upper bounds of the stochastic com- plexity. On the other hand, if the support of #(w) is compact, then a lower bound is proven Moreover, if the learning machine contains the true distribution, then holds (Watanabe, 1999b; Watanabe, 2001a). In this paper, based on algebraic geometrical methods, we prove rigorously the upper bounds of F (n) such as are constants and o(log n) is a function of n which satisfies o(log n)/ log n # Mathematically speaking, although the generalization error G(n) is equal to F (n natural number n, we can not derive the asymptotic expansion of G(n). However, in section 5, we show that, if G(n) has some asymptotic expansion, then it should satisfy the inequality for su#ciently large n, from eq.(11). The main results of this paper are the upper bounds of the stochastic complexity, however, we also discuss the behavior of the generalization errors based on eq.(12). 3 A Parametric Case In this section, we consider a parametric case when the true probability distribution q(y|x)q(x) is contained in the learning machine p(y|x, w)q(x), and show the relation between the algebraic geometrical structure of the machine and the asymptotic form of the stochastic complexity. 3.1 Algebraic Geometry of Neural Networks In this subsection, we briefly summarize the essential result of the previous paper. For the mathematical proofs of this subsection, see (Watanabe, 1999b; Watanabe, 2001a). Strictly speaking, we need assumptions that log p(y|x, w) is an analytic function of w, and that it can be analytically continued to a holomorphic function of w whose associated convergence radii is positive uniformly for arbitrary (x, y) that satisfies q(y|x)q(x) > 0 (Watanabe, 2000; Watanabe, 2001a). In this paper, we apply the result of the previous paper to the three-layer perceptron. If a three-layer perceptron is redundant to approximate the true distribution, then the set of true parameters {w; is a union of several sub-manifolds in the parameter space. In general, the set of all zero points of an analytic function is called an analytic set. If the analytic function H(w) is a polynomial, then the set is called an algebraic variety. It is well known that an analytic set and an algebraic variety have complicated singularities in general. We introduce a state density function v(t) where #(t) is Dirac's delta function and # > 0 is a su#ciently small constant. By definition, if t < 0 or t > #, then using v(t), F # (n) is rewritten as exp(-nH(w))#(w)dw dt . Hence, if v(t) has an asymptotic expansion for t # 0, then F # (n) (n #) has an asymptotic expansion for n #. In order to examine v(t), we introduce a kind of the zeta function J(z) (Sato of the Kullback information H(w) and the a priori probability density #(w), which is a function of one complex variable z, H(w) z #(w)dw (14) Then J(z) is an analytic function of z in the region Re(z) > 0. It is well known in the theory of distributions and hyperfunctions that, if H(w) is an analytic function of w, then J(z) can be analytically continued to a meromorphic function on the entire complex plane and its poles are on the negative part of the real axis (Atiyah, 1970; Bernstein, 1972; Sato & Shintani, 1974; Bj-ork, 1979). Moreover, the poles of J(z) are rational numbers (Kashiwara, 1976). Let -# 1 be the largest pole and its order of J(z), respectively. Note that eq.(15) shows J(z) (z # C) is the Mellin transform of v(t). Using the inverse Mellin transform, we can show that v(t) satisfies where c 0 > 0 is a positive constant. By eq.(13), F # (n) has an asymptotic expansion, where O(1) is a bounded function of n. Hence, by eq.(8), Moreover, if the support of #(w) is a compact set, by eq.(9), we obtain an asymptotic expansion of F (n), We have the first theorem. Theorem 1 (Watanabe, 1999b; Watanabe, 2001a) Assume that the support of #(w) is a compact set. The stochastic complexity F (n) has an asymptotic expansion, are respectively the largest pole and its order of the function that is analytically continued from H(w) z #(w)dw, where H(w) is the Kullback information and #(w) is the a priori probability density function. Remark that, if the support of #(w) is not compact, then Theorem 1 gives an upper bound of F (n). The important constants # 1 and m 1 can be calculated by an algebraic geometrical method. We define the set of parameters W # by It is proven by Hironaka's resolution theorem (Hironaka, 1964 ; Atiyah, 1970) that there exist both a manifold U and a resolution map d in an arbitrary neighborhood of an arbitrary u # U that satisfies where a(u) > 0 is a strictly positive function and {k i } are non-negative even integers Figure 2). Let be a decomposition of W # into a finite union of suitable neighborhoods W # , where By applying the resolution theorem to the function J(z), H(w) z #(w)dw H(w) z #(w)dw is given by recursive blowing-ups, the Jacobian |g # (u)| is a direct product local variables u 1 , d where c(u) is a positive analytic function and {h j } are non-negative integers. In a neighborhood U # , a(u) and #(g(u)) can be set as constant functions in calculation of the poles of J(z), because we can take each U # small enough. Hence we can set loss of generality. Then, d where both k (#) j depend on the neighborhood U # . We find that J(z) has poles {-(h (#) j }, which are rational numbers on the negative part of the real axis. Since a resolution map g(u) can be found by using finite recursive procedures of blowing-ups, # 1 and m 1 can be found algorithmically. It is also proven that # 1 # d/2 if {w; #(w) > 0, #, and that m 1 # d. Theorem 2 (Watanabe, 1999b; Watanabe, 2001a) The largest pole -# 1 and its of the function J(z) can be algorithmically calculated by Hironaka's resolution theorem. Moreover, # 1 is a rational number and m 1 is a natural number, and if {w; #(w) > 0, where d is the dimension of parameter. Note that, if the learning machine is a regular statistical model, then always # Also note that, if Je#reys' prior is employed in neural network learning, which is equal to zero at singularities, the assumption {w; #(w) > 0, is not satisfied, and then both # even if the Fisher metric is degenerate (Watanabe, 2001c). Example.1 (Regular Model) Let us consider a regular statistical model exp(-2 with the set of parameters Assume that the true distribution is exp(-2 and the a priori distribution is the uniform distribution on W . Then, For a subset S # W , we define Then We introduce a mapping Then =2 z has a pole at z = -1. We can show JW 2 (z) has the same pole just the same way as . Hence # resulting in F (n) log n. This coincides with the well known result of the Bayesian asymptotic theory of regular statistical models. The mapping in eq.(17) is a typical example of a blowing-up. Example.2 (Non-identifiable model) Let us consider a learning machine, p(y|x, a, b, c) =# 2# exp(-2 Assume that the true distribution is same as eq.(16), and that the a priori probability distribution is the uniform one on the set Then, the Kullback information is Let us define two sets of parameters, | By using blowing-ups recursively, we find a map which is defined by By using this transform, we obtain Therefore, H(w) z dw =2 z The largest pole of JW 1 (z) is -3/4 and its order is one. It is also shown that JW\W 1 (z) have largest pole -3/4 with order one. Hence # resulting that log n +O(1). 3.2 Application to Layered Perceptron We apply the theory in the foregoing subsection to the three-layer perceptron. A three-layer perceptron with the parameter defined by a k #(b k - x where y, f(x, w), and a h are N dimensional vectors, x and b h are M dimensional vectors, c h is a real number, and and K are the numbers of input units, output units, and hidden units. In this paper, we consider a machine which does not estimate the standard deviation s > 0 (s is a constant). We assume that the true distribution is That is to say, the true regression function is This is a special case, but analysis of this case is important in the following section where the true regression function is not contained in the model. Theorem 3 Assume that the learning machine given by eq.(18) and eq.(19) is trained using samples independently taken from the distribution, eq.(20). If the a priori distribution satisfies #(w) > 0 in the neighborhood of the origin (Proof of Theorem We use notations, a Then the Kullback information is (b, c) a hp a kp , where Our purpose is to find the pole of the function where Let us apply the blowing-up technique to the Kullback information H(a, b, c). Firstly, we introduce a mapping which is defined by a a Let u # be the variables of u except u 11 , in other words, where and the Jacobian |g # (u)| of the mapping g is We define a set of paramaters for # > 0 By the assumption, there exists # > 0 such that In order to obtain an upper bound of the stochastic complexity, we can restrict the integrated region of the parameter space, by using eq.(5) and (6). By the assumption #(w) > 0 in g(U(#)). In calculation of the pole of J(z), we can assume is a constant) in g(U(#)). du # db dc The pole of the function # respectively the largest poles of J(z) and Then, since H 1 does not have zero point in the interval (-# 1 , #). larger than -# 1 , then z = -NK/2 is a pole of J(z). If otherwise, then J(z) has a larger pole than -NK/2. Hence # 1 # NK/2. Secondly, we consider another blowing-up g, which is defined by Then, just the same method as the first half, there exists an analytic function which implies Therefore By combing the above two results, the largest pole -# 1 of the J(z) satisfies the inequality, which completes the proof of Theorem 3. (End of Proof). By Theorem 1, Moreover, if G(n) has an asymptotic expansion (see section 5), we obtain an inequality of the generalization error, On the other hand, it is well known that the largest pole of a regular statistical model is equal to -d/2, where d is the number of parameters. When a three-layer perceptorn with 100 input units, 10 hidden units, and 1 output unit is employed, then the regular statistical models with the same number of parameters has It should be emphasized that the generalization error of the hierarchical learning machine is far smaller than that of the regular statistical models, if we use the Bayesian estimation. When we adopt the normal distribution as the a priori probability density, we have shown the same result as Theorem 3 by a direct calculation (Watanabe, 1999a). However, Theorem 3 shows systematically that the same result holds for an arbitrary a priori distribution. Moreover, it is easy to generalize the above result to the case when the learning machine has M input units, K 1 first hidden units, K 2 second hidden units, ., K p pth hidden units, and N output units. We assume that hidden units and output units have bias parameters. Then by using same blowing-ups, we can generalize the proof of Theorem 3, Of course, this result holds only when the true regression function is the special case, However, in the following section, we show that this result is necessary to obtain a bound for a general regression function. 4 A Non-parametric Case In the previous section, we have studied a case when the true probability distribution is contained in the parametric model. In this section, we consider a non-parametric case when the true distribution is not contained in the parametric models, which is illustrated in Figure 3. Let w 0 be the parameter that minimizes H(w), which is a point C in Figure 3. Our main purpose is to clarify the e#ect of singular points such as A and B in Figure 3 which are not contained in the neighborhood of w 0 . Let us consider a case when a three-layer perceptron given by eq.(18) and eq.(19) is trained using samples independently taken from the true probability distribution, where g(x) is the true regression function and q(x) is the true probability distribution on the input space. Let E(k) be the minimum function approximation error using a three-layer perceptron with k hidden units, Here we assume that, for each 1 # k # K, there exists a parameter w that attains the minimum value. Theorem 4 Assume that the learning machine given by eq.(18) and eq.(19) is trained using samples independently taken from the distribution of eq.(21). If the a priori distribution satisfies #(w) > 0 for an arbitrary w, then { (D where (Proof of Theorem 4) By Jensen's inequality eq.(4), we have where H(w) is the Kullback distance, be natural numbers which satisfy both 0 # k 1 # K and We divide the parameter Also let # 1 and # 2 be real numbers which satisfy both # 1 > 1 and Then, for arbitrary u, v # R N , Therefore, for arbitrary (x, w), Hence we have an inequality, where we use definitions, As F (n) is an increasing function of H(w), where are some functions which satisfy Here we can choose both # 1 (w 1 ) and # 2 (w 2 ) which are compact support functions. Firstly, we evaluate F 1 (n). Let w # 1 be the parameter that minimizes H 1 (w 1 ). Then, by eq.(22) and Theorem 2, is the number of parameters in the three-layer perceptron with k 1 hidden units. Secondly, by applying Theorem 3 to F 2 (n), By combining eq.(23) with eq.(24), and by taking # 1 su#ciently close 1, we obtain { for an arbitrary given we Theorem 4. (End of Proof). Based on Theorem 4, if G(n) has an asymptotic expansion (see section 5), then G(n) should satisfy the inequalities for n > n 0 with a su#ciently large n 0 . Hence { E(k) for n > n 0 with a su#ciently large n 0 . Figure 4 illustrates several learning curves corresponding to k (0 # k # K). The generalization error G(n) is smaller than every curve. It is well known (Barron, 1994; Murata, 1996) that, if g(x) belongs to some kind of function space, then for su#ciently large k, where C(g) is a positive constant determined by the true regression function g(x). Then, { ASYMPTOTIC PROPERTY OF THE GENERALIZATION ERROR 22 If both n and K are su#ciently large, and if then, by choosing The inequality (27) holds if n is su#ciently large. If n is su#ciently large but not extensively large, then G(n) is bounded by the generalization error of the middle size model. If n becomes larger, then it is bounded by that of the larger model, and if n is extensively large, then it is bounded by that of the largest model. A complex hierarchical learning machine contains a lot of smaller models in its own parameter space as analytic sets with singularities, and chooses the appropriate model adaptively for the number of training samples, if Bayesian estimation is applied. Such a property is caused by the fact that the model is non-identifiable, and its quantitative e#ect can be evaluated by using algebraic geometry. 5 Asymptotic Property of the Generalization Er- ror In this section, let us consider the asymptotic expansion of the generalization error. By eq.(2), F (n) is equal to the accumulate generalization error, where G(0) is defined by F (1). Hence, if G(n) has an asymptotic expansion for #, then F (n) also has the asymptotic expansion. However, even if F (n) has an asymptotic expansion, G(n) may not have an asymptotic expansion. In the foregoing sections, we have proved that F (n) satisfies inequalities such as are constants determined by the singularities and the true distribu- tion. In order to mathematically derive an inequality of G(n) from eq.(30), we need an assumption. ASYMPTOTIC PROPERTY OF THE GENERALIZATION ERROR 23 Assumption (A) Assume that the generalization error G(n) has an asymptotic expansion a q s q (n) where {a q } are real constants, s q (n) > 0 is a positive and non-increasing function of n which satisfies Based on this assumption, we have the following lemma. Lemma 1 If G(n) satisfies the assumption (A) and if eq.(30) holds, then G(n) satisfies an inequality, (Proof) By the assumption (A) which shows a 1 #. If a 1 < #, then eq.(35) holds. If a ks 2 (k). By eq.(32),eq.(33), and eq.(34), t(k) # or t(k) # C (C > 0). If t(k) #, then, for arbitrary M > 0, there exists k 0 such that Hence which contradicts eq.(36). Hence t(n) # C and a 2 C #. (End of Proof Lemma 1). In this paper, we have proven the inequalities same as eq.(30) in Theorem 1, 2, 3, and 4 without assumption (A). Then, we obtain corresponding inequalities same as if we adopt the assumption (A). In other words, if G(n) has an asymptotic expansion and if eq.(30) holds, then G(n) should satisfy eq.(35). It is conjectured that natural learning machines satisfy the assumption (A). A su#cient condition for the assumption (A) is that F (n) has an asymptotic expansion R a 1). For example, if the learner is p(y|x, a) =# 2# exp(-2 where the a priori distribution of a is the standard normal distribution, and if the true distribution is }), then, it is shown by direct calculation that the stochastic complexity has an asymptotic expansion Hence G(n) has an asymptotic expansion c 2+2n It is expected that, in a general case, G(n) has the same asymptotic expansion as Assumption (A), however, mathematically speaking, the necessary and su#cient condition for it is not yet established. This is an important problem in statistics and learning theory for the future. 6 Discussion In this section, universal phenomena which can be observed in hierarchical learning machines. 6.1 Bias and variance at singularities We consider a covering neighborhood of the parameter space, where {W (w j )} are the su#ciently small neighborhood of the parameter w j which The number J in eq.(38) is finite when compact. Then, the upper-bound of the stochastic complexity can be rewritten as exp(-H(w))#(w)dw is the function approximation error of the parameter w j H(w), and V (w j ) is the statistical estimation error of the neighborhood of w j , (- log n) m(w j )-1 where c 0 > 0 is a constant. The values -#(w j ) and m(w j ) are respectively the largest pole and its multiplicity of the meromorphic function Note that B(w j ) and V (w j ) are called the bias and the variance, respectively. In the Bayesian estimation, the neighborhood of the parameter w j that minimizes is selected with the largest probability. In regular statistical models, the variance does not depend on the parameter, in other words, #(w j for an arbitrary parameter w j , hence the parameter that minimizes the function approximation error is selected. On the other hand, in hierarchical learning machines, the variance V (w j ) strongly depends on the parameter w j , and the parameter that minimizes the sum of the bias and variance is selected. If the number of training samples is large but not extensively large, parameters among the singular point A in Figure 3 that represents a middle size model, is automatically selected, resulting in the smaller generalization error. As n increases, the larger but not largest model B is selected. At last, if n becomes extensively large, then the parameter C that minimizes the bias is selected. This is a universal phenomenon of hierarchical learning machines, which indicates the essential di#erence between the regular statistical models and artificial neural networks. 6.2 Neural networks are over-complete basis Singularities of a hierarchical learning machine originate in the homogeneous structure of a learning model. A set of functions used in an artificial neural network, for example, is a set of over-complete basis, in other words, coe#cients {a(b, c)} in a wavelet type decomposition of a given function g(x), are not uniquely determined for g(x) (Chui, 1989; Murata, 1996). In practical applications, the true probability distribution is seldom contained in a parametric model, however, we adopt a model which almost approximates the true distribution compared with the fluctuation caused by random samples, a k #(b k - x If we have an appropriate number of samples and choose an appropriate learning model, it is expected that the model is in an almost redundant state, where output functions of hidden units are almost linearly dependent. We expect that this paper will be a mathematical foundation to study learning machines in such states. 7 Conclusion We considered the case when the true distribution is not contained in the parametric models made of hierarchical learning machines, and showed that the parameters among singular points are selected by the Bayesian distribution, resulting in the small generalization error. The quantitative e#ect of the singularities was clarified based on the resolution of singularities in algebraic geometry. Even if the true distribution is not contained in the parametric models, singularities strongly a#ect and improve the learning curves. This is a universal phenomenon of the hierarchical learning machines, which can be observed in almost all artificial neural networks. --R Likelihood and Bayes procedure. A universal theorem on learning curves. Four Types of Learning Curves. Neural Computation Statistical theory of learning curves under entropic loss. Resolution of Singularities and Division of Distributions. Communications of Pure and Applied Mathematics Approximation and estimation bounds for artificial neural networks. The analytic continuation of generalized functions with respect to a parameter. Mathematical methods of statistics An introduction to Wavelets. Testing in locally conic models Generalized functions. On the problem of applying AIC to determine the structure of a layered feed-forward neural network A Failure of likelihood asymptotics for normal mixtures. Mutual information Resolution of singularities of an algebraic variety over a field of characteristic zero. A statistical approaches to learning and generalization in layered neural networks. Bayesian interpolation. On the stochastic complexity of learning realizable and unrealizable rules. An integral representation with ridge functions and approximation bounds of three-layered network Bounds for predictive errors in the statistical mechanics of supervised learning. Stochastic complexity and modeling. On zeta functions associated with prehomogeneous vector space. A optimization method of layered neural networks based on the modified information criterion. On the essential di On the generalization error by a layered statistical model with Bayesian estimation. Algebraic analysis for non-regular learning machines Neural Computation Probabilistic design of layered neural networks based on their unified framework. Learning in artificial neural networks: a statistical prespective. Neural Computation A decision-theoretic extension of stochastic complexity and its applications to learning --TR Bayesian interpolation Four types of learning curves A universal theorem on learning curves An introduction to wavelets Statistical theory of learning curves under entropic loss criterion Approximation and Estimation Bounds for Artificial Neural Networks On the Stochastic Complexity of Learning Realizable and Unrealizable Rules A regularity condition of the information matrix of a multilayer perceptron network An integral representation of functions using three-layered networks and their approximation bounds Algebraic Analysis for Singular Statistical Estimation --CTR Miki Aoyagi , Sumio Watanabe, Stochastic complexities of reduced rank regression in Bayesian estimation, Neural Networks, v.18 n.7, p.924-933, September 2005 Keisuke Yamazaki , Sumio Watanabe, Singularities in mixture models and upper bounds of stochastic complexity, Neural Networks, v.16 n.7, p.1029-1038, September Sumio Watanabe , Shun-ichi Amari, Learning coefficients of layered models when the true distribution mismatches the singularities, Neural Computation, v.15 n.5, p.1013-1033, May Shun-Ichi Amari , Hiroyuki Nakahara, Difficulty of Singularity in Population Coding, Neural Computation, v.17 n.4, p.839-858, April 2005 Haikun Wei , Jun Zhang , Florent Cousseau , Tomoko Ozeki , Shun-ichi Amari, Dynamics of learning near singularities in layered networks, Neural Computation, v.20 n.3, p.813-843, March 2008 Shun-Ichi Amari , Hyeyoung Park , Tomoko Ozeki, Singularities Affect Dynamics of Learning in Neuromanifolds, Neural Computation, v.18 n.5, p.1007-1065, May 2006

resolution of singularities;generalization error;stochastic complexity;asymptotic expansion;algebraic geometry;non-identifiable model

607896

A Comparison of Static Analysis and Evolutionary Testing for the Verification of Timing Constraints.

This paper contrasts two methods to verify timing constraints of real-time applications. The method of static analysis predicts the worst-case and best-case execution times of a tasks code by analyzing execution paths and simulating processor characteristics without ever executing the program or requiring the programs input. Evolutionary testing is an iterative testing procedure, which approximates the extreme execution times within several generations. By executing the test object dynamically and measuring the execution times the inputs are guided yielding gradually tighter predictions of the extreme execution times. We examined both approaches on a number of real world examples. The results show that static analysis and evolutionary testing are complementary methods, which together provide upper and lower bounds for both worst-case and best-case execution times.

Introduction For real-time systems the correct system functionality depends on their logical correctness as well as on their temporal correctness. Accordingly, the verification of the temporal behavior is an important activity for the development of real-time systems. The temporal behavior is generally examined by performing a schedulability analysis to ensure that a task's execution can finish within specified deadlines. The models for schedulability analysis are commonly based on the assumption that the worst-case execution time (WCET) is known. Specifically, the models assume that the WCET must not exceed the task's deadline. The best-case execution time (BCET) may also be used to predict system utilization or ensure that minimum sampling intervals are met. Techniques of static analysis (SA) can be used in the course of system design in order to assess the execution times of planned tasks as pre-condition for schedulability analysis. Static timing analysis constitutes an analytical method to determine bounds on the WCET and BCET of an application. SA simulates the timing behavior at a cycle level for hardware concepts such as caches and pipelines of a given processor. The approach discussed in this paper uses the method of Static Cache Simulation followed by Path Analysis within a timing analyzer. Timing estimates are calculated without knowledge of the input and without executing the actual application. Dynamic testing is one of the most important analytical method for assuring the quality of real-time systems. It serves for the verification as well as the validation of sys- tems. An investigation of existing test methods shows that they mostly concentrate on testing the logical correctness. There is a lack of support for testing the temporal system behavior. For that reason, we developed a new approach to test the temporal behavior of real-time systems: Evolutionary Testing (ET). ET searches automatically for test data, which produces extreme execution times in order to check if the timing constraints specified for the system are vio- lated. This search is performed by means of evolutionary computation. Although SA and ET are usually applied in different phases of system development, both procedures aim at estimating the shortest and longest execution times for a sys- tem, which makes a comparison of these two methods very interesting. Both approaches are compared in this paper with the help of several examples. Chapter 2 offers a general overview of related work on SA as well as on testing. The third chapter describes the tool we employ for SA. Afterwards, chapter 4 introduces ET. Both approaches have been used to determine the minimum and maximum run times of different systems. Chapter 5 summarizes the obtained results. These are discussed in chapter 6. It will be seen that a combination of SA and ET makes a reliable definition of extreme run times possible. The most important statements are summarized in chapter 7 that also includes a short outlook on future work. 2. Related Work This section presents an overview of published work in timing analysis for real-time systems followed by a discussion of previous work on testing methods for real-time systems 2.1. Timing Analysis of Real-Time Systems Bounding the WCET of programs is a difficult task. Due to the undecidability of the halting problem, static WCET analysis is subject to constraints on the use of programming language constructs and on the underlying operating system. For instance, an upper bound on the number of loop iterations has to be known, indirect calls should not be used, and memory should not be allocated dynamically [25]. Often, recursive functions are also not allowed, although there exist outlines on treating bounded recursion similar to bounded loops [19]. Recent research in the area of predicting the WCET of programs has made a number of advances. Conventional methods for static analysis have been extended from unoptimized programs on simple CISC processors to optimized programs on pipelined RISC processors, and from uncached architectures to instruction caches [1, 16] and data caches [13, 16, 31]. Today, mainly three fundamental models for static timing analysis exist. First, a source-level oriented timing schema propagates times through a tree and handles pipelined RISC processors with first-level split caches [23, 13]. Second, a constraint-based method models architectural aspects, including caches, via integer linear programming [16]. Third, our approach uses data-flow analysis to model the cache behavior separate from pipeline simula- tion, which is handled later in a timing analyzer via path analysis [1, 11, 31]. The first and second approaches use integrated analysis of caches while our approach uses separate analysis. This allows us to deal with multi-level memory hierarchies or unified caches. Another approach using data-flow analysis to modeling caching originally used the same categorizations as our approach but a different data-flow model. Recently, the approach has been generalized to handle a number of data-flow solutions with differing complexity and accuracy [9]. In the presence of caches, non-preemptive scheduling was initially assumed to prevent undeterministic behavior due to the absence of unpredictable context switch points. If context switches occurred at arbitrary points (e.g., in a preemptive system), cache invalidations may occur resulting in unexpected cache misses when the execution of a task is resumed later on. Hardware and software approaches have been proposed to counter this problem but find little use in practice due to a loss of cache performance when caches are partitioned [14, 17]. Recently, attempts have been made to incorporate caching into rate-monotone analysis and response-time analysis [5, 15], which allows WCET predictions for non-preemptive systems to be used in the analysis of preemptively scheduled systems. This approach seems most promising since the information gathered for static timing analysis can be utilized within this extended framework for schedulability analysis. 2.2. Testing Real-Time Systems Analytical quality assurance plays an important role in ensuring the reliability and correctness of real-time systems, since a number of shortcomings still exist within the development life cycle. In practice, dynamic testing is the most important analytical method for assuring the quality of real-time systems. It is the only method that examines the run-time behavior, based on an execution in the application environment. For embedded systems, testing typically consumes 50% of the overall development effort and budget [8, 29]. It is one of the most complex and time-consuming activities within the development of real-time systems [12]. In comparison with conventional software systems the examination of additional requirements like timeliness, simul- taneity, and predictability make the test costly, and technical characteristics like the development in host-target environ- ments, the strong connection with the system environment or the frequent use of parallelism, distribution, and fault-tolerance mechanisms complicate the test. The aim of testing is to find existing errors in a system and to create confidence in the system's correct behavior by executing the test object with selected inputs. For testing real-time systems, the logical system behavior, as well as the temporal behavior of the systems, need to be examined thoroughly. An investigation of existing test methods shows that a number of proven test methods are available for examining the logical correctness of systems [22, 10]. But there is a lack of support for testing the temporal behavior of sys- tems. Only very few works deal with testing the temporal behavior of real-time systems. Braberman et al. have published an approach that is based on modeling the system design with a particular, formally defined SA/SD-RT notation that is translated into high-level timed Petri nets [4]. Out of this formal model a symbolic representation of the temporal behavior is formed, the time reachability tree. Each path from the root of the tree to its leaves represents a poten- Source Files Compiler Information Control Flow Cache Simulator Configurations I/D-Cache Interface User Analyzer Timing User Timing Requests Timing Predictions Virtual Address Information Address Calculator Addr Info and Relative Data Decls Dependent Machine Information Categorizations I/D-Caching Figure 1. Framework for Timing Predictions tial test case. The tree already becomes very extensive for small programs so that the number of test cases must be restricted according to different criteria. Results of any practical trial testing of this approach are not reported. Mandrioli et al. developed an interactive tool that enables the generation of test cases for real-time systems from formal specifications written in TRIO [18]. The language TRIO extends classical temporal logic to deal explicitly with time measures. At present, however, the applicability of the tool is restricted to small systems whose properties are specified through simple TRIO formulas. Clarke and Lee [6] as well as Dasarathy [7] describe further techniques for verifying timing constraints using timed process algebra or finite-state machines. All of these approaches demand the use of formal specification techniques. Since the use of formal methods has not yet been generally adopted in industrial practice due to the great expenditure connected with it and the lack of maturity of the existing tools, the testing approaches mentioned have not spread far in industry, particularly since the suitability of these approaches in many cases remains restricted to small systems. Accordingly, there are no specialized methods available at the moment that are suited for testing the temporal behavior of real-time systems. For that reason, testers usually go back to conventional test procedures developed originally for the examination of logical correctness, e.g., systematic black-box or white-box oriented test methods. Since the temporal behavior of complex systems is hard to comprehend and can therefore be examined only insufficiently with traditional test methods, existing test procedures must be supplemented by new methods, which concentrate on determining whether or not the system violates its specified timing constraints. Therefore we examine the applicability of evolutionary testing (ET) to test the temporal behavior of real-time systems. 3. Static Analysis (SA) Our framework of WCET prediction uses a set of tools as depicted in Figure 1. An optimizing compiler has been modified to emit control-flow information, data informa- tion, and the calling structure of functions in addition to regular target code generation. Up to now, the research compiler VPCC/VPO [3] performed this task. We are currently integrating Gnat/Gcc [26, 28] into this environment. A static cache simulator uses the control-flow information and calling structure in conjunction with the cache configuration to produce instruction and data categorizations, which describe the caching behavior of each instruction and data reference. We currently use a separate analyzer for instruction and data caches since data references require separate preprocessing via an address calculator. Current work also includes a single analyzer for unified caches and the handling of secondary caches [20]. The timing analyzer uses these categorizations and the control-flow information to perform a path analysis of the program. It then predicts the BCET and WCET for portions of the program or the entire program, depending on user requests. In the experiments described in section 5, we chose an architecture without caches for reasons also explained in section 5. Thus, only the portion of the toolset shaded grey in Figure was used in these experiments. Next, we describe the interaction of the various tools of the entire framework. The framework can be retargeted by changing the cache configurations and porting the machine description. How- ever, the largest retargeting overhead constitutes a port of the compiler. Thus, our current efforts to integrate Gnat/Gcc into the framework will greatly improve portability. 3.1. Static Cache Simulation Static cache simulation provides the means to predict the caching behavior of the instructions and data references of a program/task (see Static Cache Simulator in Figure 1). The addresses of instruction references is obtained from the control-flow information emitted by the compiler. Addresses of data references are calculated by the Address Calculator (see Figure 1) from locating data declarations for global data and obtaining offsets for relative addresses of local data, which are translated into virtual addresses by taking the context of a process into account. For both instruction and data references, the caching behavior is dis- Category 1st reference consecutive ref. Always-hit hit hit Always-miss miss miss First-hit miss hit First-miss hit miss Table 1. Categorizations for Cache Reference tinguished by the categories described in Table 1. For each category, the cache behavior of the first reference and consecutive references is distinguished. Consecutive references are strictly due to loops since we distinguish function invocations by their call sites. For data caches, an additional category, called calculated, denotes the total number of data cache misses out of all references within a loop for a memory reference. A program may consist of a number of loops, possibly nested and distributed over several functions. For each loop level, an instruction receives a distinct categorization. The timing analyzer can then derive tight bounds of execution time by inspecting the categorizations for each loop level. Since instruction categorizations have to be determined by inter-procedural analysis of the entire program, the call graph of the program has to be analyzed. The method of static cache analysis traces the origin of calls within the call graph by distinguishing function instances. Since instruction categorizations for a function are specified for each function instance, the timing analyzer can interpret different caching behaviors depending on the calling sequence to yield tighter WCET predictions. The static cache simulator determines the categories of an instruction based on a novel view of cache memories, using a variation of iterative inter-procedural data-flow analysis (DFA). The following information results from DFA: ffl The abstract cache state describes which program lines that map into certain cache blocks may potentially be cached within the control flow. ffl The linear cache state contains the analog information in the (hypothetical) absence of loop. ffl The post-dominator set describes the program lines certain to still be reached within the control flow. The above data-flow information can also be reduced with respect to certain subsets, in particular to check if the information is available within a certain loop level. A formal framework for this analysis for instruction and data caches is described in [31]. The data-flow information provides the means to derive the above categories, for example for set-associative instruction caches with multiple levels of associativity. The following categories are derived for each loop level of an instruction for the worst-case cache behavior Always-hit: (on spatial locality within the program line) or ((the instruction is in cache in the absence of loops) and ((there are no conflicting instructions in the cache state) or (all conflicts fit into the remaining associativity levels))). First-hit: (the instruction was a first-hit for inner loops) or (it is potentially cached, even without loops and even for all loop preheaders, it is always executed in the loop, not all conflicts fit into the remaining associativity levels but conflicts within the loop fit into the remaining associativity levels for the loop headers, even when disregarding loops). First-miss: the instruction was a first-miss for inner loops, it is potentially cached, conflicts do not fit into the remaining associativity levels but the conflicts within the loop do. Always-miss: This is the conservative assumption for the prediction of worst-case execution time when none of the above conditions apply. A loop header is an entry block into the loop with at least one predecessor block outside the loop, called the pre- header, and at least one predecessor block inside the loop. 3.2. Timing Analysis The timing analyzer (see Figure 1) calculates the BCET and WCET by constructing a timing tree, traversing paths within each loop level, and propagating the timing information bottom-up within the tree. During the traversal, the timing analyzer has to simulate hardware characteristics (e.g., pipelining) and the instruction categorizations have to be interpreted The timing analyzer does not have to take the cache configuration into account. Instead, the instruction categoriza- tions, as introduced above, are used to interpret the caching behavior. The approach of splitting cache analysis via static cache simulation and timing analysis makes the caching aspects completely transparent to the timing analyzer. Solely based on the instruction categorizations, the timing analyzer can derive the WCET by propagating timing predictions bottom-up within the timing tree. The timing tree represents the calling structure and the loop structure of the entire program. As seen in the context of the static cache simulator, functions are distinguished by their calling paths into function instances. This allows a tighter prediction of the WCET due to the enhanced information about the calling context. Each function instance is regarded as a loop level (with one iteration) and is represented as a node in the timing tree. Regular loops within the program are represented as child nodes of its surrounding function instance (outer-most loops) or as child nodes of another loop that they are nested in. The timing analyzer determines the BCET and WCET in a bottom-up traversal of the tree. For any node, all possible paths (sequences of basic blocks) within the current loop level have to be analyzed, which will be described in more detail for the WCET. When a child node is encountered along a path, its WCET is already calculated and can simply be added to the WCET of the current path, sometimes with small adjustments. Adjustments are necessary for transitions from first-misses to first-misses and always- misses to first-hits between loop levels [1]. For a loop with n iterations, a fix-point algorithm is used to determine the cumulative WCET of the loop along a sequence of (possibly different) paths. Once a pattern of longest paths has been established, the remaining iterations can be calculated by a closed formula. In practice, most loops have one longest path. Thus, the first iteration is needed to adjust the WCET of child loops along the path, and the second iteration represents the fix-point time for all remaining iterations. The scope of the WCET analysis can such be limited to one loop level at a time, making timing analysis very efficient compared to an exhaustive analysis of all permutations of paths within a program. See [1, 11] for a more detailed description of the timing analyzer and an analog description for the BCET. 4. Evolutionary Testing (ET) Evolutionary testing is a new testing approach, which combines testing with evolutionary computation. In first experiments the application of ET for examining the temporal behavior of real-time systems achieved promising results. In ten experiments performed ET always achieved better results compared to random testing with respect to effectiveness as well as efficiency. More extreme execution times were found by means of evolutionary computation with a less or equal testing effort than for random testing (see [30] for more details). 4.1. A Brief Introduction to Evolutionary Compu- tation Evolutionary algorithms represent a class of adaptive search techniques and procedures based on the processes of natural genetics and Darwin's theory of evolution. They are characterized by an iterative procedure and work in parallel on a number of potential solutions, the population of indi- viduals. In every individual, permissible solution values for the variables of the optimization problem are coded. Evolutionary algorithms are particularly suited for problems involving large numbers of variables and complex input do- mains. Even for non-linear and poorly understood search spaces evolutionary algorithms have been used successfully because of their robustness. The evolutionary search and optimization process is based on three fundamental principles: selection, recom- bination, and mutation. The concept of evolutionary algorithms is to evolve successive generations of increasingly better combinations of those parameters, which significantly effect the overall performance of a design. Starting with a selection of good individuals, the evolutionary algorithm achieves the optimum solution by the random exchange of information between these increasingly fit samples (recombination) and the introduction of a probability of independent random change (mutation). The adaptation of the evolutionary algorithm is achieved by the selection and reinsertion procedures since these are based on fitness. The fitness-value is a numerical value, which expresses the performance of an individual with regard to the current op- timum. The notion of fitness is essential to the application of evolutionary algorithms; the degree of success in using them may depend critically on the definition of a fitness function that changes neither too rapidly nor too slowly with the design parameters. Figure 2 gives an overview of a typical procedure of evolutionary optimization. Reinsertion Evaluation Mutation Recombination Selection Optimization criteria met? Initialization Evaluation Figure 2. The Process of Evolutionary Com- putation At first, a population of guesses to the solution of a problem is initialized, usually at random. Each individual in the population is evaluated by calculating its fitness. The results obtained will range from very poor to good. The remainder of the algorithm is iterated until the optimum is achieved or another stopping condition is fulfilled. Pairs of individuals are selected from the population and are combined in some way to produce a new guess in an analogous way to biological reproduction. Selection and combination algorithms are numerous and vary. A survey can be found in [24]. After recombination the offspring undergoes mutation. Mutation is the occasional random change of a value, which alters some features with unpredictable consequences. Mutation is like a random walk through the search space and is used to maintain diversity in the population and to keep the population from prematurely converging on one local solu- tion. Besides, mutation creates genetic material that may not be present in the current population [27]. Afterwards, the new individuals are evaluated for their fitness and replace those individuals of the original population who have lower fitness values (reinsertion). Thereby a new population of individuals develops, which consists of individuals from the previous generation and newly produced individuals. If the stopping condition remains unfulfilled, the process described will be repeated. 4.2. Applying Evolutionary Computation to Testing Temporal System Behavior The major objective of testing is to find errors. As described in section 2, real-time systems are tested for their logical correctness by standard testing techniques. The fact that the correctness of real-time systems depends not only on the logical results of computations but also on providing the results at the right time adds an extra dimension to the verification and validation of such systems, namely that their temporal correctness must be checked. The temporal behavior of real-time systems is defective when such computations of input situations exist that violate the specified timing constraints. Normally, a violation means that outputs are produced too early or their computation takes too long. The task of the tester therefore is to find the input situations with the shortest or longest execution times to check if they produce a temporal error. This search for the shortest and longest execution times can be regarded as an optimization problem to which evolutionary computation seems an appropriate solution. Evolutionary computation enables a totally automated search for extreme execution times. When using evolutionary optimization for determining the shortest and longest execution times, each individual of the population represents a test datum with which the test object is executed. In our experiments the initial population is generated at ran- dom. If test data has been obtained by a systematic test, in principle, these could also be used as initial population. Thus, the evolutionary approach benefits from the tester's knowledge of the system under test. For every test datum, the execution time is measured. The execution time determines the fitness of the test datum. If one searches for the WCET, test data with long execution times obtain high fitness values. Conversely, when searching for the BCET, individuals with short execution times obtain high fitness val- ues. Members of the population are selected with regard to their fitness and subjected to combination and mutation to generate new test data. By means of selection, it is decided what test data are chosen for reproduction. In order to retain the diversity of the population, and to avoid a rapid convergence against local optima, not only the fittest individuals are selected, but also those individuals with low fitness values obtain a chance of recombination. In our experiments stochastic universal sampling [2] was used as selection strategy. For the recombination of test data discrete recombination [21] was applied, a simple exchange of variable values between individuals (see Figure 3). The probability of mutating an individual's variables was set to be inversely proportional to its number of variables. The more dimensions one individual has, the smaller is the mutation probability for each single variable. This mutation rate has been used with success in a multitude of experiments [24, 27]. It is checked if the generated test data are in the input domain of the test object. Then, the individuals produced are also evaluated by executing the test object with them. Afterwards, the new individuals are united with the previous generation to form a new population according to the reinsertion procedures laid down. In our experiments we applied a reinsertion strategy with a generation gap of 90%. The next generation therefore contained more offspring than parents since 90% of a popula- tion's individuals were replaced by offspring. This process repeats itself, starting with selection, until a given stopping condition is reached, e.g., a certain number of generations is reached or an execution time is found, which is outside the Parents Offspring Figure 3. Discrete Recombination with four Randomly Defined Crossover Points specified timing constraints. In this case, a temporal error is detected. If all the times found meet the timing constraints specified for the system under test, confidence in the temporal correctness of the system is substantiated. In all experiments evolutionary testing was stopped after a predefined number of generations which we have specified according to the complexity of the test objects with respect to their number of input parameters and lines of code (LOC). 5. Verifying Timing Constraints: SA vs. ET We used SA and ET in five experiments to determine the BCET and WCET of different systems. Except for the last two examples described in this section, all programs tested come from typical real-time systems used in practice. The test programs were chosen since three of them cover different areas within industrial applications of the Daimler Benz company, and the remaining two programs serve as a reference to related work, where these had been used as examples for general-purpose algorithms within real-time appli- cations. The test program also cover a wide range of real-time applications within graphics, transportation, defense, numerical analysis and standard algorithms. Of course, the results are dependent on the hardware/software platform and are generally not directly transferable from one to another since the processor speed and the compiler used directly affect the temporal behavior. All the experiments that are described in the following were carried out on a SPARCstation IPX running under Solaris 2.3 with 40 MHz. The execution times in processor cycles were derived by SA and ET. We chose a SPARC IPX platform since this architecture does not have any caches. At the current stage of devel- opment, the timing analyzer for SA only supports either instruction cache categorizations or data cache categorization. We are working on an extension to support both categorizations at the same time. For ET the execution times were measured using the performance measurement tool Quan- tify, available from Rational. Quantify performs cycle-level timing though object code instrumentation. Thus, overheads of the operating system were ruled out, and the execution times reported were the same for repeated runs with identical parameters. However, Quantify does not take the effects of caching into account. Thus, we needed an uncached architecture to perform our experiments. The SA approach utilized the pipeline simulation of the timing analyzer for the experiments. The instruction execution was simulated for a five-stage pipeline with a through-put of one instruction per cycle for most cases, as commonly found in RISC architectures. Load and store instructions caused a stall of two cycles to access memory. Floating point instructions resulted in stalls with varying durations, specified for the best case and worst case of such operations. The timing analyzer calculates a conservative estimate of the number of cycles required for an execution based on path analysis. For the worst case, the estimate is guaranteed to be greater or equal than the actual WCET. Conversely, the estimate is less or equal than the actual BCET. The library of evolutionary algorithms, which we applied for ET, was a Matlab-based toolbox developed at the Daimler-Benz laboratories by Hartmut Pohlheim. It provides a multitude of different evolutionary operators for se- lection, recombination, mutation, and reinsertion [24]. For each experiment, the evolutionary algorithms were applied twice; first, to find the longest execution time, and then the shortest. The fitness was set equal to either the execution time measured in processor cycles for the longest path or its reciprocal for the shortest path. The population size was varied for the experiments according to the complexity of the test objects. Pairs of test data were chosen at random and combined using different operators like discrete recombination or double crossover depending on the representation of the individuals. The mutation probability was set inversely proportional to the length of the individuals. There is no means of deciding when an optimum path has been found, and ET was usually allowed to continue for 100 generations before it was stopped. 5.1. Test Objects The first example is a simple computer graphics function in C, which checks whether or not a line is covered by a given rectangle with its sides parallel to the axes of the co-ordinate system. The function has two input parameters: the line given by the co-ordinates of both line end points, and the rectangle, which is described using the position of its upper left corner, its width and its height. This amounts to eight atomic input variables altogether. The function has 107 LOC and contains a total of 37 statements in 16 program branches. The second application comes from the field of railroad control technology. It concerns a safety-critical application that detects discrepancies between the separate channels in a redundant system. It has 389 LOC and 512 different input parameters: binary variables, 384 variables ranging from 0 to 255 and 112 variables with a range of each from 0 to 4095. The third application concerned comes from the field of defense electronics. It is an application that extracts characteristics from images. A picture matrix is analyzed with regard to its brightness, and the signal-to-noise ratio of its brightest point and its background is established. The defense electronics program has 879 LOC and 843 integer input parameters. The first two input parameters represent the position of a pixel in a window and lie within the range 1.1200 and 1.287 respectively. The remaining 841 Program Graphics Railroad Defense Matrix Sort Method best worst best worst best worst best worst best worst SA 309 2,602 389 23,466 848 71,350 8,411,378 15,357,471 16,003 24,469,014 actual N/A N/A N/A N/A N/A N/A 10,315,619 13,190,619 20,998 11,872,718 Table 2. Execution Times [cycles] for Test Programs parameters define an array of 29 by 29 pixels representing a graphical input located around the specified position; each integer describes the pixel color and lies in the range 0.4095. The fourth sample program multiplies two integer matrices of size 50 by 50 and stores the result in a third matrix. Only integer parameters in the range between 0 and 8095 are permissible as elements of the matrices. Matrix operations are typical for embedded image processing applications The fifth test program performs a sort of an array of 500 integer numbers using the bubblesort algorithm. Arbitrary integer values can be sorted. Sorting operations are common for countless applications within and beyond the area of real-time systems. 5.2. Experiments For all test objects mentioned the shortest (best) and longest (worst) execution times were determined. The results of the experiments are summarized in Table 2 for the best case and worst case. The first row depicts the results for static analysis and the last row shows the measurements for evolutionary testing. The middle row shows the actual shortest and longest execution times for the multiplication of matrices and the bubblesort algorithm that were easily determined by applying a systematic test. Notice that the actual execution times could only be determined with certainty in the absence of caching due to hardware complexities [31]. The other examples of actual real-time systems are so complex with regard to their functionality that their extreme execution times cannot be definitely determined by a systematic test. For applications used in practice this is the normal case. For the computer graphics example SA calculated a lower bound of 309 processor cycles for the shortest execution time and an upper bound of 2602 cycles for the longest execution time. ET discovered a shortest time of 457 cy- cles, and a longest time of 2176 cycles within 24 genera- tions. The population size was set to 50. The generation of 76 additional generations with 3800 test data sets does not produce any longer or shorter execution times. Thus the shortest execution times determined vary by 32%, the longest by 16%. For the railroad technology example the population size for ET was increased to 100 because of the complex input interface of the test object with its more than 500 pa- rameters. Starting from the first generation a continuous improvement up to the 100th generation could be observed for ET. This suggests that ET would find even more extreme execution times if the number of generations was in- creased. The shortest execution time found by ET so far (508 cycles) is nearly 24% above the 389 cycles computed by SA. The longest execution time determined (22626 cy- cles) varies only by 4% from the one calculated by SA (23466 cycles). Therefore, the worst-case execution time of this example can already be defined very accurately after 100 generations. It can be guaranteed that the maximum execution time of this task lies between 22626 cycles and cycles. The defense electronics program has 843 input parame- ters. Therefore, the population size in this experiment was also set to 100. For this example, evolutionary algorithms were used to generate pictures surrounding a given position. The number of generations was increased to 300 because of the large range of the variables and the large number of input parameters. Again the longest execution time increased steadily with each new generation and asymptoted towards the current maximum of 35226 cycles when the run was terminated after 300 generations. The fastest execution time was found to be 9095 after 300 generations. Compared to the results achieved by SA significant differences could be observed. The estimates for the extreme execution times calculated by SA are 848 cycles and 71350 cycles. A closer analysis of possible causes for these deviations lead to the possibility that certain instructions were assumed to take different times for their pipeline execution. The instructions in question are multiply and divide, which account for multiple cycles during the execution stage. We are currently trying to isolate these effects for the Quantify tool to allow a proper comparison with SA. The next example in the table is the multiplication of matrices. Due to its functional simplicity the minimum and maximum run time can very easily be determined by systematic testing because they represent special input situations. The longest execution time of 13190619 cycles results if all elements of both matrices are set to the largest permissible value (8095). The shortest run time of cycles results if both matrices are fully initialized with 0. When ET is applied to the multiplication of matrices a single individual is made up of 5000 parameters (2*50*50). The resulting search space is by far the largest of the examples presented here. For each generation with 100 individuals 500000 parameter values have to be generated. Nevertheless, the number of generations for this example was increased to 2000. When searching for the longest execution time, a maximum of 13007019 cycles was found. The evolutionary algorithms had found an execution time that lies only a good 1% below the absolute maximum. The longest execution time that was determined with the help of SA (15357471 cycles) exceeds the absolute maximum by about 16%. The shortest execution time determined by the evolutionary algorithms is 12050569 cycles, which means a deviation of 17% compared to the actual shortest run time. The deviation of SA is nearly similar: the execution time of 8411378 cycles lies about 18% below the actual value. The last example is the bubblesort algorithm. Again the determination of the extreme run times is very easy with the help of a systematic test. The longest execution time for bubblesort results from the list sorted in reverse and amounts to 11872718 cycles. The shortest run time results, of course, from the sorted list, which leads to an execution time of just 20998 cycles. Once again the longest execution time found by ET (11826117 cycles) comes close to the actual maximum. It deviates by less than 1%. The upper bound (24469014 cycles) for the longest execution time that was determined by SA exceeds the actual one by more than 100%. This overestimation is caused by a deficiency of the algorithm that interpolates the execution time for loops. In particular, two loops are nested with a loop counter of the inner loop whose initial value is dependent on the counter of the outer loop. Currently, the timing analyzer estimates the number of iterations of the inner loop conservatively as is the maximum number of iterations for the outer loop. We are working on a method to handle such loop dependencies to correctly estimate the number of iterations for nested loops. In this case, the inner loop has 1n 2 iterations. As a coarse estimate, 12234507 cycles or half the estimated value should be calculated taking the actual loop overhead into account, i.e., the value would be around 3% off the actual value. Further discussions will refer to this adjusted value. The shortest execution time of the bubblesort algorithm is only insufficiently evaluated by evolutionary optimization (1464577 cycles). Although shorter run times have been continually found over 2000 generations the results are far from the absolute minimum. For that reason current work focuses on a detailed analysis of the bubblesort example and an improvement of the ET results. Also, our main focus was to bound the WCET since this provides the means to verify that deadlines cannot be missed, a very important property of real-time systems. The shortest execution time determined by SA (16003 cycles) differs by 24% from the absolute minimum. 6. Discussion The measurements of the last section show that the methods of static analysis and ET bound the actual execution times. While SA always estimates the extreme execution times in such a way that the actual run times possible for the system will never exceed them, ET provides only actually occurring execution times. For the worst case, the estimates of SA provide an upper bound while the measurements of ET give a lower bound on the actual time. Conversely, SA's estimates provide a lower bound for the best-case time while ET's measurements constitute an upper bound. In about half of the experiments, the actual execution times were bounded within \Sigma3% or better with respect to the range of execution times determined by SA. These results are directly applicable to schedulability analysis and provide a high confidence about the range for the actual WCET. In further cases, the variation between the two approaches was about \Sigma10%, which may still yield useful results for schedulability analysis. We regard \Sigma10% as a threshold for useful results in the sense that larger deviations between the two methods may not be accurate enough to guarantee enough processor utilization, even though they may be safe. For the multiplication of matrices and the defense example larger variations were detected. This indicates that both approaches need further investigation to improve their precision. The overhead for estimating the extreme execution times differs for both approaches. ET requires the execution of a test program over many generations with a large number of input data, i.e., the overhead is dependent on the actual execution times of the test object and additional delays caused by the timing. SA requires a test overhead in the order of seconds for the tested programs since one simulation suffices to predict the extreme execution times, i.e., the overhead is independent of the actual execution times. Instead, the overhead depends on the complexity of the combined call graph and control-flow graphs of the entire program and roughly increases quadratically with the program size. SA automatically yields not only timing estimates for the entire application, but also estimates for arbitrary subroutines or portions of the control flow. To obtain corresponding data with ET test objects have to be isolated. A prerequisite for performing SA is the knowledge of the cycle-level behavior for the target processor that has to be supplied in configuration files. The ET approach works for a wide range of timing methods. On one hand, hardware timers calculating wall-clock time may be used without knowledge of the actual hardware. This method is highly portable but subject to interference with hardware and software components, e.g., caches and operating systems. On the other hand, cycle-level timing information, excluding the instrumentation instructions, may be calculated as part of the program execution, as seen in the above experiments. The portability of this method is constrained by the portability of the instrumentation tool. In summary, the ET approach cannot provide safe timing guarantees. It measures the actual, running system. ET is universally applicable to arbitrary architectures and requires knowledge about the input specification. The SA approach yields conservative estimates that safely approximate the actual execution times. It requires knowledge about loop frequencies and information about the cycle-level behavior of the actual hardware. New hardware features have to be implemented in the simulator, which limits the portability of SA. If hardware details are not known, only the ET approach can be applied. If the hardware is not available yet but the specification of the hardware has been supplied, only the SA approach will yield results. We regard the two methods as complementary ap- proaches. Whenever deadlines have to be guaranteed SA should be used to yield safe estimates of the WCET. ET may be used additionally to bound the extreme execution times more precisely. Furthermore, ET may suffice if missed deadlines can be tolerated sporadically. The WCET for schedulability analysis should also be derived from SA for hard real-time environments where soft real-time environments may choose between SA and ET, or even the mean between SA and ET. In general, every real-time system should be tested for its logical as well as temporal correct- ness. Independent of the methods used during system de- sign, we recommend to apply ET to validate the temporal correctness of systems. The confidence in the application is increased since ET checks for timing violations over many input configurations. 7. Conclusion and Future Work This work introduced two methods to verify timing constraints of actual real-time applications, namely the method of static analysis and the method of evolutionary testing. Both methods were implemented and evaluated for a number of test programs with respect to their prediction of the worst-case and best-case execution times. The results show that the methods are complimentary in the sense that they bound the actual extreme execution times from opposite ends. For most of the investigated programs the actual execution times for the best and worst cases could be guaranteed with fairly high precision. They are within \Sigma10% of the mean between the results of both methods relative to the possible execution times determined by static analysis. Less precise results were obtained for few experiments indicating that further improvements for both approaches are necessary to ensure their general applicability. Current work on static analysis includes extensions to handle loop dependencies and integrate the Gnat/Gcc compiler. Current work on evolutionary testing focuses on the development of robust algorithms that reduce the probability of getting caught in local optima. Furthermore, suitable stopping criteria to terminate the test are to be defined. If the program code is available the degree of coverage achieved during evolutionary testing and the observation of the program paths executed could be an interesting aspect for deciding when to stop the test. The most promising criteria seem to be branch and path coverage because of the strong correlation between the program's control flow, the execution of its statements and the resulting execution times. The coverage reached will also be used to assess the test quality when comparing evolutionary testing with systematic functional testing another area where we want to intensify our research in the future, in order to estimate thoroughly the efficiency of different testing approaches for the examination of real-time systems' temporal behavior. In comparison, evolutionary testing should be more portable but requires extensive experimentation over many program executions. Static analysis has a lower overhead for the simulation process but requires detailed information of hardware characteristics and extensions to the simulation models for new architectural features. We recommend that the worst-case execution time for schedulability analysis be derived from static analysis for hard real-time environments where soft real-time environments may choose between static analysis and evolutionary testing. Furthermore, we suggest that evolutionary testing be used to increase the confidence in the temporal correctness of the actual, running system. --R Bounding worst-case instruction cache performance Reducing bias and inefficiency in the selection algorithm. A portable global optimizer and linker. Testing timing behavior of real-time software Adding instruction cache effect to schedulability analysis of preemptive real-time systems Testing real-time constraints in a process algebraic setting Timing constraints of real-time systems: Constructs for expressing them Testing large Applying compiler techiniques to cache behavior prediction. Integrating the timing analysis of pipelining and instruction caching. Efficient worst case timing analysis of data caching. Analysis of cache-related preemption delay in fixed-priority preemptive scheduling Cache modeling for real-time software: Beyond direct mapped instruction caches Functional test case generation for real-time systems Static Cache Simulation and its Applications. Timing predictions for multi-level caches The Art of Software Testing. Predicting program execution times by analyzing static and dynamic program paths. Genetic and evolutionary algorithm toolbox for use with matlab - documentation Calculating the maximum execution time of real-time programs Free Software Foundation The Automatic Generation of Software Test Data Using Genetic Algorithms. Testing real-time systems using genetic algorithms --TR --CTR Sibin Mohan, Worst-case execution time analysis of security policies for deeply embedded real-time systems, ACM SIGBED Review, v.5 n.1, p.1-2, January 2008 Mark Harman , Joachim Wegener, Getting Results from Search-Based Approaches to Software Engineering, Proceedings of the 26th International Conference on Software Engineering, p.728-729, May 23-28, 2004 Kiran Seth , Aravindh Anantaraman , Frank Mueller , Eric Rotenberg, FAST: Frequency-aware static timing analysis, ACM Transactions on Embedded Computing Systems (TECS), v.5 n.1, p.200-224, February 2006 Kaustubh Patil , Kiran Seth , Frank Mueller, Compositional static instruction cache simulation, ACM SIGPLAN Notices, v.39 n.7, July 2004 John Regehr, Random testing of interrupt-driven software, Proceedings of the 5th ACM international conference on Embedded software, September 18-22, 2005, Jersey City, NJ, USA Ajay Dudani , Frank Mueller , Yifan Zhu, Energy-conserving feedback EDF scheduling for embedded systems with real-time constraints, ACM SIGPLAN Notices, v.37 n.7, July 2002 Aravindh Anantaraman , Kiran Seth , Kaustubh Patil , Eric Rotenberg , Frank Mueller, Virtual simple architecture (VISA): exceeding the complexity limit in safe real-time systems, ACM SIGARCH Computer Architecture News, v.31 n.2, May Yifan Zhu , Frank Mueller, Feedback EDF Scheduling of Real-Time Tasks Exploiting Dynamic Voltage Scaling, Real-Time Systems, v.31 n.1-3, p.33-63, December 2005 Andr Baresel , David Binkley , Mark Harman , Bogdan Korel, Evolutionary testing in the presence of loop-assigned flags: a testability transformation approach, ACM SIGSOFT Software Engineering Notes, v.29 n.4, July 2004 Dennis Brylow , Jens Palsberg, Deadline Analysis of Interrupt-Driven Software, IEEE Transactions on Software Engineering, v.30 n.10, p.634-655, October 2004

evolutionary testing;real-time systems;timing analysis;testing;genetic algorithms;static timing analysis

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Data Squashing by Empirical Likelihood.

Data squashing was introduced by W. DuMouchel, C. Volinsky, T. Johnson, C. Cortes, and D. Pregibon, in Proceedings of the 5th International Conference on KDD (1999). The idea is to scale data sets down to smaller representative samples instead of scaling up algorithms to very large data sets. They report success in learning model coefficients on squashed data. This paper presents a form of data squashing based on empirical likelihood. This method reweights a random sample of data to match certain expected values to the population. The computation required is a relatively easy convex optimization. There is also a theoretical basis to predict when it will and won't produce large gains. In a credit scoring example, empirical likelihood weighting also accelerates the rate at which coefficients are learned. We also investigate the extent to which these benefits translate into improved accuracy, and consider reweighting in conjunction with boosted decision trees.

Introduction A staple problem in data mining is the construction of classication rules from data. Some data warehouses are so large, that it becomes impractical to train a classication rule using all available data. Instead a sample of the available data may be selected for training. For instance, the Enterprise Miner from the SAS Institute features the SEMMA process, an acronym in which the leading \S" stands for \sample". DuMouchel et al. (1999) introduce \data squashing" to improve upon sampling. Instead of scaling up algorithms to large data sets, one scales down the data to suit existing algorithms. And instead of relatively passive sampling from a large data set, they construct a data set in a way that should make it suitable for training algorithms on. Suppose that the original data consist of N pairs (X Here X i is a vector of predictor variables and Y i is a variable to be predicted from X i . In data squashing, one constructs a much smaller data set assigning weights w i . There is not necessarily any connection between points like x 1 and X 1 with the same index. Indeed a value like x 1 might not correspond to X i for any i. The idea is that training an algorithm on n weighted can be much faster than training on all N original data points. Large speed gains may be expected when the squashed data t in main memory. Here is an outline of this paper. Section 2 describes data squashing, presents a version using empirical likelihood weights, and points out connections between data squashing, numerical integration, and variance reduction techniques used in Monte Carlo simulation and survey sampling. Finding the empirical likelihood weights reduces to a very tractable convex optimization problem. Empirical likelihood squashing also has theoretical underpinnings that predict when it will and won't work, as outlined in Section 2. Section 3 describes a credit scoring problem. The data values in it have been simulated and distorted by obfuscating transformations, and the variable names and data source have been hidden for condentiality. But I am assured that it remains a good test case for algorithms. Section 4 applies logistic regression to small data samples, with and without empirical likelihood reweighting. The reweighting accelerates the rate at which coe-cients are learned. Section 5 replaces logistic regression with boosted decision trees. Section 6 presents our conclusions. We are less pleased with the results of squashing than are DuMouchel et al. (1999), though we describe the sort of problem where we expect squashing to add the most value. Our dierent conclusions could be due of dierences in the algorithms, dierences in the way the results are assessed, or simply because the data sets are dierent. We conclude this section with some more references. Madigan, Raghavan, DuMouchel, Nason, Posse & Ridgeway (2000) oer a likelihood based form of squashing, geared to exploit a user-specied statistical model. Bradley, Fayyad & Reina (1998) have goals similar to those of DuMouchel et al. (1999) and Madigan et al. (2000), but instead of representing the data by a weighted set of points, they employ mixture models. The elements in the mixtures include Gaussian distributions, multinomial distributions, and products thereof. Rowe (1983) describes some earlier work in this direction, but the more recent cited work is much more ambitious as bets the greater computational power available today. 2 Data Squashing We begin by outlining the data squashing method of DuMouchel et al. (1999). Then we cast some older methods in a new light, as special cases of data squashing. Our notation diers somewhat from the original. DuMouchel et al. (1999) do not distinguish predictor and response during squashing, deferring that distinction to the training stage. This allows the same squashed data set to be used for multiple prediction problems. They also choose weights w i so that so that the average weight is 1, that is training algorithms are not aected by this scaling, and in any case it is simple to alternate between these conventions. DuMouchel et al. (1999) choose (w outlined here. The rst step is to group the (X vectors into regions. They suggest several ways to construct regions. In the simplest method, the points (X region are those that share values for every discrete variable, and also share values for discretized versions of every continuous variable. For the points in each region, some low order moments of the non-categorical variables are computed. Then for each region, a set of points corresponding weights w i are chosen, so that the weighted moments on the squashed data match, or nearly match, the unweighted moments on the original data. For a function of the (X; Y ) pairs. A moment within a region corresponds to taking for g m a product of powers of non-categorical variables, multiplied by a function that is one inside that region and zero outside of it. Let Z weights would provide a perfect match, withn and Given enough moments and regions, ideal weights are not possible, and DuMouchel et al. (1999) minimize !m Zm instead. Here !m > 0 with larger values for the lower order moments. The value of (3) is minimized over w i , x i , and y i , for n. For a scalar valued variable, like a person's age, or the number of children in a household, the squashed data value need not match any of the sample values. But it is not allowed to go outside the range of the data. Thus the squashed data may have records with 2:2 children but should not have records with 3 children. 2.1 Sampling as squashing Some issues in data mining echo those of sampling. Two good references on sampling are Cochran (1977) and Lohr (1999). Simple random sampling can be cast as a trivial version of squashing. Let be a subset of n distinct a simple random sample (without replacement) of them. Take In stratied sampling, the population is partitioned into strata sample of n h values is taken from stratum h and the weight nN h =(Nn h ) makes (1) hold for functions g m that are indicators of the strata. The regression estimator is used in sampling theory to incorporate a known value of some population mean. Suppose that (1=N) Zm are known for . Then form the weights Z z)S 1 where z The regression weights satisfy (1) for . The regression estimator can be shown to subsume stratication by introducing indicator variables z m . Regression and stratication can also be combined in several ways. The regression estimator is also widely used in Monte Carlo simulation. There it is known as the method of control variates. Two general references are (Bratley, Fox & Schrage 1987, Ripley 1987). Hesterberg (1995) has a good presentation of the reweighting approach to control variates. 2.2 Empirical likelihood squashing The problem with regression weights is that they can take negative values these may be unusable in some training algorithms. If one insists that w i 0, then either there are no solutions to (1) and (2), or else there is an n M 1 dimensional family of solutions. When there are no solutions, one might either increase n or remove some of the moments from consideration. Suppose that there is an n M 1 dimensional family of solutions. It is natural to pick the one that is somehow closest to having equal weights. The empirical likelihood weights are those that maximize subject to Z. Owen (1990) describes how to compute these weights. It reduces to minimizing a convex function over a convex domain, which can be taken to be M dimensional Euclidean space. An Splus function available in http://www-stat.stanford.edu/owen computes the empirical likelihood weights. Empirical likelihood provides one way of picking the weights w i that are closest to equality. One can also use other distance measures, such as the Kullback-Liebler distance or the Hellinger distance (w 1=2 . Empirical likelihood weights have an advantage in that their computation is slightly simpler than the alternatives. Minimizing the Euclidean distance simpler still, but reduces to the regression weights (4) that may be negative (Owen 1991). 2.3 Benets of weighting Stratication, or more generally regression weighting, has the advantage of reducing the variance of associated estimators. Let h(X a function of the data cases. Let From a simple random sample, the estimate of H is h which has variance approximately 2 H =n, where The main error in this approximation is a multiplicative factor 1 n=N which we take to be virtually one for data squashing. When the eect of regression weighting is to reduce the variance to (1 2 H =n where is the correlation between h(X For M 1 the reduction factor is 1 R 2 where R 2 is proportion of variance of explained by a linear regression on Z shows that empirical likelihood reduces the asymptotic variance of estimated means by the same factor that regression estimators do. A training method that estimates means more accurately can often be shown to predict more accurately. In the simplest cases, like linear regres- sion, the prediction is constructed as a smooth function of some sample moments. In more complicated settings, like maximum likelihood estimation a parameter vector is dened by equationsN @ @ log and the estimate ^ solves the equationsn @ @ log f(x Qin & Lawless (1994) show that empirical likelihood weights produce variance reduction for compared to an unweighted estimate. The extent of the reduction depends on how well Z i are correlated with the derivatives being averaged in (5). Baggerly (1998) shows that the same reduction holds for other distance measures such as Kullback-Liebler and Hellinger. 2.4 Diminishing returns Better parameter estimates translate directly into better prediction rules, but generally there are diminishing returns. For example, consider a logistic regression in which for a parameter vector simplicity that the logistic regression is in fact accurate, so that knowing means knowing the Bayes rule. In ordinary sampling the estimate ^ approaches with an error of order n 1=2 . For weighted misclassication losses, the loss using is then typically O(n 1 ) above the Bayes loss (Wol, Stork & Owen 1996). The reason is that at the Bayes rule the derivative with respect to of the expected misclassication is zero. We expect an error of approximate form B if no squashing is used, and of the approximate form B or empirical likelihood squashing is used, with a xed list of M functions. Generally we can expect that A 0 A, but if the Bayes error B 0 dominates the estimation error An 1 , then regression or empirical likelihood squashing will bring only a small benet. If the logistic model fails to hold, then instead of taking B 0 to be the Bayes error, take it to be the best error rate available within the logistic family. The squashing method of DuMouchel et al. (1999) adjusts more than the weights. It also estimates new values x i and y i . Since these are not sampled, we cannot quote results like those for empirical likelihood squashing. But we can suppose that searching for x i , y i and w i should have the eect of matching (or approximately matching) many more than M functions for the same value of n. This is similar to the way in which a Gauss quadrature rule that adjusts both the location and weights in numerical integration (Davis & Rabinowitz 1984), integrates higher order polynomials than one that only adjust the weights or the locations. By matching more function values, it should be possible to come closer to the Bayes error rate, but not of course to reduce the Bayes error rate. Thus we could reasonably expect an error of the form Thus we expect squashing, in its various forms, to be eective in cases where the Bayes error is dominated by sampling or approximation errors. In particular, settings with a zero Bayes error may benet enormously from squashing. 2.5 When to expect benets It is reasonable to expect better model coe-cients from squashing, albeit with eventually diminishing gains in prediction accuracy. In order to realize gains in the coe-cients, they must be related to quantities that are correlated with the values of g m (X precisely, if the vector @ log f(X well approximated by a linear combination of g m (X then we can expect an improved estimate of . It helps to distinguish between local and global features of the data. A logistic regression uses global features of the data. It is reasonable to expect that these features could be highly correlated with judiciously chosen global features A nearest neighbor method uses local features, such as averages of Y i over small regions (determined by X values). It is not reasonable to expect one of these local averages to be correlated with global features of the data. Therefore squashing with global features g m will not help nearest neighbors much, and for an improvement, one must consider ways to employ a large number of local functions g m . A method like a classication tree would seem a priori to be intermediate. The rst split is a global feature of the data. The nal splits made, at least in a large tree, are very local features. Thus squashing with global g m should help on the rst splits but not the later ones. 3 Example data This data set inspired by a real commercial problem, but the problem has been disguised (from me) in order to preserve condentiality. The training data have 92000 rows and 46 columns. The data arise from a credit scoring problem, but their source is not known, and the data set has been transformed and obfuscated, as described below. Each row of data describes one credit case, and the rows are presented in random order. Each column contains one variable. The response variable is in column 41, and is a 0 or 1 describing bad and good credit outcomes respectively. It may have been possible to attribute a dollar value to each bad or good outcome, but such dollar values were not in the data I received, and indeed may not have existed. The data have roughly 85% good cases although this is not necessarily the percentage good in the population. Variables 2 through 40 and 42 through are predictor variables describing the credit history of the case. The original data values have been transformed. The original values for a given predictor were put into a vector v of 92000 elements. The transformed values are z = [(v min(v))=(max(v) min(v))] p where the power p was chosen at random, independently for each predictor variable. Missing values remained missing after transformation, and did not contribute to the min(v) and max(v). Column 1 is a score variable used to predict the response. It was constructed with the knowledge of what all the input variables mean. An unknown and possibly proprietary algorithm was used to generate this column. This custom-built score serves as a benchmark against which to compare the performance of training methods. Missing values in the original data were stored as 9999:0. The missing values are interpreted as \not available" or \not believed". There are 309262 missing values, about 8:5% of the predictor values. Column 19 was almost 97% missing. Dropping that column and 5 other columns that were more than 10% missing, left the data with 78165 missing values. Values were imputed for the other missing entries as described below. The result is 38 remaining predictors. Prior to building prediction models with this data, a transformation was applied to each column of predictor values. The non-missing values were replaced by X ij raised to a power p 0 chosen from among the values 10g. The value p 0 was chosen to maximize a normalized separation of means, means and variances over pairs (X with nonmissing X ij , and n yj is the number of Y ij is not missing. Each missing value X p ij was simply replaced by an imputed value, 0))=2. The idea was to replace the missing values by ones that were as neutral as possible regarding the classication at hand. There were 24430 observations with one or more imputed values. 4 Logistic regression The rst classication method to be applied was simple logistic regression. The training data contain cases in randomized order. Therefore a simple random sample is obtained by taking x was t to the rst n cases for 1000; 2000; 4000; 8000, and 92000. Both weighted and unweighted logistic regressions were run. For all the weights are 1:0, making the weighted and unweighted analyses identical. The weights were chosen so that for each and each y 2 f0; 1g, the weighted mean of those x ij with y matched the unweighted mean of those X ij with Y y. The reason for such a choice is as follows. Some simple global classiers are based solely on response group conditional means, variances and covariances of predictors, so it is reasonable to expect these conditional means to carry some relevant information. There are too many predictor variables to allow use of all of the conditional second moments The conditional moments can be matched by imposing equation (1) with taking The weights for are shown in Figure 1. The smallest weight is 0:35 and the largest is 3:25. As n increases the weights become more nearly equal to one. For it was not possible to reweight the data to match the conditional moments, using only positive weights. This is why is the smallest sample size we use. Figure shows how the Euclidean distance between estimated coe-cient vectors and the full data coe-cient vector decreases as n increases. The decrease is faster for the empirical likelihood weighted estimates. In terms of accuracy in estimating coe-cients, empirical likelihood weighting increases the eective sample size by roughly 4. Figure 1: Shown are the empirical likelihood weights for the credit scoring data for Increased accuracy in coe-cient estimation leads to increased accuracy in classication, but with diminishing returns. Figure 3 shows receiver operating characteristic (ROC) curves for several classiers, described below, on this data. An ROC curve can be plotted for any classier that produces a score function (X) from the predictors. The interpretation of the score Figure 2: Shown are the distances between the logistic regression coe-cients for all 92; 000 sample points and those based on subsamples. The lower line is for weighted logistic regressions using empirical likelihood weights. function is that larger values of (X) make likely. A point is classied as only if (X) > 0 , where the threshold 0 is chosen to trade o the error rates of false positive and false negative predictions. The ROC curve plots the proportion of the good versus the proportion of the bad As 0 decreases from 1 to 1 the ROC curve arcs from (0; 0) to (1; 1). The top ROC curve in Figure 3 corresponds to the customized score vector supplied with the data. The other solid lines correspond to empirical likelihood weighted logistic regressions on n points for 8000, 92000. These lines increase with increasing n. The dashed lines correspond to unweighted logistic regression for 92000, the weighted and unweighted ROC curves are the same. There is a reference point at (0:2; 0:8). This describes a hypothetical classication in which the rule accepts 80% of the good cases and only 20% of the bad ones. The custom rule is nearly this good. ROC curves tend to make performance dierences among classiers look very small. Part of the reason is that the underlying probabilities are plotted over ranges from 0% to 100%, while important distinctions among real clas- siers can be much smaller than this. For example the dierence between 75% and 80% acceptance of good cases, while small on a plot like this, is likely to be of practical importance. Despite this, it is clear that there are diminishing returns as n increases, whether weighted or unweighted. Logistic regression on 8000 cases produces an ROC curve that essentially overlaps the logistic regression on all 92000 Figure 3: Shown are ROC curves for logistic regressions and a proprietary score. The percent of good cases classied as good is plotted against the percent of bad cases classied as good. For example, the point at (0:2; 0:8) describes an unrealized setting in which 80% of good cases would be accepted, along with only 20% of bad cases. The solid curves, from top to bottom are for: a proprietary score, empirical likelihood weighted logistic regression on samples of sizes 92000, 8000, 4000, 2000, and 1000. The dashed curves, from top to bottom are for unweighted logistic regression on 8000, 4000, 2000, and 1000 cases. The curves overlap signicantly, as described in the text. cases. Empirical likelihood weighting produces such overlap at a smaller sample, perhaps Although the coe-cients keep getting better, performance tends to converge to a limit. It is reasonable to expect that better squashing techniques would get logistic regressions as good as the full data logistic regression at even smaller sample sizes than empirical likelihood weighted logistic regression does. The ROC curves in Figure 3 are computed on N points including the points used for training. But, there is little risk of overtting here. The sample sizes n are all either very large compared to 39 or small compared to (or both). As evidence that these logistic regressions do not overt, notice that logistic regression on all 92000 cases has not produced an ROC curve much better than one on 4000 cases. 5 Boosted Trees Logistic regression is by now a fairly old classication technique. More modern classication methods can also make use of observation weights. We also considered boosted classication trees. Boosted classication trees make predictions by combining a very large number of typically small classication trees. In the extreme, the individual trees have only one split. Taking a weighted sum of such stumps produces an additive model. Friedman (1999a) and Friedman (1999b) describe Multiple Additive Regression Tree, or MART, modeling for constructing boosted tree classiers. This builds on earlier work by Friedman, Hastie & Tibshirani (1999) which built in turn on Freund & Schapire (1996). ROC curves were obtained for MART using samples of size 1000, 2000, 4000, 8000 and 92000, using both empirical likelihood weighted and un-weighted analyses. When plotted, these ROC curves tend to be very hard to distinguish from each other as well as from those of logistic regression and the customized method. As in Figure 3 the curves separate the most visually, over the interval between 0:1 and 0:2 on the horizontal axis. Over that range they are roughly parallel with some crossings among close curves. Custom 0.217 0.479 0.651 0.792 0.9448 0.99217 0.99761 0.99895 0.999793 Mart 0.190 0.485 0.634 0.774 0.9433 0.99172 0.99733 0.99891 0.999871 Logistic 0.163 0.431 0.604 0.754 0.9244 0.99026 0.99720 0.99881 0.999858 Mart 4 0.188 0.456 0.626 0.770 0.9361 0.99150 0.99689 0.99877 0.999832 Mart 8 0.189 0.477 0.636 0.774 0.9419 0.99147 0.99707 0.99889 0.999871 Mart 1w 0.143 0.430 0.585 0.745 0.9238 0.98980 0.99656 0.99844 0.999651 Mart 2w 0.178 0.432 0.598 0.750 0.9274 0.98830 0.99571 0.99829 0.999625 Mart 4w 0.170 0.431 0.599 0.753 0.9326 0.98950 0.99624 0.99846 0.999754 Mart 8w 0.183 0.477 0.633 0.775 0.9435 0.99163 0.99720 0.99885 0.999819 Table 1: ROC values for boosted trees. Shown are the heights of 11 ROC curves, corresponding to 11 methods as described in the text. The ROC curves are evaluated at horizontal values given in the top row. Table show numerical values from these ROC curves. Values smaller than 0:5 are given to 3 signicant places, while values close to 1 are given so that their dierence from 1 may be computed to 3 signicant places. In the region over (0:10; 0:20) both weighted and unweighed MART models tend to do better on larger sample sizes. The use of weights sometimes helps and sometimes hurts, but does not seem to make much dierence. MART models respond to both global and local features of the data. We anticipated that weighting might help the global portion but not the local one. It does not appear that weights greatly accelerate MART. We also investigated boosted trees using an evaluation copy of Mineset. We were unable to obtain results better than logistic regression for this data, and there did not appear to be any benet to using empirical likelihood weights, even when boosting stumps (which are global in nature). 6 Discussion The results for empirical likelihood based data squashing are not as encouraging as those in the original paper by DuMouchel et al. (1999). Here we outline the dierences, and then describe where more positive results might be expected. First, they based their comparisons primarily on the quality of estimated logistic regression coe-cients. Like them, we get good results for coe- cients, but nd diminishing returns for classication performance. They also compare predicted probabilities from squashed models to predicted probabilities from the full data set. Such probabilities are deterministic functions of coe-cients and so they won't show diminishing returns the way that mis- classication rates do. A second dierence is that we report results on some local methods in addition to global ones, and found little benet there. This is an area where more ambitious squashing as described in DuMouchel et al. (1999) might be able to make a big improvement. Thirdly, it is reasonable to expect that optimistic results are entirely appropriate on one data set and not on another. More data sets will need to be investigated. Their data set had only 7 predictors while we used 38. As a consequence they were able to look at interactions, where we did not consider our sample sizes large enough for that. Nor is it reasonable to match all interaction moments in our case. Both data sets were of comparable total size, because they had 744963 records compared to our 92000. We should point out that the original motivation for squashing is speed, although much of this article stresses accuracy. The reason is that essentially the same speed gains can be achieved by sampling. So for squashing to represent a gain over sampling, it should be more accurate for the same n. The diminishing returns suggest that for some small n squashing could be much better than sampling, but for larger n the practical value will disappear. This suggests that squashing will be most useful on problems where even when one lls computer memory with data, one is undersampling. Here are some settings which maximize the promise of squashing. First, problems with near zero Bayes error might benet more from squashing. Secondly, while in classication one only needs to compute a score on the right side of a threshold, in other problems one must predict a numerical value (e.g. prot versus protable). Here the diminishing returns might set in much later. Third, when the records have only 7 or 38 predictors a very large n will t in memory. But when the records have many thousands or millions of predictors, much smaller values of n will t in memory and there could be more to gain from some form of squashing. Finally, the squashing described in DuMouchel et al. (1999) might serve as a good data obfuscation device. An organization could release a squashed training data set and a squashed test set for researchers to evaluate learning methods, without ever releasing a single condential data record. Acknowledgements I thank Bruce Hoadley for valuable discussions on data mining and Jerome Friedman for making available an early version of his MART code. This work was supported by NSF grants DMS-9704495 and DMS-0072445. --R Scaling clustering algorithms to large databases A Guide to Simulation (Second Edition) Methods of Numerical Integration (2nd Squashing at Additive logistic regression: a statistical view of boosting Stochastic Simulation --TR

reweighting;misclassification loss;MART;database abstraction;credit scoring

608041

Characterization of E-Commerce Traffic.

The World Wide Web has achieved immense popularity in the business world. It is thus essential to characterize the traffic behavior at these sites, a study that will facilitate the design and development of high-performance, reliable e-commerce servers. This paper makes an effort in this direction. Aggregated traffic arriving at a Business-to-Business (B2B) and a Business-to-Consumer (B2C) e-commerce site was collected and analyzed. High degree of self-similarity was found in the traffic (higher than that observed in general Web-environment). Heavy-tailed behavior of transfer times was established at both the sites. Traditionally this behavior has been attributed to the distribution of transfer sizes, which was not the case in B2C space. This implies that the heavy-tailed transfer times are actually caused by the behavior of back-end service time. In B2B space, transfer-sizes were found to be heavy-tailed. A detailed study of the traffic and load at the back-end servers was also conducted and the inferences are included in this paper.

Introduction The explosive popularity of Internet has propelled its usage in several commercial avenues. E-commerce, the usage of Internet for buying and selling products, has found a major presence in today's economy. E-commerce sites provide up-to-date information and services about products to users and other businesses. Services ranging from personalized shopping to automated interaction between corporations are provided by these web-sites. It has been reported that e-commerce sites generated $132 Billion in 2000, more than double of the $58 Billion reported in 1999 [1]. Even though the power of the servers hosting e-commerce sites has been increasing, e-commerce sites have been unable to improve their level of service provided to the users. It has been reported that around $420 Million has been lost in revenues due to slow processing of the transactions in 1999. Thus it is desirable and necessary to focus on the performance of the servers used in these environments. There are two main classes of e-commerce sites, Business-to-Business (B2B) and Business-to-Consumer (B2C), providing services to corporations and individual users respectively. Web sites like Delphi, which provide services to corporations like General Motors come under B2B sites, whereas sites like Amazon.com providing services to general users come under B2C sites. This work was supported in part by the National Science Foundation. In this paper, we have analyzed the characteristics of e-commerce traffic. Traffic from a B2C and a B2B site is being used for the study. The workload is initially inspected for understanding the diurnal nature of the traffic. Different load periods were identified for both the B2C and B2B en- vironments. These have been found to be complimentary in nature, which may be intuitive. A set of parameters were chosen for each site for each component which would impact the performance of the system to the maximum extent. Statistical tests are then used to prove the self-similar nature of the traffic at different scales. Two different tests are used for validating the results for each of the parameters. It has been observed that the arrival traffic is highly bursty in nature, much more than the burstiness seen in normal web-traffic [5]. The response-time distribution is found to be heavy-tailed. This has been previously attributed to the heavy-tailed nature of request and response file-sizes. But the behavior of transfer sizes is not heavy-tailed, unlike the general web-environment. The traffic arriving at the back-end servers is characterized to obtain similar statistics about the impact of burstiness on the system. Also preliminary tests have shown that the back-end utilization is more bursty than the front-end server utilization, the reasons for which are explained later. A correlation is drawn between the behavior of the front-end and the back-end servers under different load conditions. Performance implications from the results of the above experiments will give valuable information for improving e-commerce server performance. The workload characterization studied in this paper is based on one representative system from each of the environments (B2C and B2B). Considering the difficulty in obtaining this valuable and guarded information from the e-commerce sites, and the fact the sites we have considered are quite busy, the results, although preliminary, could be valuable for future studies on e-commerce workload characterization and server designs. The rest of the paper is organized as follows. Related work is outlined in Section 2. Section 3 discusses the architecture of e-commerce sites along with a description of the configuration of the sites used for this study. Section 4 discusses the behavior of the workload and the traffic and load characteristics of the front-end and back-end servers. The concluding remarks are sketched in Section 5. Note: For this study data from two popular sites (one B2C and the other B2B) was used. Due to a non-disclosure agreement (NDA), the identity of these sites is not revealed. Throughout this work the two sites are identified as B2C site and B2B site. Without the NDA, we would not have been able to acquire the data for the study. Related Work Although there have been several studies reported on the workload characetrization of general web servers [3, 7, 10, 13], only a few studies have been reported on the characterization of e-commerce traffic based on the client behavior. The main reason for this shortcoming is the unavailability of representative data. E-commerce sites have highly secure information in the traces and access logs. Due to the security implications e-commerce sites are reluctant to divulge this information for research purposes. Due to this, studies in this field are still in the preliminary stages. In [8] the authors have developed a resource utilization model for a server which represents the behavior of groups of users based on their usage of the site. It should also be noted that the existing work reported on e-commerce traffic has been done on the front-end servers only and to the best of our knowledge nothing has been reported on the back-end servers. The back-end servers are the ones which experience the maximum load in an e-commerce environment [4]. We would like to characterize the load on the back-end servers along with a study of the system characteristics collected from system logs in E-commerce sites. 3 E-Commerce Architecture A generic organization of e-commerce sites is depicted in Figure 1. E-commerce sites can be broadly classified into two different categories. Business to Business (B2B) and Business to Consumer (B2C). The main difference between the two categories of sites lies in the user population accessing these sites. B2B sites serve transactions between different businesses whereas B2C sites serve general users over the Internet. INTERNET USERS Edge router/ ISP Provider Cache m/cs Load Balancer Web Servers Firewall Database Servers Front-End Back-End Figure 1: A Generic E-Commerce Site. Business-to-Business: One of the main characteristics of this category of sites is the regularity in the arrival traffic. It was observed that heavy traffic comes between 9am to 5pm, normal business hours. Regularity does not imply the lack of heavy spikes in the traffic. There will be sustained peak load on the system either due to seasonal effects or due to the availability of different services at the site. These sites can be categorized by the high amount of buying taking place in them. It has been observed that the percentage of transactions resulting in buying are very high compared to those in B2C environment. Business-to-Consumer: B2C servers are the normal e-commerce sites where any user can get service. The security involved in B2C site is only restricted to any financial transactions involved, whereas in a B2B environment all the transactions are normally done in secure mode. One implication of this is that increased buying in a B2C environment can throttle the system since the designed system does not expect high percentage of buy transactions. Another important characteristic of a B2C site is the very low tolerance to delayed responses. This increases the need to make QoS more important than providing absolute security for all the transactions, hence security is reserved for transactions involving buying. 3.1 Front-End and Back-End Servers Typically the front-end servers are comprised of the web server, application server, server load balancer, and the secure socket layer (SSL) off-loader. Front-end web servers serve requests from the clients and are the only authorized hosts able to access the back-end database and application services as necessary. The application servers are responsible for the business logic services. The application server will be the most heavily loaded server in the B2C envi- ronment. This is due to the heavy traffic of dynamic and secure requests arriving at the server. In a large scale e-commerce site, there will be dedicated application servers, alternatively these servers can be combined with the Web Servers or the Database servers. Due to the heavy traffic seen by e-commerce servers and also due to the availability requirements, there will be a network of web servers instead of a single monolithic server at the front-end. This basically improves the scalability and fault-tolerance of the server to any bursts of busy traffic. Load balancers help increase the scalability of an e-commerce site. Load balancing works by distributing user requests among a group of servers that appear as single virtual server to the end user. SSL is a user authentication protocol developed by Netscape using RSA Data Security's encryption technology. Many commerce transaction-oriented web sites that request credit card or personal information use SSL. The SSL off-loader typically decrypts all https requests arriving at the server. The back-end servers mainly comprise of the database servers and the firewall which would protect sensitive data from being accessed by unauthorized clients. These firewalls provide security services through connection con- trol. They are predominantly used when protecting mission-critical or sensitive data is of utmost importance. The database servers reside in the back-end of the network and house the data for e-commerce transactions as well as sensitive customer information. This is commonly referred to as the data services. The clients do not directly connect to these servers, the front-end Web servers initiate connections to these servers when a client conducts a series of actions such as logging in, checking inventory, or placing an order. Most e-commerce sites scale up their database servers for scalability and implement fail-over clustering for high avail- ability. Partitioned databases, where segments of data are stored on separate database servers, are also used to enhance scalability and high availability in a scale-out fashion. 3.2 B2C Configuration A simplified configuration of the B2C site being used for the study is given in Figure 2. This site comprises of ten web servers, each one powered by a Intel Quad P-III systems with a 512MB of RAM. The web servers run IIS 4.0 HTTP server. This cluster of web servers is supported by three image servers, each one powered by a Dual P-II sys- tem. As can be seen from the figure, the image servers serve both the database servers and the front-end web servers. For the purpose of our study, the image servers were considered to be in the back-end system. The product catalog server, connected to both the front-end and the back-end, runs an NT 4.0 providing backup and SMTP services to the back-end servers. The LDAP server is connected to the back-end. Figure 2: Simplified configuration of the B2C site. 3.3 B2B Configuration In the B2B space, the design of e-commerce sites is completely different from their design in B2C space. Here the user population is known a-priori. The transactions being processed by each user arriving at the server is also known with reasonable bounds. B2B sites serve a limited population as opposed to B2C sites which aim at serving the entire Internet. These aspects enable the designers to customize the site to specific user requirements. Scalability is one of the main issue that has to be taken care of when designing such customized system. So the design is done as a cluster of B2C sites, interconnected to form a large B2B portal. The interconnections between the individual B2C components in the site determine the user population to that site and also the services provided by that site. Figure 3 shows a simplified version of the B2B site being used for the study. Each of the web servers, can be individually used as a B2C site with its own database and network connection. Figure 3: Simplified configuration of the B2B site. 4 Workload Characterization In this study we have analyzed the behavior of e-commerce servers with relation to the behavior of the incoming traf- fic. Data was collected at different levels in the system. Web Server access logs from the the front-end and the back-end servers were collected at a granularity of 1 sec. This is an application level data giving the load on the httpd. This data will give the characteristics of the traffic arriving at the system, average network bandwidth utilization, and the file transfer rate. For the system level information, data was collected from the Performance logs from all the servers present in the site. This data was collected at a granularity of 5 sec. This would give information about the I/O bandwidth used, the processor and disk utilization of the system etc. Data was collected at a constant rate of 5 sec intervals. So this data is at a higher scale than the logs from the web servers. But both the scales are below the non-stationarity time scale used for the analysis. Data was collected at the server and the performance monitor for an entire day. A weekday is used for data collection since this would represent the average traffic. Addition- ally, data for a five day period was used to study the average behavior of the traffic over a long period of time. 4.1 Characteristics of the Workload The main differences between general web and e-commerce workload are the following. 1. Presence of a high level of Online Transaction Processing (OLTP) activity is observed among the transactions at the server. This is due to the database transactions accruing for every request from the user. Due to security reasons most of the data is present in the database server which is protected by a secure firewall. This prevents the web server from responding to most of the requests without sending a query to the back-end server. 2. A large proportion of requests come in secure mode. B2C traffic has lesser secure traffic, B2B sites experience almost complete secure traffic from users. This is due to the heavy security constraints present in industry to industry transactions. Increased amount of Arrival Rate (Reqs/sec) Time Figure 4: Arrival Process at B2C site.51525 Arrival Rate (Req/sec) Time (Bucket 6 secs) Figure 5: Arrival process at B2B site. secure transactions implies heavy processing load at the front-end server. Most of the sites have SSL off- loaders, which do encryption/decryption of requests to reduce the load on the system. This process adds to the response time. Aggregating these transactions with normal transactions increases the variability in the response times observed by the user. 3. The proportion of dynamic requests (that require some amount of processing) is very high, as was expected. In fact, in most e-commerce sites almost all requests are handled as dynamic requests. 4.2 Front-End Characterization A visual inspection reveals the workload at e-commerce sites to be more bursty than normal web workload. To study this behavior, the following parameters were used, which would have the maximum impact on the behavior of the traf- fic: Arrival process, Utilization of the server, Response time, Request file sizes, and Response file sizes. 4.2.1 Arrival Process Figures 4 and 5 show the arrival process at the B2C and e-commerce sites for an entire day (12am-12am). The data shows traffic on a normal weekday with an average arrival rate of 0.65 requests/sec at the front-end web server for the B2C site and around 1 req/sec arrival rate at one of the web servers in the B2B site. A visual inspection reveals the burstiness in the arrival process. The B2C server is a 4P system with an average processor utilization of 6% per processor and disk utilization of 2% during the period starting from 9.00am till 6.00pm. The low utilization is typical of e-commerce sites since they are designed for much higher load and sustain a very minimal load during normal working periods. It is the high load periods showing bursts of orders of magnitude more than normal operating parameters which cause concern for better capacity planning and performance analysis of these systems. Figures 4 and 5 show that the sites have distinct high and low load periods during the course of a day. For the B2C site, busy period starts around 6:00pm in the evening and ends at around 11:00pm in the night. Since this is a serving general consumers, the traffic is heavy during the after-office periods. Distinctive low periods during the morning between 7:30am to 11:30am can also be ob- served. In case of the B2B site, the traffic concentration lies mostly during normal office hours, between 9:00am to 8:00pm, which is intuitive. It should be noticed that the graphs show aggregated arrival traffic for the B2B site and the averaged arrival process for the B2C site. The Arby-Veitch (AV) [11] estimator test was used for estimating the Hurst-parameter (H-parameter) [6] for the arrival time-series. This is known to be a reliable test for workloads with busy periods showing a non-stationary behavior. Hurst parameter is also calculated using the R/S plot test [6]. Reliability of this test under low time-scales for e-commerce traffic is tested by comparing the H-parameters obtained using the two methods. The H-parameter is estimated to be 0.662 using the AV test. This shows that the arrival process at the B2C site is self-similar in nature. The Hurst parameter is also estimated to be 0.662 using the R/S plot test for the B2C site, which matches the estimation made by the AV-estimator. Similar test was done for the arrival traffic at the B2B site. Using the AV-estimator the H-parameter was estimated at 0.69, whereas the R/S plot gave an estimate of 0.70 for the H- parameter, which is a good approximation. 4.2.2 Processor Utilization Figures 6 and 7 show the utilization of the front-end web server for the B2C and B2B sites respectively. As explained earlier, this data is collected between 9:00am till 5:00pm at a granularity of 5 secs for the B2C site. For the B2B site the data represents the activity between 10:00 am in the morning till 9:30 am the next day morning. The B2C server sustains a constant load throughout the day, with an average load of 7% on each of the four processors. High and low load periods can be observed on the B2C server during the course of the day. This behavior is absent in the B2B server. This is due to the a-priori knowledge of the transactions and load from users in the B2B space. B2B sites are customized for specific traffic patterns and a normal traffic would not affect the load on the system to a higher degree. Thus the load on the system appears almost constant even though there is a variation in the arrival rate at the server. The time-series obtained from the utilization was also tested for self-similar behavior. The AV-wavelet based test and the R/S plot test are used for estimating the H-parameter. The estimated H- parameter is 0.755 using the AV estimator, and 0.77 using Utilization (4P) Time (secs) Figure Utilization of the front-end web-server at B2C site10305070 Utilization Time (buckets 5 secs) Figure 7: Processor(2P) Utilization of one of the front-end web-servers at B2B site the R/S plot test for the B2C site. In the B2B space, the load on the system did not have a high degree of self-similarity. The H-parameter is estimated to be 0.66 using both the AV- estimator and the R/S plot test. Due to a balanced load on the B2B system throughout the duration, the degree of self-similarity is very low. The effect of the arrival process is not seen in the overall load sustained by the B2B server. A higher H-parameter implies an increased degree of self- similarity. Utilization is a factor of the response-time and the arrival process. The inherent burstiness in the arrival process is already established in the previous section. The higher H value can be attributed to long-range dependence in the service process. 4.2.3 Response Time In Figure 8, the response time observed by the users over the entire day period is shown for the B2C site. Previous studies [5, 12] have concentrated on the study of the heavy-tailed behavior of web response times. In this work the response-time distribution is converted into a time-series by aggregating the response-times seen for non-overlapping intervals of 5 secs. Even though the times seen are not the actual response times observed by the user, they can be used for time-series analysis. Only a multiplicative factor of 1/5 will be required to get the actual response-times. The time-series obtained is checked for self-similarity and any non-stationary behavior. The AV test and R/S plot test are used for estimating the H-parameter (A-V test As explained earlier, a good estimation of2000006000001e+061.4e+060 2000 4000 6000 8000 10000 12000 14000 16000 18000 Aggregate Response Time Time (5 sec buckets) response time Figure 8: Aggregate Response time at the front-end Web-Server (4P), B2C site H-parameter is obtained using R/S test, only when the time-series is stationary. So both the tests are used for estimating the H-parameter. Response time is one of the very important performance metrics in the design and analysis of any server system. High burstiness in the arrival traffic implies saturating server queues, leading to high response times. Studies have shown that the 90th percentile response-times can be used for predicting the mean response-time [2]. This measure cannot be used in presence of high burstiness in the response-time distribution. Figure 8 shows response times orders of magnitude higher during the high load periods in the evening. Comparing this graph with the arrival process shown in Figure 4, unmistakable correlation can be found between the different load periods. Even though the utilization of the system does not get effected, buffer queue lengths increase thereby increasing the user perceived response times. Increased burstiness impacts the overall response time of the system to a higher extent than the arrival process. This burstiness in the response time is a factor of the back-end data retrieval time and the server processing time. 4.2.4 Request/Response File Sizes The request and response file sizes in web environment [5, 9] have been studied previously. It was observed that these distributions show a heavy-tailed behavior with a tail weight of approximately bytes [5]. This was considered one of the main reasons for the heavy-tailed behavior of the web response times. In e-commerce environment, it has already been shown that transfer times have a heavy tailed behavior with In this section the behavior of transfer size distribution is studied Figures show the request and response size distribution over the observation period at the B2C server. It can be observed that the distribution of transfer sizes is fairly constant in the B2C environment. A visual inspection rules out the possibility of heavy burstiness in the aggregated time-series obtained from the transfer sizes. The distribution of request sizes is further investigated for heavy-tailed behavior using Log-Log cumulative distribution plots (LLCD) plots [5]. Figure 11 shows the log-scale plot of the cumulative probability function over the different request sizes observed. The plot appears linear after x > 2:5. A linear-regression fit to the points for requests more than 320 Bytes Request Size (Bytes) Requests arrival Figure 9: Request size distribution over time, B2C site20000600001000001400001800000 10000 20000 30000 40000 50000 Response Size (Bytes) Time (secs) Figure 10: Response size distribution over time, B2C site gives a line with slope This gives an estimate of = 4.12 thereby indicating that the request size distribution is not heavy-tailed in nature. This result refutes the previous results about web traffic. In [5] the authors found that the requests also follow a heavy tail distribution with Using similar tests, we also infer that the response file sizes do not follow a heavy tailed distribution. 4.3 Performance Implications Previous studies on web traffic and LAN traffic have attributed the self-similar behavior of network traffic to the aggregation of long-range dependent ON/OFF processes. In e-commerce space, the response-times are found to be heavy-tailed in nature even though the request and response file sizes are almost a constant. The heavy-tailed behavior of response-times in web environment was believed to be caused by the heavy-tailed behavior of the file transfer sizes in the web environment. In e-commerce environment, the transfer sizes do not follow a heavy-tailed distribution as shown earlier in this section. Heavy-tailed behavior of web transfer sizes are fundamentally caused by the inclusion of image and video files in the overall traffic. Since these files are minimized in e-commerce environment (for reducing the overhead in response times), the behavior of the transfer sizes becomes somewhat intuitive. The lack of large image and video files removes the heavy-tailed nature of e-commerce traffic. It is observed that the response time still shows a heavy-tailed behavior in both B2C and B2B space. As explained earlier this implies that the user perceived response-time can increase by orders of magnitude under load conditions. Due Log10(Request file size) Figure 11: LLCD of request size distribution, B2C site to the critical nature of e-commerce applications and also the business model (increasing criticality with the increase in load), it is imperative that the response-times are kept under normal bounds even in high load conditions. In e-commerce environment response-time is mainly dependent on the processing time and the transfer time. Since the file-sizes do not follow a heavy-tailed distribution, it can be safely assumed that the transfer time does not contribute to the variation in the response-time. This shows that the characteristic of the processing time is affecting the response-time to a higher extent than the response size. Also the effect of file- sizes appears to be negligible on the end-end response-times observed. This result contradicts the behavior of response-times for normal web traffic where the response-size of files can be assumed as a good approximation of the response- time. The difference is that, in web environment the transfer times consumes most portion of the response-time which is not the case in e-commerce environment due to the different composition of requests. 4.4 Back-End Characterization The most important and sensitive information in e-commerce servers is kept in the back-end servers. It is the back-end servers that execute the business logic for the e-commerce site and are hence the most crucial components of any e-commerce server. The parameters used for doing the characterization depend mostly on the configuration of the site and the purpose of the individual components [13] in the back-end. The composition of back-end servers is closely dictated by the business model of the site. So different parameters might be interesting for different sites. In this study processor utilization and disk accesses are used for studying the characteristics of the two sites. In the B2C site there are four different servers at the back- end. These are: Main database server, Customer database server, Image server, and LDAP server. The image server and LDAP server are not heavily loaded during the observation period. There is a single burst of traffic to and from these servers when the data is updated daily. This burst is also seen in other back-end databases and will be discussed in detail later in this chapter. The only servers that experience a sustained load throughout the day are the customer database and the main database. These two servers are used for studying the characteristics of the back-end system. Processor Utilization (4P) Time (from 9:33:13) Figure 12: Processor utilization of Catalog Server (5 secs), Processor Utilization Time (from 9:33:13) Figure 13: Processor utilization of the Main D/B server (5 secs), B2B site 4.4.1 Processor Utilization In Figures 12 and 13 the processor utilization of the two back-end servers in the B2C site is shown. It can be observed that the back-end server experiences a sustained load of 10% on average over the entire period. There is a visible peak of almost 100% utilization of the catalog server. This will be discussed later in the section. For the Main D/B server, the utilization remains at around 30% for most of the observation period. This shows that the load on back-end servers is higher than on the front-end servers, when compared with figure 6. Previous studies have speculated that the load on the back-end servers is more regulated due to the presence of the front-end server. One of the reasons for this speculation is the service time of the front-end server. This either causes a delay or reduces peak of any burst reaching the back-end servers. This behavior of the back-end servers is investigated by looking at the time-series obtained from the utilization of the servers. H-parameter values of 0.87 and 0.77 were obtained for the utilization of the main database server and the catalog server respectively. The burstiness observed at the back-end servers is more than the front-end servers results have been observed in the B2B space also. The utilization of the database server of the B2B site is shown in Figure 14. It can be observed that the load on the system reaches 100% around the 4000th bucket. This is due to the update activity which takes place periodically in most e-commerce sites. The actual time when this takes place is around 1.00 pm in the night. Similar activity can be seen in the other back-end servers. Nothing can be observed at10305070900 1000 2000 3000 4000 5000 Utilization Time (bucket 1 sec) Figure 14: Processor Utilization of the B2B Database server1000300050000 1000 2000 3000 4000 5000 6000 File Operations sec Time (5 secs) Figure 15: File Operations per second from Main DB server (5sec) the front-end servers, as the bulk of the data which needs any maintenance is present in the back-end servers only. The back-end server in B2B space is also found to be more bursty than the front-end server. This contradicts previous assumptions about burstiness at the back-end servers in web environment. 4.4.2 Disk Accesses The B2C site has four disks for the Main D/B system. Disk are used for the study instead of disk utilization. Reliable data could not be obtained for the disk utilization due to the presence of a cluster of four disks. Figures 15, shows the distribution of the file request rate at the Main D/B server. This shows the arrival rate of file requests seen by the four hard disks. Figure 16 shows the1030500 1000 2000 3000 4000 5000 6000 Avg. Queue Length Time (5 secs) Figure Queue Length at Main DB server (5sec) average queue length seen by the hard disks at the Main D/B server. The average queue length is found to be self-similar in nature with This would result in a heavy-tailed behavior in the average response-time of the hard disk. The reason for the burstiness in the queue length can be attributed to the arrival of file transfers at the hard disk. This rate is also found to be bursty in nature with H 0.83. The buffer cache does not appear to be effective since the hard disk is experiencing requests at this level of burstiness. In the previous section, the response-time at the front-end is found to be heavy-tailed in nature even though the request and response size did not follow this distribution. The burstiness in the service time at the back-end was attributed to this behavior. Here it can be seen that the heavy-tailed distribution of response-time at the back-end is due to the bursty arrival process to the hard disks, causing the queue length to be bursty. This high burstiness in queue length will remove the effect file sizes may have on the transfer times. This conclusion also supports the previous speculation that file-sizes were not a good representation of response-times in e-commerce environment. 5 Conclusion Aggregated traffic arriving at B2B and B2C e-commerce servers is characterized in this paper. Access logs from the web servers is collected for application level information, Microsoft performance logs were collected for system level information and processor counters were collected for architectural information. Information from this data was used to understand the load behavior of the traffic for a normal weekday. Only a specific set of parameters (arrival pro- cess, utilization, response-time, transfer sizes etc.) which would impact the system to the maximum extent were used for characterization of the workload. Self-similar nature of the traffic was established using Hurst-parameter as a measure of degree of self-similarity. Two different tests were used for measuring the Hurst- parameter: AV-estimator and the R/S plot. It was observed that the load behavior of the two sites was complementary in nature with traffic load shifting from one type of e-commerce site to the other during the later part of the day. Unlike previous speculation, the back-end server was found more bursty than the front-end server, this was attributed to the fractal nature of the service time at the back-end. At both the sites, the response-times were found to be heavy-tailed in nature, complying to the results found in web environ- ment. But in the B2C environment, highly bursty arrival of file requests was seen at the disks. It was found that this arrival process is causing high queuing delays at the disk reducing the impact of disk transfer time as compared to the queuing time. This increased the burstiness in the overall response-time seen at the front-end server. This work provides an understanding of the complexity of the traffic arriving at e-commerce sites while providing a preliminary workload characterization. --R "Predicting the performance of an e-commerce server: Those mean percentiles," "An admission control scheme for predictable server response time for web ac- cesses," "Cisco and microsoft e-commerce framework architec- ture." "Self-similarity in world-wide traffic : Evidence and possible causes," Introduction to computer system performance eval- uation "Server capacity planning for web traffic workload," "Resource management policies for e-commerce servers," "Web server workload characterization: The search for invariants," "Generating representative web workloads for network and server performance evaluation," "A wavelet based joint estimator for the parameters of lrd," "On multimedia networks: Self-similar traffic and network performance," Capacity Planning for Web Performance: Metrics --TR --CTR Lance Titchkosky , Martin Arlitt , Carey Williamson, A performance comparison of dynamic Web technologies, ACM SIGMETRICS Performance Evaluation Review, v.31 n.3, p.2-11, December

business-to-consumers B2C;self-similarity and web-servers;bussines-to-business B2B;traffic characterization;e-commerce servers

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Verifying lossy channel systems has nonprimitive recursive complexity.

Lossy channel systems are systems of finite state automata that communicate via unreliable unbounded fifo channels. It is known that reachability, termination and a few other verification problems are decidable for these systems. In this article we show that these problems cannot be solved in primitive recursive time.

Introduction Channel systems, also called Finite State Communicating Machines, are systems of nite state automata that communicate via asynchronous unbounded fo channels [Boc78, BZ83]. Figure 1 displays an example, where the labels c!x and c?x mean that message x (a letter) is sent to (respectively read from) channel c. Channel systems are a natural model for asynchronous channel c 1 b a a b a channel c 2 a c Figure 1: A channel system with two automata and two channels communication protocols and constitute the semantical basis for ISO protocol specication languages such as SDL and Estelle. Channel systems are Turing powerful, and no verication method for them can be general and fully algorithmic. A few years ago, Abdulla and Jonsson identied lossy channel systems as a very interesting model: in lossy channel systems messages can be lost while they are in transit, without any notication. These systems are very close to the completely specied protocols independently introduced by Finkel, and for which he showed the decidability of termination. Abdulla and Jonsson showed that reachability, safety properties over traces, and eventuality properties over states are decidable for lossy channel systems. The decidability results of [Fin94, CFP96, AJ96b] are fundamental since lossy systems are the natural model for fault-tolerant protocols where the communication channels are not supposed to be reliable (see [AKP97, ABJ98, AAB99] for applications). For lossy channel systems, the aforementioned decidability results lead to algorithms whose termination rely on Higman's Lemma (see [A CJT00, FS01] for more examples of this phe- nomenon). No complexity bound is known and, e.g., Abdulla and Jonsson stated in [AJ96b] that they could not evaluate the cost of their algorithm. In this article we show that all the above-mentioned decidable problems have nonprimitive recursive complexity, i.e., cannot be solved by algorithms with running time bounded by a primitive recursive function of their input size. This puts these problems among the hardest decidable problems. Our proof relies on a simple construction showing how lossy channel systems can weakly compute some fast growing number-theoretic functions A related to Ackermann's function and their inverses A 1 \weakly computing f" we mean that, starting from x, all values between 0 and f(x) can be obtained. This notion was used by Rabin in his proof that equality of the reachability sets of Petri nets is undecidable, a proof based on the weak computability of multivariate polynomials (see [Hac76]). Petri nets can weakly compute the A n functions (see [MM81]) but they cannot weakly compute their inverses A 1 n as lossy channel systems can do. There exist other families of systems that can weakly compute both A n and A 1 lossy counter machines of [May00], or the reset nets of [DFS98, DJS99]. Our construction can easily be adapted to show that, for these systems too, decidable problems like termination, control-state reachability, . , are nonprimitive recursive. Finally, let us observe that there does not exist many uncontrived problems that have been shown decidable but not primitive recursive. In the eld of verication, we are only aware of one instance: the \nite equivalence problem for Petri nets" 1 introduced by Mayr and Meyer [MM81]. This problem is, given two Petri nets, to decide whether they both have the same set of reachable markings and this set is nite (equivalence is undecidable without the niteness assumption). It can be argued that the verication (termination or reachability) of lossy channel systems is a less contrived problem. Channel systems, from perfect to lossy A channel system usually combines several nite-state automata that communicate through several channels. Here, and without loss of generality, we assume our systems only have one 1 See [Jan01] for a more general proof that all nite equivalence problems for Petri nets are nonprimitive recursive. automaton that uses its several channels as fo buers. Formally, a channel system is a tuple is a nite set of control states, is a nite set of channels, is a nite alphabet of messages, and Q C f?; !g Q is a nite set of transition rules (see below). A conguration of S is a tuple denoting that control is currently in state q, while channels c 1 to c k contain words w (from ). The transition rules in state how S can move from a conguration to another. For- mally, S has a \perfect" step is some hq; w 0 is some and (3.1) there is a rule (q; c has been written to the tail of c i ) or (3.2) there is a rule (q; c has been read from the head of c i ). These steps are called perfect because no message is lost. It is well known that, assuming perfect steps, channel systems can faithfully simulate Turing machines in quadratic time [BZ83] (a single channel is enough to replace a Turing machine work tape; reading and writing in the middle of the channel requires rotating the content of the channel for positioning reasons, hence the quadratic overhead). Thus all interesting verication problems are undecidable for systems with perfect channels, even when restricted to single-channel systems. 2.1 Lossy systems The most elegant and convenient way to model lossy channel systems is to see them as channel systems with an altered notion of steps [AJ96b]. We write u v v when u is a subword of v, i.e. u can be obtained by deleting any number (including of letters from v. E.g. abba v abracadabra as indicated by the underlining. The subword ordering extends to congurations: hq; w Lemma [Hig52], this gives a a well-quasi-ordering: Lemma 2.1 Every innite sequence congurations contains an innite increasing subsequence When we write 0 and say that S may evolve from to 0 by losing messages. The steps of a lossy channel system are all congurations - 0 (i.e. losses may occur before and after a perfect step is performed). Note that a perfect step is a special case of a lossy step. A run is a sequence of chained lossy steps. A perfect run is a run that uses perfect steps only. We use and 0 to denote the existence of a nite run (resp. perfect run) that goes from to We are interested in the following two problems: Given a channel system S and an initial conguration 0 , are all runs from Reachability: Given a channel system S and two congurations 0 and f , is there a run from 0 to Theorem 2.2 [Fin94, AJ96b]. Termination and reachability are decidable for lossy channel systems. In the remaining of this note we show Theorem 2.3 Termination and reachability for lossy channel systems have nonprimitive recursive complexity. Theorem 2.3 also applies to the other verication problems that are known decidable for lossy channel systems. Indeed, termination is an instance of inevitability (shown decidable in [AJ96b]). Reachability is easily reduced to control-state reachability (shown decidable in [AJ96b]). Finally, termination can be reduced to simulation with a nite-state system 2 (shown decidable in [AK95, A CJT00]). Thus we are entitled to claim that \verifying lossy channel systems has nonprimitive recursive complexity". Note that there exist many undecidable problems for lossy channel systems [AK95, CFP96, AJ96a, May00, ABPJ00, Sch01]. 3 The main construction 3.1 Ackermann's function Let be the following sequence of functions over the natural numbers: A | {z } times 2. (2) Thus A n is followed by A 3 A 4 (a tower of k 2's) (4) and so on. The A n 's are monotonic and expansive in the following sense: for any n 2 and We dene inverse functions Observe that the A 1 n 's are partial functions. Another way to understand these functions is to notice that | {z } times There exists many versions of Ackermann's function. One possible denition is Ack(n) A n (2). It is well-known that Ack(n) dominates any primitive recursive function of n. Thus it follows from classical complexity-theoretic results that halting problems for Turing machines running in time or space bounded by Ack(n) (n being the size of the input) cannot be decided in primitive recursive time or space.S terminates i it is not simulation-equivalent with a simple loop. 3.2 Weakly computing A n with expanders We construct a family . of expanders, channel systems that weakly compute A 2 (k), A 3 (k), . As illustrated in Fig. 2, E n , the nth expander, uses n channels: c 1 is the \output channel c n channel c n 1 channel c 1 Figure 2: Interface for expander E n channel" (in which the system will write the result A n (k)), c n is the \input channel" (from which the argument k is read), and channels c 2 to c n 1 are used to store auxiliary results. E n has one starting and one ending state, called s n and f n respectively. These systems use a simple encoding for numbers: k 2 N is encoded as a string dke over the alphabet 0 eg. Formally, dke k e is made of k letters \1" surrounded by one \b"egin and one \e"nd marker. For example, the channels in Fig. 2 contain respectively d0e, . , d0e, and d4e. Before explaining the construction of the expanders, we describe the simple transferring devices This illustrates the way encodings dke of numbers are used. The T n 's, dened in Fig. 3, start in state t n and end in state x n after they have transfered the contents of channel c 1 into channel c n (assuming c 1 contains at least d1e). In Fig. 3 (and in future constructions) we sometimes omit depicting all intermediate states when their names are not required in the proof. Figure 3: T n , channel system for transferring c 1 into c n (n > 1) Formally, the lossy behaviors of T n can be characterized by: Proposition 3.1 For all n 2, a 1 Proof. Omitted. Note that the design of T n ensures that it blocks when c 1 contains d0e. This property is used in the proof of Lemma 3.4. We now move to the expanders themselves. The internals of E n are given in Fig. 4. Figure 4: Expanders For subsystems. This is because E n implements equation (2): every time a 1 is consumed from c n , run and the result is transfered (by can be applied again. It is easy to convince oneself that the perfect (non-lossy) behavior of E n is to compute A n in the following formal sense: and Indeed, for n > 2 a perfect run has the following form (where congurations are displayed in vector d0e d0e dke d1e d0e dke d1e d0e eb d1e d0e eb d0e d1e eb d0e d1e eb d0e eb d0e eb d0e eb d0e eb d0e d0e When perfect behavior is not assumed, E n still computes A n , this time in a weak sense, according to the following statement: Proposition 3.2 For all n 2, a 1 A n (k) and Proof. The \(" direction is an easy consequence of (9). A detailed proof of the \)" direction can be found in the Appendix but the underlying idea is simple: First, if some 1s are lost at any time during the computation, the nal result will end up being smaller because of the monotonicity properties (5). Then, if a b or a e marker is lost, the system can never recover it and will fail to reach a conguration where all channels contain encodings of numbers. Note that the dierence between (9) and (10) does not only come from the replacement of a \= A n (k)" by a \ A n (k)": (9) guarantees that the w i s are encodings of numbers, while assumes this. 3.3 Weakly computing A 1 n with folders Folder systems F are channel systems that weakly compute the A 1 's. Here channel c 1 is the input channel and c n is the output, while channels c 2 to c n 1 store auxiliary results. The denition of the F n 's, given in Fig. 5, is based on (7). It uses transferring systems T 0 these systems are variants of the T n 's and move the contents of channel c n into channel c 1 (instead of the other way around). When possibly lossy behaviors are considered, F n weakly computes A 1 n in the following formal sense: Proposition 3.3 For all n 2, Figure 5: Folders F Proof. We prove the \)" direction in the Appendix and omit the easier \(" direction. One additional property of the construction will be useful in the following: Lemma 3.4 E n and F n have no innite run, regardless of the initial conguration. Proof. We rst deal with the E n 's by induction over n. E 2 terminates since its loop consumes from c 2 . For n > 2, a run of E n cannot visit q n innitely often (this would consume innitely from c n ) and, by ind. hyp., cannot contain an innite subrun of (obviously, the T n 1 part must terminate too). For the F n 's, we rst observe (again using induction on n) that, if c 1 contains at least one 1, then a run from s 0 n to f 0 removes strictly more 1's than it writes back. Finally, traversing must move some 1's to c 1 (or lose them). 4 The hardness results Expander and folder systems can now be used to prove Theorem 2.3. 4.1 Hardness for reachability Let's consider a Turing Machine (a TM) M that is started on a blank work tape of size m (for some m) and that never goes beyond the allocated workspace. One can build a channel system S that simulates M using as workspace channel c 1 initially lled with b1 e. We do not describe further the construction of S since it follows the standard simulation of TM's by channel systems (from [BZ83]) 3 . Since M never goes beyond the allocated workspace, the 3 And since TM's are not really required and could be replaced by perfect channel systems that preserve the size of the channel contents. transition rules of S always write exactly as many messages as they read, so that if no loss occurs the channel always contains the same number of letters. The resulting system has two types of runs: perfect runs where M is simulated faithfully, and lossy runs that do not really simulate M but where messages have been irremediably lost. Now, in order to know whether M accepts in space m, there only remains to provide S with enough workspace, and to look at runs where no message is lost. This is exactly what is done by S n , depicted in Fig. 6, where expander E n provides a potentially large dme in channel c 1 and folder F n is used to check that no message has been lost. Simulation of M using c 1 as bounded workspace start accept Figure simulating Turing machine M with huge workspace Thus any run of S n M of the form hs perfect and must visit both hstart; dAck(n)e; Hence Proposition 4.1 hs accepts in space Ack(n). Therefore, since S n M has size O(n + jM j), reachability for lossy channel systems is at least as hard as termination for TM's running in Ackermann space. Hence Corollary 4.2 Reachability for lossy channel systems has nonprimitive recursive complexity. 4.2 Hardness for termination The second hardness result uses a slight adaptation of our previous construction, and relies on the following simulation (Fig. 7). Here S 0n lls two channels with dAck(n)e: c 1 used as before as working space, and c 0 used as a countdown that ensures termination of the simulation of M . Every time one step of M is simulated, S 0n consumes one 1 from c 0 . When the accepting state of M is reached, M moves to s 0 n where it uses F n to check that c 1 does contain dAck(n)e (i.e. the simulation was faithful). If the check succeeds, S 0n can enter a loop. Therefore, a run of S 0n terminates when the simulation is not perfect, or when M does not accept in at most Ack(n) steps. Proposition 4.3 S 0n M has an innite run from hs n accepts in time Ack(n). Proof. Clearly, S 0n M can only reach the nal loop if the simulation is faithful, and halts in at most Ack(n) steps. There remains to show that the unfaithful, lossy, behaviors are deals with the E n and F n part of S 0n M , the duplication gadget (be- tween f n and start) obviously terminates, and we solve the problem for the simulation part by programming it in such a way that the rotation of the tape (necessary for simulating a (duplicating c 1 in c 0 ) start loop Simulation of M using c 1 as bounded workspace and c 0 as bounded time accept Figure 7: S 0n another simulation of M with huge workspace TM) cannot induce non-termination. One way 4 to achieve this is to use two copies (one positive and one negative) of the TM alphabet: in \+" mode, the simulation reads +-letters and writes back their -twin. Only when an actual TM step is performed does S 0n from \+" mode to \ " mode and vice versa. More details can be found in section 5 where the same trick is used. This shows that termination for lossy channel systems is at least as hard as termination for TM's runnings in Ackermann time. Hence Corollary 4.4 Termination for lossy channel systems has nonprimitive recursive complexity. 5 Systems with only one channel Our construction used several channels for clarity, not out of necessity, and our result still holds when we restrict ourselves to lossy channel systems with only one channel. This is one more application of the slogan \lossy systems with k channels can be encoded into lossy systems with one channel". The encoding given in [AJ96a, Section 4.5] preserves the existence of runs that visit a given control state innitely often. Below we give another encoding that further preserves termination and reachability. It uses standard techniques (e.g. from the study of TM's with k tapes) and the only original aspect is the lossy behavior of our systems. Consider a system that uses channels c We simulate S by a system uses one single channel c. Without loss of generality we assume a dierent subalphabet is used with every channel of S (i.e. is partitioned in disjoint alphabets The encoding uses a larger alphabet where k markers # have been added, and where every letter comes in two copies (a positive and a negative one). Formally and a pair (x; shortly x . For an occurrence of some x in c means \one x in c i " (and the polarity is only used for bookkeeping purposes). 4 An alternative solution would use c0 as a countdown for channel system steps rather than TM steps, but Prop. 4.3 would have to be reworded in a clumsy way. A k-tuple hw k is encoded as same polarity is used to label all letters. For example, hab; "; ddci is coded as under positive polarity. Fig. 8 shows how two example rules from (left-hand side) are encoded in S 0 (right-hand side). On this gure, four loops (marked by ) provide for the rotation of the contents of c c!x c?x c!x c?x Figure 8: From S to encoding k channels into one (with a change of polarity): they are shorthands for several loops as indicated by the \for all x" comment. A state q from S gives rise to two copies q + and q in positive letters and writes negative letters, thus preventing non-termination induced by the rotation loops, q does the converse, and S 0 changes polarity each time a step of S has been simulated. Now, if W and V are encodings of hw g. This extends to lossy behaviors: S terminates from hq; w Note that these equivalences only hold for W and V that are correct encodings of k-tuples, and for q; q 0 that are original states from Q. S 0 has behaviors that do not correspond to behaviors of S in the sense of (12), e.g. when it gets blocked in new states, or when it loses one of the # i markers. Corollary 5.1 Reachability and termination for lossy single-channel systems have nonprimitive recursive complexity. 6 Conclusion There exist several constructions in the literature where a problem P is shown undecidable for lossy channel systems by simulating a Turing machine in such a way that the faithfulness of the simulation can be ensured, or checked, or rewarded in some way. Our construction uses similar tricks since it rst builds a nonprimitive recursive number of messages, and later checks that none has been lost. Still, there are a number of new aspects in our construction, and this explains why it is the rst complexity result for decidable problems on lossy channel systems. Acknowledgements We are grateful to Alain Finkel who brought this topic to our attention, to Petr Jancar who suggested major improvements on an earlier draft, and to the anonymous referee who helped us improve section 5. A Appendix Proof of Prop. 3.2 The di-culty with the \)" direction of Prop. 3.2 is that we have to consider lossy behaviors that need not respect the logic of the E n systems we designed. However, when we restrict our attention to behaviors that do not lose the b and e markers, managing the problem becomes feasible. We start with the following lemma: Lemma A.1 If a run hs is such that every w i contains one b and one e, then every w i has the form b1 e, i.e. encodes a number. The same holds for a run ht Proof. Since our systems always write a b or a e after they consumed one, saying that every contains one b and one e means that losses did not concern the markers. Therefore, it only remains to prove that the pattern b1 e is respected, even when some 1's have been lost. has to consume all of da 1 e and da 2 e and what it writes in c 1 and c 2 encodes a number even after losses. The same reasoning applies to the channels c 1 and c n of T n , while the other channels are untouched (and losses there respect the b1 e pattern). For proceed by induction over n. For c n , observe that all of a n is consumed and replaced by d0e. For the other channels (c 1 to c n 1 ), they all contain a number when the run rst reaches q n and, since the lemma holds for this remains the case every time the run revisits q n , until it eventually reaches f n . Then all channels contain encodings of numbers. We are now ready to prove that a 1 A n (k); and by induction over n. The base case is left to the reader: a simple inspection of E 2 shows it weakly computes A 2 For n > 2 we consider a run and isolate the congurations where (14) visits q n and f n 1 by writing it under the form Since the b and e markers are not lost in (14), we can state that all w i n have the formk i eb (resp. 1 eb). Since the transition leaving q n for t n 1 consumes one 1 from c n , one sees that implying m k. Finally a can only be reached by consuming e from c n ) and this concludes the proof for the part that concerns c n . When we consider the other channels (c 1 to c n 1 ) the proof of Lemma A.1 shows that, i all are encodings of numbers. Furthermore, they satisfy | {z } times (1)e and w j 2: as we prove by induction on j. The base case, (W 0 ) is a consequence of the assumption (14). Then one shows that (W j ) entails (V j+1 ) using Prop. 3.1. Finally, one proves that using One concludes the proof of (H n ) by observing that m k and (Wm ) entail the right hand side of (H n ) because of the monotonicity and expansion properties of A n stated in (5). Appendix Proof of Prop. 3.3 For the \)" direction, we proceed as with the proof of Prop. 3.2, and start with a result mimicking Lemma A.1: Lemma B.1 If a run hs 0 is such that every w i contains one b and one e, then every w i encodes a number. Proof. Omitted. We now prove by induction over n. The base is easy to see. For n > 2, we consider a run isolate the congurations where it visits writing it under the form We start with the contents of c n : the w i n have the form eb1 k i and, resp., eb1 k 0 . The transition from q 0 n to s 0 one 1 to c n , so that, for l a n we deduce l a n . When we consider the contents of the other channels, the w j 's and v j 's encode numbers for Using (H 0 shows, by induction on i, that w j so that a writing so that, by monotonicity of A n 1 , | {z } l times (because nally leaving q 0 consumes d1e from c n 1 entail that A n (a n ) m, completing the proof. --R Symbolic veri Reasoning about probabilistic lossy channel systems. Undecidable veri Verifying programs with unreliable channels. Decidability of simulation and bisimulation between lossy channel systems and An improved search strategy for lossy channel systems. Finite state description of communication protocols. Reset nets between decidability and undecidability. Decidability of the termination problem for completely speci Well structured transition systems everywhere! The equality problem for vector addition systems is undecidable. Ordering by divisibility in abstract algebras. Undecidable problems in unreliable computations. The complexity of the Bisimulation and other undecidable equivalences for lossy channel systems. --TR Unreliable channels are easier to verify than perfect channels Undecidable verification problems for programs with unreliable channels The Complexity of the Finite Containment Problem for Petri Nets On Communicating Finite-State Machines Algorithmic analysis of programs with well quasi-ordered domains Nonprimitive recursive complexity and undecidability for Petri net equivalence Well-structured transition systems everywhere! Bisimulation and Other Undecidable Equivalences for Lossy Channel Systems An Improved Search Strategy for Lossy Channel Systems Boundedness of Reset P/T Nets Reset Nets Between Decidability and Undecidability Undecidable Problems in Unreliable Computations Symbolic Verification of Lossy Channel Systems Reasoning about Probabilistic Lossy Channel Systems Decidability of Simulation and Bisimulation between Lossy Channel Systems and Finite State Systems (Extended Abstract) On-the-Fly Analysis of Systems with Unbounded, Lossy FIFO Channels --CTR Giorgio Delzanno, Constraint-based automatic verification of abstract models of multithreaded programs, Theory and Practice of Logic Programming, v.7 n.1-2, p.67-91, January 2007 Alexander Rabinovich, Quantitative analysis of probabilistic lossy channel systems, Information and Computation, v.204 n.5, p.713-740, May 2006 P. A. Abdulla , N. Bertrand , A. Rabinovich , Ph. Schnoebelen, Verification of probabilistic systems with faulty communication, Information and Computation, v.202 n.2, p.141-165, 1 November, 2005 Blaise Genest , Dietrich Kuske , Anca Muscholl, A Kleene theorem and model checking algorithms for existentially bounded communicating automata, Information and Computation, v.204 n.6, p.920-956, June 2006 Roberto M. Amadio , Charles Meyssonnier, On decidability of the control reachability problem in the asynchronous -calculus, Nordic Journal of Computing, v.9 n.2, p.70-101, Summer 2002 Grard Cc , Alain Finkel, Verification of programs with half-duplex communication, Information and Computation, v.202 n.2, p.166-190, 1 November, 2005 Antonn Kuera , Philippe Schnoebelen, A general approach to comparing infinite-state systems with their finite-state specifications, Theoretical Computer Science, v.358 n.2, p.315-333, 7 August 2006 Antonn Kuera , Petr Janar, Equivalence-checking on infinite-state systems: Techniques and results, Theory and Practice of Logic Programming, v.6 n.3, p.227-264, May 2006

verification of infinite-state systems;communication protocols;program correctness;formal methods

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Combining Classifiers with Meta Decision Trees.

The paper introduces meta decision trees (MDTs), a novel method for combining multiple classifiers. Instead of giving a prediction, MDT leaves specify which classifier should be used to obtain a prediction. We present an algorithm for learning MDTs based on the C4.5 algorithm for learning ordinary decision trees (ODTs). An extensive experimental evaluation of the new algorithm is performed on twenty-one data sets, combining classifiers generated by five learning algorithms: two algorithms for learning decision trees, a rule learning algorithm, a nearest neighbor algorithm and a naive Bayes algorithm. In terms of performance, stacking with MDTs combines classifiers better than voting and stacking with ODTs. In addition, the MDTs are much more concise than the ODTs and are thus a step towards comprehensible combination of multiple classifiers. MDTs also perform better than several other approaches to stacking.

Introduction The task of constructing ensembles of classiers [8] can be broken down into two sub-tasks. We rst have to generate a diverse set of base-level classiers. Once the base-level classiers have been generated, the issue of how to combine their predictions arises. Several approaches to generating base-level classiers are possible. One approach is to generate classiers by applying dierent learning algorithms (with heterogeneous model representations) to a single data set (see, e.g., Merz [14]). Another possibility is to apply a single learning algorithm with dierent parameters settings to a single data set. Finally, methods like bagging [5] and boosting [9] generate multiple classies by applying a single learning algorithm to dierent versions of a given data set. Two dierent methods for manipulating the data set are used: random sampling with replacement (also called bootstrap sampling) in bagging and re-weighting of the misclassied training examples in boosting. Techniques for combining the predictions obtained from the multiple base-level classiers can be clustered in three combining frameworks: voting (used in bagging and boost- ing), stacked generalization or stacking [22] and cascading [10]. In voting, each base-level classier gives a vote for its prediction. The prediction receiving the most votes is the nal prediction. In stacking, a learning algorithm is used to learn how to combine the predictions of the base-level classiers. The induced meta-level classier is then used to obtain the nal prediction from the predictions of the base-level classiers. Cascading is an iterative process of combining classiers: at each iteration, the training data set is extended with the predictions obtained in the previous iteration. The work presented here focuses on combining the predictions of base-level classiers induced by applying dierent learning algorithms to a single data set. It adopts the stacking framework, where we have to learn how to combine the base-level classiers. To this end, it introduces the notion of meta decision trees (MDTs), proposes an algorithm for learning them and evaluates MDTs in comparison to other methods for combining classiers. Meta decision trees (MDTs) are a novel method for combining multiple classiers. The dierence between meta and ordinary decision trees (ODTs) is that MDT leaves specify which base-level classier should be used, instead of predicting the class value directly. The attributes used by MDTs are derived from the class probability distributions predicted by the base-level classiers for a given example. We have developed MLC4.5, a modication of C4.5 [17], for inducing meta decision trees. MDTs and MLC4.5 are described in Section 3. The performance of MDTs is evaluated on a collection of twenty-one data sets. MDTs are used to combine classiers generated by ve base-level learning algorithms: two tree- learning algorithms C4.5 [17] and LTree [11], the rule-learning algorithm CN2 [7], the k-nearest neighbor (k-NN) algorithm [20] and a modication of the naive Bayes algorithm [12]. In the experiments, we compare the performance of stacking with MDTs to the performance of stacking with ODTs. We also compare MDTs with two voting schemes and two other stacking approaches. Finally, we compare MDTs to boosting and bagging of decision trees as state of the art methods for constructing ensembles of classiers. Section 4 reports on the experimental methodology and results. The experimental results are analyzed and discussed in Section 5. The presented work is put in the context of previous work on combining multiple classiers in Section 6. Section 7 presents conclusions based on the empirical evaluation along with directions for further work. Combining Multiple Classiers In this paper, we focus on combining multiple classiers generated by using dierent learning algorithms on a single data set. In the rst phase, depicted on the left hand side of Figure 1, a set of N base-level classiers is generated by applying the learning algorithms A 1 ; A to a single training data set L. training set "! "! A A A A A A A A AN CN A A A A A A A A x new example CN CML Figure 1: Constructing and using an ensemble of classiers. Left hand side: generation of the base-level classiers by applying N dierent learning algorithms to a single training data set L. Right hand side: classication of a new example x using the base-level classiers in C and the combining method CML . We assume that each base-level classier from C predicts a probability distribution over the possible class values. Thus, the prediction of the base-level classier C when applied to example x is a probability distribution vector: is a set of possible class values and p C (c i jx) denotes the probability that example x belongs to class c i as estimated (and predicted) by classier C. The class c j with the highest class probability p C (c j jx) is predicted by classier C. The classication of a new example x at the meta-level is depicted on the right hand side of Figure 1. First, the N predictions fp C1 (x); p C2 of the base-level classiers in C on x are generated. The obtained predictions are then combined using the combining method CML . Dierent combining methods are used in dierent combining frameworks. In the following subsections, the combining frameworks of voting and stacking are presented. 2.1 Voting In the voting framework for combining classiers, the predictions of the base-level classiers are combined according to a static voting scheme, which does not change with training data set L. The voting scheme remains the same for all dierent training sets and sets of learning algorithms (or base-level classiers). The simplest voting scheme is the plurality vote. According to this voting scheme, each base-level classier casts a vote for its prediction. The example is classied in the class that collects the most votes. There is a renement of the plurality vote algorithm for the case where class probability distributions are predicted by the base-level classiers [8]. Let p C (x) be the class probability distribution predicted by the base-level classier C on example x. The probability distribution vectors returned by the base-level classiers can be summed to obtain the class probability distribution of the meta-level voting classier: The predicted class for x is the class c j with the highest class probability p CML 2.2 Stacking In contrast to voting, where CML is static, stacking induces a combiner method CML from the meta-level training set, based on L, in addition to the base-level classiers. CML is induced by using a learning algorithm at the meta-level: the meta-level examples are constructed from the predictions of the base-level classiers on the examples in L and the correct classications of the latter. As the combiner is intended to combine the predictions of the base-level classiers (induced on the entire training set L) for unseen examples, special care has to be taken when constructing the meta-level data set. To this end, the cross-validation procedure presented in Table 1 is applied. Table 1: The algorithm for building the meta-level data set used to induce the combiner CML in the stacking framework. function build combiner(L, fA stratified partition(L; m) do to N do Let C j;i be the classier obtained by applying A j to L n L i Let CV j;i be class values predicted by C j;i on examples in L i Let CD j;i be class distributions predicted by C j;i on examples in L i endfor endfor Apply AML to LML in order to induce the combiner CML return CML endfunction First, the training set L is partitioned into m disjoint sets equal size. The partitioning is stratied in the sense that each set L i roughly preserves the class probability distribution in L. In order to obtain the base-level predictions on unseen examples, the learning algorithm A j is used to train base-level classier C j;i on the training set L n L i . The trained classier C j;i is then used to obtain the predictions for examples in L i . The predictions of the base-level classiers include the predicted class values CV j;i as well as the class probability distributions CD j;i . These are then used to calculate the meta-level attributes for the examples in L i . The meta-level attributes are calculated for all N learning algorithms and joined together into set of meta-level examples. This set is one of the m parts of the meta-level data set LML . Repeating this procedure m times (once for each set L i ), we obtain the whole meta-level data set. Finally, the learning algorithm AML is applied to it in order to induce the combiner CML . The framework for combining multiple classiers used in this paper is based on the combiner methodology described in [6] and the stacking framework of [22]. In these two ap- proaches, only the class values predicted by the base-level classiers are used as meta-level attributes. Therefore, the meta level attributes procedure used in these frameworks is trivial: it returns only the class values predicted by the base-level classiers. In our approach, we use the class probability distributions predicted by the base-level classiers in addition to the predicted class values for calculating the set of meta-level attributes. The meta-level attributes used in our study are discussed in detail in Section 3. 3 Meta Decision Trees In this section, we rst introduce meta decision trees (MDTs). We then discuss the possible sets of meta-level attributes used to induce MDTs. Finally, we present an algorithm for inducing MDTs, named MLC4.5. 3.1 What are Meta Decision Trees The structure of a meta decision tree is identical to the structure of an ordinary decision tree. A decision (inner) node species a test to be carried out on a single attribute value and each outcome of the test has its own branch leading to the appropriate subtree. In a leaf node, a MDT predicts which classier is to be used for classication of an example, instead of predicting the class value of the example directly (as an ODT would do). The dierence between ordinary and meta decision trees is illustrated with the example presented in Tables 2 and 3. First, the predictions of the base-level classiers (Table 2a) are obtained on the given data set. These include predicted class probability distributions as well as class values themselves. In the meta-level data set M (Table 2b), the meta-level attributes C 1 and C 2 are the class value predictions of two base-level classiers C 1 and C 2 for a given example. The two additional meta-level attributes Conf 1 and Conf 2 measure the condence of the predictions of C 1 and C 2 for a given example. The highest class probability, predicted by a base-level classier, is used as a measure of its prediction condence. Table 2: Building the meta-level data set. a) Predictions of the base-level classiers. b) The meta-level data set M . a) Predictions of the base-level classiers Base-level attributes (x) p C1 (0jx) p C1 (1jx) pred. p C2 (0jx) p C2 (1jx) pred. . 0.875 0.125 0 0.875 0.125 0 . . . . . . . b) Meta-level data set M The meta decision tree induced using the meta-level data set M is given in Table 3a). The MDT is interpreted as follows: if the condence Conf 1 of the base-level classier C 1 is high, then C 1 should be used for classifying the example, otherwise the base-level classier should be used. The ordinary decision tree induced using the same meta-level data set M (given in Table 3b) is much less comprehensible, despite the fact that it re ects the same rule for choosing among the base-level predictions. Note that both the MDT and the ODT need the predictions of the base-level classiers in order to make their own predictions. Table 3: The dierence between ordinary and meta decision trees. a) The meta decision tree induced from the meta-level data set M (by MLC4.5). b) The ODT induced from the same meta-level data set M (by C4.5). c) The MDT written as a logic program. a) The MDT induced from M Conf1 <= 0.625: C2 Conf1 > 0.625: C1 b) The ODT induced from M | Conf1 > 0.625 | Conf1 <= 0.625: | | | | | Conf1 > 0.625: 1 | Conf1 <= 0.625: | | | | c) The MDT written as a logic program The comprehensibility of the MDT from Table 3a) is entirely due to the extended expressiveness of the MDT leaves. Both the MDT and the ODT in Table 3a) and b) are induced from the propositional data set M . While the ODT induced from M is purely propositional, the MDT is not. A (rst order) logic program equivalent to the MDT is presented in Table 3c). The predicate combine(Conf 1 , C 1 , Conf 2 , C 2 , C) is used to combine the predictions of the base-level classiers C 1 and C 2 into class C according to the values of the attributes (variables) Conf 1 and Conf 2 . Each clause of the program corresponds to one leaf node of the MDT and includes a non-propositional class value assignment in the rst and in the second clause). In the propositional framework, the only possible assignments are one for each class value. There is another way of interpreting meta decision trees. A meta decision tree selects an appropriate classier for a given example in the domain. Consider the subset of examples falling in one leaf of the MDT. It identies a subset of the data where one of the base-level classiers performs better than the others. Thus, the MDT identify subsets that are relative areas of expertise of the base-level classiers. An area of expertise of a base-level classier is relative in the sense that its predictive performance in that area is better as compared to the performances of the other base-level classiers. This is dierent from an area of expertise of an individual base-level classier [15], which is a subset of the data where the predictions of a single base-level classier are correct. Note that in the process of inducing meta decision trees two types of attributes are used. Ordinary attributes are used in the decision (inner) nodes of the MDT (e.g., attributes Conf 1 and Conf 2 in the example meta-level data set M ). The role of these attributes is identical to the role of attributes used for inducing ordinary decision trees. Class attributes (e.g., C 1 and C 2 in M ), on the other hand, are used in the leaf nodes only. Each base-level classier has its class attribute: the values of this attribute are the predictions of the base-level classier. Thus, the class attribute assigned to the leaf node of the MDT decides which base-level classier will be used for prediction. When inducing ODTs for combining classiers, the class attributes are used in the same way as ordinary attributes. The partitioning of the data set into relative areas of expertise is based on the values of the ordinary meta-level attributes used to induce MDTs. In existing studies about areas of expertise of individual classiers [15], the original base-level attributes from the domain at hand are used. We use a dierent set of ordinary attributes for inducing MDTs. These are properties of the class probability distributions predicted by the base-level classiers and re ect the certainty and condence of the predictions. However, the original base-level attributes can be also used to induce MDTs. Details about each of the two sets of meta-level attributes are given in the following subsection. 3.2 Meta-Level Attributes As meta-level attributes, we calculate the properties of the class probability (CDP) distributions predicted by the base-level classiers that re ect the certainty and condence of the predictions. First, maxprob(x; C) is the highest class probability (i.e. the probability of the predicted class) predicted by the base-level classier C for example x: Next, entropy(x; C) is the entropy of the class probability distribution predicted by the classier C for example x: Finally, weight(x; C) is the fraction of the training examples used by the classier C to estimate the class distribution for example x. For decision trees, it is the weight of the examples in the leaf node used to classify the example. For rules, it is the weight of the examples covered by the rule(s) which has been used to classify the example. This property has not been used for the nearest neighbor and naive Bayes classiers, as it does not apply to them in a straightforward fashion. The entropy and the maximum probability of a probability distribution re ect the certainty of the classier in the predicted class value. If the probability distribution returned is highly spread, the maximum probability will be low and the entropy will be high, indicating that the classier is not very certain in its prediction of the class value. On the other hand, if the probability distribution returned is highly focused, the maximum probability is high and the entropy low, thus indicating that the classier is certain in the predicted class value. The weight quanties how reliable is the predicted class probability distribution. Intuitively, the weight corresponds to the number of training examples used to estimate the probability distribution: the higher the weight, the more reliable the estimate. An example MDT, induced on the image domain from the UCI repository [3], is given in Table 4. The leaf denoted by an asterisk (*) species that the C4.5 classier is to be used to classify an example, if (1) the maximum probability in the class probability distribution predicted by k-NN is smaller than 0.77; (2) the fraction of the examples in the leaf of the tree used for prediction is larger than 0.4% of all the examples in the training set; and (3) the entropy of the class distribution predicted by C4.5 is less then 0.14. In sum, if the k-NN classier is not very condent in its prediction (1) and the C4.5 classier is very Table 4: A meta decision tree induced on the image domain using class distribution properties as ordinary attributes. knn maxprob <= 0.77147: | c45 weight <= 0.00385: KNN | c45 weight > 0.00385: | | c45 entropy <= 0.14144: C4.5 (*) | | c45 entropy > 0.14144: LTREE condent in its prediction (3 and 2), the leaf recommends using the C4.5 prediction; this is consistent with common-sense knowledge in the domain of classier combination. Another set of ordinary attributes used for inducing meta decision trees is the set of original domain (base-level) attributes (BLA). In this case, the relative areas of expertise of the base-level classiers are described in terms of the original domain attributes as in the example MDT in Table 5. Table 5: A meta decision tree induced on the image domain using base-level attributes as ordinary attributes. short-line-density-5 <= 0: | short-line-density-2 <= 0: KNN | short-line-density-2 > 0: LTREE short-line-density-5 > 0: | short-line-density-5 <= 0.111111: LTREE | short-line-density-5 > 0.111111: C45 (*) The leaf denoted by an asterisk (*) in Table 5 species that C4.5 should be used to classify examples with short-line-density-5 values larger than 0.11. MDTs based on the base-level ordinary attributes can provide new insight into the applicability of the base-level classiers to the domain of use. However, only a human expert from the domain of use can interpret an MDT induced using these attributes. It cannot be interpreted directly from the point of view of classier combination. Note here another important property of MDTs induced using the CDP set of meta-level attributes. They are domain independent in the sense that the same language for expressing meta decision trees is used in all domains once we x the set of base-level classiers to be used. This means that a MDT induced on one domain can be used in any other domain for combining the same set of base-level classiers (although it may not perform very well). In part, this is due to the fact that the CDP set of meta-level attributes is domain independent. It depends only on the set of base-level classiers C. However, an ODT built from the same set of meta-level attributes is still domain dependent for two reasons. First, it uses tests on the class values predicted by the base-level classiers (e.g., the tests in the root node of the ODT from Table 3b). Second, an ODT predicts the class value itself, which is clearly domain dependent. In sum, there are three reasons for the domain independence of MDTs: (1) the CDP set of meta-level attributes; (2) not using class attributes in the decision (inner) nodes and (3) predicting the base-level classier to be used instead of predicting the class value itself. 3.3 MLC4.5 - a Modication of C4.5 for Learning MDTs In this subsection, we present MLC4.5 1 , an algorithm for learning MDTs based on Quinlan's C4.5 [17] system for inducing ODTs. MLC4.5 takes as input a meta-level data set as generated by the algorithm in Table 1. Note that this data set consists of ordinary and class attributes. There are four dierences between MLC4.5 and C4.5: 1. Only ordinary attributes are used in internal nodes; 2. Assignments of the form class attribute) are made by MLC4.5 in leaf nodes, as opposed to assignments of the form is a class value); 3. The goodness-of-split for internal nodes is calculated dierently (as described below); 4. MLC4.5 does not post-prune the induced MDTs. The rest of the MLC4.5 algorithm is identical to the original C4.5 algorithm. Below we describe C4.5's and MLC4.5's measures for selecting attributes in internal nodes. patch that can be used to transform the source code of C4.5 into MLC4.5 is available at http://ai.ijs.si/bernard/mdts/ C4.5 is a greedy divide and conquer algorithm for building classication trees [17]. At each step, the best split according to the gain (or gain ratio) criterion is chosen from the set of all possible splits for all attributes. According to this criterion, the split is chosen that maximizes the decrease of the impurity of the subsets obtained after the split as compared to the impurity of the current subset of examples. The impurity criterion is based on the entropy of the class probability distribution of the examples in the current subset S of training examples: denotes the relative frequency of examples in S that belong to class c i . The gain criterion selects the split that maximizes the decrement of the info measure. In MLC4.5, we are interested in the accuracies of each of the base-level classiers C from C on the examples in S, i.e., the proportion of examples in S that have a class equal to the class attribute C. The newly introduced measure, used in MLC4.5, is dened as: where accuracy(C; S) denotes the relative frequency of examples in S that are correctly classied by the base-level classier C. The vector of accuracies does not have probability distribution properties (its elements do not sum to 1), so the entropy can not be calculated. This is the reason for replacing the entropy based measure with an accuracy based one. As in C4.5, the splitting process stops when at least one of the following two criteria is satised: (1) the accuracy of one of the classiers on a current subset is 100% (leading to info ML or (2) a user dened minimal number of examples is achieved in the current subset. In each case, a leaf node is being constructed. The classier with the maximal accuracy is being predicted by a leaf node of a MDT. In order to compare MDTs with ODTs in a principled fashion, we also developed a intermediate version of C4.5 (called AC4.5) that induces ODTs using the accuracy based info A measure: 4 Experimental Methodology and Results The main goal of the experiments we performed was to evaluate the performance of meta decision trees, especially in comparison to other methods for combining classiers, such as voting and stacking with ordinary decision trees, as well as other methods for constructing ensembles of classiers, such as boosting and bagging. We also investigate the use of dierent meta-level attributes in MDTs. We performed experiments on a collection of twenty-one data sets from the UCI Repository of Machine Learning Databases and Domain Theories [3]. These data sets have been widely used in other comparative studies. In the remainder of this section, we rst describe how classication error rates were estimated and compared. We then list all the base-level and meta-level learning algorithms used in this study. Finally, we describe a measure of the diversity of the base-level classiers that we use in comparing the performance of meta-level learning algorithms. 4.1 Estimating and Comparing Classication Error Rates In all the experiments presented here, classication errors are estimated using 10-fold stratied cross validation. Cross validation is repeated ten times using a dierent random reordering of the examples in the data set. The same set of re-orderings have been used for all experiments. For pair-wise comparison of classication algorithms, we calculated the relative improvement and the paired t-test, as described below. In order to evaluate the accuracy improvement achieved in a given domain by using classier C 1 as compared to using clas- sier C 2 , we calculate the relative improvement: 1 error(C 1 )=error(C 2 ). In the analysis presented in Section 5, we compare the performance of meta decision trees induced using CDP as ordinary meta-level attributes to other approaches: C 1 will thus refer to combiners induced by MLC4.5 using CDP. The average relative improvement across all domains is calculated using the geometric mean of error reduction in individual domains: The classication errors of C 1 and C 2 averaged over the ten runs of 10-fold cross validation are compared for each data set (error(C 1 ) and error(C 2 ) refer to these averages). The statistical signicance of the dierence in performance is tested using the paired t-test (exactly the same folds are used for C 1 and C 2 ) with signicance level of 95%: += to the right of a gure in the tables with results means that the classier C 1 is signicantly better/worse than C 2 . Another aspect of tree induction performance is the simplicity of the induced decision trees. In the experiments presented here, we used the size of the decision trees, measured as the number of (internal and leaf) nodes, as a measure of simplicity: the smaller the tree, the simpler it is. 4.2 Base-Level Algorithms Five learning algorithms have been used in the base-level experiments: two tree-learning algorithms C4.5 [17] and LTree [11], the rule-learning algorithm CN2 [7], the k-nearest neighbor (k-NN) algorithm [20] and a modication of the naive Bayes algorithm [12]. All algorithms have been used with their default parameters' settings. The output of each base-level classier for each example in the test set consist of at least two components: the predicted class and the class probability distribution. All the base-level algorithms used in this study calculate the class probability distribution for classied examples, but two of them (k-NN and naive Bayes) do not calculate the weight of the examples used for classication (see Section 3). The code of the other three of them (C4.5, CN2 and was adapted to output the class probability distribution as well as the weight of the examples used for classication. The classication errors of the base-level classiers on the twenty-one data sets are presented in Table 7 in Appendix A. The smallest overall classication error is achieved using linear discriminant trees induced with LTree. However, on dierent data sets, dierent base-level classiers achieve the smallest classication error. 4.3 Meta-Level Algorithms At the meta-level, we evaluate the performances of eleven dierent algorithms for constructing ensembles of classiers (listed below). Nine of these make use of exactly the same set of ve base-level classiers induced by the ve algorithms from the previous section. In brief, two perform stacking with ODTs, using the algorithms C4.5 and AC4.5 (see previous sec- three perform stacking with MDTs using the algorithm MLC4.5 and three dierent sets of meta-level attributes (CDP, BLA, CDP+BLA); two are voting schemes; Select-Best chooses the best base-level classier, and SCANN performs stacking with nearest neighbor after analyzing dependencies among the base-level classiers. In addition, boosting and bagging of decision trees are considered, which create larger ensembles (200 trees). uses ordinary decision trees induced with C4.5 for combining base-level classiers. uses ODTs induced with AC4.5 for combining base-level classiers. MDT-CDP uses meta decision trees induced with MLC4.5 on a set of class distribution properties (CDP) as meta-level attributes. MDT-BLA uses MDTs induced with MLC4.5 on a set of base-level attributes (BLA) as meta-level attributes. MDT-CDP+BLA uses MDTs induced with MLC4.5 on a union of two alternative sets of meta-level attributes (CDP and BLA). P-VOTE is a simple plurality vote algorithm (see Section 2.1). CD-VOTE is a renement of the plurality vote algorithm for the case where class probability distributions are predicted by the base-level classiers (see Section 2.1). Select-Best selects the base-level classier that performs best on the training set (as estimated by 10-fold stratied cross-validation). This is equivalent to building a single leaf MDT. SCANN [14] performs Stacking with Correspodence Analysis and Nearest Neighbours. Correspondence analysis is used to deal with the highly correlated predictions of the base-level classiers: SCANN transforms the original set of potentially highly correlated meta-level attributes (i.e., predictions of the base-level classiers), into a new (smaller) set of uncorrelated meta-level attributes. A nearest neighbor classier is then used for classication with the new set of meta-level attributes. Boosting of decision trees. Two hundred iterations were used for boosting. Decision trees were induced using J48 2 , (C4.5) with default parameters' settings for pre- and post- pruning. The WEKA [21] data mining suite implements the AdaBoost [9] boosting method with re-weighting of the training examples. Bagging of decision trees. Two hundred iterations (decision trees), were used for bagging, using J4.8 with default settings. A detailed report on the performance of the above methods can be found in Appendix A. Their classication errors can be found in Table 9. The sizes of (ordinary and meta) decision trees induced with dierent meta-level combining algorithms are given in Table 8. Finally, a comparison of the classication errors of each method to those of MDT-CDP (in terms of average relative accuracy improvement and number of signicant wins and losses) is given in Table 10. A summary of this detailed report is given in Table 6. 4.4 Diversity of Base-Level Classiers Empirical studies performed in [1, 2] show that the classication error of meta-level learning methods as well as the improvement of accuracy achieved using them is highly correlated to the degree of diversity of the predictions of the base-level classiers. The measure of the diversity of two classiers used in these studies is error correlation. The smaller the error correlation, the greater the diversity of the base-level classiers. 2 The experiments with bagging and boosting have been performed using the WEKA data mining suite which includes J48, a Java re-implementation of C4.5. The dierences between the J48 results and the C4.5 results are negligible: an average of 0.01% with a maximum relative dierence of 4%. correlation is dened by [1, 2] as the probability that both classiers make the same error. This denition of error correlation is not \normalized": its maximum value is the lower of the two classication errors. An alternative denition of error correlation, proposed in [11], is used in this paper. Error correlation is dened as the conditional probability that both classiers make the same error, given that one of them makes an error: are predictions of classiers C i and C j for a given example x and c(x) is the true class of x. The error correlation for a set of multiple classiers C is dened as the average of the pairwise error correlations: relative improvement over LTree (base-level) degree of the error correlation between base-level classifiers australian balance breast-w bridges-td car chess diabetes echocardiogram german glass heart hepatitis hypothyroid image ionosphere iris soya tic-tac-toe vote waveform wine insignificant significant relative improvement over k-NN (base-level) degree of the error correlation between base-level classifiers australian balance breast-w bridges-td car chess diabetes echocardiogram german glass heart hepatitis hypothyroid image ionosphere iris soya tic-tac-toe vote waveform wine insignificant significant Figure 2: Relative accuracy improvement achieved with MDTs when compared to two base-level classiers (LTree and k-NN) in dependence of the error correlation among the ve base-level classiers. The graphs in Figure 2 conrm these results for meta decision trees. The relative accuracy improvement achieved with MDTs over LTree and k-NN (two base-level classiers with highest overall accuracies) decreases as the error correlation of the base-level clas- siers increases. The linear regression line interpolated through the points conrms this trend, which shows that performance improvement achieved with MDTs is correlated to the diversity of errors of the base-level classiers. Table Summary performance of the meta-level learning algorithms as compared to MDTs induced using class distribution properties (MDT-CDP) as meta-level attributes: average relative accuracy improvement (in %), number of signicant wins/losses and average tree size (where applicable). Details in Tables 10 and 8 in Appendix A. Meta-level algorithm Ave. rel. acc. imp. (in MDT-BLA 14.47 8/0 66.94 MDT-CDP+BLA 13.91 8/0 73.73 Select-Best 7.81 5/2 1 SCANN 13.95 9/6 NA Boosting 9.36 13/6 NA Bagging 22.26 10/4 NA 5 Analysis of the Experimental Results The results of the experiments are summarized in Table 6. In brief, the following main conclusions that can be drawn from these results: 1. The properties of the class probability distributions predicted by the base-level clas- siers (CDP) are better meta-level attributes for inducing MDTs than the base-level attributes (BLA). Using BLA in addition to CDP worsens the performance. 2. Meta decision trees (MDTs) induced using CDP outperform ordinary decision trees and voting for combining classiers. 3. MDTs perform slightly better than the SCANN and Select-Best methods. 4. The performance improvement achieved with MDTs is correlated to the diversity of errors of the base-level classiers: the higher the diversity, the better the relative performance as compared to other methods. 5. Using MDTs to combine classiers induced with dierent learning algorithms outperforms ensemble learning methods based on bagging and boosting of decision trees. Below we look into these claims and the experimental results in some more detail. 5.1 MDTs with Dierent Sets of Meta-Level Attributes We analyzed the dependence of MDTs performance on the set of ordinary meta-level attributes used to induce them. We used three sets of attributes: the properties of the class distributions predicted by the base-level classiers (CDP), the original (base-level) domain attributes (BLA) and their union (CDP+BLA). The average relative improvement achieved by MDTs induced using CDP over MDTs induced using BLA and CDP+BLA is about 14% with CDP being signicantly better in 8 domains (see Table 6 and Table 10 in Appendix A). MDTs induced using CDP are about times smaller on average than MDTs induced using BLA and CDP+BLA (see Table 8). These results show that the CDP set of meta-level attributes is better than the BLA set. Furthermore, using BLA in addition to CDP decreases performance. In the remainder of this Section, we only consider MDTs induced using CDP. An analysis of the results for ordinary decision trees induced using CDP, BLA and CDP+BLA (only the results for CDP are actually presented in the paper) shows that the claim holds also for ODTs. This result is especially important because it highlights the importance of using the class probability distributions predicted by the base-level classiers for identifying their (relative) areas of expertise. So far, base-level attributes from the original domain have typically been used to identify the areas of expertise of base-level classiers. 5.2 Meta Decision Trees vs. Ordinary Decision Trees To compare combining classiers with MDTs and ODTs, we rst look at the relative improvement of using MLC4.5 over C-C4.5 (see Table 6, column C-C4.5 of Table 10 in Appendix A and left hand side of Figure 3). performs signicantly better in 15 and signicantly worse in 2 data sets. There is a 4% overall decrease of accuracy (this is a geometric mean), but this is entirely due to the result in the tic-tac-toe domain, where all combining methods perform very well. If we exclude the tic-tac-toe domain, a 7% overall relative increase is obtained. We can thus say that MLC4.5 performs slightly better in terms of accuracy. However, the MDTs are much smaller, the size reduction factor being over 16 (see Table 8), despite the fact that ODTs induced with C4.5 are post-pruned and MDTs are not. relative improvement over (in tic-tac-toe balance breast-w hypothyroid soya heart image vote hepatitis glass echocardiogram waveform iris german australian chess diabetes car bridges-td wine ionosphere avg. insignificant significant relative improvement over (in iris bridges-td image heart soya waveform vote breast-w hepatitis german balance diabetes echocardiogram glass ionosphere australian hypothyroid chess wine car tic-tac-toe avg. insignificant significant Figure 3: Relative improvement of the accuracy when using MDTs induced with MLC4.5 when compared to the accuracy of ODTs induced with AC4.5 and C4.5. To get a clearer picture of the performance dierences due to the extended expressive power of MDT leaves (as compared to ODT leaves), we compare MLC4.5 and C-AC4.5 (see Table 6, column C-AC4.5 in Table 10 and right hand side of Figure 3). Both MLC4.5 and AC4.5 use the same learning algorithm. The only dierence between them is the types of trees they induce: MLC4.5 induces meta decision trees and AC4.5 induces ordinary ones. The comparison clearly shows that MDTs outperform ODTs for combining classiers. The overall relative accuracy improvement is about 8% and MLC4.5 is signicantly better than C-AC4.5 in 12 out of 21 data sets and is signicantly worse in only one (ionosphere). Consider also the graph on the right hand side of Figure 3. MDTs perform better than ODTs in all but two domains, with the performance gains being much larger than the losses. Furthermore, the MDTs are, on average, more than 34 times smaller than the ODTs induced with AC4.5 (see Table 8). The reduction of the tree size improves the comprehensibility of meta decision trees. For example, we were able to interpret and comment on the MDT in Table 4. In sum, meta decision trees performs better than ordinary decision trees for combining classiers: MDTs are more accurate and much more concise. The comparison of MLC4.5 and AC4.5 shows that the performance improvement is due to the extended expressive power of MDT leaves. 5.3 Meta Decision Trees vs. Voting Combining classiers with MDTs is signicantly better than plurality vote in 10 domains and signicantly worse in 6. However, the signicant improvements are much higher than the signicant drops of accuracy, giving an overall accuracy improvement of 22%. Since performs slightly better than plurality vote, a smaller overall improvement of 20% is achieved with MDTs. MLC4.5 is signicantly better in 10 data sets and signicantly worse in 5. These results show that MDTs outperform the voting schemes for combining classiers (see Table 6 and Table 10 in Appendix A). -0.4 relative improvement over degree of the error correlation between base-level classifiers australian balance breast-w bridges-td car chess diabetes echocardiogram german glass heart hepatitis hypothyroid image ionosphere iris soya tic-tac-toe vote waveform wine insignificant significant -0.4 relative improvement over degree of the error correlation between base-level classifiers australian balance breast-w bridges-td car chess diabetes echocardiogram german glass heart hepatitis hypothyroid image ionosphere iris soya tic-tac-toe vote waveform wine insignificant significant Figure 4: Relative improvement of the accuracy of MDTs over two voting schemes in dependence of the degree of error correlation between the base-level classiers. We explored the dependence of accuracy improvement by MDTs over voting on the diversity of the base-level classiers. The graphs in Figure 4 show that MDTs can make better use of the diversity of errors of the base-level classiers than the voting schemes. Namely, for the domains with low error correlation (and therefore higher diversity) of the base-level classiers, the relative improvement of MDTs over the voting schemes is higher. However, the slope of the linear regression line is smaller than the one for the improvement over the base-level classiers. Still, the trend clearly shows that MDTs make better use of the error diversity of the base-level predictions than voting. 5.4 Meta Decision Trees vs. Select-Best Combining classiers with MDTs is signicantly better than Select-Best in 5 domains and signicantly worse in 2, giving an overall accuracy improvement of almost 8% (see Table 6 and Table in Appendix A). relative improvement over Select-Best degree of the error correlation between base-level classifiers australian balance breast-w bridges-td car chess diabetes echocardiogram german glass heart hepatitis hypothyroid image ionosphere iris soya tic-tac-toe vote waveform wine insignificant significant Figure 5: Relative improvement of the accuracy of MDTs over Select-Best method in dependence of the degree of error correlation between the base-level classiers. The results of the dependence analysis of accuracy improvement by MDTs over Select- Best on the diversity of the base-level classiers is given in Figure 5. MDTs can make slightly better use of the diversity of errors of the base-level classiers than Select-Best. The slope of the linear regression line is smaller than the one for the improvement over the voting methods. 5.5 Meta Decision Trees vs. SCANN Combining classiers with MDTs is signicantly better than SCANN in 9 domains and signicantly worse in 6 (see Table 6 and Table 10 in Appendix A). However, the signicant improvements are much higher than the signicant drops of accuracy, giving an overall accuracy improvement of almost 14%. These results show that MDTs outperform the SCANN method for combining classiers. -0.4 relative improvement over degree of the error correlation between base-level classifiers australian balance breast-w bridges-td car chess diabetes echocardiogram german glass heart hepatitis hypothyroid image ionosphere iris soya tic-tac-toe vote waveform wine insignificant significant Figure Relative improvement of the accuracy of MDTs over SCANN method in dependence of the degree of error correlation between the base-level classiers. We explored the dependence of accuracy improvement by MDTs over SCANN on the diversity of the base-level classiers. The graph in Figure 6 shows that MDTs can make a slightly better use of the diversity of errors of the base-level classiers than SCANN. The slope of the linear regression line is smaller than the one for the improvement over the voting methods. 5.6 Meta Decision Trees vs. Boosting and Bagging Finally, we compare the performance of MDTs with the performance of two state of the art ensemble learning methods: bagging and boosting of decision trees. performs signicantly better than boosting in 13 and signicantly better than bagging in 10 out of the 21 data sets. MLC4.5 performed signicantly worse than boosting in 6 domains and signicantly worse than bagging in 4 domains only. The overall relative improvement of performance is 9% over boosting and 22% over bagging (see Table 6 and Table in Appendix A). It is clear that MDTs outperform bagging and boosting of decision trees. This comparison is not fair in the sense that MDTs use base-level classiers induced by decision trees and four other learning methods, while boosting and bagging use only decision trees as a base-level learning algorithm. However, it does show that our approach to constructing ensembles of classiers is competitive to existing state of the art approaches. 6 Related Work An overview of methods for constructing ensembles of classiers can be found in [8]. Several meta-level learning studies are closely related to our work. Let us rst mention the study of using SCANN [14] for combining base-level classiers. As mentioned above, SCANN performs stacking by using correspondence analysis of the classications of the base-level classiers. The author shows that SCANN outperforms the plurality vote scheme, also in the case when the base-level classiers are highly correlated. SCANN does not use any class probability distribution properties of the predictions by the base-level classiers (although the possibility of extending the method in that direction is mentioned). Therefore, no comparison with the CD voting scheme is included in their study. Our study shows that MDTs using CDP attributes are slightly better than SCANN in terms of performance. Also, the concept induced by SCANN at the meta-level is not directly interpretable and can not be used for identifying the relative areas of expertise of the base-level classiers. In cascading, base-level classiers are induced using the examples in a current node of the decision tree (or each step of the divide and conquer algorithm for building decision trees). New attributes, based on the class probability distributions predicted by the base-level classiers, are generated and added to the set of the original attributes in the domain. The base-level classiers used in this study are naive Bayes and Linear Discriminant. The integration of these two base-level classiers within decision trees is much tighter than in our combining/stacking framework. The similarity to our approach is that class probability distributions are used. A version of stacked generalization, using the class probability distributions predicted by the base-level classiers, is implemented in the data mining suite WEKA [21]. However, class probability distributions are used there directly and not through their properties, such as maximal probability and entropy. This makes them domain dependent in the sense discussed in Section 3. The indirect use of class probability distributions through their properties makes MDTs domain independent. Ordinary decision trees have already been used for combining multiple classiers in [6]. However, the emphasis of their study is more on partitioning techniques for massive data sets and combining multiple classiers trained on dierent subsets of massive data sets. Our study focuses on combining multiple classiers generated on the same data set. Therefore, the obtained results are not directly comparable to theirs. Combining classiers by identifying their areas of expertise has already been explored in [15] and [13]. In their studies, a description of the area of expertise in the form of an ordinary decision tree, called arbiter, is induced for each individual base-level classier. For a single data set, as many arbiters are needed as there are base-level classiers. When combining multiple classiers, a voting scheme is used to combine the decisions of the arbiters. However, a single MDT, identifying relative areas of expertise of all base-level classiers at once is much more comprehensible. Another improvement presented in our study is the possibility to use the certainty and condence of the base-level predictions for identifying the classiers expertise areas and not only the original (base-level) attributes of the data set. The present study is also related to our previous work on the topic of meta-level learning [18]. There we introduced an inductive logic programming [16] (ILP) framework for learning the relation between data set characteristics and the performance of dierent (base- level) classiers. A more expressive (non-propositional) formulation is used to represent the meta-level examples (data set characteristics), e.g., properties of individual attributes. The induced meta-level concepts are also non-propositional. While MDT leaves are more expressive than ODT leaves, the language of MDTs is still much less expressive than the language of logic programs used in ILP. 7 Conclusions and Further Work We have presented a new technique for combining classiers based on meta decision trees (MDTs). MDTs make the language of decision trees more suitable for combining classiers: they select the most appropriate base-level classier for a given example. Each leaf of the MDT represents a part of the data set, which is a relative area of expertise of the base-level classier in that leaf. The relative areas of expertise can be identied on the basis of the values of the original (base-level) attributes (BLA) of the data set, but also on the basis of the properties of the class probability distributions (CDP) predicted by the base-level classiers. The latter re ect the certainty and condence of the class value predictions by the individual base-level classiers. The extensive empirical evaluation shows that MDTs induced from CDPs perform much better and are much more concise than MDTs induced from BLAs. Due to the extended expressiveness of MDT leaves, they also outperform ordinary decision trees (ODTs), both in terms of accuracy and conciseness. MDTs are usually so small that they can easily be interpreted: we regard this as a step towards a comprehensible model of combining clas- siers by explicitly identifying their relative areas of expertise. In contrast, most existing work uses non-symbolic learning methods (e.g., neural networks) to combine classiers [14]. MDTs can use the diversity of the base-level classiers better than voting: they out-perform voting schemes in terms of accuracy, especially in domains with high diversity of the errors made by the base-level classiers. MDTs also perform slightly better than the SCANN method for combining classiers and the Select-Best method, which simply takes the best single classier. Finally, MDTs induced from CDPs perform better than boosting and bagging of decision trees and are thus competitive with state of the art methods for learning ensembles. MDTs built by using CDPs are domain independent and are, in principle, transferable across domains once we x the set of base-level learning algorithms. This in the sense that a MDT built on one data set can be used on any other data set (since it uses the same set of attributes). There are several potential benets of the domain independence of MDTs. First, machine learning experts can use MDTs for domain independent analysis of relative areas of expertise of dierent base-level classiers, without having knowledge about the particular domain of use. Furthermore, an MDT induced on one data set can be used for combining classiers induced by the same set of base-level learning algorithms on other data sets. Finally, MDTs can be induced using data sets that contain examples originating from dierent domains. Exploring the above options already gives us some topics for further work. Combining data from dierent domains for learning MDTs is an especially interesting avenue for further work that would bring together the present study with meta-level learning work on selecting appropriate classiers for a given domain [4]. In this case, attributes describing individual data set properties can be added to the class distribution properties in the meta-level learning data set. Preliminary investigations along these lines have been already made [19]. There are several other obvious directions for further work. For ordinary decision trees, it is already known that post-pruning gives better results than pre-pruning. Preliminary experiments show that pre-pruning degrades the classication accuracy of MDTs. Thus, one of the priorities for further work is the development of a post-pruning method for meta decision trees and its implementation in MLC4.5. An interesting aspect of our work is that we use class-distribution properties for meta-level learning. Most of the work on combining classiers only uses the predicted classes and not the corresponding probability distributions. It would be interesting to use other learning algorithms (neural networks, Bayesian classication and SCANN) to combine classiers based on the probability distributions returned by them. A comparison of combining clas- siers using class predictions only vs. with class predictions along with class probability distributions would be also worthwhile. The consistency of meta decision trees with common sense classiers combination knowledge, as brie y discussed in Section 3, opens another question for further research. The process of inducing meta-level classiers can be biased to produce only meta-level classiers consistent with existing knowledge. This can be achieved using strong language bias within MLC4.5 or, probably more easily, within a framework of meta decision rules, where rule templates could be used. Acknowledgments The work reported was supported in part by the Slovenian Ministry of Education, Science and Sport and by the EU-funded project Data Mining and Decision Support for Business Competitiveness: A European Virtual Enterprise (IST-1999-11495). We thank Jo~ao Gama for many insightful and inspirational discussions about combining multiple classiers. Many thanks to Marko Bohanec, Thomas Dietterich, Nada Lavrac and three anonymous reviewers for their comments on earlier versions of the manuscript. --R On explaining degree of error reduction due to combining multiple decision trees. reduction through learning multiple descriptions. UCI repository of machine learning databases. Analysis of Results. Bagging Predictors. On the Accuracy of Meta-learning for Scalable Data Mining Rule induction with CN2: Some recent improvements. Experiments with a New Boosting Algorithm. Combining Classi Discriminant trees. A Linear-Bayes Classi er Integrating multiple classi Using Correspondence Analysis to Combine Classi Exploiting Multiple Existing Models and Learning Algorithms. Learning logical de A study of distance-based machine learning algorithms Data Mining: Practical Machine Learning Tools and Techniques with Java Implementations. Stacked Generalization. --TR --CTR Michael Gamon, Sentiment classification on customer feedback data: noisy data, large feature vectors, and the role of linguistic analysis, Proceedings of the 20th international conference on Computational Linguistics, p.841-es, August 23-27, 2004, Geneva, Switzerland Saso Deroski , Bernard enko, Is Combining Classifiers with Stacking Better than Selecting the Best One?, Machine Learning, v.54 n.3, p.255-273, March 2004 Efstathios Stamatatos , Gerhard Widmer, Automatic identification of music performers with learning ensembles, Artificial Intelligence, v.165 n.1, p.37-56, June 2005 Christophe Giraud-Carrier , Ricardo Vilalta , Pavel Brazdil, Introduction to the Special Issue on Meta-Learning, Machine Learning, v.54 n.3, p.187-193, March 2004 Joo Gama, Functional Trees, Machine Learning, v.55 n.3, p.219-250, June 2004 Pavel B. Brazdil , Carlos Soares , Joaquim Pinto Da Costa, Ranking Learning Algorithms: Using IBL and Meta-Learning on Accuracy and Time Results, Machine Learning, v.50 n.3, p.251-277, March Nicols Garca-Pedrajas , Domingo Ortiz-Boyer, A cooperative constructive method for neural networks for pattern recognition, Pattern Recognition, v.40 n.1, p.80-98, January, 2007 S. B. Kotsiantis , I. D. Zaharakis , P. E. Pintelas, Machine learning: a review of classification and combining techniques, Artificial Intelligence Review, v.26 n.3, p.159-190, November 2006

stacking;meta-level learning;decision trees;ensembles of classifiers;combining classifiers

608175

Polynomial Formal Verification of Multipliers.

Not long ago, completely automatical formal verification of multipliers was not feasible, even for small input word sizes. However, with Multiplicative Binary Moment Diagrams (&ast;BMD), which is a new data structure for representing arithmetic functions over Boolean variables, methods were proposed by which verification of multipliers with input word sizes of up to 256 Bits is now feasible. Unfortunately, only experimental data has been provided for these verification methods until now.In this paper, we give a formal proof that logic verification with &ast;BMDs is polynomially bounded in both, space and time, when applied to the class of Wallace-tree like multipliers. Using this knowledge online detection of design errors becomes feasible during a verification run.

Introduction Verifying that an implementation of a combinational circuit meets its specification is an important step in the design process. Often this is done by applying a set of test-input patterns to the circuit. With these patterns a simulation is performed to ensure oneself in the correct behavior of the circuit. In the last few years new methods have been proposed for a formal verification of circuits concerning the logic function represented by the circuit, e.g. [1]. These methods are based on Binary Decision Diagrams (BDDs). But all these methods fail to verify some interesting combinational circuits, such as multipliers, since all BDDs for the multiplication function are exponential in size [5]. Recently, a new type of decision diagrams the Multiplicative Binary Moment Diagram (*BMD) [2] has been introduced. *BMDs are well suited for representing linear arithmetic functions with Boolean domain. *BMDs have been successfully used to verify the behavior of combinational arithmetic circuits such as multipliers with input word sizes of up to 256 bits [3]. Motivated by these results, several extensions of this approach have been discussed [7], [6]. The verification method of [3] is based on partitioning the circuit into components with easy word-level specifications. First it is shown, that the bit-level implementation of a component Research supported by DFG grant Mo 645/2-1 and BE 1176/8-2 module implements correctly its word-level specification. Then the composition of word-level specifications according to the interconnection structure of the whole circuit is derived. This composition is an algebraic expression for which a *BMD is generated. Then the resulting *BMD is compared with the one generated from the overall circuit specification. A problem arising with this methodology is the need of high-level specifications of component modules. Taking this problem into account, Hamaguchi et.al. have proposed another method for verifying arithmetic circuits. They called their method verification by backward construction [9]. This method does not need any high-level information. In [9] only experimental data has been provided to show the feasibility of their method for some specific multiplier circuits. In this paper, we give a formal proof that verification by backward construction is polynomially bounded with respect to the input word size. We do not consider a specific multiplier circuit, but the class of Wallace-tree like multipli- ers, e.g. [11], [10]. Additionally, we consider not only the costs of intermediate results, but also the costs, that arise during the synthesis operation on the *BMDs. This leads to the possibility to interrupt the verification process in presence of a design or implementation failure. Furthermore we give classes of variable orders and we give classes of depth-first and breadth-first orderings of the circuit with that our tight bounds can be reached. 2. Multiplicative Binary Moment Diagrams In this section, we give a short introduction how to represent integer-valued functions by the proposed decision diagram type *BMD. Details can be found in [2], [3]. *BMDs have been motivated by the necessity to represent functions over Boolean variables having non-Boolean ranges, i.e. functions f mapping a Boolean vector onto integer numbers or onto rational respectively. Here, we restrict ourselves to integer functions. For this class of functions the Boole-Shannon expansion can be generalized with - multiplication and addition re- spectively. f - x is called the constant moment and f - is called the linear moment of f with respect to variable x (\Gamma denotes integer subtraction). Equation (1) is the moment decomposition of function f with respect to variable x. *BMDs are rooted, directed, acyclic graphs with nonterminal and terminal vertices. Each nonterminal vertex has exactly two successors. The edges pointing to the successors are named low- and high-edge and the successors themselves are named low- and high-successor, respectively. The nonterminal vertices are labeled with Boolean variables. The terminal vertices have no successor and are labeled with integer values. The low- (high-) edge of a nonterminal node points to the constant (linear) moment of the function represented by this node. The *BMD data structure is reduced and ordered in the common way. Addition- ally, a *BMD makes use of a common factor in the constant and linear moment. It extracts this factor and places it as a so called edge-weight on the incoming edge to the node. This could lead to smaller representations. For ex- ample, a *BMD with root vertex v labeled with x and weight m on the incoming edge to v represents the function In the following, we denote the node v, labeled with a variable x simply as node x, if the context is clear. In this paper, we will consider integer edge-weights and integer values of terminal vertices since we are interested in integer linear functions only. To make the *BMD- representation canonical, a further restriction on the edge-weights is necessary. The edge-weights on the low- and high-edge of any nonterminal vertex must have greatest common divisor 1. Additionally, weight 0 appears only as a terminal value, and if either outgoing edge of a node points to this terminal node, the weight of the other outgoing edge is 1. Note that because of reduction a terminal node with value 0 can not appear at a high-edge. 2.1. Basic Operations on *BMDs We now present in detail an algorithm, which is the basis for verification by backward construction. The method of backward construction is based on substituting variables in a *BMD for a function f by a *BMD for another function g. According to Equation (1), this substitution is based on the x . A sketch of the substitution algorithm can be seen in Figure 1. It starts with a *BMD(f) and a *BMD(g), which is a notation for the *BMD representing the function f and g, respectively. Additionally, the variable x that is to be substituted, is handed to the algorithm. If the top variable of *BMD(f) is not the variable to be substituted, the algorithm calls itself recursively with the low- and high- successor. If the variable x is found, an integer multiplication and an integer addition is performed (line 3). Note that both operations have exponential worst case behavior [8]. In the following we show, that under specific conditions, the worst case will not occur. The key to this is a *BMD with a specific structure, which we call Sum Of weighted Variables (SOV). The *BMD for the unsigned integer encoding a is an example of a *BMD in SOV-structure. Its outline is given in Figure 2. Such subst( *BMD(f), x/*BMD(g) )f 1 check for terminal cases; return Figure 1. Substitution algorithm Figure 2. SOV-structure a *BMD has the properties that there is exactly one nonterminal node for each variable and the high-edge of each non-terminal node is labeled with weight 1 pointing directly to a terminal node labeled with the value of the corresponding variable. Because of these properties the substitute algorithm simplifies considerably: Let us assume f is a *BMD in SOV, g is a *BMD not in SOV and the variable x is at an arbitrary position in the middle of f . In a first phase lines and 4 are in the middle of f . In a first phase lines 2 and 4 are executed continually until x is reached. Line 3 calls a multiplication and an addition. The multiplication returns its result immediately since high(f) is a terminal node. The complexity of the addition depends on the *BMDs f and g. But again, the SOV-structure reduces the number of execution steps, since f - y is a terminal node for all variables y. Later on in the paper g will be a *BMD of known size, so we can give a more detailed analysis. Returning from the recursive calls of line 4, line 5 does no recursive calls because of high(f). For the same reason line 7 ends immediately and the *BMD E4 in line 8 has a depth of 1. For these reasons, maintaining the SOV-structure as long as possible reduces the number of execution steps of the substitute algorithm. Since substitution is the basic operation in the method of verification by backward construction the overall number of execution steps reduces considerably, too. In addition, the SOV-structure simplifies the analysis of the complexity of the method extremely. 3. The Class of Wallace-Tree like Multipliers Let be two n-bit numbers binary representation. (If n is not a power of 2 extend n to a power of 2 and set the superfluous inputs to zero.) The multiplication of a and b is then equivalent to summing up n partial products shifted by j positions to the left for We assume in the following, that we can divide the circuit into two parts. One part calculates the partial product bits a i 1. The other part adds these partial products bits to the final result using (only) fulladder-cells. How these cells are arranged, e.g. as a 3to2 or as a 4to2 reduction, as a binary tree or as a linear list, has no influence on the proof. For a more detailed description of representatives of Wallace-tree like multipliers please see [10], [4]. 4. Complexity of Backward Construction In this section we analyze the complexity of the method of verification by backward construction, as introduced in [9]. In Subsection 4.1 we explain the principles of the method. In Subsection 4.2 the backward construction of the *BMDs for the adder part of the circuit is analyzed. In Sub-section 4.3 we take the partial product bits into account and discuss the resulting complexity. In the last subsection we put together these parts to get the final overall complexity. 4.1. The Method of Backward Construction In general, the method of verification by backward construction works as follows: First, to each primary output a distinct variable is assigned. In the next step the *BMD for the output word is constructed by weighting and summing up all *BMDs of these variables according to the given output encoding. (We assume an unsigned integer encoding of the outputs.) Note, that we obtain the SOV-structure for the resulting *BMD. Then, a cut is placed, crossing all primary outputs of the circuit. The cut is moved towards the primary inputs, such that the output lines of each gate move onto the cut according to some reverse topological order of the gates. While moving the cut towards the primary inputs of the cir- cuit, the *BMD is constructed as follows: For the next line of the circuit crossing the cut, which is also an output line of some gate of the circuit, find its corresponding variable in the *BMD constructed so far. Substitute this variable by the (output) function of the gate, delete the output line from the cut, and add all input lines of the gate to the cut. Fi- nally, the cut has been moved to the outputs of AND gates, computing the partial product bits. The very final step is the substitution of corresponding variables in the obtained *BMD by the partial product bits. After this step, we obtain the *BMD representing the multiplication of a and b if the circuit is correct. As mentioned above, the cut is moved according to a reverse topological order of the gates. In general, there exists more than one such order. As we shall see, our proofs are not based on the knowledge of a specific reverse topological order, as long as we substitute the variables representing the fulladder outputs before the variables representing the initial partial product bits. Nevertheless, we can give one specific class of variable orders, for that our statements hold even if we apply a different topological order. 4.2. Constructing the *BMD for the Adder Part In this subsection we analyze the costs of constructing the *BMD for the adder-part of the multiplier. The first lemma shows, that the substitution of the sum- and carry-output of the same fulladder in a *BMD in SOV maintains the SOV-structure. This means, that SOV is invariant against these substitutions. For that reason, we have to analyze the substitution costs for a single fullad- der (Lemma 4.2) to get the overall costs (Theorem 4.1 and Theorem 4.2). Lemma 4.1 Let F be a *BMD in SOV and let X denotes the set of variable of F . Let x be two variables in representing the sum- and carry-output of the same fulladder. The inputs to the fulladder are represented by . Independent of the order on the variable sets of the *BMDs involved in the substitution process, it holds that substituting x F by the *BMDs for the functions and: 1. F 0 is in SOV. 2. The terminal value of the high-successors of the nodes for is the same as that for the high- successor of the node for x i in F . Proof: The proof is based on a merely functional argu- ment. For the sum- and carry-output we get the following functions: \Gamma2x l xm by expressing the boolean operations \Phi and - by integer addition, subtraction and multiplication, i.e., x \Phi Let the variables x in the *BMD F represent the sum- and carry-output of the same fulladder. These variables have weights w and 2w, respectively. The substitution of x yields the following function f 0 as can easily be verified. The rest of the variables in f are not affected since x k ; x l ; xm 62 X and the weighted variables only once in f . Therefore the *BMD representing function f 0 , must be in SOV again, independent of the variable order. Furthermore all three high- edges of the nodes for x k ; x l ; xm in F 0 point to a terminal node with value w. This completes the proof. 2 Lemma 4.1 we can be sure, that the SOV-structure of the *BMD is maintained, if we have fulladders as basic cells and if we substitute the variables corresponding to both outputs of a particular fulladder directly one after another. This is independent of the chosen variable order and the chosen reverse topological order of the DAG representing the adder part of the multiplier. Additionally, at each point of the substitution process, we know exactly the size of the x x Figure 3. Representation of substituting x i by Su and x j by Ca. x x Figure 4. *BMD after substitution of variable *BMD: After having processed the next fulladder, the size is increased by one. We proceed now with calculating the costs by counting the calls of the substitute algorithm. Lemma 4.2 Let F be a *BMD in SOV with size defined as in Lemma 4.1. Substituting is bounded by O(jF with respect to time, independent of the variable order or order of substitutions. Proof: We denote the *BMDs for by Su and Ca. We consider first the substitution of x i by Su. The substitution process can be visualized in Figure 3. (Note that the order between does not matter because the sum- and carry- function are totally symmetric. Therefore, the nonterminal node labels are omitted.) The substitute-algorithm calls itself recursively until it reaches the node in F labeled with variable x i . Obviously, the number of recursive subst-calls is bounded by O(jF j). At the node labeled with x i , the operation F low(x has to be carried out, the *BMDs to which the low- and high-edge of node x i point. Since F is in SOV, F high(x i ) is a terminal node and the call to mult ends im- mediately. Since F low(x i ) is in SOV, too, and Su is of constant size, the addition is bounded by O(jF j). For all other recursive subst-calls, there will be only a constant number of mult- and add-calls because of the SOV-structure, as can easily be verified. We get an overall bound of O(jF for the substitution of x i by Su. The *BMD after this substitution can be seen in Figure 4 for a variable order For the substitution of x j by Ca, analogous arguments hold, except for the fact, that the SOV-structure of F is destroyed by Su (see Figure 4). First of all, this leads to some additional recursive subst-calls. But this will be only a constant because Su has constant depth. Furthermore, we get some additional calls to add and mult during some of the recursive subst-calls, when the nodes labeled with variables x k ; x l ; xm meet each other. But this number is bounded by a constant value too, since Su and Ca have constant depth. Therefore, since the resulting *BMD is of size jF j according to Lemma 4.1, we get a time bound of O(jF the substitution of x i by Su and x j by Ca independent of the variable order on X and (X n fx g. The proof for first substituting x j by Ca and then x i by Su is analogous. 2 Lemma 4.1 and 4.2 leads directly to the space bound for the construction of the *BMD for the adder part. Theorem 4.1 Constructing the *BMD for the adder part of the multiplication circuit using substitution is bounded by respect to space. This is independent of the chosen reverse topological order for the fulladder-cells. Proof: The *BMD F 0 with which we start has size O(n) as follows since there is one nonterminal node for each variable representing a primary output. The resulting *BMD constructed for the adder-part has size O(n 2 One nonterminal node for each variable representing an initial partial product bit. The exact size depends on the chosen realization of the multiplier. With Lemma 4.1 and 4.2 we have the space bound O(n 2 ). 2 After analyzing the space requirements for the substitu- tions, we now consider the time requirements. Theorem 4.2 Constructing the *BMD for the adder-part using substitution is bounded by O(n 4 ) with respect to time, independent of the chosen reverse topological order for the fulladder-cells. Proof: For the proof we first count the number of ful- ladder elements. Depending on the realization, the exact number of these elements differs. Asymptotically there are cells, forming the adder-part of any meaningful multiplier. Therefore the number of execution steps has an upper bound of the size of the initial *BMD according to the proof of Theorem 4.1. This sum is figured out as follows: with 4.3. Substitution of the Partial Products Up to this point we analyzed the complexity of the method of verification by backward construction for the part of the multiplier circuit that adds the initial partial product bits to obtain the result of the multiplication. In the sequence we analyze the costs of the final step, substituting the partial product bits into the *BMD. By doing so, we Figure 5. *BMD after substituting some partial product bits. Figure 6. Final *BMD for the multiplication. will destroy the SOV-structure of the *BMD. Our starting point is the *BMD constructed up to the outputs of the AND gates. It has size m since there are n 2 partial product bits. In fact, it holds with m from the proof of Theorem 4.2 and with #FA denoting the number of fulladder cells. We assume in the following a fixed order among the aand b-word, assigned to the primary inputs of the circuit. In fact, the variable order within the a i and b j is of no interest as long as all variables forming one input word are before the variables forming the second input word. Otherwise the final *BMD may be larger than 2n [2]. We define a low-path of a *BMD F as the path in F from the root to a leaf, consisting of only low-edges. Note, that there is only one such path in a *BMD. We now show, that the intermediate *BMDs have a structure like that in Figure 5. (The small box on an edge denotes the multiplicative factor.) All nodes with a terminal high-successor in the upper diagonal line, the low-path, are marked with a variable x k , assigned to a fulladder input line. These lines are also output lines of AND gates. Nodes with a nonterminal high-successor are labeled with a variable a i (also in the low-path). The high-successors of them are labeled with a variable b j . These are located in the lower diagonal line. The final *BMD is structured as shown in Figure 6. Theorem 4.3 Let F be the *BMD constructed for the adder part. Substitution of the variables of F by the *BMDs for the initial partial product bits a i \Delta b j is bounded by O(n 2 ) with respect to space and O(n 4 ) with respect to time independent of the variable order in F and independent of the chosen reverse topological order for the AND gates. Proof: Let F 0 denote an intermediate *BMD generated after the substitution of some initial partial product bits. If the node labeled with variable x has not yet been substituted, it must be on the low-path of F 0 . Now we consider the substitution of node x by the *BMD for an initial partial product bit a i Obviously, the number of recursive calls to the substitute algorithm (Figure 1) is bounded by O(jF 0 j). During the last of the recursive subst calls, i.e., at node x, the following operations have to be carried out: The call to mult ends immediately, since x has a terminal high-successor because of the SOV-structure of F at the beginning of the substitution process. For the add calls we have to distinguish two different cases. 1. No nodes have been substituted by initial partial products bits with variable a i until now. Since node x (one non-terminal node) has to be substituted by \LambdaB MD(a i nonterminal nodes) the size of F 0 increases by 1 if no node for variable b j can be shared, and it remains unchanged otherwise If variable a i comes after the predecessor of variable x in the variable order, \LambdaB MD(a i reaches its final position in *BMD F 0 by calls to add during the final substitute call, i.e.line 3 of Figure 1. The number of these calls is obviously bounded by O(jF 0 j). If variable a i comes before the predecessor of variable x in the variable order, a i reaches its final position in *BMD F 0 by calls to add during resolving previous substitute calls (line 8 of Figure 1). The number of these add calls is con- stant, as one can easily make sure. Furthermore, there is one call to MakeNode and one call to mult for each of the previous subst calls (lines 6 and 7 of Figure 1). 2. There exists already a node for variable a i , i.e., some nodes of F have already been substituted by initial partial product bits a i so far. If there exists also a node for b j , e.g. created during the substitution of an initial partial product bit a l \Delta b j , the size of F 0 during the substitution may decreases by 1, if the node b j can be shared, or remains unchanged, otherwise. The only difference to the first case is, that there are an additional number of at most n add calls, because of the position of b j in the variable order among the b i 1 (the worst case occurs, if b we have a bound on the total number of calls of O(jF Cases 1 and 2 together give a bound on the total number of algorithm calls of O(jF 0 j) for the substitution of node x by \LambdaB MD(a i we have at most an increase of n in the size of the starting BMD F (the first n initial partial product bits of the substitution process all have different a-variables) and the size of F is O(n 2 ), the size of the intermediate *BMDs during substitution is bounded by This proves the first part of the theorem. Fur- thermore, since the number of substitutions is bounded by and the size of the starting *BMD F is O(n 2 ) we get an overall time bound of This proves the second part of the theorem and we have completed the proof. 2 4.4. Complexity of Backward Construction With Theorema 4.1, 4.2 and 4.3, we conclude, that the method of backward construction applied to the class of Wallace-tree like multipliers is bounded by O(n 2 ) with respect to space and by O(n 4 ) with respect to time. These bounds do (largely) not depend on the chosen variable order during the single substitution steps. Additionally, these bounds do also not depend on the order of substitutions as long as we first substitute all fulladder-cells and afterwards the initial partial product bits. If we allow one further restriction on the variable order- ing, we attain to be totally free in the substitution ordering across both circuit parts. This means, there is a class of variable orders, with that we can substitute an initial partial product bit before having processed all fulladder cells. Nevertheless, the complexity remains polynomial. We only have to regard the reverse topological order over the whole multiplier circuit. We define an (x; a; b) variable order to be any variable order, which has at first all x-variables, at second all a- variables and at third all b-variables The x-variables denote the fulladder in- and outputs and the a- and b-variables denote the variables of the input words to the multiplier. Then we can give the following theorem: Theorem 4.4 Given an (x; a; b) variable order, the method of backward construction, applied to the class of Wallace- tree like multipliers, is bounded by O(n 2 ) with respect to space and O(n 4 ) with respect to time independent of the reverse topological order on the circuit. Proof: The *BMD decomposes into two parts. The upper part consists of x-variables only, and is in SOV. (Despite the fact, that the last high-edge points to a nonterminal node, labeled with an a-variable.) The lower part consists of aand b-variables. We have to consider two cases: 1. Substituting Su (Ca), we first have to find the substituted variable. That will be found in the upper part of the *BMD. Depending on the variable ordering within the x k , the add operation of the substitution continues downward the *BMD. Maximally, it reaches the edge, pointing to a 0 (respectively any a i , which ever is the 'smallest' a-variable so far). There, the recursive calls terminate, since x k ! a i , for all k; i. Theo- rema 4.1, 4.2 can be applied for the costs, considering only the size of the upper part of the *BMD. 2. Substituting a partial product, Theorem 4.3 can be applied One open problem is the complexity if we allow, that initial partial product bits are substituted before all fulladder- cells are processed, but with a different variable order as that from Theorem 4.4, i.e., not all x-variables are at the beginning of the variable order. We expect, that the complexity remains still polynomial, since the structure of the resulting *BMD is similar to the one used here. 5. Conclusions and Future Research In this paper we analyzed the complexity of the method of verification by backward construction applied to the class of Wallace-tree like multipliers. We gave a formal proof of polynomial upper bounds on run-time and space requirements with respect to the input word sizes for that method. Note, that until now, only experimental data has been given to show the feasibility of the method. A conclusion of our results is, that we can prematurely detect design errors by watching the *BMD, if these errors result in non correct *BMD sizes. Assume, we have processed only gates in the adder-part of the circuit. After processing the ith gate the *BMD has size jF If the actual *BMD has not that size, there is an error. Assume now, that we considered at least some initial partial product bits of the circuit. Then, the size of the *BMD depends on which initial partial product bits have been substituted. Not considering an accurate size bound, the *BMD must not be larger than jF 0 j +#FA+n or smaller than 2n, at any time. Future research directions are to remove the restrictions, e.g. for variable order we mentioned above, and take a look at integer dividers, which could not be verified by backward construction so far. --R HSIS: A BDD-Based Environment for Formal Verification Verification of Arithmetic Functions with Binary Moment Diagrams. Verification of Arithmetic Circuits with Binary Moment Diagrams. An Easily Testable Optimal-Time VLSI- Multiplier On the Complexity of VLSI Implementations and Graph Representations of Boolean Functions with Application to Integer Multiplication. Hybrid Decision Diagrams - Overcoming the Limitations of MTBDDs and BMDs Note on the Complexity of Binary Moment Diagram Representations. Efficient Construction of Binary Moment Diagrams for Verifying Arithmetic Cir- cuits Recursive Implementation of Optimal Time VLSI Integer Multipliers. A suggestion for a fast multiplier. --TR Graph-based algorithms for Boolean function manipulation On the Complexity of VLSI Implementations and Graph Representations of Boolean Functions with Application to Integer Multiplication Sequential circuit verification using symbolic model checking Symbolic Boolean manipulation with ordered binary-decision diagrams HSIS Verification of arithmetic circuits with binary moment diagrams Efficient construction of binary moment diagrams for verifying arithmetic circuits Hybrid decision diagrams PHDD Logic Synthesis and Verification Algorithms K*BMDs Verification of Arithmetic Functions with Binary Moment Diagrams

integer multipliers;formal verification;equivalence checking;backward construction;multiplicative binary moment diagrams

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A Compared Study of Two Correctness Proofs for the Standardized Algorithm of ABR Conformance.

The ABR conformance protocol is a real-time program that controls dataflow rates on ATM networks. A crucial part of this protocol is the dynamical computation of the expected rate of data cells. We present here a modelling of the corresponding program with its environment, using the notion of (parametric) timed automata. A fundamental property of the service provided by the protocol to the user is expressed in this framework and proved by two different methods. The first proof relies on inductive invariants, and was originally verified using theorem-proving assistant COQ. The second proof is based on reachability analysis, and was obtained using model-checker HYTECH. We explain and compare these two proofs in the unified framework of timed automata.

Introduction Over the last few years, an extensive amount of research has been devoted to the formal verification of real-time concurrent systems. Basically, formal proof methods belong to two different fields: theorem proving and model-checking. For all these methods, a first crucial phase is to build a formal description of the system under study. With theorem proving, the description consists of a set of formulas, and the verification is done using logical inference rules. With model-checking, the description is a graph and the verification is performed using systematic search in the graph. The theorem-proving methods apply to more general prob- lems, but often need human interaction while model-checking methods are more mechanical, but apply to a restricted class of systems. These methods thus appear as complementary ones, and several authors advocated for the need to combine them together [17, 21, 24]. This is now an exciting and ambitious trend of research, but still a very challenging issue. We believe that a preliminary useful step towards this objective is to evaluate comparatively the respective merits Partially supported by Action FORMA (French Programme DSP- STTC/CNRS/MENRT) and RNRT Project Calife and shortcomings of such methods, not only at a general abstract level, but on difficult practical examples. From such a comparison, one may hopefully draw some general lessons for combining methods in the most appropriate way. Besides the comparison may be interesting per se, as it may contribute to a better understanding of the specific treated problem. We illustrate here the latter point by performing a compared analysis of two correctness proofs obtained separately for a sophisticated real-life protocol. More precisely, we propose a comparison between two proofs of the Available Bit Rate (ABR) conformance algorithm, a protocol designed at France Telecom [15] in the context of network communications with Asynchronous Transfer Mode (ATM). The first proof [19, 18] was obtained in the theorem proving frame-work using Floyd-Hoare method of inductive invariants and the proof assistant Coq [7]. The second proof [8] was based on the method of reachability analysis, and used the model-checking tool HyTech [14]. In order to compare these methods more easily, we formulate them in the unified framework of "p-automata" [8], a variant of parametric timed automata [5]. The choice of such a framework (motivation, differences with other models, specific problems) is discussed at the end of the paper (section 7). Context and motivation of the case study. ATM is a flexible packet-switching network architecture, where several communications can be multiplexed over a same physical link, thus providing better performances than traditional circuit-switching networks. Several types of ATM connexions, called ATM Transfer Capabilities (ATC), are possible at the same time, according to the dataflow rate asked (and paid) for by the service user. Each mode may be seen as a generic contract between the user and the network. On one side, the network must guarantee the negociated quality of service (QoS), defined by a number of characteristics like maximum cell loss or transfer delay. On the other side, data packets (cells) sent by a user must conform to the negociated traffic parameters. Among other ATCs, Deterministic Bit Rate connexions operate with a constant rate, while Statistical Bit Rate connexions may use a high rate, but only for In some of the most recently defined ATCs, like Available Bit Rate (ABR), the allowed cell rate (Acr) may vary during the same session, depending on the current congestion state of the network. Such ATCs are designed for irregular sources, that need high cell rates from time to time, but may reduce their cell rate when the network is busy. A servo-mechanism is then proposed in order to let the user know whether he can send data or not. This mechanism has to be well defined, in order to have a clear traffic contract between user and network. The conformance of cells sent by the user is checked using an algorithm called GCRA (generic control of cell rate algorithm). In this way, the network is protected against user misbehaviors and keeps enough resources for delivering the required QoS to well behaved users. In fact, a new ATC cannot be accepted (as an international standard) without an efficient conformance control algorithm, and some evidence that this algorithm has the intended behavior. In this paper, we study the particular case of ABR, for which a simple "ideal" algorithm of conformance control can be given. This algorithm is very inefficient, in terms of memory space, and only approximation algorithms can be implemented in practice. The correctness proof for these approximation algorithms consists in showing that their outputs are never smaller than the outputs computed by the ideal algorithm. More precisely, we focus on an algorithm, called due to Christophe Rabadan at France-Telecom, now part of the I.371.1 standard [15]. We describe this algorithm and prove its correctness with respect to the ideal algorithm, using the two methods mentioned above. The plan of the paper is as follows: section 2 gives an informal description of the problem of ABR conformance control and section 3 presents an incremental algorithm used henceforth as a specification. Section 4 describes a general modelling framework, called "p-automata", and expresses the two different proof methods in this context; the description of ABR algorithms as p-automata is given in section 5. Verification with the two proof methods is done in section 6. A discussion on constraints linked to the modelling with p-automata follows in section 7, then a comparison between the two methods is given in section 8. We conclude in section 9. Overview of ABR update DGCRA resource management cell user network ACR data cell Fig. 1. conformance control An abstract view of the ABR protocol is given in Figure 1. The conformance control algorithm for ABR has two parts. The first one is quite simple and is not addressed here. It consists of an algorithm called DGCRA (dynamic GCRA), which is an adaptation of the public algorithm for checking conformance of cells: it just checks that the rate of data cells emitted by the user is not higher than a value which is approximately Acr, the allowed cell rate. Excess cells may be discarded by DGCRA. In the case of ABR, the rate Acr depends on time: its current value has to be known each time a new data cell comes from the user. Thus, the complexity lies in the second part: the computation of Acr(t) ("update" in Figure 1), where t represents an arbitrary time in the future, at which some data cell may arrive. The value of Acr(t) depends on successive values carried by resource management (RM) cells transmitted from the network to the user 1 . Each such value corresponds to some rate Acr, that should be reached as soon as possible. 2.1 Definition of ideal rate Acr(t) We consider the sequence of values (R carried by RM cells, ordered by their arrival time (r i). By a slight abuse of notation, the cell carrying R i will be called . The value Acr(t) depends only on cells R i whose arrival time r i occur before t. Intuitively, Acr(t) should be the last value R i received at time t, i.e. Rn with ug. Unfortunately, because of electric propagation time and various transmission mechanisms, the user is aware of this expected value only after a delay. Taking the user's reaction time - observed by the control device into consideration, that is, the overall round trip time between the control device and the user, Acr(t) should then be Rn with may vary in turn. ITU-T considers that a lower bound - 3 and an upper bound - 2 for - are established during the negociation phase of each ABR connection. Hence, a cell arriving from the user at time t on DGCRA may legitimately have been emitted using any rate R i such that i is between last(t rate less than or equal to any of these values, or, equivalently less than or equal to the maximum of them, should then be allowed. Therefore, Acr(t) is taken as the maximum of these R i . are the successive arrival times of RM cells, such that : (in other words, depend on t) and if Rm ; are the corresponding rate values, then the expected rate is The case where obtained when no new RM cell has arrived between A program of conformance control based on this specification would need to compute, at each instant s, the maximum of the rate values of all the RM Actually, RM cells are sent by the user, but only their transmission from the network to the user is relevant here; details are available in [22] cells received during interval which may be several hundreds on an ATM network with large bandwidth. ITU-T committee considered that an exact computation of Acr(t) along these lines is not feasible at reasonable cost with current technologies. 2.2 Algorithm efficient computation of an approximation of A more realistic algorithm, called due to C. Rabadan, has been proposed by France-Telecom and adopted in I.371.1. It requires the storage of no more than two RM cell rates at a time, and dynamically computes an approximation value A of Acr(t). Two auxiliary variables, Efi and Ela, are used in the program for storing these two RM cell rates and two dates, tfi and tla, are associated with Efi and Ela respectively. When the current time s reaches tfi, A is updated with value Efi, and only one RM cell rate is kept in memory (Efi:=Ela, tfi:=tla). When a new RM cell arrives auxiliary variables Efi, Ela, tfi and tla are updated according to several cases, depending on the position of r k w.r.t. tfi and tla, and R k w.r.t. Efi and Ela. The full description of B 0 is given in section 5.3 (cf. pseudocode in appendix A). Correctness of B 0 . Before being accepted as an international standard (norm ITU I-371.1), algorithm B 0 had to be proved correct with respect to Acr(t): it was necessary to ensure that the flow control of data cells by comparison with A rather than Acr(t) is never disadvantageous to the user. This means that when some data cell arrives at time t, A is an upper approximation of Acr(t). 3 Incremental computation of the rate Acr(t) As explained above, the correctness of algorithm B 0 mainly relies on a comparison between the output value A of B 0 and the ideal rate Acr(t). The initial specification of Acr(t) given in section 2.1, turns out to be inadequate for automated (and even manual) manipulation. Therefore, it is convenient to express the computation of Acr(t) under an algorithmic form as close as possible to B 0 , where, in particular, updates are performed upon reception of RM cells. Monin and Klay were the first to formulate such an incremental computation using a higher-order functional point of view [19]. We recall their algorithm, then adopt a slightly different view, which is more suited to the modelling framework described subsequently. 3.1 A higher-order functional view: algorithm F We now consider an algorithm, called F , that stores, at current time s, an estimation E of Acr, and updates it at each arrival of an RM cell. More precisely, when receiving a new cell R k at time r k , algorithm F computes the new function 0 from the former function E, depending on the situation of r k with respect to the argument t of E: It can be shown that the value E(t) at current time s is equal to the ideal rate Acr(t), as defined in section 2.1, for each t such that t - s proof of this statement can be found in [19], although a much shorter one can be deduced from the justification in section 3.2 below. Algorithm F can thus be seen as a higher order functional program which computes the (first-order) functions E. It might be implemented using the general notion of "closures". However, the functions E are constant over time intervals [a; b[, where a and b are of the form r and such that there is no value of this form between a and b. Hence, they can be encoded using well chosen finite lists of pairs (t; e), where t is a time and e is the rate E(t). Such lists can be seen as schedulers for the expected rate: when current time s becomes equal to t, the expected rate becomes e. The length of these schedulers may be several hundreds on networks with large bandwidth (entailing a high frequency of events R k ), even if only the relevant pairs (t; e) are kept. In contrast, algorithm can be considered as using a scheduler of length at most two (containing pairs (tfi,Efi), (tla,Ela)). 3.2 A parametric view: algorithm I t Another way of looking at F is to consider t as a parameter (written t) whose value is fixed but unknown, and represents a target observation time. Function F becomes an "ideal algorithm" I t , which updates a value E(t), henceforth written , upon reception of RM cells as follows: It is now almost immediate that the value E t computed by I t , is equal to the ideal rate Acr(t), as defined in section 2.1, when t. Indeed, as s increases, k takes the values with takes accordingly the values In particular, when t, we have E t The correctness property of B 0 with respect to Acr(t) can now be reformulated as follows, where A is the output value of B 0 and E t the value computed by algorithm I t when This property is referred to as U t and should be proved for all values of parameter t. Henceforth, the parameter t is left implicit and we simply write U instead of U t . Accordingly, we write E instead of E t and I instead of I t . From this point on, algorithm I plays the r-ole of specification. Modelling Framework and Proof Methods It should be clear from the informal definition of the control conformance algorithm above, that the ability to reason about real time is essential. The expressions A and E, involved in property U , denote quantities that evolve as time goes, and should be considered as functions of the current time s. In order to express and prove property U , we need a formal framework. The model of p-automata which was chosen in this paper, turns out to be sufficient for our purposes of formal description and verification of the considered system. We describe this model hereafter as well as two proof methods for verifying properties in this context. 4.1 p-automata The model of parametric timed automata, called p-automata for short, is an extension of timed automata [4] with parameters. A minor difference with the classical parametric model of Alur-Henzinger-Vardi [5] is that we have only one clock variable s and several "discrete" variables w 1 ; :::; wn while, in [5], there are several clocks and no discrete variable. One can retrieve (a close variant of) Alur- Henzinger-Vardi parametric timed automata by changing our discrete variable w into s\Gammaw i (see [12]). Alternatively, p-automata can be viewed as particular cases of linear hybrid automata [2, 3, 20], and our presentation is inspired from [14]. The main elements of a p-automaton are a finite set L of locations, transitions between these locations and a family of real-valued variables. In the figures, as usual, locations are represented as circles and transitions as labeled arrows. Variables and constraints. The variables of a p-automaton are: a tuple p of parameters, a tuple w of discrete variables and a universal clock s. These real-valued variables differ only in the way they evolve when time increases. Parameter values are fixed by an initial constraint and never evolve later on. Discrete variable values do not evolve either, but they may be changed through instantaneous updates. A universal clock is a variable whose value increases uniformly with time. A (parametric) term is an expression of the form w+ \Sigma i ff or \Sigma i ff are constants in Z. With the usual convention, an empty set of indices corresponds to a term without parameter. A convex constraint is a conjunction of (strict or large) inequalities between terms. A p-constraint is a disjunction of convex constraints. An update relation is a conjunction of inequalities between a variable and a term. It is written w 0 fi term, where w 0 is a primed copy of a discrete variable and term is a parametric term. As usual, x is implicit if x 0 does not appear in the update relation. Locations and transitions. With each location ' 2 L is associated a convex constraint called location invariant. Intuitively, the automaton control may reside in location ' only while its invariant value is true, so invariants are used to enforce progress of the executions. When omitted, the default invariant of a location is the constant true. A transition in a p-automaton is of the form: h'; '; a; '; ' 0 i, where a is the label of the transition, ' the origin location, ' 0 the target location, ' a guard and ' an update relation. The guard is a convex constraint. Guards may additionnaly contain the special expression asap. In our model, a location ' is called urgent if all transitions with origin ' contain asap in the guard (no time is allowed to pass in such a location). Otherwise, it is called stable. A sequence of transitions is called complete if it is of the form h'; are stable locations and all intermediate locations ' i are urgent locations. With the usual convention, the case corresponds to a single transition between stable locations. Executions. The executions of a p-automaton are described in terms of a transition system. A (symbolic) state q is defined by a formula (- /(s; p; w), where - is a variable ranging over the set of locations, and / is a p-constraint. We are primarily interested in states for which / implies the location invariant I ' . Such states are called admissible states. A p-zone, represented by a formula \Pi(-; s; p; w), is a finite disjunction of states. Alternatively it can be regarded as a finite set of states. The initial state is q init some p-constraint / init and initial location ' init . Initial location is assumed to be stable. Since parameter values are fixed from the initial state, we often omit the tuple p of parameters from the formulas. For a p-automaton, two kinds of moves called action moves and delay moves are possible from an admissible state - An action move uses a transition of the form h'; '; a; '; ' 0 i, reaching an admissible state equivalent to 9w /(s; w) - '(w; w 0 ). Informally, discrete variables are modified according to update relation ' and the automaton switches to target location ' 0 . This notion of action move generalizes in a natural way to the notion of action move through a complete sequence of transitions. Note that action moves are instantaneous: the value s of the clock does not evolve. - A delay move corresponds to spending time in a location '. This is possible if ' is stable and if the invariant I ' remains true. The resulting admissible state is q (nothing else is changed during this time). A successor of a state q is a state obtained either by a delay or an action move. For a subset Q of states, P ost (Q) is the set of iterated successors of the states in Q. It can be easily proved that the class of p-zones is closed under the P ost operation. As a consequence, P ost (Q) is a p-zone if Q is a p-zone and computation of P ost terminates. The notion of predecessor is defined in a similar way, using operator P re. Synchronization. From two or more p-automata representing components of a system, it is possible to build a new p-automaton by a synchronized product. Let A 1 and A 2 be two p-automata with a common universal clock s. The synchronized product (or parallel composition, see e.g. [14]) A 1 \Theta A 2 is a p-automaton with s as universal clock and the union of sets of parameters (resp. discrete variables) of A 1 and A 2 as set of parameters (resp. discrete variables). Locations of the product are pairs of locations from A 1 and A 2 respectively. A are stable. The p-constraints associated with locations (invariants, initial p-constraint) are obtained by the conjunction of the components p-constraints. The automata move independently, except when transitions from A 1 and A 2 have a common synchronization label. In this case, both automata perform a synchronous action move, the associated guard (resp. update relation) being the conjunction of both guards (resp. update relations). 4.2 Proof methods We now present two proof methods for proving a property \Pi(-; s; w) in the framework of p-automata. This property is assumed to involve only stable loca- tions, and to hold for all parameter valuations satisfying the initial p-constraint of the modeled system. The first method, based on Floyd-Hoare method of as- sertions, consists in proving that \Pi is an inductive invariant of the model (see, e.g., [27]). The second one, based on model-checking techniques, consists in characterizing the set of all the reachable states of the system, and checking that no element violates \Pi. Inductive invariants. To prove \Pi by inductive invariance, one has to prove that \Pi holds initially, and is preserved through any move of the system: either an action or a delay move. Formally, we have to prove: - For any transition h'; '; a; '; ' 0 i between two stable locations ' and A similar formula must also be proved for complete sequences of transitions. - For any stable location ': I ' (s; w) - I ' (s Reachability analysis. Since P ost (q init ) represents the set of reachable states of a p-automaton, property \Pi holds for the system if and only if P ost (q init ) is contained in the set Q \Pi of states satisfying \Pi. Equivalently, one can prove the emptiness for the zone P ost (q init is the set of states violating \Pi. Also note that the same property can be expressed using P re by re (Q:\Pi 5 Description of the system Algorithms I and B 0 will be naturally represented by p-automata. However, they are reactive programs: they react when some external events occur (viz., upon reception of an RM cell) or when current time s reaches some value (e.g., tfi). Thus, in order to formally prove correctness property U , we need to model as a third p-automaton, an appropriate environment viewed as an event generator. Finally, in the full system obtained as a synchronized product of the three au- tomata, we explain how to check the correctness property. All these p-automata share a universal clock s, the value of which is the current time s. Without loss of understanding (context will make it clear), we often use s instead of s. 5.1 A model of environment and observation As mentioned above, the p-automaton A env modeling environment (see Figure 2) generates external events such as receptions of RM cells. It also generates a "snapshot" action taking place at time t. Note that for our purpose of verification of U , it is enough to consider the snapshot as a final action of the system. The variables involved are the parameter t (snapshot time) and a discrete variable R representing the rate value carried by the last received RM cell. In the initial location W ait, a loop with label newRM simulates the reception of a new RM cell: the rate R is updated to a non deterministic positive value (written R' ? 0, as in HyTech [14]). The snapshot action has s=t as a guard, and location W ait is assigned invariant s - t in order to "force" the switch to location EndE. Wait newRM R snapshot Fig. 2. Automaton Aenv modeling arrivals of RM cells and snapshot 5.2 Algorithm I Algorithm I computes E in an incremental way as shown in the table of section 3.2. Variable E is updated at each reception of an RM cell, until current time s becomes equal to t. More precisely, algorithm I involves variable R and parameter t (in common with A env ) and, in addition: - the two parameters - 3 and - 2 (representing the lower and upper bounds of the transit time from the interface to the user and back), - the "output" variable E (which equates with the ideal rate Acr(t) when Initially, E and R are equal. Algorithm I reacts to each arrival of a new RM cell with rate value R by updating E. There are three cases, according to the position of its arrival time s with respect to t- 2 and t- 3 (see section 3.2): 1. If s - t- 2 , E is updated to the new value R: 2. If t- the new rate becomes E'=max(E,R). To avoid using function max, this computation is split into two subcases: 3. If s ? t- 3 , the rate E is left unchanged: Algorithm I terminates when the snapshot takes place (s=t). In the following, we will sometimes write the updated output value E' under the ``functional'' form I(s; R; E). Automaton AI . Algorithm I is naturally modeled as p-automaton A I (see Figure 3). Initial location is Idle, with initial p-constraint R. The reception of an RM cell is modeled as a transition newRM from location Idle to location UpdE. This transition is followed by an urgent (asap) transition from UpdE back to Idle, which updates E depending on the position of s w.r.t. t- 2 and t- 3 , as explained above. Without loss of understanding, transitions from UpdE to Idle are labeled [I1], [I2a], [I2b], [I3] as the corresponding operations. Observation of the value E corresponds to the transition snapshot from Idle to final location EndI. 5.3 Algorithm computation of an approximation We now give a full description of algorithm B 0 (cf. pseudo-code in appendix). Like I, algorithm B 0 involves parameters - 3 and - 2 and variable R. However, note that t is not a parameter for B 0 . It computes A (intended to be an approximation of E) using five specific auxiliary variables: - tfi and tla, which play the role of fi-rst and la-st deadline respectively, - Efi, which is the value taken by A when current time s reaches tfi, - Ela, which stores the rate value R carried by the last received RM cell. - Emx, a convenient additional variable, representing the maximum of Efi, Ela. Initially, s=tfi=tla, and the other variables are all equal. Algorithm B 0 reacts to two types of events: "receiving an RM cell" (which is an event in common with I), and "reaching tfi" (which is an event specific to B 0 ). Idle snapshot newRM [I3] Fig. 3. Automaton AI Receiving an RM cell. When, at current time s, a new RM cell with value R arrives, the variables are updated according to the relative positions of s+- 3 and s+- 2 with respect to tfi and tla, and those of R with respect to Emx and A. There are eight cases from [1] to [8] (with two subcases for [1]): [3] if s ! tfi and Emx != R and tfi ?= s+- 3 and A != R then [6] if s ! tfi and Emx ? R and R ?= Ela then [8] if s ?= tfi and A ? R then Reaching tfi. When the current time s becomes equal to tfi, the approximate current rate A is updated to Efi while Efi is updated to Ela and tfi is updated to tla (operation [9]): When the events "reaching tfi" (s=tfi) and "receiving an RM cell" simultaneously occur, operation [9] must be performed before operation [1], ., [8] (accounting for the RM cell reception). Like I, algorithm B 0 terminates at snapshot time (s=t). If the snapshot occurs simultaneously with reaching tfi, operation [9] must be performed before termination of B 0 . Note that the ordering of s, tfi and tla just after operation [9] depends on the respective positions of tfi and tla at the moment of performing [9]. In case (s=)tfi=tla, one still has s=tfi=tla just after performing [9], then s becomes greater than tfi=tla as time increases (until an RM cell occurs or s=t). In case (s=)tfi!tla when performing [9], one has s!tfi=tla immediately after. Automaton AB 0 . In order to implement the higher priority of operation [9] over the other operations in case of simultaneous events, it is convenient to distinguish the case where s is greater than tfi from the case where s-tfi. To that goal, we introduce two locations Greater and Less. Operation [9] always occurs at location Less, but the target location depends on whether tfi=tla (subcase [9a]) or tfi!tla (subcase [9b]). The p-automaton AB 0 is represented in Figure 4 with only the most significant guards and no update information. Like before, the same labels are used for automaton transitions and corresponding program operations. Less Greater UpdAL UpdAG s=tfi!tla snapshot snapshot s=tfi=tla newRM asap or [8] newRM asap [1] or [2] or ::: or [6] Fig. 4. Approximation automaton AB 0 Initially AB 0 is in Greater, with p-constraint: s=tfi=tla - A=Efi=Ela=Emx=R. Location Less has s-tfi as an invariant, in order to force execution of transition (if tfi!tla) or [9a] (if tfi=tla) when s reaches tfi. From Less, transition goes back to Less (since, after update, s!tfi=tla) while transition [9a] switches to Greater (since s-tfi=tla as time increases). The reception of an RM cell corresponds to a transition newRM . There are two cases depending on whether the source location is Less or Greater. From Less (resp. Greater), transition newRM goes to location UpdAL (resp. UpdAG). This transition is followed by an urgent transition from UpdAL (resp. UpdAG) back to Less, which updates the discrete variables according to operations [1],.,[6] (resp. [7],[8]), as explained above. Note that transition newRM from Less to UpdAL has an additional guard s!tfi in order to prevent an execution of newRM before [9a] or [9b] when s=tfi (which is forbidden when "reaching tfi" and newRM occur simultaneously). Like before, observation is modeled as a transition snapshot from location Less or Greater to EndB. Also note that transition snapshot from Less to EndB has guard s!tfi in order to prevent its execution before [9a] or [9b] when s=tfi (which is forbidden when "reaching tfi" and the snapshot occur simultaneously). 5.4 Synchronized product and property U The full system is obtained by the product automaton I \Theta AB 0 of the three p-automata above, synchronized by the labels newRM and snapshot. The action moves occur when the current time reaches tfi or t or upon reception of an RM cell (newRM ). In this last case, return to a stable location is obtained by a complete sequence of transitions: newRM followed by transitions [I1],[I2a],[I2b],[I3] in A I and [1],.,[6] or [7],[8] in AB 0 . Recall that property U expresses in terms of the ideal rate E computed by I, and the approximate value A computed by B 0 , as: when In our model T , snapshot action occurs as soon as s=t, and makes the automaton switch to its final location Henceforth we respectively for locations (W ait; Idle; Greater) and (W ait; Idle; Less). Actually, the property A - E does not hold in all locations of T when s=t. This is due to the necessary completion of all the actions in case of simultaneous events. Thus, at location ' \Gamma , when s=t=tfi, one may have A ! E just before treatment of [9]. However in location ' all the appropriate actions are completed. Property U states therefore as follows: Since location ' 1 is reached when s=t, and no action then occurs, an equivalent statement of U is: 6 Verification of correctness 6.1 Verification with inductive invariants In order to prove U E, we are going to prove that Inv j U - is an inductive invariant of the system, where Aux i are auxiliary properties of the system. Some of these auxiliary properties (viz., Aux 3 involve an additional variable r, which represents the reception date of the last RM cell. (Such variables, that record some history of system execution without affecting it, are called "history" variables [1, 27].) In our model, this can be easily implemented by introducing a discrete variable r in the environment automa- ton, and updating it with current time value s, whenever event newRM occurs. . Enriched automaton A env is represented in Figure 5. Wait newRM R snapshot Fig. 5. Enriched automaton Aenv modeling arrivals of RM cells and snapshot More precisely, let Aux proved by inductive invariance, i.e. by showing that it holds initially and is preserved through any transition corresponding to either a complete action move or a delay move. The stable locations are ' . The action moves starting from and leading to one of these locations are those associated with the reception of an RM cell, the reaching of tfi, or snapshot. In the case of RM cell reception, there are several subcases depending on the complete sequence of actions in AB 0 (newRM followed by [1a], [1b], and, subsidiarily, on the sequences in A I (newRM followed by [I3]g). We now give in details the statements involved in the proof. Variables of Inv are explicitly mentionned by writing Inv(-; s; w is the vector (E, Efi, Ela, Emx, A, R, tfi, tla). Note that, provided an encoding of locations given as integers (e.g., 0; 1; 2), all these statements are linear arithmetic formula over reals (involving variables -; s; can be proved just by arithmetic reasoning. This can be done automatically with an arithmetic theorem prover. Such a proof was actually done using Coq [18], then was reformulated in the present context, after encoding p-automata in Coq. Let us recall that, in Coq, the user states definitions and theorems, then he proves the latters by means of scripts made of tactics. Scripts are not proofs, but produce proofs, which are data (terms) to be checked by the kernel of the proof assistant. Some tactics used for ABR were described in [18]. The script written for the ABR is about 3500 lines long and required about 4 man-months of work. A crucial part of the human work consisted in identifying the relevant invariants. Around two hundred subproofs were then automatically produced and checked. The whole proof check takes 5 minutes on a PC 486 (33 MHz) under Linux. In order to give a flavour of the proof structure, we give now some typical statements to be proved in the case of action and delay moves. Action moves. As an example, consider the reception of an RM cell where the subsequent action of AB 0 is [1b]. This corresponds to a complete sequence of transitions from location ' \Gamma to itself. We have to prove: E) The conclusion conjunction of the 9 tla For example, let us show how to prove Aux 0 9 , i.e. E assuming tfi - . By Aux 10 , we have E - Ela (since tla - t). By Aux 8 , we have Ela - Efi (since tfi=tla). Hence, by transitivity: E- Efi. On the other hand, E'=E by [I3] (since (= tla All the other cases are proved similarly to this one, by case analysis, use of transitivity of -; !; =, and regularity of + over these relations. Delay moves. They take place at stable locations ' . The corresponding properties to be proved are respectively: These formula easily reduce to: The first (resp. second, third) implication is true because its conclusion A - E follows from the hypothesis using the Aux 1 -conjunct (resp. Aux 2 -conjunct, U - conjunct) of Inv. 6.2 Verification by reachability analysis In order to mechanically prove property U , we have to compute P ost for the product automaton T , starting from its initial state where / init is the p-constraint s=tfi=tla - R=E=A=Efi=Ela=Emx- 0!- 3 !- 2 . We then have to check that P ost (q init ) does not contain any state where the property U is violated. Recall that property U can be stated as: The state where U does not hold is then Automata A env , A I and AB 0 can be directly implemented into HyTech [14], which automatically computes the synchronized product T . The modelling of the protocol and property as p-automata and the encoding in Hytech required about 3 man-months of work. The forward computation of P ost (q init ) requires iteration steps and its intersection with q :U is checked to be empty. (This takes 8 minutes on a SUN station ULTRA-1 with 64 Megabytes of RAM memory. Appendix B for a display of the generated p-zone at ' 1 . ) This achieves an automated proof of correctness of B 0 . Such a proof first appears along these lines in [8]. Note that HyTech can provide as well a proof by backward reasoning (using P re instead of P ost ). 7 Discussion on p-automata Tools based on timed automata have been successfully used in the recent past for verifying or correcting real-life protocols (e.g., the Philips Audio Control protocol [16] and the Bang & Olufsen Audio/Video protocol [13]). Experiences with such tools are very promising. This observation led us to use here p-automata, a close variant of timed automata. The differences between p-automata and classical timed automata are two-fold. A first minor difference is that p-automata use a form of "updatable" time variables instead of traditional clocks (but see [9] for a proof of equivalence between the two classes). Second p-automata incorporate parameters, which are essential in our case study. The choice of Hytech, as an associated tool, is natural in this context. We now explain some new features that appear in the proof of the protocol using p-automata, and some difficulties encountered in the process of building the specification. 7.1 Towards p-automata We first point out some significant differences between the proof by invariants as stated by Monin and Klay [19, 18] and the corresponding proof presented here (see section 6.1). Representation of time. In the work reported in [19, 18], the formalization of time aspects, though influenced by timed automata, was performed using ad hoc devices, appealing to the reader's natural understanding. This problem is settled here, thanks to p-automata, which include a built-in notion of clock and rely on a well-understood and widely accepted notion of time. The use of p-automata, although it introduces an additional level of encoding, thus makes the effort of specification easier with this respect. Note however that the encoding of p- automata in Coq does not specify a priori the granularity of time evolution, and allows for either a continuous or discrete underlying model of time. It is based on a minimal underlying theory of arithmetic (mainly the transitivity of relations !; - and the regularity of + over them). This is not the case in the theory of timed automata and the associated tools like Hytech (or others [6, 11]) where the time domain is assumed to be continuous (Q or R), and a sophisticated package for manipulating linear constraints over real arithmetic is used. We will come back to this difference (see section 8.2). Higher-order vs. first-order specifications. As recalled in section 3, Monin and Klay [19] introduced an incremental way of computing the ideal rate Acr(t) (higher-order algorithm F). We then recast their algorithm F under a parametric form I. The latter view is probably less natural, but fits better into the first order framework of p-automata. Reformulating the proof by invariance. We also rewrote the proof by invariance of [19] in the context of p-automata, in order to assess the proofs in a uniform framework. Expressing the auxiliary properties needed in this proof required to (re)introduce history variable r accounting for the reception time of the last RM cell. Moreover, these properties only concerned stable locations of the system: expressions of the form had to be included, and only complete sequences of actions were considered. Note that the auxiliary property is different from its counterpart Aux 0 as found in [19]. Actually, Aux 0 8 is false in our model. We will subsequently explain this discrepancy (see section 8.2). 7.2 Specific modelling problems with p-automata In the process of constructing p-automata while keeping with the specification, we had to face some problems, which are listed below. Modelling the environment as a p-automaton. In addition to the automata corresponding naturally to I and B 0 , a third automaton was introduced to model the environment, thus providing a clear separation between external and internal events. Introducing urgent locations. The class of update relations in p-automata (derived from HyTech) does not allow for simultaneous updates. For instance, choosing (at random) a new identical value of R and E (with an instruction like E'=R'?0) is forbidden. In order to implement such an update relation, an urgent intermediate location, such as UpdE depicted in figure 6, had to be introduced. Idle asap update R newRM instead of Idle newRM Fig. 6. Reception and update with two locations instead of one Introducing two stable locations. In order to implement the higher priority of operation "reaching tfi" (when occuring simultaneously with other actions of the system), we were led to create two stable locations (Greater and Less) in the p-automaton representing algorithm B 0 . Note that we overlooked this priority requirement in a preliminary implementation embedding only one stable loca- tion, which entailed a violation of property U . (This was detected subsequently by running HyTech.) 8 Proof Comparison We now assess the respective merits and shortcomings of the proof methods by invariance and reachability analysis within the unified framework of p-automata, regarding the ABR conformance problem. We also explain how to cross-fertilize the results of the two methods. 8.1 Automated proof vs. readable proof It is well-known that a proof in a model-checker is more automatic, but that more insight in the algorithm is gained by doing the proof with a theorem-prover. Let us confirm this general opinion in our particular case study. As already noticed, the reachability proof was done in a fully automatic manner (via HyTech). This is an outstanding advantage over the proof by inductive invariance (which required the human discovery of several nontrivial auxiliary properties) and justifies a posteriori our effort of translating the problem into the formalism of p-automata. In particular, it becomes easier to validate other ABR conformance protocols as soon as they are formalized themselves in terms of p-automata. This is actually what was done recently in the framework of RNRT project Calife: different variants of B 0 were easily checked (or invalidated) with HyTech by reachability analysis along the lines described above. It was not possible to do the same with the inductive invariance approach because several of the original auxiliary properties became false while others had to be discovered. Nevertheless, several qualifications must be done about this positive side of model-checking. Let us first stress that the proof obtained by reachability analy- sis, merely consists of a long list of constraints (see appendix B) that represents the whole set of reachable symbolic states. This information is hardly exploitable by a human: in particular the essential fact that such a list is complete (i.e. "cov- ers" all the reachable states) is impossible to grasp by hand. In contrast, the invariance proofs as checked by a theorem prover are more human-oriented. It is instructive to inspect the case analysis that was automatically performed, and allows the reader to be convinced of the property accurateness (or a contrario of some flaws). Besides, the auxiliary properties are very important per se, and bring important information about algorithm B 0 itself. (Some of these properties are indeed part of norm ITU I.371.1, and must be henceforth fulfilled by any new ABR algorithm candidate to normalization.) We explain now how one can go beyond the limitations of each method, by using both of them in a fruitful cross-fertilizing way. 8.2 Cross-fertilizing proofs Checking the output produced by Hytech. A proof produced by Hytech, i.e. the (finite) list P ost of all the symbolic states reached from the initial one, can be seen as a fixed-point associated with the set of transitions of the product automaton. Therefore, one can verify that such a list is "complete" (covers all the reachable states) by checking that it is invariant through action and delay moves. This can be done, using the Coq system, exactly as explained in section 6.1. This gives of course an increased confidence in the model-checking proof. In addition it may give new insight about the conditions on the environment that were assumed to perform the proof: as noticed in section 7.1, in Coq, we use a very flexible model for time, assuming only that time increases (but nothing about its continuity). In fact, the correctness of algorithm B 0 holds even for a discrete time modelling. Such a feature cannot not be derived from the proof of Hytech, since it uses a priori an assumption of continuous time evolution. Checking the invariants used in Coq. In the other way around, one can check the correctness of the auxiliary properties simply by asking Hytech if all the reachable states satisfy them (i.e. P ost ' 10). The answer is always "yes", which gives us another proof of the Aux i s. Recall however that Aux 8 differs from its counterpart Aux 0in [19]: 8 is false in the p-automata model and true in the original model of Monin-Klay (see section 7.1). This discrepancy originates from the different ways two consecutive RM cells follow each other in the two models. In our p-automata model, two consecutive RM cells may arrive simultaneously while this is precluded in the model of Monin-Klay as reception times of RM cells must form a strictly increasing sequence. The model presented here is then more general than the original model of Monin-Klay, as it relaxes some assumption concerning the sequence of RM cells. As a by-product, this provides us with a better understanding of the conditions under which B 0 behaves correctly. Finally note that the model of p-automata is flexible enough to incorporate the assumption of strictly increasing sequences of RM cells, as used in [19]: it suffices to use explicitly the additional variable r mentioned in section 7.1 (date of last RM cell reception), and add guard s ? r to the newRM transition in the environment automaton of figure 5. With such a modification, property Aux 0also becomes true in the model with p-automata. 8.3 Further experiments and foreseen limits We thus claim that checking properties proved by one tool, using the other one, is very fruitful. As examplified above, it may reveal possible discrepancies, which lead in turn to discover implicit modelling assumptions. It may also of course detect real flaws, which originate from the protocol or its modelling (although it has not been the case here). In any case this proof confrontation helps the verification work, and increases the confidence of the human in mechanical proofs. One can now wonder how general are the remarks we made on this case study, given the fact that we focused on one problem (the correctness of algorithm used two specific tools as a theorem-prover (Coq) and a model-checker (Hytech). Regarding the tools, we believe that our experience with Coq and Hytech is not specific, but can be reproduced with equivalent tools as well. We have concrete indications in this sense. Actually, in the framework of project Calife, Pierre Cast'eran and Davy Rouillard, from University of Bordeaux, have performed a proof similar to the proof in Coq, using the model of p-automata and theorem prover Isabelle [10, 25]. Concerning Hytech, we do not know any other model-checking tool allowing for parameters but, as mentioned in appendix C, we did some successful experiments with Gap [12], a tool based on constraint logic programming, which works as a fixed-point engine very much as Hytech generates P ost . Concerning the studied problem, the success of the proof by model-checking comes from the fact that the computation of the P ost computation with HyTech had terminated. This can be considered a "lucky" event, since analysis of such a parametric algorithm is known to be undecidable [5]. This means that computation of P ost does not always terminate for all p-automata. (This observation leads us to propose, in Appendix C, an "approximate" version of B 0 , belonging to a subclass for which P ost is guaranteed to terminate.) Is such a termination property preserved when considering ABR conformance algorithms other than The answer is ambivalent. On the one hand, as already mentioned, our model-checking experience on B 0 was successfully reused on other (relatively close) algorithms of ABR conformance in the framework of project Calife. On the other hand, we failed to mechanically check an algorithm of ABR conformance of a different kind: the generic algorithm of Rabadan-Klay (see e.g. [26]). This algorithm involves an unbounded list of N scheduled dates (instead of 2, as in B 0 ), and cannot be modeled with p-automata due to the use of a list data type. Even for the restricted version where N is bound to a small value, e.g: 3, in which case we get a natural model with p-automata, Hytech runs out of memory and fails to generate P ost . The latter experiment recalls us some inherent limits of model-checking: if the algorithm uses not only a finite set of numeric variables, but also unbounded data structures (such as lists), then the verification process has to rely essentially on classical methods of theorem-proving; this is also true when the program can be modeled as a p-automaton, but the space of reachable symbolic states is too big to be computed by existing model-checkers. 9 Conclusion As a recapitulation, we believe that many useful informations about real-time programs can be obtained without resorting to new integrated tools, when it is possible to make a joint use of well-established theorem prover and model checker. In our case, we gained much insight about the algorithm B 0 , and important confidence in the proofs of correctness produced by Coq and Hytech, basically by using the unified framework of p-automata, and cross-fertilizing the two proofs. In particular we saw that algorithm B 0 is robust in the sense that several underlying assumptions can be relaxed: the nature of time can be discrete (instead of continuous); the (measured) time interval between two received RM cells can be null. Moreover, the basic p-automata model underlying B 0 was successfully reused for proving the correctness of some variants. To our knowledge, it is the first time that such a compared study between theorem proving and model checking has been performed on the same industrial problem. We hope that this work paves the way for further experiences on real-life examples. In the framework of project Calife, we are currently developing a two-step methodology for verifying the quality of new services provided by telecommunication networks, which exploits the synergy between the two proof methods: the first step, based on model-checking, yields a p-automaton model endowed with a collection of invariants it satisfies; in the second step, the p-automaton is recasted under an algorithmic form better suited to the end-user, and verification is done via a generic proof assistant, with the help of invariants. --R "The existence of refinement mappings" "The Algorithmic Analysis of Hybrid Systems" "Hybrid Automata: An Algorithmic Approach to the Specification and Verification of Hybrid Systems" "Automata for Modeling Real-Time Systems" "Parametric real-time reasoning" "UPPAAL - a Tool Suite for Automatic Verification of Real-Time Systems" "Are Timed Automata Updat- able?" "The Tool KRONOS" "A Closed-Form Evaluation for Extended Timed Automata" "A User Guide to HYTECH" "Traffic control and congestion control in B- ISDN" "Model-Checking for Real-Time Systems" "Beyond model checking" "Proving a real time algorithm for ATM in Coq" Correctness Proof of the Standardized Algorithm for ABR Conformance. "An Approach to the Description and Analysis of Hybrid Systems" "A Platform for Combining Deductive with Algorithmic Verification" L'ABR et sa conformit'e. "A Closed-Form Evaluation for Datalog Queries with Integer (Gap)- Order Constraints" "An Integration of Model Checking with Automated Proof Checking" "Mechanical Verification of a Generic Incremental ABR Conformance Algorithm" "An Introduction to Assertional Reasoning for Concurrent Sys- tems." --TR The existence of refinement mappings An introduction to assertional reasoning for concurrent systems A closed-form evaluation for Datalog queries with integer (gap)-order constraints Parametric real-time reasoning The algorithmic analysis of hybrid systems The tool KRONOS UPPAALMYAMPERSANDmdash;a tool suite for automatic verification of real-time systems Automata For Modeling Real-Time Systems A User Guide to HyTech Proving a Real Time Algorithm for ATM in Coq Hybrid Automata Correctness Proof of the Standardized Algorithm for ABR Conformance Automated Verification of a Parametric Real-Time Program Mechanical Verification of an Ideal Incremental ABR Conformance Are Timed Automata Updatable? Beyond Model Checking An Integration of Model Checking with Automated Proof Checking A Platform for Combining Deductive with Algorithmic Verification Model-Checking for Real-Time Systems An Approach to the Description and Analysis of Hybrid Systems Formal modeling and analysis of an audio/video protocol --CTR Patricia Bouyer , Catherine Dufourd , Emmanuel Fleury , Antoine Petit, Updatable timed automata, Theoretical Computer Science, v.321 n.2-3, p.291-345, August 2004

telecommunication protocols;theorem proving;model checking

608242

Controllability of Right-Invariant Systems on Solvable Lie Groups.

We study controllability of right-invariant control systems groups. Necessary and sufficient controllability conditions for Lie groups not coinciding with their derived subgroup are obtained in terms of the root decomposition corresponding to the adjoint operator ad B. As an application, right-invariant systems on metabelian groups and matrix groups, and bilinear systems are considered.

Introduction Control systems with a Lie group as a state space are studied in the mathematical control theory since the early 1970-ies. R.W. Brockett [1] considered applied problems leading to control systems on matrix groups and their homogeneous spaces; e.g., a model of DC to DC conversion and the rigid body control raise control problems on the group of rotations of the three-space SO(3) and on the group SO(3) \Theta R 3 respectively. The natural framework for such problems are matrix control systems of the where x(t) and A, are n \Theta n matrices. There was established the basic rank controllability test for homogeneous systems: such systems are controllable iff the Lie algebra generated by the matrices has the full dimension. This test was specified for the groups of matrices with positive determinant GL+ (n; R), the group of matrices with 1991 Mathematics Subject Classification. 93B05, 17B20. Key words and phrases. Controllability, right-invariant systems, bilinear systems, Lie groups. This work was partially supported by the Russian Foundation for Fundamental Re- search, Projects No. 96-01-00805 and No. 97-1-1a/22. The author is a recipient of the Russian State Scientific Stipend for 1997.1079-2724/97/1000-0531$09.50/0 c Publishing Corporation 532 YU. L. SACHKOV determinant one SL(n; R), the group of symplectic matrices Sp(n), and the group of orthogonal matrices with determinant one SO(n). Some controllability conditions for nonhomogeneous matrix systems were also obtained. The first systematic mathematical study of control systems on Lie groups was fulfilled by V. Jurdjevic and H. J. Sussmann [2]. They noticed that the passage from the matrix system (1) to the more general right-invariant system G; u(t) 2 R; (2) are right-invariant vectorfields on a Lie group G, "in no essential way affects the nature of the problem." The basic properties of the attainable set (the semi-group property, path-connectedness, relation with the associated Lie subalgebras determined by the vectorfields A, established. The rank controllability test was proved for system (2) in the homogeneous case and in the case of a compact group G. Sufficient controllability conditions for other cases were also given. V. Jurdjevic and I. Kupka [4] introduced a systematic tool for studying controllability on Lie groups. For the control system (2) presented in the form of the polysystem ae L (3) (where L is the Lie algebra of the group G) they considered its Lie saturation LS(\Gamma) - the largest system equivalent to \Gamma. Controllability of the system \Gamma on G is equivalent to L, and a general technique for verification of this equality was proposed. (This technique is outlined in Subsec. 4.2 and used in Subsecs. 4.3, 4.4 below.) In [4] sufficient controllability conditions for the single-input systems were obtained for simple and semi-simple groups G with the use of this technique. They were given in terms of the root decomposition of the algebra L corresponding to the adjoint operator ad B. In their preceding paper V. Jurdjevic and I. Kupka [3] presented the enlargement technique for systems on matrix groups G ae GL(n; R) and obtained sufficient controllability conditions for GL+ (n; R). These results for SL(n; R) and GL+ (n; R) were generalized by J.P. Gauthier and G. Bornard [5]. B. Bonnard, V. Jurdjevic, I. Kupka, and G. Sallet [6] obtained a characterization of controllability on a Lie group which is a semidirect product of a vector space and a compact group which acts linearly on the vector CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 533 space. The case n\Omega s SO(n) was applied to the study of Serret-Frenet moving frames. The results of [4] for simple and semi-simple Lie groups were generalized in a series of papers by J.P. Gauthier, I. Kupka, and G. Sallet [7], R.El Assoudi and J. P. Gauthier [9], [10], F. Silva Leite and P.E. Crouch [8]: analogous controllability conditions were obtained for classical Lie groups with the use of the Lie saturation technique and the known structure of real simple and semi-simple Lie algebras. In contrast to this "simple" progress, invariant systems on solvable groups seem not to be studied in the geometric control theory at all until 1993. Then a complete solution of the controllability problem for simply connected nilpotent groups G was given by V. Ayala Bravo and L. San Martin [11]. Some results on controllability of (not right-invariant) systems on Lie groups analogous to linear systems on R n were obtained by V. Ayala Bravo and J. Tirao [12]. Several results on controllability of right-invariant systems were obtained within the framework of the Lie semigroups theory [13], [14]: for nilpotent groups by J. Hilgert, K. H. Hofmann, and J. D. Lawson [15], for reductive groups by J. Hilgert [16]. For Lie groups G with cocompact radical, J. D. Lawson [17] proved that controllability of a system \Gamma ae L follows from nonexistence of a half-space in L bounded by a Lie subalgebra and containing \Gamma; if G is additionally simply connected, this condition is also necessary for controllability. This result generalizes controllability conditions for compact groups [2], nilpotent groups [15], and for semidirect products of vector groups and compact groups [6]. In [18] the author characterized controllability of hypersurface right-invariant systems, i.e., of systems \Gamma of the form (3) with the codimension one Lie subalgebra generated by the vectorfields This gave a necessary controllability condition for simply connected groups - the hypersurface principle, see its formulation for single-input systems \Gamma in Proposition 2 below. In its turn, the hypersurface principle was applied and there was obtained a controllability test for simply connected solvable Lie groups G with Lie algebra L satisfying the additional condition: for all X 2 L the adjoint operator ad X has real spectrum. The aim of this paper is to give convenient controllability conditions of single-input systems \Gamma for a wide class of Lie groups including solvable ones; more precisely, for Lie groups not coinciding with their derived subgroups. The structure of this paper is as follows. We state the problem and introduce the notation in Sec. 2. In Sec. 3 we give the necessary controllability condition for simply connected groups G not coinciding with their derived subgroup G (1) (Theorem 1 and Corollary 1). These propositions are proved in Subsec. 3.3 after the preparatory work in Subsec. 3.2. The main tools are the rank controllability 534 YU. L. SACHKOV condition (Proposition 1) and the hypersurface principle (Proposition 2). Sec. 4 is devoted to sufficient controllability conditions for the groups G 6= G (1) . We present the main sufficient results in Subsec. 4.1. Then we recall the Lie saturation technique in Subsec. 4.2 and prove preliminary lemmas in Subsec. 4.3. The main results (Theorem 2 and Corollaries 2, are proved in Subsec. 4.4. In Sec. 5 we consider several applications of our results. Controllability conditions for metabelian groups are obtained in Subsec. 5.1. Then controllability conditions for some subgroup of the group of motions of the Euclidean space are studied in detail (Subsec. 5.2) and are applied to bilinear systems (Subsec. 5.3). Finally, the clear small-dimensional version of this theory for the group of motions of the two-dimensional plane is presented in Subsec 5.4. A preliminary version of the below results was stated in [19]. 2. Problem statement and definitions Let G be a connected Lie group, L its Lie algebra (i.e., the Lie algebra of right-invariant vector fields on G), and A, B any elements of L. The single-input affine right-invariant control system on G is a subset of L of the form The attainable set A of the system \Gamma is the subsemigroup of G generated by the set of the one-parameter semigroups The system \Gamma is called controllable if G. To see the relation of these notions with the standard system-theoretical ones, let us write the right-invariant vector fields A and B as A(x) and G. Then the system \Gamma can be written in the customary form G: The attainable set A is then the set of points of the state space G reachable from the identity element of the group G for any nonnegative time. The system \Gamma is controllable iff any point of G can be reached along trajectories of this system from the identity element of the group G. By right-invariance of the fields A(x), B(x), the identity element in the previous sentence can be changed by an arbitrary one. Our aim is to characterize controllability of the system \Gamma in terms of the Lie group G and the right-invariant vector fields A and B. Now we introduce the notation we will use in the sequel. For any subset l ae L we denote by Lie (l) the Lie subalgebra of L generated by l. Closure of a set M is denoted by cl M . The signs \Phi and CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 535 direct sums of vector spaces; \Phi s and\Omega s stand for semidirect products of Lie algebras and Lie groups correspondingly. We denote by Id the identity operator or the identity matrix of appropriate dimension, sin fft cos fft ff r for t, ff, r 2 R. The square matrix with all zero entries except one unit in the ith raw and the jth column is denoted by E ij . Now we introduce the notation connected with eigenvalues and eigenspaces of the adjoint operator ad B in L: ffl the derived subalgebra and the second derived subalgebra: ffl the complexifications of L and L (i) , 2: (i)\Omega C (the tensor products over R), ffl the adjoint representations and operators: ffl spectra of the operators ad Bj L (i) , 2: \Phi a 2 C j Ker(ad c Bj L (i) c ffl real and complex eigenvalues of the operators ad Bj L (i) , 2: ffl complex eigenspaces of ad c Bj L (1) c c ffl real eigenspaces of ad Bj ffl complex root subspaces of ad c Bj L (i) c 2: c a ffl real root subspaces of ad Bj L (i) 2: c (a) 536 YU. L. SACHKOV ffl real components of L (i) , 2: Note that the subalgebras L (1) and L (2) are ideals of L, so they are (ad B)- invariant, and the restrictions ad Bj L (1) and ad Bj L (2) are well defined. In the following lemma we collect several simple statements about decomposition of the subalgebras L (1) and L (2) into sums of root spaces and eigenspaces of the adjoint operator ad B. Lemma 2.1. (2) r , (3) L (2) (a) ae L (1) (a) for any a 2 Sp (2) , r ae L (1) r , Proof. Is obtained by the standard linear-algebraic arguments. In item (5) Jacobi's identity is additionally used. Consider the quotient operator defined as follows: Analogously for a 2 Sp (1) we define the quotient operator in the quotient root space: and its complexification: ad c (a)=L (2) c (a)=L (2) c (a); (ad c (a) 8X 2 L (1) c (a): Definition 1. Let a 2 Sp (1) . We denote by j (a) the geometric multiplicity of the eigenvalue a of the operator - ad c B(a) in the vector space c (a)=L (2) c (a). CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 537 Remarks. (a) For a 2 Sp (1) the number j (a) is equal to the number of Jordan blocks of the operator ad B(a) in the space L (1) (a)=L (2) (a). (b) If an eigenvalue a 2 Sp (1) is simple, then j Suppose that assumption will be justified by Theorem below). Then by Lemma 2.1 that is why any element X 2 L can uniquely be decomposed as follows: We will consider such decomposition for the uncontrolled vector field A of the system \Gamma: We denote by ] A(a) the canonical projection of the vector A(a) 2 L (1) (a) onto the quotient space L (1) (a)=L (2) (a). Definition 2. Let We say that a vector A has the zero a-top if ad B(a) \Gamma a Id)(L (1) (a)=L (2) (a)): In the opposite case we say that A has a nonzero a-top. We use the corresponding notations: top Remark . Geometrically, if a vector A has a nonzero a-top, then the vector A(a) has a nonzero component corresponding to the highest adjoined vector in the (single) Jordan chain of the operator ad B(a). Due to nonuniqueness of the Jordan base, this component is nonuniquely determined, but its property to be zero is basis-independent. Definition 3. A pair of complex numbers (ff; fi), Re ff - Re fi, is called an N-pair of eigenvalues of the operator ad B if the following conditions hold: (2) L (2) (ff) 6ae Re a; Re (3) L (2) (fi) 6ae Re a; Re \Psi . 538 YU. L. SACHKOV Remarks. (a) In other words, to generate the both root spaces L (2) (ff) and L (2) (fi) for an N-pair (ff; fi), we need at least one root space L (1) (fl) with Re fi]. The name is explained by the fact that N-pairs can NOT be overcome by the extension process described in Lemma 4.2: they are the strongest obstacle to controllability under the necessary conditions of Theorem 1. (b) The property of absence of the real N-pairs will be used to formulate sufficient controllability conditions in Theorem 2. In some generic cases this property can be verified by Lemma 4.3. 3. Necessary controllability conditions x 3.1. Main theorem and known results. It turns out that controllability on simply connected Lie groups G with G 6= G (1) is a very strong property: it imposes many restrictions both on the group G and on the system \Gamma. Theorem 1. Let a Lie group G be simply connected and its Lie algebra L satisfy the condition L 6= L (1) . If a system \Gamma is controllable, then: (1) dimL (3) L (2) r , r , r ae The notations j (a) and top (A; a) used in Theorem 1 are explained in Definitions 1 and 2 in Sec. 2. Remarks. (a) The first condition is a characterization of the state space G but not of the system \Gamma. It means that no single-input system can be controllable on a simply connected Lie group G with dimG (1) ! dimG \Gamma 1. That is, to control on such a group, one has to increase the number of inputs. There is a general lower for the number of the controlled vectorfields necessary for controllability of the multi-input system (3) on a simply connected group G [18]. (b) Conditions (3)-(7) are nontrivial only for Lie algebras L with L (2) 6= L (1) (in particular, for solvable noncommutative L). If L then these conditions are obviously satisfied. CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 539 (c) The third condition means that j r , that is why condition (6) is nontrivial only for a 2 c . (d) By the same reason, in condition 7 the inclusion a 2 Sp (1) can be changed by a 2 c . Note that if j (a) = 0, then by the formal Definition 2 the vector A has the zero a-top. (e) The fourth and fifth conditions are implied by the third one but are easier to verify. The simple (and strong) "arithmetic" necessary controllability condition (5) can be verified by a single glance at spectrum of the operator ad Bj L (1) . (f) For solvable L under conditions (1), (2) the spectrum is the same for all homotheties. Then conditions (3)-(5) depend on L but not on B. (g) For the case of simple spectrum of the operator ad Bj L (1) the necessary controllability conditions take respectively the more simple Corollary 1. Let a Lie group G be simply connected and its Lie algebra L satisfy the condition L 6= L (1) . Suppose that the spectrum Sp (1) is simple. If a system \Gamma is controllable, then: (1) dimL r , r ae Theorem 1 and Corollary 1 will be proved in Subsec. 3.3. Remark . Now we discuss the condition L (1) 6= L essential for this work and motivated by its initial focus - solvable Lie algebras L. Consider a Levi decomposition It is well known (see, e.g., [20], Theorem 3.14.1) that the Levi decomposition of the derived subalgebra is then This means that If a Lie algebra L is semisimple (i.e., rad obviously L The converse is generally not true (although this is asserted by [21], Sec. 87, Corollary 3). For example, for the Lie algebra R 3 \Phi s so(3) (which is the Lie 540 YU. L. SACHKOV algebra of the Lie group of motions of the three-space) its derived subalgebra coincides with the algebra itself. (This example was kindly indicated to the author by A. A. Agrachev). The main tools to obtain the necessary controllability conditions given in Theorem 1 is the rank controllability condition and the hypersurface principle. The system \Gamma is said to satisfy the rank controllability condition if the Lie algebra generated by \Gamma coincides with L: Proposition 1. (Theorem 7.1, [2]). The rank controllability condition is necessary for controllability of a system \Gamma on a group G. Generally, the attainable set A lies (and has a nonempty interior, which is dense in A) in the connected subgroup of G corresponding to the Lie algebra Lie (A; B). The hypersurface principle is formulated for the system \Gamma as follows: Proposition 2. (Corollary 3.2, [18]). Let a Lie group G be simply con- nected, L, and let the Lie algebra L have a codimension one subalgebra containing B. Then the system RB is not controllable on G. The sense of this proposition is that under the hypotheses stated there exists a codimension one subgroup of the group G which separates G into two disjoint parts, is tangent to the field B, and is intersected by the field A in one direction only. Then the attainable set A lies "to one side" of this subgroup. Notice that the property of absence of a codimension one subalgebra of containing B is sufficient for controllability of \Gamma on a Lie group G with cocompact radical; if G is additionally simply connected, this condition is also sufficient (Corollary 12.6, [17]). x 3.2. Preliminary lemmas. First we obtain several conditions sufficient for existence of codimension one subalgebras of a Lie algebra L containing a vector B 2 L. Lemma 3.1. Suppose that L (1) +RB 6= L. Then there exists a codimension one subalgebra of L containing B. Proof. Denote by l the vector space L (1) +RB. We have [l; l] ae L (1) ae l, that is why l is a subalgebra; any vector space containing l is a subalgebra of L too. Since l 6= L, there exists a codimension one subspace l 1 of L containing l. Then l 1 is the required codimension one subalgebra of L containing B. CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 541 Lemma 3.2. Let L (1) \Phi r 6= L (1) r , then there exists a codimension one subalgebra of L containing B. Proof. If L (1) r 6= L (2) r , then there exists a real eigenvalue a 0 2 r such that L (1) (a 0 be a Jordan base of the operator ad B(a 0 (We suppose, for simplicity, that the eigenvalue a 0 of the operator - ad is geometrically simple, i.e., matrix of this operator is a single Jordan block; for the general case of several Jordan blocks the changes of the proof are obvious.) Consider the vector space It follows from (4), (5) that the space l 1 is (ad B)-invariant. Additionally, we have dim l Then we define the vector spaces f L (1) (a) j a 2 First, dim l that is why Third, the space l 2 is (ad B)-invariant. That is why, by virtue of (6) and (8), we obtain the chain Hence, l 3 is the required subalgebra of L: it has codimension one (see (7)) and contains the vector B (see (6)). In the following three lemmas we obtain conditions sufficient for violation of the rank controllability condition, i.e., necessary for controllability. 542 YU. L. SACHKOV Lemma 3.3. Suppose that 2 L (1) and let there exist a vector subspace l 1 ae L such that the following relations hold (1) L (2) ae l 1 ae L (1) , Then Lie Proof. By condition (1), i.e., l 1 is a Lie subalgebra. Consider the vector space l = RB \Phi l 1 . We have (in view of condition so l is a Lie subalgebra too. By condition (3) we have Lie (A; B) ae l, and condition (2) implies l 6= L. Hence, Lie (A; B) 6= L. Lemma 3.4. Let Proof. Consider the case of the complex a 0 2 c first. By the condition the quotient operator ad c B(a 0 ) has at least two cyclic spaces V; W ae L (1) c (a 0 )=L (2) c (a 0 ). That is, there are two Jordan chains and in these bases matrices of the operators - ad c B(a 0 )j V and - ad c B(a 0 )j W are the Jordan blocksB a . a 0 0 a . a 0 0 (Obviously, we can assume that the complex conjugate bases fv and Jordan chains of the operator - ad c B(a 0 ) in the complex conjugate spaces V ; W ae L (1) c (a 0 )=L (2) c (a 0 ).) CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 543 Notice that ad c B(a 0 ad c B(a 0 For the direct sum c (a 0 )=L (2) (other cyclic spaces of - ad c B(a 0 consider the decomposition (components in other cyclic spaces of - ad c B(a 0 )): We can assume that This will be proved at the end of this proof. Now suppose that condition (14) holds and, for definiteness, That is why (other cyclic spaces of - ad c B(a 0 )); c (a 0 )=L (2) and, in view of (10), (11), ad c B(a 0 Now let l ae L (1) c (a 0 ) be the canonical preimage of the space ~ l. Obviously, (ad c B) l ae l; c (a 0 Then we pass to realification: 544 YU. L. SACHKOV Finally, for the space l 1 := l r \Phi we obtain (ad all conditions of Lemma 3.3 are satisfied, and Lie (A; B) 6= L modulo the unproved condition (14). To prove this condition, suppose that A v1 6= 0 and Aw1 6= 0 in decomposition (13). In view of symmetry between V and W , we can assume that . Define the new basis in A v1 It is easy to see that f~v is a basis of V , and for the new basis f~v and the old basis fw g. Now we show that the new basis is a Jordan one. ad c B(a 0 ) ad c B(a 0 ) A v1 ad c B(a 0 ) A v1 (a A v1 A v1 ad c B(a 0 ) ad c B(a 0 ) A v1 ad c B(a 0 ) A v1 a A v1 ad c B(a 0 ) ad c B(a 0 ) A v1 ad c B(a 0 ) A v1 a A v1 CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 545 And if q ad c B(a 0 ) ad c B(a 0 ) Finally, if ad c B(a 0 ) ad c B(a 0 ) is a Jordan basis for - ad c B(a 0 )j V , and (13), as was claimed. So the lemma is proved for the case of the complex eigenvalue a 0 . And if a 0 is real, then the proof is analogous and easier: there is no need in complexification and further realification. Lemma 3.5. Let Proof. The Jordan base of the operator - ad c B(a 0 ) consists of one Jordan chain c (a 0 )=L (2) and moreover, in the decomposition we have A ad c B(a) \Gamma a 0 Id)(L (1) c (a 0 )=L (2) Then in the same way as in Lemma 3.4 we denote by l the preimage of the space under the projection L (1) c (a 0 )=L (2) and by l the complex conjugate to l in L c . Then the space satisfies all hypotheses of Lemma 3.3, that is why Lie (A; B) 6= L. x 3.3. Proofs of the necessary controllability conditions. Proof of Theorem 1. Suppose that the system \Gamma is controllable on the group G. Items (1) and (2). If dimL (1) L. It follows from Lemma 3.1 and the hypersurface principle (Proposition 2) that \Gamma is not controllable. This contradiction proves items (1) and (2), and allows to assume below in the proof that L (1) \Phi (3). If L (2) r 6= L (1) r , then it follows from Lemma 3.2 and the hyper-surface principle that \Gamma is not controllable. immediately from item (3). 546 YU. L. SACHKOV (5). From the previous item we have r . But consequently, r ae Items (6), (7) follow from Lemmas 3.4, 3.5 and from the rank controllability condition (Proposition 1). Proof of Corollary 1. If the spectrum Sp (1) is simple, then L (1) for all a 2 r or Sp (1) c respectively. Further, L (2) r is equivalent to Sp (2) r , and top (A; a) 6= 0 iff A(a) 6= 0, a 2 Sp (1) . Now Corollary 1 follows immediately from Theorem 1. 4. Sufficient controllability conditions x 4.1. Main results. Under the necessary assumptions of Theorem 1, we can give wide sufficient controllability conditions. Notice that the assumption of simple connectedness can now be removed. So the below sufficient conditions are completely algebraic; this is in contrast with the geometric assumption (the finiteness of center of G) essential for the sufficient controllability conditions for simple and semi-simple Lie groups G [4]. Theorem 2. Suppose that the following conditions are satisfied for a Lie algebra L and a system \Gamma: (1) dimL (3) L (2) r , c , c , (6) the operator adBj L (1) has no N-pairs of real eigenvalues. Then the system \Gamma is controllable on any Lie group G with the Lie algebra L. The notation top (A; a) and the notion of N-pair used in Theorem 2 are explained in Definitions 2 and 3 in Sec. 2. Remarks. (a) Conditions (1)-(3) are necessary for controllability in the case of a simply connected G 6= G (1) (by Theorem 1). (b) Conditions (4) and (5) are close to the necessary conditions (6) and (7) of Theorem 1 respectively. Notice that the fourth condition means that all complex eigenvalues of ad Bj L (1) are geometrically simple. (c) Conditions (2) and (5) are open, i.e., they are preserved under small perturbations of A and B. CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 547 (d) The most restrictive of conditions (1)-(6) is the last one. It can be shown that the smallest dimension of L (1) in which this condition is satisfied and preserved under small perturbations of spectrum of ad Bj L (1) for solvable L is (6). This can be used to obtain a classification of controllable systems \Gamma on solvable Lie groups G with small-dimensional derived subgroups G (1) . (e) The technically complicated condition (6) can be changed by more simple and more restrictive one, and sufficient conditions can be given as in Corollary 2 below. (f) Under the additional assumption of simplicity of the spectrum the sufficient controllability conditions take the even more simple form presented in Corollary 3 below. Corollary 2. Suppose that the following conditions are satisfied for a Lie algebra L and a system \Gamma: (1) dimL (3) L (2) r , c , c , 0g. Then the system \Gamma is controllable on any Lie group G with the Lie algebra L. Corollary 3. Suppose that the following conditions are satisfied for a Lie algebra L and a system \Gamma: (1) dimL (3) the spectrum Sp (1) is simple, r , c , 0g. Then the system \Gamma is controllable on any Lie group G with the Lie algebra L. Theorem 2 and Corollaries 2, 3 will be proved in Subsec. 4.4. x 4.2. Lie saturation. To prove the above sufficient conditions we use the notion of the Lie saturation of a right-invariant system introduced by V. Jurdjevic and I. Kupka. Now we recall the basic definition and properties necessary for us (see details in [4], pp. 163-165). 548 YU. L. SACHKOV Given a right-invariant system \Gamma ae L on a Lie group G, its Lie saturation ae L is defined as follows: cl A 8t 2 R+ \Psi LS(\Gamma) is the largest (with respect to inclusion) system having the same closure of the attainable set as \Gamma. Properties of Lie saturation: (2) LS(\Gamma) is a convex closed cone in L, controllability condition: x 4.3. Preliminary lemmas. In this section we assume that L 6= L (1) (this condition holds, e.g., for solvable L). In view of Theorem 1, we suppose additionally that dimL First we present a necessary technical lemma. Lemma 4.1. Let lim Z (j=2) lim Z Proof. Is obtained by the direct computation. Now we prove the proposition that plays the central role in obtaining our sufficient controllability conditions (Theorem 2). It is analogous to item (a) of Proposition 11, [4]. Lemma 4.2. Let C 2 LS(\Gamma) " L (1) . Suppose that for any a 2 c the following conditions hold : (2) top (C; a) 6= 0 or L (1) (a) ae LS(\Gamma). Suppose additionally that for the number (or CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 549 we have LS(\Gamma) oe Proof. For simplicity suppose that where there are more than two pairs of complex conjugate eigenvalues at the line f Re the proof is analogous; if there are less than two pairs, then the proof is obviously simplified. So we have (notice that if respectively, For any element D 2 L, nonnegative function g(t), and natural number consider the limit Z It follows from the properties of the cone LS(\Gamma) (see Subsec. 4.2) that if D 2 LS(\Gamma) and the limit I(D; Z if the limit exists. Introduce the notation C a Notice that Z For any bounded nonnegative function g(t) and any p 2 N we have is equal to the size of the maximal Jordan block of the operator ad c Bj L (1) c corresponding to the eigenvalues c 2 Sp (1) with Re c ! r. That is why lim Z Consequently, Z if the limit exists. Now we choose the bases f x in the spaces L (1) (a) and L (1) (b) in which matrices of the operators adBj L (1) (a) and adBj L (1) (b) are the Jordan blocksB . M r;fi 0 CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 551 where fi. In the bases f x and we have (the prime denotes transposition of vectors and matrices). Then in the base of the space L (1) (a) \Phi L (1) (b) we have C a (t) e oe ff (t)C X1 oe ff (t)(tC X1 oe ff (t) oe fi (t)C Z1 oe fi (t)(tC Z1 +CZ2 ) oe fi (t) l, then the below argument can easily be modified). (A) Now we show that span(x k ; y k According to the hypotheses of this lemma, we have L (1) (a) ae LS(\Gamma) or top (C; a) 6= 0. If L (1) (a) ae LS(\Gamma), then span(x That is why we suppose below that top (C; a) 6= 0, which means in the base that CX1 6= 0. Taking into account (21), (22), and Lemma 4.1, we obtain Z where 552 YU. L. SACHKOV By virtue of the fact that the convex conic hull of vectors (23) for the matrices M of the form (24) and (25), jjj - 1, is the plane span(x k ; y k ), we We take v(t) 2 LS(\Gamma) equal to i.e., to the component of vector (22) in the plane span(x k ; y k ), and repeat the limit passage described in (A) replacing I(C; g; and obtain span(x We repeat process (B) with I(C; g; p; v), where v(t) is the component of vector (22) in the plane span(x is decreasing from obtain the inclusion span(x l+1 ; y (D) We apply process (C) with l and using the functions g(t) of the We decrease p and repeat procedure (D) until In view of (16), the proof of the lemma is completed. We can give several sufficient conditions for an element B not to have real N-pairs of eigenvalues. These conditions can be verified simply by the picture of spectrum of the operator ad Bj L (1) in the complex plane. We need them to obtain Corollary 2. Lemma 4.3. Suppose that r . Then any one of the following conditions is sufficient for the operator ad Bj L (1) not to have real N-pairs of eigenvalues: (1) (2) (3) 0g. Proof. The first case (Sp (1) obvious as there are no real eigenvalues at all. Case 2. Let (ff; fi) be a real N -pair, 0 ! ff - fi. We have Jacobi's identity implies that [L (1) (a); L (1) (b)] ae L (1) (a + b), and the spaces direct sum, that is why L (2) (ff) ae CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 553 From the conditions a ff. That is why (26) gives L (2) (ff) ae This contradicts item (2) of Definition 3. Case 3 is considered analogously. x 4.4. Proofs of the sufficient controllability conditions. Proof of Theorem 2. We show that LS(\Gamma) oe L (1) . Introduce the following numbers and sets: Suppose that LS(\Gamma) 6oe L (1) , then Recall that we have the following decomposition of the vector A corresponding to root subspaces of the operator adBj L (1) : Define the element Notice that A 1 2 LS(\Gamma) since all terms in the right-hand side belong to LS(\Gamma). In addition, we have A 1 2 L (1) . Consider the decomposition For any a 2 Sp (1) we have Re a 2 [n; m] Re According to condition 6 of this theorem, the pair of real numbers (n; m) is not an N -pair. That is why at least one of conditions (1)-(3) of Definition 3 is violated. Now we consider these cases separately and come to a contradiction. (1) Let condition (1) of Definition 3 be violated, i.e., n 62 Sp (1) or m 62 . Suppose, for definiteness, that m 62 Apply Lemma 4.2 with m. Then we have LS(\Gamma) oe f L (1) (a) j a 2 f L (1) (a) j a 2 which is a contradiction to (28). If n 62 we come to a contradiction with (27) analogously. That is why case (1) is impossible. (2) Let now n; m 2 Sp (1) and let condition (2) of Definition 3 be violated, i.e., But for Re -; we have L (1) (-); L (1) (-) ae LS(\Gamma) (by definitions (27) and (28)), consequently, L (2) (n) ae LS(\Gamma). According to hypotheses of this theorem, L (1) that is why Consider the vector A Now we apply Lemma 4.2 with LS(\Gamma) oe f L (1) (a) j a 2 Then, by virtue of (29), we have LS(\Gamma) oe f L (1) (a) j a 2 which is a contradiction with (27). That is why case (2) is impossible, and condition (2) of Definition 3 cannot be violated. (3) We prove analogously that condition (3) of Definition 3 cannot be violated as well. Hence, all three conditions of Definition 3 hold, and (n; m) is a real N-pair of eigenvalues. This is a contradiction with condition (6) of this theorem. That is why LS(\Gamma) oe L (1) . But controllable by the controllability condition (15). Proof of Corollary 2 follows immediately from Theorem 2 and Lemma 4.3. Proof of Corollary 3 is obvious in view of Corollary 2. CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 555 5. Examples and applications x 5.1. Metabelian groups. Solvable Lie algebras L having the derived series of length 2: are called metabelian. A Lie group with a metabelian Lie algebra is also called metabelian. Our previous results make it possible to obtain controllability conditions for metabelian Lie groups. Theorem 3. Let G be a metabelian Lie group. Then the following conditions are sufficient for controllability of a system \Gamma on G: (1) dimL (3) c , c . If the group G is simply connected, then conditions (1)-(5) are also necessary for controllability of the system \Gamma on G. The notation top (A; a) used in Theorem 3 is explained in Definition 2 of Sec. 2. Proof. The sufficiency follows from Corollary 2. In order to prove the necessity for the simply connected G suppose that \Gamma is controllable. Conditions (1) and (2) follow then from items (1) and (2) of Theorem 1. Condition (3) follows from item (3) of Theorem 1 and from the metabelian property of G: Condition (4). For any a 2 c we have L (2) f0g, that is why j (a) is equal to geometric multiplicity of the eigenvalue a of the operator ad Bj L (1) (a) , i.e., to dimL c (a). By item (6) of Theorem 1, we have that is why dimL c (a) = 1. Condition (5). For any a 2 c we have j (a) = 1, then, by item (7) of Theorem 1, we obtain top (A; a) 6= 0. Example. Let l be a finite-dimensional real Lie algebra acting linearly in a finite-dimensional real vector space V . Consider their semidirect product l. It is a subalgebra of the Lie algebra of affine transformations of the space V since L ae V \Phi s gl (V ). If l is Abelian, then L is metabelian: In the following subsection we study in detail a particular case when l is one-dimensional. x 5.2. Matrix group. Now we apply the controllability conditions from the previous subsection to some particular metabelian matrix group. To begin with we describe this group. Let V be a real finite-dimensional vector space, linear operator in V . The required metabelian Lie algebra is the semi-direct product RM (compare with the example at the end of the previous subsection). Now we choose and fix a base in V , and denote the matrix of the operator M in this base by the same letter M . Then L(M ) can be represented as the subalgebra of gl (n generated by the following matrices: (Recall that E ij is the n \Theta n matrix with the only unit entry in the ith line and the jth raw.) Obviously, we have Notice also that [y and M is the matrix of the adjoint operator ad xj L (1) in the base f y g. In the sequel we consider the Lie algebra L(M in this matrix representation. Let G(M ) be the connected Lie subgroup of GL(n corresponding to L(M ). The group G(M ) can be parametrized by the matrices It is a semidirect product: n\Omega The group G(M ) is not simply connected iff the one-parameter subgroup G 1 is periodic, which occurs iff the matrix M has purely imaginary commensurable spectrum. More precisely, we say that a set of numbers (b R n is commensurable if (b CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 557 And the group G(M ) is not simply connected iff the set Im(Sp(M )) is commensurable: oe Before studying controllability conditions for the group G(M ) we present an auxiliary proposition, which translates the Kalman condition (equivalent both to controllability and to rank controllability condition for linear systems into the language of eigenvalues of the matrix A and of components of the vector b in the corresponding root spaces). We will apply this proposition below to reformulate our controllability conditions for right-invariant and bilinear systems. Lemma 5.1. Let A be a real n \Theta n matrix, b 2 R n . Then the Kalman condition rank (b; is equivalent to the following conditions: (1) the matrix A has a geometrically simple spectrum, (2) top (b; -) 6= 0 for any eigenvalue - 2 Sp(A). By analogy with Definition 2 in Sec. 2, we say that top (b; -) 6= 0 if the component b(-) of the vector b in the root space R n (-) corresponding to the eigengalue - satisfies the condition i.e., the vector b(-) has a nonzero component corresponding to the highest adjoined vector in the (single) Jordan chain of the operator A corresponding to -. To prove Lemma 5.1, we cite the following Proposition 3. (Hautus Lemma, [22], Lemma 3.3.7.) Let A be a complex n \Theta n matrix, b 2 C n . Then the Kalman condition (31) is equivalent to the condition Proof of Lemma 5.1. In view of Proposition 3, we prove that condition (32) is equivalent to conditions (1), (2) of Lemma 5.1. First, we suppose that all eigenvalues of A are real; otherwise we pass to complexification. Second, the Kalman condition (31) preserves under 558 YU. L. SACHKOV changes of base in R n . That is why we assume that the matrix A is in the Jordan normal form: A =B @ . - l 0 Then the n \Theta (n + 1) matrix in condition (32) is represented as denotes projection of the vector b onto the root space of the matrix A corresponding to the eigenvalue - l . Necessity. We assume that rank conditions (1), (2) of Lemma 5.1. 1. If spectrum of A is not geometrically simple, then - j. Then the matrix OE(- i ) has two zero columns, and rank OE(- 2. Suppose that the vector b has the zero -top for some - 2 Sp(A); for definiteness, let top (b; the first component of b in the chosen Jordan base equals to zero, and the first raw of the matrix OE(- 1 ) is zero. Hence rank OE(- 1 Sufficiency. If conditions (1), (2) of Lemma 5.1 hold, then it is easy to see from representation (33) that all matrices OE(- l ), linearly independent columns and condition (32) is satisfied. Now we obtain controllability conditions for the universal covering and for the group G(M ) itself. Theorem 4. Let M be an n \Theta n matrix, is controllable on G if and only if the following conditions hold : (1) the matrix M has a purely complex geometrically simple spectrum, CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 559 For the group G(M ) conditions (1)-(3) are sufficient for controllability; if conditions (30) are violated, then (1)-(3) are equivalent to controllability on The notation top (A; -) used in Theorem 4 is explained in Definition 2 in Sec. 2. Remark . By Lemma 5.1, conditions (1)-(3) of the above theorem are equivalent to the following ones: (1) the matrix M has a purely complex spectrum, Proof of Theorem 4. Theorem 3 (see Subsec. 5.1) is applicable to the group G(M ), and condition (1) of Theorem 3 is satisfied. Decompose the vector B 2 L using the base of L: 2 L (1) is equivalent to B x 6= 0. Moreover, in view of the metabelian property of L, By virtue of Theorem 3, the system \Gamma is controllable on G if and only if the following conditions hold: (3) the matrix M has a geometrically simple spectrum, Now the proposition of the current theorem for - For G(M ), controllability is implied by controllability on its universal covering conditions (30) are violated, then G(M Let now conditions (30) be satisfied. Then the group G(M ) is a semi-direct product of the vector group R n and the one-dimensional compact group G 1 . But controllability conditions on such semi-direct products were obtained by B. Bonnard, V. Jurdjevic, I. Kupka, and G. Sallet [6]: if the compact group has no fixed nonzero points in the vector group (which is just the case), then the controllability is equivalent to the rank controllability condition (Theorem 1, [6]). 560 YU. L. SACHKOV So we have complete controllability conditions of systems of the form on the group G(M ) and its simply connected covering In the simply connected case (i.e., when conditions (30) are violated) we have Theorem 4, and otherwise the theorem of B. Bonnard, V. Jurdjevic, I. Kupka, and G. Sallet [6] works. x 5.3. Bilinear system. Now we apply the controllability conditions for the group G(M ) and study global controllability of the bilinear system where A is a constant real n \Theta n matrix and b 2 R n . Theorem 5. The system \Sigma is globally controllable on R n if and only if the following conditions hold : (1) the matrix A has a purely complex spectrum, Remark . By Lemma 5.1, conditions (1)-(2) of this theorem can equivalently be formulated as follows: (1) the matrix A has a purely complex geometrically simple spectrum, Proof of Theorem 5. We use the hypotheses of this theorem in the equivalent form given in the above remark. Sufficiency. Consider the bilinear system where are (n matrices. It is easy to see that the system \Sigma is globally controllable on R n iff the system \Sigma is globally controllable in the n-dimensional affine plane (R Consider the matrix Lie algebra L(A) and the corresponding Lie group described in the previous subsection. We have ae L(A) is a right-invariant system on the group G(A). Theorem 4 ensures that under hypotheses (1), (2) of the current theorem the system is controllable on the group G(A). But the group G(A) acts transitively in the plane (R , and the bilinear system \Sigma is the projection of the right-invariant system \Gamma from the group G(A) onto the plane (R That is why controllability of \Gamma on G(A) implies controllability of \Sigma on (R Thus \Sigma is globally controllable on R n . CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 561 Necessity. Assume that \Sigma is globally controllable on R n . (1a) First we show that the matrix A has no real eigenvalues. Suppose there is at least one eigenvalue a 2 We choose a Jordan base of the matrix A and denote by f x the corresponding coordinates in R n . Let e k denote the maximum order root vector coresponding to the eigenvalue a: and k is the maximal possible integer. Then the system \Sigma implies where b k is the kth coordinate of the vector b in the base f e g. Now it is obvious that at least one of the half-spaces f x is positive invariant for the system \Sigma, i.e., this system is not controllable. (1b) Now we show that the spectrum Sp(A) is geometrically simple. Suppose that for some (complex) eigenvalue - 2 Sp(A) there are at least two linearly independent eigenvectors. Then we apply the same transformation of Jordan chains as in Lemma 3.4 to obtain the zero component of the vector b in the two-dimensional subspace of R n spanned by the pair of the highest order root vectors of the matrix A (see conditions (13), (14)). Now if x k , y k are the coordinates in R n in the transformed Jordan base corresponding to the above-mentioned two-dimensional subspace, then the system \Sigma yields Hence it follows that the codimension two subspace f x is (both positive and negative) invariant for the system \Sigma, and so it is not controllable. (2) Finally, we show that the vector b has a nonzero -top for any eigenvalue this is not the case, we choose any Jordan chain in the root space corresponding to -, apply the argument from item 1.b) above, and show that \Sigma is not controllable. The necessity and sufficiency are now completely proved. x 5.4. The Euclidean group in two dimensions. It is interesting to consider the work of the above general theory for the visual three-dimensional case. E(2) be the Euclidean group of motions of the plane R 2 . E(2) is connected but not simply connected. It can be represented as the group of 3 \Theta 3 matrices of the form sin t cos t s 2 562 YU. L. SACHKOV where sin t cos t The corresponding matrix Lie algebra L is spanned by the matrices Consider the system E(2) - the universal covering of E(2). A complete characterization of controllability of \Gamma on g E(2) is derived Theorem 4. Theorem 6. The system \Gamma is controllable on g E(2) if and only if the vectors A, B are linearly independent and Let us compare the controllability conditions for g E(2) with the following conditions for E(2) derived from Theorem 1, [6]: Theorem 7. The system \Gamma is controllable on E(2) if and only if the vectors A, B are linearly independent and span(A; B) 6ae span(y; z). Finally, Theorem 5 gives the following geometrically clear proposition. Theorem 8. The system is controllable on the plane R 2 if and only if : (1) the matrix A has a purely complex spectrum, Acknowledgment . The author thanks Professor G'erard Jacob and Laboratoire d'Informatique Fondamentale de Lille, Universit'e Lille I, where this paper was started, for hospitality and excellent conditions for work. The author is also grateful to Professor A. A. Agrachev for valuable discussions of the results presented in this work. CONTROLLABILITY OF RIGHT-INVARIANT SYSTEMS 563 --R System theory on group manifolds and coset spaces. Control systems on Control systems subordinated to a group action: Acces- sibility Transitivity of families of invariant vector fields on the semidirect products of Controllability of right invariant systems on real simple Controllability on classical Controllability of right invariant systems on real simple Controllability of nilpotent systems. Controllability of linear vector fields on Foundations of Notes Math. Controllability of systems on a nilpotent Lie group. Controllability on real reductive Maximal subsemigroups of Controllability of hypersurface and solvable invariant systems. Mathematical control theory: Deterministic finite dimensional systems. --TR Mathematical control theory: deterministic systems --CTR Dirk Mittenhuber, Controllability of Solvable Lie Algebras, Journal of Dynamical and Control Systems, v.6 n.3, p.453-459, July 2000 Yu. L. Sachkov, Classification of Controllable Systems on Low-Dimensional Solvable Lie Groups, Journal of Dynamical and Control Systems, v.6 n.2, p.159-217, April 2000 Dirk Mittenhuber, Controllability of Systems on Solvable Lie Groups: The Generic Case, Journal of Dynamical and Control Systems, v.7 n.1, p.61-75, January 2001

right-invariant systems;controllability;lie groups;bilinear systems

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Heuristic Methods for Large Centroid Clustering Problems.

This article presents new heuristic methods for solving a class of hard centroid clustering problems including the p-median, the sum-of-squares clustering and the multi-source Weber problems. Centroid clustering is to partition a set of entities into a given number of subsets and to find the location of a centre for each subset in such a way that a dissimilarity measure between the entities and the centres is minimized. The first method proposed is a candidate list search that produces good solutions in a short amount of time if the number of centres in the problem is not too large. The second method is a general local optimization approach that finds very good solutions. The third method is designed for problems with a large number of centres&semi; it decomposes the problem into subproblems that are solved independently. Numerical results show that these methods are efficientdozens of best solutions known to problem instances of the literature have been improvedand fast, handling problem instances with more than 85,000 entities and 15,000 centresmuch larger than those solved in the literature. The expected complexity of these new procedures is discussed and shown to be comparable to that of an existing method which is known to be very fast.

INTRODUCTION . analysis is to partition a set of entities into subsets, or clusters, such that the subsets are homogeneous and separated one another, considering measurements describing the entities. This problem already preoccupies Aristotle and appears in many practical applications. For instance, it has been studied by naturalists in the XVIII th century for classifying living species. In this paper, we propose new efficient methods for centroid clustering problems. More precisely, we are going to apply our methods to problems of the following type: given n entities e i with weights w it is searched p centres c j (j = 1, ., p) minimizing , where measures the dissimilarity between e i and c j . However, the methods are very general and may be applied to other problems or objective functions. If the entities are described by their co-ordinates in IR m , d(e i , c j ) is typically the distance or the square of the distance between e i and c j . In the last case, the problem is the well known sum-of-squares clustering (SSC) (see e.g. Ward (1963), Edwards and Cavalli-Sforza (1965), Jancey (1966), MacQueen(1967)). There are many commercial softwares that implement approximation procedures for this hard problem. For instance, the popular S-Plus statistical analysis software incorporates the k-means iterative relocation algorithm of Hartigan (1975) to try to improve the quality of 1. A former version of the article was entitled "Heuristic methods for large multi-source Weber problems". given clusters. For exact algorithms for SSC, see e. g. Koontz, Narendra and Fukunaga (1975) and Diehr (1985). In case: 1) the space is IR 2 , i. e. the Euclidean plane, 2) the centres can be placed everywhere in the dissimilarity measure is the Euclidean distance, the problem is called the multi-source Weber problem (MWP). This problem occurs in many practical applications, such as the placement of warehouses, emitter antennas, public facilities, airports, emergency services, etc. See e. g. Saaty (1972), Dokmeci (1977), Fleischmann and Paraschis (1988), Bhaskaran (1992) and Lentnek, MacPerson and Phillips (1993) that describe practical applications that need to solve MWPs with up to more than 1700 entities and 160 centres. For exact methods solving the MWP, see e. g. Rosing (1992) and Krau (1997). For a unified comparison of numerous approximation algorithms, see Brim- berg et al. (1997). In case dissimilarities between entities are given by an arbitrary n - n matrix and the centres can be placed on the entities only, the problem is called the p-median problem (PMP). The last is a well-known NP-hard problem, see e. g. Hakimi (1965), ReVelle and Swain (1970), Mirchandani and Francis (1990) and Daskin (1995). For exact methods solving the PMP, see e. g. Erlenkotter (1978), Rosing, ReVelle and Rosing-Vogelaar (1979), Beasley (1985) and Hanjoul and Peeters (1985). For an introduction to location theory and clustering see also Gordon (1981), Sp-th (1985), Wesolowsky (1993). The new methods presented in this paper, candidate list search (CLS), local optimization (LOPT) and decomposition/recombination (DEC), have been successfully applied to SSC, MWP and PMP, but they can be extended to solve other problems. For example, the CLS and LOPT methods can be applied to any location-allocation problems as soon as two appropriate procedures are available: the first one for allocating entities to centres and the second one for optimally locating a centre, given the entities allocated to it. For SSC, MWP or PMP, the allocation procedure simply consists in finding the nearest centre to each entity. For other problems, this procedure must be more elaborated (e. g. if there is a constraint limiting the sum of the weights of the entities allocated to a centre). The LOPT method proceeds by local optimization of subproblems. This is a general optimization method that can be applied to problems not directly related to clustering. For example, multi-depot vehicle routing problems can be approached along its lines: the depots being identified with the centres and the optimization of sub-problems being a procedure customized for solving relatively small multi-depot vehicle routing problems. Also, the approach of Taillard (1993) for solving large vehicle routing problems can be viewed as a special application of LOPT where the centres are identified with the centres of gravity of the vehicle tours and the optimization procedure being an efficient taboo search solving vehicle routing problems with few vehicles. In order to remain relatively concise, we are going to present applications of our methods for PMP, SSC and MWP only, but with a special attention to the under studied MWP. Indeed, while the MWP by itself does not embrace all of the problem features found in some practical applications, this model can be very useful, especially for real applications dealing with many thousands of enti- 3ties. In Figure 1, we show the decomposition into 23 clusters of a very irregular problem built on real data, involving 2863 cities of Switzerland. The large black disks are the centres while the small disks are the cities (or entities). Cities allocated to the same centres have the same colour. In this figure, we have also added the federal frontiers and the lakes. Politically, Switzerland is composed of 23 states physically, it is composed of extremely thickly populated regions (Plateau) and regions without cities (Alps, lakes). We see in Figure 1 that the positions of the centres are sensible (no centres are located outside Switzerland or on a mountain or in a lake) and that the decomposition generally respects the natural barriers (spaces without cities); there are very few entities that are separated from their centre by a chain of mountains 2 . Figure 1 has to be compared with Figure 2 showing the decomposition of Switzerland into the same number of clusters obtained by solving a PMP with dissimilarity measure being the true shortest paths (the road network having more than 30000 connections). We see in this figure that the PMP solution is very similar to the MWP one (21 centres are placed almost at the same position; the main difference is that there are less entities allocated to a centre located on the other border of a lake). However, solving this PMP is time consuming: the computation of the shortest paths matrix took 2. The expert can even identify a number of Swiss Cantons in this figure. There are however differences that could be appropriate for solving political problems, such as the union of the South part of Jura to the Canton of Jura, the separation of the German-speaking part of Valais or the union of the small primitive Cantons. Figure 1: Decomposition of Switzerland into 23 clusters by solving a multi-source Weber Problem. 100 times longer than finding a very good MWP solution. Therefore, solving an MWP in a first phase before attacking the true problem (as exemplified by a PMP or a multi-depot vehicle routing problem) can be pertinent, even with an irregular, real problem. Since the clustering problems treated in this paper are difficult, they can be solved exactly for instances of moderate size only. For solving larger instances, as often arise in practice (see the 6800 entities, 380000 network nodes instance of Hikada and Okano (1997)), it is appropriate to use heuristic methods. However, most of the methods of the literature present the same disadvantage of a large increase of the computing time as the number of centres increases and, simultaneously, a decrease in the quality of the solutions produced. The aim of this paper is to show that it is possible to partition a problem with a large number of centres into subproblems that are much smaller, in order to benefit from the advantages of the existing methods for small problems while rapidly producing solutions of good quality to the original problem. The article is structured as follows: in Section 2, we present in detail the alternate location-allocation (ALT) procedure used as a subprocedure of our candidate list search (CLS), showing how it can be implemented efficiently. ALT was first proposed by Cooper (1963) for the MWP. However, it can be generalised for any location-allocation problem as soon as a location procedure and an allocation procedure are available. In this section, we also present CLS, our basic procedure for solving the subproblems generated by partition methods. In Section 3, we present two partition methods for large problems. The first one, LOPT, can be viewed either as a generalization of the ALT procedure Figure 2: Decomposition of Switzerland into 23 clusters obtained by solving a PMP with true shortest paths. or as a restricted CLS for the post-optimization of a given solution. The second decomposition method, DEC, splits a large problem into independent subproblems and the solutions of these sub-problems are optimally mixed together to create a solution to the original problem. Section 4 analyses the computational performances of the methods proposed. 2. BASIC PROCEDURES ALT AND CLS. The procedures ALT and CLS are used as subprocedures in the decomposition methods we pro- pose. Referring to the paper of Cooper (1963) is not sufficient to understand the procedure ALT well, since certain details of this algorithm are not discussed in the original paper and the choices made for implementing the procedure can have a profound impact on its effectiveness. Moreover, we have adapted this procedure to accelerate its execution. Generalized ALT procedure . The iterative location-allocation procedure of Cooper (1963) may be sketched as follows: Cooper has designed this algorithm for the MWP. In this case, the location procedure can be implemented using a procedure like those of Weiszfeld (1937). For the SSC, the centre of gravity of the entities is the optimum location of the centre. For the PMP, the optimum location of a centre can be obtained by enumerating all possible location for the centre. The allocation procedure is very simple for SSC, PMP and MWP: each entity is allocated to its nearest centre. For other problems, this procedure can be more difficult to implement. Two steps of this algorithm have to be discussed: the choice of the initial solution at step 1, and the repositioning of centres that are not used at step 2a. For the choice of an initial solution, many variants have been tested: Position the centres on p randomly elected entities; the probability of choosing an entity being proportional to its weight. Choose the position of the centres one by one, by trying to position them on an entity and by electing the position that minimizes the objective function. The first variant takes into account the structure of the problem, i. e. the geographical and weighting spread of the entities. It produces relatively good initial solutions, especially for problems with non uniform weights. Input: Set of entities with weight and dissimilarity measure, problem specific allocation and location procedures. Choose an initial position for each centre. Repeat the following steps while the location of the centres varies: 2a) Allocate the entities given the centre locations. 2b) Given the allocation made at step 2a, locate each centre optimally. Algorithm 1:Locate-allocate procedure of Cooper, 1963, (ALT). The second variant induces the ALT procedure to produce the best solutions on the average but its computing time is high: for each of the p centres, O(n) positions have to be tried, and for each of these positions, one has to verify whether each entity is serviced by the new position. This implies a procedure that operates in O(n 2 -p) time, while the other variant can be done much faster. To reduce the complexity of this variant and to make it non deterministic 3 , we adopt the following O(n-p) greedy procedure in the spirit of those of Dyer and Frieze (1985): After having repositioned the centres at step 2b of the ALT procedure, it may happen that the allocation of the next iteration, at step 2a, does not use all the centres. The unused centres can be relocated to improve the current solution. We have adopted the following policy: Determine the centre that contributes most to the objective function and place an unused centre on its most distant entity; re-allocate the entities and repeat this as long as unused centres exist. Starting with a very bad initial solution (O(p) centres that are not used), this re-location policy could lead to a O(p 2 -n) procedure. However, our initial solution generator (as well as our CLS procedure presented below) furnish solutions to the ALT procedure that contain an unused centre only exceptionally (for the MWP, we have observed somewhat less than one occurrence in 1000, even for a large number of centres). So, the re-location policy has almost no influence on the solution quality, if one starts with a "good" initial solution as we do. Mladenovic and Brimberg (1996) have shown that the re-location policy can have a substantial effect on MWP solution quality if one starts with "bad" initial solutions. Complexity of ALT for PMP, SSC and MWP. First, let us introduce a new complexity notation: In the remaining of the paper, let -(.) denote an empirically estimated complexity, while O(.) denotes the standard worst case complexity. For example, both quick sort and bubble sort algorithms operate in O(n 2 ) time. In practice however, it is 3. In the context we use ALT, it is more interesting to have a non deterministic procedure. First, it may happen that ALT is called many times for solving the same (sub-)problem. With a non deterministic procedure, it is avoided to repeat exactely the same work. Then, let us mention that only non deterministic procedure can solve NP-hard problems in polynomial time if P - NP. Therefore, our personal view is to consider non deterministic procedure potentially more interesting than deterministic ones, even if there is no theory supporting this for the moment. Input: Set of entities with weight and dissimilarity measure. Choose an entity at random and place a centre on this entity. Allocate all entities to this centre and compute their weighted dissimilarities. 3a) Find the entity that is the farthest from a centre (weighted dissimilarities) and place the k th centre at that entity's location. 3b) For is allocated to a centre farther than centre k: Allocate entity i to centre k and update its weighted dissimilarity Algorithm 2:Initial solution generator. observed that quick sort has an -(n-log(n)) behaviour while bubble sort has an -(n 2 ) behaviour (Rapin, 1983) 4 . There are also algorithms for which the theoretical worst case complexity is not established. However, observing the average computing times by executing an algorithm on many instances can provide a good idea of its complexity in practice. The advantage of this notation is to make a distinction between practice and theory. Indeed, it is common to read that the complexity of quick sort is O(n-log(n)), which is not true, formally. Moreover, the "^" notation is often used by statisticians for estimated values. The complexity of the ALT procedure can be estimated as follows. The complexity of Step 2a (allocation of the entities to a centre) is O(p-n). Indeed, for the problems under consideration one has to allocate each entity to its nearest centre. For large values of p, this step can be substantially accelerated by observing that only the centres that have moved from one iteration to the next can modify the allocation previously made. (Compares the computing times of old and new ALT implementations in Table 3.) Step 2b can be performed in O(n) for the SSC. Indeed, each entity contributes only once in the computation of the position of each centre (independently from the number of centres). For the MWP, the optimum location can be found with a Weiszfeld-like procedure (1937) that repeats an unknown number of gradient steps. We have arbitrarily limited this number to 30. So, in our imple- mentation, Step 2b has a complexity of O(n). For small values of p, the computing time of this step dominates. For the PMP, let us suppose that O(n/p) entities are allocated to each centre (this is reasonable if the problem is relatively regular). 5 For each centre, one has to scan O(n/p) possible locations and the evaluation of one position can be performed in O(n/p). So, the total complexity of Step 2b is locating the p centres. Since p is bounded by n, the global complexity of steps 2a and 2b is bounded by O(n 2 ) for SSC, MWP and PMP. Now, we have to estimate the number of repetitions of Loop 2 which is unknown. However, in practice, we have observed that the number of iterations seems to be polynomial in n and p. Therefore, we will use an -(p a -n b ) estimation of the overall complexity of our implementation of the ALT procedure. In this study, we are mostly interested in instances with large values of p, so, we have considered instances with n/5 - p - n/3 for evaluating the a and b values for the various clustering problems. For the SSC and MWP, we have considered about 7000 instances uniformly generated with up to 9400 entities. For the PMP, we have considered about 38000 runs of the ALT procedure. The PMP instances were based on the 40 different distance matrices proposed by Beasley (1985). The number of entities for these instances ranges from 100 to 900. 4.More precisely, such a behaviour can be mathematically proven. In that case, we propose to follow the usual notation in statistics and to write for an expected running time derived from a mathematical analysis. Therefore it can be written that the complexity of quick sort is . 5.Without this assumption, the complexity is higher; with stronger assumptions (e. g. Euclidean distances), a lower complexity can be derived. O . O log For the SSC, we have estimated a @ 0.83 and b @ 1.19; for the PMP the estimation is a @ 0.70 and b @ 1.23 and for the MWP a @ 0.85 and b @ 1.34. So, if p grows linearly with n, the estimated complexity of the ALT procedure is not far from -(n 2 ) for all these problem types. The memory requirement is O(n) for the SSC and MWP and O(n 2 ) for the PMP i. e. equivalent to the data size. . 2 . Candidate list search (CLS) . CLS is based on a greedy procedure that randomly perturbs a solution that is locally optimal according to the ALT procedure. Then, ALT is applied to the perturbed solution and the resulting solution is accepted only if it is better than the initial one, otherwise one returns to the initial solu- tion. The perturbation of a solution consists in eliminating a centre and in adding another one, located on an entity. The process can be repeated until all pairs entities/centres have been scanned. This greedy procedure finds very good solutions: In Table 1, we report the quality of the solutions found when applied to the 40 PMP instances of Beasley (1985). These instances have been solved exactly and the quality of a solution is given in per cent above the optimum value. The greedy procedure was executed 20 times for each instance. For 8 instances each run found the global optimum and all instances but one were optimally solved at least once. For MWP instances with 50 (respectively 287) entities we observed that the greedy procedure finds a global optimum in more than 60% (respectively 40%) of the cases. For the SSC, we succeeded in improving all the best solutions known to 16 instances with 1060 entities and 10 to 160 centres (See Table 4). For the p-median problem, this type of perturbation has been used for a long time (c. f. Goodchild and Noronha, 1983, Whitaker, 1983, Glover, 1990, Vo-, 1996, Rolland, Schilling and Current, 1997); in this case Glover proposes an efficient way to evaluate the cost of eliminating a centre: during the allocation phase, the second closest centre is memorized - this can be done without increasing the complexity. However, evaluating the decrease of the cost due to the opening of a centre on an entity takes a time proportional to n. Therefore, finding the best possible perturbation has a complexity of O(n 2 -p), without considering the application of the ALT procedure. This complexity is too high for large instances thus we make use of a candidate list strategy scheme proposed by Glover (1990) for implementing a probabilistic perturbation mechanism. The idea is to identify the centre to close by a non deterministic but systematic approach. The entity associated with an open centre is also randomly chosen, but its weighted distance from its previously Table 1: Quality of the greedy procedure for Beasley's PMP instances (% above optimum). allocated centre must be higher than the average. The process is repeated for a number q of itera- tions, specified by the user. Algorithm 3 presents CLS into details. The most time consuming part of this algorithm is step 3e, i. e. the application of the ALT procedure to the perturbed solution. As seen above, we can estimate the complexity of this step as -(p a -n b ). Therefore, the complexity of CLS is -(q-p a -n b ). From now on, we write CLS(q) the improvement of a given solution with q iterations of the CLS procedure. 3. DECOMPOSITION METHODS. In this section, we propose two decomposition methods for solving problems with a large number p of centres. The complexity of these methods is not higher than the ALT procedure while producing solutions of much higher quality. The first decomposition technique, LOPT, starts with any solution with p centres and improves it by considering a series of subproblems involving r < p centres and the entities allocated to them. The subproblems are solved by our CLS algorithm. This method can be viewed as a local search defined on a very large neighbourhood involving up to r centres re-locations at a time. Another point of view is to consider this procedure as a generalization of ALT. Indeed, a solution produced by ALT is locally optimal if we consider any subset of entities allocated to a single centre: every entity is serviced by the nearest centre, and the centres are optimally positioned for the subset of entities they are servicing. Our procedure produces a solution that is sub-optimal (since the subproblems are solved in a heuristic way and since we do not consider all subsets of r centres) for subsets of entities allocated to r centres. A third point of view is to consider LOPT as a CLS procedure with a much smaller list of candidate moves regarding to the CLS presented above. The second decomposition method, DEC, partitions the problem into t smaller subproblems. These subproblems are then solved with our CLS for various numbers of centres. A solution to the (location of the p centres), parameter q. Generate p, a random permutation of the elements {1, ., p} and -, a random permutation of the elements the distance of the most (weighted) distant entity. 3c) While the weighted distance from entity - i to the nearest centre is lower than (d 3d) Close centre p j and open a new one located at entity - i to obtain a perturbed solution s with ALT to obtain s k '' new random permutation p. Algorithm 3:Candidate list search (CLS). initial problem is then found by combining solutions of the subproblems. To decompose the initial problem, we solve an intermediate problem with t centres with our CLS procedure. Each set of entities allocated to a centre of the intermediate problem is considered as an independent subproblem. 3 . 1 . Local optimization (LOPT) . The basic idea of LOPT is to select a centre, a few of its closest centres and the set of entities allocated to them to create a subproblem. We try to improve the solution of this subproblem with CLS. If an improved solution is found, then all the selected centres are inserted in a candidate list C, otherwise the first centre used for creating the subproblem is removed from C. Initially, all the centres are in C and the process stops when C is empty. LOPT has two parameters: r, the number of centres of the subproblems and s, the number of iterations of each call to CLS. Algorithm 4 presents more formally the LOPT method. Complexity of LOPT. To estimate the complexity of LOPT we make two assumptions. First we assume that O(n/p) entities are allocated to each centre (this hypothesis is reasonable if the problem instance is relatively uniform) and second that loop 3 is repeated -(p g -n l ) times. Empirically, we have observed that g is less than 1 and l is close to 0 (see Table 8); we estimate that the value of g is about 0.9 and l is about for the LOPT parameters we have chosen and for the MWP). Then, the complexity of LOPT can be established as follows: Steps 3a and 3d have a complexity of O(p); step 3b has a complexity of O(r-p); step 3c solves a problem with r centres and O(r-n/p) entities, this leads to a complexity of -(s-r a -(r-n/p) b ). This leads to a total complexity of -(r-p are fixed and if p grows linearly with n, the complexity of the LOPT procedure is therefore -(n l This complexity seems to be similar to that of the ALT procedure. In practice, step 3c of the LOPT procedure takes most of the computing time, even if steps 3b has a higher expected complexity for extremely large p. Indeed, for fixed n, we have always observed that the computing time diminishes as p increases, even for p larger than 10000 (see Tables 4 to 8). From now on, we denote by LOPT(r, s) the version of the LOPT procedure using parameters r and s. The memory requirement of the LOPT procedure is O(n). initial position of the p centres, parameters r and s. While C -, repeat the following steps: 3a) Randomly select a centre i - C. 3b) Let R be the subset of the r closest centres to i (i - R). 3c) Consider the subproblem constructed with the entities allocated to the centres of R and optimize this subproblem with r centres with CLS(s). 3d) If no improved solution has been found at step 3c, set else set Algorithm 4: Local optimization procedure (LOPT). 3 . 2 . Decomposition algorithm (DEC) . LOPT optimizes the position of a given number of centres dynamically, but it is also possible to proceed to a static decomposition of the entities, and solve these subproblems with a variable number of centres. A solution to the complete problem may be found by choosing the right number of centres for each subproblem. Naturally, the total number of centres must be limited to p. This re-composition may be performed efficiently and optimally with dynamic programming. The crucial phase of the algorithm is the first decomposition: if the subproblems created do not have the right structure, it is impossible to obtain a good solution at the end. The more irregular the problem is (i. e. where the entities are not uniformly distributed, or their weights differ widely), the more delicate its decomposition is. For partitioning the problem, we use our CLS procedure applied to the same set of entities but with a number t < p of centres. The subproblems created may have very different sizes: a subproblem may consist of just a few entities with very high weights or it may comprise a large number of close entities. Thus it could be difficult to evaluate the number of centres to be assigned to a subproblem. Let n i be the number of entities of subproblem i. Suppose that subproblem i is solved with and let f ij be the value of the objective function when solving subproblem i with j centres. To build a solution to the initial problem, we have to find This problem is a kind of knapsack and may be reformulated as: Thus, the problem can be decomposed and solved recursively by dynamic programming in O(t-p) time. This procedure can also produce all the solutions with t, t centres in O(t-n) time. Such a feature can be very useful when we want to solve a problem for which the number of centres is unknown and must be determined, as for example when there is an opening cost for each centre (the opening cost has just to be added in the f ij values). However, solving each subproblem with 1, ., n i centres is time consuming. If the problem is relatively uniform, one can expect that the optimum number of centres found by dynamic programming is not far from p/t for all subproblems. So, we propose to first solve the subproblems for only three different numbers of centres: -p/t - 1-p/t- and -p/t + 1-. These are solved with one less minimize such that j i p minimize minimize such that j i (respectively one more) centre when the optimum number of centres determined by dynamic programming is exactly the lower (respectively higher) number for which a solution was computed. Algorithm 5 presents our DEC procedure in details. Complexity of DEC . For analysing the complexity of DEC, we make the following assumptions: First, each subproblem has O(n/t) entities, second, each subproblem is assigned O(p/t) centres and third, the number of repetitions of loop 5 is a constant (i. e. the total number of subproblems solved with CLS in steps 3 and 5a is in O(t) ). These assumptions are empirically verified if the problem instances are relatively uniform (see Table 8). With these assumptions, the complexity of DEC can be established as follows: Step 1 is in -(u-t a -n b ); steps 3 and 5a can be performed in -(v-t 1 - a - b -p a -n b ); finally, the complexity of dynamic programming, in steps 4 and 5b is O(t-p). The overall complexity of DEC strongly depends on the parameter t. As shown in the next section, the quality of solutions produced by CLS slightly diminishes as the number of centres increases. We therefore seek to reduce the number of centres in the auxiliary problem and in subproblems as much as possible. For this pur- pose, we have chosen . The overall complexity of our implementation of DEC is are constant and p grows linearly with n, the complexity is lower than the ALT procedure. The memory requirement is O(n 3/2 ). DEC requires more memory than CLS and LOPT, but the increase is not too high and we have succeeded in implementing all the algorithms on a personal workstation. From now on, we note DEC(u, v) the use of the DEC procedure with parameters u and v. Input: Set of entities with dissimilarity measure. Solve an auxiliary problem with t centres with CLS(u). 2) The subsets of entities allocated to the same centre form t independent subproblems. For each subproblem do: Solve subproblem i with CLS(v) with centres and update the associated. Find a collection j 1 *, ., j t * of optimum number of centres to attribute to each sub-problem with dynamic programming. While 5a) For all (i, using CLS(v) and update the f iji associated. 5b) Find a new collection j 1 *, ., j t * of optimum number of centres to attribute to each subproblem with dynamic programming Algorithm 5:The decomposition procedure (DEC). 4. NUMERICAL RESULTS. 4 . 1 . Test problems . For the numerical results presented in this section, we consider six sets composed of 654, 1060, 2863, 3038, 14051 and 85900 entities respectively. The 2863 entities set is built on real data: the entities are the cities of Switzerland and the weight of each city is the number of inhabitants. This set is denoted CH2863. 9 130936.12 160 178764596.3 170 260281.77 170 1890823.0 5000 254125361 14 84807.669 250 126652250.1 300 184832.94 300 1404028.4 10000 166535699 50 29338.011 700 41923352.49 800 94301.618 800 824127.49 70 21465.436 900 28112316.53 1000 78458.720 1000 725300.72 90 17514.423 2000 475580.22 100 16083.535 U1060 2500 409677.92 Table 2: Number of centres and best solution values of the MWP instances. The other sets correspond to the travelling salesman problems that can be found under the names of P654, U1060, Pcb3038 Brd14051 and Pla85900 in the TSPLIB compiled by Reinelt (1995). For these sets, all entities are weighted to one and the dissimilarity between two entities is the Euclidean distance (for PMP and MWP) or the square of the Euclidean distance (for the SSC). From these six sets of entities, we have constructed a large collection of instances by varying p. In Table 2, we give the values of p we have considered for each set, and the best MWP solution known associated with each p. All the best solutions known have been found during the elaboration of methods presented in this paper; some have been reported earlier in Hansen, Mladenovic and Taillard (1996) or in Brim- berg et al. (1997) for P654 and U1060. For P654, we were able to find the same best solution values reported by Brimberg et al. for p - 60, and to find better values for p > 60; for U1060, we succeeded in improving all the best solution values with the exception of we got the same value. The best solutions published in Brimberg et al. were obtained by considering more than 20 different methods, and running each of them 10 times. This last reference also reports the optimum solution values of smaller problem instances with 50 and 287 entities. We were able to find all these optimum solution values with our CLS method. So, we conjecture that many of the solution values given in Table 2 for the smallest set of entities are optimal. For the larger sets, we think that small improvements can be obtained. The aim of Table 2 is to provide new MWP instances and to assert the absolute quality of our methods: Indeed we think that providing the relative quality (measured in per cent over the best solution value of Table 2) allows comparisons to be made more easily than providing absolute solution values. Sometimes, the best solutions known have been found by using sets of parameters for which results are not reported in this paper and it would be difficult to estimate the effort needed to obtain each best solution known. Consequently, we do not provide computing times in this table. Our algorithms are implemented in C++ and run on a Silicon Graphics (SG) 195MHz workstation with R10000 processor. In order to make fair comparisons with algorithms implemented by other authors and executed on a different machine, we have sometimes used another computer, clearly indicated in the tables that follow. It was not possible to report exhaustive numerical results due to the large number of problem instances (160), problem types (PMP, MWP or SSC) and methods (CLS, DEC and LOPT). We try to report representative results in a condensed form. However, let us mention that the conclusions we draw for a given method for a problem type are generally valid for another problem type. 4 . 2 . ALT and CLS . First, we want to show the efficiency of our CLS algorithm by comparing it to the results produced by one of the best methods at the present time for the MWP: MWPM, an algorithm that first solves exactly a p-median before re-locating optimally the centres in the continuous plane. This method is due to Cooper (1963) but has been forgotten for a long time before Hansen, Mladenovic and Taillard (1996) show that in fact, it is one of the most robust for small and medium size MWPs (see also Brimberg et al., 1997). We do not consider methods such as those of Bongartz, Calamai and Conn (1994) which are too slow and produce too poor solutions or those of Chen (1983) or Murtagh and Niwattisyawong (1982) which are not competitive according to Bongartz et al. Also, we do not compare our results with the HACA algorithm of Moreno, Rodr-gez and Jim-nez (1990) for two reasons: First the complexity of HACA is O(p 2 n) and requires an O(p 2 ) memory, i. e. O(n 3 ) in time and O(n 2 ) in memory if are clearly higher than those of our methods. Second, produces solutions that are not as good as MWPM. Indeed, HACA first builds a heuristic solution to the p-median instance associated to the MWP and then applies the ALT procedure to the p-median solution. The reader is referred to Brimberg et al., 1997 for a unified comparison of a large range of heuristic methods for the MWP. To show the effects of the improvements of the ALT procedure proposed in this paper, we provide the best solutions obtained over 100 repetitions of an old version of ALT that starts with different initial solutions; this method is denoted MALT(100). The results for MALT(100) and MWPM originate from Hansen, Mladenovic and Taillard (1996). In Table 3, we give the solution quality (measured in per cent above the solution value given in Table 2) of MWPM, MALT(100), CLS(100) and CLS(1000) and their respective computing times (seconds on Sun P654. The computing time of CLS(1000) is roughly 10 times that of CLS(100). We have averaged all these results for 10 independent runs of the algorithm. Where the 10 runs of CLS(100) find solutions values identical to those given in Table 2, we provide in brackets the number of iterations Quality (% above best known) Computing time [s. Sun Sparc 10] 7.3 Table 3: Comparisons of CLS(100) and CLS(1000) with MWPM and MALT(100) for MWP instances P654. required by the worst run of CLS out of 10 to find the best solution known. From this table, we can conclude that: - The new ALT procedure runs 6 to 9 times faster than the old one (both MALT(100) and call 100 times an ALT procedure, the old one for MALT, the new one for CLS). provides much better solutions than MALT(100). - CLS(1000) provides better solutions than MWPM, in a much shorter computing time. As p grows, the solution quality of all the algorithms diminishes. 4 . 3 . Decomposition methods DEC and LOPT. As LOPT requires an initial solution in input, we indicate the performances of LOPT when applied to the solution produced by the DEC procedure. Table 4 compares CLS(1000), DEC(20, 50), LOPT(10, 50) and 3 VNS variants (due to Hansen and Mladenovic (1997)) for SSC instances built on entities set U1060. This table gives: The best solution value known (found with our methods), the solution quality of the methods (per cent over best known; VNS results originate from Hansen and Mladenovic), and their respective computing times (seconds on Sun workstation). The computing time of VNS1 and VNS2 is 150 seconds for all instances. VNS3 corresponds to the best over ten executions of VNS2; therefore, its computing time is 1500 seconds. It is shown in Hansen and Mladenovic that all VNS variants are more efficient than other methods of the literature, such as the k-means algorithm of Hartigan (1975). For LOPT, we do not take into consideration the computing time of DEC to obtain the initial solution. From this table, we can conclude: - For small values of p, CLS provides better solutions than DEC, VNSs. - For the largest values of p, DEC produces fairly good solutions and their quality seems not to decrease as p increases. value known Quality (% above best known) Computing time [s. Sparc 10] CLS DEC LOPT VNS1 VNS2 VNS3 CLS DEC LOPT 50 255509536.2 0.33 7.54 0.45 30.65 1.97 0.54 114.8 10.8 44.1 90 110456793.7 1.04 7.45 0.49 46.08 1.52 0.78 122.4 12.0 25.2 100 96330296.40 1.14 7.29 0.44 44.51 2.23 1.06 125.2 12.3 24.7 Table 4: Comparison of CLS(1000), DEC(20, 50), LOPT(10, 50) and various VNSs for SSC instances U1060. - The solution quality of LOPT is always very good and seems to be somewhat correlated with the initial solution quality (obtained here with DEC). - Unexpectedly, the computing time of LOPT and DEC diminishes as p increases; this is undoubtedly due to the small number of entities of U1060. produces better solutions than VNSs in a much lower computation time. Table 5 show the effect of the parameters of DEC and LOPT by confronting DEC(20, 50), DEC(20, 200), LOPT(10, 50) (starting with the solution obtained by DEC(20, 50)) and (starting with the solution obtained by DEC(20, 200)). This table provides the solution quality and the computing times (seconds on SG) for the MWP instance CH2863; all the results are averaged over 10 runs. From this table, we can conclude: - For small values of p, the quality of DEC(20, 200) is slightly better than DEC(20, 50) but the computing times are much higher. Starting with solutions of similar quality, LOPT(10, 50) and LOPT(10, 200) produces solutions of similar quality but the computing time of LOPT(10, 200) is much higher. - For larger values of p, the quality of DEC(20, 50) slightly decreases but the quality of remains almost constant. - The computing times of DEC and LOPT diminishes as p increases. greatly improves the solution quality obtained by DEC. - The methods seems to be very robust since they provide good results for instances with a very irregular distribution of dissimilarities. Quality [%] Computation time [s. on SG] 20, 50 20, 200 10, 50 10, 200 20, 50 20, 200 10, 50 10, 200 100 3.4 3.2 0.28 0.20 28 160 56 183 170 4.4 4.1 0.24 0.14 22 126 43 163 190 4.8 4.3 0.34 0.20 24 132 45 157 43 149 300 5.3 4.3 0.47 0.20 400 5.1 4.0 0.63 0.38 17 103 36 108 500 4.8 3.4 1.02 0.39 700 5.8 3.7 1.48 0.52 19 106 900 6.6 4.0 1.59 0.71 1000 7.1 4.7 2.18 1.10 20 108 33 69 Table 5: Quality and computing time of DEC and LOPT for different parameter settings for MWP instance CH2863 In tables 6 and 7, we compare DEC + LOPT to a fast variant of VNS, called RVNS, for SSC and PMP instances built on entities set Pcb3038. RVNS results originate from Hansen and Mladenovic (1997). We have adapted the LOPT parameters in order to get comparable computation times. For all SSC, PMP and MWP instances, we succeeded in improving the best solutions published in this last reference. In Table 6, we can see that DEC + LOPT is able to find better solutions than RVNS in shorter computation times. For the PMP, RVNS seems to be faster than DEC and LOPT for the smallest number of centres. However, let us mention that our implementation derives directly from the MWP one and is not optimized for the PMP. For example we do not compute the distances only once at the beginning of the execution and store them in a (very large) matrix. For large number of centres, DEC + LOPT is again faster and better than RVNS. In Table 8, we provide computational results for our methods DEC(20, 50), DEC(20, 200) and LOPT(7, 50) (applied to the solution obtained with DEC(20, 200)) for MWP instances Brd14051 and Pla85900. We give the following data in this table: The number n of entities, the number p of centres, the solution quality obtained by DEC and LOPT (per cent over best known), the respective computing times (seconds on SG), the proportion of sub-problems solved by DEC and LOPT. For DEC, this proportion corresponds to the number of subproblems solved divided by . For LOPT, this proportion corresponds to the number of subproblems solved divided by p. The results are averaged for 5 runs for Brd14051 and the methods were executed only once for Pla85900. From Table 8, we can conclude that: - The solution quality provided by DEC slightly decreases as p increases; this is due mainly to the decrease in the solution quality provided by the CLS procedure when solving the subproblems. Quality [%] Time [s. Sparc 10] 200 21885997.1 2.44 5.55 0.90 159.9 48 44 300 13290304.8 2.50 7.22 1.44 229.3 400 9362179.2 3.35 7.56 1.70 165.0 43 27 500 7102678.4 2.85 7.47 1.73 204.4 43 25 Table Comparison of DEC(20, 50), LOPT(6, 40) and RVNS for SSC instance Pcb3038. Quality [%] Time [s. Sparc 10] 200 238432.02 1.23 4.12 0.74 107.6 106 187 500 135467.85 0.88 4.01 0.71 209.7 59 81 Table 7: Comparison of DEC(20, 50), LOPT(7, 50) and RVNS for PMP instance Pcb3038. - The computing times of DEC and LOPT diminishes as p increases. However, we can observe an increase in DEC computation times - as predicted by the complexity analysis - only for very large values of p. For LOPT we cannot observe such an increase, meaning that solving the sub-problems takes more time than finding close centres for generating the subproblems. - The solution quality of LOPT is very good, generally well below 1% over the value of the best solution known. - The proportion of subproblems solved by LOPT diminishes as p increases, showing that g is smaller than 1. - The proportion of subproblems solved by DEC seems to be constant, as assumed in the complexity analysis of section 3.2. 5. CONCLUSIONS In this article we have proposed three new methods for heuristically and rapidly solving centroid clustering problems. First, we propose CLS, a candidate list search that rapidly produces good solutions to problems with a moderate number p of centres. Second, we propose LOPT, a procedure that locally optimizes the quality of a given solution. This method notably reduces the gap between the initial solution and the best solution known. The third method proposed, DEC, is based on decomposing the initial problem into subproblems. DEC and LOPT are well adapted to solve very large Quality [%] Time [s. SG] Proportion (7, 50)100 2.4 2.38 0.39 458 1931 3109 4.4 4.4 4.4 200 1.9 2.03 0.30 336 1379 1632 4.3 4.6 3.3 300 2.2 2.08 0.30 252 1073 1055 5.0 5.0 2.6 400 2.2 2.04 0.26 214 915 885 4.9 5.2 2.4 500 2.5 2.12 0.27 195 909 838 4.9 5.8 2.5 2.4 1.99 0.26 184 799 707 5.1 5.8 2.3 700 2.5 1.93 0.23 177 829 632 5.7 6.8 2.2 800 2.5 1.97 0.24 149 760 555 4.8 6.8 2.1 900 2.6 2.00 0.26 146 736 505 6.2 7.3 2.1 1000 3.0 2.17 0.31 129 667 437 4.6 6.6 2.0 2000 4.0 2.81 0.59 88 379 227 4.7 4.8 1.8 3000 4.7 3.65 1.15 82 347 173 4.6 4.9 1.8 5000 4.4 3.59 1.28 73 309 123 4.3 4.5 1.51000 1.78 1.53 0.09 3557 9415 7634 4.2 5.2 2.9 1500 1.97 1.70 0.17 3149 7885 5343 4.3 5.7 2.8 2000 1.81 1.46 0.12 2819 6923 4750 4.1 5.1 2.5 3000 1.74 1.30 0.10 2405 5959 4532 4.1 5.2 2.6 5000 1.87 1.41 0.00 2597 5214 3423 4.1 4.3 2.0 7000 2.03 1.50 0.13 2276 4770 2526 4.3 4.3 1.9 8000 2.07 1.53 0.03 2681 4685 2344 4.1 4.6 2.0 9000 2.55 1.67 0.16 2796 4658 1992 4.1 4.3 1.8 10000 2.78 1.80 0.16 2629 4863 1813 5.1 4.5 1.8 15000 3.71 2.61 0.58 3144 5242 1552 5.3 6.0 1.8 Table 8: Computational results for DEC and LOPT for MWP instances Brd14051 and Pla85900. problems since their computing time increases more slowly with the number of entities than that of other methods in the literature. These methods can solve problems whose size is many order of magnitude larger than the problems treated up to now. Despite its speed, they produce solutions of good quality. The expected complexity of these procedures are given and experimentally verified on very large problem instances. In fact, LOPT is a general optimization method that can be considered as a new meta-heuristic. Indeed it can be adapted for solving any large optimization problem that can be decomposed into independent sub-problems. LOPT has been shown to be very efficient for centroid clustering problems and vehicle routing problems. Future works should consider to apply LOPT to other combinatorial optimization problems. The success of the methods presented in this paper could be explained as follows: solving problems with a very limited number of centres (e. g. below 15) is generally an easy task. Thanks to the use of an adequate neighbourhood, the CLS method allows problems up to 50-70 centres to be treated in a satisfactory way. DEC treats the problem at a high level and is able to determine the general structure of good solutions involving a large number of centres. Starting with a solution that has a good structure, LOPT is able to find very good solutions using a very simple improving approach. Therefore, it is interesting to remark that a very simple improving scheme can lead to a very efficient method if an initial solution with a good structure can be identified and an efficient neighbourhood is used. Indeed, the quality of the solutions obtained by the DEC + LOPT method rival what one would expect from a more elaborate meta-heuristic such as a genetic algorithm, taboo search or simulated annealing. The use of inadequate neighbourhood structures can explain the poor performances of previous implementation of such meta-heuristics. In summary, we can say that our methods open new horizons in the solution of large and hard clustering problems. Acknowledgements The author would like to thank Fred Glover for suggesting numerous improvements, Ken Rosing and Nicki Schraudolf for their constructive comments. This research was supported by the Swiss National Science Foundation, project number 21-45653.95. 6. --R "A Note on Solving Large p-Median Problems" "Identification of transshipment center locations" "A projection method for l p norm location-allocation prob- lems" "Improvements and Comparison of Heuristics for Solving the Multisource Weber Problem" "Solution of minisum and minimax location-allocation problems with Euclidean distances" "Location-allocation problems" Network and Discrete Location "Evaluation of a branch and bound algorithm for clustering" "A quantitative model to plan regional health facility systems" "A Simple Heuristic for the p-Centre Problem" "A Method for Cluster Analisis" "A dual-based procedure for uncapacitated facility location" "Solving a large scale districting problem: a case report" "Location-Allocation for Small Computers" Classification: Methods for the Exploratory Analysis of Multivariate Data "Tabu Search for the p-Median Problem" "Optimum distribution of switching centers in a communication network and some related graph theoretic problems" "An approach for solving a real-world facility location problem using digital map" "A comparison of two dual-based procedures for solving the p-median prob- lem" "Heuristic solution of the multisource Weber problem as a p-median problem" "An introduction to Variable Neighborhood Search" Clustering Algorithms "Multidimensional Group Analysis" "A Branch and bound clustering algorithm" Extensions du probl-me de Weber "Optimum producer services location" "Some Methods for Classification and Analysis of Multivariate Observations" Discrete Location Theory "Heuristic cluster algorithm for multiple facility location-allocation problem" "An efficient method for the multi-depot location-allocation problem" Cours d'informatique g-n-rale "TSPLIB95" "An efficient tabu search procedure for the p-Median Prob- lem" "An Optimal Method for Solving the (Generalized) Multi-Weber Porblem" "The p-Median and its Linear Programming Relaxation: An Approach to Large Problems" "Optimum positions for airports" Dissection and Analysis (Theory "Parallel iterative search methods for vehicle routing problems" "A reverse elimination approach for the p-median problem" "Hierarchical Grouping to Optimize an Objective Function" "Sur le point pour lequel la somme des distances de n points donn-s est minimum" "The Weber problem: history and perspectives" "A fast algorithm for the greedy interchange for large-scale clustering and median location problems" --TR --CTR Hongzhong Jia , Fernando Ordez , Maged M. Dessouky, Solution approaches for facility location of medical supplies for large-scale emergencies, Computers and Industrial Engineering, v.52 n.2, p.257-276, March, 2007 Mauricio G. C. Resende , Renato F. Werneck, A Hybrid Heuristic for the p-Median Problem, Journal of Heuristics, v.10 n.1, p.59-88, January 2004 Teodor Gabriel Crainic , Michel Gendreau , Pierre Hansen , Nenad Mladenovi, Cooperative Parallel Variable Neighborhood Search for the p-Median, Journal of Heuristics, v.10 n.3, p.293-314, May 2004

multi-source Weber problem;sum-of-squares clustering;p-median;clustering;location-allocation

608327

Satellite Image Deblurring Using Complex Wavelet Packets.

The deconvolution of blurred and noisy satellite images is an ill-posed inverse problem. Direct inversion leads to unacceptable noise amplification. Usually the problem is regularized during the inversion process. Recently, new approaches have been proposed, in which a rough deconvolution is followed by noise filtering in the wavelet transform domain. Herein, we have developed this second solution, by thresholding the coefficients of a new complex wavelet packet transform&semi; all the parameters are automatically estimated. The use of complex wavelet packets enables translational invariance and improves directional selectivity, while remaining of complexity O(N). A new hybrid thresholding technique leads to high quality results, which exhibit both correctly restored textures and a high SNR in homogeneous areas. Compared to previous algorithms, the proposed method is faster, rotationally invariant and better takes into account the directions of the details and textures of the image, improving restoration. The images deconvolved in this way can be used as they are (the restoration step proposed here can be inserted directly in the acquisition chain), and they can also provide a starting point for an adaptive regularization method, enabling one to obtain sharper edges.

Introduction The problem presented here is the reconstruction of a satellite image from blurred and noisy data. The degradation model is represented by the equation : Y is the observed data and X the original image. N is additive noise and is assumed to be Gaussian, white and stationary. The # represents a circular convolution. The Point Spread Function (PSF) h is positive, and possesses the Shannon property. We deal with a real satellite image deblurring problem, proposed by the French Space Agency (CNES). This problem is part of a simulation of the future SPOT 5 satellite. The noise standard deviation # and the PSF h are assumed known. The deconvolution problem is ill-posed because of the noise, which contaminates the data. The inversion process strongly amplifies the noise if no regularization is done. Thus, we have to deconvolve the observed image while recovering the details, but without amplifying the noise. The process used to estimate X from the degraded data Y must preserve textures, to enable the results to be visually correct. Moreover, the noise should remain small in homogeneous regions. Many methods have been proposed for regularizing this problem by introducing a priori constraints on the solution [4], [9], [23]. However, most of them do not preserve textures, since these textures are not taken into account in the regularizing model. To achieve a better deconvolution, other authors [1], [19], [21] prefer to use a wavelet-based regularizing function. But all these approaches have the drawback that they are iterative, which means that they are time consuming and not always appropriate for deconvolving satellite data, where the images can be very large. A few authors, such as Donoho et al. [6], and Mallat and Kalifa [16], proposed denoising the image after a deconvolution without regularization. The images are represented using a wavelet or wavelet packet basis, and the denoising process is done in this basis. This method is not iterative and provides a very fast implementation. A simple inversion of the observation equation (1) in frequency space gives an unacceptably noisy solution. To denoise it, a compact representation has to be chosen, in order to separate the signal from the noise as well as possible. A representation is said to be compact if it approximates the signal with a small number of parameters, which can be the coe#cients of the decomposition in a given basis. The noise amplified by the deconvolution process is colored. Furthermore, the coe#- cients of this noise are not independent in the wavelet basis. Thus, the basis must adapt to the covariance properties of the noise. The covariance should be nearly diagonal [15] w.r.t. the basis, to decorrelate the noise coe#cients as much as possible. The Fourier basis achieves such a diagonalization, but the energy of the signal is not concentrated over a small number of coe#cients (the basis vectors are not spatially localized), so the Fourier transform is not suitable for any thresholding method. A good compromise is to use a wavelet packet basis [5], since it nearly realizes the two following essential conditions, i.e. the signal representation is sparse, and the noise covariance operator is nearly diagonalized [15]. Many types of wavelets transforms can be used to construct a packet basis ; they exhibit di#erent properties depending on their spatial or frequency localization, and on their separability w.r.t. rows and columns. Decimated real wavelet transforms are e#cient for satellite image deconvolution but produce artefacts since the transform is not shift invariant. To avoid these artefacts, the resulting image has to be averaged over all possible integer translations. It is also possible to use shift invariant transforms, but the redundancy is generally very high and depends on the depth of the transform. Anyway, these two techniques have the major drawback of slowing down the algorithm. The main motivation of our work is to solve this problem in a computationally e#cient manner. There is a way to enable translation invariance without much loss of computational time, by using complex wavelets [17], [18]. Such wavelets also provide a better restoration by separating 6 directions, while real separable wavelets only take into account two directions. In order to achieve the necessary near-diagonalization of the deconvolved noise covariance, we have implemented a complex wavelet packet algorithm. Our essential contributions are the following 1. We have designed a new transform, the complex wavelet packet transform, which has better directional selectivity than the complex wavelet transform, while exhibiting the same shift and rotational invariance properties. 2. The proposed algorithm is fully automatic, since it is based on a Bayesian approach, where all the necessary parameters are estimated by Maximum Likelihood. 3. We have proposed a new hybrid technique, consisting of combining two di#erent meth- ods, regularization and wavelet thresholding, to obtain optimal deconvolution results. 4. It performs the inversion much faster than shift invariant real transforms and reconstructs features of various orientations better. The paper is organized as follows. First, in section II, we detail how to compute the proposed complex wavelet packet transform and the properties of this new transform. Then, in section III, we present the Bayesian thresholding framework used to estimate the unknown coe#cients from the observed data, and explain how to compute the variance of the deconvolved noise. Sections III-A to III-C are devoted to the presentation of the di#erent prior models put on the wavelet coe#cients - homogeneous, noninformative and inhomogeneous priors. In section III-C.2, we detail the hybrid technique used to estimate the adaptive parameters of the latter. This model is used in the algorithm proposed in section IV. Finally, we present, in section V, a comparison with classical algorithms used for satellite image deconvolution, to demonstrate the superiority of the proposed method. II. Complex wavelet packets To build a complex wavelet transform, Kingsbury [17] has developed a quad-tree al- gorithm, by noting that an approximate shift invariance can be obtained with a real biorthogonal transform by doubling the sampling rate at each scale. This is achieved by computing 4 parallel wavelet trees, which are di#erently subsampled. Thus, the redundancy is limited to 4, compared to real shift invariant transforms. The shift invariance is perfect at level 1, and approximately achieved beyond this level : the transform algorithm is designed to optimize the translation invariance. Therefore, it involves two pairs of biorthogonal filters, odd, h o and g o , and even, h e and g e . At level simply a non-decimated wavelet transform (using h are re-ordered into 4 interleaved images by using their parity. This defines the 4 trees B, C and D. For j > 1, each tree is processed separately, as a real transform, with a combination of odd and even filters depending on each tree. The transform is achieved by a fast filter bank technique, of complexity O(N ). We have extended the original transform by applying the filters h and g on the detail subbands, thus defining a complex wavelet packet (CWP) transform [13], [14]. This new transform exhibits the same invariance properties as the original complex wavelet transform. The tree corresponding to this transform is given in Fig. 1. The filter bank used for the decomposition is illustrated by Fig. 2, on which the subbands are indexed by (p, q) for each tree T . The impulse responses, shown in Fig. 3, and the related partitioning of the frequency space given in Fig. 4, demonstrate the ability to separate up to 26 directions for the chosen decomposition tree. Compared to real separable transforms, which only define two directions (rows and columns), it provides near rotational invariance and gives a selectivity which better represents strongly oriented textures (thus separating them better from the noise). Furthermore, compared to the original complex wavelet transform, which only separates 6 directions, the directional selectivity is higher. Instead of dividing an approximation space which does not define any new orientation, the wavelet packet decomposition processes the detail subbands, which are strongly oriented. Each detail subband at level 1 isolates an area of the frequency space defined by a mean direction and a dispersion, enabling one to select a range of directions around an orientation #. If the subband is decomposed into 4 new subbands, it means that the corresponding frequency area splits into 4 new areas, which can define new orientations, as shown in Fig. 4. This is not really a complex transform, since it is not based on a continuous complex mother wavelet. Nevertheless, the quad-tree transform has in practice the same properties as a complex transform w.r.t. shifting of the input image, and it is perfectly invertable and computationally e#cient. The complex coe#cients are obtained by combining the di#erent trees together. If we index the subbands by k, the detail subbands d j,k of the parallel trees A, B, C and D are combined to form complex subbands z j,k and z j,k - , by a linear transform (represented by a matrix M ), in the following way : z j,k A - d j,k z j,k Thresholding the magnitudes |z - | without modifying the phase enables us to define a nearly shift invariant filtering method. The reconstruction is done in each tree independently, by using the dual filters, and the results of the 4 trees are averaged to obtain a 0 to ensure the symmetry between the trees, thus enabling the desired shift invariance. III. Bayesian thresholding Let us denote X the deconvolved image without regularization. For each subband k, the variables x and # denote one of the CWP coe#cients corresponding respectively to the observation Y and the original image X . We suppose that H is invertible (i.e. it has no zeros in the Fourier space). Since the noise N is white and Gaussian, Equ. (1) multiplied by H -1 gives, in the CWP domain : To obtain the expression of the noise coe#cients n, let us recall Equ. (2). Each complex coe#cient is obtained by summing or subtracting the coe#cients of the 4 trees A, B, C, D. If we compute the covariance between the real and imaginary parts, we find (for the coe#cients z are the noise coe#cients for each tree. By symmetry assumptions, this covariance is null, since all covariances E[n T n T # ] between di#erent trees T and T # are equal. Then, the distribution of the noise is defined as a joint distribution of (n r , We assume that the coe#cients # are independent in a given subband, between di#erent subbands of a given scale, and also between scales. This is an approximation which enables a fast thresholding technique: we will not handle here possible correlations between subbands or neighbour coe#cients. The covariance matrix of the noise is supposed to be nearly diagonal in the chosen basis (see Fig. 6 and Fig. 7), so we consider that the noise variables are also independent in the wavelet packet basis. We assume that the noise variance is constant in each subband k. We compute # k by considering the undecimated transform of the noise N , which is performed by a linear operator (convolution with the impulse response w k , obtained by an inverse CWPT of a Dirac). F denotes the Fourier transform. We estimate the unknown coe#cients # within a Bayesian framework [22]. We have demonstrated in [14] that this approach provides slightly better results than the Minimax risk calculus on real satellite data. To compute the MAP estimate of #, we use Bayes law to calculate the expression of the posterior probability : where P (x | #) is given by the distribution of the noise : A. Homogeneous prior model Generalized Gaussian distributions have been used to model real wavelet coe#cients [20], [22]. We also propose to use them to model the CWP subbands. We have the prior probability of # : where # k is a prior parameter and p k is an exponent. However, we assume that the variables # are independent within a given subband, and also between the di#erent subbands. As shown by Fig. 5, the complex density is a bidimensional function which only depends on the magnitude (it exhibits a radial symmetry). It is generally not separable for p #= 2. We set a given density function on the magnitude, while the phase is uniformly distributed in The exponent p k can be set to a fixed value, the same for all subbands, to simplify the computation. Indeed, if this parameter is not specified it must be estimated. This is quite complex and is not justified by the improvement of the results. On large size images, the parameter # k can be computed e#ciently by various methods, such as Maximum Likelihood for example. We choose to estimate it automatically from the histogram of a given subband. Then we obtain the following expression of the MAP, using the prior law It is possible to demonstrate that it is equivalent to apply a thresholding operator # p k to the magnitude of each coe#cient [14] : For complex wavelets, this leads to the following equations : and This shows that # is obtained from x by keeping the phase, and thresholding the magnitude. It reduces to simple soft thresholding if elsewhere. Thus, we have : This classical filtering method naturally arises from the Bayesian approach. The resulting thresholding functions are smoother for is the exponent of the prior law (7)), and they become linear if 2. If p k < 1, which is more realistic for satellite images, the functions become discontinuous. Then, the thresholding functions # are numerically computed by solving equation (10) w.r.t. Fig. 8 shows the behaviour of these functions for di#erent values of p k . Experimental studies have shown that provides an e#cient model for satellite images [14]. B. Noninformative Je#rey's prior It is possible to use a di#erent approach for subband modeling. This approach has been proposed in [8] within a wavelet based image denoising framework. It is based on the following assumption : the inference procedure should be invariant under changes of amplitude and scale. It means that the prior probability law of # must keep the same behaviour even if it is rescaled, by setting for example this is not the case for classical Gaussian or Laplacian models, the author uses the following prior, which is called a noninformative prior This corresponds to an extremely heavy-tailed distribution, which approximately describes the original wavelet coe#cients #. Unfortunately it is improper : the resulting posterior density function is not integrable. Therefore, an alternative to the fully Bayesian framework is chosen. It consists of treating the real or imaginary part of each coe#cient as a zero-mean Gaussian variable, of variance s 2 . This defines an adaptive model, since each coe#cient has a di#erent variance. Then, this variance is supposed to follow Je#rey's hyperprior distribution, i.e. we have : This is equivalent to P (# 1/|#| within a Bayesian context. Thus, the model remains homogeneous, even though an adaptive Gaussian model is used intermediately to address the problem of the improper distribution. The estimation is then performed in two steps : . estimate the variance - s 2 by using the | x) . estimate the unknown coe#cient by using the To express P (s 2 | x), we need P both signal and noise are Gaussian, of respective variances s 2 and # 2 k , we have P We also have the hyperprior (13). Then we | which is equivalent to s The MAP estimate - for the inhomogeneous Gaussian model gives the following expression (see the next section for a complete proof) : By combining equations (14) and (15) we obtain the following estimate, which we call the noninformative thresholding function # J : The advantage of such a method is that there is no need for parameter estimation, since there is no parameter. However, there is a drawback, which is especially visible in homogeneous areas : the residual noise is quite apparent, its variance being higher than with the previously presented homogeneous model. This probably comes from the lack of robustness of the estimation method. We can remark that if we remove the prior law of the estimation of s 2 is done by the MLE, which gives a function like but with a threshold 2# 2 instead of 4# 2 . This is insu#cient since the magnitude of the noise has a variance equal to 2# 2 . Thus, the hyperprior makes the estimation more robust, by doubling the threshold value. It is still not su#cient to remove noise peaks in large constant areas. Hereafter, we detail a more robust method to filter the complex wavelet packet subbands. C. Inhomogeneous prior model C.1 The model The Generalized Gaussian model presented previously has been chosen because it seems to match correctly the original coe#cient distribution, which is heavy-tailed. Another possible way to capture this property is to define an inhomogeneous model, which adapts to the local characteristics of the subbands. Some approaches to spatial adaptivity in the real wavelet domain can be found in the literature, for example [3]. To simplify, let us choose a Gaussian model. The variance parameter is di#erent for each coe#cient, which enables us to di#erentiate edges or textures, which have a high intensity, from the homogeneous areas which generally correspond to very low values of the coe#cients. Since the parameters can be very di#erent from one variable to another, the histogram of a subband can have a heavy-tailed behaviour, even if the distribution of each variable is Gaussian. We denote by s 2 the variance of the real or imaginary part of an original coe#cient #, as before. We have the prior law : If the parameters of the prior distribution are known, the unknown variables are estimated by computing the MAP. This is a fully Bayesian technique (we use the property (5)). Recall the expression of the noise distribution (6), and combine it with Equ. (17) to obtain : Therefore, the MAP is given by : To compute the minimum, di#erentiate w.r.t. the real and the imaginary parts of #. This gives two equations which are recombined to form an equation with complex numbers : #r +xr This gives the inhomogeneous MAP estimate : C.2 Adaptive parameter estimation The most di#cult problem in this approach is to estimate the adaptive parameters of the model. As is shown in [12], the MLE is not robust when applied to incomplete data (i.e. when the estimation is made on noisy data). Indeed, there are as many parameters as observed data. But the robustness becomes su#cient when the estimation is made from the original image X . A good approximation of this image is still su#cient to provide useful parameter estimates. Here, we use this complete data approach to estimate the variances s 2 . Consider Equ. (17). The complete data MLE is defined by - s Assumimg independence, we obtain : where the factor 2 comes from the dimensionality of the distribution. We obviously do not have access to the original coe#cients, which is why we take the transform coe#cients of an approximate original image instead. Experiments have shown that a satisfactory approximation is provided by a nonlinear regularizing algorithm, such as RHEA, detailed in [11]. It essentially consists of a variational method based on #- functions [4] (minimization of a criterion which penalizes noisy solutions, but preserves edges) preceded by an automatic parameter estimation step to compute the hyperparameters of the regularizing model. This method is certainly not perfect and some residual noise remains. It is visible in constant areas. However, we filter this noise as well as the deconvolved noise by a thresholding technique. We choose to use the noninformative threshold of the previous section, because it does not require any additional parameter estimation. The proposed algorithm consists of obtaining the desired approximate original image using RHEA [11], filtering the CWPT of the result using Equ. (16), estimating the adaptive parameters using the complete data MLE with Equ. (20), and then estimating the unknown coe#cients by computing the MAP by Equ. (19). In addition to the computation of the deconvolved noise variance, we also need to compute the variance 2-# 2 k of the residual noise of the approximate image. Let us denote by - # the thresholded transform coe#cients of the approximate original image. - # is supposed to be su#ciently exempt from noise and to contain su#cient information to enable texture and edge recovery. Homogeneous areas and edges are fine, since we have used an edge-preserving method followed by an e#cient noise thresholding. But we still have to explain why the coe#cients related to textured areas are su#ciently high to avoid too strong an attenuation by using Equ. (19) in these areas. The variational method used does not completely remove the textures, and even if they are visually not very sharp, they are su#ciently present in the approximate image to enable a correct reconstruction using the method detailed here. Finally, if - # is known, by using Equ. (19) and (20), the estimate for the coe#cient is : | - If we compute the expression of the thresholding function which minimizes the Bayesian risk of the estimator - #, defined by denotes an expectation w.r.t. the distribution of x, we find : It has exactly the same form as Equ. (21), if we take the approximate original coe#cient # instead of #. Both Bayesian and minimum risk methods lead to the same expression, which means that the chosen model is good, since the corresponding estimator provides the minimum risk. As in the case of Wiener filtering [10], computing the MAP under Gaussian assumptions (both signal and noise have Gaussian distributions) is equivalent to minimizing the risk of a linear estimator w.r.t. the attenuation factor. Therefore, the two approaches are equivalent in the Gaussian case. IV. The deconvolution algorithm The adaptive model described by Equ. (17) provides much better results (visually and w.r.t. SNR) than the homogeneous model described by Equ. (7). Details are better preserved and constant areas are cleaner. That is why we keep this model for the final version of the proposed algorithm. We have also compared this scheme with the classical approach [6] and with minimum risk computation for various models [14]. It consistently exhibits better results. The initial deconvolution is made in the Cosine Transform space instead of the Fourier space to avoid artefacts near the borders of the image. The proposed algorithm is called COWPATH, for COmplex Wavelet Packet Automatic THresholding, and consists of the following steps (see Fig. . DCT (Discrete Cosine Transform) of the observation Y . Deconvolution : divide by F [h] (in practice, divide by F [h] is small, since some of the coe#cients of F [h] can be null) . Inverse DCT of the result, which gives X . CWP transform of X . Computation of the approximate original image - apply the RHEA algorithm [11] on Y (nonlinear regularization, with automatic parameter estimation) . CWP transform of - . Computation of # k using the known h and # (see Equ. (4)) . Computation of - # k (residual noise on the approximate original image) using the known h and # (see [14] for details) . Thresholding of the approximate image coe#cients using Je#rey's noninformative prior and - . Estimation of the parameters - s of the inhomogeneous Gaussian model (see Equ. (20)) . Coe#cient thresholding by computing the MAP (see Equ. (21)) . Inverse CWP transform, which gives the estimate - X. The variance of the residual noise in homogeneous areas, which is needed to denoise the approximate original image before using this image for parameter estimation, is computed in the same manner as the variance of the deconvolved noise (i.e. using an equation similar to Equ. (4)). For this computation, we assume that constant regions correspond to a quadratic regularization. Then it is possible to use sums in the Fourier space. We refer to [14] for more details. The main characteristic of this algorithm is the use of two di#erent methods, regularization and wavelet thresholding, to obtain a hybrid technique whose results are better than the results of a single regularization method or a wavelet thresholding. The more decorrelated the deconvolved noise and the residual noise of the approximate original image, the higher the quality of the deconvolved image. It is possible to replace the nonquadratic regularizing model of the RHEA algorithm by a simple quadratic model. The advantages are to enable a single step deconvolution in the frequency space and to make the parameter estimation step deterministic and fast (in the nonquadratic case we need a MCMC method [11]). The edges are not as sharp as with the nonquadratic model, but this approximate image is su#ciently accurate to provide a correct estimation of the inhomogeneous parameters of the subband model (the SNR di#erence between the original and accelerated algorithms is about 0.1 dB for the SPOT 5 simulation image shown in Fig. 13). The complexity of the algorithm is 115 log (operations per pixel) (1980 op/pix for a 512 - 512 image). If we use the accelerated version, based on the previously described quadratic model, it is 15 log image, which represents about 4 s on a Pentium II 400 MHz machine). V. Satellite image deblurring A. simulation Fig. 10 shows a 128x128 area extracted from the original image of N- mes (SPOT 5 simulation at 2.5 m resolution, provided by the French Space Agency (CNES)). Fig. 11 shows the PSF and Fig. 12 shows the observed image Y Fig. 13 shows the image deconvolved with the proposed algorithm. B. Comparison with di#erent methods 1. Quadratic regularization [23]. This is nearly equivalent to the parametric Wiener filter which gives the same results. It is also equivalent to isotropic di#usion. The edges are filtered as well as the noise, as seen on Fig. 14. It is therefore impossible to obtain sharp details and noisefree homogeneous areas at the same time. Thus, the SNR remains low (about 19.7 dB) because of insu#cient noise removal in these areas. 2. Nonquadratic regularization [4], [9]. The resulting image of the RHEA algorithm [11] exhibits sharp edges, compared to the result shown previously. However, some noise remains in homogeneous regions and textures are attenuated. The SNR is 22.0 dB. This result is used as the approximate original image and is illustrated by Fig. 15. 3. Real wavelet packets [15]. The proposed complex wavelet packet transform is more than two times faster than the shift invariant real wavelet packet transform (based on Symmlet-4 wavelets) and is much more directionally selective. Real wavelet packet thresholding gives a SNR equal to 21.8 dB. See Fig. 16 for an illustration. 4. The proposed method. This is faster than the other methods, and provides the highest SNR (22.2 dB). The textures and the oriented features are sharp and regular, while the homogeneous regions remain noise free, as seen on Fig. 13. VI. Conclusion We have proposed a new complex wavelet packet transform to make an e#cient satellite image deconvolution algorithm. This transform exhibits better directional and shift invariance properties than real wavelet packet transforms, for a lower computational cost. The proposed deconvolution method is superior to other competing algorithms on satellite images : it is faster, more accurate and fully automatic. The essential novelty of the proposed algorithm consists of an hybrid approach, in which two radically di#erent methods are combined to produce a deconvolved image of higher quality than the result of each method individually. Finally, if a quadratic model is used, the speed is greatly increased, which opens the path to real time processing of image sequences or to onboard image processing in satellites by using specialized chips. Furthermore, this new type of approach can be extended and the results can be improved to handle more di#cult cases (di#erent types of blur and higher noise variance). It is possible to take into account various levels of noise and di#erent convolution kernels. Indeed, the wavelet packet decomposition tree is not unique and should be adapted to the noise statistics. The case of noninvertible blur, such as motion blur, forbids the use of nonregularized inversion in the Fourier domain. Therefore, in this case, a regularized deconvolution should replace the rough inversion used in the proposed method. An improvement of the adaptive Gaussian model could also be provided by choosing a more accurate distribution, to capture the heavy-tailed distribution of the wavelet packet coe#cients. For example, adaptive Generalized Gaussian models could be investigated. Furthermore, including dependence between di#erent scales by means of multiscale hidden Markov trees could perhaps enable better separation of the small features from the deconvolved noise. These types of models would certainly better model edges which propagate across scales and enable their reconstruction more e#ciently. VII. Acknowledgements The authors would like to thank J-er-ome Kalifa (from CMAPX, at Ecole Polytechnique) for interesting discussions and Nick Kingsbury (from the Signal Processing Group, Dept. of Eng., University of Cambridge) for the complex wavelets source code and collaboration, Peter de Rivaz (same institution) for his kind remarks, and the French Space Agency (CNES) for providing the image of N- mes (SPOT 5 simulation). --R Wavelet domain image restoration using edge preserving prior models. Bayesian Theory. Spatial adaptive wavelet thresholding for image denoising. Wavelet analysis and signal processing Denoising by soft thresholding. spatial adaptation via wavelet shrinkage. Bayesian wavelet-based image estimation using noninformative priors Stochastic Relaxation Image restoration by the method of least squares. Restauration minimax et d-econvolution dans une base d'ondelettes miroirs Wavelet packet deconvolutions. The dual-tree complex wavelet transform: a new e#cient tool for image restoration and enhancement "Wavelets: the key to intermittent information?" A theory for multiresolution signal decomposition: the wavelet representation. A wavelet regularization method for di Bayesian inference in wavelet based methods Regularization of incorrectly posed problems. --TR

complex wavelet packets;satellite and aerial images;deblurring;bayesian estimation

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PAC-Bayesian Stochastic Model Selection.

PAC-Bayesian learning methods combine the informative priors of Bayesian methods with distribution-free PAC guarantees. Stochastic model selection predicts a class label by stochastically sampling a classifier according to a posterior distribution on classifiers. This paper gives a PAC-Bayesian performance guarantee for stochastic model selection that is superior to analogous guarantees for deterministic model selection. The guarantee is stated in terms of the training error of the stochastic classifier and the KL-divergence of the posterior from the prior. It is shown that the posterior optimizing the performance guarantee is a Gibbs distribution. Simpler posterior distributions are also derived that have nearly optimal performance guarantees.

INTRODUCTION A PAC-Bayesian approach to machine learning attempts to combine the advantages of both PAC and Bayesian approaches [20, 15]. The Bayesian approach has the advantage of using arbitrary domain knowledge in the form of a Bayesian prior. The PAC approach has the advantage that one can prove guarantees for generalization error without assuming the truth of the prior. A PAC-Bayesian approach bases the bias of the learning algorithm on an arbitrary prior distribution, thus allowing the incorporation of domain knowledge, and yet provides a guarantee on generalization error that is independent of any truth of the prior. PAC-Bayesian approaches are related to structural risk minimization (SRM) [11]. Here we interpret this broadly as describing any learning algorithm optimizing a tradeoff between the "complexity", "structure", or "prior probability" of the concept or model and the "goodness of fit", "description length", or "likelihood" of the training data. Under this interpretation of SRM, Bayesian algorithms that select a concept of maximum posterior probability (MAP algorithms) are viewed as a kind of SRM algorithm. Various approaches to SRM are compared both theoretically and experimentally by Kearns et al. in [11]. They give experimental evidence that Bayesian and MDL algorithms tend to over fit in experimental settings where the Bayesian assumptions fail. A PAC-Bayesian approach uses a prior distribution analogous to that used in MAP or MDL but provides a theoretical guarantee against over fitting independent of the truth of the prior. Perhaps the simplest example of a PAC-Bayesian theorem is noted in [15]. Consider a countable class of concepts f 1 , f 2 , f 3 each concept f i is a mapping from a set X to the two-valued set f0; 1g. Let P be an arbitrary "prior" probability distribution on these functions. Let D be any probability distribution on pairs hx; yi with x 2 X and y 2 f0; 1g. We do not assume any relation between P and D. Define ffl(f i ) to be the error rate of f i , i.e., the probability over selecting hx; yi according to D that f i (x) 6= y. Let S be a sample of m pairs drawn independently according to D and define to be the fraction of pairs hx; yi in S for which f i (x) 6= y. Here "ffl(f i ) is a measure of how well f i fits the training data and log 1 can be viewed as the "description length" of the concept f i . It is noted in [15] that a simple combination of Chernoff and union bounds yields that with probability at least the choice of the sample S we have the following for all f i . s 2m (1) This inequality justifies a concept selection algorithm which selects f to be the f i minimizing the description-length vs. goodness-of-fit tradeoff in the right hand side. If there happens to be a low-description-length concept that fits well, the algorithm will perform well. If, however, all simple concepts fit poorly, the performance guarantee is poor. So in practice the probabilities should be arranged so that concepts which are a-priori viewed as likely to fit well are given high probability. Domain specific knowledge can be used in selecting the distribution P . This is precisely the sense in which P is analogous to a Bayesian prior - a concept f i that is likely to fit well should be given high "prior probability" P (f i ). Note, however, that the inequality (1) holds independent of any assumption about the relation between the distributions P and D. Formula (1) is for model selection - algorithms that select a single model or concept. However, model selection is inferior to model averaging in certain applications. For example, in statistical language modeling for speech recognition one "smoothes" a trigram model with a bigram model and smoothes the bigram model with a unigram model. This smoothing is essential for minimizing the cross entropy between, say, the model and a test corpus of newspaper sentences. It turns out that smoothing in statistical language modeling is more naturally formulated as model averaging than as model selection. A smoothed language model is very large - it contains a full trigram model, a full bigram model and a full unigram model as parts. If one uses MDL to select the structure of a language model, selecting model parameters with maximum likelihood, the resulting structure is much smaller than that of a smoothed trigram model. Furthermore, the MDL model performs quite badly. A smoothed trigram model can be theoretically derived as a compact representation of a Bayesian mixture of an exponential number of (smaller) suffix tree models [18]. Model averaging can also be applied to decision trees that produce probabilities at their leaves rather than hard classifications. A common method of constructing decision trees is to first build an overly large tree which over fits the training data and then prune the tree in some way so as to get a smaller tree that does not over fit the data [19, 10]. For trees with probabilities at leaves, an alternative is to construct a weighted mixture of the subtrees of the original over fit tree. It is possible to construct a concise representation of a weighting over exponentially many different subtrees [3, 17, 9]. This paper is about stochastic model selection - algorithms that stochastically select a model according to a "posterior distribution" on the models. Stochastic model selection seems intermediate between model selection and model averaging - like model averaging it is based on a posterior distribution over models but it uses that distribution differently. Model averaging deterministically picks the value favored by a majority of models as weighted by the posterior. Stochastic model selection stochastically picks a single model according to the posterior distribution. The first main result of this paper is a bound on the performance of stochastic model selection that improves on (1) - stochastic model selection can be given better guarantees than deterministic model selection. Intuitively, model averaging should perform even better than stochastic model selection. But proving a PAC guarantee for model averaging superior to the PAC guarantees given here for stochastic model selection remains an open problem. This paper also investigates the nature of the posterior distribution providing the best performance guarantee for stochastic model selection. It is shown that the optimal posterior is a Gibbs distribution. However, it is also shown that simpler posterior distributions are nearly optimal. Section gives statements of the main results of this paper. Section 3 relates these results to previous work. The remaining sections present proofs. 2 Summary of the Main Results Formula (1) applies to a countable class of concepts. It turns out that the guarantees on stochastic model selection hold for continuous classes as well, e.g., concepts with real-valued parameters. Here we assume a prior probability measure P on a possibly uncountable (continuous) concept class C and a sampling distribution D on a possibly uncountable set of instances X . We also assume a measurable loss function l such that for any concept c and instance x we have l(c; x) 2 [0; 1]. For example, we might have that concepts are predicates on instances and there is a target concept c t such that l(c; x) is We define l(c) to be the expectation over sampling an instance x of l(c; x), i.e., E xD l(c; x). We let S range over samples of m instances each drawn independently according to distribution D. We define " l(c; S) to be 1 x2S l(c; x). If Q is a probability measure on concepts then l(Q) denotes E cQ l(c) and " l(Q; S) denotes The notation 8 signifies that the probability over the generation of the sample S of \Phi(S) is at least 1 \Gamma ffi . For countable concept classes formula (1) generalizes as follows to any loss function l with Lemma 1 (McAllester98) For any probability distribution P on a countable rule class C we have the following. 8 s As discussed in the introduction, this leads to a learning algorithm that selects the concept c minimizing the SRM tradeoff in the right hand side of the inequality. The first main result of this paper is a generalization of (1) to a uniform statement over distributions on an arbitrary concept class. The new bound involves the Kullback-Leibler divergence, denoted D(QjjP ), from distribution Q to distribution P . The quantity D(QjjP ) is defined to be dP (c) . The following is the first main result of this paper and is proved in section 4. Theorem 1 For any probability distribution (measure) on a possibly uncountable set C and any measurable loss function l we have the following where Q ranges over all distributions (measures) on C. 8 s Note that the definition of l(Q), namely E cQ l(c), is the average loss of a stochastic model selection algorithm that makes a prediction by first selecting c according to distribution Q. So we can interpret theorem 1 as a bound on the loss of a stochastic model selection algorithm using posterior Q. In the case of a countable concept class where Q is concentrated on the single concept c the quantity D(QjjP ) equals (c) and, for large m, theorem 1 is essentially the same as lemma 1. But theorem 1 is considerably stronger than lemma 1 in that it handles the case of uncountable (continuous) concept classes. Even for countable classes theorem 1 can lead to a better guarantee than lemma 1 if the posterior Q is spread over exponentially many different models having similar empirical error rates. This might occur, for example, in mixtures of decision trees as constructed in [3, 17, 9]. The second main result of this paper is that the posterior distribution minimizing the error rate bound given in theorem 1 is a Gibbs distribution. For any value of fi 0 we define Q fi to be the posterior distribution defined as follows where Z is a normalizing constant. Z For any posterior distribution Q define B(Q) as follows. s The second main result of the paper is the following. Theorem 2 If C is finite then there exists fi 0 such that Q fi is optimal, i.e., B(Q fi ) B(Q) for all Q, and where fi satisfies the following. Unfortunately, there can be multiple local minima in B(Q fi ) as a function of fi and even multiple local minima satisfying (2). Fortunately, simpler posterior distributions achieve nearly optimal performance. To simplify the discussion we consider parameterized concept classes where each concept is specified by a parameter vector \Theta 2 R n . Let l(\Theta; x) be the loss of the concept named by parameter vector \Theta on the data point x (as discussed above). To further simplify the analysis we assume that for any given x we have that l(\Theta; x) is a continuous function of \Theta. For example, we might take \Theta to be the coefficients of an nth order polynomial p \Theta and take l(\Theta; x) to be max(1; ffjp \Theta (x) \Gamma f(x)j) where f(x) is a fixed target function and ff is a fixed parameter of the loss function. Note that a two valued loss function can not be a continuous function of \Theta unless the prediction is independent of \Theta. Now consider a sample S consisting of m data points. These data points define an empirical loss " l(\Theta) for each parameter vector \Theta. This empirical loss is an average of a finite number of expressions of the form l(\Theta; x) and hence " l(\Theta) must be a continuous function of \Theta. Assuming that the prior on \Theta is given by a continuous density we then get that there exists a continuous density p( " l) on empirical errors satisfying the following where P (U) denotes the measure of a subset U of the concepts according to the prior measure on concepts. x The second main result of the paper can be summarized as the following approximate equation where B(Q ) denotes inf Q B(Q). This approximate inequality is justified by the two theorems stated below. Before stating the formal theorems, however, it is interesting to compare (3) with lemma 1. For a countable concept class we can define c to be the concept minimizing the bound in lemma 1. For large m, lemma 1 can be interpreted as follows. s Clearly there is a structural similarity between (4) and (3). However, the two formulas are fundamentally different in that (3) applies to continuous concept densities while (4) only applies to countable concept classes. Another contribution of this paper is theorems giving upper and lower bounds on B(Q ) justifying (3). First we give a simple posterior distribution which nearly achieves the performance of (3). Define " l as follows. Define the posterior distribution Q( " l ) as follows where Z is a normalizing constant. We now have the following theorem. Theorem 3 For any prior (probability measure) on a concept class where each concept is named by a vector \Theta 2 R n and any sample of m instances, if the loss function l(\Theta; x) is always in the interval [0; 1] and is continuous in \Theta, the prior on \Theta is a continuous probability density on R n and the density p( " l) is non-decreasing over the interval we have the following. All of the assumptions used in theorem 3 are quite mild. The final assumption that the density p( " l) is nondecreasing over the interval defining Q( " l ) is justified by fact that the definition of " l implies that for any differentiable density function p( " l) we must have that the density p( " l) is increasing at the Finally we show that Q( " l ) is a nearly optimal posterior. Theorem 4 For any prior (probability measure) on a concept class where each concept is named by a vector \Theta 2 R n and any sample of m instances, if the loss function l(\Theta; x) is always in the interval [0; 1] and is continuous in \Theta, and the prior on \Theta is a continuous probability density on R n , then we have the following for any posterior Q. 3 Related Work A model selection guarantee very similar to (1) has been given by Barron [1]. Assume concepts f 1 , f 2 , f 3 true and empirical error rates ffl(f i ) and "ffl(f i ) as in (1). Let f be defined as follows. s For the case of error rates (also known as 0-1 loss) Barron's theorem reduces to the following. s There are several differences between (1) and (5). When discussing (1) I will take f to be the concept f i minimizing the right hand side of (1) which is nearly the same as the definition of f in (5). Formula (1) implies the following. 8 s Note that (5) bounds the expectation of ffl(f ) while (1) is a large deviation result - it gives a bound on ffl(f ) as a function of the desired confidence level ffi. Also note that (1) provides a bound on ffl(f ) in terms of information available in the sample while (5) provides a bound on (the expectation of) ffl(f ) in terms of the unknown quantities ffl(f i ). This means that a learning algorithm based on (1) can output a performance guarantee along with the selected concept. This is true even if the concept is selected by incomplete search over the concept space and hence is different from f . No such guarantee can be computed from (5). If a bound in terms of the unknown quantities ffl(f i ) is desired, the proof method used to prove (1) yields the following. 8 Also note that (5), like (1) but unlike theorem 1, is vacuous for continuous concept classes. Various other model selection results similar to (1) have appeared in the literature. A guarantee involving the index of a concept in an arbitrary given sequence of concepts is given in [12]. A bound based on the index of a concept class in a sequence of classes of increasing VC dimension is given in [14]. Neither of these bounds handle an arbitrary prior distribution on concepts. They do, however, give PAC SRM performance guarantees involving some form of prior knowledge (learning bias). Guarantees for model selection algorithms for density estimation have been given by Yamanishi [21] and Barron and Cover [2]. The guarantees bound measures of distance between a selected model distribution and the true data source distribution. In both cases the model is assumed to have been selected so as to optimize an SRM tradeoff between model complexity and the goodness of fit to the training data. The bounds hold without any assumption relating the prior distribution to the data distribution, However, the performance guarantee is better if there exist simple models that fit well. The precise statement of these bounds are somewhat involved and perhaps less interesting than the more elegant guarantee given in formula (6) discussed below. Guarantees for model averaging have also been proved. First I will consider model averaging for density estimation. Let f 1 , f 2 , f 3 be an infinite sequence of models each of which defines a probability distribution on a set X . Let P be a "prior probability" on the densities f i . Assume an unknown distribution g on X which need not be equal to any f i . Let S be a sample of m elements of X sampled IID according to the distribution g. Let h be the natural "posterior" density on X defined as follows where Z is a normalizing constant. Note that the posterior density h is a function of the sample and hence is a random variable. Catoni [5] and Yang [23] prove somewhat different general theorems both of which have as a special case the statement that, independent of how g is selected, the expectation (over drawing a sample according to g) of the Kullback-Leibler Divergent D(gjjh) is bounded as follows. Again we have that (6) holds without any assumed relation between g and the prior P . If there happens to be a low complexity (simple) model f i such that D(gjjf i ) is small, then the posterior density h will have small divergence from g. If no simple model has small divergence from g then D(gjjh) can be large. Also not that (6), unlike theorem 1, is vacuous for continuous model classes. These observations also apply to the more general forms of appearing in [23] and [5]. Catoni [4] also gives performance guarantees for model averaging for density estimation over continuous model spaces using a Gibbs posterior. However, the statements of these guarantees are quite involved and the relationship to the bounds in this paper is unclear. Yang [22] considers model averaging for prediction. Consider a fixed distribution D on pairs hx; yi with x 2 X and y 2 f0; 1g. Consider a countable class of conditional probability rules f 1 , f 2 , f 3 each f i is a function from X to [0; 1] where f i (x) is interpreted as P (yjx; f i ). Consider an arbitrary prior on the models f i and construct the posterior given a sample S as Q(f i This posterior on the models induces a posterior h on y given x defined as follows. Let g(x) be the true conditional probability P (yjx) as defined by the distribution D. For any function g 0 from X to [0; 1] define the loss L(g 0 ) as follows where x D denotes selecting x from the marginal of D on X . Finally, define ffi i as follows. For m 2, the following is a corollary of Yang's theorem. iA This formula bounds the loss of the Bayesian model average without making any assumption about the relationship between the data distributions D and the prior distribution P . However, it seems weaker than (5) or (6) in that it does not imply even for a finite model class that for large samples the loss of the posterior converges to the loss of the best model. As with (6), the guarantee is vacuous for continuous model classes. These same observations apply to the more general statement in [22]. Weighted model mixtures are also widely used in constructing algorithms with on-line guarantees. In particular, the weighted majority algorithm and its variants can be proved to compete well with the best expert on an arbitrary sequence of labeled data [13, 6, 8, 7]. The posterior weighting used in most on-line algorithms is a Gibbs posterior Q fi as defined in the statement of theorem 2. One difference between these on-line guarantees and theorem 1 is that for these algorithms one must know the appropriate value of fi before seeing the training data. Since a-prior knowledge of fi is required, the on-line algorithm is not guaranteed to perform well against the optimal performing well against the optimal SRM tradeoff requires tuning fi in response to the training data. Another difference between on-line guarantees and either formula (1) or theorem 1 is that (1) (or theorem 1) provides a guarantee even in cases where only incomplete searches over the concept space are feasible. On-line guarantees require that the algorithm find all concepts that perform well on the training data - finding a single simple concept that fits well is insufficient. The most closely related earlier result is a theorem in [15] bounding the error rate of stochastic model selection in the case where the model is selected stochastically from a set U of models under a probability measure that is simply a renormalization of the prior on U . Theorem 1 is a generalization of this result to the case of arbitrary posterior distributions. 4 Proof of Theorem 1 The departure point for the proof of theorem 1 is the following where S is a sample of size m and \Delta(c) abbreviates Lemma 2 For any prior distribution (probability measure) P on a (possibly uncountable) concept space C we have the following. 8 4m Proof: It suffices to prove the following. 4m (7) Lemma 2 follows from (7) by an application of Markov's inequality. To prove it suffices to prove the following for any individual given concept. 4m (8) For a given concept c, the probability distribution on the sample induces a probability distribution on \Delta(c). By the Chernoff bound this distribution on \Delta satisfies the following. It now suffices to show that any distribution satisfying must satisfy (8). The distribution on \Delta satisfying (9) and maximizing Ee is the continuous density f (\Delta) satisfying which implies . So we have the following Z 1e theorem 1 we consider selecting a sample S. Lemma 2 implies that with probability at least 1 \Gamma ffi over the selection of a sample S we have the following. 4m To prove theorem 1 it now suffices to show that the constraint (10) on the function \Delta(c) implies the body of theorem 1. We are interested in computing an upper bound on the quantity S). Note that \Delta(c). We now prove the following lemma. Lemma 3 For fi ? 0, K ? 0, and Q; we have that if then s Before proving lemma 3 we note that lemmas 3 and 2 together imply theorem 1. To see this consider a sample satisfying (10) and an arbitrary posterior probability measure Q on concepts. It is possible to define three infinite sequences of vectors the conditions of lemma 3 with satisfying the following. By taking the limit of the conclusion of lemma 3 we then get E cQ \Delta(c) To prove lemma 3 it suffices to consider only those values of i for which dropping the indices where does not change the value of enlarging the feasible set by weakening the constraint (10). Furthermore, if at some point where the theorem is immediate. So we can assume without loss of generality that By Jensen's inequality we have ( . So it now suffices to prove that This is a consequence of the following lemma. 1 1 The original version of this paper [16] proved a bound of approximately the form maximizing subject to constraint 10. A Lemma 4 For fi ? 0, K ? 0, and Q; and then n To prove lemma 4 we take P and Q as given and use the Kuhn-Tucker conditions to find a vector y maximizing subject to the constraint (11). are functions from R n to R, y is a maximum of C(y) over the set satisfying f 1 (y) and C and each f i are continuous and differentiable at y, then either (at y), or there exists some f i with f i (at y), or there exists a nonempty subset of the constraints f i 1 that positive coefficients 1 such that Note that lemma 4 allows y i to be negative. The first step in proving lemma 4 is to show that without loss of generality we can work with a closed and compact feasible set. For K ? 0 it is not difficult to show that there exists a feasible point, i.e., a vector y such that Let C 0 denote an arbitrary feasible value, i.e., point y. Without loss of generality we need only consider points y satisfying 1. So we now have a constrained optimization problem with objective function set defined by the following constraints. version of theorem 1, which is of the form " l(Q; S)+ proved from this bound by an application of Jensen's inequality. The idea of maximizing i and achieving theorem 1 directly is due to Robert Schapire. Constraint (12) implies an upper bound on each y i and constraint (13) then implies a lower bound on each y i . Hence the feasible set is closed and compact. We now note that any continuous objective function on a closed and compact feasible set must be bounded and must achieve its maximum value on some point in the set. A constraint of the form f(y) 0 will be called active at y if For an objective function whose gradient is nonzero everywhere, at least one constraint must be active at the maximum. Since C 0 is a feasible value of the objective function, constraint (13) can not be active at the maximum. So by the Kuhn-Tucker lemma, the point y achieving the maximum value must satisfy the following. Which implies the following. Since constraint (12) must be active at the maximum, we have the following. So we get and the following. Since this is the maximum value of the lemma is proved. 5 Proof of Theorem 2 We wish to find a distribution Q minimizing B(Q) defined as follows where the distribution P and the empirical error " l(c) are given and fixed. s Letting K be ln(1=ffi)+ln m+2 and letting fl be objective function can be rewritten as follows where K and fl are fixed positive quantities independent of Q. s To simplify the analysis we consider only finite concept classes. Let P i be the prior probability of the ith concept and let " l i be the empirical error rate of the ith concept. The problem now becomes finding values of Q i satisfying minimizing the following. s If P i is zero then if Q i is nonzero we have that D(QjjP ) is infinite. So for minimizing B(Q) we can assume that Q i is zero if P i is zero and we can assume without loss of generality that all P i are nonzero. If all P i are nonzero then the objective function is a continuous function of a compact feasible set and hence realizes its minimum at some point in the feasible set. Now consider the following partial derivative. @ Note that if Q i is zero when P i is nonzero then @D(QjjP )=@Q This means that any transfer of an infinitesimal quantity of probability mass to Q i reduces the bound. So the minimum must not occur at a boundary point satisfying we can assume without loss of generality that is nonzero for each i where P i is nonzero - the two distributions have the same support. The Kuhn-Tucker conditions then imply that rB is in the direction of the gradient of one of the constraints In all of these cases there must exist a single value such that for all i we have @B=@Q This yields the following. Hence the minimal distribution has the following form. r This is the distribution Q fi of theorem 2. 6 Proof of Theorems 3 and 4 be the posterior distribution of theorem 3. First we note the following. dP (c) We have assumed that p( " l) is nondecreasing over the interval 1=m]. This implies the following. We also have that " theorem 3 now follows from the definition of B(Q). We now prove theorem 4. First we define a concept distribution U such that U induces a uniform distribution on those error rates " l with Let W be the subset of the values " l 2 [0; 1] such that p( " l) ? 0. Let ff denote the size of W as measured by the uniform measure on [0; 1]. Note that ff 1. Define the concept distribution U as follows. The total measure of U can be written as follows. Z dU dP dP Z Hence U is a probability measure on concepts. Now let Q be an arbitrary posterior distribution on concepts. We have the following. dP dP dU This implies the following where the third line follows from Jensen's inequality s s s min s 7 Conclusion PAC-Bayesian learning algorithms combine the flexibility prior distribution on models with the performance guarantees of PAC algorithms. PAC- Bayesian Stochastic model selection can be given performance guarantees superior to analogous guarantees for deterministic PAC-Bayesian model se- lection. The performance guarantees for stochastic model selection naturally handle continuous concept classes and lead to a natural notion of an optimal posterior distribution to use in stochastically selecting a model. Although the optimal posterior is a Gibbs distribution, it is shown that under mild assumptions simpler posterior distributions perform nearly as well. An open question is whether better guarantees can be given for model averaging rather than stochastic model selection. Acknowledgments I would like to give special thanks to Manfred Warmuth for inspiring this paper and emphasizing the analogy between the PAC and on-line settings. I would also like to give special thanks to Robert Schapire for simplifying and strengthening theorem 1. Avrim Blum, Yoav Freund, Michael Kearns, John Langford, Yishay Mansour, and Yoram Singer also provided useful comments and suggestions. --R Complexity regularization with application to artificial neural networks. Minimum complexity density estimation. Learning classification trees. Gibbs estimators. Universal aggregation rules with sharp oracle inequali- ties Warmuth How to use expert advice. Adaptive game playing using multiplicative weights. Predicting nearly as well as the best pruning of a decision tree. An experimental and theoretical comparison of model selection methods. Results on learnability and the Vapnik-Chervonenkis dimension The weighted majority algo- rithm Concept learning using complexity regulariza- tion Some pac-bayesian theorems On pruning and averaging decision trees. An efficient extension to mixture techniques for prediction and decision trees. A pac analysis of a bayesian estimator. Learning non-parametric densities in tyerms of finite-dimensional parametric hypotheses Adaptive estimation in pattern recognition by combining different procedures. Mixing strategies for density estimation. --TR --CTR Franois Laviolette , Mario Marchand, PAC-Bayes risk bounds for sample-compressed Gibbs classifiers, Proceedings of the 22nd international conference on Machine learning, p.481-488, August 07-11, 2005, Bonn, Germany Matti Kriinen , John Langford, A comparison of tight generalization error bounds, Proceedings of the 22nd international conference on Machine learning, p.409-416, August 07-11, 2005, Bonn, Germany Avrim Blum , John Lafferty , Mugizi Robert Rwebangira , Rajashekar Reddy, Semi-supervised learning using randomized mincuts, Proceedings of the twenty-first international conference on Machine learning, p.13, July 04-08, 2004, Banff, Alberta, Canada Arindam Banerjee, On Bayesian bounds, Proceedings of the 23rd international conference on Machine learning, p.81-88, June 25-29, 2006, Pittsburgh, Pennsylvania Ron Meir , Tong Zhang, Generalization error bounds for Bayesian mixture algorithms, The Journal of Machine Learning Research, 4, 12/1/2003 Matthias Seeger, Pac-bayesian generalisation error bounds for gaussian process classification, The Journal of Machine Learning Research, 3, p.233-269, 3/1/2003

gibbs distribution;model averaging;posterior distribution;PAC-Baysian learning;PAC learning

608351

Relative Loss Bounds for Temporal-Difference Learning.

Foster and Vovk proved relative loss bounds for linear regression where the total loss of the on-line algorithm minus the total loss of the best linear predictor (chosen in hindsight) grows logarithmically with the number of trials. We give similar bounds for temporal-difference learning. Learning takes place in a sequence of trials where the learner tries to predict discounted sums of future reinforcement signals. The quality of the predictions is measured with the square loss and we bound the total loss of the on-line algorithm minus the total loss of the best linear predictor for the whole sequence of trials. Again the difference of the losses is logarithmic in the number of trials. The bounds hold for an arbitrary (worst-case) sequence of examples. We also give a bound on the expected difference for the case when the instances are chosen from an unknown distribution. For linear regression a corresponding lower bound shows that this expected bound cannot be improved substantially.

Introduction Consider the following model of temporal-difference learning: Learning proceeds in a sequence of trials :, where at trial t, \Gamma the learner receives an instance vector x t 2 R n , \Gamma the learner makes a prediction " \Gamma the learner receives a reinforcement signal r t 2 R. A pair called an example. The Learner tries to predict the outcomes y t 2 R. For a fixed discount rate parameter fl 2 [0; 1), y t is the discounted sum s=t of the future reinforcement signals 1 . For example, if r t is the profit of a company in month t, then y t can be interpreted as an approximation of the company's worth at time t. The discounted sum takes into account that profits in the distant future are less important than short term profits. Note that the outcome y t as defined in (1.1) is well-defined if the reinforcement signals are bounded. In an episodic setting one can also define the outcomes y t as finite discounted sums. This is discussed briefly in Section 7. A strategy that chooses predictions for the learner is called an on-line learning algorithm. The quality of a prediction is measured with the square loss: The loss of the learner at trial t is (y and the loss of the learner at trials 1 through T is We want to compare the loss of the learner against the losses of linear functions. A linear function is represented by a weight vector and the loss of w at trial t is (y Ideally we want to bound the additional loss of the learner over the loss of the best linear predictor for arbitrary sequences of examples, i.e. we want to bound for arbitrary T and arbitrary sequences of examples The first sum in (1.2) is the total loss of the learner at trials 1 through T . The argument of the infimum is the total loss of the linear function Alternatively y t could be defined as (1 \Gamma fl) s=t rs , which makes y t a convex combination of the reinforcement signals r t , r t+1 , . The alternate definition amounts to a simple rescaling of the outcomes and predictions. Relative Loss Bounds for Temporal-Difference Learning 3 w at trials 1 through T . Thus (1.2) is the additional total loss of the learner over the total loss of the best linear function. Following Vovk (1997) we also examine the more general problem of bounding for a fixed constant a 0. Here akwk 2 is a measure of the complexity of w, i.e. the infimum in (1.3) includes a charge for the complexity of the linear function. For larger values of a it is obviously easier to show bounds on (1.3). Bounds on (1.2) or (1.3) that hold for arbitrary sequences of examples are called relative loss bounds. Relative loss bounds for the temporal-difference learning setting were first shown by Schapire and Warmuth (1996). An overview of their results is given in Section 4. They also show how algorithms that minimize (1.2) can be used for value function approximation of Markov processes. This is an important problem in reinforcement learning which is also called policy evaluation. The Markov processes can have continuously many states which are represented by real vectors. If one wants to predict state values then our instances x t correspond to states of the environment, if action values are predicted then an instance corresponds to both a state of the environment and an action of the agent. For an introduction to reinforcement learning see Sutton and Barto (1998). The paper is organized as follows. We discuss previously known relative loss bounds for linear regression and for temporal-difference learning in Sections 3 and 4. In Section 5, we propose a new second order learning algorithm for temporal-difference learning (the TLS algorithm), and we prove relative loss bounds for this algorithm in Section 6. In Section 7, we adapt the TLS algorithm to the episodic case, where the trials are divided into episodes and where an outcome is a discounted sum of the future reinforcement signals from the same We discuss previous second order algorithms for temporal-difference learning in Section 8, and give lower bounds on the relative loss in Section 9. 2. Notation and preliminaries For is the set of n-dimensional real vectors. For m;n 2 N, R m\Thetan is the set of real matrices with m rows and n columns. In this paper vectors x 2 R n are column vectors and x 0 denotes the transpose 4 J. FORSTER AND M. K. WARMUTH of x. The scalar product of two vectors w;x 2 R n is w the Euclidean norm of a vector x 2 R n is We recall some basic facts about positive (semi-)definite matrices: n\Thetan is called positive definite if x holds for all vectors x 2 R n n f0g. n\Thetan is called positive semi-definite if \Gamma The sum of two positive semi-definite matrices is again positive semi-definite. The sum of a semi-definite matrix and a positive definite matrix is positive definite. Every positive definite matrix is invertible. \Gamma For matrices A; B 2 R n\Thetan we write A B if B \Gamma A is positive semi-definite. In this case x 0 Ax x 0 Bx for all vectors x 2 R n . \Gamma The Sherman-Morrison formula (see Press, Flannery, Teukolsky, holds for every positive definite matrix A 2 R n\Thetan and every vector For example, the unit matrix I 2 R n\Thetan is positive definite and for every vector x 2 R n the matrix xx 0 2 R n\Thetan is positive semi-definite. To find a vector w 2 R n that minimizes the term that appears in the relative loss (1.3) we define k=s Because (2.2) is convex in w, it is minimal if and only if its gradient is zero, i.e. if and only if A t . If a ? 0, then A t is invertible and is the unique vector that minimizes (2.2). If a Relative Loss Bounds for Temporal-Difference Learning 5 A t might not be invertible and the equation A t might not have a unique solution. The solution with the smallest Euclidean norm is t is the pseudoinverse of A t . For the definition of the pseudoinverse of a matrix see, e.g., Rektorys (1994). There it is also shown how the pseudoinverse of a matrix can be computed with the singular value decomposition. We give a number of properties of the pseudoinverse A t in the appendix. If a ? 0, then applying the Sherman-Morrison formula (2.1) to A t shows that A A A 3. Known relative loss bounds for linear regression First note that linear regression is a special case of our setup since when t. The standard algorithm for linear regression is the ridge regression algorithm which predicts x t at trial t. Relative loss bounds for this algorithm (similar to the bounds given in the below two theorems) have been proven in Foster (1991), Vovk (1997) and Azoury and Warmuth (1999). The bounds obtained for ridge regression are weaker than the ones proven for a new algorithm developed by Vovk. We will give a simple motivation of this algorithm and then discuss the relative loss bounds that were proven for it. We have seen in Section 2 that the best linear functions for trials 1 through t (i.e. the linear function that minimizes (2.2)) would make the prediction b 0 x t at trial t. Note that via b t and A t this prediction depends on the examples However, only the examples and the instance x t are known to the learner when it makes the prediction for trial t. If we set the unknown outcome y t to zero we get the prediction b 0 This prediction was introduced in Vovk (1997) using a different motivation. The above motivation follows Azoury and Warmuth (1999). Forster (1999) gives an alternate game theoretic motivation. Vovk proved the following bound on (1.3) for his prediction algorithm. THEOREM 3.1. Consider linear regression any sequence of examples in R n with the predictions " y 6 J. FORSTER AND M. K. WARMUTH a A T Y a an where x t;i is the i-th component of the vector x t and where In Vovk's version of Theorem 3.1 the term X 2 n is replaced by the larger is a bound on the supremum norms of the instances . The last inequality in Theorem 3.1 follows from Y a an -z an where the first inequality holds because the geometric mean is always smaller than the arithmetic mean. Azoury and Warmuth (1999) and Forster (1999) give the following refined bound for Vovk's linear regression algorithm. There only the case a ? 0 is considered. We will show that the case a = 0 of Theorem 3.2 (i) follows from the case a ? 0 by letting a go to zero. THEOREM 3.2. Consider linear regression any sequence of examples in R n \Theta R. (i) If a 0, then with the predictions " (ii) If a ? 0, then a A T for all vectors x Relative Loss Bounds for Temporal-Difference Learning 7 Proof. We only have to show that the equality in (i) also holds for the case a = 0. We do this by showing that both sides of the equality are continuous in a 2 [0; 1). Because of Lemma A.2 we only have to check that for t 2 Tg the term is continuous in a 2 [0; 1). If x t 2 X t\Gamma1 , this again follows from Lemma A.2. Otherwise x . Then by Lemma A.3, the expression (3.1) is zero for a = 0. We have to show that (3.1) converges to zero for a & 0. For a ? 0, we can rewrite (3.1) by applying (2.5): (b 0 The factor (b 0 converges because of Lemma A.2. Because of Lemma A.4, the term x 0 goes to zero as a & 0. Together this shows that (3.1) indeed goes to zero as a & The learning algorithm of Theorem 3.1 and Theorem 3.2 is a second order algorithm in that it uses second derivatives. There is a simpler first order algorithm called the Widrow-Hoff or Least Mean Square algorithm (Widrow & Stearns, 1985). This algorithm maintains a weight vector w t 2 R n and predicts with " . The weight vector is updated by gradient descent. That is, w for some learning rates A method for setting the learning rates for the purpose of obtaining good relative loss bounds is given in Cesa-Bianchi, Long and Warmuth (1996) and in Kivinen and Warmuth (1997). In this method the learner needs to know an upper bound X on the Euclidean norms of the instances and needs to know parameters W , K such that there is a vector w 2 R n with norm kwk W and loss For any such vector w the bound holds. As noted by Vovk (1997) this bound is incomparable to the bounds of Theorem 3.1 (See also the next section). The bound (3.3) also holds for the ridge regression algorithm (Hassibi, Kivinen, & War- muth, 1995). There the parameter a is set depending on W and K. We believe that with a proper tuning of a such bounds also hold for Vovk's 8 J. FORSTER AND M. K. WARMUTH 4. Known relative loss bounds for temporal-difference learning For the case that the discount rate parameter fl is not assumed to be zero, Schapire and Warmuth (1996) have given a number of different relative loss bounds for the learning algorithm TD The first order algorithm TD () is essentially a generalization of the Widrow-Hoff algorithm. It is a slight modification of the learning algorithm TD() proposed by Sutton (1988). Schapire and Warmuth show that the loss of TD (1) with a specific setting of its learning rate is (where c and that the loss of the algorithm TD (0) with a specific setting of its learning rate is for every vector w 2 R n with kwk W and The setting of the learning rate depends on an upper bound X on the Euclidean norms of the instances and on W and K. The learner needs to know these parameters in advance. The loss of the best linear function will often grow linearly in T , e.g. if the examples are corrupted by Gaussian noise. In this case the relative loss bounds in (3.3), (4.1) and (4.2) will grow like T . The second order learning algorithm we propose for temporal-difference learning has the advantage that the relative loss bounds we can prove for it grow only logarithmically in T . Also, our algorithm does not need to know parameters like K and W . However, it needs to know an upper bound Y on the absolute values of the outcomes y t . TD () can be sensitive to the choice of . Another advantage of our algorithm is that we do not have to choose a parameter like the of the TD () algorithm. 5. A new second order algorithm for temporal-difference learning In this section we propose a new algorithm for the temporal-difference learning setting. We call this algorithm the temporal least squares algorithm, or shorter the TLS algorithm. Relative Loss Bounds for Temporal-Difference Learning 9 We assume that the absolute values of the outcomes y t are bounded by some constant Y , i.e. and assume that the bound Y , the discount rate parameter fl, and the parameter a are known to the learner. Knowing Y the learner can "clip" a real number y 2 R using the function 5.1. Motivation of the TLS algorithm The new second-order algorithm for temporal-difference learning is given in Table I. We call this algorithm the Temporal Least Squares (TLS) algorithm. The motivation for the TLS algorithm is the same as the motivation from Azoury and Warmuth (1999) that we gave for Vovk's prediction for linear regression in Section 3. We will use the equality k=s that holds for all s t. The best linear function for trials 1 through t that minimizes (2.2) would make the prediction k=s at trial t. We set the unknown outcome y t to zero and get the prediction e k=s The TLS algorithm predicts with " clipping function assures that the prediction lies in the bounded range In the following we will show that the relative loss (1.3) of TLS is at most J. FORSTER AND M. K. WARMUTH Table I. The temporal least squares (TLS) algorithm. At trial t, the learner knows ffl the parameters Y , fl, a, ffl the instances ffl the reinforcement signals TLS predicts with where k=s rk and CY given by (5.2) clips the prediction to the interval [\GammaY; Y ]. If A t is not invertible, then the inverse A \Gamma1 of A t must be replaced by the pseudoinverse A and we will use this result to get worst and average case relative loss bounds that are easier to interpret. 5.2. Implementation of the TLS algorithm For the case a ? 0 a straightforward implementation of the TLS algorithm would need O(n 3 ) arithmetic operations at each trial to compute the inverse of the matrix A t . In Table II we give an implementation that only needs O(n 2 ) arithmetic operations per trial. This is achieved by computing the inverse of A t iteratively using the Sherman-Morrison formula (2.5). This implementation makes the correct predictions because at the end of each FOR-loop This follows from the Sherman-Morrison formula (2.5) and from the equality Relative Loss Bounds for Temporal-Difference Learning 11 Table II. Implementation of TLS for a ? 0. A inv := 1 a I 2 R n\Thetan z Receive instance vector x t 2 R n A inv := A inv \Gamma t A inv x t Predict with " y Receive reinforcement signal r t 2 R z 6. Relative loss bounds for the TLS algorithm In the temporal-difference learning setting we do not know the outcomes when we need to predict at trial t. So we cannot run Vovk's linear regression algorithm which uses these outcomes in its pre- diction. The TLS algorithm approximates Vovk's prediction by setting the future reinforcement signals to zero (It also clips the prediction into the range [\GammaY; Y ]). We will show that the loss of the TLS algorithm is not much worse than the loss of Vovk's algorithm for which good relative loss bounds are known. We start by showing two lemmas. The first is a technical lemma. We use it for proving the second lemma in which we bound the absolute values of the differences between Vovk's prediction b 0 and the unclipped prediction e of the TLS algorithm. LEMMA 6.1. If a 0, then s t and any vectors x Proof. From it follows that If a ? 0, then the pseudoinverses are inverses and the Sherman-Morrison formula (2.5) shows that A J. FORSTER AND M. K. WARMUTH Thus A \Gamma1 s and x 0 This proves the lemma for a ? 0. For the case a = 0 we use Lemma A.2 and let a go to 0. 2 LEMMA 6.2. Proof. Note that Thus Lemma 6:1 YT -z -z now show our main result. THEOREM 6.1. Consider temporal-difference learning with and a 0. Let any sequence of examples in R n \Theta R such that the outcomes y lie in the real interval [\GammaY; Y ]. Then with the predictions " the TLS algorithm, Relative Loss Bounds for Temporal-Difference Learning 13 Proof. Let . Because of y we know that (y i.e. the relative loss bound of Theorem 3.2 (i) also holds for the clipped predictions C Y (p t ). Thus it suffices to show that This holds because 4Y Lemma 6:2 :In the next two subsections we apply Theorem 6.1 to show relative loss bounds for the worst case and for the average case. 6.1. Worst case relative loss bound COROLLARY 6.1. Consider temporal-difference learning with [0; 1) and a ? 0. Let any sequence of examples in R n \Theta R such that the outcomes y lie in the real interval [\GammaY; Y ]. Then with the predictions " the TLS algorithm, a A T an 14 J. FORSTER AND M. K. WARMUTH Proof. The first inequality follows from Theorem 6.1 and Theorem 3.2 (ii). The second follows from Theorem 3.1. 2 6.2. Average case relative loss bound If we assume that the outcomes y lie in [\GammaY; Y ] and that the instances are i.i.d. with some unknown distribution on R n , we can show an upper bound on the expectation of the relative loss (1.2) for trials 1 through T that only depends on n; Y; fl; T . In particular we do not need the term akwk 2 that measures the complexity of the vector w in the relative loss (1.3), and we do not need to assume that the instances are bounded. To show this result we will use Theorem 6.1 and will then bound sums of terms x 0 with the following theorem (Tr(A) is the trace of a square matrix A and dim(X) is the dimensionality of a vector space X). THEOREM 6.2. For any t vectors x linear span Proof. We first look at the case a = choose any orthonormal basis e em of X t . Then The above can also be written as This means that if we interpret the matrix n\Thetan as a linear function from R n to R n , it is the identity function on X t . The assertion for the case a = 0 now follows from 1i;jm 1i;jm 1i;jm Relative Loss Bounds for Temporal-Difference Learning 15 1jm 1jm 1jm 1jm To prove the theorem for the case a ? 0 we choose an arbitrary orthonormal basis e apply the result for a = 0 to the vectors x 1\Gamman := First note that s=1\Gamman x s x 0 . From the case we have s=1\Gamman x 0 since the vectors fx 1\Gamman have rank n. The equality for a ? 0 now follows from s=1\Gamman x 0 a t ). The first inequality for the case a ? 0 follows from aTr(A \Gamma1 COROLLARY 6.2. Consider the temporal-difference learning setting with Assume that the instances x are i.i.d. with unknown distribution on R n and that the outcomes y given by (1.1) lie in the real interval [\GammaY; Y ]. Then with the predictions of the TLS algorithm, the expectation of (1.2) is Proof. Because of Theorem 6.1 the expected relative loss is at most Because are i.i.d. and because of Theorem 6.2: This proves the first inequality of Corollary 6.2. The second follows from J. FORSTER AND M. K. WARMUTH Table III. The temporal least squares (TLS) algorithm for episodic learning. At trial t, TLS predicts with where rk CY given by (5.2) clips the prediction to the interval [\GammaY; Y ] and start(k) is the first trial in the same episode to which trial k belongs. If A t is not invertible, then the inverse A \Gamma1 t of A t must be replaced by the pseudoinverse A t . 7. Episodic learning Until now we studied a setting where the outcomes y s=t are discounted sums of all future reinforcement signals. If we use our algorithm for policy evaluation in reinforcement learning, this corresponds to looking at continuing tasks (see Sutton and Barto (1998)). For episodic tasks the trials are partitioned into episodes of finite length. Now an outcome depends only on reinforcement signals that belong to the same episode. Let t be a trial. The first trial that is in the same episode as trial t is denoted by start(t) and the last by end(t). With this notation and with a discount rate parameter fl 2 [0; 1], the outcome y t in the episodic setting is defined as s=t This replaces the definition of y t given in (1.1) for the continuous setting. The definitions of the relative loss (1.2) and (1.3) remain un- changed. Note that the continuous setting is essentially the episodic setting with one episode of infinite length. With the same motivation as in Section 5 we get the TLS algorithm for episodic learning which is presented in Table III. An implementation of this algorithm is given in Table IV. We can check the correctness of this implementation by verifying that Relative Loss Bounds for Temporal-Difference Learning 17 Table IV. Implementation of TLS for episodic learning. A inv := 1 a I 2 R n\Thetan If a new episode starts at trial t, then set z := 0 2 R n Receive instance vector x t 2 R n A inv := A inv \Gamma t A inv x t Predict with " y Receive reinforcement signal r t 2 R z hold after every iteration of the FOR-loop. This follows from (2.5) and the equality A note for the practitioner: If a = 0 and if t is small, then the matrix A t might not be invertible and our algorithm uses the pseudoinverse of A t . In practice we suggest to use a ? 0 and tune this parameter. Then A t is always invertible and the calculation of pseudoinverses can be avoided. We also conjecture that the clipping is not needed for practical data. We now show relative loss bounds for episodic learning. Again we assume that a bound Y on the outcomes is known to the learner in advance. The proof of the following theorem is very similar to the proof of Theorem 6.1. THEOREM 7.1. Consider temporal-difference learning with episodes of length at most ' and let a 0. Let sequence of examples in R n \Theta R such that the outcomes y given by (7.1) lie in the real interval [\GammaY; Y ]. Then with the predictions of the TLS algorithm of Table III, J. FORSTER AND M. K. WARMUTH If a ? 0, then (7.2) is bounded by an If the instances x are i.i.d. with some unknown distribution on R n , then the expectation of (7.2) is at most A lower bound corresponding to (7.3) is shown in Theorem 9.2. Note that Theorem 7.1 does not exploit the fact that different episodes have varying length. Related theorems that depend on the actual lengths of the episodes can easily be developed. 8. Other second order algorithms Second order algorithms for temporal-difference learning have been proposed by Bradtke and Barto (1996) and Boyan (1999). We compare the algorithms in the episodic setting. Their algorithms maintain weight vectors w t and predict with " at trial t. Bradtke and Barto's Least-squares TD, or shorter LSTD, algorithm uses the weight vectors r s x s Boyan's LSTD() algorithm (he only considers the case a parameter 2 [0; 1] like the TD() algorithm. It uses the weight vectors z s where In contrast our TLS algorithm uses the weight vectors s Relative Loss Bounds for Temporal-Difference Learning 19 and the prediction of the TLS algorithm at trial t is " the above formulas the inverse must be replaced by the pseudoinverse if the matrix is not invertible.) Note that the TLS algorithm does not have a parameter like TD() or LSTD() and that for the case the algorithms LSTD and are identical. An important difference of the TLS algorithm to TD() and LSTD() is that it does not use differences in the definition of the "covariance" matrix. Bradtke, Barto and Boyan experimentally compare their algorithms to TD(). Under comparatively strong assumptions Bradtke and Barto also show that the w t of their algorithm converge asymptotically. The TLS algorithm we proposed in this paper was designed to minimize the relative loss (1.3), and our relative loss bounds show that TLS does this well. We do not know whether similar relative loss bounds hold for the LSTD and LSTD() algorithms. An experimental comparison would be useful. 9. Lower bounds In this section we give lower bounds for linear regression and for episodic temporal-difference learning. First consider the case of linear regression, i.e. In this case the outcomes y t are equal to the reinforcement signals. If the outcomes y t lie in [\GammaY; Y ], then Corollary 6.2 gives the upper bound of on the expected relative loss (1.2). Here the examples are i.i.d. with respect to an arbitrary distribution. In the next theorem we show that the bound (9.1) cannot be improved substantially. Our proof is very similar to the proof of Theorem 2 of Vovk (1997). However, if the dimension n of the instances is greater than one, the examples in his proof are generated by a stochastic strategy and are not i.i.d. THEOREM 9.1. Consider linear regression there is a constant C and a distribution D on the set of examples R n \Theta [\GammaY; Y ] such that for all T and for every learning algorithm the expectation of the relative loss (1.2) is where the examples are i.i.d. with distribution D. J. FORSTER AND M. K. WARMUTH Proof. For a fixed parameter ff 1 we generate a distribution D on the examples with the following stochastic strategy: A vector ' 2 [0; is chosen from the prior distribution Beta(ff; ff) n , i.e. the components of ' are i.i.d. with distribution Beta(ff; ff). Then is the distribution for which the example n and the example n . Here e are the unit vectors of R n . In each trial the examples are generated i.i.d. with D ' . We can calculate the Bayes optimal learning algorithm for which the expectation of the loss in trials 1 through T is minimal. The expectation of the relative loss of this algorithm gives the lower bound of Theorem 9.1. Details of the proof are given in the appendix. 2 Now consider the setting discussed in Section 7. Here the trials are partitioned into episodes and the outcomes are given by y s=t . The following lower bound is proven by a reduction to the previous lower bound for linear regression. THEOREM 9.2. Consider episodic temporal-difference learning where all episodes have fixed length '. Let [\GammaY; Y ] be a range for the outcomes. Then for every " ? 0 there is a constant C and a stochastic strategy that generates instances x 1 , x reinforcement signals r 1 , such that the outcomes lie in [\GammaY; Y ] and for all T divisible by ' and for every learning algorithm the expectation (over the stochastic choice of the examples) of the relative loss (1.2) is Proof. We modify the stochastic strategy used in the proof of Theorem 9.1. When this strategy generates an instance x t and an outcome y t in trial t, we now generate a whole episode of ' trials with instances reinforcement signals . The outcomes for this episode are fl Consider just the q-th trials from each episode fore some 1 q '. In these trials the learner essentially processes the scaled examples the lower bound of Theorem 9.1 applies with a factor of fl 2('\Gammaq) . All ' choices of q lead to a factor of 1 the lower bound. 2 Relative Loss Bounds for Temporal-Difference Learning 21 10. Conclusion and open problems We proposed a new algorithm for temporal-difference learning, the TLS algorithm. Contrary to previous second order algorithms, the new algorithm does not use differences in the definition of its "covariance matrix", see discussion in Section 8. The main question is whether these differences are really helpful. We proved worst and average case relative loss bounds for the TLS algorithm. It would be interesting to know how tight our bounds are for some practical data. In our bounds the class of linear functions serves as a comparison class. We use a second order algorithm and its additional loss over the loss of the best comparator is logarithmic in the number of trials. We conjecture that even for linear regression there is no first order algorithm with adaptive learning rates for which the additional loss is logarithmic in the number of trials. The algorithms analyzed here can be applied to the case when the instances are expanded to feature vectors and the dot product between two feature vectors is given by a kernel function (see Saunders, Gam- merman, & Vovk, 1998). Also Fourier or wavelet transforms can be used to extract information from the instances, see Walker (1996) and Graps (1995). With these linear transforms one can reduce the dimensionality of the comparison class which leads to smaller relative loss bounds. So far we compared the total loss of the on-line algorithm to the total loss of the best linear predictor on the whole sequence of examples. Now suppose that the comparator is produced by partitioning the data sequence into k segments and picking the best linear predictor for each segment. Again we aim to bound the total loss of the on-line algorithm minus the total loss of the best comparator of this form. Such bound have been obtained by Herbster and Warmuth (1998) for the case of linear regression using first-order algorithms. We would like to know whether there is a simple second-order algorithm for linear regression that requires O(n 2 ) update time per trial and for which the additional loss grows with the sums of the logs of the section lengths. Most of our paper focused on continuous learning, where each outcome is an infinite discounted sum of future reinforcement signals. In Section 7 we discussed how the TLS algorithm can be adapted to the setting. Here the outcomes only depend on reinforcement signals from the same episode: s=t For some applications it might make more sense to let the outcomes y t be convex combinations of the future reinforcement signals of the 22 J. FORSTER AND M. K. WARMUTH episode and define s=t In the case when each outcome would be the average of the future reinforcement signals. We do not know of any relative loss bounds for the case when y t is defined as (10.1). On a more technical level we would like to know if it is really necessary to clip the predictions of the temporal-difference algorithm we proposed. Our proofs are reductions to the previous proofs for linear regression. Direct proofs might avoid clipping. Another open technical question is discussed at the end of Section 3. We conjecture that the parameter a in Vovk's linear regression algorithm can be tuned to obtain bounds of the form (3.3) proven for the (first order) Widrow-Hoff algorithm. Similarly we believe that the parameter a in the new (second order) learning algorithm of the paper can be tuned to obtain the bound (4.1) proven for the (first order) TD () algorithm of Schapire and Warmuth. Finally, note that we do not have lower bounds for the continuous setting with fl ? 0. It should be possible to show a lower bound of on the expected relative loss (1.2). (See Theorem 9.2 for a corresponding lower bound in the episodic case.) Acknowledgements Jurgen Forster was supported by a "DAAD Doktorandenstipendium im Rahmen des gemeinsamen Hochschulsonderprogramms III von Bund und Landern". Manfred Warmuth was supported by the NSF grant CCR-9821087. Thanks to Nigel Duffy for valuable comments. --R Relative loss bounds for on-line density estimation with the exponential family of distributions Linear analysis: An introductory course. On relative loss bounds in generalized linear regression. Prediction in the worst case. An introduction to wavelets. Unpublished manuscript. Department of Computer Science Tracking the best regressor. Additive versus exponentiated gradient updates for linear prediction Numerical recipes in pascal. Survey of applicable mathematics Ridge regression learning algorithm in dual variables. On the Worst-case Analysis of Temporal-Difference Learning Algorithms Learning to predict by the methods of temporal differences. Reinforcement learning: An introduction. Competitive on-line linear regression Fast Fourier transforms Adaptive signal processing. --TR

temporal-difference learning;relative loss bounds;on-line learning;machine learning

608352

Polynomial-Time Decomposition Algorithms for Support Vector Machines.

This paper studies the convergence properties of a general class of decomposition algorithms for support vector machines (SVMs). We provide a model algorithm for decomposition, and prove necessary and sufficient conditions for stepwise improvement of this algorithm. We introduce a simple rate certifying condition and prove a polynomial-time bound on the rate of convergence of the model algorithm when it satisfies this condition. Although it is not clear that existing SVM algorithms satisfy this condition, we provide a version of the model algorithm that does. For this algorithm we show that when the slack multiplier C satisfies \sqrt{1/2} C mL, where m is the number of samples and L is a matrix norm, then it takes no more than 4LC2m4/&epsi; iterations to drive the criterion to within &epsi; of its optimum.

Introduction The soft margin formulation in (Cortes & Vapnik, 1995) has the advantage that it provides a design criterion for support vector machines (SVMs) for both separable and nonseparable data while maintaining a convex programming problem. To maintain a computationally feasible approach across all kernels, algorithms are developed for the Wolfe Dual Quadratic Program (QP) problem whose size is independent of the dimension of the ambient space. The Gram matrix for the Wolfe Dual is is the number of data samples. For large m the storage requirements for this matrix can be excessive, thereby preventing the application of many existing QP solvers. This barrier can be overcome by decomposing the original QP problem into smaller QP problems and employing algorithmic strategies that solve a sequence of these smaller QP problems. For the class of algorithms considered here these smaller QP problems are restrictions of the original QP problem where optimization is allowed over a subset of the data called the working set. The key is to select working sets that guarantee progress toward the original problem solution at each step. Such algorithms are commonly referred to as decomposition algorithms, and many existing SVM algorithms fall into this class (Cristianini & Shawe-Taylor, 2000; Joachims, 1998; Keerthi, Shevade, Bhattacharyya, 1998). In this paper we provide a model algorithm for decomposition and prove necessary and su-cient conditions for stepwise improvement of this algorithm. These conditions require that each working set contain a certifying pair (dened in section 3). Computation of a certifying pair takes O(m) time. We dene a simple \rate certifying" condition on certifying pairs that enables the proof of a polynomial-time bound on the rate of convergence. It is not clear that the working sets chosen by existing SVM algorithms contain certifying pairs that satisfy this condition. On the other hand, we provide an O(m log m) algorithm for determining a certifying pair that does. The next section sets the stage for our development by providing a formal denition of the problem and establishing some of its basic properties. Preliminaries be a nite set of observations from a two-class pattern recognition problem where x 1g. The Support Vector Machine (SVM) maps the space of covariates X to a Hilbert space H of higher dimension (possible innite), and ts an optimal linear classier in H. It does so by choosing a map in such a way that known and easy to evaluate function K. Su-cient conditions for the existence of such a map are provided by Mercer's theorem (Vapnik, 1998). Let z so that A linear classier in H is given by LANL Technical Report: LA{UR{00-3800 2 Preliminaries In the soft margin formulation of (Cortes & Vapnik, 1995) the optimal is given by optimizes the Wolfe Dual quadratic programming problem, s.t. where The choice of the unspecied parameter C > 0 has been investigated but we do not address that here. Once has been determined the optimal value of b is given by ~ v() low b ~ v() high where ~ v() low and ~ v() high are dened in section 3. This paper is concerned with the analysis of a class of algorithms for WD(S) that are motivated by situations where m is so large that direct storage of Q is prohibitive. Let WD(S) denote an instance of the Wolfe Dual dened by the sample set S. Let (S) represent the set of feasible solutions for WD(S), Note that (S) is both convex and compact. Denote the Wolfe Dual criterion by and let (S) represent the set of optimal solutions for WD(S), R()g: verifying that Q is symmetric and positive semi-denite. Thus, R() is a concave function over (S) and R unique. The Lagrangian for WD(S) takes the form LANL Technical Report: LA{UR{00-3800 3 Optimality using Certifying Pairs Then the Karush-Kuhn-Tucker (KKT) conditions (e.g. see (Avriel, 1976), p.96) for WD(S) take the form where we have made use of the relation There are three regimes for i ; two where it equals a bound, and one where it falls between the bounds. Combining the conditions above with these three regimes we obtain a simpler set of conditions that are equivalent to the KKT conditions It is possible to use the satisfaction of these equations as a stopping condition for optimization algorithms, but they involve . An alternative set of optimality conditions were introduced in (Keerthi et al., 2001; Keerthi & Gilbert, 2000) that do not use . In the next section we present these conditions and use them to develop a simple optimality test. 3 Tests for Optimality using Certifying Pairs We dene a partition of the index set of S based upon the data I low I I and low g and let i2I low i2I high where the sup and inf of the empty set are dened as 1 and 1 respectively. LANL Technical Report: LA{UR{00-3800 3 Optimality using Certifying Pairs Denition 1. is properly ordered for S if jV int low v high or jV int low V int v We now prove a result rst stated by Keerthi and Gilbert (Keerthi & Gilbert, 2000). Theorem 1. (Keerthi and Gilbert) A feasible for the Wolfe dual problem WD(S) is optimal if and only if is properly ordered for S. Proof. The optimality conditions (7) can be rewritten as low Now suppose that is optimal. Then equations (12) imply that low low The rst equation implies that jV int and the second equation implies that v low v high . When jV int the second and third equations imply that low V int v high and so is properly ordered. On the other hand, suppose is properly ordered. Then jV int By the denitions of v low and v high it is clear that low low and we can choose to be any point in [v low high so that the conditions (12) are satised. Consequently, is optimal if and only if it is properly ordered for S. Tests for proper ordering can be simplied if we dene ~ I low = I low [ I int ~ I high = I high [ I int and i2 ~ I low I high Then is properly ordered for WD(S) if and only if low ~ v The proof of this statement follows directly from the proof of Theorem 1. Lack of optimality can be determined by the existence of a certifying pair. Denition 2. A certifying pair for 2 (S) is a pair of indices i and j in the index set of S whose values (v are su-cient to prove that is not properly ordered for S. We note that Keerthi et. al. (Keerthi & Gilbert, 2000) refer to this as a violating pair. However, because we later dene rate certifying pair we decided not to adopt this terminology. Theorem 2. is not properly ordered for S if and only if there exists a certifying pair. A certifying pair can be obtained by making at most one pass through the data while making two comparisons. Proof. Suppose that is not properly ordered for S. Then there exists indices i 2 ~ I high and I low such that v i < v j . Choose any such pair. To determine a certifying pair make one pass through the data while keeping track of indices that represent high and ~ v low . Stop at the rst point where ~ high < ~ v low . 4 A General Decomposition Algorithm Algorithmic solutions for the Wolfe dual must consider the fact that when m is large the storage requirements for Q can be excessive. This barrier can be overcome by decomposing the original QP problem into smaller QP problems. Suppose we partition the index set of into a working set W and a non-working set W c . Note that W indexes a subset of the data. Then are partitioned accordingly and Q is partitioned as follows QW c W QW c W c W . Then (3) can be written xed this becomes a QP problem of size dim(W ) with the same generic properties as the original. This motivates algorithmic strategies that solve a sequence of QP problems over dierent working sets. The key is to select a working set at each step that will guarantee progress toward the original problem solution. Theorem 3. Consider the subset constrained Wolfe dual problem dened as follows. Consider a feasible . Dene a subset W of the index space of S with complement W c . Optimize the Wolfe dual criterion with respect to subject to the constraint that = on W c . Let denote a solution to this constrained problem. Then, R( ) > R(), if and only if W contains a certifying pair for . Proof. Since R is concave, is non-optimal for WD(S) if and only if there is a feasible innitesimal _ at such that > 0: (16) Further, the solution to the constrained Wolfe dual produces an increase in R if and only if there is a feasible constrained _ (with nontrivial components on W only) such that dR() _ Consequently, to prove the theorem it is su-cient to show that a feasible _ W exists that satises if and only if W contains a certifying pair. The derivative of R is given by _ . The feasible directions _ satisfy _ when In terms of d these conditions become d high , and low . Decompose components under the subsets dened by I high , I low , and I int . Then (16) can be written d high v high low v low and the feasibility constraints are d high 1 low low 0; d int free: (18) Assume that W contains a certifying pair. Then it must satisfy one of the following inequalities, low low In all four cases we can verify (17)-(18) by choosing d for the certifying pair and so that The proof of \if" is nished. Now assume that there is a feasible _ W for which dR() _ W > 0. Then (17)-(18) are be the restrictions of V int (I int ) to the indices of W . If jV int (W )j > 1 then any two components certifying pair. If jV int (W d high v high low v low Combining with (18) gives d high (v high v low (v low v 1) < 0; d high 0; d low 0 For this inequality to hold at least one of the two terms must be negative. To make the rst term negative at least one component of (v high v 1) must be negative. Similarly, to make the second term negative at least one component of (v low v 1) must be positive. Either case gives a certifying pair. Finally, if jV int (W d high v high low v low < 0 d high low 1; d high 0; d low 0 Without loss of generality let the components of d high and d low be normalized so that i2I high i2I low Then (d high v high low v low ) is the dierence between convex combinations of V high (W ) and convex combinations of V low (W ). For this dierence to be negative the two convex hulls must overlap. This implies a certifying pair. This nishes the \only if" part, so the proof is nished. Theorem 3 motivates a class of algorithms of the form Algorithm A 1 below. Members from this class solve a sequence of decomposed QP problems of the form in (15) over working sets that can vary in size from 2 to jSj and contain at least one certifying pair. The initialization ensures that W (0) contains at least one certifying pair. The QPSolve routine on line 11 solves the QP problem restricted to the current working set W (k 1). Line 14 chooses a certifying pair for inclusion in the next working set. The algorithm terminates when a certifying pair no longer exists. The AnySubset routine on line chooses a subset of samples to be included with the certifying pair in the next working set. This subset is irrelevant to the issue of guaranteed improvement, but is likely to have an eect on the rate of convergence. e Algorithm Decomposition Algorithm. 1: 2: 3: 4: OUTPUT: 5: 7: 8: ~ I low 9: ~ I 10: W (0) subset of I S with at least one sample from each class: 12: loop 13: 14: Update membership in ~ I low ; ~ I high for samples in W (k 1) I low I high ; and v i > v j 17: if 19: end if 22: end loop Convergence In general, the stepwise improvement of Algorithm A 1 is not su-cient to guarantee convergence. Indeed, Keerthi and Ong (Keerthi & Ong, 2000) provide an example where each working set contains a certifying pair but Algorithm A 1 does not converge to the optimal solution. However, convergence results have been proved for some special cases, e.g. see (Keerthi & Gilbert, 2000), (Chang, Hsu, & Lin, 2000), (Lin, 2000). The convergence result in (Keerthi & Gilbert, 2000) denes to be -optimal if it satises ~ v low < ~ high It then shows that the generalized SMO (GSMO) algorithm converges to a -optimal solution in a nite number of steps. The GSMO algorithm is a special case of Algorithm A 1 where the AnySubset function returns the empty set. The analysis in (Keerthi & Gilbert, 2000) leaves open the question of accuracy with respect to the optimal solution, that is it provides no bound on jR( ) R j or (Chang et al., 2000) give a proof of convergence for a special case of Algorithm A 1 where the working set is dened to be the indices corresponding to the nontrivial components of d in e the solution to the optimization problem s.t. d where q 2. Their proof shows that, with this choice of working set, Algorithm A 1 produces a sequence f(k)g whose limit point is optimal for WD(S). More recently (Lin, 2000) has provided a similar proof of convergence for SV M light where the working set is dened by Joachims (Joachims, 1998) to be the indices corresponding to the nontrivial components of d in the solution to a slightly dierent optimization problem s.t. d where q 2. The analysis in (Chang et al., 2000) and (Lin, 2000) is asymptotic and therefore leaves open the question of nite step convergence to the optimum. In the following section we provide a nite step convergence proof for a special case of Algorithm A 1 that corresponds to \chunking". 5.1 Finite Step Convergence for Chunking Chunking (as described in (Cristianini & Shawe-Taylor, 2000)) is a decomposition method in which each working set contains all support vectors from the current solution plus an additional set of samples that violate an \optimality condition". If the optimality condition is chosen so that the additional set always contains at least one certifying pair 1 then the resulting algorithm takes the form of Algorithm A 1 where the AnySubset routine returns, at a minimum, the indices for all samples with i > 0. The following theorem holds for this class of chunking algorithms. Theorem 4. Let S be a nite set of observations containing at least one sample from each class. Consider Algorithm A 1 where the AnySubset routine returns any set that contains the indices for all samples with i > 0. This algorithm converges to a solution of WD(S) for nite k. Proof. Algorithm A 1 terminates only when there are no certifying pairs, and if it terminates then 2 (S). We assume that QPSolve provides an exact solution to the constrained Wolfe dual. Then Theorem 3 guarantees that when we are not at a solution the criterion for WD(S) is strictly increased from one step to the next, i.e. R((k all nontrivial contribution to R is made by the working set. Thus, no working set is revisited, and since there are a nite number of working sets, and R is unique, termination in nite k is guaranteed. This requires a slight modication to the chunking algorithm in (Cristianini & Shawe-Taylor, 2000). e We now show that with the proper choice of certifying pair we can provide polynomial-time bounds on the run time of Algorithm A 1 . 5.2 Convergence Rate In this section we give a nite step convergence result for Algorithm A 1 when each working set contains a rate certifying pair (dened below). We also provide bounds on the convergence rate. More specically we give a polynomial bound on the number of iterations required to drive jR() R j to within of its optimum. Note that the criterion has a strong dependence on the size of the sample set m. In general R becomes unbounded as m ! 1. Consequently the development of convergence rates requires the normalization of R in terms of the number of samples. For example, in empirical risk minimization it is standard to divide the number of training errors by the number of samples to obtain the fraction of training errors. However at present we know of no natural normalization for R. Therefore to allow for the incorporation of an appropriate normalization we implicitly denote the error tolerance as a function of m through the notation m . Let be an optimal parameter value and R the optimal criterion value. Because of concavity, which can be rewritten as If we dene we obtain Let denote a parameter value which diers from in at most two places and dene When () () for some 0 < < 1 then we can bound the distance to the optimum by ()=. We use this to determine a bound on the convergence rate for Algorithm A 1 . Let k denote the value of the state at the k-th iteration and let k denote a parameter that diers from k in at most two indices. We note that in previous sections the subscripted k was used for the k-th component of the vector and the parenthetic (k) was used for the state of the algorithm at the k-th iteration. However, in the present analysis we need no components of the vector and feel the use of k for the state at the k-th iteration is a better notation for this section. Let R e and Denition 3. Algorithm A 1 is a rate certifying algorithm if there exists an such that the certifying pair chosen on line 14 satises for all k. A rate certifying pair is a pair of indices in the index set of S for which at iteration k of a rate certifying algorithm. Chang, et. al. (Chang et al., 2000) establish a relationship of this type for a particular choice of rate certifying pair with use it to prove asymptotic convergence. The following theorem gives a bound on the number of iterations that are su-cient to drive the criterion to within m of its optimum for a rate certifying algorithm. Theorem 5. Let (k) denote the sequence of states generated by Algorithm A 1 . If it is a rate certifying algorithm then R BL iterations, where (R R( 0 and L is the maximum of the norms of the 2 by 2 matrices determined by restricting Q to indices. In words, if we wish to get an accuracy of m , then it is su-cient to performq BL Proof. Let fi; jg W (k) denote the indices of a rate certifying pair in the working set such that Following (Dunn, 1979) we consider the following auxiliary equations. Let k dier from k in the two indices dr k ( e we have which can be written We show by induction that k B k as follows. We now control k . Plugging the denition of ! k in equation (26) into equation (27) for k we obtain In the latter case j Putting the two equations from (28) together we obtain where since k Therefore, by (Dunn, 1979) equations (29) and (30) imply that but going back through the relations L and k B k implies Consequently, when BL then and The proof is nished. 5.3 E-cient Computation of a Rate Certifying Pair In the previous section we determined that k k is su-cient to establish (Chang et al., 2000) show that a certifying pair always exists such that They do this by considering the solution to a linear programming (LP) problem (similar to the LP problem for k ), and then restricting this solution to two indices. In this section we show how to solve this LP to produce a rate certifying pair in O(m log m) operations. be the current solution and dene Let be the solution to the linear program s.t. Note that the solution to this problem and (19) are related by . As in section 3, dene ~ I low I low I I high I and choose low high From (Chang et al., 2000) we know that the certifying pair (i; j) given by is a rate certifying pair with rate . The following lemma establishes that this pair can be determined in a computationally e-cient manner. Lemma 1. Given y, the rate certifying pair (i; can be computed in O(m log m) time. Proof. We describe an algorithm that computes this pair in O(m log m) time. Our algorithm solves the LP in (31) and then computes the two indices using (32)-(33). Once the LP is solved it is straightforward to implement (32)-(33) in O(m) steps, so we describe only the LP solution. Consider the LP in (31). Recall that dR() . The Karush-Kuhn-Tucker conditions for the solution are with i 0, i 0 and These equations can be written high low int where I low I I e To solve these equations, x and determine to satisfy high low v i 0: For example, if v i > , then set To determine we use the constraint Written out this becomes i2I low | {z } i2I high | {z } i2I int Our strategy is to choose so that it splits the samples into I low and I high in such a way that the rst and second sums cancel as closely as possible. When they do not cancel exactly we shift so that the split occurs on a value v i , thereby placing samples with this value into I int and allowing us to choose their parameters i to satisfy the equality. More specically we sort the values of v in increasing order and use k to index the sorted list (i.e. v k v k+1 ). As increases from 1 to 1, jumping over values where being determined as above, the value of y is monotonically increasing and must pass from negative to positive. In fact it is easy to see that y increases by C each time an individual sample is jumped. Suppose that this increasing function achieves the value 0 on the interval (v k ; v k+1 ). Then we let be any value in this interval and since I int is empty and was chosen to satisfy (35) we have a solution. Suppose this increasing function skips the value 0 and jumps from a < 0 to b > 0 at and there are a total of M 1 samples with this value of v (i.e. Then set place the rst of these samples in I low (the rest remain in I high ). If a=C is integral then this gives and we have a solution once again (with M of the samples satisfying (35) with equality and I before). If a=C is not integral then its remainder is used to determine k+M 1 , the component of corresponding to v k+M 1 . This gives and places this sample in I int , and again we have a solution. Note that there are many solutions to these equations. This construction gives and , both of which are necessary to implement (32)-(33). It takes O(m log m) steps to sort the v, followed by an additional pass through the list to initialize , placing all samples in I high and yielding (0) y. Since y begins at (0) y and increases by C each time is increased past a data point, the components of for all the points up to k C c are changed by C placing them in I low . Then, if (0)y C is not integral its remainder is used to determine the component of for the which is moved to I int . Updating in this way requires at most one complete pass through the list. This completes the proof. Algorithm 5.3 computes a rate certifying pair using the method described in the proof above. In addition to the sort, this algorithm makes a total of four passes through the list. The number of computations in this procedure can sometimes be reduced. Let i; j be a rate certifying pair. Then v i and v j are on opposite sides of , and since i; j is also a certifying e must lie between ~ high and ~ low (dened in (13) and (14)). This means that the sorting operation required in our search for can be restricted to the v i in this interval. Since the sorting operation dominates the run time this can lead to a substantial savings when the number of samples in this interval is small. Algorithm Certifying Pair Algorithm. INPUTS: y, v, and (at the current iteration) fsample indices for a rate certifying pairg fL is an ordered list of indices in nondecreasing order of fv i g so that v L(l) v L(l+1) g finitially place all samples in I high and compute (0) yg do if (y else if (y end for fdetermine split point index and move samples into I low g l bEtaDotY=Cc for l do end for fif needed, move sample into I int g if (EtaDotY < l l else value in [v L(l use v L(i fdetermine indices for rate certifying pairg LANL Technical Report: LA{UR{00-3800 6 Discussion 5.4 Summary of Rates If we use Algorithm A 2 to choose a rate certifying pair then 2 and by theorem 5 Algorithm A 1 will drive the criterion to within m of its optimum in no more than iterations. Further, with so that neglecting lower order terms, the number of iterations simplies to In the case where the working sets are of size two we can use this result to establish a worst case overall run time for Algorithm A 1 . At each iteration we must solve a 2 by 2 QP problem, update the v i (k), and determine the next certifying pair. The time to solve the 2 by 2 QP problem is a constant, and it takes order m operations to update the v i (k). If we add m log m operations to determine the certifying pair, the worst case run time is of order Now consider our choice for m obtained through an appropriate normalization of R (see discussion at the beginning of this section). Because R tends to increase with m, m will be an increasing function of m. Although the form of this function is not yet known it will clearly improve the run-time bounds presented above. For example, if then the order of the polynomial in these bounds is reduced by p. 6 Discussion This paper considers a class of algorithms for support vector machines that decompose the original Wolfe Dual QP problem into a sequence of smaller QP problems dened on subsets of the data. Following the work of Keerthi et al. (Keerthi & Gilbert, 2000; Keerthi et al., 2001) we provide a scalar condition that is necessary and su-cient for optimality of the QP problem. This leads naturally to the introduction of certifying pairs as a necessary and su-cient condition for stepwise improvement, and motivates the use of Algorithm A 1 as a model algorithm for this problem. By leveraging the results of Chang, et al. (Chang et al., 2000) we have developed Algorithm A 2 for selecting the certifying pair in Algorithm A 1 . Theorem 5 shows that the number of iterations for this instantiation of Algorithm A 1 is O(m 4 ) and the overall run time is O(m 5 log m). Many existing SVM algorithms are either special cases of Algorithm A 1 or can be made so through slight modication. For example, Platt's Sequential Minimal Optimization (SMO) algorithm, which chooses working sets of size two, is designed to choose a pair that give a strict increase in R at each step (Platt, 1998). The original algorithm however, contains a aw that LANL Technical Report: LA{UR{00-3800 References can lead to improper behavior (Keerthi et al., 2001; Keerthi & Gilbert, 2000). This behavior can be traced to its inability to guarantee a certifying pair in each working set. By forcing each working set to contain a certifying pair the corrected algorithm not only has guaranteed convergence, but also improved performance (Keerthi et al., 2001). The SV M light algorithm in (Joachims, 1998) uses a modication of Zoutendijk's method (Zoutendijk, 1970) to choose working sets of size q 2. This choice can be shown to contain the q=2 largest v i from ~ I low and the q=2 smallest v i from ~ I high , thus guaranteeing at least one certifying pair. The chunking algorithm described in (Cristianini & Shawe-Taylor, 2000) and the decomposition algorithm of (Osuna et al., 1997) both attempt to ensure improvement in R by choosing working sets that include support vectors from the current solution plus a subset of samples that violate an \optimality condition" with respect to this solution. A strict implementation of the algorithms described in these papers can lead to undesirable behavior because they cannot guarantee a certifying pair in their working sets. However, such a guarantee can be achieved with a slight modication (as we did for the chunking algorithm in section 5.1). It is not clear that the algorithms above satisfy the rate certifying condition in Denition 3, nor that this is necessary to establish rates for them. We have described a new SVM algorithm that satises the rate certifying condition and has polynomial-time rates. It is not yet clear how this algorithm will compare with existing algorithms in practice. Note that Keerthi's GSMO algorithm (Keerthi et al., 2001) and Jochamin's SV M light algorithm (Joachims, 1998) require O(m) time to determine a certifying pair while A 2 requires O(m log m) time. However, we know of no bounds on the rates of convergence for GSMO and SV M light (although they seem to work well in practice), but can guarantee a polynomial convergence rate when we use A 2 . Finally we note that the polynomial-time bound on the number of iterations scales as 4 , which is unattractive. We leave open the issue of the tightness of this bound, although we suspect that it may be loose. A closely related issue is the determination of a proper normalization for R that would give rise to an explicit functional dependence of on m. This is likely to improve the rate. --R Nonlinear Programming: Analysis and Methods (1st edition). The analysis of decomposition methods for support vector machines. An Introduction to Support Vector Machines and Other Kernel-based Learning Methods (1st edition) Rates of convergence for conditional gradient algorithms near singular and non-singular extremals Convergence of a generalized SMO algorithm for SVM classi Improvements to Platt's SMO algorithm for SVM classi On the convergence of the decomposition method for support vector machines. Support vector machines: training and applications. Fast training of support vector machines using sequential minimal optimiza- tion Statistical Learning Theory. Methods of Feasible Directions: A study in linear and non-linear pro- gramming --TR --CTR Hong Qiao , Yan-Guo Wang , Bo Zhang, A simple decomposition algorithm for support vector machines with polynomial-time convergence, Pattern Recognition, v.40 n.9, p.2543-2549, September, 2007 Tobias Glasmachers , Christian Igel, Maximum-Gain Working Set Selection for SVMs, The Journal of Machine Learning Research, 7, p.1437-1466, 12/1/2006 Rong-En Fan , Pai-Hsuen Chen , Chih-Jen Lin, Working Set Selection Using Second Order Information for Training Support Vector Machines, The Journal of Machine Learning Research, 6, p.1889-1918, 12/1/2005 Thorsten Joachims, Training linear SVMs in linear time, Proceedings of the 12th ACM SIGKDD international conference on Knowledge discovery and data mining, August 20-23, 2006, Philadelphia, PA, USA Nikolas List , Hans Ulrich Simon, General Polynomial Time Decomposition Algorithms, The Journal of Machine Learning Research, 8, p.303-321, 5/1/2007 Don Hush , Patrick Kelly , Clint Scovel , Ingo Steinwart, QP Algorithms with Guaranteed Accuracy and Run Time for Support Vector Machines, The Journal of Machine Learning Research, 7, p.733-769, 12/1/2006 Cheng-Ru Lin , Ken-Hao Liu , Ming-Syan Chen, Dual Clustering: Integrating Data Clustering over Optimization and Constraint Domains, IEEE Transactions on Knowledge and Data Engineering, v.17 n.5, p.628-637, May 2005 Luca Zanni , Thomas Serafini , Gaetano Zanghirati, Parallel Software for Training Large Scale Support Vector Machines on Multiprocessor Systems, The Journal of Machine Learning Research, 7, p.1467-1492, 12/1/2006

support vector machines;decomposition algorithms;polynomial-time algorithms

608356

Scenario Reduction Algorithms in Stochastic Programming.

We consider convex stochastic programs with an (approximate) initial probability distribution P having finite support supp P, i.e., finitely many scenarios. The behaviour of such stochastic programs is stable with respect to perturbations of P measured in terms of a Fortet-Mourier probability metric. The problem of optimal scenario reduction consists in determining a probability measure that is supported by a subset of supp P of prescribed cardinality and is closest to P in terms of such a probability metric. Two new versions of forward and backward type algorithms are presented for computing such optimally reduced probability measures approximately. Compared to earlier versions, the computational performance (accuracy, running time) of the new algorithms has been improved considerably. Numerical experience is reported for different instances of scenario trees with computable optimal lower bounds. The test examples also include a ternary scenario tree representing the weekly electrical load process in a power management model.

Introduction Many stochastic decision problems may be formulated as convex stochastic programs of the form minf Z where X R m is a given nonempty closed convex set, a closed subset of R s , the function f 0 from R m to R is continuous with respect to ! and convex with respect to x, and P is a xed Borel probability measure on P( ). For instance, this formulation covers (convex) two- and multi-stage stochastic programs with recourse. Typical integrands f 0 (; x), x 2 X , in convex stochastic programming problems are nondierentiable, but locally Lipschitz continuous on . In the fol- lowing, we assume that there exist a continuous and nondecreasing function nondecreasing function and some xed element s such that Heitsch, Romisch for each x 2 X , where the function c R is given by c(!; ~ This means that the function h(k ! 0 describes the growth of the local Lipschitz constants of f 0 (; x) in balls around ! 0 with respect to some norm k on R s . An important particular case is that the function h grows poly- nomially, i.e., 1. For instance, it is shown in [11] that the choice is appropriate for two-stage models with stochasticity entering prices and right-hand sides. It is shown in [4,11] that the model (1) behaves stable with respect to small perturbations in terms of the probability metric c (P; Q) := sup Z Z where F c is the class of continuous functions dened by !) for all !; ~ and probability measures P and Q in the set P( Z (see also the earlier work in [13]). The distance c is a probability metric on with -structure and also called a Fortet-Mourier (type) metric. In this generality, it is introduced in [14] and further studied in [10,12]. In particular, the metric c has dual representations in terms of the Kantorovich-Rubinstein functional (cf. Section 5.3 in [10] and [12]). An important instance is that the initial probability measure P is itself discrete with nitely many atoms (or scenarios) or that a good discrete approximation of P is available. Its support may be very large so that, for reasons of computational complexity and time limitation, this probability measure is further approximated by a probability measure Q carried by a (much) smaller subset of scenarios. In this case the distance c (P; Q) represents the optimal value of a nite-dimensional linear program. More precisely, from (4) we obtain for c where J P( denotes the Dirac measure placing unit mass at !. In particular, the metric c can be used to evaluate distances of specic probability measures obtained during a scenario-reduction process. Various reduction rules appear in the literature in the context of recent large-scale real-life applications. We refer to the corresponding discussion in [4], to Scenario Reduction Algorithms in Stochastic Programming 3 the recent work [3] on scenario generation and reduction, and to the paper [9], in which an approach to scenario generation based on Fortet-Mourier distances is given. In the present paper, we follow the approach for reducing scenarios of a given discrete probability measure developed in [4]. It consists in determining an index set J of given cardinality #J and a probability measure Q such that Problem (7) may be reformulated as a mixed-integer program. In Section 2 we derive bounds for (7) and develop two new algorithms (fast forward selection and simultaneous backward reduction), which constitute heuristics for solving (7). We study their complexity and their relations to the algorithms in [4]. Indeed, the fast forward selection algorithm turns out to be an ecient implementation of the forward selection procedure of [4], producing the same reduced probability measures. In order to compare the performance of the algorithms we provide, in Section 3, explicit formulas for the minimal distances (7) in case that P is a regular (binary or ternary) scenario tree (i.e., a tree having a specic structure) and Q is a reduced tree with xed cardinality n. In Section 4 we report on numerical experience for the reduction of regular binary and ternary scenario trees. The test trees also include the ternary scenario tree representing the weekly electrical load process in a power management model which was considered in [4]. It turns out that the new implementation of the fast forward selection algorithm is about 10-100 times faster than the earlier version. Furthermore, fast forward selection is the best algorithm when comparing accuracy. The results of the simultaneous backward reduction algorithm are more accurate than the backward reduction variant of [4] in most cases, but at the expense of higher running times. When comparing running times, fast forward selection (simultaneous backward reduction) is preferable in case that n < N Reduction We consider the stochastic program (1) and select the function c of form (3) such that the Lipschitz condition (2) is satised. Let the initial probability distribution P be discrete and carried by nitely many scenarios ! iwith weights and consider the probability measure 4 Heitsch, Romisch i.e., compared to P , the measure reduced by deleting all by assigning new probabilistic weights q j to each scenario . The optimal reduction concept described above recommends to consider the probability distance depending on the index set J and q. The optimal reduction concept (7) says that the index set J and the optimal weight q are selected such that D(J ; q Jg. First we recall the following bounds for min q D(J ; when the index set J is xed ([4], Theorem 3.1). Theorem 2.1 (redistribution) For any index set J where C := max Furthermore, we have equality in (9) and, hence, optimality of q if h 1. For convenience of the reader the proof of Theorem 2.1 is displayed in the Appendix . The interpretation of the formula (10) is that the new probability of a preserved scenario is equal to the sum of its former probability and of all probabilities of deleted scenarios that are closest to it with respect to c. If h 1, we call (10) the optimal redistribution rule. Next we discuss the optimal choice of an index set J for scenario reduction with xed cardinality #J . Theorem 2.1 motivates to consider the following formulation of the optimal reduction problem for given n 2 N, n < minfD J := ng (11) Problem (11) means that the set has to be covered by two sets such that J has xed cardinality N n and the cover has minimal cost D J . Thus, (11) represents a set-covering problem. It may be formulated as a 0-1 integer program (cf. [7]) and is NP-hard. Since e-cient solution algorithms are hardly available in general, we are looking Scenario Reduction Algorithms in Stochastic Programming 5 for (fast) heuristic algorithms exploiting the structure of the costs D J . In the specic cases of solving (11) becomes quite easy. In case that #J = 1, the problem (11) takes the form min l2f1;:::;Ng l min If the minimum is attained at l i.e., the scenario ! l is deleted, the redistribution rule (10) yields the probability distribution of the reduced measure Q. If j 2 arg min j 6=l c(! l holds that q and q g. Of course, the optimal deletion of a single scenario may be repeated recursively until a prescribed number N n of scenarios is deleted. This strategy recommends a conceptual algorithm called backward reduction. In case that #J = N 1, the problem (11) is of the form min u2f1;:::;Ng If the minimum is attained at u only the scenario ! u is kept and the redistribution rule (10) provides strategy provides the basic concept of a second conceptual algorithm called forward selection. First, we take a closer look at the backward reduction strategy. A backward type algorithm was already suggested in [4,6]. It determines a reduced scenario set by reducing N n scenarios from the original set of scenarios as follows. Let the indices l i be selected such that l min It can be shown that is a lower bound of the optimal value of (11). Furthermore, it holds that the index set is a solution of (11) if for all n, the set is nonempty ([4,6]). This property motivates the following algorithm. In the rst step, an index n 1 with determined using formula (14) such that J is a solution of (11) for Next, the redistribution rule of Theorem 2.1 is used. This leads to the reduced probability measure P 1 containing all scenarios indexed by reduced by deleting all scenarios belonging to some index set J 2 with #J and n which is obtained in the same way using formula (14). This procedure is continued until, in step r, we have n 6 Heitsch, Romisch Finally, the redistribution rule (10) is used again for the index set J . This algorithm is called backward reduction of scenario sets. There are still many degrees of freedom to choose the next scenario in each step. Often there exist several candidates for deletion. In Section 4 we use one particular implementation of backward reduction of scenario sets. Another particular variant consists in the case that #J This variant (without the nal redistribution) was already announced in [2,5]. However, numerical tests have shown that the backward reduction of scenario sets provides slightly more accurate results compared to backward reduction of single scenarios. Next we are going to present a new modication of the backward reduction principle. The major dierence is to include all deleted scenarios into each backward step simultaneously. Namely, the next index l i is determined as a solution of the optimization problem A more detailed description of the whole algorithm, which will be called simultaneous backward reduction, is given in Algorithm 2.2 (simultaneous backward reduction) Sorting of fc c [1] ll := min z [1] l := p l c [1] l2f1;:::;Ng z [1] Step i: c [i] kl := min z [i] l := z [i] Step N-n+1: Redistribution by (10): Algorithm 2.2 allows the following interpretation. Its rst step corresponds to the optimal deletion of only one scenario. For i > 1, l i is chosen such that where D J [i 1] [flg is dened in (11). Hence, the index l i is dened recursively such that the index set is optimal subject to the constraint Scenario Reduction Algorithms in Stochastic Programming 7 that the previous indices are xed. Since running times are important characteristics of scenario reduction al- gorithms, we study the computational complexity, i.e., the number of elementary arithmetic operations, of Algorithm 2.2. It is shown in [6] that a proper implementation (without sorting) of backward reduction of scenario sets requires a complexity of O(N 2 ) operations which holds uniformly with respect to n. When comparing formulas (14) and (16), one notices an increase of complexity in the cost structure of (16) for determining l i . More precisely, step i requires the computation of N sums each consisting of i summands and N i+1 comparisons. Each summand represents a product of two numbers. One of these factors requires about 2 operations for determining the minimum. The sorting process in step 1 requires O(N 2 log N) operations ([1], Chapter 1). When excluding the complexity of evaluating the function c and of the redistribution rule, altogether we obtain where a(N) := N 3+O(N 2 log N) operations for selecting a subset of n scenarios. Hence, we have Proposition 2.3 The computational complexity for reducing a set of N 2 N scenarios to a subset containing n 2 consists of b N (n) (see (18)) operations when using simultaneous backward reduction. Hence, the complexity of simultaneous backward reduction is increasing with decreasing n and is, of course, minimal at This result corresponds to the running time observations of our numerical tests reported in Section 4. Next, we describe a strategy that is just the opposite of backward reduction. Its conceptual idea is based on formula (13) and consists in the recursive selection of scenarios that will not be deleted. The basic concept of such an algorithm is given in [4] and called forward selection. Forward selection determines an index set fu un g such that g. The rst step of this procedure coincides with solving problem (13). After the last step, the optimal redistribution rule has to be used to determine the reduced probability measure. Formula (19) allows the same interpretation as in case of simultaneous backward reduction. It is again closely related to the structure of D J in (11). Now, let us consider the following algorithm, which is easy to implement and is called fast forward selection. 8 Heitsch, Romisch Algorithm 2.4 (fast forward selection) z [1] u2f1;:::;Ng z [1] Step i: c [i] z [i] z [i] Step n+1: Redistribution by (10): Theorem 2.5 The index set fu un g determined by Algorithm 2.4 is a solution of the forward selection principle, i.e., u i satises condition (19) for each holds for each D J [i] is dened in (11). Proof: For the result is immediate. For holds that z [i] Hence, the index u i satises condition (19) and it holds that z [i] The conditions (17) and (20) show that both algorithms are based on the same basic idea for selecting the next (scenario) index. The only dierence Scenario Reduction Algorithms in Stochastic Programming 9 consists in the use of backward and forward strategies, respectively, i.e., in determining the sets of deleted and remaining scenarios, respectively. As in the case of backward reduction, the computational complexity of Algorithm 2.4 is of interest. Step i requires operations for computing c [i] operations for z [i] operations for determining u i . Altogether, we obtain operations for selecting a subset of n scenarios. Hence, we have Proposition 2.6 The computational complexity of fast forward selection for reducing a set of N 2 N scenarios to a subset containing n 2 scenarios consists of fN (n) (see (21)) operations. Hence, the complexity of fast forward selection is increasing with increasing n and is maximal if . Thus, the use of fast forward selection is recommendable if the number n of remaining scenarios satises the condition (n). The number n such that fN (n is a zero of a polynomial of degree 3 which depends nonlinearly on N . It turns out that n Nfor large N . 3 Minimal distances of scenario trees All algorithms discussed in the previous section provide only approximate solutions of (11) in general. Since error estimates for these algorithms are not available, we need test examples of discrete original and reduced measures of dierent scale with known (optimal) c -distances. Because of their practical importance, we consider probability measures with scenarios exhibiting a tree structure. In particular, we derive optimal distances of certain regularly structured original scenario trees and of their reduced trees containing different numbers of scenarios. We consider a scenario tree that represents a discrete parameter stochastic process with a parameter set f0; and with scenarios (or paths) branching at each parameter k 2 f0; branching degree d (i.e., each node of the tree has d successors). In case of the tree is called binary and ternary, respectively. Hence, the tree consists of N := d K scenarios , and has d K as its root node. Furthermore, we let all scenarios have equal probabilities . Such a scenario tree is called regular if, for each k 2 there exist symmetric sets V k := f- k d g R such that Heitsch, Romisch where a (K + 1)-tuple of indices (i corresponds to each index We say that In case of means that the sets V k are of the form V respectively, for some Clearly, it holds that - 0 trees. Figure 1 c c c c c c c c c c c @ @ @ @ @ @ @ @ @ @ @ @ Figure 1: Binary scenario tree shows an example of a regular binary scenario tree with scenarios. We specify the function c in (3) by setting h 1 and by choosing the maximum norm k k1 on R K+1 , i.e., c(!; ~ Our rst result provides an explicit formula for the minimal distance between a regular binary tree and reduced subtrees with at least Proposition 3.1 (3/4-solution) Let a regular binary scenario tree with it holds for each n 2 N with N Proof: We use the representation (22) of each scenario ! i for Let the corresponding (K + 1)-tuples of indices. Let l 2 Kg be such that l 1. Then we obtain (- r jr )j l (- r Scenario Reduction Algorithms in Stochastic Programming 11 Hence, it holds for each J that It remains to show that there exists an index set J such that #J and that the lower bound is attained, i.e., D . To this end, we consider the index set I := fi I . Let T r denote the tree consisting of all I . Figure 2 illustrates a detail of T r starting at a node r r r r r r @ @ @ @ @ @ Figure 2: Detail of the subtree T r at level k 0 1 and ending at level k 2. Hence, for the cardinality of I and J we obtain that 4 and #J Now we want to show that for each j 2 J , there exists an index i 2 I such that be the related scenario. Let us consider the behaviour of ! j on the branching levels k 2. we have to distinguish three cases each for - k0 Case Case Case (3): - k0+1 Now, we consider the following (K all k 62 fk 12 Heitsch, Romisch denote the corresponding index. Clearly, i 2 I and, consequently, it holds for the distance between ! i and ! j that (- r jr )j (- r jr )j case (1) case (2) maxfj2- k0 j; j2- k0 2- k0+1 jg , in case (3) The latter equation holds due to the assumption that - k0 maxf- 2- k0 . Hence, D . By considering subsets of J having cardinality in [1; 3 the result follows for the general case, too. 2 The second result provides a similar formula for the minimal distance between a regular ternary tree and reduced subtrees containing n 2 9 N scenarios. Proposition 3.2 (7/9-solution) Let a regular ternary scenario tree with Then it holds for each n 2 N with 2 Proof: Similarly as in Proposition 3.1 we obtain for all for each subset J of Again we have to show that there exists an index set J such that #J and that the lower bound N n attained with D J . We consider the index set I := fi _ (- k0 denote the tree consisting of all I . Figure 3 illustrates a detail of T r starting at a Scenario Reduction Algorithms in Stochastic Programming 13 r r r r r r @ @ @ @ @ @ @ @ Figure 3: Detail of the subtree T r node at level k 0 1 and ending at level k 2. We obtain for the cardinality of I and J that 9 3 9 N and #J Similarly as in Proposition 3.1 it can be shown that for each j 2 J , there exists an index i 2 I such that it holds that k! 9 - k0 . By considering subsets of J having cardinality in 9 N ], the result follows for the general case, too. 2 Similar results are available under additional assumptions in case of the Euclidean norm instead of the maximum norm (see also [6]). The aim of this section is to report on numerical experience of testing and comparing the algorithms described in Section 2, namely, backward reduction of scenario sets, simultaneous backward reduction, fast forward selection. All algorithms were implemented in C. The test runs were performed on an HP 9000 (780/J280) Compute-Server with 180 MHz frequency and 768 MByte main memory under HP-UX 10.20, i.e., the same conguration as for the numerical tests in [4]. We consider the situation where the function c is dened by c(!; ~ !) and the original discrete probability measure P is given in scenario tree form. More precisely, we use a test battery of three binary and ternary scenario trees, respectively. All test trees are regular and, thus, the results of Section 3 apply. They provide minimal distances of P to reduced measures supported by n scenarios when n is not too small. Example 4.1 (binary scenario tree) (0:5; 0:6; 0:7; 0:9; 1:1; 1:3; 1:6; 1:9; 2:3; 2:7). Figure 4 illustrates the original scenario tree. Proposition 3.1 applies with k N holds for each N= 256 n < N . Example 4.2 (ternary scenario tree) (0:7; 0:9; 1:2; 1:5; 2:6; 3:3). The tree is shown in Figure 5. Proposition 3.2 applies with N holds for each 2N 14 Heitsch, Romisch Example 4.3 (ternary load scenario tree) We consider the scenario tree construction in Section 4 of [4] for the weekly electrical load process of a German power utility (see also [5,8] for a description of a stochastic power management model and its solution by Lagrangian relaxation). The original construction is based on an hourly discretization of the weekly time horizon with branching points at t and on a piecewise linear interpolation between the t k . The corresponding mean shifted tree is illustrated in Figure 6. For a moment, we disregard all non-branching points of the time discretization and consider the corresponding mean shifted tree. The latter tree is a regular ternary scenario tree with denotes the standard deviation of the stochastic load process at time t. Since in this case t increases with increasing t, Proposition 3.2 applies with k it holds that D min N for Finally, it remains to remark that, due to the piecewise linear structure of the scenarios and the choice of the maximum norm for dening c, the minimal distance D min n does not change when including all non-branching points. The original scenario trees of the Examples 4.1{4.3 were reduced to trees containing n scenarios by using all 3 reduction algorithms. The corresponding tables contain the relative accuracy and the running time of each algorithm needed to produce a reduced tree with n scenarios. In addition, the tables provide the (relative) lower bound (15) and the (relative) minimal distance n in percent if available. Here, \relative" always means that the corresponding quantity is divided by the minimal c -distance of P and one of its scenarios endowed with unit mass. In particular, the relative accuracy is dened as the quotient of the c -distance of the original measure P and the reduced measure Qn (having n scenarios) and of the c -distance of P and the rel c denotes the set of scenarios of P and ! i is dened by c c Our numerical experience shows that all algorithms work reasonably well. All algorithms reduce 50% of the scenarios of P in an optimal way. As ex- pected, simultaneous backward reduction and fast forward selection produce more accurate trees than backward reduction of scenario sets at the expense of higher running times. Our results also indicate that fast forward selection is slightly more accurate than simultaneous backward reduction, although both backward reduction variants are sometimes competitive. Fast forward selection works much faster than the implementation of forward selection in Scenario Reduction Algorithms in Stochastic Programming 15 Figure 4: Original binary scenario tree Number Backward of Simultaneous Fast Lower Minimal n of Scenario Sets Backward Forward Bound Distance Scenarios rel c Time rel c Time rel c Time Table 1: Results of binary scenario tree reduction Heitsch, Romisch Figure 5: Original ternary scenario tree Number Backward of Simultaneous Fast Lower Minimal n of Scenario Sets Backward Forward Bound Distance Scenarios rel c Time rel c Time rel c Time Table 2: Results of ternary scenario tree reduction Scenario Reduction Algorithms in Stochastic Programming 17 -50050024 48 72 96 120 144 168 Figure Original load scenario tree Number Backward of Simultaneous Fast Lower Minimal n of Scenario Sets Backward Forward Bound Distance Scenarios rel c Time rel c Time rel c Time Table 3: Results of load scenario tree reduction Heitsch, Romisch [4]. For instance, fast forward selection required 35 seconds to determine a load scenario subtree (Example 4.3) containing 600 scenarios instead of 8149 seconds reported in [4]. Especially, in the case of deeply reduced trees, fast forward selection works very fast and accurately. Furthermore, it turned out that the lower bound is very good (even optimal) for large n, but extremely pessimistic for small n. Another observation is that the reduction of half of the scenarios implies only a loss of about 10% of the relative accuracy. For instance, in case of Example 4.2 it is possible to determine a subtree containing just 6 out of the originally 729 scenarios that still carries about 50% of the relative accuracy. Finally, we take a closer look at the numerical results of the load scenario tree reduction. In particular, we compare the running times of simultaneous backward reduction and fast forward selection in this case. Figure 7 displays10300 100 200 300 400 500 600 Time in seconds Number of scenarios fast forward simultaneous backward Figure 7: Running time for reducing the load scenario tree the running times of both algorithms and shows clearly their opposite algorithmic strategies. It re ects the corresponding theoretical complexity results (Propositions 2.3 and 2.6) and shows that the running time of fast forward selection is smaller if n N(approximately). This conrms again that the forward selection concept is favourable if n is small. Figures 8, 9 and 10 show the reduced load trees with 15 scenarios obtained by all algorithms. The gures display the scenarios with line width proportional to scenario probabilities. Scenario Reduction Algorithms in Stochastic Programming 19 -50050024 48 72 96 120 144 168 Figure 8: Backward reduction / load tree -50050024 48 72 96 120 144 168 Figure 9: Simultaneous backward reduction / load tree -50050024 48 72 96 120 144 168 Figure 10: Fast forward selection / load tree Heitsch, Romisch Acknowledgement This research was partially supported by the BMBF project 03-ROM5B3. Appendix Proof (Theorem 2.1): Let J I := be an arbitrary index set. We set c ij := c(! programming duality implies for any feasible q In particular, we consider It holds ki and I . Hence, we obtain Next, we set u i := min for each i 2 I . Noting that u all for all Hence, we obtain for any feasible q that and the proof is complete. 2 --R Introduction to Algorithms Reduktion von Szenariob in: Handbooks in Operations Research and Management Science Probability Metrics and the Stability of Stochastic Models Theory of Probability and its Applications 28 --TR Introduction to algorithms Integer programming Quantitative Stability in Stochastic Programming

scenario reduction;electrical load;probability metric;scenario tree;stochastic programming

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A statistical approach to case based reasoning, with application to breast cancer data.

Given a large set of problems and their individual solutions case based reasoning seeks to solve a new problem by referring to the solution of that problem which is "most similar" to the new problem. Crucial in case based reasoning is the decision which problem "most closely" matches a given new problem. A new method is proposed for deciding this question. The basic idea is to define a family of distance functions and to use these distance functions as parameters of local averaging regression estimates of the final result. Then that distance function is chosen for which the resulting estimate is optimal with respect to a certain error measure used in regression estimation. The method is illustrated by simulations and applied to breast cancer data.

Introduction Assume that one is interested in the solution of a problem where the latter is described by a number of observable variables. It is not necessary that these variables determine the solution completely, but we assume that there is a correlation between the observable variables and the solution. Furthermore, we assume that a list of problems of the same type is available for which the values of the observable variables and the solutions are known. Instead of the categories problem/solution one may also think of premise/conclusion, cause/effect, current state/final result, or, more concretely, health condition/observed survival time. Case based reasoning seeks to solve a new problem by refering to the solution of that problem which is "most similar" to the new problem. The basic assumption is that the solution of the problem in the data base which is "most similar" to the new problem is close to the unknown solution of the new problem. This is different to rule based reasoning where the knowledge is represented in the rules rather than in a data base built up by previous experience. The phrases case based reasoning and rule based reasoning were coined in the field of artificial intelligence (see, e.g., Menachem and Kolodner (1992)). Crucial in case based reasoning is the decision, which cases "most closely" match a given new case. In this article we propose a new method for deciding this question. The basic idea is to define a family of distance functions, to use these distance functions as parameters of local averaging regression estimates of the final outcome, and to choose that distance function for which the resulting estimate is optimal with respect to a certain error measure used in regression estimation. In this article we apply case based reasoning in the context of prediction of survival times of breast cancer patients treated with different therapies. For each therapy a list of cases is given which includes observed survival time and variables describing the case. Examples for the latter are size of primary tumor, number of affected lymph nodes and menopausal status. These categorial variables are part of the tumor classification system TNM. From each of these lists one selects those cases which are "most similar" to a given new case. Then a physician tries to gain information about an appropriate therapy for the new patient by considering the therapy and the final result A statistical approach to case based reasoning 3 of these cases. A naive approach to decide which cases "most closely" match a new case is to use only those cases which have the same values in all variables as the new case. Unfortunately, if the number of observable variables is large, then there will be usually none or only very few such cases. Therefore this naive approach is in general not useful. Another approach was used in the context of predicting survival times of breast cancer patients by Mariuzzi et al. (1997). There the range of each observable variable is divided into (four) quartiles. The values of the variables are coded by 1, 2, 3 or 4 according to the quartile to which they belong to. Then the distance of observable variables (x (1) which are coded by (i l ), is defined by d l and those cases are chosen which are (with respect to this distance function) most closely to the new case. The main drawback of this distance is that it does not reflect a possibly different influence of each individual variable on the final result (for instance, the values of x (1) might influence the survival time much more than the value of x (2) ). In this article we propose a new method to determine that cases of a data base which "most closely" match the new case. In Section 2 we give a short introduction to nonparametric regression. The proposed method is described in detail in Section 3. It is illustrated with some simulated data in Section 4 and applied to breast cancer data in Section 5. 2. Nonparametric regression independent identically distributed random variables, where X is R d -valued, Y is real-valued and EY 2 ! 1. In regression analysis one wants to predict Y after having observed X, i.e., one wants to find a function m : R d ! R such that m (X) is "close to" Y . If closeness is measured by the mean squared error, one is interested in a function m which 4 J. Dippon et al. minimizes the so-called L 2 risk Efjm Introduce the regression function xg. For an arbitrary (measurable) function R one has Z where - denotes the distribution of X. Therefore m and the L 2 risk of an arbitrary function f is close to the optimal value if and only if the (squared) L 2 error The optimal predictor m depends on the distribution of (X; Y ). In applications this distribution will be usually unknown, thus m will be unknown, too. But often it is possible to observe independent copies Based on the data the nonparametric regression problem asks for an estimate mn of m such that the L 2 error R jmn 2.1. Local averaging estimates The response variables Y i can be rewritten as 0: Hence one can consider the Y i 's as the sum of the function m evaluated at X i and a random error with zero mean. This motivates to construct an estimate of m(x) by averaging over those Y i 's where X i is close to x (hopefully then m(X i ) is close to m(x) and the average of the ffl i is close to zero). Such an local averaging estimate can be represented as are nonnegative weights which sum up to one. A statistical approach to case based reasoning 5 The most popular local averaging estimate is the Nadaraya-Watson kernel estimate (Nadaraya (1964) and Watson (1964)) with weights hn hn R is the so-called kernel function, is the so-called bandwidth, and for hn Often one uses spherical symmetric kernels K which satisfy for a univariate kernel function R! R, which we denote again by K, where jxj is the euclidean norm of x. For such kernel functions one gets hn hn Usually kernel functions are chosen such that K(0) ? 0, If the so-called naive kernel are used, then m(x) is estimated by the average of those Y i where X i is in a ball with radius h (1) and center x. For more general K : R! R+ , e.g. the Gaussian kernel one uses a weighted average of the Y i 's as an estimate of m(x) where more weight is given to those are close to x. 2.2. Choice of bandwidth The choice of the bandwidth hn is crucial for the kernel estimate. If the components of hn are too small, then only very few of the Y i 's have a weight not close to zero and the estimate is determined by only these very few Y i 's which induces a high variance in the estimate. On the other hand, if the components of hn are too big, then there will be Y i 's which are far away from x and which have a big weight. But if X i is far away from x then even if m is smooth m(X i ) might be far away from m(x) which induces again a large error into the estimate. 6 J. Dippon et al. For the quality of the estimate it is important to apply a method which chooses hn close to the optimal value (which depends on the usually unknown distribution of (X; Y using only the data Dn . A well known method which tries to do this is cross-validation, which will be described in the sequel. For further information see H-ardle (1990), e.g. Recall that the aim in nonparametric regression is to construct mn such that the L 2 risk is small. So the bandwidth of a kernel estimate should be chosen such that (2) is minimal compared to other choices of the bandwidth. In an application this is not possible because (2) depends on the unknown distribution of (X; Y ). What we will propose is to estimate the L 2 risk (2) and to choose the bandwidth by minimizing the estimated L 2 risk. For a fixed function f : R d ! R the L 2 risk can be estimated by the so-called empirical If we use (3) with (which depends on Dn ) to estimate (2), then this so-called resubstitution estimate is a too optimistic estimate of the L 2 risk and minimization of it leads to estimates which are well adapted to Dn but which are not suitable to predict new data independent from Dn . This can be avoided by splitting the data Dn into two parts, learning and testing data, by computing the kernel estimate with the learning data and by choosing the bandwidth such that the empirical risk on the testing data is minimal. The resulting bandwidth depends on the way the data is splitted. This is a drawback if one wants to use this bandwidth for a kernel estimate which uses the whole data Dn (and thus has nothing to do with the way the data is splitted). It can be avoided by repeating this procedure for several (e.g. randomly chosen) splits of the sample and by choosing the bandwidth such that the average empirical L 2 risk on the testing data is minimal. For k-fold cross-validation the splits are chosen in a special deterministic way. Let 1 - k - n. For notational simplicity we assume that n k is an integer. Divide the data into k groups of equal A statistical approach to case based reasoning 7 size n k and denote the set consisting of all groups except the lth one by D n;l : For each data set D n;l and bandwidth h 2 R d construct a kernel estimate mn\Gamma n ;h (x; D n;l k l+1;:::;ng K k l+1;:::;ng K Choose the bandwidth such thatk l=1n l is minimal and use this bandwidth h as bandwidth hn for the kernel estimate (1). n-fold cross-validation is denoted as cross-validation. In this case D n;l is the whole sample is minimized with respect to h. 2.3. Curse of dimensionality If d is large, then estimating a regression function is especially difficult. The reason for this is that in this case it is in general not possible to densely pack the space of X with finitely many sample points, even if the sample size n is very large. This fact is often referred to as curse of dimensionality, a phrase which is due to Bellman (1961). It will be illustrated by an example; further examples can be found in Friedman (1994). independent identically distributed R d -valued random variables with X uniformly distributed in the hypercube [0; 1] d . Consider the expected supremum norm distance of X to its closest neighbor in X 1 ae min oe (where k(x For ae min oe 8 J. Dippon et al. thus Z 1P ae min oe For instance, d1 (10; 1000) - 0:22 and d1 (20; 10000) - 0:28. Thus, for d large, even for a large sample size n, the supremum norm distance of X to its closest neighbor in the sample is not close to zero (observe that the supremum norm distance between any two points in the above sample is always less than or equal to one). 3. Basic idea In this section we will apply the concepts of nonparametric regression introduced in the previous section to case based reasoning. To do this, we will consider the prediction of the final result in case based reasoning as a regression estimation problem. Hence we assume that the final result is a real valued random variable and that the observable variables are components of a R d -valued random variable In applications often some of the observable variables are categorical random variables. How to handle such variables will be described in Section 5 below. Recall that our aim in case based reasoning is to determine those cases of a given list which "most closely" match a new case. This can be done by defining a distance function which determines a distance d(x; x i ) between two cases with observable variables x and x i . Given such a distance function one can define the k "most similar" cases to a new case as those k cases among whose distances to the new case x belongs to the k smallest occuring distances. The basic idea to determine such a distance function is to define a regression estimate depending on this distance function and then to choose that distance function which minimizes (an estimate) of the L 2 risk of the resulting regression estimate. As regression estimates we use kernel estimates introduced in the previous section. For define a distance function dh : R d \Theta R d ! R+ by dh =@ d A A statistical approach to case based reasoning 9 Then the kernel estimate can be written as so the family of distance functions \Psi corresponds to a family of regresion estimates is the kernel function. In our simulations in Section 4 and in our application in Section 5 we use the Gaussian kernel Our aim is to choose the distance function such that the corresponding regression estimate has minimal L 2 risk. As already explained in the previous section, this is not possible because the L 2 risk depends on the unknown distribution of (X; Y ). What we do instead is to estimate the L 2 risk by cross-validation and to choose the distance function dh (h 2 R d which the estimated L 2 risk of m n;h is minimal. Our method can be summarized as follows: Estimate the L 2 risk of a kernel estimate m n;h by cross-validation, determine h 2 R d such that this estimated L 2 risk is minimal (compared to all other choices h 2 R d use the corresponding distance function dh . 3.1. Subset selection We have seen in the previous section that estimating the regression function is very difficult if the dimension d of X is large. This problem occurs for every estimate, hence for the kernel estimate which we use in this paper as well. The only way to handle this problem is to make assumptions on the underlying distibution (e.g. to assume that the regression function is additive, cf. Stone (1985), Stone (1994). In this paper we assume that, even if d is large (say 20), the regression function mainly depends on a few (say 3 to 6) of the components of X. If this assumption holds, then a distance function can be determined by applying our method to this small subset of the observable variables (which leads to a regression estimation problem with small dimension of X). Of course, in applications we will not know on which of the observable variables the regression function mainly depends. In order to check this we consider all (or if d is too big all small) subsets of the observable variables, apply our method to each of these subsets and choose that subset for which the via cross-validation estimated L 2 risk of the "optimal" estimate is minimal. J. Dippon et al. 3.2. Simplification of computation For each subset of the observable variables and for each variable of this subset the method described above requires computation of a vector of scaling factors which leads to a multivariate minimization problem. In order to avoid solving many such minimization problems, we use the following simplification: In a first step determine for each of the d observable variables a univariate scaling n by computing (as described above) the "optimal" bandwidth of a univariate kernel estimate fitting the data (X (i) This yields scaling factors h (1) In a second step we choose for each subset fi d g of dg the distance function h@ d on R d , such that the (via cross-validation) estimated L 2 risk of is minimal with respect to h 2 R+ . Finally we choose that subset fi d g of which the (via cross-validation) estimated L 2 risk of the corresponding optimal kernel estimate is minimal. Observe that minimization of the estimated L 2 risk of m n;h (h 2 R+ ) is only a univariate minimization problem. 3.3. Robustification of the bandwidth estimation It is well known that bandwidth selection by cross-validation is often highly variable (see Simonoff (1996)). Some authors suggested alternatives such as plug-in methods and claimed that these are superior to ordinary cross-validation. But Loader argues that no theoretical results support these claims and that the plug-in methods itself depends sensitively on a pilot estimate of the bandwidth (see Ch. 10 in Loader (1999)). Let m n;h be a kernel estimate with bandwith h. In order to robustify the cross-validation bandwidth selection given by A statistical approach to case based reasoning 11 with CV function cv(h) as defined in (4), we suggest the following heuristic rule: where and is specified below. The idea behind the rule is the following. Often the graph of the CV function resembles a valley with possibly several local minima at the bottom or with a flat bottom where the position of the global minimum seems to be accidental. Then an estimate of the middle of the valley appears to be a less variable measure. Choose two parameters We define a level r p;q in dependence of For the regression estimate equals the mean of the responses fY coincides with the empirical variance 1 Yng. In our simulations we first used but for this choice it might happen that r p;q ? sup h?h0 cv(h). In order to avoid this we set r p;q := cv(h For values of p and q we used in this paper. Our robustification implies that we possibly dispense with explaining a fraction of the empirical variance of fY g. This doesn't seem to be too harmful, since we are more interested in finding similarity neighborhoods than in prediction. J. Dippon et al. 4. Application to Simulated Data 4.1. Simulated Model Let us consider the defined by where a 2 (0; 1] is a fixed constant, and the model with independent random variables X - U ([0; 1). Apparently, the third component X (3) has no influence on Y . If a is small, the first component X (1) of the random vector X has a larger influence on the response Y than the second component X (2) . In other words, for a small constant a the regression function m is almost constant with respect to X (2) ; in any case m is constant with respect to X (3) (Figure 1). We simulate realizations of (X independent copies of (X; Y ). Then, as described in Subsections 3.2 and 3.3, for each i 2 f1; 2; 3g a robustified estimate e h (i) 0 of the "optimal" bandwidth h (i) 0 of the univariate kernel estimator based on the sample computed by cross-validation. If a is significantly less than 1, then we expect the relation 1. Furthermore, the corresponding estimated L 2 risks should be in ascending order, too (Figure 2). For Our model selection procedure defines different distance functions on [0; 1] 3 as given by (5). For each distance function we compute the minimal estimated L 2 risk of the multivariate kernel estimator (6) with respect to h ? 0 and consider that distance function as best for which the estimated L 2 risk is minimal. If inclusion of a further component reduces the estimated L 2 risk only "slightly", the distance function with the smaller set of components but slightly larger corresponding estimated L 2 risk will be preferred. We use the resulting distance functions to define neighborhoods A statistical approach to case based reasoning 130.20.61 Component 1 of X0.20.61 Component 2 of X Fig. 1. Graph of the regression function m for (considered as function of the first two components of the predictor variable X) and scatter plot of simulated realizations of (X For each ffi ? 0 we consider every point in N (x; ffi ) to be more similar to x than every point outside of N (x; ffi ). In this sense those realizations of X are the k most similar ones to x whose distances belongs to the k smallest among 4.2. Results of the Simulations For each pair (a; n) of parameters a 2 f0:3; 0:5; 1g and n 2 f100; 200g the simulation is repeated 20 times but with different seeds of the generator producing the (pseudo) random numbers. Below we discuss the results of the simulations performed with parameter set (a; n) = (0:5; 100) in some detail. As suggested in Section 3 we compute bandwidths e h (1) 0 . In all but one of the twenty runs the procedure found 0 . In most cases the CV curves and the univariate regression estimates corresponding to the minimal value of the CV curve look as in 14 J. Dippon et al. Figure 2. Furthermore, the optimal value h (i)and the robustified value e h (i)differ only slightly. However, in some cases the situation appears as in Figure 3. There the smooths related to e h (i)seem to be appropriate. Only in one of the twenty runs the method fails to detect that the first component has a stronger influence on the function values of m than the second component, see Figure 4. To compare multivariate kernel estimates (6) with distance functions selecting different subsets of prediction variables we compute the minimum cv(h 0 of the estimate cv(h) (given by (4)) of the L 2 risk of (6) and compare the ratios cv(h 0 )=cv(1). Table 1 shows that for each run the subset selection procedure favors fX (1) ; X (2) g or fX (1) g. But in cases is preferred the gain is too small to be significant. Hence in all 20 runs we choose subset fX (1) ; X (2) g. The relation of e h (1) determines the geometry of the neighborhoods which will be used to characterize "similar cases". For the parameter pair (a; n) = (0:5; 100) the ratios of the the computed bandwidths e h (1) turned out to be 2, 0.16, 0.54, 0.45, 0.60, 0.50, 0.23, 0.57, 0.34, 0.76 From the second row of Table 2 one can extract minimum, maximum, median and interquartile range of this ratios. To visualize the variability of the resulting distance functions neighborhoods projected on the first two components are plotted around center 0:5). The parameter chosen in such a way that the area of these sets equals 1=10, see Figure 5. Comparing the results for sample sizes indicates that larger sample sizes lead to less variable neighborhoods. A statistical approach to case based reasoning 15 y Component 2 y y Fig. 2. Determination of univariate bandwidths e h (1); e h (2); e h (3)for the 6th simulation run. Left column: Cross-validated L2 risk of the regression estimate of E(Y jX (i) ) depending on the bandwidth h (i) . The optimal bandwidth h (j)and the robustified bandwidth e h (j)are indicated by tick marks (if within the range of the x- axis). Right column: Univariate kernel estimate of E(Y jX (i) ) using the optimal bandwidth h (i)(dotted) and the robustified bandwidth e h (i)(solid), i 2 f1; 2; 3g. The finding that the curves related to h (i)and e h (i)can be hardly distinguished is true for most of the simulated samples. J. Dippon et al. Component 1 y Component 2 y Component 3 y Fig. 3. Determination of univariate bandwidths e h (1); e h (2); e h (3)for the 3th simulation run. Left column: Cross-validated L2 risk of the regression estimate of E(Y jX (i) ) depending on the bandwidth h (i) . The optimal bandwidth h (j)and the robustified bandwidth e h (j)are indicated by tick marks (if within the range of the x- axis). Right column: Univariate kernel estimate of E(Y jX (i) ) using the optimal bandwidth h (i)(dotted) and the robustified bandwidth e h (i)(solid), i 2 f1; 2; 3g. Despite the fact that the bandwidths h (i)minimizes the CV error criterion, these bandwidths are not useful to compare the dependency of the random variable Y on . For this sample the robustified bandwidth e h (i)seems to be more appropriate. A statistical approach to case based reasoning 17 Component 1 y Component 2 y y Fig. 4. Determination of univariate bandwidths e h (1); e h (2); e h (3)for the 11th simulation run. Left column: Cross-validated L2 risk of the regression estimate of E(Y jX (i) ) depending on the bandwidth h (i) . The optimal bandwidth h (j)and the robustified bandwidth e h (j)are indicated by tick marks (if within the range of the x-axis). Right column: Univariate kernel estimate of E(Y jX (i) ) using the optimal bandwidth h (i)(dotted) and the robustified bandwidth e h (i)(solid), i 2 f1; 2; 3g. This is the only of all 20 samples for which the suggested method fails to detect that the first component has a stronger influence on the function values of m than the second component. J. Dippon et al. Table 1. Ratio cv(h0)=cv(1) of estimated L2 risk of the multivariate kernel estimate with the specified set of components. Parameters of the simulated model were a Run Selected Subset 9 0.40 0.97 1.00 0.19 0.43 0.98 0.23 A statistical approach to case based reasoning 19 Table 2. Statistics of the ratios e h (1)= e h (2)each computed from 20 simulation runs. Parameters Statistics of e h (1)= e h (2)a n Min. 1st Quart. Median Mean 3rd Quart. Max. 1.0 200 0.14 0.93 1.10 1.02 1.16 1.43 Component 1 of X Componentof Component 1 of X Componentof Component 1 of X Componentof Component 1 of X Componentof Component 1 of X Componentof Component 1 of X Componentof Fig. 5. These plot show level lines of the regression function m for parameter a 2 f0:3; 0:5; 1g and (cutted) ellipsoidal neighborhoods around the point (0:5; 0:5). Each neighborhood set is computed from one of the simulated samples of size contains 1/10th of the unit square. J. Dippon et al. 5. Application to Breast Cancer Data We compute a distance function on the space of covariates as suggested in Section 3 to determine for a given new breast cancer patient (with unknown survival time) "similar" cases among patients in a database with known (censored) survival time. The data were made available by the Robert- Bosch-Krankenhaus in Stuttgart, Germany. They are collected between the years 1987 and 1991 with a follow-up of 80 months. Each of the cases is described by a 10-dimensional parameter vector. We consider the nine first parameters as predictor variables. It includes age at diagnosis AGE (in years), menopause status MS (which equals 1 and 2 for pre and post menopause, respectively), histological type of the breast cancer HT (with values in number of affected lymph nodes PN (grouped into four classes with values in size PT (grouped into four classes occurence of metastases PM (with values 0 and 1 for no and yes, respectively), grading of tumor GR (with values in f1; 2; 3g), estrogene status ES (with values 1 and 2 for positive and negative, resp.) and progesterone status PS (with values 1 and 2 for positive and negative, resp. The last component of the parameter vector describes the observed (censored) survival time OST (in years) and is considered as reponse variable. In many cases the actual survival time can't be observed, since the patient is still alive after the end of the study or because of the patient's withdrawal from the study. Hence the observed survival time can be understood as the minimum of the actual survival time T (time elapsed from date of diagnosis to date of death) and a censoring time C (time between date of diagnosis and a date at which the patient is known to be alive). In our approach we estimated the censored survival time instead of the more complicated and often unknown uncensored survival time in order to simplify the problem. We hope that this simplification doesn't affect our result too much, because we used the estimate to construct similarity neighborhoods rather than to estimate the survival time as a function of a covariate vector (as to the latter compare Carbonez et al. (1995)). AGE and OST are continuously distributed r.v.'s, PT , PN , GR are ordered categorical r.v.'s, and MS, HT , PM , ES and PS are nominal r.v.'s. Values in the predictor variables are allowed A statistical approach to case based reasoning 21 Table 3. Robustified bandwidths e h (j)for the the univariate regression problems. The products 1= e h (j)times the spread of component j allow to compare the maximal influence of component j on the distance function. component AGE MS HT PT PN PM GR ES PS to be missing. Assume that is the covariate vector of a new patient and z = (z is the covariate of a patient in the database fz related to a continuous or ordered categorical r.v., we define the distance function d (j) h by d (j) z (j) are not missing k and z (j) l are not missingg otherwise is related to a nominal r.v., we define the distance function d (j) by d (j) z (j) and both x (j) and z (j) are not missing z (j) or x (j) or z (j) is missing As described in Section 3, for each j 2 we compute by cross-validation "optimal" 0 for the univariate regression estimates n;h where y i is a realization of the response variable Y i . Figures 6 and 7 show the CV function cv and the regression estimates related to the robustified estimate e h (j) 0 as displayed in Table 3. Now, on the range X of the covariate variable X we define for each l 2 for each 22 J. Dippon et al. 50 100 150 20011.82AGE Observed Survival Time Observed Survival Time 1.0 1.2 1.4 1.6 1.8 2.02610 12.6 . Observed Survival Time Observed Survival Time PN Observed Survival Time A statistical approach to case based reasoning 23 Observed Survival Time Observed Survival Time Observed Survival Time 1.0 1.2 1.4 1.6 1.8 2.02610h PS Observed Survival Time J. Dippon et al. Table 4. For each number l 2 of covariates that subset is displayed which possesses the smallest estimated L2 risk. l Best subset cv(h l 6 AGE, PT, PN, PM, ES, PS 0.7894 7 AGE, HT, PT, PN, PM, ES, PS 0.7894 8 AGE, MS, HT, PT, PN, PM, ES, PS 0.7932 9 AGE, MS, HT, PT, PN, PM, GR, ES, PS 0.8281 subset dg the distance function d where h is a fixed positive number, and we obtain the multivariate regression estimate compute its CV function cv as a function of h ? 0, and determine that subset J l := with smallest global minimum cv(h l ) of the related CV function. Table 4 shows the results of this model selection step. Together with Table 4 the scree plot in Figure 8 suggests to choose that distance function which includes the variables PN , PM , ES. By adding a fourth component the relative improvement of the ratio cv(h 4 )=cv(1) compared to cv(h 3 )=cv(1) is less than 0:05. Hence, we propose to use the distance d, defined by d (PN) 0:71 d (PM) 1:42 d (ES) 0:79 Now, for a given new case with covariate x compute r i := d(x; z i the z 0 according to increasing values of the r 0 s: fz g. Then, for fixed k 2 A statistical approach to case based reasoning 25 Number of Components Fig. 8. Scree plot of (l; cv(h l 9g. the subset fz of the database consists of that k cases which are "most similar" to the new case with respect to the distance d. Their history logs may help the physician in choosing an appropriate therapy for the new patient. 6. Discussion As it is often the case with nonparametric estimators in a multivariate setting, the suggested method has its deficiencies and limitations, too. For instance, assume that the random vector 26 J. Dippon et al. takes on the values (0; 0), (1; 0), (0; 1) and (1; 1) each with probability 1=4, and let Y be defined by 0, if x 2 f(0; 0); (1; 1)g, and by 1 otherwise. Then E(Y jX (1) but m is far from being a constant. Hence, in this case, the proposed method will fail. Adjusting the so-called resubstitution estimate of the L 2 risk by penalizing functions leads to various other bandwidth selection procedures such as generalized cross-validation, Shibata's model selector, Akaike's information criterion, Akaike's finite prediction error and Rice's T (see, e.g., H-ardle (1990), H-ardle (1991). These may be used instead of the least squares cross-validation criterion. Further research should improve the regression estimate. For instance, dimension reduction as handled in projection pursuit regression is adapted to find structures in linear subspaces of the covariate space. Additionally, the fact of censored observation times should be taken into account. In our application concerning survival times of breast cancer patients the influence of the chosen therapy was ignored. This might lead to an underestimation of the influence of known predictor variables. The reason is that the choice of the therapy usually depends on some of the predictor variables and that the therapy has an influence on the survival time. There isn't a problem, if one has data for which the choice of the therapy is independent of the predictor variables. Such data does not exist for the predictors we used. But this requirement is fulfilled for any new predictor which was unknown in the past or was considered to be unimportant and therefore was not used for choosing the therapy. A distance function which is additionally based on such a new predictor allows to judge the importance of the new predictor. 7. Acknowledgements This research was partly supported by the Robert Bosch Foundation, Stuttgart. We are grateful to Prof. H. Walk, Stuttgart, for stimulating discussions. Readers wishing to obtain the breast cancer data or the S-Plus code of the algorithms should contact the authors. A statistical approach to case based reasoning 27 --R Adaptive Control Processes. An overview of predictive learning and function approximation. Theory and Pattern Recognition Applications. Applied Nonparametric Regression. Smoothing Techniques with Implementations in S. Local Regression and Likelihood. On estimating regression. Smoothing Methods in Statistics. Additive regression and other nonparametric models. The use of polynomial splines and their tensor products in multivariate function estimation. Smooth regression analysis. --TR Random approximations to some measures of accuracy in nonparametric curve estimation

robustness;case based reasoning;nonparametric multivariate regression estimation;kernel estimation;band-width selection

608626

Making Complex Articulated Agents Dance.

We discuss the tradeoffs involved in control of complex articulated agents, and present three implemented controllers for a complex task: a physically-based humanoid torso dancing the Macarena. The three controllers are drawn from animation, biological models, and robotics, and illustrate the issues of joint-space vs. Cartesian space task specification and implementation. We evaluate the controllers along several qualitative and quantitative dimensions, considering naturalness of movement and controller flexibility. Finally, we propose a general combination approach to control, aimed at utilizing the strengths of each alternative within a general framework for addressing complex motor control of articulated agents.

Introduction Control of humanoid agents, dynamically simulated or physical, is an extremely difficult problem due to the high dimensionality of the control space, i.e., the many degrees of freedom (DOF) and the redundancy of the system. In robotics, methods have been developed for simpler manipulators and have been gradually scaled up to more complex arms (Paul 1981, Brady, Hollerbach, Johnson, Lozano-Perez & Mason 1982) and recently to physical human-like arms (Schaal 1997, Williamson 1996). Anthropomorphic control has also found an application area in realistic, physically-based animation, where dynamic simulations of human characters, involving realistic physical models, matches the complexity of the robotics problem (Pai 1990, Hodgins, Wooten, Brogan & O'Brien 1995, van de Panne & Lamouret 1995). In this paper, we present three controller implementations to address the tradeoffs involved in different approaches to articulated control, including joint-space control and Cartesian control, and their relevance to the different application areas, including biological models, robotics, and animation. The three controllers are implemented on a physics-based humanoid torso simulation, and applied to the task of performing a continuous sequence of smooth movements. The movement sequence chosen is the popular dance "Macarena", which provides a non-trivial, well-defined task M. Matari'c et al. for comparison. The particular controllers are: joint-space torque control, joint-space force-field control, and Cartesian impedance control. The paper describes each approach, and compares its performance with human data. The speed and smoothness of the resulting motions are evaluated, along with other qualitative and quantitative measures. The rest of the paper is organized as follows. Section 2 gives the relevant background and related work in manipulator control, including biological, robotics, and animation issues. Section 3 describes Adonis, our humanoid simulation test bed. Section 4 gives a detailed specification of our task. Section 5 describes a joint-space torque controller and Section 6 describes the joint-space force-field-based controller. Section 7 contrasts those methods with a Cartesian impedance controller. Section 8 presents a detailed performance analysis and comparison of the methods along several criteria including qualitative and quantitative naturalness of appearance and controller use and flexibility. Section 9 describes our continued work toward a combination approach to articulated control, and Section 10 concludes the paper. 2. Background and Related Work Computer animation and robotics are two primary areas of research into motion for artificial agents. This section briefly reviews each, and then introduces some biological inspiration for the types of control we will discuss. 2.1. Control in Robotics In robotics, manipulator control has been largely, but not exclusively, addressed for point-to-point reaching. Position control of manipulators is a mature area of research offering a variety of standard techniques. A review of robotics methods can be found in Craig (1989), Paul (1981), and Brady et al. (1982). Solving the inverse kinematics (IK), or finding the relevant joint angles to obtain a desired end-point position and orientation for a given manipulator, is a difficult task, especially when the structure is redundant (Baker & Wampler II 1988). Rather than solving the inverse kinematics analytically, some techniques linearize the system kinematics about the operating point, using either the Jacobian (Salisbury 1980), or the inverse Jacobian (Whitney 1969) to achieve position control. The uses of the pseudo-inverse of the Jacobian for redundant systems has also been explored (Klein Huang 1983). Control methods which were originally used for force control such as hybrid position/force control (Raibert & Craig 1981), inspired work on stiffness control (Salisbury 1980) and the more general impedance control (Hogan 1985) which can be used to control the end-point position (see Section 7). Nearly all of these techniques have been augmented to include models of the robot's dynamics in order to improve the accuracy of control. The most common example is the computed torque method, where the inverse dynamics of the manipulator are solved to provide feed-forward torques during a motion (Luh, Walker & Paul 1980). In addition, learning methods, using a variety of techniques (neural networks, fuzzy logic, adaptive control, etc.) have also been explored and continue to be applied to these problems (Atkeson 1989, Schaal & Atkeson 1994, Slotine & Li 1991, Jordan & Rumelhart 1992). 2.2. Control in Computer Animation In computer graphics, 3D character animation has traditionally been created by hand, but recent- ly, physical modeling has been used to automatically generate realistic motion. Current techniques for physical modeling can be classified by their level of automation; some methods minimize user-specified constraints with an automatic solver while others rely on controllers that require stronger KLUWER STYLE FILE 3 user intervention. For example, Witkin & Kass (1988) presented a constraint-based approach with specified start and end conditions that generated motion containing characteristics such as anticipation and determination. Cohen (1992) extended this approach with higher DOF systems and more complex constraints. Ngo & Marks (1993) introduced a constraint approach to creating behaviors automatically using genetic algorithms. Hand-tuned control of dynamic simulations has been applied successfully to more complex systems such as articulated full-body human figures. Dynamic simulation has been used to generate graphical motion by applying dynamics to physically-based models and using forward integration. Simulation ensures physically plausible motion by enforcing the laws of physics. Pai (1990) simulated walking gaits, drawing strongly from robotics work. His torso and legs use a controller based on high-level time-varying constraints. Raibert & Hodgins (1991) demonstrated rigid body dynamic simulations of legged creatures. Their hand-tuned controllers consist of state machines that cycle through rule-based constraints to perform different gaits. Hodgins et al. (1995) extended this work to human characters, suggesting a toolbox of techniques for controlling articulated human-like systems to generate athletic behaviors such as 3D running, diving, and bicycling. van de Panne used search techniques to find balancing controllers for human-like character locomotion, aiming at more automatic control of such simulated agents. Other methods for generating animation automatically exist as well, including motion capture and procedural animation, but are not as relevant to the controller work presented here. For a more complete review of control in computer animation, see Badler, Barsky & Zeltzer (1991). 2.3. Control with Biological Motivation The flexibility and efficiency of biological motion provides a desirable model for complex agent control. Our work is inspired by a specific principle derived from evidence in neuroscience. Mussa-Ivaldi Giszter (1992), Giszter, Mussa-Ivaldi & Bizzi (1993) and related work on spinalized frogs and rats suggests the existence of force-field motor primitives that converge to single equilibrium points and produce high-level behaviors such as reaching and wiping. When a particular field is activated, the frog's leg executes a behavior and comes to rest at a position that corresponds to the equilibrium point; when two or more fields are activated, either a linear superposition of the fields is obtained, or one of the fields dominates (Mussa-Ivaldi, Giszter & Bizzi 1994). This suggests an elegant organizational principle for motor control, in which entire behaviors are coded with low-level force-fields, and may be combined into higher-level, more complex behaviors. The idea of supplying an agent with a collection of basis behaviors or primitives representing force-fields, and combining those into a general repertoire for complex motion, is very appealing. Our previous work (Matari'c 1995, Matari'c 1997), inspired by the same biological results, has successfully applied the idea of basis behaviors to control of planar mobile agents/robots. This paper extends the notion to agents with more DOF. The work most similar to ours was performed by Williamson (1996) and Marjanovi'c, Scassellati & Williamson (1996), who presented a controller for reaching with a 6-DOF robot arm, based on the same biological evidence. The system used superposition to interpolate between three reaching primitives, and one resting posture. Another inspiration comes from psychophysical data describing what people fixate on when observing human movement. Matari'c & Pomplun (1998) and Matari'c & Pomplun (1997) demonstrate that when presented with videos of human finger, hand, and arm movements, observers focus on the hand, yet when asked to imitate the movements, subjects are able to reconstruct complete trajectories (even for unnatural movements involving multiple DOF) in spite of having attended to the end-point. This could suggest some form of internal models of complete behaviors or primitives for movement, which effectively encapsulate the details of low-level control. Given an appropriately designed motor controller, tasks could be specified largely by end-point positions and a few addi- 4 M. Matari'c et al. tional constraints, and the controller could generate the appropriate corresponding postures and trajectories. 3. Adonis: The Dynamic Humanoid Torso Simulation Our chosen implementation test bed, Adonis, is a rigid-body simulation of a human torso, with static graphical legs (Figure 1), consisting of eight rigid links connected with revolute joints of one and three DOF, totaling 20 DOF. The dynamic model for Adonis was created by using methods described in Hodgins et al. (1995). Mass and moment-of-inertia information is generated from the graphical body parts and human density estimates. Equations of motion are calculated using a commercial solver, SD/Fast (Hollars, Rosenthal & Sherman 1991). The simulation acts under gravity, accepts other external forces from the environment. No collision detection, with itself or its environment, or joint limits are used in the described implementations; we have implemented these extensions in subsequent work. 8 Rigid Body Sections 20 Degrees of Freedom Wrist Wrist Waist Y Z 3 DOF Neck 8 Rigid Body Sections 20 Degrees of Freedom Wrist Wrist Waist Y Z 3 DOF Neck Figure 1. The Adonis dynamic simulation test bed consisting of eight rigid links connected with revolute joints of one and three DOF, totaling 20 DOF. Adonis is particularly well-suited for testing and comparing different motor control strategies; the simulation is fairly stable and the static ground alleviates the need for explicit balance control. In addition, virtual external forces may be applied to the end-points without explicit calculation of the inverse kinematics (IK) of the arms. This, in turn, enables us to implement and evaluate experimental controllers for human-like movement more easily, while having the simulation software handle the issues of IK and dynamics. The next section introduces the task used to compare different control approaches on Adonis. 4. Task Specification Natural, goal-driven movement relies on precise specification and coordination, and realistic con- straints. As a test task should be challenging to control but familiar enough to evaluate, we chose KLUWER STYLE FILE 5 the Macarena, a popular dance which involves a sequence of coordinated movements that constitute natural sub-tasks. We used a verbal description of the Macarena, found on the Web at http://www.radiopro.com/macarena.html, and aimed at teaching people the dance. Omitting the hip and whole-body sub-tasks at the end, the description is given in Table I. Table I. The 12 sub-tasks of the Macarena. 1. Extend right arm straight out, palm down 2. Extend left arm straight out, palm down 3. Rotate right hand (palm up) 4. Rotate left hand (palm up) 5. Touch right hand to top of your left shoulder 6. Touch left hand to top of your right shoulder 7. Touch right hand to the back of your head 8. Touch left hand to the back of your head 9. Touch right hand to the left side of your ribs 10. Touch left hand to the right side of your ribs 11. Move right hand to your right hip 12. Move left hand to your left hip This description, given as a set of sub-tasks, was used directly as the formal specification of the task. No task-level planning or sequencing was necessary because the order is provided by the dance specification. It is interesting that the individual sub-tasks are not specified in a consistent frame of reference. The first four deal with a defined posture of the whole arm, perhaps best expressed in joint angles, while the rest define the hand position, and are thus better described in an ego-centric Cartesian reference frame. As mentioned above (Section 2), people watching movement do not appear to pay active attention to the whole arm, but rather focus on the hand. However, hand position alone does not sufficiently constrain the rest of the arm, whose other joints also require specification; thus a mixture of coordinate frames is needed. This type of heterogeneous task specification is common in natural language descriptions, and control systems must satisfy each of the different goals regardless of the underlying representation. To address the issue of controller representation, we used the same Macarena specification to implement three different alternatives, described next. 5. The Joint-Space PD-Servo Approach Joint-space controllers command torques for all actuated joints in a manipulator, and have been used successfully as low-level controllers to generate behaviors for a variety of systems (Pai 1990, Raibert Hodgins 1991, Hodgins et al. 1995, van de Panne & Lamouret 1995). We implemented the Macarena by calculating the torques for each joint as a function of angular position and velocity errors between the feedback state and desired state, i.e., by using a hand-tuned PD-servo controller: actual where k is the stiffness of the joint, k d the damping, ' desired ; ' desired are the desired angles and velocities for the joints, and ' actual ; ' actual are the actual angles and velocities. To generate the Macarena controller, the desired angles used for the feedback error are interpolated from hand-picked target postures. The postures are derived from the task specification, each corresponding to one of the 12 sub-tasks enumerated in Section 4 above. Intermediate postures between sub-tasks were used as via points to help guide the joint trajectories through difficult tran- sitions. For example, a via point was needed for swinging the hands around the head to prevent 6 M. Matari'c et al. a direct yet unacceptable path through the head. The incremental desired angles use a spline to smoothly interpolate between the postures and via points. Gains for the PD-servo are chosen by hand and remain constant through the whole Macarena. The PD-servo approach allows direct control of each actuated joint in the system, giving the user local control of the details of each behavior. However, the controller in turn requires a complete set of desired angles at all times. Specifying that information can be tedious, especially for joints such as the neck that are less important to the behavior being generated. Interpolating between postures is a reasonable method for reducing the required amount of information. The control of actuated joints may be individually modified using their respective desired angles, thus allowing localized control over the generated motion. All desired postures are specified as a set of angles in joint-space. In the Macarena, position constraints such as "hands behind the head", can be satisfied with user-level feedback. However, precise Cartesian space constraints, like "finger on the tip of the nose", would be difficult to control with hand-tuning using joint-space errors directly. For these cases an inverse kinematics solver could be used to generate desired angles from position constraints. 6. The Joint-Space Force-Field Approach The second implemented controller we describe is a non-linear force-field approach based on the recent work by Mussa-Ivaldi (1997), inspired by the biological data described in Section 2. In earlier work, Mussa-Ivaldi & Giszter (1992) showed that a small number of force-field primitives could be used to generate a wide range of force fields at the frog's foot. By combining the primitives using superposition, the end-point of a simulated leg could be moved to different parts of the workspace. However, the actual path taken by the leg under the influence of the field is not straight or natural looking. Subsequently, Mussa-Ivaldi (1997) showed how combinations of primitives can be used to move from one point to another in a straight line. In that work, the primitives were weighted using step and pulse functions: steps to achieve a target position, and pulses to control the trajectory of the motion. To apply this approach to the Macarena task, stable joint-space potential fields with single static equilibrium points are combined to generate control for each sub-task. These primitives are combined with weighting functions such that step functions move the agent to its sub-task target position and pulse functions dictate desired trajectories for the arm motion, such as moving the hand to avoid the head. Torque non linear linear Figure 2. Graph showing the difference between the linear and non-linear joint-space controllers. The torque due to the non-linear controllers drops off at high errors. KLUWER STYLE FILE 7 Each primitive or force-field is specified as a torque-angle relationship at each joint of the arm: where is the joint torque, and OE is a torque-angle relationship primitive depending on time, actual angle ' actual and its derivative ' actual . A primitive OE i for a particular joint with stiffness k, damping k d , and desired angle ' desired , is calculated as: desired )e \Gammak(' actual \Gamma' desired actual (3) This defines a non-linear relationship, which is the derivative of a Gaussian potential centered at ' desired . The non-linear response of this controller is similar to a linear PD-servo for small errors desired ). However, with large errors, the torque calculated by the primitive drops off exponentially, as shown in Figure 2. Mussa-Ivaldi & Giszter (1992) suggest that this behavior is consistent with biological muscle, and that the non-linearity of the controller increases the richness of behavior that can be produced. We specified each sub-task of the Macarena with two such non-linear primitives combined to create the whole motion. The two primitives perform different tasks: the static position, defined by a force-field OE i weighted by a step function ! i (t), and the path between sub-tasks, manipulated using another force-field / i , itself weighted by a pulse function AE i (t): actual The step function is defined by: which yields a smooth transition in the control corresponding to movement toward a particular final posture defined by ' desired . The pulse function is defined by: ae which creates a smooth adjustment in the trajectory allowing separate control of the path taken in the movement. Our implementation differs from Mussa-Ivaldi (1997) in a number of ways. Mussa-Ivaldi uses a set of arbitrarily chosen primitives, and solves a least squares optimization problem to determine the sizes of the steps and pulses. Rather than select arbitrary primitives, we chose ours to correspond to the positions of the arm at each sub-task, thus simplifying the weighting. This is a pragmatic decision; it is unclear how well the optimization method scales from the 2 DOF system implemented in the Mussa-Ivaldi paper, to the full 20 DOF Adonis simulation. Finally, in the Mussa-Ivaldi work the primitives are defined as a Gaussian potential in the full joint-space, coupling the joints, while in our implementation they are treated independently. 1 This force-field-based joint-space controller (heretofore referred to as the torque-field controller) is similar to the PD-servo joint-space controller described in the previous section, in that they both rely on torque-angle relationships at the joints to determine the arm motion. The main difference (\Gammak joints actual \Gamma' desired 'actual which couples the joints through the exponential term. 8 M. Matari'c et al. actual F desired Figure 3. Impedance Control: The virtual force F is computed by attaching a virtual spring and damper from the hand position x to the desired position xe . The torques at the joints are then calculated to produce this desired force at the end of the arm, and thus move it to the desired position. is that the torque-field approach uses non-linear controllers at the joints, as opposed to the linear PD-servos. This non-linearity allows the controller to simply switch set-points for a new task, rather than interpolate as in the linear case, and to use pulse functions to manipulate the trajectory, rather than define explicit via points. 7. The Cartesian Impedance Control Approach In contrast to the first two, our third implemented controller acts in the Cartesian frame of reference, which allows for a more intuitive interface for the user, as the Cartesian position of the hand is easier to visualize than the angles of all the joints. The approach is based on the principle of impedance control, introduced by Hogan (1985), has been applied to robot manipulation. The general principle is to modulate the mechanical impedance of the end-point of an arm by altering the torques at the arm's joints. Mechanical impedance for an object is defined as the relationship between an imposed disturbance and a generated force. For example, a compressed spring exerts a force proportional to the displacement. The impedance of such a system is constant and equal to the stiffness of the spring. For a more complicated mechanism like a robot arm, the mechanical impedance is determined by the control at the joint level. For example, a mechanical arm can be made to appear as if a virtual spring and damper are connected to some equilibrium point; moving the point will drag the arm around, and the arm will automatically return to its equilibrium position if disturbed. Arranging the control of the arm in this way has advantages in terms of stability, especially when interacting with different environments (Colgate & Hogan 1988). Our impedance controller calculates the force F from the virtual spring and damper, as illustrated in Figure 3, given by: where x actual is the 6-D vector defining the position and orientation of the end-point (hand) in space, x actual is a vector of velocities, and x desired and x desired are 6-D vectors of desired posi- tions/orientations and velocities. K and B are stiffness and damping matrices. This desired force is implemented by applying torques at the joints, which are calculated using the Jacobian J(' actual ), using the following simple relation (Craig 1989): KLUWER STYLE FILE 9 Applying this equation results in stable control of the position and orientation of the hand over the workspace of the arms. However, it does not constrain the final orientation of the whole arm, or prevent the arm from violating joint limits or moving through the body. To further constrain the arm, a second impedance controller was added to control the elbow motion. This allows the positions of the elbow and the hand to be moved, which is an intuitively sensible method of constraining the arm motion. Experiments showed that the best way to control the elbow was to specify a desired orientation for the upper arm, rather than specifying the elbow position. 2 The control is calculated in a similar manner to Equation 8, although the Jacobian is defined for the transformations between the elbow and 3D shoulder joint, and the force F is only due to desired rotations. Other terms added to the impedance control include compensation for the effect of gravity on the links of the arms (g(' actual )), and some extra damping at the shoulder joint (b shoulder ), making the final torque applied to the joints: hand F hand elbow F elbow shoulder To perform each sub-task of the Macarena, we specify the desired position and orientation of the hand, and the desired orientation of the upper arm. The control scheme then calculates the torques at the joints in order to move the arm to that position, and maintain it there. Low-level PD-servos, as described previously, control the waist and neck. To move between sub-tasks, a linear interpolation scheme is used to gradually shift the desired positions. As with the PD-servo controller, extra via points are used to avoid collisions with the head. The method has several advantages over position control techniques using inverse kinematics (Baker & Wampler II 1988). It is computationally simple, requiring only the forward kinematics and the Jacobian (Whitney 1982), and it is stable both when moving freely, and during contact with surfaces (Hogan 1985). In addition, the general formulation of impedance control provides a simple merging mechanism for different control strategies (Beccari & Stramigioli 1998). The main difficulty encountered when implementing this scheme was finding a compact and intuitive way to specify the orientations of the elbow and hand. The orientation of the hand was specified using a single angle relative to the lower arm, while the orientation of the upper arm was specified by aligning the x-axis of the segment with a desired vector. In addition, the scheme produces straight-line motions of the hand which are not always the most natural. For example, when moving the hand from straight out (sub-task 3) to touching the shoulder (sub-task 5), the most natural motion is for the hand to come up and over, rather than moving directly in a straight line. A curved solution is possible with this controller, but would require a more detailed specification of the desired trajectory. As an alternative to impedance control, the simulation system allows arbitrary forces to be applied to the end-point of the arm. Thus a force could be calculated as in Equation 7, and directly applied to the hand. A variant of this approach was experimented with, applying the following force: desired )jx actual \Gamma x desired j (10) where v desired is the desired velocity, v actual is the actual velocity, x defined as above, and c is a gain constant. For carefully chosen values of c, this controller has the effect of moving the hand to the desired Cartesian position x desired . Although simpler to implement than the impedance controller, this controller has a number of disadvantages. Since the force is only applied at the hand, high damping has to be used to constrain the rest of the arm, which results in unnatural motion. The 2 This is due to the fact that under impedance control, the arm moved under the influence of the applied virtual springs and dampers at the hand and elbow. The effect of two forces on the arm can be unintuitive for arbitrary positioning of the set-points. Specifying the orientation of the upper arm, as well as the position and orientation of the hand, makes the system much more predictable and easy to operate. M. Matari'c et al. Figure 4. An example of Adonis performing the Macarena, shown as a series of snap-shots, in this case using the joint-space torque PD-servo controller. impedance controller was also found to be less sensitive to singular configurations of the arms (such as in sub-task 1, where the arm is straight). For these reasons, we chose not to use this final control method for evaluation; for more details on this implementation, see Matari'c, Zordan & Mason (1998b). 8. Performance Analysis and Comparisons Analysis and evaluation of complex behavior is an open research challenge. As synthetic behaviors for agents in animation, robotics, and AI become more complex, the issue of analysis becomes increasingly acute. In this section, we explore several evaluation criteria, both qualitative and quantitative, and make observations about differences between the different controllers performing the same task, consistencies from task to task for a single controller, and similarities between human and synthetic motion. 8.1. Naturalness of Movement: Qualitative Judging the naturalness of movement is an important aspect of both robotic and animation eval- uation, but aesthetic judgment is difficult to quantify. Qualitative judgments of motion require real-time playbacks of recorded behaviors; for the three controllers we implemented, those are available from: http://www-robotics.usc.edu/ agents/macarena.html Figure 5 shows a time-lapse image for sub-task 10 with the goal of facilitating a qualitative comparison of the arm trajectory generated by each of the three controllers. The impedance controller is shown on the left, torque-field controller in the middle, and the PD-servo controller on the right. While the beginning and end postures are very similar for all three, and all paths are valid KLUWER STYLE FILE 11 Figure 5. A time-lapse image of sub-task 10, showing the trajectories the hand takes using the different controllers: impedance on the left, torque-field in the middle, and PD-servo on the right. in that they avoid body collisions and unnatural postures, the paths themselves vary significantly. The motion generated by the PD-servo is smooth but contains an exaggerated curve, due to the joint-space spline interpolation between the chosen via points. The torque-field movement is also smooth, resulting from the Gaussian controllers. In contrast, the impedance controller motion is more jerky because its set-point moves along straight lines. Many differences between human movement and that of our simulated agents are due to the underlying dynamics of our chosen test bed; the qualitative features caused by the limitations of the dynamic simulation must be separated from those dictated by the underlying controller. Rigid body simulation imposes limitations that cannot be overcome by control. For instance, Adonis's unactuated spine necessarily appears stiff. Furthermore, dynamic simulation constrains motion to be physically plausible but not necessarily natural. For example, since the simulation does not constrain joint limits or avoid collisions, the controllers must handle these limitations directly. Because the controllers have no knowledge of body boundaries, avoiding self-collisions was accomplished through the user's choice of desired positions and/or angles, resulting in conservative, unnatural trajectories. This can be improved with direct collision prediction and avoidance, as well as by built-in joint limits. In contrast to limitations caused by the simulation, some qualitative differences are caused by the controllers directly. For example, the joint-space torque method interpolated postures with splines to smooth the resulting motion. It also included small head and hand movements that produce more natural appearance for the overall motion. Qualitative differences between controllers are often aesthetic, and thus difficult to quantify. Some metrics, such as comfort, can be applied, but even those vary under different dynamics and involve some observer/performer bias. To avoid this problem, the next section addresses two approaches to a more quantitative evaluation of the controllers. 8.2. Naturalness of Movement: Quantitative The whole arm path, analyzed qualitatively in the previous section, is still too complex to easily compare in a quantitative fashion without introducing external metrics. To focus, we consider only the end-effector motion, particularly the velocity and jerk of the dominant or active hand during individual sub-tasks. As a base-case or control in this analysis, we use hand positions recorded from a human performing the Macarena. M. Matari'c et al. 8.2.1. Comparison of End-Effector Speed subtask 21.03.05.0 hand speed (m/s) human torque-field PD-servo impedance subtask 81.03.0hand speed (m/s) subtask 101.03.05.07.0time time Figure 6. A comparison of the hand velocity profiles in four sub-tasks: sub-task 2 (extending the arm to straight (moving from straight out to touching the shoulder), sub-task 8 (moving from shoulder to the back of the head), and sub-task 10 (moving from the back of the head to the ribs), and human data. Hand position data of a person performing the Macarena were recorded with a commercial Flock of Birds electro-magnetic tracking system and used to compute the hand velocities. These are compared to the velocities of the three controllers we implemented; Figure 6 shows the velocities for the analyzed controllers and for a human performing the dance. To evaluate an individual controller performing a given sub-task, we consider the overall shape and smoothness of the velocity profile as well as the peak speed. Since the human motion data was recorded at fairly low variable sample rates (about 5 samples/sec), it produces stair-step velocity profiles; we assume the effect would be smoothed with higher frequency samples. An analysis of peak velocities shows that the joint-space PD-servo torque controller generated unnaturally fast hand movements while the other two controllers more closely matched the human peak speeds. In contrast, the same controller generated the smoothest and most symmetric hand profiles; natural human movement has been categorized as having such symmetric properties (Morasso 1981, Atkeson Hollerbach 1985). Furthermore, in the movements not requiring collision avoidance (sub-tasks 2 and 6), the impedance controller produced motion that closely matches the shape of the human velocity profile. Differences in hand movements from task to task indicate how a controller performs over a variety of sub-tasks and suggest the potential generality of that controller for use in new tasks. Task variability exercises the controller by forcing it to perform in a variety of conditions. Notably, sub-tasks 8 and 10 require more sophisticated paths in order to avoid head/arm collisions. The PD- servo and impedance controllers use via points to avoid this collision. The effect of these postures can be seen most dramatically in the speed profile for sub-task 8, noting the change in speed corresponding to the posture change at about 0.5 seconds. However, the torque-field controller uses KLUWER STYLE FILE 13 an initial pulse to control the overall trajectory and it remains more consistent across these tasks. Although the via points help achieve the goal of collision avoidance, the resulting velocity profiles indicate the need for a more sophisticated approach. 8.2.2. Comparison of End-Effector Jerk Minimal jerk of hand position has been proposed by Flash & Hogan (1985) as a metric for describing human arm movements in the plane. Inspired by their work in planar motion, we propose a 3D evaluation metric, according to the following index: where @ 3 x=@t 3 is the third derivative of x, y and z positions with respect to time. We chose jerk as an evaluation metric over other measures such as minimum torque change (Uno, Kawato & Suzuki or energy (Nelson 1983), because it is much easier to record from a human subject and is also a good measure of smoothness.10100000 human impedance PD-servo torque-field square jerk sub-tasks Figure 7. A comparison of the jerk values for the different controllers (PD-servo, torque-field, and impedance), and for human data. The lines connecting the data points do not correspond to actual data, since the sub-tasks are calculated independently, and map to left and right hand movements. The calculated jerk values of the three different controllers and the human data are shown in the graph (Figure 7) corresponding to the square jerk for the active hand (e.g., in sub-task 1 the right arm, in sub-task 2 the left arm, and so on) over the length of the task. We do not expect a correspondence between the controllers and the human jerk values, but instead focus on trends across sub-tasks. As expected, movements that involve collision avoidance with the head (i.e., sub-tasks 7 through 10) have high jerk values overall, reflecting their complexity. Since jerk is based solely on Cartesian movement, it is low for movements that are primarily specified by joint constraints (i.e., sub-tasks 3 and 4 which command "turn the hand palm up"). Finally, low jerk also results from movements over short distances between Cartesian goals (i.e., sub-tasks 11 and 12, moving the hand from one hip to the other). Jerk is a sensitive measure that varies strongly from task to task and from controller to con- thus the log scale. Furthermore, the motion-capture system used to gather human data 14 M. Matari'c et al. can suffer from marker slippage, adding further noise into the evaluation. We made no effort to create correspondence between the paths taken by the human and the different controllers, and thus variability in arm path is unaccounted for. Finally, timing has an effect on the jerk; slower movements have less jerk than faster ones. The movements shown do not all have the same timing and, although we tried various methods to normalize according to the timing, the data shown do not account for these differences explicitly, i.e., are not normalized. Therefore, the exact values in this graph are less reliable than the general trends they indicate, and it is remarkable to see the obvious correlations between the different data sets. 8.3. Controller Use and Flexibility In addition to evaluating the success of the controllers in creating a life-like Macarena, we have also evaluated the controllers from the user's point of view. In this section we consider issues such as the amount of information required by each controller, the ease with which that information is input to the simulation, the simplicity with which the final motion is tuned for the various cases, and the actual computational complexity of the controllers themselves. Once the gains and other constants have been fixed, there is not a great difference in the amount of information required by the three different controllers. The torque-field controller has the lowest overhead, requiring 14 values per arm per sub-task (7 for the step function, and 7 for the pulse). The PD-servo controller requires only 7 values per arm, but these need to be input at every time-step of the simulation, thus calling for an extra interpolation routine. The impedance controller also requires 7 values, including the hand position, orientation and the elbow orientation; like the PD-servo, it also uses an interpolation routine. Rather more important than the number of parameters needed to specify a particular position is the ease with which that information is determined. For the PD-servo and torque-field controllers, this information is input in joint-space, so the user needs to solve the inverse kinematics of the arm manually, usually by trying different angles and adjusting. This is straight-forward if a little tedious, due to the fact that the joints are in an articulated chain, making the effect of any one joint on the arm motion dependent on the angles of all the others. The impedance controller works in Cartesian space, which makes the specification of hand positions much easier. Specifying the orientations of the elbow and hand is slightly more difficult, however, mainly due to the awkwardness of specifying three-dimensional rotations. This illustrates the fundamental tradeoff between the two types of control; the joint-space controllers are awkward to use but have explicit control over all the joints, while the Cartesian space controller is easier to use, but has less control over the individual degrees of freedom. A third factor is the influence of the dynamics of the arm. While dancing the Macarena, the arm is moving quickly enough for dynamics to be important, making the choice of set-points, and particularly via points, quite important. For the torque-field controller, the pulse torque-field requires hand-tuning to create the motion, while for the other controllers, the positions of the via points requires hand-tuning. Since the motion of the arm is not wholly determined by the positions of these points, it is difficult to map from an error in the arm path to changes in a specific parameter. This difficulty is apparent in both reference frames, for the same reasons as described previously. A final evaluation can be made in terms of the complexity of the implementation. The most computationally simple controller is the PD-servo method, followed closely by the torque-field controller. The impedance controller is considerably more complex, requiring a 7-by-6 and a 7- Jacobian matrix to be calculated at each time-step, as well as numerous vector operations for gravity compensation. However, this is still considerably less complex than any explicit inverse kinematics algorithm. The increased complexity of the impedance controller presents a trade-off in return for the ease of specifying positions in Cartesian space. KLUWER STYLE FILE 15 9. Continuing Work: The Combination Approach The three controller implementations we presented all involve unavoidable tradeoffs, because each uses only a single, consistent approach to generating movement. However, different reference frames appear even in the simplest task specifications, resulting in unnatural and challenging transformations between the specification and the implementation. From the stand-point of the user, as well as the appearance of the final synthesized behavior, it would be preferable to have a means of flexibly combining the different control alternatives, so as to always utilize the approach most suited for a given task or sub-task. We are currently working on developing just such an approach to control. Our approach is implemented within the behavior-based framework (Matari'c 1997, Brooks 1991), which uses behaviors as abstractions for encapsulating low-level control details within each prim- itive. Consequently, we can implement generic primitives such as get-posture and go-to-point and parameterize them with the specific goals of each sub-task, as it is assigned. One of the benefits of the behavior decomposition is not only that there are different ways of structuring a given system (i.e., different types of controllers), but also that once a behavior decomposition is achieved, the specific behavior controllers can themselves vary, depending on the available sensors and effectors. For example, get-posture can be implemented with different types of joint-space controllers, and, anal- ogously, go-to-point can use different Cartesian controllers, if desired. Furthermore, other behavior types can be added, such as an oscillator-based primitives for movements such as bouncing, waving, swinging, etc (Williamson 1998). In an early demonstration of this approach, Matari'c, Williamson, Demiris & Mohan (1998a) employed the notion of different types of motor primitives as behaviors to generate the same Macarena sub-tasks. There, the sub-tasks were assigned different types of controllers: PD-servo joint-space control for posture-related sub-tasks (such as sub-tasks 1 through 4), and impedance Cartesian control, for extrinsic or body-centered movements (such as sub-tasks 5 through 12). Our implementation executed each sub-task sequentially, thus eliminating interference between the different controllers. Besides sequencing, however, behaviors/primitives can also be co-activated, i.e., executed in parallel. For example, our implementation included an avoid-collisions primitive executed concurrently with any get-posture or go-to-point primitive, in order to generate safe, collision-free movement. Concurrent behavior combination is more complex than sequencing, however, and requires consistent output representations between the controllers being combined (Matari'c 1997). Using different types of primitives assumes that either the user or some intelligent automated method can subdivide the overall task into sub-tasks, and assign those to the most appropriate types of behaviors/primitives. We believe that these are not unreasonable assumptions. Human-generated specifications are typically sequential and presented in a step-wise fashion. Sub-task breaks can also be generated directly from observing movement, such as for example using zero-velocity breaks for each end-point. Automatically assigning sub-tasks to primitives is more complex; it could be coarsely approximated using parsing and key-word search of the textual task specification, which provides strong hints in the form of references body parts and joints. In such a combination control system, individual behaviors may utilize different representations, coordinate frames, and underlying computation, but their use and performance can be seamlessly integrated by sequencing and co-activation. An effective means of encapsulating generic behaviors would also allow the integration of control schemes from different users. As complex articulated agents become more prevalent, such a modular approach to control could use its "open architecture" to combine the advantages of various successful approaches. M. Matari'c et al. 10. Conclusion We have compared a set of three approaches for control of anthropomorphic agents, including PD- servo control, torque-field control, and impedance control, implemented on the same dynamic torso simulation, Adonis, and tested on the same Macarena sub-tasks. We compared the three controllers against one another and against human data, using qualitative and quantitative metrics, including naturalness of appearance, hand velocity and jerk, and controller use and flexibility. To facilitate a realistic comparison, the controllers and the human data were generated indepen- dently. However, various techniques can be implemented to generate a closer fit between the data, if that is desired. Specifically, human hand positions could be used to select goal positions for the impedance controller. Similarly, an IK solver could be used to compute postures for the joint-space controllers that achieve these hand positions. Timing taken from human motion could be used to generate simulated motion that more closely fits the human performance. Lastly, minimization techniques could be applied to the controller parameters to find movements that minimize jerk and/or match other performance metrics. The fundamental tradeoff between believability and control effort still remains, as the approaches produce different results depending on sub-task specification. In order to address these tradeoffs, we proposed a combination framework which allows different types of movement primitives (under different reference frames and representations) to be used for different types of sub-tasks, in order to maximize the match between the description of the task and the controller that achieves it. Acknowledgments This work is supported by the NSF Career Grant IRI-9624237 to M. Matari'c. The authors thank Nancy Pollard for help with the jerk calculations, Stefan Schaal and Jessica Hodgins for sharing expertise and providing many insightful comments. The Adonis simulation was developed by Jessica Hodgins at Georgia Institute of Technology. --R Making Them Move: Mechanics Impedance Control as Merging Mechanism for a Behaviour-Based Architecture Robot Motion: Planning and Control Intelligence Without Reason Interactive Spacetime Control for Animation Introduction to Robotics: Mechanics and Control 'Convergent force fields organized in the frog's spinal cord' SD/Fast User's Manual What do People Look at When Watching Human Movement? Nonlinear force Fields: A Distributed System of Control Primitives for Representing and Learning Movements Spacetime Constraints Revisited Programming Anthropoid Walking: Control and Simulation Robot Manipulators: Mathematics Animation of Dynamic Legged Locomotion Active Stiffness Control of a Manipulator in Cartesian Coordinates Learning from demonstration Applied nonlinear control Guided Optimization for Balanced Locomotion The mathematics of coordinated control of prosthetic arms and manipulators Postural Primitives: Interactive Behavior for a Humanoid Robot Arm Rhythmic robot control using oscillators Spacetime Constraints --TR --CTR Maja J. Mataric, Getting Humanoids to Move and Imitate, IEEE Intelligent Systems, v.15 n.4, p.18-24, July 2000 Z. M. Ruttkay , D. Reidsma , A. Nijholt, Human computing, virtual humans and artificial imperfection, Proceedings of the 8th international conference on Multimodal interfaces, November 02-04, 2006, Banff, Alberta, Canada Michael Neff , Eugene Fiume, Modeling tension and relaxation for computer animation, Proceedings of the 2002 ACM SIGGRAPH/Eurographics symposium on Computer animation, July 21-22, 2002, San Antonio, Texas Ajo Fod , Maja J. Matari , Odest Chadwicke Jenkins, Automated Derivation of Primitives for Movement Classification, Autonomous Robots, v.12 n.1, p.39-54, January 2002 Aude Billard , Maja J. Matari, A biologically inspired robotic model for learning by imitation, Proceedings of the fourth international conference on Autonomous agents, p.373-380, June 03-07, 2000, Barcelona, Spain Maja J. Mataric, Sensory-motor primitives as a basis for imitation: linking perception to action and biology to robotics, Imitation in animals and artifacts, MIT Press, Cambridge, MA, 2002 David A. Forsyth , Okan Arikan , Leslie Ikemoto , James O'Brien , Deva Ramanan, Computational studies of human motion: part 1, tracking and motion synthesis, Foundations and Trends in Computer Graphics and Vision, v.1 n.2, p.77-254, July 2006

animation;motor control;robotics;articulated agent control

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Defining Open Software Architectures for Customized Remote Execution of Web Agents.

Agent-based solutions promise to ameliorate Web services, by promoting the modular construction of Web servers, relieving the network from transferring useless data, supporting user mobility, etc. However, existing Web servers do not favor the hosting of agents. This paper proposes a description of agent behavior in terms of its requirements regarding resource utilization (e.g. memory, and disk space), functional services (e.g. system calls), and non-functional properties (e.g. degree of replication, and access control). When formally expressed, these requirements can be used in an automated decision process, which is based on software specification matching techniques. Upon the acceptance of an agent, the host uses these requirements to construct an environment customized to agent's execution. We discuss the benefits of this approach, and how it can be used to promote existing agent-based solutions in the Web framework.

Introduction Since its appearance in the early 90's, Web has been greatly evolved. While it was initially intended for the transfer of hypertext documents, its success and world-wide acceptance led the focus of research interest into other directions like distributed computing, electronic commerce, etc. However, from the perspective of intensive use, Web's architecture was not designed for such employment. Consequently, today Web is confronted with a number of problems ranging from server saturation and heavy network tra-c, to access control and verication of client requests. One of the most prominent solutions for coping with those problems is the use of software agents, a concept introduced to distributed systems from the eld of articial intelligence [1]. In common sense, an agent can be anyone who acts on behalf or in the interest of somebody else. Consequently, a software agent is a piece of code endowed with capabilities that allow it to perform some task in the place of a person or some other piece of code. In the context of this document we use the term agent to signify an autonomous piece of code c 1999 Kluwer Academic Publishers. Printed in the Netherlands. V. Issarny and T. Saridakis with mobile characteristics, that can be executed independently from its originator. Agents permit computations to be executed close to the data, which results in a decrease of both the network bandwidth utilization, and the response time. By relaxing the restrictions of permanent location of computations, they promote distribution of workload among the components of a distributed system. Finally, agents support the construction of modular servers capable of adapting to a large variety of client requests. On the other hand, agents raise major problems on their potential hosts. Such problems are related to host's capabilities to provide all the indispensable primitives for supporting an agent's execution, and to the compatibility of agent's requirements with host's policies. These execution primitives and policies are the execution properties that fully describe the agent's behavior in the host environment. Accepting an agent without supporting all required execution properties might not be acceptable for the agent's originator, while accepting an agent whose execution properties are in con ict with host's policies might not be acceptable for the host. Hence, before accepting an agent, the host system should be informed of agent's properties, analyze them and decide whether or not the given agent can be properly hosted. This paper presents a framework, in which a host decides on the acceptance of an agent based on the explicit description of agent's execution properties. Based on means provided by the software architecture eld, we describe an agent's execution properties as an open architecture, which abstractly but unambiguously characterizes the host environment expected by the agent. Using this architectural description as the agent's specication, the host is able to detect conicts with its own policies and to decide whether or not to accept the agent. In case of agent acceptance, the host also has the information to customized the execution environment to meet agent's requirements. The remainder of the paper is structured as follows: next section gives an overview of the results in the eld of software architecture, which we use as a base ground for our proposal. Section 3 addresses the description of open software architectures so as to characterize Web agents with respect to the execution properties that are expected from the host environment. The instantiation of a host environment customized to a given agent, is further addressed in Section 4. Related work is discussed in Section 5, and we conclude in Section 6 by summarizing our contribution. Customized Remote Execution of Web Agents 3 2. Software Architecture Research in the software architecture domain aims at reducing costs of developing complex software systems [18, 21]. Towards that goal, formal notations are being provided to describe software architectures, replacing their usual informal description in terms of box-and-line di- agrams. These notations are generically referred to as Architecture Description Languages (Adls). An Adl allows the developer to describe the gross organization of a system in terms of coarse-grained architectural elements, abstracting away their implementation details. Prominent elements of a software architecture are subdivided into the following categories: Components, that characterize a unit of computation or a data store. Connectors, that characterize a unit of interaction. Congurations, that describe a specic, possibly generic, software architecture through the composition of a set of components via connectors. Existing Adls dier depending on the software architecture aspects to which they are targeted. We identify at least two research di- rections, which dene dierent Adls: (i) the architecture analysis, which provides the formal specication of an architecture's behavior (e.g. [14, 10]), and (ii) the architecture implementation, which delivers the implementation of an application its architectural description (e.g. [20, 8]). Both research directions instigate the design and implementation of CASE tools, based on technologies such as model checking, theorem proving, and type checking [12]. 2.1. Example To illustrate the use of Adls, we take as an example a primitive Distributed File System (Dfs). The Dfs is composed of a client interacting with a (possibly distributed) le server for performing le accesses. We further assume that the interaction protocol between components is Rpc-like. Using a simplied version of the Adl used in the Aster project 1 , the Dfs architecture is given as a conguration made of the client and le server components, which are bound together using a connector describing an Rpc protocol. Figure 1 contains the corresponding declarations. 4 V. Issarny and T. Saridakis component client = port client (typeFormat format); server (typeFormat format); functional typeInt close (typeDsc fd); interaction open, close, read, write: client; key, state: server; non-functional state: CheckPoint; component port server (typeFormat format); functional typeInt close (typeDsc fd); read (typeDsc fd, interaction open, close, read, write: server; non-functional read, write: FailureAtomicity; connector role client (typeFormat format); server (typeFormat format); port client (typeFormat format); functional interaction state: client; non-functional Secure, Reliable; components C: client; connectors com: RPC; binding functional C.open: FS.open; C.close: FS.close; C.read: FS.read; C.write: FS.write; com.key: com.state: C.state; interaction C.client: com.client; com.server: Figure 1. The architectural description of the Dfs. Customized Remote Execution of Web Agents 5 The architectural elements described in Figure 1 dene the gross organization of the Dfs in an abstract manner, and give the associated execution properties. More precisely, the execution properties of the Dfs are subdivided into: Functional properties, which dene the operations that are provided and called by the architectural elements. The fact that a functional property is either provided or called by the element declaring it, is given by the semantics of the port through which the corresponding interaction is performed. For instance, the client component calls the open() operation of the leServer component. Non-functional properties, which characterize the resource management policies that are provided by the architectural elements. For instance, the state() and key() operations of the client component provide the CheckPoint and Authentication properties respectively; the RPC connector provides Reliable and Secure communications for all the interactions using it. Interaction properties, which characterize the communication protocols that are used for performing the interactions among com- ponents. In the Dfs example, interactions are achieved using an Rpc protocol. Until this point, we have considered the declaration of execution properties in terms of operation signatures and \names" associated to the declared operations (e.g. key: Authentication). The type-checking process typically employed in distributed programming environments to verify the correctness of bindings among declared operations is based on pattern matching. However, using pattern matching techniques to verify the satisfaction of the execution properties requirements placed on bindings relies on an informal description of the execution properties. Obviously, we cannot rely on pattern matching performed on \names" describing the execution properties (e.g. Authentication, RPC, etc), to verify the correctness of bindings; a slightly dierent interpretation of the same \name" by each of the interacting sides may cause communication problems di-cult to track down and resolve. To overcome this problem, we associate to a \name" a set of formal specications that serve as its denition. This allows both clean interface declarations and execution properties requirements resolution based on specica- tion matching. The next subsection gives an overview of work done in the area of formally specifying execution properties. For the sake of conciseness, we do not give examples of formal specications in the following. The interested reader may refer to [7] for more details on this topic. 6 V. Issarny and T. Saridakis 2.2. Formal specifications of execution properties Formal specication of functional properties amounts to specifying the behavior of the operations required and provided by the architectural elements. the behavior of operations is specied in terms of pre- and post-conditions using Hoare's logic (e.g. [17, 23]). In addition to the straightforward benet of formally specifying functional properties for verifying the correctness of component interconnection, it further favors software reuse and evolution. A component may be retrieved from a component database using the component specication. The correctness of component substitution within a conguration can be checked with respect to the specications of involved components. The aforementioned verications lie in the denition of relations over specications. These relations dene, in terms of specication match- ing, correctness conditions for software interconnection, re-use, and substitution [17, 13, 23]. A non-functional property characterizes a resource management policy that is implemented by the underlying execution platform. Similar to functional properties, non-functional ones are specied in terms of rst order logic (e.g. [9]). These specications refer to operations that are not explicitly stated in the conguration description of a software system, but which are provided by the execution platform or the middleware in a way transparent to the software system. This allows us to specify any non-functional property in terms of a unique predicate instead of pre- and post-conditions. Practically, the formal specication of non-functional properties provide for the verication, with respect to the declared non-functional properties, of the correctness regarding the bindings among components. In addition, it enables the systematic customization of middleware with respect to properties required by the architectural elements (e.g. [8, 24]). Brie y stated, middleware components providing non-functional properties are retrieved in a systematic way through specication matching between required and provided properties. The retrieved components are integrated with the application components using base connectors. Formal specication of interaction properties has been examined in [2], where a Csp-like notation is introduced for the formal specication of the behaviors of components and connectors with respect to their communication patterns. This allows the correctness verication of a conguration with respect to the communication protocols that are used, using the notion of renement given in Csp. In that framework, a component description embeds a set of port processes that are the component interaction points, and a coordination process dening the coordination among ports. The behavior of a component is described Customized Remote Execution of Web Agents 7 by the parallel composition of port and coordination processes. Simi- larly, a connector is dened in terms of a set of role processes, which realize communications among components, and a coordination process that species the coordination among roles. The behavior of a connector is described by the parallel composition of role and coordination processes. 3. Web Agent in an Open Software Architecture A Web agent denes a software component, which interacts with components from the host environment. Each of these interactions is characterized by the functional, non-functional, and interaction properties associated to it. Thus, a software agent can be specied inside an open software architecture, which denes a conguration made of the agent component and the set of open components and connectors that must be provided by the host environment. Then, a host may safely accept and execute an agent if the former's capabilities cover the latter's requirements agent's requirements. The agent's requirements on the host environment can be precisely dened by providing the following information for the elements of the agent's open architecture. The agent component denes: the functional properties it expects from the host environment together with the associated interaction properties, and the functional, interaction, and non-functional properties it provides. Each connector characterizes the interaction and non-functional properties that are to be made available by the host environment for communication among the agent and the host's components. Each open component characterizes a software component with which the agent is willing to interact. In other words, the agent component abstractly denes the agent's behavior by exposing how it interfaces with the host. The open components and connectors provide a description of the execution properties that should be provided by the host environment. 8 V. Issarny and T. Saridakis agent component { Same as client in Figure 1 { open component hostFileServer = { Same as leServer in Figure 1 { connector { Same as RPC in Figure 1 { components agent: X; FS: hostFileServer; connectors Svce: service; binding { Same as DFS binding in Figure 1 { Figure 2. Description of an agent accessing a le server in the host environment. 3.1. Example For illustration, let us give the denition of an open architecture for an agent that accesses a le server in the host system. Figure 2 gives a description of the resulting architecture, which is close to the Dfs architecture discussed in Subsection 2.1 except from the client component that now corresponds to the agent. Given this description of the open architecture, the host can safely accept and execute the agent, if the host is able to instantiate the open hostFileServer component and the service connector, without violating its own security policies. 3.2. Interpreting agent specifications So far we have been arguing that architectural descriptions like the one given in Figure 2 provide su-cient information about an agent's requirements and a host's guarantees. Based on this information, one can decide whether an agent conforms with some host's policies, and whether a host can support all the requirements of some agent. To make practical use of this approach, we need to associate agents and hosts with interfaces that describe their execution properties. In addition, hosts must be provided with a framework that allows them to interpret such interfaces. This framework should support the analysis of agent's execution properties, and the reasoning on their combination with host's policies (e.g. see [3] for a study on the combination of properties describing security policies). Customized Remote Execution of Web Agents 9 Ideally, all three types of execution properties (i.e. functional, non- functional, and interaction properties) should be described using formal specications. However, common practice has shown that informal specications provide su-cient guarantees for correct reasoning on functional properties, like in the case of Omg's Object Transaction Service (see chapter 16 in [6]). In a similar manner, informal specication has been shown su-cient for interaction properties, due to the well-known and widely accepted interpretations of common communication protocols. In contrast to the above, formal specication is necessary for non-functional properties, since there does not exist a precise common understanding of what they represent. Consider for example the case of an Rpc system: designers have a common interpretation of the client-server interaction, but the interpretation of the associated non-functional properties, like the at-most-once failure semantics often dier. Given the architectural description that serves as the specication of an agent, the analysis and reasoning that should be performed by a host prior to the agent's acceptance, are formally dened as follows: 8C a 2 O(Agent): MatchPorts(P (C a ); P (C h 8I a 2 I(Agent): MatchFunc(I a ; I h MatchNonFunc(I a ; I h MatchInter(I a ; I h ) The functions and the symbols used in the above expressions, are dened as follows: O(Agent) and C(Host, Agent) denote respectively the open components declared by Agent, and the components available by Host to Agent, based on trust issues (i.e. the origin of the agent and the associated level of trust). I(Agent) and I(Host, Agent) denote respectively the connectors declared by Agent, and the connectors available by Host to Agent, based on trust issues. denotes the set of ports dened in C. evaluates to true if the set of ports P (which are requested by the agent) is a subset of the set of ports P 0 (which are provided by the host). This function can be implemented using pattern matching techniques. V. Issarny and T. Saridakis evaluates to true if the functional properties of C 0 match those of C. This function can be implemented using pattern matching, if functional properties are specied in terms of operation signatures. Otherwise, if functional properties are formally specied, this function can rely on a theorem prover. evaluates to true if the non-functional properties declared by C 0 match those declared by C. This function should be implemented using a theorem prover, since we have argued that specication matching of non-functional properties is mandatory. MatchInter(I, I 0 ) evaluates to true if for each interaction property in I there is at least one interaction property in I 0 that matches it. Similarly to the functional properties, this function can be implemented using pattern matching. However, for greater robustness and exibility formal specications of interaction properties should be employed. For instance, this may be achieved using a Csp-based process algebra as proposed in [2]. Then, interaction properties match if the processes declared in I 0 rene those in I. Let us remark here that the matching function may be automated using a tool like Fdr [5]. 4. Instantiating Customized Hosts for Web Agents Given an agent's specication, we have seen how the host environment can analyze the agent's requirements and reason on their compatibility with host's policies. In this section we report on the work we have carried out on using the outcome of this analysis to instantiate an execution environment customized to the agent's needs. The instantiation process is based on searching and retrieving host's architectural elements that match the open components and connectors declared by an agent. In addition to the execution properties that have been mentioned thus far, we explicitly take under consideration the usage of host resources requested by an agent, so as to guarantee that the agent will be able to execute to completion [19]. For that purpose, we include an additional clause, named resource, in the conguration description, in which the resources required for the agent's execution are declared. Figure 3 gives an example of an explicit declaration of agent requirements concerning host's resources. Customized Remote Execution of Web Agents 11 components agent: X; connectors Svce: service; binding { Same as DFS binding in Figure 1 { resource Figure 3. Explicit declaration of agent's requirements regarding host's resources. 4.1. Customizing the host environment Upon the reception of an agent specication, the host rst evaluates whether it is able to execute the agent based on the agent's originator and on available architectural elements and resources. If so, the host noties the agent's originator, which then sends the agent's code. If the host does not accept the agent, the reason of rejection is sent to agent's originator. Possible reasons include: insu-cient level of trust for performing the requested operations, unavailability of architectural elements, and unavailability of some resources. The agent's originator may then revise its initial requirements by modifying the open software architecture constraints, and make a new request to the host. Considering the agent's specication, instead of treating it as a hostile entity facilitates its acceptance by the host. However, until now, we have not considered issues related to the safety of the host, which in the Web community is considered much more important than the acceptance of an agent. By accepting an agent relying on the declared properties, we risk to accept an agent that actually exhibits a dierent behavior than the one it declares. In that case, the agent may cause damages to the execution environment. Obviously, for our approach to be viable, situations like the above should never raise. To assure this, the execution environment for a given agent is built by the host according to the agent's specication, which describe the exact interaction points with the host and their properties. Except from the allocated resources (i.e. private memory and disk space) which the agent can access without any restrictions, the only way for an agent to access the host is to pass through the declared bindings conforming to the associated execution properties. Hence, the execution environment is V. Issarny and T. Saridakis safe for the host since it does not allow the agent to perform any actions other than those declared in its interface. Although the host guarantees the correct execution of an agent whose behavior conforms with the one declared in its interface, no guarantees exist for the execution of agents whose behaviors deviate from the declared ones. In some cases the customized environment may provide \stronger" execution properties than those requested by an agent. this may occur under two conditions: (i) if the host does not possess a component providing exactly the property requested by the agent, but it does possess a component providing a \stronger" property, and (ii) if the agent is still accepted by the host when the stronger property replaces the originally requested one in the agent's specication. Such cases include the allocation of a bigger portion of resources than the one required, the support for 32-bit encryption keys while only 16-bit keys were requested, the use of a component that implements failure atomicity and retry-on-error while only the rst was requested, etc. Hence, an agent that requested 15.5KB of memory but actually uses 16KB may nally execute to completion, although this is not a priori guaranteed. 4.2. Prototype implementation To experiment the practicality of our approach, we implemented a prototype for agents written in the Java programming language. The prototype is a client-server system, where clients contact the server through Cgi to request the remote execution of one of their agents. The client-server interaction is decomposed as follows: The client rst sends the agent's specication to the server, and waits for approval or rejection notication from the server. Upon reception of an agent's specication, the server checks whether it is able to host the agent based on available architectural elements and resources, as presented in the previous subsection. Architectural elements available on the host are stored in a software repository, which is organized so as to be able to identify the subset of elements that can safely be made available to an agent with respect to its originator. In the current prototype, we distinguish between two kinds of agents: those originated from the same Intranet to which the server belongs, and those originate outside this Intranet. Availability of architectural elements simply relies on pattern matching for functional and interaction properties. On the other hand, checks regarding non-functional properties are done using specication matching and relies on the tool we developed for Customized Remote Execution of Web Agents 13 middleware customization [8]. If the server can safely execute the agent, it computes a unique key for it and reserves the requested resources; the key is sent to the client together with the acceptance notication. If the agent cannot be run, the reason for rejection is notied to the client. Once the client receives the notication message from the server, it checks whether the agent has been accepted or not. In the former case, it sends the agent's code to the server together with the associated key. In the latter case, the client re-issues the requests later on, if the reason of rejection is the temporary resource un- availability. The client may issue a new agent specication if the reason of rejection is some other execution property. Upon the reception of the agent's code, the server checks the agent's identity using the associated key. Once the agent is authen- ticated, it runs in the customized execution environment, which only allows the agent to behave in the way that was described in its specication. In addition, the customized execution environment provides a private address space for the agent's execution, in order to conne the consequences of an agent's crash on the failed agent alone. To assure that an agent respects the execution properties it has declared in its interface, we have used the SecurityManager Java class to build the AgentSecurityManager. The AgentSecurity- Manager surveys the execution of an agent in terms of le system and network accesses, and resource consumption, and causes an agent to abort if the agent attempts some unauthorized action. The prototype is su-cient for evaluating the practicality of our approach, but it needs to be enhanced from the standpoint of performance and scalability of the host instantiation process. The software repository of architectural elements available on the host is actually composed of a small set of elements, enabling sequential search. Further work is needed so as to experiment with a large software repository. In addition, our prototype implementation would obviously benet from enhanced products based on the Java technology. Products like SUN's may provide for substantial improvements to our prototype. 14 V. Issarny and T. Saridakis 5. Related Work The implicit host-agent interaction model that underlies our approach is similar to the conventional Web agent interaction model, where the agent describes a process and the host provides an execution environment (e.g. HTTP-based Mobile Agents [11]). Yet, in our case the execution environment is prepared by the host according to the agent's execution properties, which implies that the execution environment is explicitly dened by the agent, and that an agent and its execution environment are strongly coupled. From this standpoint our Web agents resemble Mobile Ambients [4], which dene the mobile code and its bounded execution environment as a single entity that can move across Web's administrative domains. However, a mobile ambient is a modeling entity used as the structural element of a system model described in the ambient calculus, whereas our Web agent is a set of execution properties describing the behavior of a piece of software. The conceptual similarity of mobile ambients and our Web agents suggests that the ambient calculus can be employed to provide a formal model of the host-agent interactions described in this paper. At the practical level, our prototype deals with the safety problems stemming from inappropriate resource use by mapping an agent to an individual process. Although this approach assures that problems like memory address violation will aect only the execution of the agent that caused them, it restricts severely the number of agents that can execute concurrently in a given host. A dierent approach is suggested by Proof-Carrying Code, or PCC for short, which has been used to support safe execution of mobile agents [15]. In PCC agents carry a proof that they conform to the host's policies and the host is capable of verifying the validity of this proof. Hence, agents can be safely mapped to threads which share their address space with other agents or even host threads. In this sense, PCC provides an elegant alternative to the heavy hosting scheme used by our prototype, for the extra cost of building a proof at the agent's originator and verifying its validity at the host. However, PCC does not provide any support for nonfunctional execution properties, nor support of any type for the agent's requirements from the host. Hence, it is not directly comparable to our approach. Rather, it should be considered as a alternative approach for resource management in prototype. In general, our approach for hosting Web agents does not aim at replacing existing environments supporting mobile agents. Instead, it aims at providing a unifying way for describing agents' behavior in terms of their functional and nonfunctional execution properties, and considering their hosting based on that behavior. In our framework, Customized Remote Execution of Web Agents 15 issues related to code mobility and its remote execution in foreign hosts can be uniformly expressed as execution properties. Execution properties integrate agent characteristics regarding both mobility support primitives (e.g. attach, move, and clone [11]), and non-functional requirements (e.g. authentication, availability, secrecy, and integrity [22]). Hence, approaches like Mobile Assistant Programming [16] that provides the underlying framework for creating agents and moving them to dierent hosts, can gain signicant benets from the presented approach, in terms of exible agent hosting. 6. Conclusion In this paper, we have proposed a framework for reasoning on the acceptance of agents and for constructing execution environments customized to agents' requirements. The proposed framework conveniently adapts existing technology from the elds of distributed systems and formal specications to the needs of Web agents. Our approach is based on the description of open software architectures characterizing the execution properties provided by agents and expected from the host environment. The proposed framework suggests that hosting an agent consists of verifying the compatibility of the agent's requirements with the host's policies, and then customizing the host environment to meet the approved requirements. Our proposal is easy to use in the simple case (i.e. the interface declaration for agents that do not have any non-functional requirements is equally easy to the declaration of a Corba while it supports the declaration of complex execution properties without sacricing the functionality, performance, or safety of the host (i.e. the customized execution environment guarantees both agent's requirements and host's constraints). Besides the benets for the agent, the presented approach provides support for modular constructions of host environments, which results in exible and scalable Web servers. The customization of the agent's execution environment permits the use of resource management algorithms which allow the host to concurrently serve more than one agent without signicant impacts on its performance. Moreover, modular constructions allow the Web servers to apply dierent hosting poli- cies, according to various criteria based on agent characteristics. As a consequence, a number of other research issues can be addressed in the Web agent framework, including how a host can use its current state for deciding to accept an agent, how to advertise, dispose of, and cost its resources, what combinations of non-functional properties should V. Issarny and T. Saridakis the host accept, how to assure fair treatment of agents with equivalent requirements, etc. --R Intelligent Agents. A Formal Basis for Architectural Connection. Dealing with Multi-Policy Security in Large Open Distributed Systems Mobile Ambients. Failures Divergence Re Common Object Services Speci Achieving Middleware Customization in a Con Characterizing Coordination Architectures According to Their Non-Functional Execution Properties Exposing the Skeleton in the Coordination Closet. A Framework for Classifying and Comparing Architecture Description Languages. Correct Architecture Re- nement Untrusted Agents using Proof-Carrying Code Mobile Assistant Programming for E-cient Information Access on the WWW The Inscape Environment. Foundations for the Study of Software Architecture. Customized Remote Execution of Web Agents. Abstraction for Software Architectures and Tools to Support Them. Software Architecture: Perspectives on an Emerging Discipline. A Framework for Systematic Synthesis of Transactional Middleware. --TR

customized execution;remote execution;mobility;execution properties;agent;software architecture;specification matching

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Verifying Compliance with Commitment Protocols.

Interaction protocols are specific, often standard, constraints on the behaviors of autonomous agents in a multiagent system. Protocols are essential to the functioning of open systems, such as those that arise in most interesting web applications. A variety of common protocols in negotiation and electronic commerce are best treated as commitment protocols, which are defined, or at least analyzed, in terms of the creation, satisfaction, or manipulation of the commitments among the participating agents.When protocols are employed in open environments, such as the Internet, they must be executed by agents that behave more or less autonomously and whose internal designs are not known. In such settings, therefore, there is a risk that the participating agents may fail to comply with the given protocol. Without a rigorous means to verify compliance, the very idea of protocols for interoperation is subverted. We develop an approach for testing whether the behavior of an agent complies with a commitment protocol. Our approach requires the specification of commitment protocols in temporal logic, and involves a novel way of synthesizing and applying ideas from distributed computing and logics of program.

Introduction Interaction among agents is the distinguishing property of multiagent sys- tems. However, ensuring that only the desirable interactions occur is one of the most challenging aspects of multiagent system analysis and design. This is especially so when the given multiagent system is meant to be used as an open system, for example, in web-based applications. Because of its ubiquity and ease of use, the web is rapidly becoming the platform of choice for a number of important applications, such as trading, supply-chain management, and in general electronic commerce. However, the web can enforce few constraints on the agents' behavior. Current approaches to security on the web emphasize how the different parties to a transaction may be authenticated or how their data may be encrypted to prevent unauthorized access. Even with authentication and controlled access, the parties would have support beyond conventional protocol techniques (such as finite state machine models) neither to specify the desired interactions nor to detect any violation. However, authentication and access control are conceptually orthogonal to ensuring that the parties behave and interact correctly. Even when the parties are authenticated, they may act undesirably through error or Venkatraman & Singh malice. Conversely, the parties involved may resist going through authentica- tion, but may be willing to be governed by the applicable constraints. The web provides an excellent infrastructure through which agents can communicate with one another. But the above problems are exacerbated when agents are employed in the web. In contrast with traditional programs and in- terfaces, neither their behaviors and interactions nor their construction is fixed or under the control of a single authority. In general, in an open system, the member agents are contributed by several sources and serve different inter- ests. Thus, these agents must be treated as autonomous-with few constraints on behavior, reflecting the independence of their users, and heterogeneous-with few constraints on construction, reflecting the independence of their designers. Effectively, the multiagent system is specified as a kind of standard that its member agents must respect. In other words, the multiagent system can be thought of as specifying a protocol that governs how its member agents must act. For our purposes, the standard may be de jure as created by a standards body, or de facto as may emerge from practice or even because of the arbitrary decisions of a major vendor or user organization. All that matters for us is that a standard imposes some restrictions on the agents. Consider the fish-market protocol as an example of such a standard protocol [14]. Example 1. In the fish-market protocol, we are given agents of two roles: a single auctioneer and one or more potential bidders. The fish-market protocol is designed to sell fish. The seller or auctioneer announces the availability of a bucket of fish at a certain price. The bidders gathered around the auctioneer can scream back Yes if they are interested and No if they are not; they may also stay quiet, which is interpreted as a lack of interest or No. If exactly one bidder says Yes, the auctioneer will sell him the fish; if no one says Yes, the auctioneer lowers the price; if more than one bidder says Yes, the auctioneer raises the price. In either case, if the price changes, the auctioneer announces the revised price and the process iterates. Because of its relationship to protocols in electronic commerce and because it is more general than the popular English and Dutch auctions, the fish-market protocol has become an important one in the recent multiagent systems liter- ature. Accordingly, we use it as our main example in this paper. Because of the autonomy and heterogeneity requirements of open sys- tems, compliance testing can be based neither on the internal designs of the agents nor on concepts such as beliefs, desires, and intentions that map to internal representations [16]. The only way in which compliance can be tested aamas.tex; 20/02/1999; 16:40; no v. Compliance with Commitment Protocols 3 is based on the behavior of the participating agents. The testing may be performed by a central authority or by any of the participating agents. However, the requirements for behavior in multiagent systems can be quite subtle. Thus, along with languages for specifying such requirements, we need corresponding techniques to test compliance. 1.1. COMMITMENTS IN AN OPEN ARCHITECTURE There are three levels of architectural concern in a multiagent system. One deals with individual agents; another deals with the systemic aspects of how different services and brokers are arranged. Both of these have received much attention in the literature. In the middle is the multiagent execution architec- ture, which has not been as intensively studied within the community. An execution architecture must ultimately be based on distributed computing ideas albeit with an open flavor, e.g., [1, 5, 11]. A well-defined execution functionality can be given a principled design, and thus facilitate the construction of robust and reusable systems. Some recent work within multiagent systems, e.g., Ciancarini et al. [8, 9] and Singh [18], has begun to address this level. Much of the work on this broad theme, however, focuses primarily on co- ordination, which we think of as the lowest level of interaction. Coordination deals with how autonomous agents may align their activities in terms of what they do and when they do it. However, there is more to interaction in gen- eral, and compliance in particular. Specifically, interaction must include some consideration of the commitments that the agents enter into with each other. The commitments of the agents are not only base-level commitments dealing with what actions they must or must not perform, but also metacommitments dealing with how they will adjust their base-level commitments [20]. Commitments provide a layer of coherence to the agents' interactions with each other. They are especially important in environments where we need to model any kind of contractual relationships among the agents. Such environments are crucial wherever open multiagent systems must be composed on the fly, e.g., in electronic commerce of various kinds on the Internet. The addition of commitments as an explicit first-class object results in considerable flexibility of how the protocols can be realized in changing situations. We term such augmented protocols commitment protocols. Example 2. We informally describe the protocol of Example 1 in terms of commitments. When a bidder says Yes, he commits to buying the bucket of fish at the advertised price. When the auctioneer advertises a price, he commits that he will sell fish at that price if he gets a unique Yes. Neither commitment is irrevocable. For example, if the fish are spoiled, the auctioneer releases the bidder from paying for them. Specifying all possibilities in terms of irrevocable commitments would complicate each commitment, but would still fail to capture the practical meanings of such a protocol. For instance, the auctioneer may not honor his offering price if a sudden change in weather indicates that fishing will be harder for the next few days. 1.2. COMPLIANCE IN OPEN SYSTEMS The existence of standardized protocols is necessary but not sufficient for the correct functioning of open multiagent systems. We must also ensure that the agents behave according to the protocols. This is the challenge of compliance. However, unlike in traditional closed systems, verifying compliance in open systems is practically and even conceptually nontrivial. Preserving the autonomy and heterogeneity of agents is crucial in an open environment. Otherwise, many applications would become infeasible. Con- sequently, protocols must be specified as flexibly as possible without making untoward requirements on the participating agents. Similarly, an approach for testing compliance must not require that the agents are homogeneous or impose stringent demands on how they are constructed. Consequently, in open systems, compliance can be meaningfully expressed only in terms of observable behavior. This leads to two subtle consid- erations. One, although we talk in terms of behavior, we must still consider the high-level abstractions that differentiate agents from other active objects. The focus on behavior renders approaches based on mental concepts ineffective [16]. However, well-framed social constructs can be used. Two, we must clearly delineate the role of the observer who assesses compliance. 1.3. CONTRIBUTIONS The approach developed here treats multiagent systems as distributed sys- tems. There is an underlying messaging layer, which delivers messages asynchronously and, for now, reliably. However, the approach assumes for simplicity that the agents are not malicious and do not forge the timestamps on the messages that they send or receive. The compliance testing is performed by any observer of the system- typically, a participating agent. Our approach is to evaluate temporal logic specifications with respect to locally constructed models for the given ob- server. The model construction proposed here employs a combination of the notion of potential causality and operations on social commitments (both described below). Our contributions are in incorporating potential causality in the construction of local models identifying patterns of messages corresponding to different operations on commitments showing how to verify compliance based on local information. Compliance with Commitment Protocols 5 Our approach also has important ramifications on agent communication in general, which we discuss in Section 4. Organization. The rest of this paper is organized as follows. Section 2 presents our technical framework, which combines commitments, potential causality, and temporal logic. Section 3 presents our approach for testing (non-)compliance of agents with respect to a commitment protocol. Section 4 concludes with a discussion of our major themes, the literature, and the important issues that remain outstanding. 2. Technical Framework Commitment protocols as defined here are a multiagent concept. They are far more flexible and general than commitment protocols in distributed computing and databases, such as two-phase commit [12, pp. 562-573]. This is because our underlying notion of commitment is flexible, whereas traditional commitments are rigid and irrevocable. However, because multiagent systems are distributed systems and commitment protocols are protocols, it is natural that techniques developed in classical computer science will apply here. Accordingly, our technical framework integrates approaches from distributed computing, logics of program, and distributed artificial intelligence. 2.1. POTENTIAL CAUSALITY The key idea behind potential causality is that the ordering of events in a distributed system can be determined only with respect to an observer [13]. If event e precedes event f with respect to an observer, then e may potentially cause f . The observed precedence suggests the possibility of an information flow from e to f , but without additional knowledge of the internals of the agents, we cannot be sure that true causation was involved. It is customary to define the local time of an agent as the number of steps it has executed. A vector clock is a vector, each of whose elements corresponds to the local time of each communicating agent. A vector v is considered later than a vector u if v is later on some, and not sooner on any, element. Definition 1. A clock over n agents is an n-ary vector natural numbers. The starting clock is ~ 0 4 Notice that the vector representation is just a convenience. We could just as well use pairs of the form hagent-id, local-timei, which would allow us to model systems of varying membership more easily. Definition 2. Given n-ary vectors u and v, u OE v if and only if 6 Venkatraman & Singh Each agent starts at ~ 0. It increments its entry in that vector whenever it performs a local event [15]. It attaches the entire vector as a timestamp to any message it sends out. When an agent receives a message, it updates its vector clock to be the element-wise maximum of its previous vector and the vector timestamp of the message it received. Intuitively, the message brings news of how far the system has progressed; for some agents, the recipient may have better news already. However, any message it sends after this receive event will have a later timestamp than the message just received.000000000111111111000000000000000000000000000111111111111111111111111111000000000000000000111111111111111111000000000000000000111111111111111111000000000111111111000000000111111111111111111 Auctioneer A Bidder B1 Bidder B2 [1,1,0] [3,2,0] money fish Figure 1. Vector clocks in the fish-market protocol. Example 3. Figure 1 illustrates the evolution of vector timestamps for one possible run of the fish-market protocol. In the run described here, the auctioneer (A) announces a price of 50 for a certain bucket of fish. Bidders B1 and B2 both decline. A lowers the price to 40 and announces it. This time leading A to transfer the fish to B1 and B1 to send money to A. For uniformity, the last two steps are also modeled as communications. The messages are labeled m i to facilitate reference from the text. 2.2. TEMPORAL LOGIC The progression of events, which is inherent in the execution of any protocol, suggests the need for representing and reasoning about time. Temporal logics aamas.tex; 20/02/1999; 16:40; no v. Compliance with Commitment Protocols 7 provide a well-understood means of doing so, and have been applied in various subareas of computer science. Because of their naturalness in expressing properties of systems that may evolve in more than one possible way and for the efficiency of reasoning that they support, the branching-time logics have been especially popular in this regard [10]. Of these, the best known is Computation Tree Logic (CTL), which we adapt here in our formal language L. Conventionally, a model of CTL is expressed as a tree. Each node in the tree is associated with a state of the system being considered; the branches of the tree or paths thus indicate the possible courses of events or ways in which the system's state may evolve. CTL provides a natural means by which to specify acceptable behaviors of the system. The following Backus-Naur Form (BNF) grammar with a distinguished start symbol L gives the syntax of L. L is based on a set \Phi of atomic propo- sitions. Below, slant typeface indicates nonterminals; \Gamma! and j are meta- symbols of BNF specification; - and AE delimit comments; the remaining symbols are terminals. As is customary in formal semantics, we are only concerned with abstract syntax. L1. L \Gamma! Prop -atomic propositions: members of \Phi AE L2. L3. L \Gamma! L - L -conjunctionAE L4. L \Gamma! A P -universal quantification over pathsAE L5. L \Gamma! E P -existential quantification over pathsAE L6. P \Gamma! L U L -until: operator over a single pathAE The meanings of formulas generated from L are given relative to a model and a state in the model. The meanings of formulas generated from P are given relative to a path and a state on the path. The boolean operators are standard. Useful abbreviations include false j (p -:p), for any p 2 \Phi, true j :false, q. The temporal operators A and E are quantifiers over paths. Informally, pUq means that on a given path from the given state, q will eventually hold and p will hold until q holds. Fq means "eventually q" and abbreviates trueUq. Gq means "always q" and abbreviates :F:q. Therefore, EpUq means that on some future path from the given state, q will eventually hold and p will hold until q holds. is a formal model for L. S is a set of states; S \Theta S is a partial order indicating branching time, and I : S 7! P (\Phi) is an interpretation, which tells us which atomic propositions are true in a given state. For t 2 S, P t is the set of paths emanating from t. at t" and M p at t along path P ." M1. M M2. M M3. M M4. M M5. M q and The above is an abstract semantics. In Section 3.3, we specify the concrete form of \Phi, S, !, and I, so the semantics can be exercised in our computations. 3. Approach In their generic forms, both causality and temporal logic are well-known. However, applying them in combination and in the particular manner suggested here is novel to this paper. Temporal logic model checking is usually applied for design-time reasoning [10, pp. 1042-1046]. We are given a specification and an implementation, i.e., program, that is supposed to meet it. A model is generated from the pro- gram. A model checking algorithm determines whether the specification is true in the generated model. However, in an open, heterogeneous environ- ment, a design may not be available at all. For example, the vendors who supply the agents may consider their designs to be trade secrets. By contrast, ours is a run-time approach, and can meaningfully apply model checking even in open settings. This is because it uses a model generated from the joint executions of the agents involved. Model checking in this setting simply determines whether the present execution satisfies the specifi- cation. If an execution respects the given protocol, that does not entail that all executions will, because an agent act inappropriately in other circumstances. However, if an execution is inappropriate, that does entail that the system does not satisfy the protocol. Consequently, although we are verifying specific executions of the multiagent system, we can only falsify (but not verify) the correctness of the construction of the agents in the system. Model checking of the form introduced above may be applied by any observer in the multiagent system. A useful case is when the observer is one of the participating agents. Another useful case is when the observer is some aamas.tex; 20/02/1999; 16:40; no v. Compliance with Commitment Protocols 9 agent dedicated to the task of managing or auditing the interactions of some of the agents in the multiagent system. Potential causality is most often applied in distributed systems to ensure that the messages being sent in a system satisfy causal ordering [3]. Causality motivates vector clocks and vector timestamps on messages, which help ensure correct ordering by having the messaging subsystem reorder and re-transmit messages as needed. This application of causality can be important, but is controversial [4, 6], because its overhead may not always be justifiable. In our approach, the delivery of messages may be noncausal. However, causality serves the important purpose of yielding accurate models of the observations of each agent. These are needed, because in a distributed system, the global model is not appropriate. Creating a monolithic model of the execution of the entire system requires imposing a central authority through which all messages are routed. Adding such an authority would take away many of the advantages that make distributed systems attractive in the first place. Consequently, our method of constructing and reasoning with models should not require a centralized message router \Gamma work from a single vantage of observation, but be able to handle situations where some agents pool their evidence. Such a method turns out to naturally employ the notion of potential causality. 3.1. MODELS FROM OBSERVATIONS The observations made by each agent are essentially a record of the messages it has sent or received. Since each message is given a vector timestamp, the observations can be partially ordered. In general, this order is not total, because messages received from different agents may be mutually unordered. Example 4. Figure 2 shows the models constructed locally from the observations of the auctioneer and a bidder in the run of Example 3. Although a straightforward application of causality, the above example shows how local models may be constructed. Some subtleties are discussed next. As remarked above, commitments give the core meaning of a protocol. Our approach builds on a flexible and powerful variety of social commit- ments, which are the commitments of one agent to another [20]. These commitments are defined relative to a context, which is typically the multiagent system itself. The debtor refers to the agent that makes a commitment, and the creditor to the agent who receives the commitment. Thus we have the following logical form. [1,0,0] [3,2,0] [1,0,0] [1,2,0] Auctioneer A Bidder B1 start start Figure 2. Observations for auctioneer and a bidder in the fish-market protocol. Definition 4. A commitment is an expression C(x; G; p), where x is the debtor, y the creditor, G the context, and p the condition committed to. The expression c is considered true in states where the corresponding commitment exists. Definition 5. A commitment G; p) is base-level if p does not refer to any other commitments; c is a metacommitment if p refers to a base-level commitment (we do not consider higher-order commitments here). Intuitively, a protocol definition is a set of metacommitments for the different roles (along with a mapping of the message tokens to operations on commit- ments). In combination with what the agents communicate, these lead to base-level commitments being created or manipulated, which is primarily how a commitment may be referred to within a protocol. The violation of a base-level commitment can give us proof or the "smoking gun" that an agent is noncompliant. The following operations on commitments define how they may be created or manipulated. When we view commitments as an abstract data type, the operations are methods of that data type. Each operation is realized through a simple message pattern, which states what messages must be communicated among which of the participants and in what order. For the operations on commitments we consider, the patterns aamas.tex; 20/02/1999; 16:40; no v. Compliance with Commitment Protocols 11 are simple. As described below, most patterns require only a single message, but some require three messages. Obeying the specified patterns ensures that the local models have the information necessary for testing compliance. That the given operation can be performed at all depends on whether the proto- col, through its metacommitments, allows that operation. However, when an operation is allowed, it affects the agents' commitments. For simplicity, we assume that the operations on commitments are given a deterministic inter- pretation. Here z is an agent and G; p) is a commitment. O1. Create(x; c) instantiates a commitment c. Create is typically performed as a consequence of the commitment's debtor promising something contractually or by the creditor exercising a metacommitment previously made by the debtor. Create usually requires a message from the debtor to the creditor. O2. Discharge(x; c) satisfies the commitment c. It is performed by the debtor concurrently with the actions that lead to the given condition being sat- isfied, e.g., the delivery of promised goods or funds. For simplicity, we treat the discharge actions as performed only when the proposition p is true. Thus the discharge actions are detached, meaning that p can be treated as true in the given moment. We model the discharge as a single message from the debtor to the creditor. O3. Cancel(x; c) revokes the commitment c. It can be performed by the debtor as a single message. At the end of this action, :c usually holds. However, depending on the existing metacommitments, the cancel of one commitment may lead to the create of other commitments. O4. Release(G; c) or release(y; c) essentially eliminates the commitment c. This is distinguished from both discharge and cancel, because release does not mean success or failure, although it lets the debtor off the hook. At the end of this action, :c usually holds. The release action may be performed by the context or the creditor of the given commitment, also as a single message. Because release is not performed by the debtor, different metacommitments apply than for cancel. O5. Delegate(x; z; c) shifts the role of debtor to another agent within the same context, and can be performed by the (old) debtor (or the context). G; p). At the end of the delegate action, c 0 - :c holds. To prevent the risk of miscommunication, we require the creditor to also be involved in the message pattern. Figure 3(l) shows the associated pat- tern. The first message sets up the commitment c from x to y and is not part of the pattern. When x delegates the commitment c to z, x tells both y and z that the commitment is delegated. z is now committed to y. Later aamas.tex; 20/02/1999; 16:40; no v. delegate(x,z,c) y z x y z assign(y,z,c) create(x,c) assign(y,z,c) delegate(x,z,c) discharge(x,c) discharge(x,c) x create(x,c) Figure 3. Message pattern for delegate (l) and assign (r). z may discharge the commitment. The two delegate messages constitute the pattern. O6. Assign(y; z; c) transfers a commitment to another creditor within the same context, and can be performed by the present creditor or the con- text. Let c G; p). At the end of the assign action, c 0 -:c holds. Here we require that the new creditor and the debtor are also involved as shown in Figure 3(r). The figure shows only the general pattern. Here x is committed to y. When y assigns the commitment to z, y tells both x and z (so z knows it is the new creditor). Eventually, x should discharge the commitment to z. A potentially tricky situation is if x discharges the commitment c even as y is assigning c to z (i.e., the messages cross). In this case, we require y to discharge the commitment to z-essentially by forwarding the contents of the message from x. Thus the worst case requires three messages. We write the operations as propositions indicating successful execution. Based on the applicable metacommitments, each operation may entail additional operations that take place implicitly. Definition 6. A commitment c is resolved through a release, discharge, can- cel, delegate, or assign performed on c. c ceases to exist when resolved. How- ever, a new commitment is created for delegate or assign. (New commitments created because of some existing metacommitment are not included in the definition of resolution. Theorem 1 states that the creditor knows the disposition of any commitments due to it. This result helps establish that the creditor can always determine compliance of others relative to what was committed to it. Compliance with Commitment Protocols 13 Theorem 1. If message m i creates commitment c and message m j resolves c, then the creditor of c sees both m i and m j . Proof. By inspection of the message patterns constructed for the various operations on commitments. Definition 7. A commitment c is ultimately resolved through a release, dis- charge, or cancel performed on c, or through the ultimate resolution of any commitments created by the delegate or assign of c. Theorem 2 essentially states that the creation and ultimate resolution of a commitment occur along the same causal path. This is important, because it legitimizes a significant optimization below. Indeed, we defined the above message patterns so we would obtain Theorem 2. Theorem 2. If message m i creates commitment c and message m j ultimately resolves Proof. By inspection of the message patterns constructed for the various operations on commitments. 3.2. SPECIFYING PROTOCOLS We first consider the coordination and then the commitment aspects of com- pliance. A skeleton is a coarse description of how an agent may behave [18]. A skeleton is associated with each role in the given multiagent system to specify how an agent playing that role may behave in order to coordinate with others. Coordination includes the simpler aspects of interaction, e.g., turn-taking. Coordination is required so that the agents' commitments make sense. For instance, a bidder should not make a bid prior to the advertise- ment; otherwise, the commitment content of the bid would not even be fully defined. The skeletons may be constructed by introspection or through the use of a suitable methodology [19]. No matter how they are created, the skeletons are the first line of compliance testing, because an agent that does not comply with the skeleton for its role is automatically in violation. So as to concentrate on commitments in this paper, we postulate that a "proxy" object is interposed between an agent and the rest of the system and ensures that the agent follows the dictates of the skeleton of its role. We now define the syntax of the specification language through the following whose start symbol is Protocol. The braces f and g indicate that the enclosed item is repeated 0 or more times. L7. Protocol \Gamma! fMetag fMessageg aamas.tex; 20/02/1999; 16:40; no v. 14 Venkatraman & Singh L8. Message \Gamma! Token: Commitment -messages correspond to L9. Meta \Gamma! C(Debtor, Creditor, Context, MetaProp) L11. Bool \Gamma! -Boolean combinations ofAE Act j Commitment j Dom L12. Act \Gamma! Operation(Agent, Commitment) L13. Operation \Gamma! -the six operations of Section 3.1AE L14. Commitment \Gamma! Meta j C(Debtor, Creditor, Context, AFDom) L15. Dom \Gamma! -domain-specific conceptsAE The above language embeds a subset of L. Our approach is to detach the outer actions and commitments, so we can process the inner L part as a temporal logic. By using commitments and actions on them, instead of simple domain propositions, we can capture a variety of subtle situations, e.g., to distinguish between release and cancel both of which result in the given commitment being removed. Example 5 applies the above language on the fish-market protocol. Example 5. The messages in Figure 1 can be given a content based on the following definitions. Here FM is the fish-market context. proposition meaning the fish is delivered proposition meaning that the appropriate money is paid (subscripted to allow different prices) an abbreviation for ))])-meaning the bidder promises to pay money i if given the fish an abbreviation for AFfish))])-meaning the auctioneer offers to deliver the fish if he gets a bid for money i an abbreviation for ))-meaning that at least two bidders have bid for the fish at price i proposition meaning the fish is spoiled Armed with the above, we can now state the commitments associated with the different messages in the fish market protocol. Compliance with Commitment Protocols 15 Payment of i from Delivering fish to \Gamma Yes from B j (for price i): create(B \Gamma No from B j (for price i): true Further, the protocol includes metacommitments that are not associated with any single message. In the present protocol, these metacommitments are of the context itself to release a committing party under certain circumstances. For practical purposes, we could treat these as metacommitments of the creditor Bad In addition, in a monotonic framework, we would also need to state the completion requirements to ensure that only the above actions are performed. The auctioneer does not commit to a price if no bid is received. If more than one bid is received, the auctioneer is released from the commitment. Notice that the auctioneer can exit the market or adjust the price in any direction if a unique Yes is not received for the current price money i . It would neither be rational for the auctioneer to raise the price if there are no takers at the present price, nor to lower the price if takers are available. However, the protocol per se does not legislate against either behavior. The No messages have no significance on commitments. They serve only to assist in the coordination so the context can determine if enough bids are received. The lower-level aspects of coordination are not being studied in this paper. Now we can see how the reasoning takes place in a successful run of the protocol. Example 6. The auctioneer sends out an advertisement, which commits the auctioneer to supplying the fish if he receives a suitable bid. This commitment will be discharged if AG[Bid i (B holds. When Bid i (B j ) is sent by B j , the bidder is committed to the bid, which is discharged if AG[fish ! holds. To discharge the adver- tisement, the auctioneer must eventually create a commitment to eventually supply the fish. If he does not create this commitment, he is in violation. If he aamas.tex; 20/02/1999; 16:40; no v. Venkatraman & Singh creates it, but does not supply the fish, he is still in violation. If he supplies the fish, the bidder is then committed to eventually forming a commitment to supply the money. If the bidder does so, the protocol is executed successfully. 3.3. REASONING WITH THE CONCRETE MODEL Now we explain the main reasoning steps in our approach and show that they are sound. The main reasoning with models applies the CTL model-checking algorithm on a model and a formula denoting the conjunction of the specifi- cations. The algorithm evaluates whether the formula holds in the initial state of the model. Thus a concrete version of the model M (see Section 2.2) is es- sential. For the purposes of the semantics, we must define a global model with respect to which commitment protocols may be specified. Intuitively, a protocol specification tells us which behaviors of the entire system are correct. Thus, it corresponds naturally to a global model in which those behaviors can be defined. Our specific concrete model identifies states with messages. Recall that the timestamp of a message is the clock vector attached to it. The states are ordered according to the timestamps of the messages. The proposition true in a state is the one corresponding to the operation that is performed by the message. Definition 8. Definition 9. For s; Definition 10. For s 2 Q, operations executed by message sg The structure is a quasimodel. (Here and below, we assume that ! and I are appropriately projected to the available states.) MQ is structurally a model, because it matches the requirements of Definition 3. How- ever, MQ is not a model of the computations that may take place, because the branches in MQ are concurrent events and do not individually correspond to a single path. A quasimodel can be mapped to a model, M with an initial state ~ 0, by including all possible interleavings of the transitions. That is, S would include a distinct state for every message in each possible ordering of the messages in Q that is consistent with the temporal order ! of MQ . The relation ! can be suitably defined for M S . However, there is potentially an exponential blowup in that the size of S may be exponentially greater than the size of Q. Theorem 3 shows that naively treating a quasimodel as if it were a model is correct. Thus, the above blowup can be eliminated entirely. Our construction aamas.tex; 20/02/1999; 16:40; no v. Compliance with Commitment Protocols 17 ensures that all the events relevant to another event are totally ordered with respect to each other. Notice that, as showing in Figure 3, the construction may appear to require one more message than necessary for the assign and delegate operations. This linear amount of extra work (for the entire set of messages), however, pays off in reducing the complexity of our reasoning algorithm. In the following, p refers to the proposition (of the form AG[q ! AFr]) of a metacommitment, which becomes true when the metacommitment is discharged. Definition 11. For a proposition p, p T is the proposition obtained by substituting EF for AF in p. Theorem 3. MQ p. Proof. From Theorem 2 and the restricted structure of MQ . The above results show that compliance can be tested and without blowing up the model unnecessarily. However, we would like to test for compliance based on local information-so that any agent can decide for itself whether it has been wronged by another. For this reason, we would like to be able to project the global model onto local models for each agent, while ensuring that the local models carry enough information that they are indeed usable in isolation from other local models. Accordingly, we can define the construction of local models corresponding to an agent's observations. This is simply by defining a subset of S for a given agent a. Definition 12. S a is a message from or to ag. M a = hS a ; !; Ii. Theorem 4 shows that if we restrict attention to commitments that the given agent can observe, then the projected quasimodel yields all and only the correct conclusions relative to the global quasimodel. Thus, if the interested party is vigilant, it can check if anyone else violated the protocol. Theorem 4. M a only if MQ that a sees all the commitments mentioned in p. Proof. From Theorem 2 and the construction of M a . Example 7. If one of the bidders backs down from a successful bid, the auctioneer immediately can establish that he is cheating, because the auctioneer is the creditor for the bidder's commitment. However, a bidder cannot ordinarily decide whether the auctioneer is noncompliant, because the bidder does not see all relevant commitments based on which the auctioneer may be released from a commitment to the bidder. Theorem 5 lifts the above results to sets of agents. Thus, a set of agents may pool their evidence in order to establish whether a third party is noncompliant. Venkatraman & Singh Thus, in a setting with two bidders, a model that includes all their evidence can be used to determine whether the auctioneer is noncompliant. Ordinarily, the bidders would have to explicitly pool their information to do so. However, in a broadcast-based or outcry protocol (like a traditional fish market in which everyone is screaming), the larger model can be built by anyone who hears all the messages. Let A be a set of agents. Definition 13. a2A S a . Theorem 5. Let the commitments observed by agents in A include all the commitments in p. Then MA Proof. From Theorem 2 and the construction of MA . Information about commitments that have been resolved, i.e., are not pending, is not needed in the algorithm, and can be safely deleted from each observer's model. This is accomplished by searching backward in time whenever something is added to the model. Pruning extraneous messages from each observer's model reduces the size of the model and facilitates reasoning about it. This simplification is sound, because the CTL specifications do not include nested commitments. Mapping from an event-based to a state-based representation, we should consider every event as potentially corresponding to a state change. This approach would lead to a large model, which accommodates not only the occurrence of public events such as message transmissions, but also local events. Such an approach would thus capture the evolution of the agent's knowledge about the progress of the system, which would help in accommodating unreliable messaging. Our approach, as described above, loses some of the agents' knowledge by not separating events and states, but has all the details we need to assess compliance assuming reliable messaging. 4. Discussion Given the autonomy and heterogeneity of agents, the most natural way to treat interactions is as communications. A communication protocol involves the exchange of messages with a streamlined set of tokens. Traditionally, these tokens are not given any meaning except through reference to the beliefs or intentions of the communicating agents. By contrast, our approach assigns public, i.e., observable, meanings in terms of social commitments. Viewed in this light, every communication protocol is a commitment protocol. Formulating and testing compliance of autonomous and heterogeneous agents is a key prerequisite for the effective application of multiagent systems in open environments. As asserted by Chiariglione, minimal specifications based on external behavior will maximize interoperability [7]. The research Compliance with Commitment Protocols 19 community has not paid sufficient attention to this important requirement. A glaring shortcoming of most existing semantics for agent communication languages is their fundamental inability to allow testing for the compliance of an agent [16, 22]. The present approach shows how that might be carried out. While the purpose of the protocols is to specify legal behavior, they should not specify rational behavior. Rational behavior may result as an indirect consequence of obeying the protocols. However, not adding rationality requirements leads to more succinct specifications and also allows agents to participate even if their rationality cannot be established by their designers. The compliance checking procedure can be used by any agent who participates in, or observes, a commitment protocol. There are two obvious uses. One, the agent can track which of the commitments made by others are pending or have been violated. Two, it might track which of its own commitments are pending or whose satisfaction has not been acknowledged by others. The agent can thus use the compliance checking procedure as an input to its normal processes of deliberation to guide its interactions with other agents. We have so far discussed how to detect violations. Once an agent detects a violation, as far as the above method is concerned, it may proceed in any way. However, some likely candidates are the following. The wronged agent may inform the agents who appeared to have violated their commitments and ask them to respect the applicable metacommitments inform the context, who might penalize the guilty parties, if any; the context may require additional information, e.g., certified logs of the messages sent by the different agents, to establish that some agents are in violation. agents in an attempt to spoil the reputation of the guilty parties. 4.1. LITERATURE Some of the important strands of research of relevance to commitment protocols have been carried out before. However, the synthesis, enhancement, and application of these techniques on multiagent commitment protocols is a novel contribution of this paper. Interaction (rightly) continues to draw much attention from researchers. Still, most current approaches do not consider an explicit execution architecture (however, there are some notable exceptions, e.g., [8, 9, 18]). Other approaches lack a formal underpinning; still others focus primarily on monolithic finite-state machine representations for proto- cols. Such representations can capture only the lowest levels of a multiagent Venkatraman & Singh interaction, and their monolithicity does not accord well with distributed execution and compliance testing. Model checking has recently drawn much attention in the multiagent community, e.g., [2, 17]. However, these approaches consider knowledge and related concepts and are thus not directly applicable for behavior-based compliance. 4.2. FUTURE DIRECTIONS The present approach highlights the synergies between distributed computing and multiagent systems. Since both fields have advanced in different direc- tions, a number of important technical problems can be addressed by their proper synthesis. One aspect relates to situations where the agents may suffer a Byzantine failure or act maliciously. Such agents may fake messages or deny receiving them. How can they be detected by the other agents? Another aspect is to capture additional structural properties of the interactions so that noncompliant agents can be more readily detected. Alternatively, we might offer an assistance to designers by synthesizing skeletons of agents who participate properly in commitment protocols. Lastly, it is well-known that there can be far more potential causes than real causes [15]. Can we analyze conversations or place additional, but reasonable, restrictions on the agents that would help focus their interactions on the true relationships between their respective computations? We defer these topics to future research. Acknowledgements This work is supported by the National Science Foundation under grants IIS- 9529179 and IIS-9624425, and IBM corporation. 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Singh, Modeling exceptions via commitment protocols, Proceedings of the fourth international joint conference on Autonomous agents and multiagent systems, July 25-29, 2005, The Netherlands Lai Xu, A multi-party contract model, ACM SIGecom Exchanges, v.5 n.1, p.13-23, July, 2004 Pinar Yolum, Towards design tools for protocol development, Proceedings of the fourth international joint conference on Autonomous agents and multiagent systems, July 25-29, 2005, The Netherlands Jeremy Pitt , Lloyd Kamara , Marek Sergot , Alexander Artikis, Formalization of a voting protocol for virtual organizations, Proceedings of the fourth international joint conference on Autonomous agents and multiagent systems, July 25-29, 2005, The Netherlands Pnar Yolum, Design time analysis of multiagent protocols, Data & Knowledge Engineering, v.63 n.1, p.137-154, October, 2007 Chrysanthos Dellarocas , Mark Klein , Juan Antonio Rodriguez-Aguilar, An exception-handling architecture for open electronic marketplaces of contract net software agents, Proceedings of the 2nd ACM conference on Electronic commerce, p.225-232, October 17-20, 2000, Minneapolis, Minnesota, United States Frank Guerin, Applying game theory mechanisms in open agent systems with complete information, Autonomous Agents and Multi-Agent Systems, v.15 n.2, p.109-146, October 2007 Peter McBurney , Simon Parsons, Posit spaces: a performative model of e-commerce, Proceedings of the second international joint conference on Autonomous agents and multiagent systems, July 14-18, 2003, Melbourne, Australia Lalana Kagal , Tim Finin, Modeling conversation policies using permissions and obligations, Autonomous Agents and Multi-Agent Systems, v.14 n.2, p.187-206, April 2007 Pnar Yolum , Munindar P. Singh, Reasoning about Commitments in the Event Calculus: An Approach for Specifying and Executing Protocols, Annals of Mathematics and Artificial Intelligence, v.42 n.1-3, p.227-253, September 2004 Munindar P. Singh, Synthesizing Coordination Requirements for Heterogeneous Autonomous Agents, Autonomous Agents and Multi-Agent Systems, v.3 n.2, p.107-132, June 2000 Phillipa Oaks , Arthur Hofstede, Guided interaction: A mechanism to enable ad hoc service interaction, Information Systems Frontiers, v.9 n.1, p.29-51, March 2007 Chihab Hanachi , Christophe Sibertin-Blanc, Protocol Moderators as Active Middle-Agents in Multi-Agent Systems, Autonomous Agents and Multi-Agent Systems, v.8 n.2, p.131-164, March 2004

protocols;formal methods;commitments;causality;temporal logic

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Semantic Issues in the Verification of Agent Communication Languages.

This article examines the issue of developing semantics for agent communication languages. In particular, it considers the problem of giving a verifiable semantics for such languagesa semantics where conformance (or otherwise) to the semantics could be determined by an independent observer. These problems are precisely defined in an abstract formal framework. Using this framework, a number of example agent communication frameworks are defined. A discussion is then presented, of the various options open to designers of agent communication languages, with respect the problem of verifying conformance.

Introduction One of the main reasons why multi-agent systems are currently a major area of research and development activity is that they are seen as a key enabling technology for the Internet-wide electronic commerce systems that are widely predicted to emerge in the near future [20]. If this vision of large-scale, open multi-agent systems is to be realised, then the fundamental problem of inter-operability must be addressed. It must be possible for agents built by dierent organisations using dierent hardware and software platforms to safely communicate with one-another via a common language with a universally agreed semantics. The inter-operability requirement has led to the development of several standardised agent communication languages (acls) [30, 19]. However, to gain acceptance, particularly for sensitive applications such as electronic commerce, it must be possible to determine whether or not any system that claims to conform to an acl standard actually does so. We say that an acl standard is veriable if it enjoys this property. Unfortunately, veriability has to date received little attention by the standards community (although it has been recognised as an issue [19, p46]). In this article, we establish a simple formal framework that allows us to precisely dene what it means for an acl to be veriable. This framework is dened in section 3, following a brief discussion of the background to this work. We then formally dene what it means for an acl to be veriable in section 4. The basic idea is to show how demonstrating conformance to an acl semantics can be seen as a verication problem in the standard software engineering sense [7]. Demonstrating c 1999 Kluwer Academic Publishers. Printed in the Netherlands. Michael Wooldridge that a program semantically complies to a standard involves showing that the program satises the specication given by the semantics. If the semantics are logical, then demonstrating compliance thus reduces to a proof problem. We discuss the practical implications of these definitions in section 4.1. In section 5, we give examples of some acls, and show that some of these are veriable, while others are not. In section 6, we discuss an alternative approach to verication, in which verication is done via model checking rather than proof. Finally, in section 7, we discuss the implications of our results, with emphasis on future directions for work on veriable acls. 2. Background Current techniques for developing the semantics of acls trace their origins to speech act theory. In this section, we give a brief overview of this work. 2.1. Speech Acts The theory of speech acts is generally recognised as having begun in the work of the philosopher John Austin [4]. Austin noted that a certain class of natural language utterances | hereafter referred to as speech acts | had the characteristics of actions, in the sense that they change the state of the world in a way analogous to physical actions. It may seem strange to think of utterances changing the world in the way that physical actions do. If we pick up a block from a table (to use an overworked but traditional example), then the world has changed in an obvious way. But how does speech change the world? Austin gave as paradigm examples declaring war and saying \I now pronounce man and wife". Stated in the appropriate circumstances, these utterances clearly change the state of the world in a very tangible way 1 . Austin identied a number of performative verbs, which correspond to various dierent types of speech acts. Examples of such performative verbs are request, inform, and promise. In addition, Austin distinguished three dierent aspects of speech acts: the locutionary act, or act of making an utterance (e.g., saying \Please make some tea"), the illocutionary act, or action performed in saying something (e.g., \He requested me to make some tea"), and perlocution, or eect of the act (e.g., \He got me to make tea"). 1 Notice that when referring to the eects of communication, we are ignoring \pathological" cases, such as shouting while on a ski run and causing an avalanche. Similarly, we will ignore \microscopic" eects (such as the minute changes in pressure or temperature in a room caused by speaking). Semantic Issues in Agent Communication 3 Austin referred to the conditions required for the successful completion of performatives as felicity conditions. He recognized three important felicity conditions: 1. a) There must be an accepted conventional procedure for the performative b) The circumstances and persons must be as specied in the procedure. 2. The procedure must be executed correctly and completely. 3. The act must be sincere, and any uptake required must be com- pleted, insofar as is possible. Austin's work was rened and considerably extended by Searle, in his 1969 book Speech Acts [38]. Searle identied several properties that must hold for a speech act performed between a hearer and a speaker to succeed, including normal I/O conditions, preparatory conditions, and sincerity conditions. For example, consider a request by speaker to hearer to perform action: 1. Normal I/O conditions. Normal I/O conditions state that hearer is able to hear the request, (thus must not be deaf, . ), the act was performed in normal circumstances (not in a lm or play, . ), etc. 2. Preparatory conditions. The preparatory conditions state what must be true of the world in order that speaker correctly choose the speech act. In this case, hearer must be able to perform action, and speaker must believe that hearer is able to perform action. Also, it must not be obvious that hearer will do action anyway. 3. Sincerity conditions. These conditions distinguish sincere performances of the request; an insincere performance of the act might occur if speaker did not really want action to be performed. Searle also gave a ve-point typology of speech acts: 1. Representatives. A representative act commits the speaker to the truth of an expressed proposition. The paradigm case is informing. 2. Directives. A directive is an attempt on the part of the speaker to get the hearer to do something. Paradigm case: requesting. 3. Commissives. Commit the speaker to a course of action. Paradigm case: promising. 4 Michael Wooldridge 4. Expressives. Express some psychological state (e.g., gratitude). Paradigm case: thanking. 5. Declarations. Eect some changes in an institutional state of aairs. Paradigm case: declaring war. 2.2. Speech Acts in Artificial Intelligence In the late 1960s and early 1970s, a number of researchers in arti- cial intelligence (ai) began to build systems that could plan how to autonomously achieve goals [2]. Clearly, if such a system is required to interact with humans or other autonomous agents, then such plans must include speech actions. This introduced the question of how the properties of speech acts could be represented such that planning systems could reason about them. Cohen and Perrault [15] gave an account of the semantics of speech acts by using techniques developed in ai planning research [18]. The aim of their work was to develop a theory of speech acts: \[B]y modelling them in a planning system as operators dened . in terms of speakers and hearers beliefs and goals. Thus speech acts are treated in the same way as physical actions". [15] The formalism chosen by Cohen and Perrault was the strips nota- tion, in which the properties of an action are characterised via pre-and post-conditions [18]. The idea is very similar to Hoare logic [24]. Cohen and Perrault demonstrated how the pre- and post-conditions of speech acts such as request could be represented in a multi-modal logic containing operators for describing the beliefs, abilities, and wants of the participants in the speech act. Consider the Request act. The aim of the Request act will be for a speaker to get a hearer to perform some action. Figure 1 denes the Request act. Two preconditions are stated: the \cando.pr" (can-do pre- conditions), and \want.pr" (want pre-conditions). The cando.pr states that for the successful completion of the Request , two conditions must hold. First, the speaker must believe that the hearer of the Request is able to perform the action. Second, the speaker must believe that the hearer also believes it has the ability to perform the action. The want.pr states that in order for the Request to be successful, the speaker must also believe it actually wants the Request to be performed. If the pre-conditions of the Request are fullled, then the Request will be successful: the result (dened by the \eect" part of the denition) will be that the hearer believes the speaker believes it wants some action to be performed. Semantic Issues in Agent Communication 5 Preconditions Cando.pr (S BELIEVE (H CANDO Want.pr (S BELIEVE (S WANT requestInstance)) Effect (H BELIEVE (S BELIEVE (S WANT ))) Preconditions Cando.pr Want.pr Effect Figure 1. Denitions from the Plan-Based Theory of Speech Acts While the successful completion of the Request ensures that the hearer is aware of the speaker's desires, it is not enough in itself to guarantee that the desired action is actually performed. This is because the denition of Request only models the illocutionary force of the act. It says nothing of the perlocutionary force. What is required is a mediating act. Table 1 gives a denition of CauseToWant , which is an example of such an act. By this denition, an agent will come to believe it wants to do something if it believes that another agent believes it wants to do it. This denition could clearly be extended by adding more pre-conditions, perhaps to do with beliefs about social relationships or power structures. Using these ideas, and borrowing a formalism for representing the mental state of agents that was developed by Robert Moore [31], Douglas Appelt was able to implement a system that was capable of planning to perform speech acts [3]. 2.3. Speech Acts as Rational Action While the plan-based theory of speech acts was a major step forward, it was recognised that a theory of speech acts should be rooted in a more general theory of rational action. This observation led Cohen and Levesque to develop a theory in which speech acts were modelled as actions performed by rational agents in the furtherance of their intentions [13]. The foundation upon which they built this model of rational action was their theory of intention, described in [12]. The for- 6 Michael Wooldridge mal theory is too complex to describe here, but as a avour, here is the Cohen-Levesque denition of requesting, paraphrased in English [13, p241]: A request is an attempt on the part of spkr , by doing e, to bring about a state where, ideally, (i) addr intends , (relative to the spkr still having that goal, and addr still being helpfully inclined to spkr ), and (ii) addr actually eventually does , or at least brings about a state where addr believes it is mutually believed that it wants the ideal situation. Actions in the Cohen-Levesque framework were modelled using techniques adapted from dynamic logic [23]. 2.4. Agent Communication Languages: KQML and FIPA Throughout the 1980s and 1990s, interest in multi-agent systems developed rapidly [6, 41]. An obvious problem in multi-agent systems is how to get agents to communicate with one-another | the inter-operability issue referred to in the introduction. To this end, in the early 1990s, the darpa Knowledge Sharing Eort (kse) began to develop the Knowledge Query and Manipulation Language (kqml) and the associated Knowledge Interchange Format (kif) as a common frame-work via which multiple expert systems (cf. agents) could exchange knowledge [33, 30]. kqml is essentially an \outer" language for messages: it denes a simple lisp-like format for messages, and 41 performatives, or message types, that dene the intended meaning of a message. Example kqml performatives include ask-if and tell. The content of messages was not considered part of the kqml standard, but kif was also dened, to express such content. kif is essentially classical rst-order predicate logic, recast in a lisp-like syntax. To better understand the kqml language, consider the following example [30, p354]: (ask-one :content (PRICE IBM ?price) :receiver stock-server :language LPROLOG :ontology NYSE-TICKS The intuitive interpretation of this message is that the sender is asking about the price of ibm stock. The performative is ask-one, which an agent will use to ask a question of another agent where exactly one reply Semantic Issues in Agent Communication 7 is needed. The various other components of this message represent its attributes. The most important of these is the :content eld, which species the message content. In this case, the content simply asks for the price of ibm shares. The :receiver attribute species the intended recipient of the message, the :language attribute species that the language in which the content is expressed is called LPROLOG (the recipient is assumed to \understand" LPROLOG), and the nal :ontology attribute denes the terminology used in the message. Formal denitions of the syntax of kqml and kif were developed by the kse, but kqml lacked any formal semantics until Labrou and Finin's [26]. These semantics were presented using a pre- and post-condition closely related to Cohen and Perrault's plan-based theory of speech acts [15]. These pre- and post-conditions were specied by Labrou and Finin using a logical language containing modalities for belief, knowledge, wanting, and intending. However, Labrou and Finin recognised that any commitment to a particular semantics for this logic itself would be contentious, and so they refrained from giving it a semantics. However, this rather begs the question of whether their semantics are actually well-founded. We return to this issue later. The take-up of kqml by the multi-agent systems community was signicant. However, Cohen and Levesque (among others) criticized kqml on a number of grounds [14], the most important of which being that, the language was missing an entire class of performatives | commissives, by which one agent makes a commitment to another. As Cohen and Levesque point out, it is di-cult to see how many multi-agent scenarios could be implemented without commissives, which appear to be important if agents are to coordinate their actions with one-another [25]. In 1995, the Foundation for Intelligent Physical Agents (fipa) began its work on developing standards for agent systems. The centrepiece of this initiative is the development of an acl [19] 2 . This acl is supercially similar to kqml: it denes an \outer" language for messages, it denes 20 performatives (such as inform) for dening the intended interpretation of messages, and it does not mandate any specic language for message content. In addition, the concrete syntax for fipa acl messages closely resembles that of kqml. Here is an example of a fipa acl message (from [19, p10]): (inform :sender agent1 simply refer to their acl as \acl", which can result in confusion when discussing acls in general. To avoid ambiguity, we will always refer to \the fipa acl". 8 Michael Wooldridge :receiver agent2 :content (price good2 150) :language sl :ontology hpl-auction Even a supercial glance conrms that the fipa acl is similar to kqml; the relationship is discussed in [19, pp68{69]. The fipa acl has been given a formal semantics, in terms of a Semantic Language (sl). The approach adopted for dening these semantics draws heavily on [13], but in particular on Sadek's enhancements to this work [9]. sl is a quantied multi-modal logic, which contains modal operators for referring to the beliefs, desires, and uncertain beliefs of a- gents, as well as a simple dynamic logic-style apparatus for representing agent's actions. The semantics of the fipa acl map each acl message to a formula of sl, which denes a constraint that the sender of the message must satisfy if it is to be considered as conforming to the fipa acl standard. fipa refer to this constraint as the feasibility condition. The semantics also map each message to an sl-formula which denes the rational eect of the action. The rational eect of a messages is its purpose: what an agent will be attempting to achieve in sending the message (cf. perlocutionary act). However, in a society of autonomous agents, the rational eect of a message cannot (and should not) be guaranteed. Hence conformance does not require the recipient of a message to respect the rational eect part of the acl semantics | only the feasibility condition. To illustrate the fipa approach, we give an example of the semantics of the fipa inform performative [19, p25]: (1) The B i is a modal connective for referring to the beliefs of agents (see e.g., [21]); Bif is a modal connective that allows us to express whether an agent has a denite opinion one way or the other about the truth or falsity of its parameter; and U is a modal connective that allows us to represent the fact that an agent is \uncertain" about its parameter. Thus an agent i sending an inform message with content ' to agent j will be respecting the semantics of the fipa acl if it believes ', and it it not the case that it believes of j either that j believes whether ' is true or false, or that j is uncertain of the truth or falsity of '. fipa recognise that \demonstrating in an unambiguous way that a given agent implementation is correct with respect to [the semantics] Semantic Issues in Agent Communication 9 is not a problem which has been solved" [19, p46], and identify it as an area of future work. (Checking that an implementation respects the syntax of an acl like kqml or fipa is, of course, trivial.) If an agent communication language such as fipa's acl is ever to be widely used | particularly for such sensitive applications as electronic commerce | then such conformance testing is obviously crucial. However, the problem of conformance testing (verication) is not actually given a concrete denition in [19], and no indication is given of how it might be done. In short, the aim of the remainder of this article is to unambiguously dene what it means for an agent communication language such as that dened by fipa to be veriable, and then to investigate the issues surrounding such verication. 3. Agent Communication Frameworks In this section, we present an abstract framework that allows us to precisely dene the veriable acl semantics problem. First, we will assume that we have a set Ag ng of agent names | these are the unique identiers of agents that will be sending messages to one another in a system. We shall assume that agents communicate using a communication language L C . This acl may be kqml together with kif [26], it may be the fipa-97 communication language [19], or some other proprietary language. The exact nature of L C is not important for our purposes. The only requirements that we place on L C are that it has a well- dened syntax and a well-dened semantics. The syntax identies a set w (L C ) of well-formed formulae of L C | syntactically acceptable constructions of L C . Since we usually think of formulae of L C as being messages, we use (with annotations: to stand for members of w (L C ). The semantics of L C are assumed to be dened in terms of a second language L S , which we shall call the semantic language. The idea is that if an agent sends a message, then the meaning of sending this message is dened by a formula of L S . This formula denes what fipa [19, p48] refer to as the feasibility pre-condition | essentially, a constraint that the sender of the message must satisfy in order to be regarded as being \sincere" in sending the message. For example, the feasibility pre-condition for an inform act would typically state that the sender of an inform must believe the content of the message, otherwise the sender is not being sincere. Michael Wooldridge The idea of dening the semantics of one language in terms of another might seem strange, but the technique is common in computer science: when Hoare-logic style semantics are given for programming lan- guages, the semantics of a program written in, for example, pascal or c are dened in terms of a second language | that of classical rst-order logic [24]; an increasingly common approach to dening the semantics of many programming languages is to give them a temporal seman- tics, whereby the semantics of a program in a language such as c or pascal are dened as a formula of temporal logic [28]. Note that in this article we are not concerned with the eects that messages have on recipients. This is because although the \rational eect" of a message on its recipient is the reason that the sender will send a message (e.g., agent i informs agent j of ' because i wants j to believe '), the sender can have no guarantee that the recipient will even receive the message, still less that it will have the intended eect. The key to our notion of semantics is therefore what properties must hold of the sender of a message, in order that it can be considered to be sincere in sending it. Formally, the semantics of the acl L C are given by a function which maps a single message of L C to a single formula of L S , which represents the semantics of . Note that the \sincerity condition" acts in eect like a specication (in the software engineering sense), which must be satised by any agent that claims to conform to the semantics. Verifying that an agent program conforms to the semantics is thus a process of checking that the program satises this specication. To make the idea concrete, recall the fipa semantics of inform messages, given in (1), above. In our framework, we can express the fipa semantics as It should be obvious how this corresponds to the fipa denition. In order that the semantics of L C be well-dened, we must also have a semantics for our semantic language L S itself. While there is no reason in principle why we should not dene the semantics of L S in terms of a further language L S 0 , (and so on), we assume without loss Semantic Issues in Agent Communication 11 of generality that the semantics of L S are given with respect to a class logical models for L S . More precisely, the semantics of L S will be dened via a satisfaction relation \j= S ", where By convention, if M 2 mod(L S ) and ' 2 w (L S ) then we write M ' to indicate that ('; M ) 2 then we read this as \' is satised (or equivalently, is true) in M ". The meaning of a formula ' of L S is then the set of models in which ' is satised. We dene a function such that if ' 2 w (L S ), then is the set of models in which ' is Agents are assumed to be implemented by programs, and we let stand for the set of all such agent programs. For each agent i 2 Ag , we assume that i 2 is the program that implements it. For our purposes, the contents of are not important | they may be java, c, or c++ programs, for example. At any given moment, we assume that a program i may be in any of a set L i of local states. The local state of a program is essentially just a snapshot of the agent's memory at some instant in time. As an agent program i executes, it will perform operations (such as assignment statements) that modify its state. Let i2Ag L i be the set of all local states. We use l (with annotations: to stand for members of L. One of the key activities of agent programs is communication: they send and receive messages, which are formulae of the communication language L C . We assume that we can identify when an agent emits such a message, and write send( l ) to indicate the fact that agent implemented by program i 2 , sends a message 2 L C when in state l 2 L i . We now dene what we mean by the semantics of an agent program. Intuitively, the idea is that when an agent program i is in state l , we must be able to characterise the properties of the program as a formula of the semantic language L S . This formula is the theory of the program. In theoretical computer science, the derivation of a program's theory is the rst step to reasoning about its behaviour. In particular, a program theory is the basis upon which we can verify that the program satises its specication. Formally, a program semantics is a function that maps a pair consisting of an agent program and a local state to a formula Michael Wooldridge Programs/state | L language | L Sn Communication language | L C Model structures for L S | mod(L S )1 Figure 2. The components of an agent communication framework. L S of the semantic language. Note that the semantics of must be dened in terms of the same semantic language that was used to dene the semantics of L C | otherwise there is no point of reference between the two. Formally then, a semantics for agent program/state pairs is a function The relationships between the various formal components introduced above are summarised in Figure 2. We now collect these various components together and dene what we mean by an agent communication framework. DEFINITION 1. An agent communication framework is a (2n tuple: ng is a non-empty set of agents, i 2 is an agent program, L i is the set of local states of i , L communication language, L is a semantic language, and is a semantics for . We let F be the set of all such agent communication frameworks, and use f (with annotations: f to stand for members of F . Semantic Issues in Agent Communication 13 4. Veriability Dened We are now in a position to dene what it means for an agent program, in sending a message while in some particular state, to be respecting the semantics of a communication framework. Recall that a communication language semantics denes, for each message, a constraint, or specication, which must be satised by the sender of the message if it is to be considered as satisfying the semantics of the communication language. The properties of a program when in some particular state are given by the program semantics, This leads to the following denition. DEFINITION 2. Suppose is an agent communication framework, and that send( to respect the semantics of framework f (written Note that the problem could equivalently have been phrased in terms of logical consequence: is an L S -logical consequence of . If we had a sound and complete proof system ' S for L S , then we could similarly have phrased it as a proof problem: . The rst approach, however, is probably the most general. Using this denition, we can dene what it means for a communication framework to have a veriable semantics. DEFINITION 3. An agent communication framework is veriable i it is a decidable question whether arbitrary i , l , . The intuition behind veriability is as follows: if an agent communication framework enjoys this property, then we can determine whether or not an agent is respecting the framework's communication language semantics whenever it sends a message. If a framework is veriable, then we know that it is possible in principle to determine whether or not an agent is respecting the semantics of the framework. But a framework that is veriable in principle is not necessarily veriable in practice. This is the motivation behind the following denition. 14 Michael Wooldridge DEFINITION 4. An agent communication framework f 2 F is said to be practically veriable i it is decidable whether polynomial in jf j jj jj jl j. If we have a practically veriable framework, then we can do the verication in polynomial time, which implies that we have at least some hope of doing automatic verication using computers that we can envisage today. Our ideal, when setting out an agent communication should clearly be to construct f such that it is practically veriable. However, practical veriability is quite a demanding proper- ty, as we shall see in section 5. In the following subsection, we examine the implications of these denitions. 4.1. What does it mean to be Verifiable? If we had a veriable agent communication framework, what would it look like? Let us take each of the components of such a framework in turn. First, our set Ag of agents, implemented by programs i , (where these programs are written in an arbitrary programming language). This is straightforward: we obviously have such components today. Next, we need a communication language L C , with a well-dened syntax and semantics, where the semantics are given in terms of L S , a semantic language. Again, this is not problematic: we have such a language L C in both kqml and the fipa-97 language. Taking the fipa case, the semantic language is sl, a quantied multi-modal logic with equality. This language in turn has a well dened syntax and semantics, and so next, we must look for a program semantics . At this point, we encounter problems. Put simply, the fipa semantics are given in terms of mental states, and since we do not understand how such states can be systematically attributed to programs, we cannot verify that such programs respect the semantics. More precisely, the semantics of sl are given in the normal modal logic tradition of Kripke (possible worlds) semantics, where each agent's \attitudes" (belief, desire, . ) are characterised as relations holding between dierent states of aairs. Although Kripke semantics are attractive from a mathematical perspective, it is important to note that they are not connected in any principled way with computational systems. That is, for any given a java program), there is no known way of attributing to that program an sl formula (or, equivalently, a set of sl models), which characterises it in terms of beliefs, desires, and so on. Because of this, we say that sl (and most similar logics with Kripke semantics) are ungrounded | they have no concrete computational interpretation. In other words, if the semantics of L S are ungrounded (as they are in the fipa-97 sl case), Semantic Issues in Agent Communication 15 then we have no semantics for programs | and hence an unveriable communication framework. Although work is going on to investigate how arbitrary programs can be ascribed attitudes such as beliefs and desires, the state of the art ([8]) is considerably behind what would be required for acl verication. Other researchers have also recognised this di-culty [39, 34]. Note that it is possible to choose a semantic language L S such that a principled program semantics can be derived. For example, temporal logic has long been used to dene the semantics of programming languages [29]. A temporal semantics for a programming language denes for every program a temporal logic formula characterising the meaning of that program. Temporal logic, although ultimately based on Kripke semantics, is rmly grounded in the histories traced out by programs as they execute | though of course, standard temporal logic makes no reference to attitudes such as belief and desire. Also note that work in knowledge theory has shown how knowledge can be attributed to computational processes in a systematic way [17]. However, this work gives no indication of how attitudes such as desiring or intending might be attributed to arbitrary programs. (We use techniques from knowledge theory to show how a grounded semantics can be given to a communication language in Example 2 of section 5.) Another issue is the high computational complexity of the veri- cation process itself [32]. Ultimately, determining whether an agent implementation is respecting the semantics of a communication frame-work reduces to a logical proof problem, and the complexity of such problems is well-known. If the semantic language L S of a framework f is equal in expressive power to rst-order logic, then f is of course not veriable. For quantied multi-modal logics, (such as that used by fipa to dene the semantics of their acl), the proof problem is often much harder than this | proof methods for quantied multi-modal logics are very much at the frontiers of theorem-proving research (cf. [1]). In the short term, at least, this complexity issue is likely to be another signicant obstacle in the way of acl verication. To sum up, it is entirely possible to dene a communication language LC with semantics in terms of a language L S . However, giving a program semantics for a semantic language (such as that of fipa-97) with ungrounded semantics is a serious unsolved problem. 5. Example Frameworks To illustrate the idea of verication, as introduced above, in this section we will consider a number of progressively richer agent communica- Michael Wooldridge tion frameworks. For each of these frameworks, we discuss the issue of veriability, and where possible, characterise the complexity of the verication problem. 5.1. Example 1: Classical Propositional Logic. For our rst example, we dene a simple agent communication frame-work f 1 in which agents communicate by exchanging formulae of classical propositional logic. The intuitive semantics of sending a message ' is that the sender is informing other agents of the truth of '. An agent sending out a message ' will be respecting the semantics of the language if it \believes" (in a sense that we precisely dene below) that ' is true. An agent will not be respecting the semantics if it sends a message that it \believes" to be false. We also assume that agent programs exhibit a simple behaviour of sending out all messages that they believe to be true. We show that framework f 1 is veriable, and that in fact every agent program in this framework respects the semantics of f 1 . Formally, we must dene the components of a framework These components are as follows. First, Ag is some arbitrary non-empty set | the contents are not signicant. Second, since agents communicate by simply exchanging messages that are simply formulae of classical propositional logic, L 0 , we have L . Thus the set w (L 0 ) contains formulae made up of the proposition symbols combined into formulae using the classical connectives \:" (not), \^" (and), \_" (or), and so on. We let the semantic language L S also be classical propositional logic, and dene the L C semantic function simply as the identity function: . The semantic function the usual propositional denotation function | the denition is entirely standard, and so we omit it in the interests of brevity. An agent i 's state l i is dened to be a set of formulae of propositional logic, hence L is assumed to simply implement the following rule: In other words, an agent program i sends a message when in state l i is present in l . The semantics of agent programs are then dened as follows: Semantic Issues in Agent Communication 17 In other words, the meaning of a program in state l is just the conjunction of formulae in l . The following theorem sums up the key properties of this simple agent communication framework. THEOREM 1. 1. Framework f 1 is veriable. 2. Every agent in f 1 does indeed respect the semantics of f 1 . Proof. For (1), suppose that send( g. Then i is respecting the semantics for f 1 i which by the f 1 denitions of reduces to But this is equivalent to showing that is an L 0 -logical consequence of logical consequence is obviously a decidable problem, we are done. For (2), we know from equation (2) that l . Since is clearly a logical consequence of l if , we are done. An obvious next question is whether f 1 is practically veriable, i.e., whether verication can be done in polynomial time. Here, observe that verication reduces to a problem of determining logical consequence in which reduces to a test for L 0 -validity, and hence in turn to L 0 - unsatisability. Since the L 0 -satisability problem is well-known to be np-complete, we can immediately conclude the following. THEOREM 2. The f 1 verication problem is co-np-complete. Note that co-np-complete problems are ostensibly harder than merely np-complete problems, from which we can conclude that practical verication of f 1 is highly unlikely to be possible 3 . 5.2. Example 2: Grounded Semantics for Propositional Logic. One could argue that Example 1 worked because we made the assumption that agents explicitly maintain databases of L 0 formulae: checking whether an agent was respecting the semantics in sending a message ' 3 In fact, f 2 will be practically veriable if and only if which is regarded as extremely unlikely [32]. Michael Wooldridge amounted to determining whether ' was a logical consequence of this database. This was a convenient, but, as the following example trates, unnecessary assumption. For this example, we will again assume that agents communicate by exchanging formulae of classical propositional logic L 0 , but we make no assumptions about their programs or internal state. We show that despite this, we can still obtain a veriable semantics, because we can ground the semantics of the communication language in the states of the program. There is an impartial, objective procedure we can apply to obtain a declarative representation of the \knowledge" implicit within an arbitrary program, in the form of Fagin- Halpern-Moses-Vardi knowledge theory [17]. To check whether an agent is respecting the semantics of the communication language, we simply check whether the information in the message sent by the agent is a logical consequence of the knowledge implicit within the agent's state, which we obtain using the tools of knowledge theory. In what follows, we assume all sets are nite. As in Example 1, we set both the communication language L C and the semantic language L S to be classical propositional logic L 0 . We require some additional denitions (see [17, pp103{114] for more details). Let the set G of global states of a system be dened by We use g (with annotations: to stand for members of G . We assume that we have a vocabulary primitive propositions to express the properties of a system. In addition, we assume it is possible to determine whether or not any primitive proposition p 2 is true of a particular global state or not. We write g j= p to indicate that p is true in state g . Next, we dene a relation i G L i for each agent Ag to capture the idea of indistinguishability. The idea is that if an agent i is in state l 2 L i , then a global state indistinguishable from the state l that i is currently in (written g i l ) . Now, for any given agent program i in local state l , we dene the positive knowledge set of i in l , (written ks to be the set of propositions that are true in all global states that are indistinguishable from l , and the negative knowledge set of i in l , (written ks to be the set of propositions that are false in all global states that are indistinguishable from l . Formally, ks ks Readers familiar with epistemic logic [17] will immediately recognise that this construction is based on the denition of knowledge in distributed systems. The idea is that if p 2 ks ks given the information that i has available in state l , p must necessarily be true (respectively, false). Thus ks Semantic Issues in Agent Communication 19 represents the set of propositions that the agent i knows are true when it is in state l ; and ks represents the set of propositions that i knows are false when it is in state l . The L C semantic function is dened to be the identity function again, so For the program semantics, we dene The formula thus encodes the knowledge that the program i has about the truth or falsity of propositions when in state l . The L S semantic function is assumed to be the standard L 0 semantic func- tion, as in Example 1. An agent will thus be respecting the semantics of the communication framework if it sends a message such that this message is guaranteed to be true in all states indistinguishable from the one the agent is currently in. This framework has the following property. THEOREM 3. Framework f 2 is veriable. Proof. Suppose that send( arbitrary i , , l . Then i is respecting the semantics for f 2 i which by the f 2 denitions of reduces to Computing G can be done in time O(jL 1 L n j); computing i can be done in time O(jL i j jG j); and given G and i , computing ks ks can be done in time O(jj jG j). Once given ks ks (; l ), determining whether reduces to the L 0 logical consequence problem This problem is obviously decidable. verication reduces to L 0 logical consequence checking, we can use a similar argument to that used for Theorem 2 to show the problem is in general no more complex than f 1 verication: Michael Wooldridge THEOREM 4. The f 2 verication problem is co-np-complete. Note that the main point about this example is the way that the semantics for programs were grounded in the states of programs. In this example, the communication language was simple enough to make the grounding easy. More complex communication languages with a similarly grounded semantics are possible. We note in closing that it is straightforward to extend framework f 2 to allow a much richer agent communication language (including requesting, informing, and commissives) [40]. 5.3. Example 3: The fipa-97 acl. For the nal example, consider a framework f 3 in which we use the fipa-97 acl, and the semantics for this language dened in [19]. Following the discussion in section 4.1, it should come as no surprise that such a framework is not veriable. It is worth spelling out the reasons for this. First, since the semantic language sl is a quantied multi-modal logic, with greater expressive power than classical rst order logic, it is clearly undecidable. (As we noted above, the complexity of the decision problem for quantied modal logics is often much harder than for classical predicate logic [1].) So the f 3 verication problem is obviously undecidable. But of course the problem is worse than this, since as the discussion in section 4.1 showed, we do not have any idea of how to assign a program semantics for semantic languages like sl, because these languages have an ungrounded, mentalistic semantics. 6. Verication via Model Checking The problem of verifying whether an agent implements the semantics of a communication language has thus far been presented as one of determining logical consequence, or, equivalently, as a proof problem. Readers familiar with verication from theoretical computer science will recognise that this corresponds to the \traditional" approach to verifying that a program satises a specication. Other considerations aside, a signicant drawback to proof theoretic verication is the problem of computational complexity. As we saw above, even if the semantic language is as impoverished as classical propositional logic, verica- tion will be co-np-complete. In reality, logics for verication must be considerably more expressive than this. Problems with the computational complexity of verication logics led researchers in theoretical computer science to investigate other approaches to formal verication. The most successful of these is model Semantic Issues in Agent Communication 21 checking [27, 22, 10]. The idea behind model checking is as follows. Recall that in proof theoretic verication, to verify that a program i has some property ' when in state l , we derive the theory of that program and attempt to establish i.e., that property ' is a theorem of the theory In temporal semantics, for example [28, 29], is a temporal logic formula such that the models of this formula correspond to all possible runs of the program In contrast, model checking approaches work as follows. To determine whether or not i has property ' when in state l , we proceed as follows: , and from them generate a model M i ;l that encodes all the possible computations of . Determine whether or not M i ;l whether the formula ' is valid in M i ;l ; the program i has property ' in state l just in case the answer is \yes". In order to encode all computations of the program, the model generated in the rst stage will be a branching time temporal model [16]. Intuitively, each branch, (or path), through this model will correspond to one possible execution of the program. Such a model can be generated automatically from the text of a program in a typical imperative programming language. The main advantage of model checking over proof theoretic ver- ication is in complexity: model checking using the branching time temporal logic ctl [11] can be done in time O(j'j jM j), where j'j is the size of the formula to be checked, and jM j is the size of the model (i.e., the number of states it contains) [16]. Model-checking approaches have recently been used to verify nite-state systems with up to 10 120 states [10]. Using a model checking approach to conformance testing for acls, we would dene the program semantics as a function which assigns to every program/state pair an L S -model, which encodes the properties of that program/state pair. Verifying that would involve checking whether whether the sincerity condition was valid in model The comparative e-ciency of model checking is a powerful argument in favour of the approach. Algorithms have been developed for (propositional) belief-desire-intention logics that will take a model and 22 Michael Wooldridge a formula and will e-ciently determine whether or not the formula is satised in that model [35, 5]. These belief-desire-intention logics are closely related to those used to give a semantics to the fipa-97 acl. However, there are two unsolved problems with such an approach. The rst problem is that of developing the program semantics We have procedures that, given a program, will generate a branching temporal model that encode all computations of that program. Howev- er, these are not the same as models for belief-desire-intention logics. Put simply, the problem is that we do not yet have any techniques for systematically assigning beliefs, desires, intentions, and uncertainties (as in the fipa-97 sl case [19]) to arbitrary programs. This is again the problem of grounding that we referred to above. As a consequence, we cannot do the rst stage of the model checking process for acls that have (ungrounded) fipa-like semantics. The second problem is that model checking approaches have been shown to be useful for systems that can be represented as nite state models using propositional temporal logics. If the verication logic allows arbitrary quantication, (or the system to be veried is not nite state), then a model checking approach is unlikely to be practicable. To summarise, model checking approaches appear to have considerable advantages over proof-theoretic approaches to verication with respect to their much reduced computational complexity. However, as with proof-theoretic approaches, the problem of ungrounded acl semantics remains a major problem, with no apparent route of attack. Also, the problem of model checking with quantied logics is an as-yet untested area. Nevertheless, model checking seems a promising direction for acl conformance testing. 7. Discussion If agents are to be as widely deployed as some observers predict, then the issue of inter-operation | in the form of standards for communication languages | must be addressed. Moreover, the problem of determining conformance to these standards must also be seriously considered, for if there is no way of determining whether or not a system that claims to conform to a standard does indeed conform to it, then the value of the standard itself must be questioned. This article has given the rst precise denition of what it means for an agent communication framework to be veriable, and has identied some problematic issues for veriable communication language semantics, the most important of which being that: Semantic Issues in Agent Communication 23 We must be able to characterise the properties of an agent program as a formula of the language L S used to give a semantics to the communication language. L S is often a multi-modal logic, referring to (in the fipa-97 case, for example) the beliefs, desires, and uncertainties of agents. We currently have very little idea about systematic ways of attributing such mentalistic descriptions to programs | the state of the art is considerably behind what would be needed for anything like practical verication, and this situation is not likely to change in the near future. The computational complexity of logical verication, (particularly using quantied multi-modal languages), is likely to prove a major obstacle in the path of practical agent communication language verication. Model checking approaches appear to be a promising alternative. In addition, the article has given examples of agent communication frameworks, some of which are veriable by this denition, others of which, (including the fipa-97 acl [19]), are not. The results of this article could be interpreted as negative, in that they imply that verication of conformance to acls using current techniques is not likely to be possible. However, the article should emphatically not be interpreted as suggesting that standards | particularly, standardised acls | are unnecessary or a waste of time. If agent technology is to achieve its much vaunted potential as a new paradigm for software construction, then such standards are important. However, it may well be that we need new ways of thinking about the semantics and verication of such standards. A number of promising approaches have recently appeared in the literature [39, 34, 40]. One approach that can work eectively in certain cases is mechanism design [36]. The basic idea is that in certain multi-agent scenarios (auctions are a well-known example), it is possible to design an interaction protocol so that the dominant strategy for any participating agent is to tell the truth. Vickrey's mechanism is probably the best-known example of such a technique [37]. In application domains where such techniques are feasible, they can be used to great eect. However, most current multi-agent applications do not lend themselves to such techniques. While there is therefore great potential for the application of mechanism design in the long term, in the short term it is unlikely to play a major role in agent communication standards. Michael Wooldridge --R Readings in Planning. Planning English Sentences. How to Do Things With Words. Readings in Distributed Arti The Correctness Problem in Computer Science. Reasoning About Knowledge. The Temporal Logic of Reactive and Concurrent Systems. Temporal Veri Computational Complexity. Rules of Encounter: Designing Conventions for Automated Negotiation among Computers. Speech Acts: An Essay in the Philosophy of Language. --TR --CTR Peter McBurney , Simon Parsons, Locutions for Argumentation in Agent Interaction Protocols, Proceedings of the Third International Joint Conference on Autonomous Agents and Multiagent Systems, p.1240-1241, July 19-23, 2004, New York, New York Guido Boella , Rossana Damiano , Joris Hulstijn , Leendert van der Torre, Role-based semantics for agent communication: embedding of the 'mental attitudes' and 'social commitments' semantics, Proceedings of the fifth international joint conference on Autonomous agents and multiagent systems, May 08-12, 2006, Hakodate, Japan Paurobally , Jim Cunningham , Nicholas R. Jennings, A formal framework for agent interaction semantics, Proceedings of the fourth international joint conference on Autonomous agents and multiagent systems, July 25-29, 2005, The Netherlands Stefan Poslad , Patricia Charlton, Standardizing agent interoperability: the FIPA approach, Mutli-agents systems and applications, Springer-Verlag New York, Inc., New York, NY, 2001 Ulle Endriss , Nicolas Maudet, On the Communication Complexity of Multilateral Trading: Extended Report, Autonomous Agents and Multi-Agent Systems, v.11 n.1, p.91-107, July 2005 Peter McBurney , Simon Parsons, Posit spaces: a performative model of e-commerce, Proceedings of the second international joint conference on Autonomous agents and multiagent systems, July 14-18, 2003, Melbourne, Australia Peter McBurney , Simon Parsons, A Denotational Semantics for Deliberation Dialogues, Proceedings of the Third International Joint Conference on Autonomous Agents and Multiagent Systems, p.86-93, July 19-23, 2004, New York, New York Yuh-Jong Hu, Some thoughts on agent trust and delegation, Proceedings of the fifth international conference on Autonomous agents, p.489-496, May 2001, Montreal, Quebec, Canada Peter McBurney , Simon Parsons , Michael Wooldridge, Desiderata for agent argumentation protocols, Proceedings of the first international joint conference on Autonomous agents and multiagent systems: part 1, July 15-19, 2002, Bologna, Italy Peter McBurney , Simon Parsons, Games That Agents Play: A Formal Framework for Dialoguesbetween Autonomous Agents, Journal of Logic, Language and Information, v.11 n.3, p.315-334, Summer 2002 Rogier M. Van Eijk , Frank S. De Boer , Wiebe Van Der Hoek , John-Jules Ch. Meyer, A Verification Framework for Agent Communication, Autonomous Agents and Multi-Agent Systems, v.6 n.2, p.185-219, March Peter Mcburney , Rogier M. Van Eijk , Simon Parsons , Leila Amgoud, A Dialogue Game Protocol for Agent Purchase Negotiations, Autonomous Agents and Multi-Agent Systems, v.7 n.3, p.235-273, November Brahim Chaib-Draa , Marc-Andr Labrie , Mathieu Bergeron , Philippe Pasquier, DIAGAL: An Agent Communication Language Based on Dialogue Games and Sustained by Social Commitments, Autonomous Agents and Multi-Agent Systems, v.13 n.1, p.61-95, July 2006 N. Maudet , B. Chaib-Draa, Commitment-based and dialogue-game-based protocols: new trends in agent communication languages, The Knowledge Engineering Review, v.17 n.2, p.157-179, June 2002

semantics;conformance testing;verification;agent communication languages;standards

608645

Emergent Properties of a Market-based Digital Library with Strategic Agents.

The University of Michigan Digital Library (UMDL) is designed as an open system that allows third parties to build and integrate their own profit-seeking agents into the marketplace of information goods and services. The profit-seeking behavior of agents, however, risks inefficient allocation of goods and services, as agents take strategic stances that might backfire. While it would be good if we could impose mechanisms to remove incentives for strategic reasoning, this is not possible in the UMDL. Therefore, our approach has instead been to study whether encouraging the other extrememaking strategic reasoning ubiquitousprovides an answer.Toward this end, we have designed a strategy (called the p-strategy) that uses a stochastic model of the market to find the best offer price. We have then examined the collective behavior of p-strategy agents in the UMDL auction. Our experiments show that strategic thinking is not always beneficial and that the advantage of being strategic decreases with the arrival of equally strategic agents. Furthermore, a simpler strategy can be as effective when enough other agents use the p-strategy. Consequently, we expect the UMDL is likely to evolve to a point where some agents use simpler strategies and some use the p-strategy.

Introduction When building a multiagent system, a designer (or a group of designers) has to worry about two issues: mechanism design (which dictates the way that agents interact), and individual-agent design. Of course, these two design issues are interdependent; a well-designed mechanism can simplify the design of individual agents (and vice versa). For instance, Vickery's auction mechanism (Vickery, 1961) makes rational agents bid their true reservation prices such that even self-interested agents, if they are rational, will behave honestly (and not try in vain to outsmart other However, designing a good mechanism that exhibits certain properties-which is called incentive engineering or mechanism design (Rosenschein & Zlotkin, 1994)-is difficult, especially for dynamic systems where the participants and their interactions evolve over time. The example of such dynamic systems that we use throughout this paper is the University of Michigan Digital Library (UMDL). In the UMDL project, we aim to provide an infrastructure for rendering library services in a networked information environment (Durfee et al., 1998). We have designed the UMDL as a multiagent system, where agents (representing users, collections, and services of the digital library) sell and buy information goods and services through auctions. While supporting flexibility and scalability, the open multiagent architecture and market infrastructure create dynamics (agents participating in an auction change, matches between buyers and sellers vary, and auctions themselves evolve), which adds additional complexity to mechanism design. As system architects, we strive for an efficient system. Although the UMDL market allows the self-interested agents to seek profits, we do not want strategic agents to undermine the overall system performance (efficiency in market), nor such agents to reap profits against other agents (efficiency in allocation). That is, we want an incentive-compatible mechanism that makes strategic reasoning unnecessary (Zlotkin & Rosenschein, 1996; Wellman, 1993). Unfortunately, we do not have such a mechanism yet for the UMDL system; we fully expect strategic agents who try to take advantage of other agents to emerge in the system. So, does it mean the UMDL will become inefficient? (and should) a UMDL agent spend much of its computational power trying to outsmart other agents? What happens if all the agents behave strategically? In this paper, we answer the above questions, by studying the properties of the UMDL with strategic agents. Instead of developing a auction mechanism may be inappropriate for certain settings. For limitations of Vickery auctions, see the work by Sandholm (Sandholm, 1996). mechanism that prevents strategic thinking (which is hard), we use a bottom-up approach: we design a strategy that the UMDL agents may use and experiment with such strategic agents to learn about the system properties. In particular, we are interested in whether making strategic reasoning ubiquitous (instead of preventing it) reduces its negative effects. In the following, we review some of the previous work on multiagent system design issues and briefly describe the target system, the UMDL service market society. We explain a strategy called p-strategy and demonstrate its advantages over other simpler strategies. Then, by experimenting with multiple p-strategy agents, we investigate some emergent properties of the UMDL system. Related Work A multiagent system can be designed to exhibit certain desirable properties. A single designer (or a group of designers sharing common goals) can calibrate the system to follow its goals (Briggs & Cook, 1995; Shoham & Tennenholtz, 1995). For instance, agents and mechanism can be built in certain ways (e.g., share information, be honest, and so on) conducive to cooperation. Social laws and conventions, however, are unsuitable for the UMDL where agent designers do not share common goals and system architects cannot impose limitations on the individual, self-interested agents. When designing multiagent systems with self-interested agents, many researchers have turned to game theory to lead systems to desired behaviors (e.g., discourage agents from spending time and computation trying to take advantage of others) (Sandholm & Lesser, 1995; Brafman 1996). For instance, Rosenschein and Zlotkin have identified two building blocks of a multiagent system, protocol and strategy (which are mechanism and individual agent in our terminology, respectively), and focused on designing a protocol which ".motivates agents towards telling the truth." (Rosenschein & Zlotkin, 1994). Unfortunately, game theory tends to be applied to highly abstract and simplified settings. We do not use game theory because the UMDL is too complex to model and because we cannot assume rationality in agents. Another approach to designing multiagent systems with self-interested agents is to let manipulation from agents happen and live with it. This may sound like bad engineering, but preventing strategic behavior of individual agents is unrealistic for many complex systems. Instead, by studying how the individual, strategic agents impact the overall system behavior, we gain insights on properties of agent societies, such as characterizing the types of environments and agent populations that foster social and anti-social behavior (Vidal & Durfee, 1996; Hu & Wellman, 1996). Our approach falls into this category, and our goal is to explain how strategic agents affect the UMDL in terms of market and allocation efficiency. The UMDL Service Market Society The UMDL Service Market Society (SMS) is a market-based multiagent system where agents buy and sell goods and services from each other. Instead of relying solely on internally-designed agents, UMDL can attract outside agents to provide new services, who are motivated by the long-term profit they might accrue by participating in the system. Since the UMDL is open, we treat all agents as selfish. Selling and buying of services are done through auction markets, operated by auction agents. Figure 1 shows an example of the UMDL auction, where User Interface Agents (UIAs) want to find sources of information for some topic (say, science) on behalf of certain kinds of users (say, high school), and some Query Planning Agents (QPAs) sell the services for finding such collections. Due to space limitations, we ignore the issues of how to describe what agents buy or sell, how to locate the right auction to participate in, when and how to create an auction, and so on. Interested readers may refer to (Durfee et al., 1998). At present, the UMDL uses the AuctionBot software (Mullen & Wellman, 1996) parameterized for double Sellers Buyers QPA-n UIA-m Auction Auction for QPA-hs-sci sell-offers buy-offers Figure 1: The UMDL auction for "QPA-service_high-school_science" auctions where agents post both sell prices and buy prices. Compared to double auctions used in some economic models (Friedman and Rust, 1993), the UMDL auction is a continuously-clearing double auction, well-suited for frequent, timely transactions needed in information economies. In the UMDL auction, a transaction is completed as soon as a buy offer and a sell offer cross (without waiting for the remaining agents to submit their bids), and the clearing price is determined for each transaction (rather than being set at some medium price among bids). In the UMDL, the auction agent continuously matches the highest buyer to the lowest seller, given that the buy price is greater than the sell price. The clearing price is based on the seller's offer price (i.e., consumers receive all the surplus). Since buyers (sellers) with bid prices higher (lower) than any standing sell (buy) offer get matched, the buyers in the auction (if any) always have lower offer prices than the sellers (if any). That is, standing offers ordered by lowest to highest bid are always in a (bbb.bsss.s) sequence, as shown in Figure 2. To manage the size of the auction, we may limit the standing offers in the auction. The auction used in our experiments limits the number of buy and sell offers not to exceed five each. When an additional buyer (seller) arrives, the seller (buyer) with the highest (lowest) offer will be kicked out first. A Strategy based on Stochastic Modeling Agents placed in the UMDL SMS want to maximize their profits by increasing the possible matches and the profit per match. In this section, we present a bidding strategy that agents may use to maximize their profits, and examine the performance of the seller 2 with such a strategy against other types of sellers. The p-strategy We have developed an agent strategy (called p-strategy) that finds the best offer price for the multiagent auction 1996). The four-step p- strategy is as follows. First, the agent models the auction process using a Markov chain (MC) with two absorbing states (success and failure). Secondly, it computes the sellers have a somewhat stronger incentive to be strategic in the UMDL, as they set the clearing price and this affects their tradeoff between the probability of trading and the profit earned, we use the p-strategy seller (instead of buyer). transition probabilities between the MC states. Thirdly, it computes the probabilities and the payoffs of success and failure. Finally, it finds the best offer price to maximize its expected utility. The main idea behind the p-strategy is to capture the factors which influence the expected utility in the MC model of the auction process. For instance, the seller is likely to raise its offer price when there are many buyers or when it expects more buyers to come. The MC model takes those factors into account in the MC states and the transition probabilities. The number of buyers and sellers at the auction, the arrival rates of future buyers and sellers, and the distribution of buy and sell prices are among the identified factors. Each state in the MC model represents the status of the auction. The (bbss*) state, for example, represents the case where there are 2 standing buy offers and 2 standing sell offers and the sell offer of the p-strategy agent (represented as s*) is higher than the other seller's offer. If we assume that offers arrive at most one at a time, the auction can go to any of the following states from the (bbss*) state (see Figure 3). arrives during the clearing interval. . (bbs*): A buy offer arrives, and it is matched with the lowest seller. . (bbbss*): Because of no match, a new buy offer becomes a standing offer. . (bss*): A sell offer arrives, and it is matched with the highest buyer. . (bbss*s): Because of no match, a new sell offer becomes the highest standing offer. . (bbsss*): A new sell offer becomes a standing offer, but the p-strategy agent's offer is still the highest. bbss* s*: my offer bbs* bbss*s bbsss* bss* bbbss* Figure 3: Transitions from the (bbss*) state s2 comes b2 comes price b1 s2 (s1 and b2 matched at s1) (bs) (bss) (bs) Figure 2: Standing offers in the auction Figure 4 depicts the MC model for the UMDL auction with a maximum size of five buyers and sellers each 3 . In this paper, we skip how to define the exact transition probabilities between the MC states (step 2), how to compute the probabilities and the payoffs of success and failure (step 3), and how to find the best payment (step 4). Readers may refer to (Park, Durfee & Birmingham, 1996). Using the MC model and its transition probabilities, the p-strategy agent is able to capture various factors that influence the utility value and tradeoffs associated with the factors. Figure 5 shows an example of tradeoffs between the number of buyers and sellers. In general, the seller raises its offer price when there are more buyers (to increase the profit of a possible match). When the number of sellers is five (at the right end of the graph), however, the p-strategy seller bids a lower price when there is one buyer than when there is no buyer. That is, the p-strategy seller lowers its offer to increase the probability of a match (instead of increasing the profit of a match). Offering a higher price in this case would have served to price it out of the auction when it might otherwise have been able to trade profitably. Intuitively, agents with complete models of other agents will always do better, but without repeated encounters complete models are unattainable. In the UMDL, an agent in its lifetime meets many different agents, and as a result its model of other agents is incorrect, imprecise, and incomplete. Instead of modeling individual agents, the p- 3 The number of MC states increases with the size of the auction. When the maximum number of standing offers is limited to m buyers and n sellers, the number of MC states is (m+1)((n+1)n)/2+2. Of course, one may shrink the size of the MC model, while sacrificing the accuracy of the model. strategy uses the model of the auction process, which is effective, especially for very dynamic systems. The p-strategy is an optimal strategy, provided that the MC model represents the auction process correctly and that all the agents have the same level of knowledge. Obviously, the p-strategy is not optimal when its model is incorrect (i.e., incorrect knowledge), and the p-strategy agent can be exploited by competing agents who know about the p-strategy (i.e., higher-level knowledge). Advantages of p-strategy In this section, we demonstrate the advantages of the p- strategy in the UMDL auction, comparing the profit of the p-strategy seller (p-QPA) with three different types of sellers. They are: Offer price Number of sellers Number of buyers Figure 5: Tradeoffs between the number of buyers and sellers at the auction bs*ssss bs*sss bbbbbs*ss bbbbbs*s bs*ssss bbbbs*sss bbbbs*ss bbbbs*s bbbs*ssss bbbs*sss bbbs*ss bbbs*s bbs*ssss bbs*sss bbs*ss bbs*s bs*ssss bs*sss bs*ss bs*s s*ssss bbbbs* bbbs* bbbbbs* s*s s*ss s*sss bss*sss bbbbbss*ss bbbbbss*s bbbbss*s ss bbbbss*ss bbbbss*s bbbss*sss bbbss*ss bbbss*s bbss*sss bbss*ss bbss*s bss*sss bss*ss bss*s bbbbss* bbbss* bbbbbss* bss* ss* bbss* ss*s ss*ss ss*sss bbbbbsss*ss bbbbbsss*s bbbbsss*ss bbbbss*s bbbsss*ss bbbsss*s bbsss*ss bbsss*s bsss*ss bsss*s sss*ss bbbbsss* bbbsss* bbbbbsss* bsss* sss* bbsss* sss*s bbbbbssss*s bbbbsss*s bbbssss*s bbssss*s bssss*s bbbbssss* bbbssss* bbbbbssss* bssss* ssss* bbssss* ssss*s bbbbsssss* bbbsssss* bbbbbsssss* bsssss* sssss* bbsssss* FailureFailure s* bbs* bs* Success Figure 4: The MC model for the UMDL auction . A seller who bids its cost plus some fixed markup . A seller who bids its cost plus some random markup . A seller who bids the clearing price of the next transaction (CP-QPA). The experimental settings are as follows. . Auction The auction clears every 3 seconds. The value of the clearing interval does not affect the results from our experiments, provided that on average at most one offer arrives at the auction each interval. . Buyers A single agent simulates multiple buyers by submitting multiple bids. In our experiments, every 6 seconds the buyer submits its bid with a probability of 0.8. By adjusting the offer interval and the offer rate of the single buyer, we can change the arrival rate of buy offers to the auction 5 . The buyer offers valuations drawn randomly from a uniform distribution between 10 and 30. . Sellers For each experiment, we compare the profits of two sellers: p-QPA and the opponent (either FM-QPA, ZI- QPA, or CP-QPA). Both sellers submit their bids every 24 seconds on average. In addition, similar to the buyer case, a single agent simulates all the other sellers at the auction. Its offer interval and offer rate are set at 12 and 0.8, respectively. The costs of all the sellers are based on their loads, which are computed from the message traffic and the current workload. That is, cost = a (number of messages per (number of matches per minute). The cost function represents economies of scale. Since the workload from matches should be higher than that from 4 ZI stands for zero-intelligence. The ZI-QPA is a "budget- constrained zero-intelligence trader" who generates random bids subject to a no-loss constraint (Gode & Sunder, 1993). 5 Arrival-rate-of-buy-offers @ (offer-rate/offer-interval) clearing-interval. The arrival rate varies, however, since the agent is allowed to submit a new offer immediately after a match even when it hasn't reached the next offer interval. communication, we set a to 1, and b to 5 for our experiments. In the first set of experiments, we have compared the profits of the p-QPA and the FM-QPA. When competing with FM-QPAs with various markups, the p-QPA always gets a higher profit. This is not surprising, since the p-QPA is able to use extra information about the auction. Figure 6-(a) shows the accumulated profits of the p-QPA and the FM-QPA who bids its cost plus 7 as its offer (i.e., fixed- Figure 6-(b) shows the profits of the p-QPA and the ZI- QPA. The ZI-QPA works poorly against the p-QPA, which indicates the randomization strategy does not work. The ZI-QPA can be thought of as an extremely naive strategy that fails to take advantage of a given situation (Gode & Sunder, 1993). In the final experiment, we have compared the p-QPA with the CP-QPA. The CP-QPA receives a price quote from the auction-the clearing price were the auction to clear at the time of the quote-and submits it as its offer as long as it is higher than its cost. Note that the clearing price is a hypothetical one, so it will change when new offer(s) arrive during the time between the clearing-price quote and the CP-QPA's offer. As shown in Figure 6-(c), the p-QPA usually gets a higher profit when competing with the CP-QPA. Since the CP-QPA gets more matches than the p-QPA (but its profit per match is smaller) on average, however, the CP-QPA works better when getting more matches does not impact its cost much (e.g., when b is 0). Bidding the next clearing price may seem like a good heuristic, but the profit of the CP-QPA decreases rapidly when there is another CP-QPA, since it no longer gets as many matches as when there is a single CP-QPA. The p-strategy works well in the UMDL auction due to its dynamics. No agent can have a complete, deterministic view about the current and future status of the auction, and naturally, an agent strategy should be able to take into account the dynamics and the resulting uncertainties. In our experiments, the p-strategy which models the auction process stochastically receives higher profit than the other FM- QPA ZI- QPA QPA QPA (a) FM-QPA and p-QPA (b) ZI-QPA and p-QPA (c) CP-QPA and P-QPA Figure Comparison of the profits of the sellers In our previous paper (Park, Durfee, and Birmingham, 1996), we have shown the advantages of the p-strategy in the domain of multiagent contracts with possible retraction. In this paper, we have demonstrated that the p-strategy works well in the new domain. Note that we have used the p-strategy seller (compared to the p-strategy buyer in the previous paper). Collective Behavior of p-strategy Agents in the UMDL Auction Given that the p-strategy is effective in the UMDL auction (from the previous section), nothing prohibits any self-interested agent from adopting the p-strategy. We expect many p-strategy agents to coexist in the UMDL, and thus are interested in the collective behavior of such agents. In this section, therefore, we investigate (1) how the absolute and relative performance of a p-strategy agent changes against other p-strategy agents, and (2) how the UMDL is affected by multiple p-strategy agents. Experimental Setting Figure 7 shows six experimental settings with 7 buyers and 7 sellers. Although we fix the number of buyers and sellers to seven each, by changing the offer rates and the offer intervals, we can simulate a large number of agents and different levels of activities in the auction. By increasing the offer rates or decreasing the offer intervals, for instance, we can simulate a more dynamic auction. In our experiments, we deliberately set supply to be higher than demand to emphasize competition among sellers; each buyer submits its bid every seconds with a probability of 0.5, while the offer interval and the offer rate of each seller are set to 30 seconds and a probability of 0.7, respectively. The buyers bid their true valuations, while the sellers bid their sell prices based on their strategies. In Session 1, all seven sellers bid their true costs. Since traders honestly report their reservation prices, Session 1 gets the most matches and serves as a benchmark for comparing market efficiency. From Session 2 through Session 6, we introduce more p-strategy agents into the auction. Efficiency of the system is measured in two ways. First, we measure the efficiency of allocation, by comparing the p-strategy agent's absolute and relative profits. Second, we measure the efficiency of the market, using the number of matches made and the total profit generated. Experimental Results Although we have shown that the p-strategy agent has an upper hand over other-strategy agents, this observation may not hold in the presence of other p-strategy agents. To test this, we compare the profits of Seller 7 (p-strategy agent) across Sessions 2 to 6. As shown in Figure 8, the marginal profit of the p-strategy (smart) agent decreases as the number of p-strategy (equally smart) agents increases.1 00 Figure 8: Profits of p-QPA Now that we have established the fact that the profit of the p-strategy agent decreases as more agents use p- strategy, another question arises. How will a simpler strategy agent perform in the presence of multiple p- strategy agents? In Figure 9, by replacing Seller 1 with the fixed-markup QPA (with markup = 5), we measure the relative performance of the FM-QPA and the p-QPA. The FM-QPA's profit is generally less than that of the p-QPA, but the difference decreases with the increase of p-strategy agents 6 . That is, the disadvantages of being less smart decreases as the number of smart agents increases. The result indicates that an agent may want to switch between using p-strategy and using a simpler strategy depending on what the other agents are doing. By dynamically switching to a simpler strategy, an agent can 6 The FM-QPA's profit exceeds that of the p-QPA in Session 5 (in Figure 9), but at present we cannot conclude whether this is statistically significant. Bidding strategy Session Description Seller1 Seller2 Seller3 Seller4 Seller5 Seller6 Seller7 (1) All competitive sellers C C C C C C C Figure 7: Experimental setting achieve a similar profit (to that of using the p-strategy) while exerting less effort (time and computation) on computing FM-QPA p-QPA Figure 9: Profits of FM-QPA and p-QPA In terms of market efficiency, we first measure the number of total transactions made at the auction, as shown in Figure 10. When the number of p-strategy agents increases, the number of matches decreases, since the p- strategy agents usually get fewer matches but more profit per match.100300500SessionSessionSessionSessionSess ionSess ion Figure 10: Efficiency measured by the number of transactions In addition, we measure the market efficiency using the total profit generated from buyers and sellers (see Figure 11). The total profit eventually decreases with increasing numbers of p-strategy agents, as the market becomes inefficient due to strategic misrepresentation of p-strategy agents (and therefore missed opportunities of matches). The total profit, however, does not decrease as sharply as one might expect due to the inherent inefficiency of the UMDL auction mechanism (it in fact increases slightly up to Session 4). If an auction waits for the bids from all agents and decides on the most efficient clearing, inefficiency due to the auction mechanism may not occur. However, this kind of periodic, clearing-house style double auction is unrealistic for the UMDL system where the participants of the auction change and matches should be made quickly. We conjecture that having strategic sellers poses interesting tradeoffs between strategic inefficiency and surplus extraction. By misrepresenting their true costs, the strategic sellers miss out on possible transactions. By anticipating the future arrival of buyers, on the other hand, they are able to seize more surplus.500150025003500 rs Buye rs ' p r Figure Efficiency measured by the total profit Lessons Learned A conventional way of designing a system that exhibits certain properties is to engineer it. Incentive engineering, however, is unsuccessful in developing the UMDL system because of its complexity and dynamics. Instead, by making the p-strategy available to the agents, we have studied the effects of strategic agents in the UMDL system. We summarize the following observations. First, although a self-interested agent in the UMDL system has the capability of complex strategic reasoning, our experiments show that such reasoning is not always beneficial. As shown in Figure 8, the advantage of being smart decreases with the arrival of equally smart agents. Second, if all the other agents use the p-strategy, an agent with a simple strategy (e.g., fixed markup) can do just as well, while incurring less overhead to gather information and compute bids. An agent may want to switch between a complex strategy and a simple one depending on the behavior of other agents. As the overhead of complex reasoning becomes more costly, an adaptive strategy that dynamically decides on which strategy to use will be more desirable. Third, we expect the UMDL is likely to evolve to a point where some agents use simpler strategies while some use more complex strategies that use more knowledge (such as the p-strategy). It follows from the above observation that if enough other agents use complex reasoning, an agent can achieve additional profit even when it continues using a simple strategy. Finally, the market efficiency of the UMDL (represented by the total profit) will not decrease as sharply as one might expect. As shown in Figure 11, having multiple p-strategy agents increases the market efficiency slightly up to a certain point. Moreover, the profit-seeking behavior of self-interested agents will keep the UMDL agent population mixed with agents of various strategies. Even though the market efficiency eventually decreases with the increase in the number of p-strategy agents, because of its mixed agent population, the UMDL will not suffer market inefficiency of the worst case. Conclusion In this paper, we have used the p-strategy to examine the collective behavior of strategic agents in the UMDL system. In particular, we have examined market and allocation efficiency with varying numbers of p-strategy agents. The findings are useful to both system designers and agent designers. It is reassuring from the system designers' viewpoints that the market efficiency of the UMDL does not decrease as sharply as one might expect and that the worst-case market inefficiency is less likely to be realized (since even though self-interested agents have the capability of complex strategic reasoning, not all of them will behave so). At present, we cannot determine the exact demographics of the agent population for the best market efficiency, but we are continuing experiments on many different types of agent populations to get a better understanding of the overall system behavior. From the allocation-efficiency perspective, on the other hand, agent designers learn that using the p-strategy does not always pay off and that a simple strategy is sometimes as effective. We are currently developing an adaptive p- strategy to dynamically determine when to use the p- strategy and when not to. An adaptive p-strategy will be beneficial not only to a self-interested agent but also to the overall system efficiency. Acknowledgments This research has been funded in part by the joint NSF/DARPA/NASA Digital Libraries Initiative under CERA IRI-9411287. The first author is partially supported by the Horace H. Rackham Barbour scholarship. --R On Partially Controlled Multi-Agent Systems Flexible Social Laws. The Dynamics of the UMDL Service Market Society. The Double Auction Market. Lower Bounds for Efficiency of Surplus Extraction in Double Auctions. Rules of Encounter: Designing Conventions for Automated Negotiation among Computers: The MIT Press. Advantages of Strategic Thinking in Multiagent Contracts Limitations of the Vickery Auction in Computational Multiagent Systems. Advantages of a Leveled Commitment Contracting Protocol. Social Laws for Artificial Agent Societies: Off-line Design The Impact of Nested Agent Models in an Information Economy. A Market-Oriented Programming Environment and its Application to Distributed Multicommodity Flow Problems Mechanism Design for Automated Negotiation and its Application to Task Oriented Domains. --TR Rules of encounter On social laws for artificial agent societies Mechanism design for automated negotiation, and its application to task oriented domains Online learning about other agents in a dynamic multiagent system Price-war dynamics in a free-market economy of software agents Conjectural Equilibrium in Multiagent Learning The Dynamics of the UMDL Service Market Society Dynamics of an Information-Filtering Economy Emergent Properties of a Market-based Digital Library with Strategic Agents --CTR Samuel P. M. Choi , Jiming Liu, A dynamic mechanism for time-constrained trading, Proceedings of the fifth international conference on Autonomous agents, p.568-575, May 2001, Montreal, Quebec, Canada Sascha Ossowski , Andrea Omicini, Coordination knowledge engineering, The Knowledge Engineering Review, v.17 n.4, p.309-316, December 2002 Hiranmay Ghosh , Santanu Chaudhury, Distributed and Reactive Query Planning in R-MAGIC: An Agent-Based Multimedia Retrieval System, IEEE Transactions on Knowledge and Data Engineering, v.16 n.9, p.1082-1095, September 2004 Ricardo Buttner, A Classification Structure for Automated Negotiations, Proceedings of the 2006 IEEE/WIC/ACM international conference on Web Intelligence and Intelligent Agent Technology, p.523-530, December 18-22, 2006

multi-agent systems;digital libraries;emergent behavior;strategic reasoning

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The Gaia Methodology for Agent-Oriented Analysis and Design.

This article presents Gaia: a methodology for agent-oriented analysis and design. The Gaia methodology is both general, in that it is applicable to a wide range of multi-agent systems, and comprehensive, in that it deals with both the macro-level (societal) and the micro-level (agent) aspects of systems. Gaia is founded on the view of a multi-agent system as a computational organisation consisting of various interacting roles. We illustrate Gaia through a case study (an agent-based business process management system).

Introduction Progress in software engineering over the past two decades has been made through the development of increasingly powerful and natural high-level abstractions with which to model and develop complex systems. Procedural abstraction, abstract data types, and, most recently, objects and components are all examples of such abstractions. It is our belief that agents represent a similar advance in abstraction: they may be used by software developers to more naturally understand, model, and develop an important class of complex distributed systems. If agents are to realise their potential as a software engineering paradigm, then it is necessary to develop software engineering techniques that are specifically tailored to them. Existing software development techniques (for example, object-oriented analysis and design [2, 6]) are unsuitable for this task. There is a fundamental mismatch between the concepts used by object-oriented developers (and indeed, by other mainstream software engineering paradigms) and the agent-oriented view [32, 34]. In particular, extant approaches fail to adequately capture an agent's flexible, autonomous problem-solving behaviour, the richness of an agent's interactions, and the complexity of an agent system's organisational structures. For these reasons, this article introduces a methodology called Gaia, which has been specifically tailored to the analysis and design of agent-based systems 1 . The remainder of this article is structured as follows. We begin, in the following sub- section, by discussing the characteristics of applications for which we believe Gaia is ap- propriate. Section 2 gives an overview of the main concepts used in Gaia. Agent-based analysis is discussed in section 3, and design in section 4. The use of Gaia is illustrated by means of a case study in section 5, where we show how it was applied to the design of a real-world agent-based system for business process management [20]. Related work is discussed in section 6, and some conclusions are presented in section 7. Domain Characteristics Before proceeding, it is worth commenting on the scope of our work, and in particular, on the characteristics of domains for which we believe Gaia is appropriate. It is intended that Gaia be appropriate for the development of systems such as ADEPT [20] and ARCHON [19]. These are large-scale real-world applications, with the following main characteristics: ffl Agents are coarse-grained computational systems, each making use of significant computational resources (think of each agent as having the resources of a UNIX process). ffl It is assumed that the goal is to obtain a system that maximises some global quality measure, but which may be sub-optimal from the point of view of the system compo- nents. Gaia is not intended for systems that admit the possibility of true conflict 2 . ffl Agents are heterogeneous, in that different agents may be implemented using different programming languages, architectures, and techniques. We make no assumptions about the delivery platform. ffl The organisation structure of the system is static, in that inter-agent relationships do not change at run-time. ffl The abilities of agents and the services they provide are static, in that they do not change at run-time. ffl The overall system contains a comparatively small number of different agent types (less than 100). Gaia deals with both the macro (societal) level and the micro (agent) level aspects of de- sign. It represents an advance over previous agent-oriented methodologies in that it is neutral with respect to both the target domain and the agent architecture (see section 6 for a more detailed comparison). 2. A Conceptual Framework Gaia is intended to allow an analyst to go systematically from a statement of requirements to a design that is sufficiently detailed that it can be implemented directly. Note that we view the requirements capture phase as being independent of the paradigm used for analysis and design. In applying Gaia, the analyst moves from abstract to increasingly concrete concepts. Each successive move introduces greater implementation bias, and shrinks the space of possible systems that could be implemented to satisfy the original requirements statement. (See [21, pp216-222] for a discussion of implementation bias.) Analysis and design can be thought of as a process of developing increasingly detailed models of the system to be constructed. The main models used in Gaia are summarised in Figure 1 ANALYSIS AND DESIGN 3 requirements roles model services model agent model model acquaintance statement model interactions design analysis Figure 1. Relationships between Gaia's models Gaia borrows some terminology and notation from object-oriented analysis and design, (specifically, FUSION [6]). However, it is not simply a naive attempt to apply such methods to agent-oriented development. Rather, it provides an agent-specific set of concepts through which a software engineer can understand and model a complex system. In partic- ular, Gaia encourages a developer to think of building agent-based systems as a process of organisational design. The main Gaian concepts can be divided into two categories: abstract and concrete; abstract and concrete concepts are summarised in Table 1. Abstract entities are those used during analysis to conceptualise the system, but which do not necessarily have any direct realisation within the system. Concrete entities, in contrast, are used within the design process, and will typically have direct counterparts in the run-time system. 3. Analysis The objective of the analysis stage is to develop an understanding of the system and its structure (without reference to any implementation detail). In our case, this understanding is captured in the system's organisation. We view an organisation as a collection of roles, that stand in certain relationships to one another, and that take part in systematic, institutionalised patterns of interactions with other roles - see Figure 2. The most abstract entity in our concept hierarchy is the system. Although the term "sys- tem" is used in its standard sense, it also has a related meaning when talking about an Table 1. Abstract and concrete concepts in Gaia Abstract Concepts Concrete Concepts Roles Agent Types Permissions Services Responsibilities Acquaintances Protocols Activities Liveness properties Safety properties agent-based system, to mean "society" or "organisation". That is, we think of an agent-based system as an artificial society or organisation. The idea of a system as a society is useful when thinking about the next level in the concept hierarchy: roles. It may seem strange to think of a computer system as being defined by a set of roles, but the idea is quite natural when adopting an organisational view of the world. Consider a human organisation such as a typical company. The company has roles such as "president", "vice president", and so on. Note that in a concrete realisation of a company, these roles will be instantiated with actual individuals: there will be an individual who takes on the role of president, an individual who takes on the role of vice president, and so on. However, the instantiation is not necessarily static. Throughout the company's lifetime, many individuals may take on the role of company president, for example. Also, there is not necessarily a one-to-one mapping between roles and individuals. It is not unusual (particularly in small or informally defined organisations) for one individual to take on many roles. For example, a single individual might take on the role of "tea maker", "mail fetcher", and so on. Conversely, there may be many individuals that take on a single role, e.g., "salesman" 3 . A role is defined by four attributes: responsibilities, permissions, activities, and proto- cols. Responsibilities determine functionality and, as such, are perhaps the key attribute associated with a role. An example responsibility associated with the role of company president might be calling the shareholders meeting every year. Responsibilities are divided into two types: liveness properties and safety properties [27] 4 . Liveness properties intuitively state that "something good happens". They describe those states of affairs that an agent must bring about, given certain environmental conditions. In contrast, safety properties are invariants. Intuitively, a safety property states that "nothing bad happens" (i.e., that an acceptable state of affairs is maintained across all states of execution). An example might be "ensure the reactor temperature always remains in the range 0-100". In order to realise responsibilities, a role has a set of permissions. Permissions are the "rights" associated with a role. The permissions of a role thus identify the resources that are available to that role in order to realise its responsibilities. In the kinds of system that we have typically modelled, permissions tend to be information resources. For example, a role might have associated with it the ability to read a particular item of information, ANALYSIS AND DESIGN 5 properties safety system responsibilities roles interactions permissions liveness properties Figure 2. Analysis Concepts or to modify another piece of information. A role can also have the ability to generate information. The activities of a role are computations associated with the role that may be carried out by the agent without interacting with other agents. Activities are thus "private" actions, in the sense of [28]. Finally, a role is also identified with a number of protocols, which define the way that it can interact with other roles. For example, a "seller" role might have the protocols "Dutch auction" and "English auction" associated with it; the Contract Net Protocol is associated with the roles "manager" and "contractor" [30]. Thus, the organisation model in Gaia is comprised of two further models: the roles model (section 3.1) and the interaction model (section 3.2). 3.1. The Roles Model The roles model identifies the key roles in the system. Here a role can be viewed as an abstract description of an entity's expected function. In other terms, a role is more or less identical to the notion of an office in the sense that "prime minister", "attorney general of the United States", or "secretary of state for Education" are all offices. Such roles (or offices) are characterised by two types of attribute ffl The permissions/rights associated with the role. 6 WOOLDRIDGE, JENNINGS, AND KINNY A role will have associated with it certain permissions, relating to the type and the amount of resources that can be exploited when carrying out the role. In our case, these aspects are captured in an attribute known as the role's permissions. ffl The responsibilities of the role. A role is created in order to do something. That is, a role has a certain functionality. This functionality is represented by an attribute known as the role's responsibilities. Permissions The permissions associated with a role have two aspects: ffl they identify the resources that can legitimately be used to carry out the role - intu- itively, they say what can be spent while carrying out the role; ffl they state the resource limits within which the role executor must operate - intuitively, they say what can't be spent while carrying out the role. In general, permissions can relate to any kind of resource. In a human organisation, for example, a role might be given a monetary budget, a certain amount of person effort, and so on. However, in Gaia, we think of resources as relating only to the information or knowledge the agent has. That is, in order to carry out a role, an agent will typically be able to access certain information. Some roles might generate information; others may need to access a piece of information but not modify it, while yet others may need to modify the information. We recognise that a richer model of resources is required for the future, although for the moment, we restrict our attention simply to information. Gaia makes use of a formal notation for expressing permissions that is based on the FUSION notation for operation schemata [6, pp26-31]. To introduce our concepts we will use the example of a COFFEEFILLER role (the purpose of this role is to ensure that a coffee pot is kept full of coffee for a group of workers). The following is a simple illustration of the permissions associated with the role COFFEEFILLER: reads coffeeStatus // full or empty changes coffeeStock // stock level of coffee This specification defines two permissions for COFFEEFILLER: it says that the agent carrying out the role has permission to access the value coffeeStatus, and has permission to both read and modify the value coffeeStock. There is also a third type of permission, generates, which indicates that the role is the producer of a resource (not shown in the example). Note that these permissions relate to knowledge that the agent has. That is, coffeeStatus is a representation on the part of the agent of some value in the real world. Some roles are parameterised by certain values. For example, we can generalise the COFFEEFILLER role by parameterising it with the coffee machine that is to be kept refilled. This is specified in a permissions definition by the supplied keyword, as follows: reads supplied coffeeMaker // name of coffee maker coffeeStatus // full or empty changes coffeeStock // stock level of coffee ANALYSIS AND DESIGN 7 Table 2. Operators for liveness expression Operator Interpretation x:y x followed by y occurs x x occurs 0 or more times x+ x occurs 1 or more times x w x occurs infinitely often [x] x is optional Responsibilities The functionality of a role is defined by its responsibilities. These responsibilities can be divided into two categories: liveness and safety responsibilities. Liveness responsibilities are those that, intuitively, state that "something good happens". Liveness responsibilities are so called because they tend to say that "something will be done", and hence that the agent carrying out the role is still alive. Liveness responsibilities tend to follow certain patterns. For example, the guaranteed response type of achievement goal has the form "a request is always followed by a response". The infinite repetition achievement goal has the form "x will happen infinitely often". Note that these types of requirements have been widely studied in the software engineering literature, where they have proven to be necessary for capturing properties of reactive systems [27]. In order to illustrate the various concepts associated with roles, we will continue with our running example of the COFFEEFILLER role. Examples of liveness responsibilities for the COFFEEFILLER role might be: ffl whenever the coffee pot is empty, fill it up; ffl whenever fresh coffee is brewed, make sure the workers know about it. In Gaia, liveness properties are specified via a liveness expression, which defines the "life- cycle" of the role. Liveness expressions are similar to the life-cycle expression of FUSION [6], which are in turn essentially regular expressions. Our liveness expressions have an additional operator, "w", for infinite repetition (see Table 2 for more details). They thus resemble w-regular expressions, which are known to be suitable for representing the properties of infinite computations [32]. Liveness expressions define the potential execution trajectories through the various activities and interactions (i.e., over the protocols) associated with the role. The general form of a liveness expression is: where ROLENAME is the name of the role whose liveness properties are being defined, and expression is the liveness expression defining the liveness properties of ROLENAME. The atomic components of a liveness expression are either activities or protocols. An activity is somewhat like a method in object-oriented terms, or a procedure in a PASCAL-like language. It corresponds to a unit of action that the agent may perform, which does not involve interaction with any other agent. Protocols, on the other hand, are activities that do require interaction with other agents. To give the reader some visual clues, we protocol names in a sans serif font (as in xxx), and use a similar font, underlined, for activity names (as in yyy). To illustrate liveness expressions, consider again the above-mentioned responsibilities of the COFFEEFILLER role: InformWorkers: CheckStock: AwaitEmpty) w This expression says that COFFEEFILLER consists of executing the protocol Fill, followed by the protocol InformWorkers, followed by the activity CheckStock and the protocol AwaitEmpty. The sequential execution of these protocols and activities is then repeated infinitely often. For the moment, we shall treat the protocols simply as labels for interactions and shall not worry about how they are actually defined (this matter will be discussed in section 3.2). Complex liveness expressions can be made easier to read by structuring them. A simple example illustrates how this is done: InformWorkers: CheckStock: AwaitEmpty The semantics of such definitions are straightforward textual substitution. In many cases, it is insufficient simply to specify the liveness responsibilities of a role. This is because an agent, carrying out a role, will be required to maintain certain invariants while executing. For example, we might require that a particular agent taking part in an electronic commerce application never spends more money than it has been allocated. These invariants are called safety conditions, because they usually relate to the absence of some undesirable condition arising. Safety requirements in Gaia are specified by means of a list of predicates. These predicates are typically expressed over the variables listed in a role's permissions attribute. Returning to our COFFEEFILLER role, an agent carrying out this role will generally be required to ensure that the coffee stock is never empty. We can do this by means of the following safety expression: By convention, we simply list safety expressions as a bulleted list, each item in the list expressing an individual safety responsibility. It is implicitly assumed that these responsibilities apply across all states of the system execution. If the role is of infinitely long duration (as in the COFFEEFILLER example), then the invariants must always be true. It is now possible to precisely define the Gaia roles model. A roles model is comprised of a set of role schemata, one for each role in the system. A role schema draws together the ANALYSIS AND DESIGN 9 Role Schema: name of role Description short English description of the role Protocols and Activities protocols and activities in which the role plays a part Permissions "rights" associated with the role Responsibilities Liveness liveness responsibilities Safety safety responsibilities Figure 3. Template for Role Schemata Role Schema: COFFEEFILLER Description: This role involves ensuring that the coffee pot is kept filled, and informing the workers when fresh coffee has been brewed. Protocols and Activities: Fill, InformWorkers, CheckStock, AwaitEmpty Permissions: reads supplied coffeeMaker // name of coffee maker coffeeStatus // full or empty changes coffeeStock // stock level of coffee Responsibilities Liveness: InformWorkers: CheckStock: AwaitEmpty) w Safety: Figure 4. Schema for role COFFEEFILLER various attributes discussed above into a single place (Figure 3). An exemplar instantiation is given for the COFFEEFILLER role in Figure 4. This schema indicates that COFFEEFILLER has permission to read the coffeeMaker parameter (that indicates which coffee machine the role is intended to keep filled), and the coffeeStatus (that indicates whether the machine is full or empty). In addition, the role has permission to change the value coffeeStock. 3.2. The Interaction Model There are inevitably dependencies and relationships between the various roles in a multi-agent organisation. Indeed, such interplay is central to the way in which the system func- tions. Given this fact, interactions obviously need to be captured and represented in the CoffeeFiller Fill CoffeeMachine Fill coffee machine supplied coffeeMaker coffeeStock Figure 5. The Fill Protocol Definition analysis phase. In Gaia, such links between roles are represented in the interaction model. This model consists of a set of protocol definitions, one for each type of inter-role interac- tion. Here a protocol can be viewed as an institutionalised pattern of interaction. That is, a pattern of interaction that has been formally defined and abstracted away from any particular sequence of execution steps. Viewing interactions in this way means that attention is focused on the essential nature and purpose of the interaction, rather than on the precise ordering of particular message exchanges (cf. the interaction diagrams of OBJECTORY [6, pp198-203] or the scenarios of FUSION [6]). This approach means that a single protocol definition will typically give rise to a number of message interchanges in the run time system. For example, consider an English auction protocol. This involves multiple roles (sellers and bidders) and many potential patterns of interchange (specific price announcements and corresponding bids). However at the analysis stage, such precise instantiation details are unnecessary, and too premature. A protocol definition consists of the following attributes: ffl purpose: brief textual description of the nature of the interaction (e.g., "information "schedule activity" and "assign task"); ffl initiator: the role(s) responsible for starting the interaction; ffl responder: the role(s) with which the initiator interacts; ffl inputs: information used by the role initiator while enacting the protocol; ffl outputs: information supplied by/to the protocol responder during the course of the ffl processing: brief textual description of any processing the protocol initiator performs during the course of the interaction. As an illustration, consider the Fill protocol, which forms part of the COFFEEFILLER role Figure 5). This states that the protocol Fill is initiated by the role COFFEEFILLER and involves the role COFFEEMACHINE. The protocol involves COFFEEFILLER putting coffee in the machine named coffeeMaker, and results in COFFEEMACHINE being informed about the value of coffeeStock. We will see further examples of protocols in section 5. ANALYSIS AND DESIGN 11 3.3. The Analysis Process The analysis stage of Gaia can now be summarised: 1. Identify the roles in the system. Roles in a system will typically correspond to: ffl individuals, either within an organisation or acting independently; ffl departments within an organisation; or organisations themselves. Output: A prototypical roles model - a list of the key roles that occur in the system, each with an informal, unelaborated description. 2. For each role, identify and document the associated protocols. Protocols are the patterns of interaction that occur in the system between the various roles. For example, a protocol may correspond to an agent in the role of BUYER submitting a bid to another agent in the role of SELLER. Output: An interaction model, which captures the recurring patterns of inter-role inter-action 3. Using the protocol model as a basis, elaborate the roles model. Output: A fully elaborated roles model, which documents the key roles occurring in the system, their permissions and responsibilities, together with the protocols and activities in which they participate. 4. Iterate stages (1)-(3). 4. Design The aim of a "classical" design process is to transform the abstract models derived during the analysis stage into models at a sufficiently low level of abstraction that they can be easily implemented. This is not the case with agent-oriented design, however. Rather, the aim in Gaia is to transform the analysis models into a sufficiently low level of abstraction that traditional design techniques (including object-oriented techniques) may be applied in order to implement agents. To put it another way, Gaia is concerned with how a society of agents cooperate to realise the system-level goals, and what is required of each individual agent in order to do this. Actually how an agent realises its services is beyond the scope of Gaia, and will depend on the particular application domain. The Gaia design process involves generating three models (see Figure 1). The agent model identifies the agent types that will make up the system, and the agent instances that will be instantiated from these types. The services model identifies the main services that are required to realise the agent's role. Finally, the acquaintance model documents the lines of communication between the different agents. Table 3. Instance Qualifiers Qualifier Meaning there will be exactly n instances m::n there will be between m and n instances there will be 0 or more instances there will be 1 or more instances 4.1. The Agent Model The purpose of the Gaia agent model is to document the various agent types that will be used in the system under development, and the agent instances that will realise these agent types at run-time. An agent type is best thought of as a set of agent roles. There may in fact be a one-to-one correspondence between roles (as identified in the roles model - see section 3.1) and agent types. However, this need not be the case. A designer can choose to package a number of closely related roles in the same agent type for the purposes of convenience. Efficiency will also be a major concern at this stage: a designer will almost certainly want to optimise the design, and one way of doing this is to aggregate a number of agent roles into a single type. An example of where such a decision may be necessary is where the "footprint" of an agent (i.e., its run-time requirements in terms of processor power or memory space) is so large that it is more efficient to deliver a number of roles in a single agent than to deliver a number of agents each performing a single role. There is obviously a trade-off between the coherence of an agent type (how easily its functionality can be understood) and the efficiency considerations that come into play when designing agent types. The agent model is defined using a simple agent type tree, in which leaf nodes correspond to roles, (as defined in the roles model), and other nodes correspond to agent types. If an agent type t 1 has children t 2 and t 3 , then this means that t 1 is composed of the roles that make up t 2 and t 3 . We document the agent instances that will appear in a system by annotating agent types in the agent model (cf. the qualifiers from FUSION [6]). An annotation n means that there will be exactly n agents of this type in the run-time system. An annotation m::n means that there will be no less than m and no more than n instances of this type in a run-time system n). An annotation means that there will be zero or more instances at run-time, and means that there will be one or more instances at run-time (see Table 3). Note that inheritance plays no part in Gaia agent models. Our view is that agents are coarse grained computational systems, and an agent system will typically contain only a comparatively small number of roles and types, with often a one-to-one mapping between them. For this reason, we believe that inheritance has no useful part to play in the design of agent types. (Of course, when it comes to actually implementing agents, inheritance may be used to great effect, in the normal object-oriented fashion.) ANALYSIS AND DESIGN 13 4.2. The Services Model As its name suggests, the aim of the Gaia services model is to identify the services associated with each agent role, and to specify the main properties of these services. By a service, we mean a function of the agent. In OO terms, a service would correspond to a method; however, we do not mean that services are available for other agents in the same way that an object's methods are available for another object to invoke. Rather, a service is simply a single, coherent block of activity in which an agent will engage. It should be clear there every activity identified at the analysis stage will correspond to a service, though not every service will correspond to an activity. For each service that may be performed by an agent, it is necessary to document its properties. Specifically, we must identify the inputs, outputs, pre-conditions, and post-conditions of each service. Inputs and outputs to services will be derived in an obvious way from the protocols model. Pre- and post-conditions represent constraints on services. These are derived from the safety properties of a role. Note that by definition, each role will be associated with at least one service. The services that an agent will perform are derived from the list of protocols, activ- ities, responsibilities and the liveness properties of a role. For example, returning to the coffee example, there are four activities and protocols associated with this role: Fill, InformWorkers, CheckStock, and AwaitEmpty. In general, there will be at least one service associated with each protocol. In the case of CheckStock, for example, the service (which may have the same name), will take as input the stock level and some threshold value, and will simply compare the two. The pre- and post-conditions will both state that the coffee stock level is greater than 0. This is one of the safety properties of the role COFFEEFILLER. The Gaia services model does not prescribe an implementation for the services it doc- uments. The developer is free to realise the services in any implementation framework deemed appropriate. For example, it may be decided to implement services directly as methods in an object-oriented language. Alternatively, a service may be decomposed into a number of methods. 4.3. The Acquaintance Model The final Gaia design model is probably the simplest: the acquaintance model. Acquaintance models simply define the communication links that exist between agent types. They do not define what messages are sent or when messages are sent - they simply indicate that communication pathways exist. In particular, the purpose of an acquaintance model is to identify any potential communication bottlenecks, which may cause problems at run-time (see section 5 for an example). It is good practice to ensure that systems are loosely coupled, and the acquaintance model can help in doing this. On the basis of the acquaintance model, it may be found necessary to revisit the analysis stage and rework the system design to remove such problems. An agent acquaintance model is simply a graph, with nodes in the graph corresponding to agent types and arcs in the graph corresponding to communication pathways. Agent acquaintance models are directed graphs, and so an arc a ! b indicates that a will send 14 WOOLDRIDGE, JENNINGS, AND KINNY messages to b, but not necessarily that b will send messages to a. An acquaintance model may be derived in a straightforward way from the roles, protocols, and agent models. 4.4. The Design Process The Gaia design stage can now be summarised: 1. Create an agent model: ffl aggregate roles into agent types, and refine to form an agent type hierarchy; ffl document the instances of each agent type using instance annotations. 2. Develop a services model, by examining activities, protocols, and safety and liveness properties of roles. 3. Develop an acquaintance model from the interaction model and agent model. 5. A Case Study: Agent-Based Business Process Management This section briefly illustrates how Gaia can be applied, through a case study of the analysis and design of an agent-based system for managing a British Telecom business process (see [20] for more details). For reasons of brevity, we omit some details, and aim instead to give a general flavour of the analysis and design. The particular application is providing customers with a quote for installing a network to deliver a particular type of telecommunications service. This activity involves the following departments: the customer service division (CSD), the design division (DD), the legal division (LD) and the various organisations who provide the out-sourced service of vetting customers (VCs). The process is initiated by a customer contacting the CSD with a set of requirements. In parallel to capturing the requirements, the CSD gets the customer vetted. If the customer fails the vetting procedure, the quote process terminates. Assuming the customer is satisfactory, their requirements are mapped against the service portfolio. If they can be met by a standard off-the-shelf item then an immediate quote can be offered. In the case of bespoke services, however, the process is more complex. DD starts to design a solution to satisfy the customer's requirements and whilst this is occurring LD checks the legality of the proposed service. If the desired service is illegal, the quote process ter- minates. Assuming the requested service is legal, the design will eventually be completed and costed. DD then informs CSD of the quote. CSD, in turn, informs the customer. The business process then terminates. Moving from this process-oriented description of the system's operation to an organisational view is comparatively straightforward. In many cases there is a one to one mapping between departments and roles. CSD's behaviour falls into two distinct roles: one acting as an interface to the customer ( CUSTOMERHANDLER, Figure 6), and one overseeing the process inside the organisation ( QUOTEMANAGER, Figure 7). Thus, the VC's, the LD's, and the DD's behaviour are covered by the roles CUSTOMERVETTER (Figure 8), respectively. The final role is that of the CUSTOMER (Figure 11) who requires the quote. ANALYSIS AND DESIGN 15 Role Schema: CUSTOMERHANDLER (CH) Description: Receives quote request from the customer and oversees process to ensure appropriate quote is returned. Protocols and Activities: AwaitCall, ProduceQuote, InformCustomer Permissions: reads supplied customerDetails // customer contact information supplied customerRequirements // what customer wants quote // completed quote or nil Responsibilities Liveness: Safety: ffl true Figure 6. Schema for role CUSTOMERHANDLER With the respective role definitions in place, the next stage is to define the associated interaction models for these roles. Here we focus on the interactions associated with the QUOTEMANAGER role. This role interacts with the CUSTOMER role to obtain the customer's requirements ( GetCustomerRequirements protocol, Figure 12c) and with the CUSTOMERVETTER role to determine whether the customer is satisfactory ( VetCustomer protocol, Figure 12a). If the customer proves unsatisfactory, these are the only two protocols that are enacted. If the customer is satisfactory then their request is costed. This costing involves enacting activity CostStandardService for frequently requested services or the CheckServiceLegality Figure 12b) and CostBespokeService (Figure 12d) protocols for non-standard requests. Having completed our analysis of the application, we now turn to the design phase. The first model to be generated is the agent model (Figure 13). This shows, for most cases, a one-to-one correspondence between roles and agent types. The exception is for the CUSTOMERHANDLER and QUOTEMANAGER roles which, because of their high degree of interdependence are grouped into a single agent type. The second model is the services model. Again because of space limitations we concentrate on the QUOTEMANAGER role and the Customer Service Division Agent. Based on the QUOTEMANAGER role, seven distinct services can be identified (Table 3). From the GetCustomerRequirements protocol, we derive the service "obtain customer requirements". This service handles the interaction from the perspective of the quote manager. It takes the customerDetails as input and returns the customerRequirements as output (Figure 12c). There are no associated pre- or post-conditions. The service associated with the VetCustomer protocol is "vet customer". Its inputs, derived from the protocol definition (Figure 12a), are the customerDetails and its outputs are creditRating. This service has a pre-condition that an appropriate customer vetter must be Role Schema: QUOTEMANAGER (QM) Description: Responsible for enacting the quote process. Generates a quote or returns no quote (nil) if customer is inappropriate or service is illegal. Protocols and Activities: VetCustomer, GetcustomerRequirements, CostStandardService, CheckServiceLegality, CostBespokeService Permissions: reads supplied customerDetails // customer contact information supplied customerRequirements // detailed service requirements creditRating // customer's credit rating serviceIsLegal // boolean for bespoke requests generates quote // completed quote or nil Responsibilities Liveness: CostService CheckServiceLegality k CostBespokeService) Safety: Figure 7. Schema for role QUOTEMANAGER available (derived from the TenderContract interaction on the VetCustomer protocol) and a post condition that the value of creditRating is non-null (because this forms part of a safety condition of the QUOTEMANAGER role). The third service involves checking whether the customer is satisfactory (the creditRating safety condition of QUOTEMANAGER). If the customer is unsatisfactory then only the first branch of the QuoteRespose liveness condition (Figure 7) gets executed. If the customer is satisfactory, the CostService liveness route is executed. The next service makes the decision of which path of the CostService liveness expression gets executed. Either the service is of a standard type (execute the service "produce standard costing") or it is a bespoke service in which case the CheckServiceLegality and CostBespokeService protocols are enacted. In the latter case, the protocols are associated with the service "produce bespoke costing". This service produces a non-nil value for quote as long as the serviceIsLegal safety condition (Figure 7) is not violated. The final service involves informing the customer of the quote. This, in turn, completes the CUSTOMERHANDLER role. ANALYSIS AND DESIGN 17 Service Inputs Outputs Pre-condition Post-condition obtain customer re- quirements customerDetails customerRequirements true true vet customer customerDetails creditRating customer vetter available creditRating nil check customer creditRating continuationDecision continuationDecision nil continuationDecision nil check service type customerRequirements serviceType creditRating bad serviceTypefstandard;bespokeg produce standard ser- vice costing serviceType, customerRequirements quote serviceType= standard- quote= nil quote nil produce bespoke ser- vice costing serviceType, customerRequirements quote, serviceIsLegal serviceType bespoke quote serviceIsLegal (quote (quote nil-:serviceIsLegal) inform customer customerDetails, quote true customers know quote Table 3. The services model Role Schema: CUSTOMERVETTER (CV) Description: Checks credit rating of supplied customer. Protocols and Activities: VettingRequest, VettingResponse Permissions: reads supplied customerDetails // customer contact information customerRatingInformation // credit rating information generates creditRating // credit rating of customer Responsibilities Liveness: Safety: Figure 8. Schema for role CUSTOMERVETTER Role Schema: LEGALADVISOR (LA) Description: Determines whether given bespoke service request is legal or not. Protocols and Activities: LegalCheckRequest, LegalCheckResponse Permissions: reads supplied customerRequirements // details of proposed service generates serviceIsLegal // true or false Responsibilities Liveness: Safety: ffl true Figure 9. Schema for role LEGALADVISOR The final model is the acquaintance model, which shows the communication pathways that exist between agents (Figure 14). ANALYSIS AND DESIGN 19 Role Schema: NETWORKDESIGNER (ND) Description: Design and cost network to meet bespoke service request requirements. Protocols and Activities: CostingRequest, ProduceDesign, ReturnCosting Permissions: reads supplied customerRequirements // details of proposed service serviceIsLegal // boolean generates quote // cost of realising service Responsibilities Liveness: Safety: Figure 10. Schema for role NETWORKDESIGNER Role Schema: CUSTOMER (CUST) Description: Organisation or individual requiring a service quote. Protocols and Activities: MakeCall, GiveRequirements Permissions: generates customerDetails // Owner of customer information customerRequirements // Owner of customer requirements Responsibilities Liveness: Safety: ffl true Figure 11. Schema for role CUSTOMER 6. Related Work In recent times there has been a surge of interest in agent-oriented modelling techniques and method- ologies. The various approaches may be roughly grouped as follows: QM CostingRequest ask for costing ReturnCosting CH, QM design network and cost solution customerRequirements customerRequirements quote TenderContract QM select which CV to award contract to vettingRequirements VettingRequest QM CV customer ask for vetting of customerDetails VettingResponse perform vetting and return credit rating customerDetails customerRatingInfo creditRating LegalCheckRequest LA service's legality ask for check of customerRequirements customerRequirements LegalCheckResponse LA QM, ND check service legality serviceIsLegal (a) (b) (c) (d) QM RequirementsRequest CUST requirements details of customer's customerDetails CUST QM details provide service customerRequirements GiveRequirements Figure 12. Definition of protocols associated with the QUOTEMANAGER role: (a) VetCustomer, (b) CheckServiceLegality, (c) GetCustomerRequirements, and (d) CostBespokeService. CustomerAgent Customer CustomerHandler QuoteManager CustomerServiceDivisionAgentCustomerVetter VetCustomerAgentNetworkDesignerAgent NetworkDesignerLegalAdvisorAgent Figure 13. The agent model ffl those such as [4, 24] which take existing OO modelling techniques or methodologies as their basis, seeking either to extend and adapt the models and define a methodology for their use, or to directly extend the applicability of OO methodologies and techniques, such as design patterns, to the design of agent systems, ANALYSIS AND DESIGN 21 CustomerServiceDivisionAgent CustomerAgent NetworkDesignAgent LegalAdvisorAgent VetCustomerAgent Figure 14. The acquaintance model ffl those such as [3, 17] which build upon and extend methodologies and modelling techniques from knowledge engineering, providing formal, compositional modelling languages suitable for the verification of system structure and function, ffl those which take existing formal methods and languages, for example Z [31], and provide definitions within such a framework that support the specification of agents or agent systems [26], and ffl those which have essentially been developed de novo for particular kinds of agent systems. CASSIOPEIA [7], for example, supports the design of Contract Net [29] based systems and has been applied to Robot Soccer. These design methodologies may also be divided into those that are essentially top-down approaches based on progressive decomposition of behaviour, usually building (as in Gaia) on some notion of role, and those such as CASSIOPEIA that are bottom-up approaches which begin by identifying elementary agent behaviours. A very useful survey which classifies and reviews these and other methodologies has also appeared [16]. The definition and use of various notions of role, responsibility, interaction, team and society or organization in particular methods for agent-oriented analysis and design has inherited or adapted much from more general uses of these concepts within multi-agent systems, including organization- focussed approaches such as [14, 9, 18] and sociological approaches such as [5]. However, it is beyond the scope of this article to compare the Gaia definition and use of these concepts with this heritage. Instead, we will focus here on the relationship between Gaia and other approaches based that build upon OO techniques, in particular the kgr approach [24, 23]. But it is perhaps useful to begin by summarizing why OO modelling techniques and design methodologies themselves are not directly applicable to multi-agent system design. 6.1. Shortcomings of Object Oriented techniques The first problem concerns the modelling of individual agents or agent classes. While there are superficial similarities between agents and objects, representing an agent as an object, i.e., as a set of attributes and methods, is not very useful because the representation is too fine-grained, operating at an inappropriate level of abstraction. An agent so represented may appear quite strange, perhaps exhibiting only one public method whose function is to receive messages from other agents. Thus an object model does not capture much useful information about an agent, and powerful OO concepts such as inheritance and aggregation become quite useless as a result of the poverty of the representation There are several reasons for this problem. One is that the agent paradigm is based on a significantly stronger notion of encapsulation than the object paradigm. An agent's internal state is 22 WOOLDRIDGE, JENNINGS, AND KINNY usually quite opaque and, in some systems, the behaviours that an agent will perform upon request are not even made known until it advertises them within an active system. Related to this is the key characteristic of autonomy: agents cannot normally be created and destroyed in the liberal manner allowed within object systems and they have more freedom to determine how they may respond to messages, including, for example, by choosing to negotiate some agreement about how a task will be performed. As the underlying communication model is usually asynchronous there is no predefined notion of flow of control from one agent to another: an agent may autonomously initiate internal or external behaviour at any time, not just when it is sent a message. Finally, an agent's internal state, including its knowledge, may need to be represented in a manner that cannot easily be translated into a set of attributes; in any case to do so would constitute a premature implementation bias. The second problem concerns the power of object models to adequately capture the relationships that hold between agents in a multi-agent system. While the secondary models in common use in OO methodologies such as use cases and interaction diagrams may usefully be adapted (with somewhat different semantics), the Object Model, which constitutes the primary specification of an OO system, captures associations between object classes that model largely static dependencies and paths of accessibility which are largely irrelevant in a multi-agent system. Only the instantiation relationship between classes and instances can be directly adopted. Important aspects of relationships between agents such as their repertoire of interactions and their degree of control or influence upon each other are not easily captured. The essential problem here is the uniformity and static nature of the OO object model. An adequate agent model needs to capture these relationships between agents, their dynamic nature, and perhaps also relationships between agents and non-agent elements of the system, including passive or abstract ones such as those modelled here as resources. Both of these are problems concerning the suitability of OO modelling techniques for modelling a multi-agent system. Another issue is the applicability of OO methodologies to the process of analyzing and designing a multi-agent system. OO methodologies typically consist of an iterative refinement cycle of identifying classes, specifying their semantics and relationships, and elaborating their interfaces and implementation. At this level of abstraction, they appear similar to typical AO methodologies, which usually proceed by identifying roles and their responsibilities and goals, developing an organizational structure, and elaborating the knowledge and behaviours associated with a role or agent. However, this similarity disappears at the level of detail required by the models, as the key abstractions involved are quite different. For example, the first step of object class identification typically considers tangible things, roles, organizations, events and even interactions as candidate objects, whereas these need to be clearly distinguished and treated differently in an agent-oriented approach. The uniformity and concreteness of the object model is the basis of the problem; OO methodologies provide guidance or inspiration rather than a directly useful approach to analysis and design. 6.2. Comparison with the KGR approach The KGR approach [24, 23] was developed to fulfill the need for a principled approach to the specification of complex multi-agent systems based on the belief-desire-intention (BDI) technology of the Procedural Reasoning System (PRS) and the Distributed Multi-Agent Reasoning System (DMARS) [25, 8]. A key motivation of the work was to provided useful, familiar mechanisms for structuring and managing the complexity of such systems. The first and most obvious difference between the approach proposed here and KGR is one of scope. Our methodology does not attempt to unify the analysis and abstract design of a multi-agent system with its concrete design and implementation with a particular agent technology, regarding the output of the analysis and design process as an abstract specification to which traditional lower-level design methodologies may be applied. KGR, by contrast, makes a strong architectural commitment ANALYSIS AND DESIGN 23 to BDI architectures and proposes a design elaboration and refinement process that leads to directly executable agent specifications. Given the proliferation of available agent technologies, there are clearly advantages to a more general approach, as proposed here. However, the downside is that it cannot provide a set of models, abstractions and terminology that may be used uniformly throughout the system life cycle. Furthermore, there may be a need for iteration of the AO analysis and design process if the lower-level design process reveals issues that are best resolved at the AO level. A re-search problem for our approach and others like it is whether and how the adequacy and completeness of its outputs can be assessed independently of any traditional design process that follows. A second difference is that in this work a clear distinction is made between the analysis phase, in which the roles and interaction models are fully elaborated, and the design phase, in which agent, services and acquaintance models are developed. The KGR approach does not make such a distinc- tion, proposing instead the progressive elaboration and refinement of agent and interaction models which capture respectively roles, agents and services, and interactions and acquaintances. While both methodologies begin with the identification of roles and their properties, here we have chosen to model separately abstract agents (roles), concrete agents and the services they provide. KGR, on the other hand, employs a more uniform agent model which admits both abstract agents and concrete agent classes and instances and allows them to be organized within an inheritance hierarchy, thus allowing multiple levels of abstraction and the deferment of identification of concrete agent classes until late in the design process. While both approaches employ responsibilities as an abstraction used to decompose the structure of a role, they differ significantly as to how these are represented and developed. Here responsibilities consist of safety and liveness properties built up from already identified interactions and activities. By contrast, KGR treats responsibilities as abstract goals, triggered by events or interactions, and adopts a strictly top-down approach to decomposing these into services and low level goals for which activity specifications may be elaborated. There are similarities however, for despite the absence of explicit goals in our approach, safety properties may be viewed as maintenance goals and liveness properties as goals of achievement. The notion of permissions, however, is absent from the KGR approach, whereas the notion of protocols may be developed to a much greater degree of detail, for example as in [22]. There protocols are employed as more generic descriptions of behaviour that may involve entities not modelled as agents, such as the coffee machine. To summarize the key differences, the KGR approach, by making a commitment to implementation with a BDI agent architecture, is able to employ an iterative top-down approach to elaborating a set of models that describe a multi-agent system at both the macro- and micro-level, to make more extensive use of OO modelling techniques, and to produce executable specifications as its final output. The approach we have described here is a mixed top-down and bottom-up approach which employs a more fine-grained and diverse set of generic models to capture the result of the analysis and design process, and tries to avoid any premature commitment, either architectural, or as to the detailed design and implementation process which will follow. We envisage, however, that our approach can be suitably specialized for specific agent architectures or implementation techniques; this is a subject for further research. 7. Conclusions and Further Work In this article, we have described Gaia, a methodology for the analysis and design of agent-based systems. The key concepts in Gaia are roles, which have associated with them responsibilities, permissions, activities, and protocols. Roles can interact with one another in certain institutionalised ways, which are defined in the protocols of the respective roles. There are several issues remaining for future work. ffl Self-Interested Agents. Gaia does not explicitly attempt to deal with systems in which agents may not share common goals. This class of systems represents arguably the most important application area for multi-agent systems, and it is therefore essential that a methodology should be able to deal with it. ffl Dynamic and open systems. Open systems - in which system components may join and leave at run-time, and which may be composed of entities that a designer had no knowledge of at design-time - have long been recognised as a difficult class of system to engineer [15, 13]. ffl Organisation structures. Another aspect of agent-based analysis and design that requires more work is the notion of an organisational structure. At the moment, such structures are only implicitly defined within Gaia - within the role and interaction models. However, direct, explicit representations of such structures will be of value for many applications. For example, if agents are used to model large organisations, then these organisations will have an explicitly defined structure. Representing such structures may be the only way of adequately capturing and understanding the organisa- tion's communication and control structures. More generally, the development of organisation design patterns might be useful for reusing successful multi-agent system structures (cf. [12]). ffl Cooperation Protocols. The representation of inter-agent cooperation protocols within Gaia is currently somewhat im- poverished. In future work, we will need to provide a much richer protocol specification frame-work ffl International Standards. Gaia was not designed with any particular standard for agent communication in mind (such as the FIPA agent communication language [11]). However, in the event of widescale industrial takeup of such standards, it may prove useful to adapt our methodology to be compatible with such standards. ffl Formal Semantics. Finally, we believe that a successful methodology is one that is not only of pragmatic value, but one that also has a well-defined, unambiguous formal semantics. While the typical developer need never even be aware of the existence of such a semantics, it is nevertheless essential to have a precise understanding of what the concepts and terms in a methodology mean [33]. Acknowledgments This article is a much extended version of [35]. We are grateful to the participants of the Agents 99 conference, who gave us much useful feedback. Notes 1. In Greek mythology, Gaia was the mother Earth figure. More pertinently, Gaia is the name of an influential hypothesis put forward by the ecologist James Lovelock, to the effect that all the living organisms on the Earth can be understood as components of a single entity, which regulates the Earth's environment. The theme of many heterogeneous entities acting together to achieve a single goal is a central theme in multi-agent systems research [1], and was a key consideration in the the development of our methodology. 2. To be more precise, we believe such systems will require additional models over and above those that we outline in the current version of the methodology. ANALYSIS AND DESIGN 25 3. The third case, which we have not yet elaborated in the methodology, is that a single role represents the collective behaviour of a number of individuals. This view is important for modelling cooperative and team problem solving and also for bridging the gap between the micro and the macro levels in an agent-based system. 4. The most widely used formalism for specifying liveness and safety properties is temporal logic, and in previous work, the use of such formalism has been strongly advocated for use in agent systems [10]. Although it has undoubted strengths as a mathematical tool for expressing liveness and safety properties, there is some doubt about its viability as a tool for use by everyday software engineers. We have therefore chosen an alternative approach to temporal logic, based on regular expressions, as these are likely to be better understood by our target audience. 5. For the moment, we do not explicitly model the creation and deletion of roles. Thus roles are persistent throughout the system's lifetime. In the future, we plan to make this a more dynamic process --R Readings in Distributed Artificial Intelligence. Formal specification of multi-agent systems: a real-world case Models and methodologies for agent-oriented analysis and design Commitments: from individual intentions to groups and organizations. Agent oriented design of a soccer robot team. A formal specification of dMARS. 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methodologies;analysis and design;agent-oriented;software engineering

608666

Hierarchical Wrapper Induction for Semistructured Information Sources.

With the tremendous amount of information that becomes available on the Web on a daily basis, the ability to quickly develop information agents has become a crucial problem. A vital component of any Web-based information agent is a set of wrappers that can extract the relevant data from semistructured information sources. Our novel approach to wrapper induction is based on the idea of hierarchical information extraction, which turns the hard problem of extracting data from an arbitrarily complex document into a series of simpler extraction tasks. We introduce an inductive algorithm, STALKER, that generates high accuracy extraction rules based on user-labeled training examples. Labeling the training data represents the major bottleneck in using wrapper induction techniques, and our experimental results show that STALKER requires up to two orders of magnitude fewer examples than other algorithms. Furthermore, STALKER can wrap information sources that could not be wrapped by existing inductive techniques.

Introduction With the Web, computer users have gained access to a large variety of comprehensive information repositories. However, the Web is based on a browsing paradigm that makes it difficult to retrieve and integrate data from multiple sources. The most recent generation of information agents (e.g., WHIRL (Cohen, 1998), Ariadne (Knoblock et al., 1998), and Information Manifold (Kirk et al., 1995) ) address this problem by enabling information from pre-specified sets of Web sites to be accessed via database-like queries. For instance, consider the query "What seafood restaurants in L.A. have prices below $20 and accept the Visa credit-card?" Assume that we have two information sources that provide information about LA restaurants: the Zagat Guide and LA Weekly (see Figure 1). To answer this query, an agent could use Zagat's to identify seafood restaurants under $20 and then use LA Weekly to check which of these accept Visa. c Publishers. Printed in the Netherlands. Ion Muslea, Steven Minton, Craig A. Knoblock Information agents generally rely on wrappers to extract information from semistructured Web pages (a document is semistructured if the location of the relevant information can be described based on a concise, formal grammar). Each wrapper consists of a set of extraction rules and the code required to apply those rules. Some systems, such as tsimmis (Chawathe et al., 1994) and araneus (Atzeni et al., 1997) depend on humans to write the necessary grammar rules. However, there are several reasons why this is undesirable. Writing extraction rules is tedious, time consuming and requires a high level of expertise. These difficulties are multiplied when an application domain involves a large number of existing sources or the format of the source documents changes over time. In this paper, we introduce a new machine learning method for wrapper construction that enables unsophisticated users to painlessly turn Web pages into relational information sources. The next section presents a formalism describing semistructured Web documents, and then Sections 3 and 4 present a domain-independent information extractor that we use as a skeleton for all our wrappers. Section 5 describes stalker, a supervised learning algorithm for inducing extraction rules, and Section 6 presents a detailed example. The final sections describe our experimental results, related work and conclusions. 2. Describing the Content of a Page Because Web pages are intended to be human readable, there are some common conventions for structuring HTML documents. For instance, the information on a page often exhibits some hierarchical structure; furthermore, semistructured information is often presented in the form of lists of tuples, with explicit separators used to distinguish the different elements. With these observations in mind, we developed the embedded catalog (EC) formalism, which can describe the structure of a wide-range of semistructured documents. The EC description of a page is a tree-like structure in which the leaves are the items of interest for the user (i.e., they represent the relevant data). The internal nodes of the EC tree represent lists of k-tuples (e.g., lists of restaurant descriptions), where each item in the k-tuple can be either a leaf l or another list L (in which case L is called an embedded list). For instance, Figure 2 displays the EC descriptions of the LA-Weekly and Zagat pages. At the top level, an LA-Weekly page is a list of 5-tuples that contain the name, address, phone, review, and an embedded list of credit cards. Similarly, a Zagat document can be seen as a 7-tuple that includes a list of addresses, Hierarchical Wrapper Induction for Semistructured Information Sources 3 Figure 1. LA-Weekly and Zagat's Restaurant Descriptions name address phone review credit_card ZAGAT Document name food decor service cost LIST( Addresses ) review street city area-code phone-number Figure 2. EC description of LA-Weekly and ZAGAT pages. where each individual address is a 4-tuple street, city, area-code, and phone-number. 3. Extracting Data from a Document In order to extract the items of interest, a wrapper uses the EC description of the document and a set of extraction rules. For each node in the tree, the wrapper needs a rule that extracts that particular node from its parent. Additionally, for each list node, the wrapper requires a list iteration rule that decomposes the list into individual tuples. Given the EC tree and the rules, any item can be extracted by simply determining the path P from the root to the corresponding node and by successively extracting each node in P from its parent. If the parent of a node x is a list, the wrapper applies first the list iteration rule and then it applies x's extraction rule to each extracted tuple. In our framework a document is a sequence of tokens S (e.g., words, tags, etc). It follows that the content of the root node in the EC tree is the whole sequence S, while the content of each of paper.tex; 19/11/1999; 16:12; p. 4 Ion Muslea, Steven Minton, Craig A. Knoblock 1: !p? Name: !b? Yala !/b?!p? Cuisine: Thai !p?!i? 2: 4000 Colfax, Phoenix, AZ 85258 (602) 508-1570 3: !/i? !br? !i? 4: 523 Vernon, Las Vegas, NV 89104 (702) 578-2293 5: !/i? !br? !i? 7: !/i? Figure 3. A simplified version of a Zagat document. its children is a subsequence of S. More generally, the content of an arbitrary node x represents a subsequence of the content of its parent p. A key idea underlying our work is that the extraction rules can be based on "landmarks" (i.e., groups of consecutive tokens) that enable a wrapper to locate the content of x within the content of p. For instance, let us consider the restaurant descriptions presented in Figure 3. In order to identify the beginning of the restaurant name, we can use the rule which has the following meaning: start from the beginning of the document and skip everything until you find the !b? landmark. More formally, the rule R1 is applied to the content of the node's parent, which in this particular case is the whole document; the effect of applying consists of consuming the prefix of the parent, which ends at the beginning of the restaurant name. Similarly, one can identify the end of a node's content by applying a rule that consumes the corresponding suffix of the parent. For instance, in order to find the end of the restaurant name, one can apply the rule from the end of the document towards its beginning. The rules R1 and are called start and end rules, and, in most of the cases, they are not unique. For instance, instead of R1 we can use or Hierarchical Wrapper Induction for Semistructured Information Sources 5 R3 has the meaning "ignore everything until you find a Name landmark, and then, again, ignore everything until you find !b?", while R4 is interpreted as "ignore all tokens until you find a 3-token landmark that consists of the token Name, immediately followed by a punctuation symbol and an HTML tag." As the rules above successfully identify the start of the restaurant name, we say that they match correctly. By contrast, the start rules SkipTo(:) and SkipTo(!i?) are said to match incorrectly because they consume too few or too many tokens, respectively (in stalker terminology, the former is an early match, while the later is a late match). Finally, a rule like SkipTo(!table?) fails because the landmark !table? does not exist in the document. To deal with variations in the format of the documents, our extraction rules allow the use of disjunctions. For example, if the names of the recommended restaurants appear in bold, while the other ones are displayed as italic, one can extract all the names based on the disjunctive start and end rules either SkipTo(!b?) or SkipTo(!i?) and either SkipTo(!/b?) or SkipTo(Cuisine)SkipTo(!/i?) Disjunctive rules, which represent a special type of decision lists (Rivest, 1987), are ordered lists of individual disjuncts. Applying a disjunctive rule is a straightforward process: the wrapper succesively applies each disjunct in the list until it finds the first one that matches (see more details in the next section's footnote). To illustrate how the extraction process works for list members, consider the case where the wrapper has to extract all the area codes from the sample document in Figure 3 . In this case, the wrapper starts by extracting the entire list of addresses, which can be done based on the start rule SkipTo(!p?!i?) and the end rule SkipTo(!/i?). Then the wrapper has to iterate through the content of list of addresses (lines 2-6 in Figure and to break it into individual tuples. In order to find the start of each individual address, the wrapper starts from the first token in the parent and repeatedly applies SkipTo(!i?) to the content of the list (each successive rule-matching starts at the point where the previous one ended). Similarly, the wrapper determines the end of each Address tuple by starting from the last token in the parent and repeatedly applying the end rule SkipTo(!/i?). In our example, the paper.tex; 19/11/1999; 16:12; p. 6 Ion Muslea, Steven Minton, Craig A. Knoblock list iteration process leads to the creation of three individual addresses that have the contents shown on the lines 2, 4, and 6, respectively. Then the wrapper applies to each address the area-code start and end rule (e.g., SkipTo( '(' ) and SkipTo( ')' ), respectively). Now let us assume that instead of the area codes, the wrapper has to extract the ZIP Codes. The list extraction and the list iteration remain unchanged, but the ZIP Code extraction is more difficult because there is no landmark that separates the state from the ZIP Code. Even though in such situations the SkipTo() rules are not sufficiently expressive, they can be easily extended to a more powerful extraction language. For instance, we can use to extract the ZIP Code from the entire address. The argument of SkipUntil() describes a prefix of the content of the item to be extracted, and it is not consumed when the rule is applied (i.e., the rule stops immediately before its occurrence). The rule R5 means "ignore all tokens until you find the landmark ',', and then ignore everything until you find, but do not consume, a number". Rules like R5 are extremely useful in practice, and they represent only variations of our SkipTo() rules (i.e., the last landmark has a special meaning). In order to keep the presentation simple, the rest of the paper focuses mainly on SkipTo() rules. When necessary, we will explain the way in which we handle the construct. The extraction rules presented in this section have two main advan- tages. First of all, the hierarchical extraction based on the EC tree allows us to wrap information sources that have arbitrary many levels of embedded data. Second, as each node is extracted independently of its siblings, our approach does not rely on there being a fixed ordering of the items, and we can easily handle extraction tasks from documents that may have missing items or items that appear in various orders. Consequently, in the context of using an inductive algorithm that generates the extraction rules, our approach turns an extremely hard problem into several simpler ones: rather then finding a single extraction rule that takes into account all possible item orderings and becomes more complex as the depth of the EC tree increases, we create several simpler rules that deal with the easier task of extracting each item from its EC tree parent. Hierarchical Wrapper Induction for Semistructured Information Sources 7 4. Extraction Rules as Finite Automata We now introduce two key concepts that can be used to define extraction rules: landmarks and landmark automata. In the rules described in the previous section, each argument of a SkipTo() function is a landmark, while a group of SkipTo() functions that must be applied in a pre-established order represents a landmark automaton. In our frame- work, a landmark is a sequence of tokens and wildcards (a wildcard represents a class of tokens, as illustrated in the previous section, where we used wildcards like Number and HtmlTag). Such landmarks are interesting for two reasons: on one hand, they are sufficiently expressive to allow efficient navigation within the EC structure of the documents, and, on the other hand, as we will see in the next section, there is a simple way to generate and refine them. Landmark automata (LAs) are nondeterministic finite automata in which each transition S i (i 6= j) is labeled by a landmark l i;j that is, the transition l i;j takes place if the automaton is in the state S i and the landmark l i;j matches the sequence of tokens at the input. Linear landmark automata are a class of LAs that have the following properties: - a linear LA has a single accepting state; - from each non-accepting state, there are exactly two possible transi- tions: a loop to itself, and a transition to the next state; - each non-looping transition is labeled by a landmarks; looping transitions have the meaning "consume all tokens until you encounter the landmark that leads to the next state". The extraction rules presented in the previous section are ordered lists of linear LAs. In order to apply such a rule to a given sequence of tokens S, we apply the linear LAs to S in the order in which they appear in the list. As soon as we find an LA that matches within S, we stop the matching process 1 . Disjunctive iteration rules are applied in a slightly different manner. As we already said, iteration rules are applied repeatedly on the content of the whole list. Consequently, by blindly selecting the first matching disjunct, there is a risk of skipping over several tuples until we find the first tuple that can be extracted based on that particular disjunct! In order to avoid such problems, a wrapper that uses a disjunctive iteration rule R applies the first disjunct D in R that fulfills the paper.tex; 19/11/1999; 16:12; p. 8 Ion Muslea, Steven Minton, Craig A. Knoblock E2: 90 Colfax, !b? Palms !/b?, Phone: ( 818 ) 508-1570 E3: 523 1st St., !b? LA !/b?, Phone: 1-!b? 888 !/b?-578-2293 E4: 403 La Tijera, !b? Watts !/b?, Phone: ( 310 ) 798-0008 Figure 4. Four examples of restaurant addresses. In the next section we present the stalker inductive algorithm that generates rules that identify the start and end of an item x within its parent p. Note that finding a start rule that consumes the prefix of p with respect to x (for short Prefix x (p)) is similar to finding an end rule that consumes the suffix of p with respect to x (i.e., Suffix x (p)); in fact, the only difference between the two types of rules consists of how they are actually applied: the former starts by consuming the first token in p and goes towards the last one, while the later starts at the last token in p and goes towards the first one. Consequently, without any loss of generality, in the rest of this paper we discuss only the way in which stalker generates start rules. 5. Learning Extraction Rules The input to stalker consists of sequences of tokens representing the prefixes that must be consumed by the induced rule. To create such training examples, the user has to select a few sample pages and to use a graphical user interface (GUI) to mark up the relevant data (i.e., the leaves of the EC tree); once a page is marked up, the GUI generates the sequences of tokens that represent the content of the parent p, together with the index of the token that represents the start of x and uniquely identifies the prefix to be consumed. Before describing our rule induction algorithm, we will present an illustrative example. Let us assume that the user marked the four area codes from Figure 4 and invokes stalker on the corresponding four training examples (that is, the prefixes of the addresses E1, E2, E3, and E4 that end immediately before the area code). stalker, which is a sequential covering algorithm, begins by generating a linear LA following two criteria. First, D matches within the content of the list. Second, any two disjuncts D1 and D2 in R that are applied in succession either fail to match, or match later than D (i.e., one can not generate more tuples by using a combination of two or more other disjuncts). Hierarchical Wrapper Induction for Semistructured Information Sources 9 (remember that each such LA represents a disjunct in the final rule) that covers as many as possible of the four positive examples. Then it tries to create another linear LA for the remaining examples, and so on. Once stalker covers all examples, it returns the disjunction of all the induced LAs. In our example, the algorithm generates first the rule which has two important properties: it accepts the positive examples in E2 and E4; - it rejects both E1 and E3 because D1 can not be matched on them. During a second iteration, the algorithm considers only the uncovered examples E1 and E3, based on which it generates the rule As there are no other uncovered examples, stalker returns the disjunctive rule either D1 or D2. To generate a rule that extracts an item x from its parent p, stalker invokes the function LearnRule() (see Figure 5). This function takes as input a list of pairs (T i each sequence of tokens T i is the content of an instance of p, and T i is the token that represents the start of x within p. Any sequence S ::= T (i.e., any instance of Prefix x (p)) represents a positive example, while any other sub-sequence or super-sequence of S represents a negative example. stalker tries to generate a rule that accepts all positive examples and rejects all negative ones. stalker is a typical sequential covering algorithm: as long as there are some uncovered positive examples, it tries to learn a perfect disjunct (i.e., a linear LA that accepts only true positives). When all the positive examples are covered, stalker returns the solution, which consists of an ordered list of all learned disjuncts. The ordering is performed by the function OrderDisjuncts() and is based on a straightforward heuristic: the disjuncts with fewer early and late matches should appear in case of a tie, the disjuncts with more correct matches are preferred to the other ones. The function LearnDisjunct() is a greedy algorithm for learning perfect disjuncts: it generates an initial set of candidates and repeatedly selects and refines the best refining candidate until it either finds a perfect disjunct, or runs out of candidates. Before returning a learned disjunct, stalker invokes PostProcess(), which tries to improve the quality of the rule (i.e., it tries to reduce the chance that the disjunct will match a random sequence of tokens). This step is necessary because during the refining process each disjunct is kept as general as possible in order to potentially cover a maximal number of examples; once the paper.tex; 19/11/1999; 16:12; p. Ion Muslea, Steven Minton, Craig A. Knoblock al be an empty rule aDisjunct =LearnDisjunct(Examples) - remove all examples covered by aDisjunct add aDisjunct to RetV al return OrderDisjuncts(RetV al) Examples be the shortest example return PostProcess(BestSolution) consist on the consecutive landmarks l 1 , l 2 , \Delta \Delta \Delta, l n be the number of tokens in l i - FOR EACH token t in Seed DO - in a copy Q of C, add the 1-token landmark t between l i and l i+1 create one such rule for each wildcard that matches t - add all these new rules to T opologyRefs in Seed DO in P , replace l i by t 0 l i - in Q, replace l i by l i t m+1 create similar rules for each wildcard that matches t 0 and t m+1 - add both P and Q to LandmarkRefs Figure 5. The stalker algorithm. Hierarchical Wrapper Induction for Semistructured Information Sources 11 refining ends, we post-process the disjunct in order to minimize its potential interference with other disjuncts 2 . Both the initial candidates and their refined versions are generated based on a seed example, which is the shortest uncovered example (i.e., the example with the smallest number of tokens in Prefix x (p)). For each token t that ends the seed example and for each wildcard w i that "matches" t, stalker creates an initial candidate that is a 2-state LA. In each such automaton, the transition S 0 labeled by a landmark that is either t or one of the wildcards w i . The rationale behind this choice is straightforward: as disjuncts have to completely consume each positive example, it follows that any disjunct that consumes a t-ended prefix must end with a landmark that consumes the trailing t. Before describing the actual refining process, let us present the main intuition behind it. If we reconsider now the four training examples in Figure 4, we see that stalker starts with the initial candidate SkipTo((), which is a perfect disjunct; consequently, stalker removes the covered examples (E2 and E4) and generates the new initial candidate R0::=SkipTo(!b?). Note that R0 matches early in both uncovered examples E1 and E3 (that is, it does not consume the whole (p)), and, even worse, it also matches within the two already covered examples! In order to obtain a better disjunct, stalker refines R0 by adding more terminals to it. During the refining process, we search for new candidates that consume more tokens from the prefixes of the uncovered examples and fail on all other examples. By adding more terminals to a candidate, we hope that its refined versions will eventually turn the early matches into correct ones, while the late matches 3 , together with the ones on the already covered examples, will become failed matches. This is exactly what happens when we refine R0 into the new rule does not match anymore on E2 and E4, and R0's early matches on E1 and E3 become correct matches for R2. We perform three types of post processing operations: replacing wildcards with tokens, merging landmarks that match immediately after each other, and adding more tokens to the short landmarks (e.g., SkipTo(!b?) is likely to match in most html documents, while SkipTo(Maritime Claims : !b?) matches in significantly fewer). The last operation has a marginal influence because it improves the accuracies of only three of the rules discussed in Section 7. 3 As explained in Section 3, a disjunct D that consumes more tokens than Prefixx(p) is called a late match on p. It is easy to see that by adding more terminals to D we can not turn it into an early or a correct match (any refined version of D is guaranteed to consume at least as many tokens as D itself). Consequently, the only hope to avoid an incorrect match of D on p is to keep adding terminals until it fails to match on p. 12 Ion Muslea, Steven Minton, Craig A. Knoblock The Refine() function in Figure 5 tries to obtain (potentially) better disjuncts either by making its landmarks more specific (landmark refinements), or by adding new states in the automaton (topology re- finements). In order to perform a refinement, stalker uses a refining terminal, which can be either a token or a wildcard (besides the nine predefined wildcards Anything, Numeric, AlphaNumeric, Alphabetic, Capitalized, AllCaps, HtmlTag, NonHtml, and Punctuation, stalker can also use domain specific wildcards that are defined by the user). A straightforward way to generate the refining terminals consists of using all the tokens in the seed example, together with the wildcards that match them. 4 . Given a disjunct D, a landmark l from D, and a refining terminal t, a landmark refinement makes l more specific by concatenating t either at the beginning or at the end of l. By contrast, a topology refinement adds a new state S and leaves the existing landmarks unchanged. For instance, if D has a transition A l (i.e., the transition from A to B is labeled by the landmark l), then given a refining terminal t, a topology refinement creates a new disjunct in which the transition above is replaced by A l As one might have noted already, LearnDisjunct() uses different heuristics for selecting the best refining candidate and the best current solution, respectively. This fact has a straightforward explanation: as long as we try to further refine a candidate, we do not care how well it performs the extraction task. In most of the cases, a good refining candidate matches early on as many as possible of the uncovered ex- amples; once a refining candidate extracts correctly from some of the training examples, any further refinements are used mainly to make it fail on the examples on which it still matches incorrectly. Both sets of heuristics are described in Figure 6. As we already said, GetBestRefiner() prefers candidates with a larger potential coverage (i.e., as many as possible early and correct matches). At equal coverage, it prefers a candidate with more early matches because, at the intuitive level, we prefer the most "regular" features in a document: a candidate 4 In the current implementation, stalker uses a more efficient approach: for the refinement of a landmark l, we use only the tokens from the seed example that are located after the point where l currently matches within the seed example. Hierarchical Wrapper Induction for Semistructured Information Sources 13 Prefer candidates that have: Prefer candidates that have: larger coverage - more correct matches more early matches - more failures to match more failed matches - fewer tokens in SkipU ntil() - fewer wildcards - fewer wildcards shorter unconsumed prefixes - longer end-landmarks fewer tokens in SkipU ntil() - shorter unconsumed prefixes longer end-landmarks Figure 6. The stalker heuristics. that has only early matches is based on a regularity shared by all examples, while a candidate that also has some correct matches creates a dichotomy between the examples on which the existing landmarks work perfectly and the other ones. In case of a tie, stalker selects the disjunct with more failed matches because the alternative would be late matches, which will have to be eventually turned into failed matches by further refinements. All things being equal, we prefer candidates that have fewer wildcards (a wildcard is more likely than a token to match by pure chance), fewer unconsumed tokens in the covered prefixes (after all, the main goal is to fully consume each prefix), and fewer tokens from the content of the slot to be extracted (the main assumption in wrapper induction is that all documents share the same underlying structure; consequently, we prefer extraction rules based on the document template to the ones that rely on the structure of a particular slot). Finally, the last heuristic consists of selecting the candidate that has longer landmarks closer to the item to be extracted; that is, we prefer more specific "local context" landmarks. In order to pick the best current solution, stalker uses a different set of criteria. Obviously, it starts by selecting the candidate with the most correct matches. If there are several such disjuncts, it prefers the one that fails to match on most of the remaining examples (remem- ber that the alternatives, early or late matches, represent incorrect matches!). In case of a tie, for reasons similar to the ones cited above, we prefer candidates that have fewer tokens from the content of the item, fewer wildcards, longer landmarks closer to the item's content, and fewer unconsumed tokens in the covered prefixes (i.e., in case of incorrect match, the result of the extraction contains fewer irrelevant tokens). Finally, stalker can be easily extended such that it also uses constructs. The rule refining process remains unchanged (after all, SkipUntil() changes only the meaning of the last landmark paper.tex; 19/11/1999; 16:12; p. 14 Ion Muslea, Steven Minton, Craig A. Knoblock in a disjunct), and the only modification involves GenerateInitial- Candidates(). More precisely, for each terminal t that matches the first token in an instance of x (including the token itself), stalker also generates the initial candidates SkipUntil(t). 6. Example of Rule Induction Let us consider again the restaurant addresses from Figure 4. In order to generate an extraction rule for the area-code, we invoke stalker with the training examples fE1, E2, E3, E4g. During the first iteration, LearnDisjunct() selects the shortest prefix, E2, as seed example. The last token to be consumed in E2 is "(", and there are two wildcards that match it: Punctuation and Anything; consequently, stalker creates three initial candidates: As R1 is a perfect disjunct 5 , LearnDisjunct() returns R1 and the first iteration ends. During the second iteration, LearnDisjunct() is invoked with the uncovered training examples fE1, E3g; the new seed example is E1, and stalker creates again three initial candidates: As all three initial candidates match early in all uncovered examples, stalker selects R4 as the best possible refiner because it uses no wild-cards in the landmark. By refining R4, we obtain the three landmark refinements Anything !b?) Hierarchical Wrapper Induction for Semistructured Information Sources 15 R10: SkipT o(Venice) SkipT o(!b?) R17: SkipTo(Numeric) SkipTo(!b?) R12: SkipT o(:) SkipTo(!b?) R19: SkipTo(HtmlTag) SkipTo(!b?) R13: SkipT o(-) SkipTo(!b?) R20: SkipTo(AlphaNum) SkipT o(!b?) R14: SkipT o(,) SkipTo(!b?) R21: SkipTo(Alphabetic) SkipTo(!b?) R15: SkipT o(Phone) SkipT o(!b?) R22: SkipTo(Capitalized) SkipTo(!b?) R24: SkipTo(Anything) SkipTo(!b?) Figure 7. All 21 topology refinements of R4. along with the 21 topology refinements shown in Figure 7. At this stage, we have already generated several perfect disjuncts: R7, R11, R12, R13, R15, R16, and R19. They all match correctly on E1 and E3, and fail to match on E2 and E4; however, stalker dismisses R19 because it is the only one using wildcards in its land- marks. Of the remaining six candidates, R7 represents the best solution because it has the longest end landmark (all other disjuncts end with a 1-token landmark). Consequently, LearnDisjunct() returns R7, and because there are no more uncovered examples, stalker completes its execution by returning the disjunctive rule either R1 or R7. 7. Experimental Results In order to evaluate stalker's capabilities, we tested it on the information sources that were used as application domains by wien (Kush- merick, 1997), which was the first wrapper induction system 6 . To make the comparison between the two systems as fair as possible, we did not use any domain specific wildcards, and we tried to follow the exact experimental conditions used by Kushmerick. For all 21 sources for which wien had labeled examples, we used the exact same data; for the remaining 9 sources, we worked closely with Kushmerick to reproduce the original wien extraction tasks. Furthermore, we also used wien's experimental setup: we start with one randomly chosen training example, learn an extraction rule, and test it against all the unseen examples. We repeated these steps times, and we average the number of test examples that are correctly extracted. Then we 5 Remember that a perfect disjunct correctly matches at least one example (e.g., E2 and E4) and rejects all other ones. 6 All these collections of sample documents, together with a detailed description of each extraction task, can be obtained from the RISE repository, which is located at http://www.isi.edu/muslea/RISE/index.html. Ion Muslea, Steven Minton, Craig A. Knoblock repeated the same procedure with 2, 3, . , and 10 training examples. As opposed to wien, we do not train on more than 10 examples because we noticed that, in practice, a user rarely has the patience of labeling more than 10 training examples. This section has four distinct parts. We begin with an overview of the performance of stalker and wien over the test domains, and we continue with an analysis of stalker's ability to learn list extraction and iteration rules, which are key components in our approach to hierarchical wrapper induction. Then we compare and contrast stalker and wien based on the number of examples required to wrap the sources, and we conclude with the main lessons drawn from this empirical evaluation. 7.1. Overall Comparison of stalker and wien The data in Table I provides an overview of the two systems' performance over the sources. The first four columns contain the source name, whether or not the source has missing items or items that may appear in various orders, and the number of embedded lists in the EC tree. The next two columns specify how well the two systems performed: whether they wrapped the source perfectly, imperfectly, or completely failed to wrap it. For the time being, let us ignore the last two columns in the table. In order to better understand the data from Table I, we have to briefly describe the type of wrappers that wien generates (a more technical discussion is provided in the next section). wien uses a fairly simple extraction language: it does not allow the use of wildcards and disjunctive rules, and the items in each k-tuple are assumed to be always present and to always appear in the same order. Based on the assumptions above, wien learns a unique multi-slot extraction rule that extracts all the items in a k-tuple at the same time (by contrast, stalker generates several single-slot rules that extract each item independently of its siblings in the k-tuple). For instance, in order to extract all the addresses and area codes from the document in Figure 3, a hypothetical wien rule does the following: it ignores all characters until it finds the string "!p?!i?" and extracts as Address everything until it encounters a "(". Then it immediately starts extracting the AreaCode, which ends at ")". After extracting such a 2-tuple, the rule is applied again until it does not match anymore. Out of the sources, wien wraps perfectly of them, and completely fails on the remaining 12. These complete failures have a straight-forward explanation: if there is no perfect wrapper in wien's language (because, say, there are some missing items), the inductive algorithm paper.tex; 19/11/1999; 16:12; p. Hierarchical Wrapper Induction for Semistructured Information Sources 17 Table I. Test domains for wien and stalker: a dash denotes failure, while p and ' mean perfectly and imperfectly wrapped, respectively. SRC Miss Perm Embd wien stalker ListExtr ListIter S5 - Ion Muslea, Steven Minton, Craig A. Knoblock does not even try to generate an imperfect rule. It is important to note that wien fails to wrap all sources that include embedded lists (remember that embedded lists are at least two levels deep) or items that are missing or appear in various orders. On the same test domains, stalker wraps perfectly 20 sources and learns 8 additional imperfect wrappers. Out of these last 8 sources, in 4 cases stalker generates "high quality" wrappers (i.e., wrappers in which most of the rules are 100% accurate, and no rule has an accuracy below 90%). Finally, two of the sources, S21 and S29, can not be wrapped by stalker. 7 In order to wrap all 28 sources, stalker induced 206 different rules, out of which 182 (i.e., more than had 100% accuracy, and another were at least 90% accurate; in other words, only six rules, which represents 3% of the total, had an accuracy below 90%. Furthermore, as we will see later, the perfect rules were usually induced based on just a couple of training examples. 7.2. Learning List Extraction and Iteration Rules As opposed to wien, which performs an implicit list iteration by repeatedly applying the same multi-slot extraction rule, stalker learns explicit list extraction and iteration rules that allow us to navigate within the EC tree. These types of rules are crucial to our approach because they allow us to decompose a difficult wrapper induction problem into several simpler ones in which we always extract one individual item from its parent. To estimate stalker's performance, we have to analyze its performance at learning the list extraction and list iteration rules that appeared in the 28 test domains above. The results are shown in the last two columns of Table I, where we provide the number of training examples and the accuracy for each such rule. Note that there are some sources, like S16, that have no lists at all. At the other end of the spectrum, there are several sources that include two lists 8 . 7 The documents in S21 are difficult to wrap because they include a heterogeneous list (i.e., the list contains elements of several types). As each type of element uses a different kind of layout, the iteration task is extremely difficult. The second source, raises a different type of problem: some of the items have just a handful of occurrences in the collection of documents, and, furthermore, about half of them represent various types of formatting/semantic errors (e.g., the date appearing in the location of the price slot, and the actual date slot remaining empty). Under these circumstances, we decided to declare this source unwrappable by stalker. 8 For sources with multiple lists, we present the data in two different ways. If all the learned rules are perfect, the results appear on the same table line (e.g., for S7, the list extraction rules required 6 and 1 examples, respectively, while the list iteration rules required 2 and 7 examples, respectively). If at least one of the rules Hierarchical Wrapper Induction for Semistructured Information Sources 19 The results are extremely encouraging: only one list extraction and two list iteration rules were not learned with a 100% accuracy, and all these imperfect rules have accuracies above 90%. Furthermore, out of the 72 rules, 50 of them were learned based on a single training example! As induction based on a single example is quite unusual in machine learning, it deserves a few comments. stalker learns a perfect rule based on a single example whenever one of the initial candidates is a perfect disjunct. Such situations are frequent in our framework because the hierarchical decomposition of the problem makes most of the subproblems (i.e., the induction of the individual rules) straightforward. In final analysis, we can say that independently of how difficult it is to induce all the extraction rules for a particular source, the list extraction and iteration rules can be usually learned with a 100% accuracy based on just a few examples. 7.3. Efficiency Issues In order to easily compare wien's and stalker's requirements in terms of the number of training examples, we divided the sources above in three main groups: - sources that can be perfectly wrapped by both systems (Table II) - sources that can be wrapped perfectly only by one system (Tables III and IV) - sources on which wien fails completely, while stalker generates imperfect wrappers (Table V). For each source that wien can wrap (see Tables II and IV), we provide two pieces of information: the number of training pages required by wien to generate a correct wrapper, and the total number of item occurrences that appear in those pages. The former is taken from (Kushmerick, 1997) and represents the smallest number of completely labeled training pages required by one of the six wrapper classes that can be generated by wien. The latter was obtained by multiplying the number above by the average number of item occurrences per page, computed over all available documents. For each source that stalker wrapped perfectly, we report four pieces of informations: the minimum, maximum, mean, and median number of training examples (i.e., item occurrences) that were required has an accuracy below 100%, the data for the different lists appear on successive lines (see, for instance, the source S9). 20 Ion Muslea, Steven Minton, Craig A. Knoblock Table II. Sources Wrapped Perfectly by Both Systems. SRC wien stalker (number of examples) Docs Exs Min Max Mean Median S5 2.0 14.4 1.0 3.0 1.5 1.0 S8 2.0 43.6 1.0 2.0 1.2 1.0 S22 2.0 200.0 1.0 1.0 1.0 1.0 5.3 15.9 1.0 9.0 2.4 1.0 to generate a correct rule 9 . For the remaining 8 sources from Tables IV and V, we present an individual description for each learned rule by providing the reached accuracy and the required number of training examples. By analyzing the data from Table II, we can see that for the sources that both systems can wrap correctly, stalker requires up to two orders of magnitude fewer training examples. stalker requires no more than 9 examples for any rule in these sources, and for more than half of the rules it can learn perfect rules based on a single example (similar observations can be made for the four sources from Table III). 9 We present the empirical data for the perfectly wrapped sources in such a compact format because it is more readable than a huge table that provides detailed information for each individual rule. Furthermore, as 19 of the 20 sources from Tables II and III have a median number of training examples equal to one, it follows that more than half of the individual item data would read "item X required a single training example to generate a 100% accurate rule." Hierarchical Wrapper Induction for Semistructured Information Sources 21 Table III. Source on which wien fails com- pletely, while stalker wraps them perfectly. SRC wien stalker (number of examples) Min Max Mean Median Table IV. Sources on which wien outperforms stalker. SRC wien stalker Docs Exs Task Accuracy Exs Product 92% 10 Manufacturer 100% 3 As the main bottleneck in wrapper induction consists of labeling the training data, the advantage of stalker becomes quite obvious. Table IV reveals that despite its advantages, stalker may learn imperfect wrappers for sources that pose no problems to wien. The explanation is quite simple and is related to the different ways in which the two systems define a training example: wien's examples are entire doc- uments, while stalker uses fragments of pages (each parent of an item paper.tex; 19/11/1999; 16:12; p. 22 Ion Muslea, Steven Minton, Craig A. Knoblock Table V. Sources on which wien fails, and stalker wraps imperfectly. SRC Task Accur. Exs SRC Task Accur. Exs ListIter 100% 1 ZIP 100% 1 Price 100% 1 Country 100% 1 Airline 100% 1 Phone 100% 1 Flight 100% 1 ArriveCode 100% 2 ListExtr 100% 1 DepartTime 100% 3 ListIter 100% 8 Alt. Name 100% 1 Image 100% 6 Price 97% 10 Translat. Artist 100% 1 Hierarchical Wrapper Induction for Semistructured Information Sources 23 is a fragment of a document). This means that for sources in which each document contains all possible variations of the main format, wien is guaranteed to see all possible variations! On the other hand, stalker has practically no chance of having all these variations in each randomly chosen training set. Consequently, whenever stalker is trained only on a few variations, it will generate an imperfect rule. In fact, the different types of training examples lead to an interesting trade-off: by using only fragments of documents, stalker may learn perfect rules based on significantly fewer examples than wien. On the other hand, there is a risk that stalker may induce imperfect rules; we plan to fix this problem by using active learning techniques (RayChaudhuri and Hamey, 1997) to identify all possible types of variations. Finally, in Table V we provide detailed data about the learned rules for the six most difficult sources. Besides the problem mentioned above, which leads to several rules of 99% accuracy, these sources also contain missing items and items that may appear in various orders. Out of the 62 rules learned by stalker for these six sources, 42 are perfect and another 14 have accuracies above 90%. Sources like S6 and S9 emphasize another advantage of the stalker approach: one can label just a few training examples for the rules that are easier to learn, and than focus on providing additional examples for the more difficult ones. 7.4. Lessons Based on the results above, we can draw several important conclusions. First of all, compared with wien, stalker has the ability to wrap a larger variety of sources. Even though not all the induced wrappers are perfect, an imperfect, high accuracy wrapper is to be preferred to no wrapper at all. Second, stalker is capable of learning most of the extraction rules based on just a couple of examples. This is a crucial feature because from the user's perspective it makes the wrapper induction process both fast and painless. Our hierarchical approach to wrapper induction played a key role at reducing the number of examples: on one hand, we decompose a hard problem into several easier ones, which, in turn, require fewer examples. On the other hand, by extracting the items independently of each other, we can label just a few examples for the items that are easy to extract (as opposed to labeling every single occurrence of each item in each training page). Third, by using single-slot rules, we do not allow the harder items to affect the accuracy of the ones that are easier to extract. Consequently, even for the most difficult sources, stalker is typically capable of learning perfect rules for several of the relevant items. Ion Muslea, Steven Minton, Craig A. Knoblock Last but not least, the fact that even for the hardest items stalker usually learns a correct rule (in most of the cases, the lower accuracies come from averaging correct rules with erroneous ones) means that we can try to improve stalker's behavior based on active learning techniques that would allow the algorithm to select the few relevant cases that would lead to a correct rule. 8. Related Work Research on learning extraction rules has occurred mainly in two con- texts: creating wrappers for information agents and developing general purpose information extraction systems for natural language text. The former are primarily used for semistructured information sources, and their extraction rules rely heavily on the regularities in the structure of the documents; the latter are applied to free text documents and use extraction patterns that are based on linguistic constraints. With the increasing interest in accessing Web-based information sources, a significant number of research projects depend on wrappers to retrieve the relevant data. A wide variety of languages have been developed for manually writing wrappers (i.e., where the extraction rules are written by a human expert), from procedural languages (Atzeni and Mecca, 1997) and Perl scripts (Cohen, 1998) to pattern matching (Chawathe et al., 1994) and LL(k) grammars (Chidlovskii et al., 1997). Even though these systems offer fairly expressive extraction lan- guages, the manual wrapper generation is a tedious, time consuming task that requires a high level of expertise; furthermore, the rules have to be rewritten whenever the sources suffer format changes. In order to help the users cope with these difficulties, Ashish and Knoblock (Ashish and Knoblock, 1997) proposed an expert system approach that uses a fixed set of heuristics of the type "look for bold or italicized strings." The wrapper induction techniques introduced in wien (Kushmerick, 1997) are a better fit to frequent format changes because they rely on learning techniques to generate the extraction rules. Compared to the manual wrapper generation, Kushmerick's approach has the advantage of dramatically reducing both the time and the effort required to wrap a source; however, his extraction language is significantly less expressive than the ones provided by the manual approaches. In fact, the wien extraction language can be seen as a non-disjunctive stalker rules that use just a single SkipTo() and do not allow the use of wildcards. There are several other important differences between stalker and wien. First, as wien learns the landmarks by searching common prefixes at the character level, it needs more training examples than stalker. Hierarchical Wrapper Induction for Semistructured Information Sources 25 Second, wien cannot wrap sources in which some items are missing or appear in various orders. Last but not least, stalker can handle EC trees of arbitrary depths, while wien's approach to nested documents turned out to be impractical: even though Kushmerick was able to manually write 19 perfect "nested" wrappers, none of them could be learned by wien. SoftMealy (Hsu and Dung, 1998) uses a wrapper induction algorithm that generates extraction rules expressed as finite transducers. The SoftMealy rules are more general than the wien ones because they use wildcards and they can handle both missing items and items appearing in various orders. Intuitively, SoftMealy's rules are similar to the ones used by stalker, except that each disjunct is either a single SkipTo() or a SkipTo()SkipUntil() in which the two landmarks must match immediately after each other. As SoftMealy uses neither multiple SkipTo()s nor multiple SkipUntil()s, it follows that its extraction rules are strictly less expressive than stalker's. Finally, SoftMealy has one additional drawback: in order to deal with missing items and various orderings of items, SoftMealy may have to see training examples that include each possible ordering of the items. In contrast to information agents, most general purpose information extraction systems are focused on unstructured text, and therefore the extraction techniques are based on linguistic constraints. However, there are three such systems that are somewhat related to stalker: whisk (Soderland, 1999), Rapier (Califf and Mooney, 1999), and 1998). The extraction rules induced by Rapier and srv can use the landmarks that immediately precede and/or follow the item to be extracted, while whisk is capable of using multiple landmarks. But, similarly to stalker and unlike whisk, Rapier and srv extract a particular item independently of the other relevant items. It follows that whisk has the same drawback as SoftMealy: in order to handle correctly missing items and items that appear in various orders, whisk must see training examples for each possible ordering of the items. None of these three systems can handle embedded data, though all use powerful linguistic constraints that are beyond stalker's capabilities. 9. Conclusions and Future Work The primary contribution of our work is to turn a potentially hard problem - learning extraction rules - into a problem that is extremely easy in practice (i.e., typically very few examples are required). The number of required examples is small because the EC description of a page simplifies the problem tremendously: as the Web pages are paper.tex; 19/11/1999; 16:12; p. 26 Ion Muslea, Steven Minton, Craig A. Knoblock intended to be human readable, the EC structure is generally reflected by actual landmarks on the page. stalker merely has to find the landmarks, which are generally in the close proximity of the items to be extracted. In other words, the extraction rules are typically very small, and, consequently, they are easy to induce. We plan to continue our work on several directions. First, we plan to use unsupervised learning in order to narrow the landmark search-space. Second, we would like to use active learning techniques to minimize the amount of labeling that the user has to perform. Third, we plan to provide PAC-like guarantees for stalker. Acknowledgments This work was supported in part by USC's Integrated Media Systems Center (IMSC) - an NSF Engineering Research Center, by the National Science Foundation under grant number IRI-9610014, by the U.S. Air Force under contract number F49620-98-1-0046, by the Defense Logistics Agency, DARPA, and Fort Huachuca under contract number DABT63-96-C-0066, and by research grants from NCR and General Dynamics Information Systems. The views and conclusions contained in this paper are the authors' and should not be interpreted as representing the official opinion or policy of any of the above organizations or any person connected with them. --R Journal of Intelligent Systems --TR Cut and paste A Web-based information system that reasons with structured collections of text Modeling Web sources for information integration Information extraction from HTML Generating finite-state transducers for semi-structured data extraction from the Web Learning Information Extraction Rules for Semi-Structured and Free Text Relational learning of pattern-match rules for information extraction Learning Decision Lists Wrapper Generation for Internet Information Sources Wrapper induction for information extraction --CTR Exploiting structural similarity for effective Web information extraction, Data & Knowledge Engineering, v.60 n.1, p.222-234, January, 2007 Retrieving and Semantically Integrating Heterogeneous Data from the Web, IEEE Intelligent Systems, v.19 n.3, p.72-79, May 2004 Sneha Desai , Craig A. 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information agents;information extraction;wrapper induction;supervised inductive learning

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Reliable Communication for Highly Mobile Agents.

The provision of a reliable communication infrastructure for mobile agents is still an open research issue. The challenge to reliability we address in this work does not come from the possibility of faults, but rather from the mere presence of mobility, which complicates the problem of ensuring the delivery of information even in a fault-free network. For instance, the asynchronous nature of message passing and agent migration may cause situations where messages forever chase a mobile agent that moves frequently from one host to another. Current solutions rely on conventional technologies that either do not provide a solution for the aforementioned problem, because they were not designed with mobility in mind, or enforce continuous connectivity with the message source, which in many cases defeats the very purpose of using mobile agents.In this paper, we propose an algorithm that guarantees delivery to highly mobile agents using a technique similar to a distributed snapshot. A number of enhancements to this basic idea are discussed, which limit the scope of message delivery by allowing dynamic creation of the connectivity graph. Notably, the very structure of our algorithm makes it amenable not only to guarantee message delivery to a specific mobile agent, but also to provide multicast communication to a group of agents, which constitutes another open problem in research on mobile agents. After presenting our algorithm and its properties, we discuss its implementability by analyzing the requirements on the underlying mobile agent platform, and argue about its applicability.

Introduction Mobile agent systems currently provide an increasing degree of sophistication in the abstractions and mechanisms they support, well beyond the purpose of achieving agent migration. However, it is questionable whether the features that are being added on top of plain agent migration are really focused on the needs of application developers, or they really address the problems that are peculiar to mobility. A good example of the gap between what is provided and what is needed is the problem of providing a communication infrastructure for mobile agents. This aspect is often overlooked or misunderstood in the context of mobile agent research. For instance, significant efforts are being devoted to the problem of enabling communication among mobile agents by defining a common semantic layer for the exchange of in- formation, as in KQML [6]. Despite their relevance, the questions posed by researchers in this area are not particularly affected by the presence of mobility, and focus on the problem of communication at a completely different, and much higher, abstraction level than the one we are concerned with in this paper. Even if we assume that the problem of ensuring a proper semantic level for agent communication is somehow solved, we are still left with the problem of reliably delivering a message to a mobile agent whose patterns of mobility are potentially unknown a priori. This is the problem we address in this paper. The challenge to reliable communication persists even under the assumption of an ideal transport mechanism that guarantees a correct delivery of information in the presence of faults in the underlying communication link or in the communicating nodes. It is the sheer presence of mobility, and not the possibility of faults, that undermines the notion of reliability. If mobile agents are allowed to move freely from one host to another according to some a priori unknown migration pattern, the challenge in delivering information properly is caused by the difficulty in determining where the mobile agent is, and in ensuring that the information reaches the mobile agent before it moves again. By and large, currently available mobile agent systems rely either on conventional communication facilities like sockets and remote procedure (or method) call [1, 8, 13], or implement their own message passing facility [10]. To our knowledge, none of them satisfactorily addresses the aforementioned problem. They require knowledge about the location of the mobile agent, which is obtained either by overly restricting the freedom of mobility or by assuming continuous connectivity-assumptions that in many cases defeat the whole purpose of using mobile agents. In this paper, we propose an algorithm that guarantees message delivery to highly mobile agents in a fault-free network. We focus on message passing as the communication mechanism that we want to adapt to mobility, because it is a fundamental and well understood form of communication in a distributed system. This incurs no loss of generality because more complex mechanisms like remote procedure call and method invocation are easily built on top of message passing. Our algorithm does not assume knowledge about the location of agents, and constrains the movement of agents only in its most enhanced form and only for a limited amount of time. Furthermore, its structure makes it inherently amenable to an extension that provides multicast communication to a group of agents dispersed in the network, another problem for which satisfactory solutions do not yet exist. The paper is structured as follows. Section 2 discusses the motivation for this work, and the current state of the art in the field. Section 3 presents our al- gorithm, starting with the underlying assumptions and illustrating subsequent refinements of the original key idea. Section 4 discusses the applicability and implementability of a communication mechanism embodying our algorithm in a mobile agent platform. Finally, Section 5 provides some concluding remarks. 2. Motivation and Related Work The typical use of a mobile agent paradigm is for bypassing a communication link and exploiting local access to resources on a remote server [7]. Thus, one could argue that, all in all, communication with a remote agent is not important and a mobile agent platform should focus instead on the communication mechanisms that are exploited locally, i.e., to get access to the server or to communicate with the agents that are co-located on the same site. Many mobile agent systems provide mechanisms for local communication, either using some sort of meeting abstraction as initially proposed by Telescript [18], event notification for group communication [1, 10], or, more recently, tuple spaces [4, 16]. Nevertheless, there are several common scenarios that provide counterarguments to the statement above. Some of them are related with the issue of managing mobile agents. Imagine a "master" agent spawning a number of "slave" mobile agents that are subsequently injected in the network to perform some kind of co-operative computation, e.g., find a piece of informa- tion. At some point, the master agent may want to actively terminate the computation of the slave agents, e.g., because the requested information has been found by one of them and thus is desirable to prevent un-necessary resource consumption. Or, it may want to change some parameter governing the behavior of the agents, because the context that determined their creation has changed in the meanwhile. Or, in turn, the slave agents may want to detect whether the master agent is still alive by performing some sort of orphan detection, which requires locating the master agent if this is itself allowed to be mobile. Other examples are related to the fact that mobile agents are just one of the paradigms available to designers of a distributed application, which can then use a mixture of mobile agent and message passing to achieve different functionalities in the context of the same ap- plication. For instance, a mobile agent could visit a site and perform a check on a given condition. If the condition is not satisfied, the agent could register an event listener with the site. This way, while the mobile agent is visiting other sites and before reporting its results, it could receive notifications of changes in the state of the sites it has already visited and decide whether they are worth a second visit. The scenarios above require the presence of a message passing mechanism for mobile agents. However, a highly desirable requirement for such a mechanism is the guarantee that the message is actually delivered (at least once) to the destination, independently from the relative movement of the source and target of communi- cation. Mobility heavily complicates matters. Typical delivery schemes suffer from the fundamental problem that an agent in transit during the delivery can easily be missed. To illustrate the issue, we discuss two strawman approaches to message delivery: broadcast and forwarding. A simple broadcast scheme assumes a spanning tree of the network nodes through which a message may be sent by any node. This node then broadcasts the message to its neighbors, which broadcast the message to their neighbors, and so on until the leaf nodes are reached. This, however, does not guarantee delivery of the message when an agent is traveling in the reverse direction with respect to the propagation of the message, as depicted in Figure 1. If the agent is be- (b) Forwarding. (a) Spanning tree broadcasting. sender agent message sender home agent A A A retransmission A Figure 1. The problem: Missing delivery in simplistic broadcast and forwarding schemes. ing transferred at the same instant when the message is propagating in the other direction, the agent and the message will cross in the channel, and delivery will never occur. A simple forwarding scheme maintains a pointer to the mobile agent at a well-known location, which is called home agent in the Mobile IP protocol [14] where this idea enables physical mobility of hosts. Upon mi- gration, the mobile agent must inform the home agent of its new location, in order to enable further com- munication. However, any messages sent between the migration and the update are lost, as the agent basically ran away from the messages before they could be delivered. Even if retransmission to the new location is attempted, the agent can move again and miss the retransmission, thus effectively preventing guaranteed delivery, as depicted in Figure 1. Furthermore, forwarding has an additional drawback in that it requires communication to the home agent every time the agent moves. In some situations, this may defeat the purpose of using mobile agents by reintroducing centralization. For instance, in the presence of many highly mobile agents spawned from the same host, this scheme may lead to a considerable traffic overhead generated around the home agent, and possibly to much slower performance if the latency between mobile and home agent is high. Finally, because of this umbilical cord that must be maintained with the home agent, this approach is intrinsically difficult to apply when disconnected operations are required. The mobile agent systems currently available employ different communication strategies. The OMG MASIF standard [11] specifies only the interfaces that enable the naming and locating of agents across different platforms. The actual mechanisms to locate an agent and communicate with it are intentionally left out of the scope of the standard, although a number of location techniques are suggested that by and large can be regarded as variations of broadcast and forwarding. Some systems, notably Aglets [10] and Voyager [13], employ forwarding by associating to each mobile component a proxy object which plays the role of the home agent. Some others, like Emerald [9], the precursor of mobile objects, use forwarding and resort to broadcast when the object cannot be found. Others, e.g., Mole [1], simply prevent the movement of a mobile agent while engaged in communication. Mole exploits also a different forwarding scheme that does not keep a single home agent, rather it maintains a whole trail of pointers from the source to destination, for faster communication. However, this is employed only in the context of a protocol for orphan detection [2]. Finally, some systems, e.g., Agent Tcl [8], provide mechanisms that are based on common remote procedure call, and leave to the application developer the chore of handling a missed delivery. A related subject is the provision of a mechanism for reliable communication to a group of mobile agents. Group communication is a useful programming abstraction for dealing with clusters of mobile agents that are functionally related and to which a same piece of information must be sent. Many mobile agent systems, notably Telescript, Aglets, and Voyager, provide the capability to multicast messages only within the context of a single runtime support. Finally, Mole [1] provides a mechanism for group communication that, how- ever, still assumes that agents are stationary during a set of information exchanges. The approach we propose provides a reasonable solution to the problem of guaranteeing delivery to a single mobile agent, and has the nice side effect of providing a straightforward way to achieve group communication as well. The details of our algorithm are discussed in the next section. 3. Enabling Reliable Communication As discussed earlier, simplistic message delivery mechanisms such as spanning tree broadcasting and forwarding have the potential for failure when agents are in transit or are rapidly moving. To address these shortcomings we note that, in general, we must flush the agents out of the channels and into regions where they cannot escape without receiving a copy of the message. For instance, in the aforementioned broadcast mechanism, we look at the case where the agent is moving in the opposite direction from the message on a bidirectional channel. In this case, if the message was still present at the destination node of the chan- nel, it could be delivered when the agent arrived at the node. This leads to a solution where the message is stored at the nodes until delivery completes. Although this simple extension would guarantee delivery, it is not reasonable to expect the nodes to store messages for arbitrary lengths of time. Therefore, we seek a solution that has a tight bound on the storage time for any given message at a node. We must also address the situation where a message is continually forwarded to the new location of the mobile agent, but never reaches it because the agent effectively is running away and the message never catches up. Again, we could store the message at every node in the network until it was de- livered, but a better solution would involve trapping the agent in a region of the graph so that wherever it moves, it cannot avoid receiving the message. The first algorithm we present for guaranteed message delivery to mobile agents is a direct adaptation of previous work done by the first author in the area of physical mobility [12]. This work assumes that the network of nodes and channels is known in advance, and further assumes that only one message is present in the system at a time. In this setting, exactly-once delivery of the message is guaranteed without modifying the agents behavior either with respect to movement or message acceptance. Next, we extend this basic algorithm to allow multiple messages to be delivered concurrently. To achieve this enhancement we must relax the exactly-once semantics to become at-least-once, meaning that duplication of messages is acceptable, but we still prevent an agent from missing a message. Although these algorithms provide reliable message delivery, the assumption that the entire network graph is known in advance is often unreasonable in situations where mobile agents are used. Therefore, we enhance our algorithm by allowing the graph to grow dynamically as agents move, and still preserve the at-least-once semantics for message delivery. For simplicity of pre- sentation, we will present this latter enhancement in two stages: first assuming that all messages originate from a single node and then allowing any node to initiate the processing needed to send a message. 3.1. Model The logical model we work with is the typical net-work graph where the nodes represent the nodes willing to host agents and the edges represent directional, FIFO channels along which agents can migrate and messages can be passed. The FIFO assumption is critical to the proper execution of our algorithm and its implications on the underlying mobile agent platform are discussed further in Section 4. We assume a con- Figure 2. A connected network with connected sub- networks. Agents can enter and leave the subnetworks only by going through the gateway servers. nected network graph (i.e., a path exists between every pair of nodes), but not necessarily fully connected (i.e., a channel does not necessarily exist between each pair of nodes). In a typical IP network, all nodes are logically connected directly. However, this is not always the case at the application level, as shown in Figure 2. There, a set of subnetworks are connected to one another through an IP network, but an agent can enter or leave a subnetwork only by passing through a gateway server, e.g., because of security reasons. We also assume that the mobile agent server keeps track of which agents it is currently hosting, and that it provides some fundamental mechanism to deliver a message to an agent, e.g., by invoking a method of the agent object. Finally, we assume that every agent has a single, globally unique identifier, which can be used to direct a message to the agent. These latter assumptions are reasonable in that they are already satisfied by the majority of mobile agent platforms. 3.2. Delivery in a Static Network Graph We begin the description of our solution with a basic algorithm which assumes a fixed network of nodes. For simplicity, we describe first the behavior of the algorithm under the unrealistic assumption of a single message being present in the system, and then show how this result can be extended to allow concurrent delivery of multiple messages. Single message delivery. Previous work by the first author in the physical mobility environment approached reliable message delivery by adapting the notion of distributed snapshots [12]. In snapshot algo- rithms, the goal is to record the local state of the nodes and the channels in order to construct a consistent global state. Critical features of these algorithms include propagation of the snapshot initiation, the flush- 1: pre: no incoming channels open action: 2: pre: message j arrives " action: if 3: pre: message j finished processing action: 4: pre: message i arrives " action: buffer message i OPEN Figure 3. State transitions and related diagram for multiple message delivery in a static network graph. ing of the channels to record all messages in transit, and the recording of every message exactly once. Our approach to message delivery uses many of the same ideas as the original snapshot paper presented by Chandy and Lamport [5]. However, instead of spreading knowledge of the snapshot using messages, we spread the actual message to be delivered; instead of flushing messages out of the channels, we flush agents out of the channels; and instead of recording the existence of the messages, we deliver a copy of the message. The algorithm works by associating a state, flushed or open, with each incoming channel of a node. Initially all channels are open. When the message arrives on a channel, the state is changed to flushed, implying that all the agents on that channel ahead of the message have been forced out of the channel (by the FIFO assumption). When the message arrives for the first time at a node, it is stored locally and propagated on all outgoing channels, starting the flushing process on those channels. The message is also delivered to all agents present at the node. All the agents that arrive through an open channel on a node storing the message must receive a copy of it. When all the incoming channels of a node are flushed, the node is no longer required to deliver the message to any arriving agents, therefore the message copy is deleted and all of the channels are atomically reset to open. Multiple message delivery. A simplistic adaptation of the previous algorithm to multiple message delivery would require a node to wait for the termination of the current message delivery and to coordinate with the other nodes before initiating a new one, in order to ensure that only one message is present in the system. However, this unnecessarily constrains the behavior of the sender and requires knowledge of non-local state. We propose instead an approach where multiple messages can be present in the system, as long as the node where the message originates tags the message with a sequence number unique to the node. In prac- tice, the sequence number allows the nodes to deal with multiple instantiations of the algorithm running con- currently, thus encompassing the case of a single source transmitting a burst of messages without waiting as well as the case of multiple sources transmitting at the same time. To allow concurrent message delivery to take place, we must address the issue of a new message arriving during the processing of the current one. In this case, the channel is already flushed, but not all other channels are flushed. To handle this case, we introduce a new state, buffering, as shown in Figure 3, in which any messages arriving on a flushed channel are put into a buffer to be processed at a later time (transition 4). A channel in the buffering state is not considered when determining the transition from flushed to open. When this transition is finally made (1), all buffering channels are also transitioned to open (3), and the messages in the buffer queues are treated as if they were messages arriving on the channel at that moment, and thus processed again. It is possible that, after the processing of the first message, the next message causes another transition to buffering, but the fact that the head of the channel is processed ensures eventual progress through the sequence of messages to be delivered. Although we force messages to be buffered, agent arrival is not restricted. The agent is being allowed to move ahead of any messages it originally followed along the channel. Effectively, the agent may move itself back into the region of the network where the message has not yet been delivered. Therefore, duplicate delivery is possible, although duplicates can be discarded easily by the runtime support or by the agent itself based on the sequence number. 3.3. Delivery in a Dynamic Network Graph Although the solutions proposed so far provide delivery guarantees in the presence of mobility, the necessity of knowing the network of neighbors a priori is sometimes unreasonable in the dynamic environment of mobile agents. Furthermore, the delivery mechanism is insensitive to which nodes have been active, and delivers the messages also to regions of the network that have not been visited by agents. Therefore, our goal is still to flush channels and trap agents in regions of the network where the messages will propagate, but also to allow the network graph used for the delivery process to grow dynamically as the agents migrate. A channel will only be included in the message delivery if an agent has traversed it, and therefore, a node will be included in the message delivery only if an agent has been hosted there. We refer to a node or channel included in message delivery as active. Our presentation is organized in two phases. First, we show a restricted approach where all the messages must originate from a single, fixed source. This is reasonable for monitoring or master-slave scenarios where all communication flows from a fixed initiator to the agents in the system. Then, we extend this initial solution to enable direct inter-agent messaging by allowing any node to send messages, without the need for a centralized source. Single message source. First, we identify the problems that can arise when nodes and channels are added dynamically, due to the possible disparity between the messages processed at the source and destination nodes of a channel when it becomes active. We initially present these issues by example, then develop a general solution. Destination ahead of source. Assume a network as shown in Figure 4(a). X is the sender of all messages and is initially the only active node in the system. The graph is extended when X sends an agent to Y , causing Y and (X; Y ) to become active. Suppose X sends a burst of messages 1::4, which are processed by Y , and later a second sequence of messages 5::8. This second transfer is immediately followed by the migration of a new agent to node Z, which makes Z and (X; Z) active. Before message 5 arrives at Y , an agent is sent from Y to Z, thus causing the channel (Y; Z) to be added to the active graph. A problem arises if the agent decides to immediately leave Z, because the messages 5::8 have not yet been delivered to it. Furthermore, what processing should occur when these messages arrive at Z along the new channel (Y; Z)? If the messages are blindly forwarded on all Z's outgoing channels, message ordering is possibly lost and messages can possibly keep propagating in the network without ever being deleted. Our solution is to hold the agent at Z until the messages 5::8 are received and, when these messages ar- rive, to deliver them only to the detained agent, i.e., without broadcasting them to the neighboring nodes. Therefore, no messages are lost and the system wide processing of messages is not affected. Notably, although we do inhibit the movement of the agent until these messages arrive, this takes place only for a time proportional to the diameter of the network, and even more important, only when the topology of the network is changing. Source ahead of destination. To uncover another potential problem, we use the same scenario just presented for nodes X , Y , and Z, except that instead of assuming an agent moving from Y to Z, we assume it is moving from Z to Y , making (Z; Y ) active ure 4(b)). Although the agent will not miss any messages in this move, two potential problems exist. First, by making (Z; Y ) active, Y will wait for Z to be flushed or buffering before proceeding to the next message. However, message 5 will never be sent from Z. Our solution is to delay the activation of channel catches up with Z. In this example, we delay until 8 is processed at Y . Second, if message 9 is sent from X and propagated along channel (Z; Y ), it must be buffered until it can be processed in order. Given this, we now present a solution that generalizes the previous one. We describe in detail the channel states and the critical transitions among these states, using the state diagram in Figure 5. ffl closed: Initially, all channels are closed and not active in message delivery. ffl open: The channel is waiting to participate in a message delivery. When an agent arrives through an open channel on a node that is storing a message destined to that agent, the agent should receive a copy of such message. ffl flushed: The current message being delivered has already arrived on this channel, and therefore this channel has completed the current message delivery. Agents arriving on flushed channels need no special processing. of source. of destination. (b) Source ahead (a) Destination ahead Z a 8 a Figure 4. Problems in managing a dynamic graph. Values shown inside the nodes indicate the last message processed by the node. The subscripts on agent a indicate the last message processed by the source of the channel being traversed by a right before a migrated. ffl buffering(j): The source is ahead of the desti- nation. Messages arriving on buffering channels are put into a FIFO buffer. They are processed after the node catches up with the source by processing message j. ffl holding(j): The destination is ahead of the source. Messages with identifiers less than or equal to j which arrive on holding channels are delivered to all held agents. Agents arriving on holding channels, and whose last received message has identifier less than j, are held until j arrives. The initial transition of a channel from closed to an active state is based on the current state of the destination node and on the state of the source as carried by the agent. The destination node can either still be inactive or it can have finished delivering the same message as the source (9), it can still be still processing such message (8), it can be processing an earlier message (10), or it can be processing a later message (7). Based on this comparison, the new active state is assigned. Once a channel is active, all state transitions occur in response to the arrival of a message. Because we have already taken measures to ensure that all messages will be delivered to all agents, our remaining concerns are that detained agents are eventually released and that at every node, the next message is eventually processed. Whether an agent must be detained or not is determined by comparing the identifier of the latest message received by the agent, carried as part of the agent state, and the current state of the destination node. Only agents that are behind the destination are actually detained. If an agent is detained at a channel in state holding(j), it can be released as soon as j is processed along this channel. By connectivity of the network graph, we are guaranteed that j will eventually arrive. When it does, the destination node will either still be processing j, or will have completed the processing. In both cases the agent is released. In the former case, the channel transitions to flushed (6) to wait for the rest of the channels to catch up, while in the latter case the channel transitions to open (5) to be ready to process the next message. To argue that eventually all messages are delivered, we must extend the progress argument presented in Section 3.2 to include the progress of the holding channels as well as the addition of new channels. As noted in the previous paragraph, message j is guaranteed to eventually arrive along the holding channel, thus ensuring progress of this channel. Next, we assert that there is a maximum number of channels that can be added as incoming channels, bounded by the num- 1: pre: no incoming channels open " no incoming channels holding action: 2: pre: message j arrives " action: if 3: pre: message j finished processing action: 4: pre: message i arrives " action: buffer message i 5: pre: message j arrives " action: deliver to held agents, release held agents pre: message j arrives " action: deliver to held agents, release held agents 7: pre: agent arrives " D ahead of S " action: 8: pre: agent arrives " curMsg S and D processing same message action: 9: pre: agent arrives " (D not active (S and D processing same message " action: 10: pre: agent a j arrives " S ahead of D action: OPEN FLUSHED CLOSED Figure 5. State transitions and related diagram for multiple message delivery with a single source in a dynamic network graph. The state transitions refer to a single channel (S; D). ber of nodes in the system. We are guaranteed that if channels are continuously added, eventually this maximum will be reached. By the other progress properties, eventually all these channels will be either flushed or buffering, in which case processing of the next message (if any) can begin. Multiple message sources. Although the previous solution guarantees message delivery and allows the dynamic expansion of the graph, the assumption that all messages originate at the same node is overly restric- tive. To extend this algorithm to allow a message to originate at any node, we effectively superimpose multiple instances of the same algorithm on the network, by allowing their concurrent execution. For the purposes of explanation, let n be the number of nodes in the system. Then: ffl The state of an incoming channel is represented by a vector of size n where the state of each node is recorded. Before the channel is added to the active graph, the channel is considered closed. Once the channel is active, if no messages have been received from a particular node, the state of the element in the vector corresponding to that node is set to open. ffl Processing of each message is done with respect to the channel state associated with the node where the message originated. ffl Nodes can deliver n messages concurrently, at most one for each node. As before, if a second message arrives from the same node, it is buffered until the prior message completes its processing. ffl An agent always carries a vector containing, for each message source, the identifier of the last message received. Moreover, when an agent traverses a new outgoing channel, it carries another vector that contains, for each message source, the identifier of the last message processed by the source of the new channel right before the agent departed. ffl An incoming agent is held only as long as, for each message source, the identifier of the last message received is greater than the corresponding holding value (if any) of the channel the agent arrived through. ffl To enable any node to originate a message, we must guarantee that the graph remains connected. To maintain this property we make all links bidi- rectional. In the case where an agent arrives and the channel in the opposite direction is not already an outgoing channel, a fake agent message is sent to S with the state information of D. This message effectively makes the reverse channel active. Again we must argue that detained agents are eventually released and that progress is made with respect to the messages sent from each node. Assume that message i is the smallest message identifier from any node which has not been delivered by all nodes. There must exist a path from a copy of i to every node where i has not arrived, and every node on this path is blocked until arrives. By connectivity of the network graph, i will propagate to every node along every channel and will complete delivery in the system. No node will buffer because it is the minimum message identifier which is being waited for. When i has completed delivery, the next message is the new minimum and will make progress in a similar manner. Because the buffering of messages is done with respect to the individual source nodes and not for the channel as a whole, the messages from each node make independent progress. Holding agents requires coordination among the nodes. The j value with respect to each node for which the channel is being held, e.g., holding(j), is fixed when the first agent arrives. Because the messages are guaranteed to make progress, we are guaranteed that eventually j will be processed and the detained agents will be released. 3.4. Multicast Message Delivery In all the algorithms described so far, we exploited the fact that a distributed snapshot records the state of each node exactly once, and modified the algorithm by substituting message recording with message delivery to an agent. Hence, one could describe our algorithm by saying that it attempts to deliver a message to every agent in the system, and only the agents whose identifier match the message target actually accept the mes- sage. With this view in mind, the solution presented can be adapted straightforwardly to support multicast. The only modification that must be introduced is the notion of a multicast address that allows a group of agents to be specified as recipients of the message-no modification to the algorithm is needed. 4. Discussion and Future Work In this section we analyze the impact of our communication mechanism on the underlying mobile agent platform, argue about its applicability, and discuss possible extensions and future work on the topic. 4.1. Implementation Issues A fundamental assumption that must be preserved in order for our mechanism to work is that the communication channels must be FIFO-a legacy of the fact that the core of our schema is based on a distributed global snapshot. The FIFO property must be maintained for every piece of information traveling through the channel, i.e., messages, agents, and any combination of the two. This is not necessarily a requirement for a mobile agent platform. A common design for it is to map the operations that require message or agent delivery on data transfers taking place on different data streams, typically through sockets or some higher-level mechanism like remote method invocation. In the case where these operations insist on the same destination, the FIFO property may not be preserved, since a data item sent first through one stream can be received later than another data item through another stream, depending on the architecture of the underlying runtime support. Nevertheless, the FIFO property can be implemented straightforwardly in a mobile agent server by associating a queue that contains messages and agents that must be transmitted to a remote server. This way, the FIFO property is structurally enforced by the server architecture, although this may require non-trivial modifications in the case of an already existing platform. Our mechanism assumes that the runtime support maintains some state about the network graph and the messages being exchanged. In the most static form of our solution, this state is constituted only by the last message received, which must be kept until de- livered. In a system with bidirectional channels, this means for a time equal to the maximum round trip delay between the node and its neighbors. On the other hand, in the most dynamic variant of our algorithm, each server must maintain a vector of identifiers for the active (outgoing and incoming) channels and, for each channel, a vector containing the messages possibly being buffered. The size of the latter is unbounded, but each message must be kept in the vector only for a time proportional to the diameter of the network. 4.2. Applicability It is evident that the algorithm presented in this work generates a considerable overall traffic overhead if compared, for instance, to a forwarding scheme. This is a consequence of the fact that our technique involves contacting the nodes in the network that have been visited by at least one agent in order to find the message recipient, and thus generates an amount of traffic that is comparable to a broadcast. Unfortunately, this price must be paid when both guaranteed delivery and frequent, unconstrained agent movement are part of the application requirements, since simpler and more lightweight schemes do not provide these guarantees, as discussed in Section 2. Hence, the question whether the communication mechanism we propose is a useful addition to mobile agent platforms will be ultimately answered by practical mobile agent applications, which are still largely missing and will determine the requirements for communication. In any case, we do not expect our mechanism to be the only one provided by the runtime support. To make an analogy, one does not shout when the party is one step away; one resorts to shouting under the exceptional condition that the party is not available, or not where expected to be. Our algorithm provides a clever way to shout (i.e., to broadcast a message) with precise guarantees and minimal constraints, and should be used only when conventional mechanisms are not applicable. Hence, the runtime support should leave to the programmer the opportunity to choose different communication mechanisms, and even different variants of our algorithm. For instance, the fully dynamic solution described in Section 3.3 is not necessarily the most convenient in all situations. In a network configuration such as the one depicted in Figure 2, where the graph is structured in clusters of nodes, the best tradeoff is probably achieved by using our fully dynamic algorithm only for the "gateway" servers that sit at the border of each cluster, and a static algorithm within each cluster, thus leveraging off of the knowledge of the internal network configuration. Along the same lines, it should also be possible to exploit hybrid schemes. For instance, in the common case where the receipt of a message triggers a reply, bandwidth consumption can be reduced by encoding the reply destination in the initial message and using a conventional mechanism, as long as the sending agent is willing to remain stationary until the reply is received. 4.3. Enhancements and Future Work In this work, we argued that the problem of reliable message delivery is inherently complicated by the presence of mobility even in the absence of faults in the links or nodes involved in the communication. In practice, however, these faults do happen and, depending on the execution context, they can be relevant. If this is the case, the techniques traditionally proposed for coping with faults in a distributed snapshot can be applied to our mechanism. For instance, a simple technique consists of periodically checkpointing the state of the system, recording the state of links, keeping track of the last snapshot, and dumping an image of the agents hosted. (Many systems already provide checkpointing mechanisms for mobile agents.) This information can be used to reconcile the state of the faulty node with the neighbors after a fault has occurred. A related issue is the ability not only to dynamically add nodes to the graph, but also to remove them. This could model faults, or model the fact that a given node is no longer willing to host agents, e.g., because the mobile agent support has been intentionally shut down. A simple solution would consist of "short cir- cuiting" the node to be removed, by setting the in-coming channels of its outgoing neighbors to point to the node's incoming neighbors. However, this involves running a distributed transaction and thus enforces an undesirable level of complexity. In this work, we disregarded the problem for a couple of reasons. First of all, while it is evident that the ability of adding nodes dynamically enables a better use of the communication resources by limiting communication to the areas effectively visited by agents, it is unclear whether a similar gain is obtained in the case of removing nodes, especially considering the aforementioned implementation complexity. Second, very few mobile agent systems provide the ability to start and stop dynamically the mobile agent runtime support: most of them assume that the runtime is started offline and operates until the mobile agent application terminates. We are currently designing and implementing a communication package based on the algorithm described in this paper, to be included in the Code [15] mobile code toolkit. The goal of this activity is to gain a hands-on understanding of the design and implementation issues concerned with the realization of our scheme, and to provide the basis for a precise quantitative characterization of our approach, especially in comparison with traditional ones. 5. Conclusions In this work we point out how the sheer presence of mobility makes the problem of guaranteeing the delivery of a message to a mobile agent inherently difficult, even in absence of faults in the network. To our knowledge, this problem has not been addressed by the research community. Currently available mobile agent systems employ techniques that either do not provide guarantees, or overly constrain the movement or connectivity of mobile agents, thus to some extent reducing their usefulness. In this work, we propose a solution based on the concept of a distributed snapshot. Several extensions of the basic idea allow us to cope with different levels of dynamicity and, along the way, provide a straightforward way to implement group communication for mobile agents. Our communication mechanism is meant to complement those currently provided by mobile agent systems, thus allowing the programmer to trade reliability for bandwith consumption. Further work will address fault tolerance and exploit an implementation of our mechanism to evaluate its tradeoffs against those of conventional mechanisms. Acknowledgments This paper is based upon work supported in part by the National Science Foundation (NSF) under grant No. CCR-9624815. Any opinions, findings and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of NSF. --R Communication Concepts for Mobile Agent Systems. The Shadow Ap- proach: An Orphan Detection Protocol for Mobile Agents Software Agents. Reactive Tuple Spaces for Mobile Agent Coordination. Distributed Snap- shots: Determining Global States of Distributed Sys- tems KQML as an agent communication language. Understanding Code Mobility. Agent Tcl. Programming and Deploying Mobile Agents with Aglets. An exercise in formal reasoning about mobile computations. ObjectSpace Inc. IP mobility support. Lime: Linda Meets Mobility. Mobile Agents: 2 nd Int. Telescript Technology: Mobile Agents. --TR --CTR Mosaab Daoud , Qusay H. Mahmoud, Reliability analysis of mobile agent-based systems, Proceedings of the 2005 ACM symposium on Applied computing, March 13-17, 2005, Santa Fe, New Mexico Scalable Platform for Highly Mobile Agents in Distributed Computing Environments, Proceedings of the 2006 International Symposium on on World of Wireless, Mobile and Multimedia Networks, p.633-637, June 26-29, 2006 Elena Gmez-Martnez , Sergio Ilarri , Jos Merseguer, Performance analysis of mobile agents tracking, Proceedings of the 6th international workshop on Software and performance, February 05-08, 2007, Buenes Aires, Argentina Hojjat Jafarpour , Nasser Yazdani , Navid Bazzaz-zadeh, A scalable group communication mechanism for mobile agents, Journal of Network and Computer Applications, v.30 n.1, p.186-208, January 2007

communication;snapshot;mobile agents

608696

Pricing in Agent Economies Using Multi-Agent Q-Learning.

This paper investigates how adaptive software agents may utilize reinforcement learning algorithms such as Q-learning to make economic decisions such as setting prices in a competitive marketplace. For a single adaptive agent facing fixed-strategy opponents, ordinary Q-learning is guaranteed to find the optimal policy. However, for a population of agents each trying to adapt in the presence of other adaptive agents, the problem becomes non-stationary and history dependent, and it is not known whether any global convergence will be obtained, and if so, whether such solutions will be optimal. In this paper, we study simultaneous Q-learning by two competing seller agents in three moderately realistic economic models. This is the simplest case in which interesting multi-agent phenomena can occur, and the state space is small enough so that lookup tables can be used to represent the Q-functions. We find that, despite the lack of theoretical guarantees, simultaneous convergence to self-consistent optimal solutions is obtained in each model, at least for small values of the discount parameter. In some cases, exact or approximate convergence is also found even at large discount parameters. We show how the Q-derived policies increase profitability and damp out or eliminate cyclic price wars compared to simpler policies based on zero lookahead or short-term lookahead. In one of the models (the Shopbot model) where the sellers' profit functions are symmetric, we find that Q-learning can produce either symmetric or broken-symmetry policies, depending on the discount parameter and on initial conditions.

Introduction Reinforcement Learning (RL) procedures have been established as powerful and practical methods for solving Markov Decision Problems. One of the most significant and actively investigated RL algorithms is Q-learning (Watkins, 1989). Q-learning is an algorithm for learning to estimate the long-term expected reward for a given state-action pair. It has the nice property that it does not need a model of the environment, and it can be used for on-line learning. Strong convergence of Q-learning to the exact optimal value functions and policies has been proven when lookup table representations of the Q-function are used (Watkins and Dayan, 1992); this is feasible in small state spaces. In large state spaces where lookup tables are infeasible, RL methods can be combined with function approximators to give good practical performance despite the lack of theoretical guarantees of convergence to optimal policies. Most real-world problems are not fully Markov in nature - they are often non-stationary, history-dependent and/or not fully observable. In order for RL methods to be more generally useful in solving such problems, they need to be extended to handle these non-Markovian properties. One important application domain where the non-Markovian aspects are paramount is the area of multi-agent systems. This area is expected to be increasingly important in the future, due to the potential rapid emergence of "agent economies" consisting of large populations of interacting software agents engaged in various forms of economic activity. The problem of multiple agents simultaneously adapting is in general non-Markov, because each agent provides an effectively non-stationary environment for the other agents. Hence the existing convergence guarantees do not hold, and in general, it is not known whether any global convergence will be obtained, and if so, whether such solutions are optimal. Some progress has been made in analyzing certain special case multi-agent problems. For example, the problem of "teams," where all agents share a common objective function, has been studied, for example, in (Crites and Barto, 1996). Likewise, the purely competitive case of zero-sum objective functions has been studied in (Littman, 1994), where an algorithm called "minimax-Q" was proposed for two-player zero-sum games, and shown to converge to the optimal value function and policies for both players. Sandholm and Crites studied simultaneous Q-learning by two players in the Iterated Prisoner's Dilemma game (Sandholm and Crites, 1995), and found that the learning procedure generally converged to stationary solutions. However, the extent to which those solutions were "optimal" was unclear. Recently, Hu and Wellman proposed an algorithm for calculating optimal Q-functions in two-player arbitrary-sum games (Hu and Wellman, 1998). This algorithm is an important first step. However, it does not yet appear to be useable for practical problems, because it assumes that policies followed by both players will be Nash equilibrium policies, and it does not address the "equilibrium coordination" problem, i.e. if there are multiple Nash equilibria, how do the agents decide which equilibrium to choose? We suspect that this may be a serious problem, since according to the "folk theorem of iterated games" (Kreps, 1990), there can be a proliferation of Nash equilibria when there is sufficiently high emphasis on future rewards, i.e., a large value of the discount parameter fl. Furthermore, there may be inconsistencies between the assumed Nash policies, and the policies implied by the Q-functions calculated by the algorithm. In this paper, we study simultaneous Q-learning in an economically motivated two-player game. The players are assumed to be two sellers of similar or identical products, who compete against each other on the basis of price. At each time step, the sellers alternately take turns setting prices, taking into account the other seller's current price. After the price has been set, the consumers then respond instantaneously and deterministically, choosing either seller 1's product or seller 2's product (or no product), based on the current price pair (p 1 leading to an instantaneous reward or profit (R 1 given to sellers 1 and 2 respectively. We assume initially that the sellers have full knowledge of the expected consumer demand for any given price pair, and in fact have full knowledge of both profit functions. Our work builds on prior research reported in (Tesauro and Kephart, 1998; Tesauro and Kephart, 1999). Those papers examined the effect of including fore- sight, i.e. an ability to anticipate longer-term consequences of an agent's current action. Two different algorithms for agent foresight were presented: (i) a generalization of the minimax search procedure in two-player zero-sum games; (ii) a generalization of the Policy Iteration method from dynamic programming, in which both players' policies are simultaneously improved, until self-consistent policy pairs are obtained that optimize expected reward over two time steps. It was found that including foresight in the agents' pricing algorithms generally improved overall agent profitability, and usually damped out or eliminated the pathological behavior of unending cyclic "price wars," in which long episodes of repeated undercutting amongst the sellers alternate with large jumps in price. Such price wars were found to be rampant in prior studies of agent economy models (Kephart, Hanson and Sairamesh, 1998; Sairamesh and Kephart, 1998) when the agents use "myopically optimal" or "myoptimal" pricing algorithms that optimize immediate reward, but do not anticipate the longer-term consequences of an agent's current price setting. Our motivation for studying simultaneous Q-learning in this paper is three- fold. First, if Q-functions can be learned simultaneously and self-consistently for both players, the policies implied by those Q-functions should be self-consistently optimal. In other words, an agent will be able to correctly anticipate the longer-term consequences of its own actions, the other agents' actions, and will correctly model the other agents as having an equivalent capability. Hence the classic problem of infinite recursion of opponent models will be avoided. In contrast, in other approaches to adaptive multi-agent system, these issues are more problematic. For example, (Vidal and Durfee, 1998) propose a recursive opponent modeling scheme, in which level-0 agents do no opponent modeling, level-1 agents model the opponents as being level-0, level-2 agents model the opponents as being level-1, etc. In both of these approaches, there is no effective way for an agent to model other agents as being at an equivalent level of depth or complexity. The second advantage of Q-learning is that the solutions should correspond to deep lookahead: in principle, the Q-function represents the expected reward looking infintely far ahead in time, exponentially weighted by a discount parameter In contrast, the prior work of (Tesauro and Kephart, 1999) was based on shallow finite lookahead. Finally, in comparison to directly modeling agent policies, the Q-function approach seems more extensible to the situation of very large economies with many competing sellers. Our intuition is that approximating Q-functions with nonlinear function approximators such as neural networks is more feasible than approximating the corresponding policies. Fur- thermore, in the Q-function approach, each agent only needs to maintain a single Q-function for itself, whereas in the policy modeling approach, each agent needs to maintain a policy model for every other agent; the latter seems infeasible when the number of sellers is large. The remainder of this paper is organized as follows. Section 2 describes the structure and dynamics of our model two-seller economy, and presents three economically-based models of seller profit (Price-Quality, Information-Filtering, and Shopbot) which are known to be prone to price wars when agents myopically optimize their short-term payoffs. We deliberately choose parameters to place each of these systems in a price-war regime. In section 3, we describe details of how we implement Q-learning in these model economies. As a first step, we examine the simple case of ordinary Q-learning, where one of the two sellers uses Q-learning and the other seller uses a fixed pricing policy (the myopically opti- mal, or "myoptimal" policy). We then examine the more interesting and novel situation of simultaneous Q-learning by both sellers. Finally, section 5 summarizes the main conclusions and discusses promising directions and challenges for future work. Model agent economies Real agent economies are likely to contain large numbers of agents, with complex details of how the agents behave and interact with each other on multiple time scales. Our approach toward modeling and understanding such complexity is to begin by making a number of simplifying assumptions. We first consider the simplest possible case of two competing seller agents offering similar or identical products to a large population of consumer agents. The sellers compete on the basis of price, and we assume that prices are discretized and can lie between a minimumand maximum price, such that the number of possible prices is at most a few hundred. This renders the state space small enough that it is feasible to use lookup tables to represent the agents' pricing policies and expected profits. Time in the simulation is also discretized; at each time step, we assume that the consumers compare the current prices of the two sellers, and instantaneously and deterministically choose to purchase from at most one seller. Hence at each time step, for each possible pair of seller prices, there is a deterministic reward or profit given to each seller. The simulation can iterate forever, and there may or may not be a discounting factor for the present value of future rewards. It is worth noting that the consumers are not regarded as "players" in the model. The consumers have no strategic role: they behave according to an extremely simple, fixed, short-term greedy rule (buy the lowest priced product at each time step), and are regarded as merely providing a stationary environment in which the two sellers can compete in a two-player game. This is clearly a simplifying first step in the study of multi-agent phenomena, and in future work, the models will be extended to include strategic and adaptive behavior on the part of the consumers as well. This will change the notion of "desirable" system behavior. In the present model, desirable behavior would resemble "collusion" between the two sellers in charging very high prices, so that both could obtain high profits. Obviously this is not desirable from the consumers' viewpoint. Regarding the dynamics of seller price adjustments, we assume that the sellers alternately take turns adjusting their prices, rather than simultaneously setting prices (i.e. the game is extensive-form rather than normal-form). Our choice of alternating-turn dynamics is motivated by two considerations: (a) As the number of sellers becomes large and the model becomes more realistic, it seems more reasonable to assume that the sellers will adjust their prices at different times rather than at the same time, although they probably will not take turns in a well-defined order. (b) With alternating-turn dynamics, we can stay within the normal Q-learning framework where the Q-function implies a deterministic optimal policy: it is known that in two-player alternating turn games, there always exists a deterministic policy that is as good as any non-deterministic policy (Littman, 1994). In contrast, in games with simultaneous moves (for example, rock-paper-scissors), it is possible that no deterministic policy is optimal, and that the existing Q-learning formalism for MDPs would have to be modified and extended so that it could yield non-deterministic optimal policies. We study Q-learning in three different economic models that have been described in detail elsewhere (Sairamesh and Kephart, 1998; Kephart, Hanson and Sairamesh, 1998; Greenwald and Kephart, 1999). The first model, called the "Price-Quality" model (Sairamesh and Kephart, 1998), models the sellers' products as being distinguished by different values of a scalar "quality" parameter, with higher-quality products being perceived as more valuable by the consumers. The consumers are modeled as trying to obtain the lowest-priced product at each time step, subject to threshold-type constraints on both quality and price, i.e., each consumer has a maximum allowable price and a minimum allowable qual- ity. The similarity and substitutability of seller products leads to a potential for direct price competition; however, the "vertical" differentiation due to differen- ing quality values leads to an asymmetry in the sellers' profit functions. It is believed that this asymmetry is responsible for the unending cyclic price wars that emerge when the sellers employ myoptimal pricing strategies. The second model is an "Information-Filtering" model described in detail in (Kephart, Hanson and Sairamesh, 1998). In this model there are two competing sellers of news articles in somewhat overlapping categories. In contrast to the vertical differentiation of the Price-Quality model, this model contains a horizontal differentiation in the differing article categories. To the extent that the categories overlap, there can be direct price competition, and to the extent that they differ, there are asymmetries introduced that again lead to the potential for cyclic price wars. The third model is the so-called "Shopbot" model described in (Greenwald and Kephart, 1999), which is intended to model the situation on the Internet in which some consumers may use a Shopbot to compare prices of all sellers offering a given product, and select the seller with the lowest price. In this model, the sellers' products are exactly identical and the profit functions are symmetric. Myoptimal pricing leads the sellers to undercut each other until the minimum price point is reached. At that point, a new price war cycle can be launched, due to buyer asymmetries rather than seller asymmetries. The fact that not all buyers use the Shopbot, and some buyers instead choose a seller at random, means that it can be profitable for a seller to abandon the low-price competition for the bargain hunters, and instead maximally exploit the random buyers by charging the maximum possible price. An example profit function that we study, taken from the Price-Quality model, is as follows: Let p 1 and p 2 represent the prices charged by seller 1 and seller 2 respectively. Let q 1 and q 2 represent their respective quality parameters, with cost to a seller of producing an item of quality q. Then assuming the particular model of consumer behavior described in (Sairamesh and Kephart, 1998) one can show analytically that in the limit of infinitely many consumers, the instantaneous profits per consumer R 1 and R 2 obtained by seller 1 and seller 2 respectively are given by: ae (1) ae (2) A plot of the profit landscape for seller 1 as a function of prices p 1 and p 2 is given in figure 1, for the following parameter settings: q q). (These specific parameter settings were chosen because they are known to generate harmful price wars when the agents use myopic optimal pricing.) We can see in this figure that the myopic optimal price for seller 1 as a function of seller 2's price, p (p 2 ), is obtained for each value of p 2 by sweeping across all values of p 1 and choosing the value that gives the highest profit. We can see that for small values of p 2 , the peak profit is obtained at whereas for larger values of p 2 , there is eventually a discontinuous shift to the other peak, which follows along the parabolic-shaped ridge in the landscape. An analytic expression for the myopic optimal price for seller 1 as a function of p 2 is as follows (defining x Similarly, the myopic optimal price for seller 2 as a function of the price set by seller 1, p (p 1 ), is given by the following formula (assuming that prices are discrete and that ffl is the price discretization interval): We also note in passing that there are similar profit landscapes for each of the sellers in the Information-Filtering model and in the Shopbot model. In all three Fig. 1. Sample profit landscape for seller 1 in Price-Quality model, as a function of seller 1 price p1 and seller 2 price p2 . models, it is the existence of multiple, disconnected peaks in the landscapes, with relative heights that can change depending on the other seller's price, that leads to price wars when the sellers behave myopically. Regarding the information set that is made available to the sellers, we have made a simplifying assumption as a first step that the players have essentially perfect information. They can model the consumer behavior perfectly, and they also have perfect knowledge of each other's costs and profit functions. Hence our model is thus a two-player perfect-information deterministic game that is very similar to games like chess. The main differences are that the profits in our model are not strictly zero-sum, and that there are no terminating or absorbing nodes in our model's state space. Also in our model, payoffs are given to the players at every time step, whereas in games such as chess, payoffs are only given at the terminating nodes. As mentioned previously, we constrain the prices set by the two sellers to lie in a range from some minimum to maximum allowable price. The prices are discretized, so that one can create lookup tables for the seller profit functions Furthermore, the optimal pricing policies for each seller as a function of the other seller's price, p (p 2 ) and p (p 1 ), can also be represented in the form of table lookups. 3 Single-agent Q-learning We first consider ordinary single-agent Q-learning in the above two-seller economic models. The procedure for Q-learning is as follows. Let Q(s; a) represent the discounted long-term expected reward to an agent for taking action a in state s. The discounting of future rewards is accomplished by a discount parameter fl such that the value of a reward expected at n time steps in the future is discounted by fl n . Assume that the Q(s; a) function is represented by a lookup table containing a value for every possible state-action pair, and assume that the table entries are initialized to arbitrary values. Then the procedure for solving for Q(s; a) is to infinitely repeat the following two-step loop: 1. Select a particular state s and a particular action a, observe the immediate reward r for this state-action pair, and observe the resulting state s 0 . 2. Adjust Q(s; a) according to the following equation: \DeltaQ(s; where ff is the learning rate parameter, and the max operation represents choosing the optimal action b among all possible actions that can be taken in the successor state s 0 leading to the greatest Q-value. A wide variety of methods may be used to select state-action pairs in step 1, provided that every state-action pair is visited infinitely often. For any stationary Markov Decision Problem, the Q-learning procedure is guaranteed to converge to the correct values, provided that ff is decreased over time with an appropriate schedule. We first consider using Q-learning for one of the two sellers in our economic models, while the other seller maintains a fixed pricing policy. In the simulations described below the fixed policy is in fact the myoptimal policy p represented for example in the Price-Quality model by equations 3 and 4. In our pricing application, the distinction between states and actions is somewhat blurred. We will assume that the "state" for each seller is sufficiently described by the other seller's last price, and that the ``action'' is the current price decision. This should be a sufficient state description because no other history is needed either for the determination of immediate reward, or for the calculation of the myoptimal price by the fixed-strategy player. We have also modified the concepts of immediate reward r and next-state s 0 for the two-agent case. We define s 0 as the state that is obtained, starting from s, of one action by the Q-learner and a response action by the fixed-strategy opponent. Likewise, the immediate reward is defined as the sum of the two rewards obtained after those two actions. These modifications were introduced so that the state s 0 would have the same player to move as state s. possible alternative to this, which we have not investigated, is to include the side-to-move as additional information in the state-space description.) In the simulations reported below, the sequence of state-action pairs selected for the Q-table updates were generated by uniform random selection from amongst all possible table entries. The initial values of the Q-tables were generally set to the immediate reward values. (Consequently the initial Q-derived policies corresponded to myoptimal policies.) The learning rate was varied with time according to: where the initial learning rate ff(0) was usually set to 0.1, and the constant when the simulation time t was measured in units of N 2 , the size of the Q-table. (N is the number of possible prices that could be selected by either player.) A number of different values of the discount parameter fl were studied, ranging from Results for single-agent Q-learning in all three models indicated that Q-learning worked well (as expected) in each case. In each model, for each value of the discount parameter, exact convergence of the Q-table to a stationary optimal solution was found. The convergence times ranged from a few hundred sweeps through each table element, for smaller values of fl, to at most a few thousand updates for the largest values of fl. In addition, once Q-learning converged, we then measured the expected cumulative profit of the policy derived from the Q-function. We ran the Q-policy against the other player's myopic policy from 100 random starting states, each for 200 time steps, and averaged the resulting cumulative profit for each player. We found that, in each case, the seller achieved greater profit against a myopic opponent by using a Q-derived policy than by using a myopic policy. (This was true even for due to the redefinition of Q updates summing over two time steps, the case effectively corresponds to a two-step optimization, rather than the one-step optimization of the myopic policies.) Furthermore, the cumulative profit obtained with the Q-derived policy monotonically increased with the increasing fl (as expected). It was also interesting to note that in many cases, the expected profit of the myopic opponent also increased when playing against the Q-learner, and also improved monotonically with increasing fl. The explanation is that, rather than better exploiting the myopic opponent, as would be expected in a zero-sum game, the Q-learner instead reduced the region over which it would participate in a mutually undercutting price war. Typically we find in these models that with myopic vs. myopic play, large-amplitude price wars are generated that start at very high prices and persist all the way down to very low prices. When a Q- learner competes against a myopic opponent, there are still price wars starting at high prices, however, the Q-learner abandons the price war more quickly as the prices decrease. The effect is that the price-war regime is smaller and confined to higher average prices, leading to a closer approximation to cooperative or collusive behavior, with greater expected utilites for both players. An illustrative example of the results of single-agent Q-learning is shown in figure 2. Figure 2(a) plots the average profit for both sellers in the Shopbot model, when one of the sellers is myopic and the other is a Q-learner. (As the model is symmetric, it doesn't matter which seller is the Q-learner.) Figure 2(b) plots the myopic price curve of seller 2 against the Q-derived price curve (at of seller 1. We can see that both curves have a maximum price of 1 and a minimum price of approximately 0.58. The portion of both curves lying vs. Q; Shopbot Model Myopic vs. Myopic Average profit pFig. 2. Results of single-agent Q-learning in the Shopbot model. (a) Average profit per time step for Q-learner (seller 1, filled circles) and myopic seller (seller 2, open circles) vs. discount parameter fl. Dashed line indicates baseline expected profit when both sellers are myopic. (b) Cross-plot of Q-derived price curve (seller 1) vs. myopic price curve (seller 2) at Dashed line and arrows indicate a temporal price-pair trajectory using these policies, starting from filled circle. along the diagonal indicates undercutting behavior, in which case the seller will respond to the opponent's price by undercutting by ffl, the price discretization interval. The system dynamics for the state (p 1 figure 2(b) can be obtained by alternately applying the two pricing policies. This can be done by a simple iterative graphical construction, in which for any given starting point, one first holds moves horizontally to the p 1 (p 2 ) curve, and then one holds moves vertically to the p 2 (p 1 ) curve. We see in this figure that the iterative graphical construction leads to an unending cyclic price war, whose trajectory is indicated by the dashed line. Note that the price-war behavior begins at the price pair (1, 1), and persists until a price of approximately 0.83. At this point, seller 1 abandons the price war, and resets its price to 1, leading once again to another round of undercutting. The amplitude of this price war is diminished compared to the situation in which both players use a myopic policy. In that case, seller 1's curve would be a mirror image of seller 2's curve, and the price war would persist all the way to the minimum price point, leading to a lower expected profit for both sellers. Multi-agent Q-learning We now examine the more interesting and challenging case of simultaneous training of Q-functions and policies for both sellers. Our approach is to use the same formalism presented in the previous section, and to alternately adjust a random entry in seller 1's Q-function, followed by a random entry in seller 2's Q-function. As each seller's Q-function evolves, the seller's pricing policy is correspondingly updated so that it optimizes the agent's current Q-function. In modeling the two-step payoff r to a seller in equation 5, we use the opponent's current policy as implied by its current Q-function. The parameters in the experiments below were generally set to the same values as in the previous section. In most of the experiments, the Q-functions were initialized to the instantaneous payoff values (so that the policies corresponded to myopic policies), although other initial conditions were explored in a few experiments. vs. Q; PQ Model Myopic vs. Myopic (1) Myopic vs. Myopic (2) Average profit vs. Q; PQ model (any g) Fig. 3. Results of simultaneous Q-learning in the Price-Quality model. (a) Average profit per time step for seller 1 (solid diamonds) and seller 2 (open diamonds) vs. discount parameter fl. Dashed line indicates baseline myopic vs. myopic expected profit. Note that seller 2's profit is higher than seller 1's, even though seller 2 has a lower quality parameter. (b) Cross-plot of Q-derived price curves (at any fl). Dashed line and arrows indicate a sample price dynamics trajectory, starting from the filled circle. The price war is eliminated and the dynamics evolves to a fixed point indicated by an open circle. For simultaneous Q-learning in the Price-Quality model, we find robust convergence to a unique pair of pricing policies, independent of the value of fl, as illustrated in figure 3(b). This solution also corresponds to the solution found by generalized minimax and by generalized DP in (Tesauro and Kephart, 1999). We note that repeated application of this pair of price curves leads to a dynamical trajectory that eventually converges to a fixed-point located at (p 0:4). A detailed analysis of these pricing policies and the fixed-point solution is presented in (Tesauro and Kephart, 1999). In brief, for sufficiently low prices of seller 2, it pays seller 1 to abandon the price war and to charge a very high price, 0:9. The value of then corresponds to the highest price that seller 2 can charge without provoking an undercut by seller 1, based on a two-step lookahead calculation (seller 1 undercuts, and then seller 2 replies with a further undercut). We note that this fixed point does not correspond to a Nash equilibrium, since both players have an incentive to deviate, based on a one-step lookahead calculation. It was conjectured in (Tesauro and Kephart, 1999) that the solution observed in figure 3(b) corresponds to a subgame-perfect equilibrium (Fudenberg and Tirole, 1991) rather than a Nash equilibrium. The cumulative profits obtained by the pair of pricing policies are plotted in figure 3(a). It is interesting that seller 2, the lower-quality seller, actually obtains a significantly higher profit than seller 1, the higher-quality seller. In contrast, with myopic vs. myopic pricing, seller 2 does worse than seller 1. vs. Q; Shopbot Model Myopic vs. Myopic Average profit Fig. 4. Results of simultaneous Q-learning in the Shopbot model. (a) Average profit per time step for seller 1 (solid diamonds) and seller 2 (open diamonds) vs. discount parameter fl. Dashed line indicates baseline myopic vs. myopic expected profit. (b) Cross-plot of Q-derived price curves at the solution is symmetric. Dashed line and arrows indicate a sample price dynamics trajectory. (c) Cross-plot of Q-derived price curves at 0:9; the solution is asymmetric. In the Shopbot model, we did not find exact convergence of the Q-functions for each value of fl. However, in those cases where exact convergence was not found, we did find very good approximate convergence, in which the Q-functions and policies converged to stationary solutions to within small random fluctua- tions. Different solutions were obtained at each value of fl. We generally find that a symmetric solution, in which the shapes of are iden- tical, is obtained at small fl, whereas a broken symmetry solution, similar to the Price-Quality solution, is obtained at large fl. We also found a range of fl values, between 0.1 and 0.2, where either a symmetric or asymmetric solution could be obtained, depending on initial conditions. The asymmetric solution was counter-intuitive to us, because we expected that the symmetry of the two sell- ers' profit functions would lead to a symmetric solution. In hindsight, we can apply the same type of reasoning as in the Price-Quality model to explain the asymmetric solution. A plot of the expected profit for both sellers as a function of fl is shown in figure 4(a). Plots of the symmetric and asymmetric solution, obtained at respectively, are shown in figures 4(b) and 4(c). Myopic vs. Myopic (1) Myopic vs. Myopic (2) Average profit vs. Q; IF model (g=0.5) pFig. 5. Results of multi-agent Q-learning in the Information-Filtering model. (a) Average profit per time step for seller 1 (solid diamonds) and seller 2 (open diamonds) vs. discount parameter fl. (The data points at Q-functions and policies.) Dashed lines indicates baseline expected profit when both sellers are myopic. (b) Cross-plot of Q-derived price curves at Finally, in the Information-Filtering model, we found that simultaneous Q-learning produced exact or good approximate convergence for small values of 0:5). For large values of fl, no convergence was obtained. The simultaneous Q-learning solutions yielded reduced-amplitude price wars, and montonically increasing profitability for both sellers as a function of fl, at least up to 0:5. A few data points were examined at fl ? 0:5, and even though there was no convergence, the Q-policies still yielded greater profit for both sellers than in the myopic vs. myopic case. A plot of the Q-derived policies and system dynamics for shown in figure 5(b). The expected profits for both players as a function of fl is plotted in figure 5(a). Conclusions We have examined single-agent and multi-agent Q-learning in three models of a two-seller economy in which the sellers alternately take turns setting prices, and then instantaneous profits are given to both sellers based on the current price pair. Such models fall into the category of two-player, alternating-turn, arbitrary-sum Markov games, in which both the rewards and the state-space transitions are deterministic. The game is Markov because the state space is fully observable and the rewards are not history dependent. In all three models (Price-Quality, Information-Filtering, and Shopbot), large-amplitude cyclic price wars are obtained when the sellers myopically optimize their instantaneous profits without regard to longer-term impact of their pricing policies. We find that, in all three models, the use of Q-learning by one of the sellers against a myopic opponent invariably results in exact convergence to the optimal Q-function and optimal policy against that opponent, for all allowed values of the discount parameter fl. The use of the Q-derived policy yields greater expected profit for the Q-learner, with monotonically increasing profit as fl increases. In many cases, it has a side benefit of also enhancing the welfare of the myopic opponent. This comes about by reducing the amplitude of the undercutting price-war regime, or in some cases, eliminating it completely. We have also studied the more interesting and challenging situation of simultaneously training Q-functions for both sellers. This is more difficult because as each seller's Q-function and policy change, it provides a non-stationary environment for adaptation of the other seller. No convergence proofs exist for such simultaneous Q-learning by multiple agents. Nevertheless, despite the absence of theoretical guarantees, we do find generally good behavior of the algorithm in our model economies. In two of the models (Shopbot and Price-Quality), we find exact or very good approximate convergence to simultaneously self-consistent Q-functions and optimal policies for any value of fl, whereas in the Information-Filtering model, simultaneous convergence was found for fl - 0:5. In the Information-Filtering and Shopbot models, monotonically increasing expected profits for both sellers were also found for small values of fl. In the Price-Quality model, simultaneous Q-learning yields an asymmetric solution, corresponding to the solution found in (Tesauro and Kephart, 1999), that is highly advantageous to the lesser-quality seller, but slightly disadvantageous to the higher-quality seller, when compared to myopic vs. myopic pricing. A similar asymmetric solution is also found in the Shopbot model for large fl, even though the profit functions for both players are symmetric. For each model, there exists a range of discount parameter values where the solutions obtained by simultaneous Q-learning are self-consistently optimal, and outperform the solutions obtained in (Tesauro and Kephart, 1999). This is presumably because the previously published methods were based on limited lookahead, whereas the Q-functions in principle look ahead infinitely far, with appropriate discounting. It is intruiging that simultaneous Q-learning works well in our models, despite the lack of theoretical convergence proofs. Sandholm and Crites also found that simultaneous Q-learning generally converged in the Iterated Prisoner's Dilemma game. These empirical findings suggest that a deeper theoretical analysis of simultaneous Q-learning may be worth investigating. There may be some underlying theoretical principles that can explain why simultaneous Q-learning works, for at least certain classes of arbitrary-sum profit functions. Several important challenges will also be faced in extending our approach to larger-scale, more realistic simulations. While there are some economic situations in the real world where there are only two dominant sellers, in general the number of sellers can be much greater. The situation that we foresee in agent economies is that the number of competing sellers will be very large. In this case, the seller profits and pricing functions will have such high input dimensionality that it will be infeasible to use lookup table state-space representations, and most likely some sort of compact representation combined with a function approximation scheme will be necessary. Furthermore, with many sellers, the concept of sellers taking turns adjusting their prices in a well-defined order becomes problematic. This could lead to an additional combinatorial explosion, if the mechanism for calculating expected reward has to anticipate all possible orderings of opponent responses. Furthermore, while our economic models have a moderate degree of realism in their profit functions, they are unrealistic in the assumptions of knowledge and dynamics. In the work reported here, the state space was fully observable infinitely frequently at zero cost and with zero propagation delays. The expected consumer demand for a given price pair was instantaneous, deterministic and fully known to both players. Indeed, the players' exact profit functions were fully known to both players. It was also assumed that the players would alternately take turns equally often in a well-defined order in adjusting their prices. Under such assumptions of knowledge and dynamics, one could hope to develop an algorithm that could calculate in advance something like a game-theoretic optimal pricing algorithm for each agent. However, in realistic agent economies, it is likely that agents will have much less than full knowledge of the state of the economy. Agents may not know the details of other agents' profit functions, and indeed an agent may not know its own profit function, to the extent that buyer behavior is unpredictable. The dynamics of buyers and sellers may also be more complex, random and unpredictable than what we have assumed here. There may also be information delays for both buyers and sellers, and part of the economic game may involve paying a cost in order to obtain information about the state of the economy faster and more frequently, and in greater detail. Finally, we expect that buyer behavior will be non-stationary, so that there will be a more complex co-evolution of buyer and seller strategies. While such real-world complexities are daunting, there are reasons to believe that learning approaches such as Q-learning may play a role in practical solu- tions. The advantage of Q-learning is that one does not need a model of either the instantaneous payoffs or of the state-space transitions in the environment. One can simply observe actual rewards and transitions and base learning on that. While the theory of Q-learning requires exhaustive exploration of the state space to guarantee convergence, this may not be necessary when function approximators are used. In that case, after training a function approximator on a relatively small number of observed states, it may then generalize well enough on the unobserved states to give decent practical performance. Several recent empirical studies have provided evidence of this (Tesauro, 1995; Crites and Barto, 1996; Zhang and Dietterich, 1996). Acknowledgements The authors thank Amy Greenwald for helpful discussions regarding the Shopbot model. --R "Improving elevator performance using reinforcement learning." Game Theory. "Shopbots and pricebots." "Multiagent reinforcement learning: theoretical frame-work and an algorithm." "Price-war dynamics in a free-market economy of software agents." A Course in Microeconomic Theory. "Markov games as a framework for multi-agent reinforcement learn- ing," "Dynamics of price and quality differentiation in information and computational markets." "On multiagent Q-Learning in a semi-competitive domain." "Temporal difference learning and TD-Gammon." "Foresight-based pricing algorithms in an economy of software agents." "Foresight-based pricing algorithms in agent economies." "Learning nested agent models in an information economy," "Learning from delayed rewards." "Q-learning." "High-performance job-shop scheduling with a time-delay TD(-) network." --TR --CTR Prithviraj (Raj) Dasgupta , Yoshitsugu Hashimoto, Multi-Attribute Dynamic Pricing for Online Markets Using Intelligent Agents, Proceedings of the Third International Joint Conference on Autonomous Agents and Multiagent Systems, p.277-284, July 19-23, 2004, New York, New York Simon Parsons , Michael Wooldridge, Game Theory and Decision Theory in Multi-Agent Systems, Autonomous Agents and Multi-Agent Systems, v.5 n.3, p.243-254, September 2002 Leigh Tesfatsion, Agent-Based Computational Economics: Growing Economies From the Bottom Up, Artificial Life, v.8 n.1, p.55-82, March 2002 Cooperative Multi-Agent Learning: The State of the Art, Autonomous Agents and Multi-Agent Systems, v.11 n.3, p.387-434, November 2005

agent economies;reinforcement learning;adaptive multi-agent systems;machine learning

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Unified Interprocedural Parallelism Detection.

In this paper, we outline a new way of detecting parallelism interprocedurally within a program. Our method goes beyond mere dependence testing, to embrace methods of removing dependences as well, namely privatization, induction recognition and reduction recognition. This method is based on a combination of techniques: a universal form for representing memory accesses within a section of code (the Linear Memory Access Descriptor), a technique for classifying memory locations based on the accesses made to them by the code (Memory Classification Analysis), and a dependence test (the Access Region Test). The analysis done with Linear Memory Access Descriptors is based on an intersection operation, for which we present an algorithm. Linear Memory Access Descriptors are independent of any declarations that may exist in a program, so they are subroutine- and language-independent. This makes them ideal for use in interprocedural analysis. Our experiments indicate that this test is highly effective for parallelizing loops containing very complex subscript expressions.

Introduction Modern computer architectures, with ever-faster processors, make it increasingly important for parallelizing compilers to do their analysis interprocedurally. A compiler that parallelizes only intraprocedurally is conned to parallelizing loops in the leaf nodes of a call graph. There are, quite often, not enough operations in the leaf nodes to make parallelization pay o. For loop parallelization within the shared memory model, the compiler should parallelize at the highest level in the call graph where parallelization is possible, to overcome parallel loop overhead costs. In addition, interprocedural dependence analysis is essential for producing SPMD message passing code from a serial program. This work is supported in part by Army contract DABT63-95-C-0097; Army contract N66001-97-C-8532; NSF contract MIP-9619351; and a Partnership Award from IBM and the KAIST seed grant program. This work is not necessarily representative of the positions of the Army or the Government. c 2000 Kluwer Academic Publishers. Printed in the Netherlands. Traditional dependence testing has been developed without regard to its applicability across procedure boundaries. All pairs of memory references which may access the same memory location within a loop are compared. These memory references occur at discrete points within the loop, thus we say that these methods are point-to-point dependence tests. Point-to-point tests require O(n 2 ) comparisons where n is the number of memory references to a particular array within the loop. Obviously, as the position within the call graph gets further from the leaves, the number n can grow, and this growth can cause interprocedural point-to-point dependence testing to get unwieldy. These considerations have motivated us to take a dierent approach to dependence testing: the memory accesses within a program section are summarized, then the summaries are intersected to determine dependence between the sections. Through a series of experiments, we have found that this approach not only reduces the number of comparisons for dependence testing, but also allows us to handle very complex array subscript expressions. This paper is organized as follows. After discussing previous related work in Section 2, we will continue our discussion in Section 3 by showing how we summarize the memory access activity of an arbitrary program section. Then, in Section 4 , we will describe a novel notation which is practical for summarizing various complex array accesses encountered in many scientic programs, and in Section 5, we show how to use access summaries stored in the this notation to perform multiple-subscript, interprocedural, summary-based dependence testing. To evaluate the eectiveness of our dependence test, we implemented it in the Polaris [4] compiler and experimented with actual codes from Perfect, SPEC, and NASA benchmark suites. The experimental results presented in Section 6 show that our test holds promise for better detection of parallelism in actual codes than other tests. 2. Previous Work 2.1. Intraprocedural Dependence Testing Most point-to-point dependence testing methods rely on an equation- solving paradigm, where the pair of subscript expressions for two array reference sites being checked for dependence are equated. Then an attempt is made to determine whether the equation can have a solution, subject to constraints on the values of variables in the program, such as loop indices. In the general case, a system of linear relations is built and a solution is attempted with a linear system solver to determine if the same memory location can be accessed in dierent loop iterations. Two of the earliest point-to-point dependence tests were the GCD Test and the Banerjee Test [2]. In practice, these were simple, e-cient, and successful at determining dependence, since most subscript expressions occurring in real scientic programs are very simple. However, the simplicity of these tests results in some limitations. For instance, they are not eective for determining dependence for multidimensional arrays with coupled subscripts, as stated in [9]. Several multiple-subscript tests have been developed to overcome this limitation: the multidimensional GCD Test [2], the -test [12], the Power Test [21], and the Delta Test The above tests are exact in commonly occurring special cases, but in some cases are still too conservative. The Omega Test [16] provides a more general method, based on sets of linear constraints, capable of handling dependence problems as integer programming problems. All of the just-mentioned tests have the common problem that they cannot handle subscript expressions which are non-a-ne. Non-a-ne subscript expressions occur in irregular codes (subscripted-subscript access), in FFT codes (subscripts frequently involving 2 I ), and as a result of compiler transformations (induction variable closed forms and inline expansion of subroutines). To solve this problem, Pugh, et al.[17] enhanced the Omega test with techniques for replacing the non-a-ne terms in array subscripts with symbolic variables. This technique does not work in all situations, however. The Range Test [3, 4] was built to provide a better solution to this problem. It handles non-a-ne subscript expressions without losing accuracy. Overall, the Range Test is almost as eective as the Omega Test and sometimes out-performs it, due mainly to its accuracy for non-a-ne expressions [3]. One critical drawback of the Range Test is that it is not a multiple-subscript test, and so is not eective for handling coupled-subscripts. 2.2. Interprocedural Summarization Techniques Interprocedural dependence testing demands new capabilities from dependence tests. Point-to-point testing becomes unwieldy across procedure boundaries, and so has given way to dependence testing using summaries of the accesses made in subroutines. The idea of using access summaries for dependence analysis was previously proposed by several researchers such as Balasundaram, et al.[1] and Tang [18]. Also, the Range Test, though it is a point-to-point test, uses summarized range information for variables, obtained through abstract interpretation of the program. To perform accurate dependence analysis with access summaries, the compiler needs some standard notation in which the information about array accesses is summarized and stored for its dependence analyzer. Several notations have been developed and used for dependence analysis techniques. Most notable are triplet notation [3, 10, 19] and sets of linear constraints [1, 7, 18]. However, as indicated in [11], existing dependence analysis techniques have deciencies directly traceable to the notations they used for access summarization. Triplet notation is simple to work with, but not rich enough to store all possible access patterns. Linear constraints are more general, but can not precisely represent the access patterns due to non-a-ne subscript expressions, and require much more complex operations. So, clearly there is room for a new dependence test and a new memory access representation, to overcome the limitations of existing techniques. 2.3. Parallelism Detection While dependence testing has been studied exhaustively, a topic which has not been adequately addressed is a unied method of parallelism detection, which not only nds dependences, but categorizes them for easy removal with important compiler transformations. Eigenmann, et al [8] studied a set of benchmark programs and determined that the most important compiler analyses needed to parallelize them were array privatization, reduction and induction (idiom) analysis, and dependence analysis for non-a-ne subscript expressions, and that all of those must be done in the presence of strong symbolic interprocedural analysis. The need for improved analysis and representational techniques prompts us to go back to rst principles, rethink what data dependence means and ask whether dependence analysis can be done with compiler transformations in mind. The key contribution of this paper is the description of a general interprocedural parallelism detection technique. It includes a general dependence analysis technique, described in Section 5, called the Access Region Test (ART). The ART is a multiple-subscript, interprocedural, summary-based dependence test, combining privatization and idiom recognition. It represents memory locations in a novel form called an Access Region Descriptor (ARD)[11], described in Section 4, based on the Linear Memory Access Descriptor of [14]. 3. Memory Classication Analysis In this section, we formulate data dependence analysis in terms of a scheme of classifying memory locations, called Memory Classication Analysis (MCA), based on the order and type of the accesses within a section of code. The method of classifying memory locations is a general one, based on abstract interpretation[5, 6] of a program, and may be used for purposes other than dependence analysis. The traditional notion of data dependence is based on classifying the relationship between two accesses to a single memory location. The operation done (Read or Write), and the order of the accesses determines the type of the dependence. A data dependence arc is a directed arc from an earlier instruction (the source) to a later instruction (the sink), both of which access a single memory location in a program. The four types of arcs are determined as shown in Table I. Table I. Traditional data dependence denition. Dependence Type Input Flow Anti Output Earlier access Read Write Read Write Later access Read Read Write Write Input dependences can be safely ignored when doing parallelization. Anti and output dependences (also called memory-related dependences) can be removed by using more memory, usually by privatizing the memory location involved. Flow dependences (also called true depen- dences) can sometimes be removed by transforming the original code through techniques such as induction variable analysis and reduction analysis [20]. A generalized notion of data dependence between arbitrary sections of code can be built by returning to rst principles. Instead of considering a single instruction as a memory reference point, we can consider an arbitrary sequence of instructions as an indivisible memory referencing unit. The only thing we require is that the memory referencing unit be executed entirely on a single processor. We refer to this memory referencing unit as a dependence grain. DEFINITION 1. A section of code representing an indivisible, sequentially executed unit, serving as the source or sink of a dependence arc in a program, will be called a dependence grain. This denition of dependence grain corresponds to the terms coarse- and ne-grained analysis, which refer to using large and small dependence grains, respectively. If we want to know whether two dependence grains may be executed in parallel, then we must do dependence analysis between the grains. Since a single grain may access the same memory address many times, we must summarize the accesses in some useful way and relate the type and order of the summaries to produce a representative dependence arc between the two grains. DEFINITION 2. A representative dependence arc is a single dependence arc showing the order in which two dependence grains must be executed to preserve the sequential semantics of the program. A single representative dependence arc summarizes the information which would be contained in multiple traditional dependence arcs between single instructions For medium- and coarse-grain parallelization, there can be many accesses to a single memory location within each dependence grain. Instead of keeping track of the dependences between all possible pairs of references which have a reference site in each grain (as in point- to-point testing), it is desired to represent the dependence relationship between the two grains, for an individual memory location, with a single representative dependence arc. There are many possible ways to summarize memory accesses. The needs of the analysis and the desired precision determine which way is best. To illustrate this idea, the next two sections show two ways of summarizing accesses: the simple, but low-precision read-only summary scheme and the more useful write-order summary scheme. 3.1. The Read-only Summary Scheme It is possible to dene a representative dependence such that it carries all of the dependence information needed for the potential parallelization of the two grains. When no dependence exists between any pair of memory references in the two grains, neither should a representative dependence exist. When two or more accesses to a memory location exist in a grain, we must simply nd a way to assign an aggregate access type to the group, so that we can determine the representative dependence in a way which retains the information we need for making parallelization decisions. Consider two grains which execute in the serial form of a program, one before the other. One consistent way to summarize dependence (for a single memory location) between the two grains is to determine whether the accesses are read-only in each grain, and dene dependence as in Table II. We call this the read-only summary scheme. Table II. One possible representative dependence deni- tion - the read-only summary scheme. Dependence Type: Input Flow Anti Output later Read-Only? Figure dependence summarization with the read-only summary scheme. When an input dependence exists between two grains, it can be ignored. When a ow dependence exists between grains, in general the grains must be serialized. Earlier Grain Later Grain Flow Dependence Input Dependence Earlier Grain Later Grain Flow Dependence Input Dependence Output dependence Anti dependence Output Dependence Between Grains Flow Dependence Between Grains Figure 1. Dependence between grains depends on whether the two grains are read-only. The situation on the right shows a case where A can be privatized in the later grain, eliminating the output dependence. When an anti dependence exists between grains, it means that only reads happen in one grain, followed by at least one write in the other. An output dependence means that at least one write occurs in both grains. In both anti and output dependence situations, if a write to the location is the rst access in the later dependence grain, then it would be possible to run the grains in parallel by privatizing the variable in the later grain. However, in the read-only summary scheme we don't keep enough information in the summary to determine whether a write happened rst in the later grain or not, so we would miss the opportunity to parallelize by privatization. This shows that while read-only summarizing can detect dependences, it does not classify the dependences clearly enough to allow us to eliminate the dependence by compiler transformations. We will derive a better scheme in the next section. 3.2. The Write-order Summary Scheme When the dependence grains are loop iterations, there exists a special case of the more general problem in that a single section of code represents all dependence grains. This fact can be used to simplify the dependence analysis task. If we were still using read-only summarization and doing loop-based dependence testing, there would no longer be four cases, just two. The iteration is either read-only or it is not. However, to be able to dierentiate between the anti and output dependences which can be removed by privatization and those which cannot, the case where it is not read-only can also be divided into two cases: one where a write is the rst access to the location (WriteFirst) and one where a read is the rst access (ReadWrite). This gives three overall classes, shown in Table III. Table III. Loop-based representative dependence table. Access ReadOnly ReadWrite WriteFirst Dependence Type Input Flow Anti/Output When an iteration only reads the location, dependence can be characterized as an Input dependence (and ignored). When the iteration reads the location, then writes it, the variable cannot be privatized. This results in a dependence which cannot be ignored and cannot be removed by privatization, so it will be called a Flow dependence. When an iteration writes the location rst, any value in the location when the iteration starts is immediately over-written, so the variable can be privatized. Since these dependences can be removed by privatization, they will be called memory-related dependences. Since privatization can be done in the memory-related dependence case, and that case is signaled when a write is the rst access, all we need to do to identify these cases is to keep track of the case when a location is written rst. The input and ow dependence cases are characterized by a read happening rst, and dierentiated by whether a occurs later or not. We call this the write-order summary scheme. It makes sense to use the write-order summarization scheme for the general case as well as for loops. Any locations which are read-only in both grains would correspond to an input dependence, those which are write-rst in the later grain would correspond to a memory related dependence (since it is written rst in the later grain, the later grain need not wait for any value from the earlier grain), and all others would correspond to a ow dependence. This is illustrated in Table IV. Table IV. A more eective way to classify dependences between two arbitrary dependence grains, using the classes ReadOnly, WriteFirst and ReadWrite - the write-order summary scheme. later ReadOnly later WriteFirst later ReadWrite earlier ReadOnly Input Anti/Output Flow earlier WriteFirst Flow Anti/Output Flow earlier ReadWrite Flow Anti/Output Flow So, the read-only summary scheme could serve as a dependence test, while the write-order summary scheme can detect dependence as well as provide the additional information necessary to remove dependences by a privatization transformation. As we will see in Section 5.4, a few simple tests added to the write-order summary scheme can collect enough information to allow some dependences to be removed by induction and reduction transformations. 3.3. Establishing an Order Among Accesses Knowing the order of accesses is crucial to the write-order summarization scheme, so we must establish an ordering of the accesses within the program. If a program contained only straight-line code, establishing an ordering between accesses would be trivial. One could simply sweep through the program in \execution-order", keeping track of when the accesses happen. But branching statements and unknown variables make it more di-cult to show that one particular access happens before another. For example, take the loop in Figure 2. The write to A(I) happens before the read of A(I) only if both P and Q are true. But if Q is true and P is false, then the read happens without the write having happened rst. If P and Q have values which are unrelated, then the compiler has no way of knowing the ordering of the accesses to A in this loop. On the other hand, if the compiler can show that P and Q are related and that in fact Q being true implies that P must have also been true, the compiler can know that the write happened rst. So, for code involving conditional branches, the major tool the compiler has in determining the ordering of the accesses is logical implication. To facilitate the use of logical implication to establish execution order, the representation of each memory reference must potentially for if (P) f Figure 2. Only through logical implication can the compiler determine the ordering of accesses to array A in the I-loop. have an execution predicate attached to it. In fact, the access in Figure 2 could be classied as ReadOnly with the condition f:P^Qg, WriteFirst with condition fPg and ReadWrite otherwise. DEFINITION 3. The execution predicate is a boolean-valued ex- pression, attached to the representation of a memory reference, which species the condition under which the reference actually takes place. An execution predicate P will be denoted as fPg. 3.4. Using Summary Sets to Store Memory Locations We can classify a set of memory locations according to their access type by adding a symbolic representation of them to the appropriate summary set. DEFINITION 4. A summary set is a symbolic description of a set of memory locations. We have chosen to use Access Region Descriptors (ARDs), described in Section 4, to represent memory accesses within a summary set. To represent memory accesses for use in the write-order summary scheme, according to Table III, requires three summary sets for each dependence grain: ReadOnly (RO), ReadWrite (RW), and WriteFirst (WF). 3.5. Classification of Memory References Each memory location referred to in the program must be entered into one of these summary sets, in a process called classication. A program is assumed to be a series of nested elementary contexts : procedures, If the programming language does not force this through its structure, then the program will have to be transformed into that form through a normalization process. ReadOnly ReadOnly to ReadWrite new_writefirst new_writefirst Figure 3. The intersection of earlier ReadOnly accesses with later WriteFirst accesses - the result is a ReadWrite set. simple statements, if statements, loops, and call statements. Thus, at every point in the program, there will be an enclosing context and an enclosed context. The contexts are traversed in \execution order". The summary sets of the enclosing context are built by (recursively) calculating the summary sets for each enclosed context and distributing them into the summary sets of the enclosing context. We can determine memory locations in common between summary sets by an intersection operation, as illustrated in Figure 3.5. Classication takes as input the current state of the three summary sets for the enclosing context (RO, WF, and RW) and the three new summary sets for the last enclosed context which was processed (RO n , produces updated summary sets for the enclosing context. The sets for the enclosed context are absorbed in a way which maintains proper classication for each memory location. For example, a memory location which was RO in the enclosing context (up to this point) and is WF or RW in the newly-calculated enclosed context becomes RW in the updated enclosing context. The steps of classication can be expressed in set notation, as shown in Figure 4. 3.5.1. Program Context Classication Simple statements are classied in the obvious way, according to the order of their reads and writes of memory. All statements within an if context are classied in the ordinary way, except that the if-condition P is applied as an execution predicate to the statements in the if block and :P is applied to the statements in the else block. Descriptors for the if and else blocks are then intersected and their execution predicates are or'ed together, to produce the result for the whole if context. Classifying the memory accesses in a loop is a two-step process. First, the summary sets for a single iteration of the loop must be collected by a scan through the loop body in execution order. They contain the symbolic form of the accesses, possibly parameterized by the index of the loop. Next, the summary sets must be expanded by RO RO (t WF \ RO) t RO ROn RO RO RO (t RW \ RO) Write First: Memory references of prior code Memory references of new code Result of classifying new references000000000000111111111 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 0000 0000 0000 0000 0000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 Read Only: Read Write: Figure 4. Classication of new summary sets ROn , WFn , and RWn into the existing summary sets RO, WF, and RW, and a pictorial example of adding new summary sets to existing summary sets. the loop index so that at the end of the process, the sets represent the locations accessed during the entire execution of the loop. The expansion process can be illustrated by the following loop: do do For a single iteration of the surrounding loop, the location A(I) is classied WriteFirst. When A(I) is expanded by the loop index I, the representation A(1:100) results. Summary sets for while loops can be expanded similarly, but we must use a basic induction variable as a loop index and represent the number of iterations as \unknown". This expansion process makes it possible to avoid traversing the back-edges of loops for classication. Classication for a call statement involves rst the calculation of the access representation for the text of the call statement itself, calculation of the summary sets for the procedure being called, matching formal with actual parameters, and nally translating the summary sets involved from the called context to the calling context (described further for ARDs in Section 4.3). 4. The Access Region Descriptor To manipulate the array access summaries for dependence analysis, we needed a notation which could precisely represent a collection of memory accesses. As brie y mentioned in Section 2, our previous study [11] gave us a clear picture of the strengths and weaknesses of existing notations. It also gave us the requirements the notation should meet to support e-cient array access summarization. Complex array subscripts should be represented accurately. In particular, non-a-ne expressions should be handled because time-critical loops in real programs often contain array references with non-a-ne subscripts. The notation should have simplication operations dened for it, so that complex accesses could be changed to a simpler form. To facilitate fast and accurate translation of access summaries across procedure boundaries, non-trivial array reshaping at a procedure boundary should be handled e-ciently and accurately. The notation should provide a uniform means for representing accesses to memory, regardless of the declared shape of the data structures in the source code. To meet these requirements, we introduced a new notation, called the Access Region Descriptor, which is detailed in the previous literature [11]. The ARD is derived from the linear memory access descriptor introduced in [13] and [14]. To avoid repetition, this section will only brie y discuss a few basics of the ARD necessary to describe our dependence analysis technique in Section 5. 4.1. Representing the Array Accesses in a Loop Nest If an array is declared as an m-dimensional array: then referenced in the program with an array name followed by a list of subscripting expressions in a nested loop, as in Figure 5, d Figure 5. An m-dimensional array reference in a d-loop nest. then implicit in this notation is an array subscripting function Fm which translates the array reference into a set of osets from a base address in memory: refers to the set of loop indices for the surrounding nested loops, refers to a set of constants determined by the rules of the programming language. As the nested loop executes, each loop index i k moves through its set of values, and the subscripting function Fm generates a sequence of osets from the base address, which we call the subscripting oset sequence: The isolated eect of a single loop index on Fm is a sequence of osets which can always be precisely represented in terms of its starting value, the expression representing the dierence between two successive values, and the total number of values in the sequence. For example, consider even a non-a-ne subscript expression: real A(0:*) do do The subscripting oset sequence is: I The dierence between two successive values can be easily expressed. To be clear, the dierence is dened to be the expression to be added to the Ith member of the sequence to produce the I +1th member of the sequence: There are N members of the subscripting oset sequence, they start at 2, and the dierence between successive members is 2 I . 4.2. Components of an ARD We refer to the subscripting oset sequence generated by an array reference, due to a single loop index, as a dimension of the access. We call this a dimension of an ARD. DEFINITION 5. A dimension of an ARD is a representation of the subscripting oset sequence for a set of memory references. It contains a starting value, called the base oset a dierence expression, called the stride, and the number of values in the sequence, represented as a dimension index, taking on all integer values between 0 and a dimension- index bound value. Notice that the access produced by an array reference in a nested loop has as many dimensions as there are loops in the nest. Also, the dimension index of each dimension may be thought of as a normalized form of the actual loop index occuring in the program when the ARD is originally constructed by the compiler from the program text. In addition to the three expressions described above for an ARD dimension, a span expression is maintained, where possible, for each dimension. The span is dened as the dierence between the osets of the last and rst elements in the dimension. The span is useful for doing certain operations and simplications on the ARD (for instance detecting internal overlap, as described in Section 4.4), however it is only accurate when the subscript expressions for the array access are monotonic. A single base oset is stored for the whole access. An example of an array access, its access pattern in memory, and its LMAD may be seen in Figure 6. The ARD for the array access in Figure 5 is written as A d with a series of d comma-separated strides (- 1 - d ) as superscripts to the variable name and a series of d comma-separated spans ( 1 d ) as subscripts to the variable name, with a base oset () written to the right of the descriptor. The dimension index is only included in the written form of the LMAD if it is needed for clarity. In that case, [index dimension-bound] is written as a subscript to the appropriate stride. 4.3. Interprocedural Translation of an ARD A useful property of the ARD is the ease with which it may be translated across procedure boundaries. Translation of array access information across procedure boundaries can be di-cult if the declaration of a formal array parameter diers from the declaration of its corresponding actual parameter. Array access representations which depend A 3, 14, 26 . A(K+26*(I-1), J) . DO K=1, 10, 3 END DO END DO END DO REAL A(14, *) Figure 6. A memory access diagram for the array A in a nested loop and the Access Region Descriptor which represents it. on the declared dimensionality of an array (as most do) are faced with converting the representation from one dimensionality to another when array reshaping occurs. This is not always possible without introducing complicated mapping functions. This is the array reshaping problem. Table V indicates that signicant array reshaping occurs in many scientic applications, as published in [11]. Table V. The gures in each entry indicate percentages of calls doing reshaping in various benchmark programs from Perfect, SPEC and NASA. They were computed from a static examination of the programs mentioned. trfd arc2d tfft2 flo52 turb3d ocean mdg bdna tomcatv swim We refer to a memory access representation that is independent of the declared dimensionality of an array as a \universal" represen- tation, because it becomes procedure independent and even language independent. A universal representation eliminates the array reshaping problem because it need not be translated to a new form (a potentially dierent dimensionality) when moving to a dierent execution context. The ARD is an example of a universal representation. When a subroutine is called by reference, the base address of a formal array parameter is set to be whatever address is passed in the actual argument list. Any memory accesses which occur in the subroutine would be represented in the calling routine in their ARD form, relative to that base address. Whenever it is desired to translate the ARD for a formal argument into the caller's context, we simply translate the formal argument's variable name into the actual argument's name, and add the base oset of the actual parameter to that of ARD for the formal parameter. For example, if the actual argument in a Fortran code is an indexed array, such as call then the oset from the beginning of the array A for the actual argument is 2I. Suppose that the matching formal parameter in the subroutine X is Z and the LMAD for the access to Z in X is Z 10;200 When the access to Z, in subroutine X, is translated into the calling routine, then the LMAD would be represented in terms of variable A as follows: A 10;200 which results from simply adding the oset after the renaming of Z to A. Notice that A now has a two-dimensional access even though it is declared to be one-dimensional. 4.4. Properties of ARDs This subsection brie y describes several basic properties of ARDs that are useful for our dependence analysis based on access summary sets. DEFINITION 6. Given an ARD with a set of stride/span pairs, we call the sum of the spans of the rst k dimensions the k-dimensional width of the ARD, dened by 4.4.0.1. Internal Overlap of an ARD The process of expanding an ARD by a loop can cause overlap in the descriptor. For example, in the following Fortran do-loop do I=1,10 do J=1,5 . A(I*4+J) . do do the ARD for A in the inner loop is A 1 1. When the ARD is expanded for the outer loop, it becomes A 1;4 exhibits an overlap due to the outer loop. This is because the access due to the outer loop does not stride far enough to get beyond the array elements already touched by the inner loop. This property may be determined by noticing that the stride of the n-th dimension is not greater than the dimensional width of the ARD. 4.4.1. Zero-span dimensions A dimension whose span is zero adds no data elements to an access pattern. This implies that whenever such a dimension appears in an ARD (possibly through manipulation of the ARD), it may be safely eliminated without changing the access pattern represented. Likewise, it implies that at any time a new dimension may be introduced with any desired stride and a zero-span, without changing the access pattern. Simplication operations exist for eliminating dimensions within an ARD, for eliminating ARDs which are found to be covered by other ARDs, and for creating a single ARD which represents the accesses of several ARDs. Since these operations are not needed for the exposition of this paper, they will not be described here, but the reader is referred to [11, 14, 13]. 5. The Access Region Test In this section, we rst describe the general dependence analysis method, based on intersecting ARDs. The general method can detect data dependence between any two arbitrary sections of code. Then, we show a simplication of the general method, the Access Region Test, which works for loop parallelization, and show a multi-dimensional, recursive intersection algorithm for ARDs. 5.1. General Dependence Testing with Summary Sets Given the symbolic summary sets RO 1 , WF 1 , and RW 1 (as discussed in Section 3.4), representing the memory accesses for an earlier (in the sequential execution of the program) dependence grain, and the sets RO 2 , WF 2 , and RW 2 for a later grain, it can be discovered whether any locations are accessed in both grains by nding the intersection of the earlier and later sets, and by consulting Table IV. Any non-empty intersection represents a dependence between grains. However, some of those dependences may be removed by compiler transformations. The intersections that must be done for each variable are: If all of these intersections are empty for all variables, then no cross-iteration dependences exist between the two dependence grains. If any of the following are non-empty: RO 1 or RO 1 \RW 2 , then they represent dependences which can be removed by privatizing the intersecting regions. If RW 1 \RW 2 is non-empty, and all references involved are in either induction form or reduction form, then the dependence may be removed by induction or reduction transformations. This will be discussed in more detail in Section 5.4. If any of the other intersections: WF 1 \RO 2 , WF 1 \RW 2 , or RW 1 \ RO 2 are non-empty, then they represent non-removable dependences. 5.2. Loop Dependence Testing with the ART The Access Region Test (ART) is used within the general framework of Memory Classication Analysis, doing write-order summarization. This means that the entire program is traversed in execution order, using abstract interpretation, with summary sets being computed for the nested contexts of the program and stored in ARDs. ARDs are used as the semantic elements for the abstract interpretion. The interpretation rules are exactly those rules described for the various program contexts in Section 3.5.1. Whenever loops are encountered, the ART is applied to the ARDs to determine whether the loops are parallel, or parallelizable by removing dependences through compiler transformations. As stated in Section 3.2, dependence testing between loop iterations is a special case of general dependence testing, described in the last section. Loop-based dependence testing considers a loop iteration to be a dependence grain, meaning all dependence grains have the same summary sets. Once we expand the summary sets by the loop index (Section 3.5.1), cross-iteration dependence can be noticed in three ways: within one LMAD, between two LMADs of one summary set, or between two of the summary sets. 5.2.1. Overlap within a Single ARD Internal overlap due to expansion by a loop index is described in Section 4.4. When overlap occurs, it indicates a cross-iteration dependence. This condition can be easily checked during expansion and agged in the ARD, so no other operation is required to detect this. 5.2.2. Intersection of ARDs within a Summary Set Even though two LMADs in the same summary set do not intersect initially, expansion by a loop index could cause them to intersect. Such an intersection would represent a cross-iteration dependence. Such an intersection within RO would be an input dependence, so this summary set need not be checked. Internal intersections for both WF and RW must be done, however. In Figure 7, for example, when the two writes to array A are rst assigned to a summary set, they do not overlap. The two write-rst ARDs are initially A 0 the base osets are dierent, the intersection is assumed to not overlap (the conservative assumption). This causes them to be separately assigned to the WF set. After expansion for I, (and creation of the dimension index I 0 ), the normalized ARDs both become A 1 do intersect, indicating a dependence. This intersection would be found by attempting to intersect the ARDs within WF. do do Figure 7. Example illustrating the need for internal intersection of the summary sets. 5.2.3. Intersection of Two Summary Sets There are only three summary sets to consider in loop dependence testing, instead of six (because there is only one dependence grain), so there are only three intersections to try, instead of the eight required in Section 5.1. After expansion by the loop index, the following intersections must be done: RO \ WF RO \ RW WF \ RW An intersection between any pair of the sets RO, WF, and RW involves at least one read and one write operation, implying a dependence 5.3. The Loop-based Access Region Test Algorithm For each loop L in the program, and its summary sets RO, WF, and RW, the ART does the following: Expand the ARDs in all summary sets by the loop index of L. Check for internal overlap, due to the loop index of L, of any ARD in WF or RW. Any found within WF can be removed by privatization. Any found in RW is removed if all references involved are in either induction or reduction form. Once overlap for an ARD is noted, its overlap ag is reset. Check for non-empty intersection between any pair of ARDs in WF (removed by privatization) or RW (possibly removed by induction or reduction). For all possible pairs of summary sets, from the group RO, WF, and RW, check for any non-empty intersection between two ARDs, each pair containing ARDs from dierent sets. Any intersection found here is noted as a dependence and moved to RW. If no non-removable dependences are found, the loop may be declared parallel. Wherever uncertainty occurs in this process, demand-driven deeper analysis can be triggered in an attempt to remove the uncertainty, or run-time tests can be generated. 5.4. Detecting Reduction and Induction Patterns As stated in Section 2.3, idiom recognition is very important for parallelizing programs. Inductions and reductions both involve an assignment with a linear recurrence structure: The forms dier slightly, as shown in the following (I represents an integer variable and R represents a oating point Induction oating point expression Reduction Each of these patterns originally presents itself as a dependence, but compiler transformations [15, 4] can remove the dependence. Three levels of tests, done within the write-order summary scheme structure, can be used to positively identify reductions and inductions. 5.4.0.1. Level 1 The rst test is for the linear recurrence structure of the assignment statement. When the pattern is found, the ARD for the statement is marked as passing the Level 1 test, plus the operator the type of the (integer constant or oating point expression) are stored. The ARD is marked with an idiom type of possible induction if it is an integer variable and the expression is an integer constant. Otherwise it is marked as a possible reduction. 5.4.0.2. Level 2 During intersection of the ARDs of a particular variable within the ReadWrite summary set (as part of the ART, described in Section 5.2.2), if one is marked as having passed Level 1, then if any other ARD in RW for the variable did not pass Level 1, with the same idiom type and operator, it fails Level 2. If there are any ARDs for the variable in either ReadOnly or WriteFirst and the ARD is a possible reduction, then it fails Level 2. If the ARD is a possible induction and any ARDs exist for the variable in WriteFirst, then it fails Level 2. Otherwise the ARD passes Level 2. 5.4.0.3. Level 3 To pass Level 3, an ARD marked as passing Level 2 must be marked as having internal overlap due to expansion by the loop index of an outer loop. This means that there is a dependence due to the access, carried by the outer loop. An ARD marked as having passed Level 3 can be considered an idiom of the stored type, and appropriate code can be generated for it. This three-level process will nd inductions and reductions interproce- durally, because of the interprocedural nature of the ART. 5.5. Generality of the ART The Access Region Test is a general, conservative dependence test. By design, it can discern three types of dependence: input, ow, and memory-related. It cannot distinguish between anti and output depen- dence, but that is because for our purposes it was considered unimportant both types of dependence can be removed by privatization transformations. For other purposes, the general MCA mechanism can be used to formulate a mechanism, with the appropriate summary sets, to produce the required information, much as data ow analysis can be formulated to solve various data ow problems. In Sections 3.1 and 3.2, we showed two dierent formulations of MCA for doing dependence analysis. The read-only summary scheme is simply a dependence test, while the write-order summary scheme provides enough information to test for dependence, and also remove some dependences through compiler transformations. 5.6. Loop-carried Dependence Handled by the ART Any dependence within an inner loop is essentially ignored with respect to an outer loop because of the fact that after expansion by a loop index, any intersecting portions of two ARDs are represented as a single ARD and moved to the RW summary set. If there are intersecting portions, they are counted as cross-iteration dependences for that loop, but because they are reduced to a single ARD, no longer will be found to intersect for outer loops. Intersections due to outer loops will be solely due to expansions for outer loop indices. This process is illustrated in Figure 8. RO: A I, J END DO END DO WF RO dependence RO WF 100(50-2)A 100(50-2) expand I expand I expand I 50-1, 100(50-3) I loop : no intersection indicates independence RO RW WF Figure 8. How the ART handles loop-carried dependence. 5.7. A Multi-Dimensional Recursive Intersection Algorithm Intersecting two arbitrary ARDs is very complex and probably in- tractable. But if the two ARDs to be compared have the same strides (we call these stride-equivalent), or the strides of one are a subset of the strides of the other (we call these semi-stride-equivalent), which has been quite often true in our experiments, then they are similar enough to make the intersection algorithm tractable. We present Algorithm Intersect in Figure 9. Input: Two ARDs, with properly nested, sorted dimensions : d (such that 0 ), k: the number of the dimension to work on (0 k d) Output:List of ARDs returns ARD List if (D == 0 ) then ARD rlist1 ARD scalar() add to list(ARD List; ARD rlist1 ) endif return ARD List endif c // periodic intersection on the left remove dim(ARD right ; k; 0 add to list(ARD List; ARD rlist1 ) (R intersection on the right remove dim(ARD left ; k; add to list(ARD List; ARD rlist2 ) endif else // intersection at the end remove dim(ARD right ; k; 0 add to list(ARD List; ARD rlist1 ) endif return ARD List intersect remove dim(ARD in ; k; new // Construct and return a new ARD equivalent to ARD in , // except without access dimension k and with new as the new base oset. // Construct and return a new access dimension with stride - and span . add dim(ARD in ; dimnew ; new // Construct and return a new ARD equivalent to ARD in , // except with new dimension dimnew and with new as the new base oset. add to list(ARD list, ARD) f // Add ARD to the list of ARDs ARD list. Figure 9. The algorithm for nding the intersection of multi-dimensional ARDs. For clarity, the removal of the intersection from the two input ARDs, the use of the conservative direction ag, and the use of the execution predicates are not shown, although all these things can be added to the algorithm in a straightforward way. The algorithm accepts two stride-equivalent ARDs. If the two ARDs are semi-stride-equivalent, then zero-span dimension(s) can be safely inserted into the ARD with fewer dimensions (as discussed in Section 4.4.1) to make them stride-equivalent. The algorithm is also passed a conservative direction ag. The ag has two possible values: under-estimation and over-estimation, which tells the algorithm what to do when the result is imprecise. For over-estimation, the result is enlarged to its maximum value and likewise, under-estimation causes the result to be reduced to its minimum value. If two ARDs are to be intersected and they are not stride-equivalent, then the result is formed, based on the conservative direction. The algorithm takes as input two ARDs which have all dimensions precisely sorted, ARD d and the number of the dimension, d, to work on. ARD left has a base oset which is less than that of ARD right . The algorithm compares the overall extent of dimension d for each ARD, as shown in Figure 10(A). If the extents do not overlap in any way, it can safely report that the intersection is empty. If they do overlap, then the algorithm calls itself recursively, specifying the next inner dimension, as shown in Figure 10(B). Intersection Intersection A Figure 10. The multi-dimensional recursive intersection algorithm, considering the whole extent of the two access patterns (A), then recursing inside to consider the next inner dimension (B). This process continues until it can either be determined that no overlap occurs, or until the inner-most dimension is reached, as shown in Figure 11, where it can make the nal determination as to whether there is an intersection between the two, considering only one-dimensional accesses. The resulting ARD for the intersection is returned, and as each recursion returns, a dimension is added to the resulting ARD. Figure 11. The multi-dimensional recursive intersection algorithm, considering the inner-most dimension, nding no intersection. For simplicity, in this description, it is assumed that the two ARDs have dimensions which are fully sorted, so that dimension i of one ARD corresponds to dimension i of the other, and that - d > - d 1 > > - 1 . 6. Experiments The Access Region Test has an advantage over other tests discussed in Section 1, in three ways: Reducing dependence analysis to an intersection operation does not restrict the ART from handling certain types of subscripting expressions, such as coupled subscripts, which are a problem for the Range Test, and non-a-ne expressions, which are a problem for most other tests. Use of the ARD provides precise access summaries for all array subscripting expressions. The test is implicitly interprocedural since ARDs may be translated precisely across procedure boundaries. To separate the value of the ART from the value of the ARD, it is instructive to consider the question of whether other dependence tests might be as powerful as the ART if they represented memory accesses with the ARD notation. The answer to this question is \no". Take as an example the Omega Test. The mechanism of the Omega Test is only dened for a-ne expressions. The user of the Omega Test must extract the linear coe-cients of the loop-variant values (loop indices, etc), plus provide a set of constraints on the loop-variants. The ARDs partially ll the role of the constraints, but if non-a-ne expressions are used, there is no way to extract linear coe-cients for the non-a-ne parts. A technique for replacing the non-a-ne parts of an expression with uninterpreted function symbols has been developed [17], but it is not general enough to work in all situations. So even using the ARD, the Omega Test could not handle non-a-ne subscript expressions because its mechanism is simply not well-dened for such expressions. Likewise, if the Range Test were to use the ARD to represent value ranges for variables, that still would not change its basic mechanism, which makes it a single-subscript test, unable to handle coupled-subscripts. The mechanism of the Range Test forces it to consider the behavior of the subscript expression due to a single subscript at a time, whereas the ART compares access patterns instead of subscript expressions. A simple example in Figure 12 shows the advantage of comparing the patterns. It shows two loop nests which display identical access patterns, yet dierent subscripting expressions. The top accesses can be determined independent by the Range Test, but the bottom accesses cannot. do do J=1, M do do do do J=1, M do do I I I Figure 12. The Range Test can determine the accesses of the top loop to be inde- pendent, but not those of the bottom loop. The ART can nd both independent, since it deals with access patterns instead of just subscript expressions. Figure 13 shows another example, from the tfft2 benchmark code, which neither the Omega Test nor the Range Test can nd independent REAL U(1), X(1), Y(1) DO I=0,2*(M/2)-1 U(1+3*2*(1+M)/2), END DO REAL U(1), X(1), Y(1) DO L0=1, (M+1)/2 END DO REAL U(*), X(*), DO I=0,2*(M-L)-1 DO K=0,2*(L-1)-1 END DO END DO Figure 13. A simplied excerpt from the benchmark program tt2, which the ART can determine to be free of dependences. due to the apparent complexity of the non-a-ne expressions involved, yet the ART can nd them independent interprocedurally at the top-most loop, due to its reliance on the simple intersection operation, its ability to translate ARDs across procedure boundaries, and the powerful ARD simplication operations which expose the simple access patterns hidden inside complex subscript expressions. As we continued to develop the ART, we needed to evaluate the ART on real programs. Therefore, we implemented a preliminary version of the ART in Polaris [4], a parallelizing compiler developed at Illinois, and experimented with ten benchmark codes. In these experiments, it was observed that the ART is highly eective for programs with complex subscripting expressions, such as ocean, bdna, and tfft2. Table VI shows a summary of the experimental results that were obtained at the time we prepared this paper. Careful analytical study conrms that the ART theoretically subsumes the Range Test. This implies that the ART can parallelize all the loops that the Range Test can, even though in this experiment, the ART failed to parallelize a few loops in flo52 and arc2d due to several implementation dependent problems reported in [11]. The numbers of loops additionally parallelized by the ART are small, but some of these loops are time-critical loops which contain complex array subscripting expressions. Our previous experiments reported in [13] also showed that the ART applied by hand increased the parallel speedup for tfft2 by factor of 7.4 on the Cray T3D 64 processors. As can be expected, Table VI shows that neither the ART, the Omega Test, nor the Range Test make a dierence in the performance for the codes with only simple array subscripting expressions, such as tomcatv, arc2d and swim. Table VI. A comparison of the number of loops parallelized by a current version of the ART with other techniques. The rst line shows the number of loops that the ART could parallelize and the Range Test could not. The second shows the number of loops that the Range Test could parallelize and the Omega Test could not. The third shows the number of loops that the Omega Test could parallelize and the Range Test could not. All other loops in the codes were parallelized identically by all tests. The data in the second and third lines are based on the previous work on Polaris. tfft2 trfd mdg flo52 hydro2d bdna arc2d tomcatv swim ocean Previous techniques based on access summaries did not show experimental results with real programs in their papers [1, 18]. Thus, it is not possible for us to determine how eective their techniques would be for actual programs. 7. Conclusion and Future Work This paper presents a technique for unifying interprocedural dependence analysis, privatization and idiom recognition in a single frame- work. This technique eliminates some of the limitations which encumber the loop-based, linear system-solving data dependence paradigm, and expands the notion of a dependence test to include a way of classifying the dependences found, so that a compiler can eliminate them using code transformations. The framework is built on a general scheme for classifying memory locations (Memory Classication Analysis), based on the order and type of accesses to them. This framework can be reformulated and used for many purposes. The read-only and write-order summarization schemes were presented, but many other schemes are possible for a variety of purposes. The multi-dimensional, recursive intersection algorithm for ARDs was introduced. It allows us to calculate a precise intersection between two stride-equivalent ARDs. This algorithm forms the core of the dependence analysis calculation. Heuristics can be added to this algorithm to handle cases in which the ARDs are not stride-equivalent. The more precise this intersection algorithm becomes, the more precise data dependence analysis becomes. We believe that the exibility and generality aorded by this reformulation of data dependence will make it very useful for many purposes within a compiler. In the future, we intend to use the MCA framework for other analyses, which will automatically extend them interprocedu- rally. In addition, we intend to formalize our methods by an analysis in terms of the abstract semantic elements and rules within the abstract interpretation framework. --R A Technique for Summarizing Data Access and its Use in Parallelism Enhancing Transformations. Dependence Analysis. Symbolic Analysis Techniques for E Parallel Programming with Polaris. Semantic foundations of program analysis. Abstract interpretation: A uni Interprocedural Array Region Analyses. On the Automatic Parallelization of the Perfect Benchmarks. An Implementation of Interprocedural Bounded Regular Section Analysis. Automatic Parallelization for Distributed Memory Machines Based on Access Region Analysis. Induction Variable Substitution and Reduction Recognition in the Polaris Parallelizing Compiler. A Practical Algorithm for Exact Array Dependence Analysis. Nonlinear Array Dependence Analysis. Exact Side E Gated SSA-Based Demand-Driven Symbolic Analysis for Parallelizing Compilers High Performance Compilers for Parallel Computing. The Power Test for Data Dependence. --TR Interprocedural dependence analysis and parallelization A technique for summarizing data access and its use in parallelism enhancing transformations Practical dependence testing A practical algorithm for exact array dependence analysis Exact side effects for interprocedural dependence analysis Nonlinear array dependence analysis Gated SSA-based demand-driven symbolic analysis for parallelizing compilers On the Automatic Parallelization of the Perfect BenchmarksMYAMPERSAND#174 Simplification of array access patterns for compiler optimizations Nonlinear and Symbolic Data Dependence Testing Abstract interpretation Dependence Analysis Parallel Programming with Polaris An Efficient Data Dependence Analysis for Parallelizing Compilers An Implementation of Interprocedural Bounded Regular Section Analysis The Power Test for Data Dependence Interprocedural Array Region Analyses Symbolic analysis techniques for effective automatic parallelization Interprocedural parallelization using memory classification analysis --CTR Y. Paek , A. Navarro , E. Zapata , J. Hoeflinger , D. Padua, An Advanced Compiler Framework for Non-Cache-Coherent Multiprocessors, IEEE Transactions on Parallel and Distributed Systems, v.13 n.3, p.241-259, March 2002 Thi Viet Nga Nguyen , Franois Irigoin, Efficient and effective array bound checking, ACM Transactions on Programming Languages and Systems (TOPLAS), v.27 n.3, p.527-570, May 2005

privatization;dependence analysis;parallelization;compiler

608747

Path Analysis and Renaming for Predicated Instruction Scheduling.

Increases in instruction level parallelism are needed to exploit the potential parallelism available in future wide issue architectures. Predicated execution is an architectural mechanism that increases instruction level parallelism by removing branches and allowing simultaneous execution of multiple paths of control, only committing instructions from the correct path. In order for the compiler to expose and use such parallelism, traditional compiler data-flow and path analysis needs to be extended to predicated code. In this paper, we motivate the need for renaming and for predicates that reflect path information. We present Predicated Static Single Assignment (PSSA) which uses renaming and introduces Full -Path Predicates to remove false dependences and enable aggressive predicated optimization and instruction scheduling. We demonstrate the usefulness of PSSA for Predicated Speculation and Control Height Reduction. These two predicated code optimizations used during instruction scheduling reduce the dependence length of the critical paths through a predicated region. Our results show that using PSSA to enable speculation and control height reduction reduces execution time from 12 to 68%.

Introduction The Explicitly Parallel Instruction Computing (EPIC) architecture has been put forth as a viable architecture for achieving the instruction level parallelism (ILP) needed to keep increasing future processor performance [8, 17]. Intel's application of EPIC architecture technology can be found in their IA-64 architecture whose first instantiation is the Itanium processor [1] An EPIC architecture issues wide instructions, similar to a VLIW architecture, where each instruction contains many operations. One of the new features of the EPIC architecture is its support for predicated execution [24], where each operation is guarded by one of the predicate registers available in the architecture. An operation is committed only if the value of its guarding predicate is true. One advantage of predicated execution is that it can eliminate hard-to-predict branches by combining both paths of a branch into a single path. Another advantage comes from using predication to combine several smaller basic blocks into one larger hyperblock [22]. This provides a larger pool from which to draw ILP for EPIC architectures. A significant limitation to ILP is the presence of control-flow and data-flow dependences. Static Single Assignment (SSA) is an important compiler transformation used to remove false data dependences across basic block boundaries in a control flow graph [12]. Removing these false dependences reveals more ILP, allowing better performance of optimizations like instruction scheduling. Without performing SSA, the benefit of many optimizations on traditional code is limited. Eliminating false dependences is equally important and a more complex task for predicated code, since multiple control paths are merged into a single predicated region. However, the control-flow and data-flow analysis needed to support predicated compilation is different than traditional analysis used in compilers for superscalar architectures. A sequential region of predicated code contains not only data dependences, but also predicate dependences. A predicate dependence exists between every operation and the definition(s) of its guarding predicate. Our technique introduces a chain of predicate dependences which represents a unique control path through the original code. We describe a predicate-sensitive implementation of SSA called Predicated Static Single Assignment (PSSA). PSSA introduces Full-Path Predicates to extend SSA to handle predicate dependences and the multiple control paths that are merged together in a single predicated region. We demonstrate that PSSA allows effective predicated scheduling by (1) eliminating false dependences along paths via renaming, (2) creating full-path predicates, and (3) providing path-sensitive data-flow analysis. We show the benefit of using PSSA to perform Predicated Speculation and Control Height Reduction during instruction schedul- b>a b=rand() if true // b=random number P2,P3 cmpp.un.uc b>a if true // if b>a then P2=true,P3=false else P2=false, P3=true b=q if P2 // if P2 is true, b=q else nullify d=b+3 if P3 // if P3 is true, d=b+3 else nullify statement f=b*2 if true // f=b*2 a) Original Control Flow Graph b) Predicated Hyperblock Figure 1: Short code example showing the transformation from non-predicated code to predicated hyperblock ing. Using PSSA allows these two optimizations, when applied together, to schedule all operations at their earliest schedulable cycle. In our implementation, the earliest schedulable cycle takes into consideration true data dependences and load/store constraints. In this paper we expand upon work we presented in [11] by including additional benchmarks and by motivating the need for renaming and for predicates that reflect path information above and beyond what is available from traditional If-converted code. The paper is organized as follows. Section 2 describes predicated execution. Section 3 motivates the need for predicate-sensitive analysis and full-path predicates. Section 4 presents Predicated Static Single Assignment. Section 5 shows how PSSA can enable aggressive Predicated Speculation and Control Height Reduction. Section 6 reports the increased ILP and reduced execution times achieved by applying our algorithms to predicated code. Section 7 summarizes related work. Section 8 discusses using PSSA within the IA-64 framework, and Section 9 describes our future work. Finally, Section 10 summarizes the contributions of this paper. Predicated Execution Predicated execution is a feature designed to increase ILP and remove hard-to-predict branches. It has also been used to support software pipelining[14, 25]. Machines with hardware to support predicated code include an additional set of registers called predicate registers. The process of predication replaces branches with compare operations that set predicate registers to either true or false based on the comparison in the original branch. Each operation is then associated with one of these predicate registers which will hold the value of the operation's guarding predicate. The operation will be committed only if its guarding predicate is true 1 . This process of replacing branches with compare operations and associating operations with a predicate defined by that compare is called If-Conversion [5, 24]. Our work uses the notion of a hyperblock [22]. A hyperblock is a predicated region of code consisting of a straight-line sequence of instructions with a single entry point and possibly multiple exit points. Branches with both targets in the hyperblock are eliminated and converted to predicate definitions using If-conversion. All remaining branches have targets outside the hyperblock. Consequently, there are no cyclic control-flow or data-flow dependences within the hyperblock. The selection of instructions to be included in the hyperblock is based on program profiling of the original basic blocks which includes information such as execution frequency, basic block size, operation latencies, and other characteristics [22]. A typical code section to include in a hyperblock is one that contains a hard-to-predict (unbiased) branch [21], as shown in Figure 1. After If-conversion, the Control Flow Graph (CFG) in Figure 1(a), which is comprised of four basic blocks, results in the predicated hyperblock shown in Figure 1(b). All operations in the hyperblock are now guarded, either by a predicate register set to the constant value of true, or by a register that can be defined as either true or false by a cmpp (compare and put (result) in predicate) operation. Operations guarded by the constant true, such as the operation f=b*2 in Figure 1, will be executed and committed regardless of the path taken. Operations guarded by a predicate register, such as the operation b=q, will be put into the pipeline, but only committed if the value of the operation's guarding predicate (P2 for this operation) is determined to be true. In what follows, we describe three types of operations that can be included in a hyperblock - cmpp operations, the predicate OR operation, and normal (non-predicate-defining) operations. As defined in the Trimaran System [2] (which supports EPIC computing via the Playdoh ISA [19]), guarding predicates are assigned their values via cmpp operations [8]. Consider an operation B,C cmpp.un.ac a?c if A as an example. The cmpp operation can define one or two predicates. This operation will define predicates B and C. The first tag (.un) applies to the definition of the first predicate B and the second tag (.ac) to C. The first character of a tag defines how the predicate is to be defined. The character u means that the predicate will unconditionally get a value, whether the guarding predicate in this case) is true or false. If A is false, then B is set to false. Otherwise, A is true and the value of B depends upon the evaluation of a?c. The character a in the second tag (.ac) indicates that the full definition of the related predicate C is contingent on the value of A, the evaluation of a?c, AND the prior value of C. If A is false, the value of predicate C does not change. If A is true and C has previously been set false then C remains false. One exception is the unconditional definition of a predicate. This is discussed later in the section. Additionally, the second character of a tag defines whether the normal (n) result of the condition (a?c) or the complement (c) of the condition must be true to make the related predicate true. If A is true and C is true and !(a?c) is true then the new value of C will be true 2 . For a complete definition of cmpp statements see the Playdoh architecture specification [19]. In our implementation of PSSA, we introduce a new OR operation currently not defined by Trimaran. The predicate OR operation defines block predicates by taking the logical OR of multiple predicates. For example, consider an operation G = OR(A, B, C) if true (where A, B and C are predicates, each defining a unique path to G). If any one of them has the value of true, G will receive a value of true, otherwise G will be assigned false. When scheduling, we make the reasonable assumption that the definition of a predicate is available for use as a source for another operation, or as a guard to a subsequent cmpp operation in the cycle following its definition. When used as a guard for all other operations, the predicate definition is available for use in the same cycle as it is defined. We refer to all other operations, which do not define predicates, as normal operations. Normal operations include assignments, arithmetic operations, branches, and memory operations. 3 Motivation for Predicate-Sensitive Analysis A major task for the scheduler of a multi-issue machine is to find independent instructions. Unfortu- nately, predication introduces additional dependences that traditional code doesn't have to consider. In Figure 1(b), there is a dependence between the definition of the guarding predicate P2 and its use in the statement b=q if P2. Since predication combines multiple basic blocks, it introduces false dependences between disjoint paths. For example, in Figure 1(b), in the absence of predicate dependence information, we would infer a dependence between the definition of b in b=q if P2 and the use of b in d=b+3 if P3. However, these two statements are guarded by disjoint predicates. Therefore, only one of the predicates (P2 or P3) can possibly be true; only one of the statements will actually be committed and no dependence does in fact exist. Johnson et. al. [18] devised a scheme to determine the disjointness of predicates using the predicate partition graph. This analysis allowed more effective register allocation as live ranges across predicated code could be more accurately determined [15]. Their approach was limited to describing disjointness with restricted path information. Path information that extended across join points was not collected. In Conversely, if A is true and C is true and !(a?c) is false, then the new value of C will be false. Figure 2, the predicate partition graph would determine that the following pairs of predicates are disjoint: G and H, B and C, D and !D. However, no information regarding the relationship between D and G or D and H would be available. This "cross-join" information is needed to provide the scheduler full flexibility in scheduling statements such as y=t+r. If path information is not available, then y=t+r is guarded on true and the scheduler correctly assumes this statement is dependent on t=rand(), t=t-s, r=5+x, and r=x+8. However, since there are two possible definitions of each operand, there are 4 combinations of operands that could in fact cause the definition of y - each executable via one (or more) paths of execution through the region. If 4 versions of this statement could be made (one for each combination of the operands), then each could be scheduled at the minimum dependence length for that version. While disjointness information can maintain information regarding paths since the most recent join, we will need to combine path information across joins to remove unnecessarily conservative scheduling dependences. Figure 2(b) shows the cross-join path information that would be needed to guard each assignment of y so that the scheduler can know the precise dependences for each copy. This will allow the most flexibility in scheduling each statement. Although precise dependence information can be determined from guarding predicate relations, we will also show that renaming techniques can be of additional use to achieve greater scheduling flexibility. By renaming variables that have more than one definition in a region, we will maintain path information even after optimizations which change the guarding predicate of a statement have been applied. 4 Predicated Static Single Assignment (PSSA) Techniques such as renaming [4] and Static Single Assignment (SSA) [13, 12] have proved useful in eliminating false dependences in traditional code [31]. Removing false dependences allows more flexibility in scheduling since data independent operations can move past each other during instruction scheduling. In non-predicated code, SSA assigns each target of an assignment operation a unique variable. At join nodes a OE-function may need to be inserted if multiple definitions of a variable reach the join. The OE-functions determine which version of the variable to use and assign it to an additional renamed version. This new variable is used to represent the merging of the different variable names. Figure 3 shows the simple example from Figure 1 in SSA form. In the assignment b3-?OE(b1,b2), the variable b3 represents the reaching definition of b which is to be used after the join of definition b1 or b2. As discussed in section 3, eliminating false dependences is equally important and a more complex task for predicated code, since multiple control paths are merged. To address this problem we developed a t>r t=t-s (D) z=w+5 if true r=5+x if G B,C t>r if G B,C t>r if H L, w>2 if B t=t-s if D br out if L y=t+r if !D&G y=t+r if !D&H y=t+r if D&G y=t+r if D&H v=y+5 if true (a) Original Control Flow Graph (b) Predicated Hyperblock with Paths First join block Second join block Third join block Br out Figure 2: Code is duplicated when more than one definition reaches a use to maintain maximum flexibility for the scheduler. In (b), the statement y=t+r is duplicated for each pair of definitions that may reach this statement. Each copy is guarded by the predicates that defined the path along which those definitions would occur. predicate-sensitive implementation of SSA called Predicated Static Single Assignment (PSSA). PSSA seeks to accomplish the same objectives as SSA for a predicated hyperblock. First, it must assign each target of an assignment operation in the hyperblock a unique variable. Second, at points in the hyperblock where multiple paths come together it must summarize under what conditions each of the multiple definitions of a variable reaches that join. The second objective is accomplished through the creation of full-path predicates and path-sensitive analysis. Consider the sample predicated code shown in Figure 4 using traditional hyperblock predication [22]. b>a (a) Control Flow Graph (b) Code in SSA form if b1>a else d1=b1+3 Figure 3: Static Single Assignment Br out z=w+5 if true r=5+x if G B,C t>r if true L, w>2 if B t=t-s if D br out if L y=t+r if true v=y+5 if true (a) Original Control Flow Graph (b) Predicated Hyperblock t>r t=t-s (D) Figure 4: Extended example of transformation from non-predicated CFG to predicated hyperblock t2=t1-s F=true if true 1 AGF,AHF cmpp.un.uc z1>7 if F 3 r1=5+x if AGF 3 BAGF,CAGF cmpp.un.uc t1>r1 if AGF 5 BAHF,CAHF cmpp.un.uc t1>r2 if AHF 5 LBAGF,EBAGF cmpp.un.uc w1>2 if BAGF 6 LBAHF,EBAHF cmpp.un.uc w1>2 if BAHF 6 ECAGF, EDCAGF cmpp.un.uc t1>7 if CAGF 6 ECAHF, EDCAHF cmpp.un.uc t1>7 if CAHF 6 t2=t1-s if D 7 br out if L 7 (AGF) (AHF) t1= rand () Br out (BAGF) (CAGF) (BAHF) (CAHF) (AGF) (AHF) (D) (a) PSSA dependence graph (b) PSSA-transformed code Figure 5: The PSSA dependence graph shows the flow of data and control through the PSSA-transformed code. Blocks labeled with full-path predicates (indicated by multiple letters) contain statements that are only executed along that path. Blocks labeled with block predicates (single letters) contain statements that will be executed along several paths. In this predicated example, all branches have been replaced (except the one leaving the hyperblock) with predicate-defining operations using If-conversion. The predicates that are defined in this example correspond to the two edges exiting each conditional branch in the CFG in Figure 4. Figure 5 shows this example after PSSA has been applied and displays a graph showing the post-PSSA dependence relationships. The PSSA transformation has 2 phases: pre- and post-optimization. Hyperblocks are converted to PSSA form before optimization. After optimization, PSSA inserts clean-up code on edges leaving the hyperblock, copying renamed variables back to their original names and then removes any unused predicate definitions. 4.1 Converting to PSSA Form When converting to PSSA form, each operation is processed in turn beginning at the top of the hyperblock and proceeding to the end. Control PSSA is applied to predicate-defining operations, and Normal PSSA is applied to all other operations. We first describe Normal PSSA. If the operation is an assignment, the variable defined is renamed. The third operation in Figure 5(b), z1=w1+5, is an example. All operands are adjusted to reflect previously renamed variables (e.g. w becomes w1). If the operation is part of a join block, multiple versions of the operands may be live. The first operation (y=t+r) in the third join block of Figure 2(a) provides an example. Here, the operation will be duplicated for each path leading to the join and the correct operand versions for each path will be used in the duplicate statement as seen in Figure 5 (in the multiple definitions of y1). The duplicates are guarded by the full-path predicate (described below) associated with the path along which the operands are defined. Though there are 6 definitions of y1 (only 4 are unique), there is only one definition of y1 on any given path. These definitions are predicated on disjoint predicates; only one of them can possibly be true, and only one of them will be committed. We next describe Control PSSA. The single cmpp operation that defined one or two block predicates (such as the definitions of B and C in Figure 4) is replaced by one or more cmpp operations, each associated with a particular path leading to that block. As can be seen in Figure 5(b) there are now two cmpp operations: one defining BAGF and CAGF, and one defining BAHF and CAHF. These new predicates are called full-path predicates (FPPs). Each FPP definition has the appropriate operand versions for its path and each is guarded by the FPP that defined the path prior to reaching the new block. For example, the cmpp defining BAGF and CAGF is predicated on AGF. A FPP specifies the unique path along which an operation is valid for execution, enabling PSSA to provide correct guarding predicates for the duplicate statements previously described. In the example in Figure 2 we pointed out that the definitions of y1 needed guarding predicates that captured information about paths of execution. The first definition of y1 needed to be guarded by a predicate representing a path of execution through block G but not block D. In addition, the predicate needs to reflect that the execution actually reached the block of the statement in question (E in this case). Register y1 would be incorrectly modified if, for example, the branch out of the hyperblock is taken and block E is never reached. The new FPP EBAGF represents the precise conditions for correct execution. In addition to the cmpp statements added to define FPPs, cmpp statements are included to rename join blocks whose statements were originally predicated on true. A and E and their associated FPPs are examples. The operations in Figure 4(b) predicated on true, are predicated on F, A and E in the PSSA version of the code shown in Figure 5. This is necessary to maintain exact path information. Clearly, this has the potential to cause an exponential amount of code duplication. It might seem more reasonable to follow the example of SSA and insert OE-functions at join points to resolve multiple definitions. For example, an implementation of OE-functions resolving r and t in the definition of y1 could be: (1) r=r1 if G (3) t=t1 if true While this would have the advantage of decreasing duplication, it does not eliminate the need for predicate-sensitive analysis. Predicate relationship information is still needed to determine the reaching definitions and associated predicates, and to determine the order of the copy operations. For example, both of the statements (3) and (4) defining t in the previous sequence could be committed. The literal predicate true is always true, and predicate D could be true as well. For the use of t in (5) to get the correct definition, statement (4) cannot be executed before statement (3). Moreover, other side effects that degrade performance are introduced. Most important is that the insertion of OE-functions adds data dependences. For example, a true dependence is introduced between the definition of t1 and its use in (3). In addition, false dependences are re-introduced. An example is the output dependence between the two definitions of t. Thus, SSA and the usual OE-function implementation does not give the desired scheduling flexibility. Block predicates are also important to the PSSA transformation. PSSA uses predicate OR statements to redefine the block predicates as the union of the FPPs associated with the paths that reach the block. PSSA does not simply duplicate every path through the hyperblock. Duplication only occurs when necessary to remove false dependences. When there is only one version of all operands reaching a statement, only one version of the statement is required. This is the case with v1=y1+5 in Figure 5. The variable y1 is the only version live in node E. This statement is guarded by E, a block predicate created by taking the logical of EBAGF, EBAHF, ECAGF, ECAHF, EDCAGF, and EDCAHF. As long as control reaches node E, regardless of the path taken, we will execute and commit the statement v1=y1+5. 4.2 Post-Optimization Clean-up After optimization is applied to code in PSSA form, a clean-up phase is run to remove unnecessary code and to assure consistent code outside of the hyperblock. The PSSA implementation described in this paper generates cmpp statements for every path and block. These are entered into the PSSA data structure that maintains information about the relationships between the predicates they define, which provides maximum flexibility during optimization. However, some of these FPP definitions may not be used, and the corresponding cmpp operations will be discarded, reducing the code size significantly. Finally, to assure correct execution following the hyperblock, PSSA inserts copy operations assigning the original variable names to all renamed definitions that are live out of the hyperblock. These are placed on the appropriate exit of the hyperblock. For example, the exit branch guarded by L in Figure 4 would include live out of the hyperblock at this exit. 5 Hyperblock Scheduling Optimizations In this section, we describe how PSSA enables Predicated Speculation (PSpec) and Control Height Reduction (CHR) for aggressive instruction scheduling. PSpec allows operations to be executed before their guarding predicates are determined and CHR allows the guarding predicates to be determined as soon as possible, reducing the number of operations that need to be speculated. Used together with PSSA, we demonstrate that we can schedule the code at its earliest schedulable cycle, assuming a machine with unlimited resources. F=true if true 1 AGF,AHF cmpp.un.uc z1>7 if F 3 BAGF,CAGF cmpp.un.uc t1>r1 if AGF 4 BAHF,CAHF cmpp.un.uc t1>r2 if AHF 4 LBAGF,EBAGF cmpp.un.uc w1>2 if BAGF 5 LBAHF,EBAHF cmpp.un.uc w1>2 if BAHF 5 ECAGF, EDCAGF cmpp.un.uc t1>7 if CAGF 5 ECAHF, EDCAHF cmpp.un.uc t1>7 if CAHF 5 t2=t1-s if true 2 br out if LBAGF 5 br out if LBAHF 5 Figure Extended code example after PSpec optimization has been applied. Statements (other than first statement) predicated on true have been speculated. 5.1 Predicated Speculation This section describes how to perform speculation on PSSA-transformed code. In general, speculation is used to relieve constraints which control dependences place on scheduling. One can speculatively execute operations from the likely-taken path of a highly-predictable branch, by scheduling those operations before their controlling branch [20]. Similarly, Predicated Speculation (PSpec) will schedule a normal operation above the cmpp operation it is dependent upon, optimizing a hyperblock's execution time. PSpec handles placement of the speculated predicated operation in a uniform manner. PSpec schedules a normal operation at its earliest schedulable cycle. When speculating an operation, the operation is scheduled earlier than the operation it is control dependent on, and is predicated on true. We assume that any exceptions raised by the speculated operations will be taken care of using architecture features such as poison bits [7]. PSpec(normal op) f if (normal op.guarding predicate not defined by normal op.earliest schedulable cycle) f if (multiple defs of normal op.target exist f rename(normal op.target); update uses(normal op.target); normal op.schedule(earliest schedulable cycle); normal op.set predicate(true); else f normal op.schedule(earliest schedulable cycle); Figure 7: Basic PSpec Algorithm. 5.1.1 Instruction Scheduling with Speculation To demonstrate the usefulness of PSSA in enabling PSpec, Figure 6 shows the code from Figure 5 after the PSpec optimization has been applied. The assignments to r1 and r2 are examples of speculated operations. Notice that based on dependences, they could both be scheduled at cycle one which would have been impossible without renaming. During predicated speculation, each operation is considered sequentially, beginning with the first instruction in the hyperblock. If it is a normal, non-store operation, PSpec compares its earliest schedulable cycle with the cycle in which its guarding predicate is currently defined. If the operation can be scheduled earlier than its guarding predicate, the operation is predicated on true and scheduled at its earliest schedulable cycle. Recall that PSSA has not performed full renaming, so further renaming may be required by PSpec. An example is the definition of y1 in Figure 5. If we speculate any of the definitions of y1 by predicating them on true without renaming, incorrect code can result. Consequently, we must rename the operations being speculated. The results of applying this to the 6 definitions of y1 (now y1, y2, y3, y4, y5,and y6) appear in Figure 6. Speculation and renaming may require the duplication of operations using the definition being speculated, since there may now be multiple reaching definitions. When speculating y1, the operation v1=y1+5 had to be duplicated and guarded on the appropriate FPP (though in Figure 6 these statements are shown after they, too, have been speculated). This is possible because PSSA previously created all the necessary FPPs and path information. If the guarding predicate has been defined by the operation's earliest schedulable cycle, we do not apply PSpec. It is again scheduled at its earliest schedulable cycle, but guarded by the guarding predicate assigned by PSSA. The instruction z1=w1+5 is an example. The algorithm for PSpec instruction scheduling is shown in Figure 7. Using PSpec, the hyperblock can now be scheduled in 6 cycles as compared to 9 cycles in Figure 5. Since PSpec is applied whenever the definition of the operation's guarding predicate occurs later than the earliest schedulable cycle of the operation, we could reduce the number of operations that need to be speculated by moving the definition of the guarding predicates earlier. The goal of the next optimization, Control Height Reduction, is to allow predicates to be defined as early as possible. 5.1.2 Branches and Speculation We chose not to PSpec branches. Therefore, a branch statement's earliest schedulable cycle is the one in which its guarding predicate is known. However, if a branch has been predicated on its block predicate by PSSA (because it does not have multiple operand versions reaching it) then it may be unnecessarily delayed in scheduling by waiting for that block predicate to be computed. As shown in Figure 6, we may choose to duplicate this statement, much as we do in normal PSpec, and guard the execution of these duplicates on their respective FPPs, instead of predicating the single instruction on its block predicate. 5.2 Control Height Reduction Control Height Reduction (CHR) eases control constraints between multiple control statements. CHR allows successive control operations on the control path to be scheduled in the same cycle, effectively reducing control dependence height. For example, in the code in Figure 6, the control comparisons for z1?7 and t1?r1 are scheduled in cycles 3 and 4, respectively. However, the second comparison is only waiting for the definition of its guarding predicate AGF. To schedule it earlier, consider the PSSA dependence graph in Figure 5. The definition of BAGF (defined by the condition t1?r1), is control dependent on the definition of AGF (defined by the condition z1?7). We could define BAGF directly as the logical AND of the conditions z1?7 and t1?r1 removing the dependence on the definition of AGF. This AND expression could be scheduled in cycle 3. Control Height Reduction was proposed in [27]. It was successfully used to reduce the height of control recurrences found in loops when applied to superblocks. A superblock is a selected trace of basic blocks through the control flow graph containing only one path of control [26]. The path-defining aspects of PSSA F=true if true 1 AGF,A HF cmpp.un.uc z1>7 if F 3 BAGF, CAGF cmpp.an.an z1>7 if true 3 BAGF, CAGF cmpp.an.ac t1>r1 if true 3 BAHF, CAHF cmpp.ac.ac z1>7 if true 3 BAHF, CAHF cmpp.an.ac t1>r2 if true 3 LBAGF,EBAGF cmpp.an.an z1>7 if true 3 LBAGF,EBAGF cmpp.an.an t1>r1 if true 3 LBAGF,EBAGF cmpp.an.ac w1>2 if true 3 LBAHF,EBAHF cmpp.ac.ac z1>7 if true 3 LBAHF,EBAHF cmpp.an.an t1>r2 if true 3 LBAHF,EBAHF cmpp.an.ac w1>2 if true 3 ECAGF,EDCAGF cmpp.an.an z1>7 if true 3 ECAGF,EDCAGF cmpp.ac.ac t1>r1 if true 3 ECAGF, EDCAGF cmpp.an.ac t1>7 if true 3 ECAHF,EDCAHF cmpp.ac.ac z1>7 if true 3 ECAHF,EDCAHF cmpp.ac.ac t1>r2 if true 3 ECAHF, EDCAHF cmpp.an.ac t1>7 if true 3 t2=t1-s if true 2 br out if LBAGF 3 br out if LBAHF 3 Figure 8: Extended example after PSpec and CHR optimizations have been applied. Cmpp instructions displayed in italics define predicates that are not used after optimization. Therefore, the statements can be removed from the final code. allow our algorithm to effectively apply CHR to predicated hyperblocks, since the full-path predicates expose all of the original, separate paths throughout the hyperblock. Schlansker et. al. [28] recently expanded on their previous research, applying speculation prior to attempting height reduction. Speculation is needed to remove dependences between the branch conditions that need to be combined to accomplish the reduction. However, in that work, speculation was limited to operations that would not overwrite a live register or memory value if speculated, since they did not use renaming. In Figure 5, the cmpp operation defining BAGF and CAGF is shown scheduled at cycle 5 due to dependences on t1 and r1. PSSA allows us to apply PSpec and schedule these definitions in cycle 1, making the cmpp available for CHR as shown in Figure 8. 5.2.1 Instruction Scheduling with PSpec and CHR During instruction scheduling, PSpec is performed as described in Section 5.1.1. During the same sequential pass through instructions, for each control operation (cmpp), CHR is performed if possible. Recall that the operations in Figure 5 are scheduled in the order given in the PSSA hyperblock. Like PSpec, CHR compares an operation's earliest schedulable cycle with when it must be scheduled if it waited for its guarding predicate to be defined. If it does not need to wait on the definition of its guarding predicate, it is simply scheduled at its earliest schedulable cycle. Without Pspec, the definition of BAGF was waiting on the definition of t1 and r1. With Pspec, it is only waiting on the definition of its guarding predicate. Therefore, it is beneficial to control height reduce. By ANDing the condition of the current definition with the condition that defined its guarding predicate, we can schedule this definition earlier. If the definition of the guarding predicate involved conditions that were ANDed as well, all of the conditions must be included, so the number of cmpp statements needed to define the current operation increases. The .a tag on each of these cmpp statements indicates that all of them are required for the final definition. Consider the operations z1?7, t1?r1 and t1?7 in Figure 5. We control height reduce these operations in Figure 8, since they are all schedulable in cycle 3 based on our scheduling constraints. The definition of ECAGF now describes the combination of z1?7 being true AND t1?r1 having a value of false AND t1?7 having a value of true. We implement this logical AND using the .ac and .an qualifiers. The definition of ECAGF requires that the conditions z1?7 and t1?7 and the condition !(t1?r1) evaluate to true for the FPP to get a value of true. If any one of the requirements are not met, the FPP will be set to false. The compares can be performed in the same cycle [19], allowing multiple links in a control path to be defined simultaneously. The algorithm for CHR is found in Figure 9. f if (cmpp op.guarding pred defined by cmpp op.earliest schedulable cycle) f cmpp op.schedule(cmpp op.earliest schedulable cycle) f Apply Control Height Reduction */ else f while (more stmts defining(cmpp op.guarding pred)) f next def=next defining stmt(cmpp op.guarding pred) copy=duplicate(next def) copy.schedule(next def.get scheduling time()) copy.predicate on(next def.get guarding pred()) copy.set define(cmpp op.get pred defined()) copy.set tag to(a) cmpp op.schedule(next def.get scheduling time()) cmpp op.predicate on(next def.get guarding pred()) cmpp op.set tag to(a) Figure 9: Basic Control Height Reduction Algorithm. Using PSpec and CHR on PSSA-transformed code results in the 4 cycle schedule shown in Figure 8. Note that the operations shown in italics can be removed in a post-pass because these operations define predicates that are never used. Using predicated speculation and control height reduction together on PSSA-transformed code allows every operation to be scheduled at its earliest schedulable cycle. 6 Results We have implemented algorithms to perform PSSA, CHR and PSpec on hyperblocks in the Trimaran System (Version 2.00). We collect profile-based execution weights for operations in the codes and schedule operations with an assumed one-cycle latency in order to calculate execution time. Additionally, we conservatively assume that a load is dependent on all prior stores along a given path, and that a store is dependent on prior stores as well. We also ensure that all instructions along a path leading to a branch out of the hyperblock are executed prior to exiting the hyperblock. Figure shows normalized execution time when applying our optimizations for several Trimaran benchmarks: fib, mm, wc, fir, wave, nbradar (a Trimaran media benchmark), qsort, alvinn (from Percent Execution Time of Original 16-way Original 16-way Original infinite Optimized 16-way Optimized infinite Figure 10: Executed cycles normalized to the number of cycles to execute the original code produced by Trimaran for a 16 issue machine. SPECFP92), compress (from SPECINT95), and li (from SPECINT95). These codes are described in the Trimaran Benchmark Certification [2]. The original execution times are created from the default Trimaran settings, with the exception that the architecture issue rate is set to 16. Execution time is estimated by summing together the frequency of execution of each hyperblock multiplied by the number of cycles it takes to execute the hyperblock assuming a perfect memory system. Infinite results do not restrict the number of operations issued per cycle. 16-way results are obtained by dividing each cycle which has been scheduled with more than 16 operations into ceiling(total operations scheduled in cycle / 16) cycles. The results are normalized to the original schedule generated by Trimaran for a 16-issue machine and scheduled 16-way. The optimized results show the performance after applying PSSA, PSpec, and CHR. The results show that using PSSA with PSpec and CHR results in a significant reduction in executed cycles. Figure 11 shows the average number of operations executed per cycle for the configurations examined in Figure 10. In comparing the two graphs for the 16-way results, 3-4 times as many instructions are issued per cycle after applying PSSA, PSpec, and CHR, and this resulted in a reduction in execution time ranging from 12% to 68%. Since PSpec and CHR as applied to PSSA code have the effect of removing the restrictions of control dependence, the optimized infinite results provide a picture of "best case" instruction level parallelism. Inspection of the optimized infinite results of alvinn, compress, and li show that, given current hyperblock formation, peak IPC is somewhat limited. The renaming required by PSSA and PSpec also significantly increases register pressure. Trimaran's ISA (Playdoh) supports 4 register files: general purpose, floating point, branch, and predicate [2, 19]. Average instructions per cycle Original 16-way Original infinite Optimized 16-way Optimized infinite Figure Weighted average number of operations scheduled per cycle for hyperblocks when using PSSA with Predicated Speculation and Control Height Reduction. Note that several of the "Optimized infinite" results are greater than 16 - the issue width simulated in these experiments.10305070 Average Live Registers Original Optimized fib mm wc fir wave nbradar qsort compress alvinn li Figure 12: Weighted average register pressure in hyperblocks when using PSSA with Predicated Speculation and Control Height Reduction. Shown from left to right for each benchmark is the general purpose file, predicate file, branch file, and floating point file (zero utilization for some benchmarks). Percent Code Expansion Static Dynamic Figure 13: Static and Dynamic Code Expansion normalized to original code size. Dynamic code expansion indicates an increase in the working set size to be supported by the instruction cache. Figure 12 shows the average number of live registers for the original code and the optimized code using PSSA, PSpec and CHR. The average live register results are weighted by the frequency of hyperblock execution. For example, matrix multiply has on average 17 live general purpose registers in the original code, and 54 live general purpose registers after optimization. Though the increase in utilization of all these register files is notable, the weighted average utilization mostly still remains within the reported register file sizes (128 general purpose, 128 floating point, 8 branch, and 64 predicate) [3]. Additionally, PSSA combined with aggressive PSpec and CHR significantly increases code size - both static and dynamic. Aggressive and resource insensitive application of CHR and PSpec aims to reduce cycles required to schedule at the cost of duplicated code specialized for particular paths (in the case of PSpec) or duplicated code for faster computation of predicates (in the case of CHR). Figure 13 shows both the static and dynamic code expansion of the PSSA, PSpec, and CHR optimized code over the original code. We calculate static code expansion by comparing the number of static operations in the optimized code with the number of static operations in the original code. Dynamic code expansion is measured similarly, with the exception that each static operation is weighted by the number of times that it is executed (as calculated by Trimaran's profile-based region weights). This ``dynamic code expansion'' is intended to capture the run-time effect that the introduced duplicated code will have on the memory system. Dynamic code expansion indicates an increase in the working set size to be supported by the instruction cache. 7 Related Work Predicated execution presents challenges and prospects that researchers have addressed in a variety of ways. Mahlke et. al. [21] showed that predicated execution can be used to remove an average of 27% of the executed branches and 56% of the branch mispredictions. Tyson also found similar results and correlated the relationship between predication and branch prediction [29]. In an effort to relieve some of the difficulties related to applying compiler techniques to predicated code, Mahlke et. al. [22] defined the hyperblock as a single-entry, multiple-exit structure to help support effective predicated compilation. These hyperblocks are formed via selective If-conversion [5, 24] - a technique that replaces branches with predicate define instructions. The success of predicated execution can depend greatly on the region of the code selected to be included in the predicated hyperblock. August et. al. [9] relates the pitfalls and potentials of hyperblock formation heuristics that can be used to guide the inclusion or exclusion of paths in a hyperblock. Warter et. al. [30] explore the use of Reverse If-conversion for exposing scheduling opportunities in architectures lacking support for predicated execution as well as for re-forming hyperblocks to increase efficiency for predicated code [9, 30]. The challenges of doing data-flow and control-flow analysis on hyperblocks have also been addressed. Since hyperblocks include multiple paths of control in one block, traditional compiler techniques are often too conservative or inefficient when applied to them. Methods of predicate-sensitive analysis have been devised to make traditional optimization techniques more effective for predicated code [15, 18]. The work presented in [11] (and expanded upon in this work) extended the localized predicate-sensitive analysis presented in [15, 18] to complete path analysis through the hyperblock. Path-sensitive analysis has previously been found useful for traditional data-flow analysis [6, 10, 16]. We use this specialized path information to accomplish PSSA (a predicate-sensitive form of SSA [13, 12]) which enables Predicated Speculation and Control Height Reduction for hyperblocks that have previously been examined only in the presence of the single path of control found in superblocks [26, 27, 28]. Moon and Ebcioglu [23] have implemented selective scheduling algorithms, which can schedule operations at their earliest possible cycle for non-predicated code. Our work extends theirs for predicated code, by allowing earliest possible cycle scheduling using predicated renaming with full-path predicates. Implementing PSSA in IA-64 Implementing PSSA using the IA-64 ISA [3] would be straightforward with the exception of the predicate statement we introduced. We found this OR statement to be very useful in efficiently combining path information in order to eliminate unnecessary code expansion. If this instruction were not explicitly added to IA-64 then it could be implemented by transferring the predicate register file into a general register using the move from predicate instruction in IA-64. The general purpose masking instruction would then be used to mask all but the bits corresponding to the sources of the predicate OR instruction. A result of zero evaluates to false, and anything else evaluates to true. IA-64, unlike the Playdoh ISA, places limits on compare instructions. For example, conditions that are included in logical AND compare statements can only compare a variable to zero. Specifically, the statement LBAGF,EBAGF cmpp.an.an t1?r1 if true in Figure 8 would not be permitted. In implementing CHR, we would have to transform the prior expression into the following 2 statements (expressed in IA-64 9 Future Work When constructing a hyperblock schedule for a specific processor implementation, resource limits will mandate how many operations can be performed in each cycle. Architectural characteristics such as issue width, resource utilization, number of available predicate registers, and number of available rename registers all need to be considered when creating an architecture-specific schedule. The goal of a hyperblock scheduler is to reduce the execution-height while taking these architectural features into consideration. In this paper, our goal was to show that PSSA provided an efficient form of renaming and precise path information to allow all operations to be scheduled at their earliest schedulable cycle. We are currently examining different PSSA representations to reduce code duplication and the number of full-path predicates created. Since various control paths through a hyperblock may have different true data dependence heights, it may provide no advantage to speculate operations that are not on the critical path through the hyperblock. PSSA could concentrate on only the critical paths through the hyperblock, reducing code duplication. For non-critical paths, it may be advantageous in PSSA to implement OE-functions combining different variable names, instead of maintaining renamed variables for each full-path in the hyperblock. At a point in the hyperblock where all paths join, copy operations could be used to return renamed definitions to original names. Path definitions could then be restarted at this point. This would reduce the amount of duplication required for a given operation to use correctly renamed variables. Our future research concentrates on these issues and creating a more efficient implementation of PSSA. Conclusions This paper extended [11], where Predicated Static Single Assignment was first introduced. It motivated the need for renaming and for predicate analysis that extends across all paths of the hyperblock. It demonstrated how Predicated Static Single Assignment (PSSA), a predicate-sensitive implementation of SSA that implements renaming using full-path predicates, can be used to eliminate false dependences for predicated code. We showed the benefit of using PSSA to enable Predicated Speculation (PSpec) and Control Height Reduction (CHR) during scheduling. Predicated Speculation allows operations to be executed at their earliest schedulable cycle, even before their guarding predicates are determined. Control Height Reduction allows guarding predicates to be defined as soon as possible, reducing the amount of speculation needed. By maintaining information about each of the original control paths in a hyperblock, PSSA can provide information that allows precise placement of renamed and speculated code, and allows the correct, renamed values to be propagated to subsequent operations. The renaming used by PSSA allows more aggressive speculation, as overwriting live values is no longer a concern. In addition, PSSA supports Control Height Reduction along every control path using full-path predicates, reducing control dependence depth throughout the hyperblock. Our experiments show that PSSA is an effective tool for optimizing predicated code. We gave extended experiments that show using PSSA with PSpec and CHR results in a reduction in executed cycles ranging from 12% to 68% for a 16 issue machine. Acknowledgments We would like to thank the Compiler and Architecture Research Group at Hewlett Packard, University of Illinois' IMPACT Group, and New York University's ReaCT-ILP Group for providing Trimaran. We specifically appreciate the time and patience of Rodric Rabbah, Scott Mahlke, Vinod Kathail, and Richard Johnson in answering many questions regarding the Trimaran system. In addition, we would like to thank Scott Mahlke for providing useful comments on this paper. This work was supported in part by NSF CAREER grant No. CCR-9733278, a National Defense Science and Engineering Graduate Fellowship, a research grant from Intel Corporation, and equipment support from Hewlett Packard and Intel Corporation. --R Merced processor and IA-64 architecture Conversion of control dependence to data dependence. Improving data-flow analysis with path profiles Integrated predicated and speculative execution in the IMPACT EPIC architecture. The IMPACT EPIC 1.0 Architecture and Instruction Set reference manual. A framework for balancing control flow and predication. Efficient path profiling. Predicated static single assignment. An efficient method of computing static single assignment form. Efficiently computing static single assignment form and the control dependence graph. Compiling for the Cydra 5. Global predicate analysis and its application to register allocation. Path profile guided partial dead code elimation using predication. HP make EPIC disclosure. Analysis techniques for predicated code. HPL PlayDoh architecture specification: Version 1.0. The Multiflow Trace Scheduling compiler. Characterizing the impact of predicated execution on branch prediction. Effective compiler support for predicated execution using the hyperblock. Parallelizing nonnumerical code with selective scheduling and software pipelining. On Predicated Execution. The Cydra 5 departmental supercomputer. Critical path reduction for scalar programs. Height reduction of control recurrences for ILP processors. Control CPR: A branch height reduction optimization for EPIC architectures. The effects of predicated execution on branch prediction. Reverse if-conversion High Performance Compilers for Parallel Computing. --TR --CTR Fubo Zhang , Erik H. D'Hollander, Using Hammock Graphs to Structure Programs, IEEE Transactions on Software Engineering, v.30 n.4, p.231-245, April 2004 Mihai Budiu , Girish Venkataramani , Tiberiu Chelcea , Seth Copen Goldstein, Spatial computation, ACM SIGARCH Computer Architecture News, v.32 n.5, December 2004

renaming;static single assignment;instruction scheduling;predicated execution

608786

Achieving Scalable Locality with Time Skewing.

Microprocessor speed has been growing exponentially faster than memory system speed in the recent past. This paper explores the long term implications of this trend. We define scalable locality, which measures our ability to apply ever faster processors to increasingly large problems (just as scalable parallelism measures our ability to apply more numerous processors to larger problems). We provide an algorithm called time skewing that derives an execution order and storage mapping to produce any desired degree of locality, for certain programs that can be made to exhibit scalable locality. Our approach is unusual in that it derives the transformation from the algorithm's dataflow (a fundamental characteristic of the algorithm) instead of searching a space of transformations of the execution order and array layout used by the programmer (artifacts of the expression of the algorithm). We provide empirical results for data sets using L2 cache, main memory, and virtual memory.

Introduction The widening gap between processor speed and main memory speed has generated interest in compile-time optimizations to improve memory locality (the degree to which values are reused while still in cache [WL91]). A number of techniques have been developed to improve the locality of \scientic programs" (programs that use loops to traverse large arrays of data) [GJ88, WL91, Wol92, MCT96, Ros98]. These techniques have generally been successful in achieving good performance on modern architectures. However, the possibility that processors will continue to outpace memory systems raises the question of whether these techniques can be \scaled" to produce ever higher degrees of locality. We say a calculation exhibits scalable locality if its locality can be made to grow at least linearly with the problem size while using cache memory that grows less than linearly with problem size. In this article, we show that some calculations cannot exhibit scalable locality, while others can (typically these require tiling). We then discuss the use of compile-time optimizations to produce scalable locality. We identify a class of calculations for which existing techniques do not generally produce scalable locality, and give an algorithm for obtaining scalable locality for a subset of this class. Our techniques make use of value-based dependence relations [PW93, Won95, PW98], which provide information about the ow of values in individual array elements among the iterations of a calculation. We initially ignore issues of cache interference and of spatial locality, and return to address these issues later. We dene the balance of a calculation (or compute balance) as the ratio of operations performed to the total number of values involved in the calculation that are live at the start or end. This ratio measures the This is supported by NSF grant CCR-9808694 for (int for (int Figure 1: Three-Point Stencil in Single-Assignment Form initialize C to zero for (int for (int for (int Figure 2: Matrix Multiplication degree to which values are reused by a calculation, and thus plays a role in determining the locality of the running code. It is similar to McCalpin's denition of machine balance as the ratio of a processor's sustained oating point operation rate to a memory system's sustained rate of transferring oating point numbers [McC95]. In some cases, limits on the total numbers of operations performed and values produced place absolute limits on compute balance, and therefore on the locality that could be achieved if that code were run in isolation. For example, if the entire array A is live at the end of the loop nest shown in Figure 1, the balance of this nest is approximately 3 (N values are live at entry, and N T are live at exit, after 3 N T operations). If all the values produced had to be written to main memory, and all of the values that are live come from main memory, then we must generate one unit of memory tra-c (one oating point value read or written) for every three calculations performed. For other codes, such as matrix multiplication (Figure 2), the balance grows with the problem size. Thus, for large matrices, we may in principle achieve very high cache hit rates. Note that compute balance depends on information about what values are live. For example, if only A[T][*] is live at the end of Figure 1, then the balance of this code is 3T . This raises the hope that we can achieve scalable locality if we do not store the other values in main memory. Compute balance also depends on the scope of the calculation we are considering. If all elements of A are killed in a second loop nest that follows the code in Figure 1, and that nest produces only N values, the balance of these two nests could be higher than the balance of Figure 1 alone. Once again, this raises the hope of improving locality, this time by keeping values in cache between the two nests. One way to achieve locality proportional to compute balance would be to require a fully associative cache large enough to hold all the intermediate values generated during a calculation. Our denition of scalable locality explicitly rules out this approach. To achieve scalable locality, we must divide the calculation into an ordered sequence of \stripes" such that (a) executing the calculation stripe-by-stripe produces the same result, (b) each stripe has balance proportional to the problem size, (c) the calculations in each stripe can be executed in an order such that the number of temporary values that are simultaneously live is \small". A value is considered temporary if its lifetime is contained within the stripe, or if it is live on entry to the entire calculation but all uses are within the stripe. By \small", we mean to capture the idea of data that will t in cache, without referring to initialize C to zero for (int for (int for (int for (int for (int Figure 3: Tiled Matrix Multiplication (from [WL91]) for (int for (int for (int Figure 4: Time-Step Three Point Stencil any particular architecture. In particular, we wish to avoid cache requirements that grow linearly (or worse) with the size of the problem. In some cases, such as the simple stencil calculations discussed in [MW98] we can limit these sizes to functions of the machine balance; in other cases, such as the TOMCATV benchmark discussed in Section 4, the cache requirement grows sublinearly with the problem size (for TOMCATV it grows with the square root of the size of the input, as well as with the balance). Existing techniques can produce scalable locality on some codes. For example, tiling matrix multiplication produces scalable locality. Figure 3 shows the resulting code, for tile size s. In our terminology, the jb and kb loops enumerate s 2 stripes, each of which executes n tiles of size s 2 , and has a balance of 2ns 2 Within each stripe, the total number of temporaries that are live simultaneously does not exceed s (for one tile of B and one column of a tile of A and C). Thus, by increasing s to match the machine balance, we could achieve the appropriate locality using a cache of size O(s 2 ) (ignoring cache interference). However, there are calculations for which current techniques cannot produce scalable locality, as we will see in the next section. The remainder of this paper is devoted to a discussion of achieving scalable locality for a class of calculations we call time-step calculations. Section 2 denes this class of calculations, and shows how to produce scalable locality for a simple example via time skewing [MW98]. Section 3 generalizes time skewing beyond the limited class of problems discussed in [MW98]. Section 4 presents empirical studies on benchmark codes. Section 5 discusses other techniques for improving locality, and Section 6 gives conclusions. Calculations and Time Skewing We say that a calculation is a time-step calculation if it consists entirely of assignment statements surrounded by structured if's and loops (possibly while loops or loops with break statements), and all loop-carried value based ow dependences come from the previous iteration of the outer loop (which we call the time loop). For example, the three point stencil calculation in Figure 4 is a time-step calculation. It computes a new value of for (int for (int Figure 5: Three Point In-Place Stencil (from [WL91]) cur[i] from the values of cur[i-1.i+1] in the previous iteration of t. This ow of values is essentially the same as that shown in Figure 1. In contrast, the value computed in iteration [t; i] of the \in-place" stencil shown in Figure 5 is used in iteration [t; so we do not call this loop nest a time-step calculation. If only the values from the last time step are live after the end of the time loop, and all values that are live on entry to the calculation are read in the rst time step, the balance of a time-step calculation is proportional to the number of time steps. Thus, we may be able to achieve scalable locality for such calculations (by producing stripes that combine several time steps). The techniques presented by Wolf and Lam [WL91, Wol92] can be used to achieve scalable locality for Figure 5, but they cannot be applied to calculations with several loop nests, such as the time-step calculation in Figure 4. In [MW98], we describe the time skewing transformation, which can be used to achieve scalable locality for Figure 4. As originally formulated, this transformation could only be applied to time-step stencil calculations (in which each array element is updated using a combination of the element's neighbors), and only if the stencil has only one statement that performs a calculation (statements that simply move values, such as the second assignment in Figure 4, are allowed). In the next section, we describe a more general form of time skewing. The remainder of this section reviews the original formulation of time skewing, as it applies to Figure 4. The essential insight into understanding time skewing is that it applies a fairly conventional combination of skewing and tiling to the set of dependences that represent the ow of values (rather than memory aliasing). For example, in Figure 4, the value produced in iteration [t; i] of the calculation is used in iterations [t+1; i 1], we had a single loop nest with this dependence pattern, the algorithm of Wolf and Lam would skew the inner loop with respect to the time loop, producing a fully permutable nest. It would then tile this loop nest to achieve the appropriate degree of locality. We can perform this skewing and tiling if we rst expand the cur array and forward substitute the value of old. In the resulting code, there are O(B) values produced in each tile for consumption in the next tile (where B is the tile size). A stripe of N such tiles produces O(N) values while performing O(N B) operations on O(N B) temporaries, O(B) of which are live simultaneously. Thus, we may hope to achieve scalable locality. Unfortunately, expanding the cur array causes each temporary to be placed in a unique memory location, which does not yield an improvement in memory locality. To improve locality, we must either recompress the expanded array in a manner that is compatible with the new order of execution, or perform the skewing and tiling on the original imperfect loop nest (this requires a combination of unimodular and non-unimodular transformations). Both of these approaches are discussed in detail in [MW98]. In the code that results from the rst, all iterations except those on the borders between tiles do all their work with a single array that is small enough to t in cache. This array should reside entirely in cache during these iterations, allowing us to ignore issues of spatial locality and cache interference. Furthermore, this technique lets us optimize code that is originally presented in single-assignment like Figure 1. However, this approach taxes the code generation system that we use to its limits, even for single-statement stencils. This causes extremely long compile times and produces code with a great deal of additional integer math overhead due to the loop structure [SW98]. 3 A General Algorithm for Time Skewing In this section, we present a more general algorithm for time skewing time-step calculations, using as an example the code from the TOMCATV program of the SPEC95 benchmark set (shown in Figure 6). We begin by giving the domain of our algorithm and our techniques to coerce aberrant programs into this domain, and then present the algorithm itself. 3.1 The Domain of Our Algorithm As with any time-step calculation, all loop-carried data ow must come from the previous iteration of the time loop (in TOMCATV, the t loop). Note that the j and i loops of the second nest (the \determine maximum" nest) carry reduction dependences [Won95], and do not inhibit time skewing. The j loops in the fourth and sixth nests (the two dimensional loops under \solve tridiagonal") do carry data ow, so these at rst appear to prevent application of time skewing. However, we can proceed with the algorithm if we treat each column of each array as a single vector value, and do not attempt to block this dimension of the iteration space (this will have consequences for our cache requirements, as we shall see below). It may be possible to extend our algorithm to handle cases in which there is a loop that carries a dependence in only one direction along such a vector, but we have not investigated this possibility. In this case, the fourth nest carries information forward through the j dimension, and the sixth carries it backward along this dimension, which rules out skewing in this dimension. Our algorithm is restricted to the subset of time-step calculations that meet the following criteria. 3.1.1 A-ne control ow All loop steps must be known, and all loop bounds and all conditions tested in if statements must be a-ne functions of the outer loop indices and a set of symbolic constants. This makes it possible to describe the iteration spaces with a set of a-ne constraints on integer variables, which is necessary because we use the Omega Library [KMP + 95] to represent and transform these spaces. We do allow one exception to this rule, however: conditions controlling the execution of the outer loop need not be a-ne. Such conditions may occur due to breaks, while loops, or simply very complicated loop bounds. These are all handled in the same way, though we present our discussion in terms of break statements, since this is what occurs in TOMCATV. 3.1.2 Uniform loop depth and restricted intra-iteration data ow Every statement within the time loop must be nested within the same number of loops, and the ow of information within an iteration of the time loop must connect identical indices of all loops surrounding the denition and use. For example, consider the value produced by the last statement in the rst nest (\nd residuals"). The value produced in iteration [t; j; i] (and stored in ry(i,j)) is used in iteration [t; j; i] of the fourth nest. Note that the reference to ry(i,j-1) in this statement does not cause trouble because we have already given up on skewing in the j dimension. In some cases we may be able to convert programs into the proper form by simply reindexing the iteration space. For example, if the rst i loop ran from 1 to n-2 and produced ry(i+1,j), we could simply bump the i loop by 1. If the calculation involves nests of dierent depths, we add single-iteration loops around the shallower statements. The third loop nest of TOMCATV (the rst in the \solve tridiagonal" set) has only two dimensions, so we add an additional j loop from 2 to 2 around the i loop. We perform this reindexing by working backwards from the values that are live at the end of an iteration of the time loop (in this example, rxm(t) and rym(t), which are used later, and x(*) and y(*), which do // find residuals of iteration t do do 2. 2. 2. 2. // determine maximum values rxm, rym of residuals do do // solve tridiagonal systems (aa,dd,aa) in parallel, lu decomposition do do do do do do // add corrections of t iteration do do if Figure benchmark from SPEC95 are used in iteration t + 1). We tag the loops containing the writes that produce these values as \xed", and follow the data ow dependences back to their source iterations (for example, the data ow to iteration [t; j; i] of the write to y in the last nest comes from iteration [t; j; i] of the sixth nest when j < n 1, and [t; i] of the fth nest when We then adjust the iteration spaces of the loops we reach in this way, and \x" them. The sixth nest is simply xed, and the fth has a j loop from n-1 to n-1 wrapped around it (since iteration [t; n 1; i] reads this value). If we ever need to adjust a xed loop, the algorithm fails (at least in one dimension). We then follow the data ow from the statements we reached in this nest, and so on, until we have explored all data ow arcs that do not cross iterations of the time loop. Any loops that are not reached by this process are dead and may be omitted. 3.1.3 Finite inter-iteration data ow dependence distances For our code generation system to work, we must know the factor by which we will skew. This means that we must be able to put known (non-symbolic) upper and lower bounds on the dierence between inner loop indices for each time-loop-carried ow of values. For example, the rst loop nest of TOMCATV reads x(i+1,j), which was produced in iteration [t our upper bound on the dierence in the i dimension must be at least 1. We cannot apply our current algorithm to code with coupled dependences (e.g., if the rst nest read x(i+j,j)). 3.2 Time Skewing Consider a calculation from the domain described above, or a calculation such as TOMCATV for which certain dimensions are within this domain. All values used in an iteration of the time loop come from within a xed distance - l of the same iteration of loop l in the previous time step. Therefore, the ow of information does not interfere with tiling if we rst skew each loop by a factor of - l . This is the same observation we made for Figure 4 in Section 2. In fact, if we think if all the j dimensions of all seven arrays as a single 7 by N matrix value, the data ow for TOMCATV is identical to that of Figure 4. We therefore proceed to skew and tile the loops as described above. This requires that we fuse various loop nests that may not have the same size. Fortunately, this is relatively straightforward with the code generation system [KPR95] of the Omega Library. We simply need to provide a linear mapping from the old iteration spaces to the new. The library will then generate code to traverse these iteration spaces in lexicographical order. Given g loops l 1 that are within our domain, and e loops e that are not, we produce the iteration space t; (l where nn is the number of the nest in the original ordering, and B is the size of the tiles we wish to produce. This is the same as the formulation given in [MW98], with the addition of the constant levels and levels not within the domain. For TOMCATV, we transform the original set of iteration spaces from [ t; nn; where nn ranges from 1 to 8 (the initialization of rxm and rym counts as a nest of 0 loops). The resulting g outer loops traverse a set of stripes, each of which contains T B g iterations that run each statement through all the e loops. The g + e inner loops that constitute a tile perform O(B g E) operations on O(B g E) oating point values, where E is the total size of the \matrix" that constitutes the value produced by the e loops executing all for statements. All but O(B g 1 E) values are consumed by the next tile, providing stripes with O(B) balance, and, as long as E grows less than linearly with the size of the problem, the hope of scalable locality. (In TOMCATV, E is a set of seven arrays of size N , while the problem involves arrays of size N 2 , so our cache requirement will grow with the square root of the problem size). As at the end of Section 2, we are left with the question of how to store these values without either corrupting the result of the calculation (e.g. if we use the original storage layout) or writing temporaries to main memory in su-cient quantities to inhibit scalable locality (e.g. by fully expanding all arrays). In principle, if we know E, we could apply the layout algorithm given in [MW98], producing an array of temporaries that will t entirely in cache. It may even be possible to develop an algorithm to perform this operation with B and E as symbolic parameters. However, in the absence of major improvements to the implementation of the code generation systems of [KPR95] and [SW98], this method is impractical. Instead, we simply expand each array by a factor of two, and use t%2 as the subscript in this new dimension. This causes some temporaries to be written out to main memory, but the number is proportional to the number of non-temporary values created, not the number of operations performed, so this does not inhibit scalable locality. This method also forces us to contend with cache interference (which we simply ignore at this point, though we could presumably apply algorithms for reducing interference to the code we generate). Finally, our code may or may not traverse memory with unit stride. If possible, the arrays should be transposed so that a dimension corresponding to the innermost loop scans consecutive memory locations. 3.2.1 Break statements We apply our algorithm to a calculation involving a break that is guarded with a non-a-ne condition as follows: we create an array of boolean values representing the value of the condition in each iteration, and convert the statement into an expression that simply computes and saves this value. At the end of each iteration of the outer loop (which steps through blocks of iterations in the original time loop), we scan this array to determine if a break occurred in any time step in the time block we have just completed. If it has, we record the number of the iteration in which the break occurred, roll back the calculation to the beginning of the time block, and restart with the upper bound on the time loop set to the iteration of the break. We can preserve the data that are present at the end of each time block by using two arrays for the values that are live at the ends of time blocks (one for even blocks, and one for odd blocks). This can double the total memory usage, but will not aect the balance or locality of the calculation (except to the degree that it changes interference eects). If it is possible to determine that a break does not aect the correctness of the result, we can avoid the overhead of the above scheme by simply stopping the calculation at the end of the time block in which the break occurred. Unfortunately, we know of no way to determine the purpose of a break statement without input from the programmer (possibly in the form of machine-readable comments within the program itself). 4 Empirical Results REVIEWERS: At this time, I only have results for the TOMCATV benchmark running with virtual memory. For the nal version, I also expect to have results on one or more workstations, such as a Sun Ultra/60 and several SGI machines, and include a larger set of benchmarks. Based on experiences with stencil calculations, I expect that these machines will show either a small gain or a small loss in performance, but that getting peak performance will require manually hoisting some loop-invariant expressions (or getting a better complier. To verify the value of time skewing in compensating for extremely high machine balance, we tested it using the virtual memory of a Dell 200MHz Pentium system running Linux. This system has 64 M of main memory, 128K of L2 cache, and 300M of virtual memory paged to a swap partition on a SCSI disk. This test was designed to test the value of time skewing on a system with extremely high balance. We transformed the TOMCATV benchmark according to the algorithm given in the previous section, except for the break statement (this break is not taken during execution with the sample data). We also increased the array sizes from 513 by 513 to 1340 by 1340, to ensure that all seven arrays could not t into main memory (the seven arrays together use about 96 Megabytes). The original code required over 9 minutes per time step (completing a run with T=8 in 4500 seconds, and a run with T=12 in 6900 seconds). For the time skewed code, we increased T to 192 to allow for a su-ciently large block size. This code required under 20 seconds per time step (completing all 192 iterations in 3500 seconds). Thus, for long runs, performance was improved by more than a factor of 30. The loop nests produced by the time skewing transformation may be more complicated than the original loops, so the transformed code may be slower than the original for small problems. For example, when the original TOMCATV data set is used, the entire data set ts in main memory, but any tile size greater than 2 exceeds the size of the L2 cache. In this case, the time skewed code is slower by a factor of two. 5 Related Work Most current techniques for improving locality [GJ88, WL91, Wol92, MCT96] are based on the search for groups of references that may refer to the same cache line, assuming that each value is stored in the address used in the original (unoptimized) program. They then apply a sequence of transformations to try to bring together references to the same address. However, their transformation systems are not powerful enough to perform the time skewing transformation: the limits of the system used by Wolf and Lam are given in Section 2.7 of [Wol92]; McKinley, Carr, and Tseng did not apply loop skewing, on the grounds that Wolf and Lam did not nd it to be useful in practice. Thus, these transformation systems may all be limited by the bandwidth of the loops they are able to transform. For example, without the time loop, none of the inner loops in TOMCATV exhibits scalable locality. Thus, there are limits to the locality produced by any transformation of the body of the time loop. Recent work by Pugh and Rosser [Ros98] uses iteration space slicing to nd the set of calculations that are used in the production of a given element of an array. By ordering these calculations in terms of the nal array element produced, they achieve an eect that is similar to a combination of loop alignment and fusion. For example, they can produce a version of TOMCATV in which each time step performs a single scan through each array, rather than the ve dierent scans in the original code. However, their system transforms the body of the time loop, without reordering the iterations of the time loop itself, and is thus limited by the nite balance of the calculation in the loop body. Work on tolerating memory latency, such as that by [MLG92], complements work on bandwidth issues. Optimizations to hide latency cannot compensate for inadequate memory bandwidth, and bandwidth optimizations do not eliminate problems of latency. However, we see no reason why latency hiding optimizations cannot be used successfully in combination with time skewing. 6 Conclusions For some calculations, such as matrix multiplication, we can achieve scalable locality via well-understood transformations such as loop tiling. This means that we should be able to obtain good performance for these calculations on computers with extremely high machine balance, as long as we can increase the tile size to provide matching compute balance. However, current techniques for locality optimization cannot, in general, provide scalable locality for time-step calculations. The time skewing transformation described here can be used to produce scalable locality for many such calculations, though it increases the complexity of the loop bounds and subscript expressions. For systems with extremely high balance, locality issues dominate, and time skewing can provide signicant performance improvements. For example, we obtained a speedup of a factor of when running the TOMCATV benchmark with arrays that required virtual memory. --R Strategies for cache and local memory management by global program transformation. The Omega Library interface guide. Code generation for multiple mappings. Memory bandwidth and machine balance in current high performance computers. Improving data locality with loop transformations. Design and evaluation of a compiler algorithm for prefetching. Time skewing: A value-based approach to optimizing for memory locality An exact method for analysis of value-based array data dependences Code generation for memory mappings. A data locality optimizing algorithm. Improving Locality and Parallelism in Nested Loops. --TR --CTR Guohua Jin , John Mellor-Crummey, Experiences tuning SMG98: a semicoarsening multigrid benchmark based on the hypre library, Proceedings of the 16th international conference on Supercomputing, June 22-26, 2002, New York, New York, USA Armando Solar-Lezama , Gilad Arnold , Liviu Tancau , Rastislav Bodik , Vijay Saraswat , Sanjit Seshia, Sketching stencils, ACM SIGPLAN Notices, v.42 n.6, June 2007 Kristof Beyls , Erik H. D'Hollander, Intermediately executed code is the key to find refactorings that improve temporal data locality, Proceedings of the 3rd conference on Computing frontiers, May 03-05, 2006, Ischia, Italy Michelle Mills Strout , Larry Carter , Jeanne Ferrante , Barbara Kreaseck, Sparse Tiling for Stationary Iterative Methods, International Journal of High Performance Computing Applications, v.18 n.1, p.95-113, February 2004 Chen Ding , Maksim Orlovich, The Potential of Computation Regrouping for Improving Locality, Proceedings of the 2004 ACM/IEEE conference on Supercomputing, p.13, November 06-12, 2004 Zhiyuan Li , Yonghong Song, Automatic tiling of iterative stencil loops, ACM Transactions on Programming Languages and Systems (TOPLAS), v.26 n.6, p.975-1028, November 2004 Chen Ding , Ken Kennedy, Improving effective bandwidth through compiler enhancement of global cache reuse, Journal of Parallel and Distributed Computing, v.64 n.1, p.108-134, January 2004

machine balance;compute balance;memory locality;storage transformation;scalable locality

608853

Deterministic Built-in Pattern Generation for Sequential Circuits.

We present a new pattern generation approach for deterministic built-in self testing (BIST) of sequential circuits. Our approach is based on precomputed test sequences, and is especially suited to sequential circuits that contain a large number of flip-flops but relatively few controllable primary inputs. Such circuits, often encountered as embedded cores and as filters for digital signal processing, are difficult to test and require long test sequences. We show that statistical encoding of precomputed test sequences can be combined with low-cost pattern decoding to provide deterministic BIST with practical levels of overhead. Optimal Huffman codes and near-optimal Comma codes are especially useful for test set encoding. This approach exploits recent advances in automatic test pattern generation for sequential circuits and, unlike other BIST schemes, does not require access to a gate-level model of the circuit under test. It can be easily automated and integrated with design automation tools. Experimental results for the ISCAS 89 benchmark circuits show that the proposed method provides higher fault coverage than pseudorandom testing with shorter test application time and low to moderate hardware overhead.

Table 1 illustrates the Huffman code for an example test set TD with four unique patterns out of a total of eighty. Column 1 of Table 1 lists the four patterns, lists the corresponding number of occurrences fi of each pattern Xi , and column 3 lists the corresponding probability of occurrence pi , given by fi =jTDj. Finally, column 4 gives the corresponding Huffman code for each unique pattern. Note that the most common pattern X1 is encoded with a single 0 bit; that is e.X1/ D 0, where e.X1/ is the codeword for X1. Since no codeword appears as a prefix of a longer codeword (the prefix-free property), if a sequence of encoded test vectors is treated as a serial bit-stream, decoding can be done as soon as the last bit of a codeword is read. This property is essential since variable-length codewords cannot be read from memory as words in the usual fashion. The Huffman code illustrated in Table 1 can be con- structedbygeneratingabinarytree(Huffmantree)with Table 1. Test set encoding for a simple example test sequence of test patterns. Unique Occur- Probability Huffman Comma patterns rences of occurrence codeword codeword 100 Iyengar, Chakrabarty and Murray Fig. 2. An example illustrating the construction of the Huffman code. edges labeled either 0 or 1 as illustrated in Fig. 2. Each unique pattern Xi of Table 1 is associated with a (leaf) node of the tree, which initially consists only of these unmarked nodes. The Huffman coding procedure iteratively selects two nodes vi and vj with the lowest probabilities of occurrence, marks them, and generates a parent node vij for vi and vj . If these two nodes are not unique, then the procedure arbitrarily chooses two nodes with the lowest probabilities. The edges (vij;vi) and (vij;vj) are labeled 0 and 1. The 0 and 1 labels are chosen arbitrarily, and do not affect the amount of compression [18]. The node vij is assigned a probability of occurrence pij D pi C pj. This process is continued until there is only one unmarked node left in the tree. Each codeword e.Xi / is obtained by traversing the path from the root of the Huffman tree to the corresponding leaf node vi . The sequence of 0-1 values on the edges of this path provides the e.Xi /. The Huffman coding procedure has a worst case complexity of O.m2 log m/, thus the encoding can be donein reasonable time. The average number of bits per patternPlH (average length of a codeword) is given by lH D imD1 wi pi , where wi is the length of the code-word corresponding to test pattern Xi . The average length of a codeword in our example is therefore given by lH D 1 0:5625 C 2 0:1875 C 3 0:1875 C 3 0:0625 D 1:68 bits. We next compare Huffman coding with equal-length coding. Let lH .lE/ be the average length of a code-word for Huffman coding (equal-length coding). Since Huffman coding is optimal, it is clear that lH lE.We next show that lH D lE under certain conditions. Theorem 1. If all unique patterns have the same probability of occurrence and the number of unique patterns m is a power of 2; then lH D lE . Proof: If all the unique patterns in TD have the same probability of occurrence p D 1=m, the entropy H.TD/ D imD1 pi log2 pi D log2 m bits. For equal-length encoding, lE Ddlog2 me, and if m is a power of then lE D log2 m, which equals the entropy bound. Therefore, l DlE for this case. The above theorem can be restated in a more general form in terms of the structure of the Huffman tree. Theorem 2. If the Huffman tree is a full binary tree; then lH D lE . Proof: A full binary tree with k levels has 2k 1 vertices, out of which 2k1 are leaf vertices. Therefore if the Huffman tree is a full binary tree with m leaf vertices, then m must be a power of 2 and the number of levels must be log m C1. It follows that every pathfrom the root to a leaf vertex is then of length log m. If all unique patterns in TD have the same probability of occurrence and m is a power of 2, then the Huffman tree is indeed a full binary tree; Theorem 2 therefore implies Theorem 1. Note that Theorem 2 is sufficient but not necessary for lH to equal lE. Figure 3 shows a Huffman tree for which lH D lE D 2, even though it is not a full binary tree. The practical implication of Theorem 1 is that Huffman encoding will be less useful when the probabilities of all of the unique test patterns are similar. This tends to happen, for instance, when the ratio of the number of flip-flops to the number of primary inputs in the CUT, denoted in Section 4, is low. Theorem 2 suggeststhatevenwhen ishigh, theprobabilitydistri- bution of the unique test patterns should be analyzed to Fig. 3. An example of a non-full binary tree with l D lE. determine if statistical encoding is worthwhile. How- ever, in all cases we analyzed where was high, statistical encoding was indeed effective. Comma Codes Although Huffman codes provide optimal test set com- pression, they do not always yield the lowest-cost decoder circuit. Therefore, we also employ a non-optimal code, namely the Comma code, which often leads to more efficient decoder circuits. The Comma code, also prefix-free, derives its name from the fact that it contains a terminating symbol, e.g. 0, at the end of each codeword. The Comma encoding procedure first sorts the unique patterns in decreasing order of probability of occurrence, and encodes the first pattern (i.e., the most probable pattern) with a 0, the second with a 10, the third with a 110, and so on. The procedure encodes each pattern by addinga1tothebeginning of the previous codeword. The codeword for the ith unique pattern Xi is thus given by a sequence of .i 1/ 1s followed by a 0. Comma codewords for the unique patterns in the example test set of Section 2.1 are listed in Column 5 of Table 1. This procedure has complexity O.m log m/and is simpler than the Huffman encoding procedure. The Comma code also requires a substantially simpler decoder DC than the Huffman code. Since each Comma codeword is essentially a sequence of 1s followed by a zero, the decoder only needs to maintain a count of the number of 1s received before a 0 signi- fies the end of a codeword. The 1s count can then be mapped to the corresponding test pattern. For a given test sequence TD with m unique patterns having probabilities of occurrence p1 p2 pm, the aPverage length of a Comma codeword is given by lC D imD1 ipi. Since the code is non-optimal, lC lH. However, the Comma code provides near-optimal compression, i.e., limm!1.lC lH/ D 0, if TD satisfies certain properties. These hold for typical test sequences that have a large number of repeated patterns. We first present the condition under which Comma codes are near-optimal and then the property of TD required to satisfy the condition. A binary tree with leaf nodes X1; X2;:::;Xm is skewed if the distance di of Xi from the root is given by di D (1) Deterministic Built-in Pattern Generation 101 For instance, the Huffman tree of Fig. 2 is a skewed binary tree with four leaf nodes. Theorem 3. Let p1 p2 pm be the probabilities of occurrence of the m unique patterns in TD. Let lH and lC be the average length of the codewords for Huffman and Comma codes; respectively. If the Huffman tree for TD is skewed then lC lH D pm and limm!1.lC lH / D 0. Proof:PIf the Huffman tree for TD is skePwed then lH D imD11 ipi C .m 1/pm and lC D imD1 ipi. Therefore, lC lH D pm. We also know that p1 p2 pm. Therefore, 0 pm 1=m, which implies that limm!1 pm D 0. Hence lC lH is vanishingly small for a skewed Huffman tree and the Comma code is near-optimal. Next we derive a necessary and sufficient condition that TD must satisfy in order for its Huffman tree to be skewed. Theorem 4. Let p1 p2 pm be the probabilities of occurrence of the unique patterns in TD. The Huffman tree for TD is skewed if and only if; for the probabilities of occurrence satisfy the condition Xm Proof: We prove sufficiency of the theorem. The necessity can be proven similarly. Generate the Huffman tree for the m patterns X1; X2;:::;Xm in TD whose probabilities of occurrence satisfy (2). Let the leaf node corresponding to the ith pattern be vi . The two leaf nodes vm and vm1 corresponding to the patterns Xm and Xm1 with the lowest probabilities pm and pm1 are first selected, and a parent node vm.m1/, with the probability .pm C pm1/, is generatePd for them. Now, pm3 pm2, andfrom(2), pm3 mkDm1 pk.Thus, pm3 pm1 C pm. Therefore, the leaf node vm2 and vm.m1/ are now the two nodes with the lowest probabilities, and a parent vm.m1/.m2/ with probability .pm C pm1 C pm2/ is generated for them. Similarly, a parent vm.m1/i is generated for nodes vi and vm.m1/.iC1/;i 2f.m3/1g. The process terminates when the root vm.m1/1 is generated for leaf node v1 and vm.m1/2. The distance d1 of v1 to the root is therefore 1. Similarly di D i, i 2f2;3;:::;m1g. Leaf nodes vm and vm1 are equidistant from the root, Iyengar, Chakrabarty and Murray since they share a common parent vm.m1/, thus dm D m 1. Therefore, di satisfies (1) for i 2f1;2;:::;mg, and the Huffman tree is skewed. We next determine the relationship between jTDj and m, the number of unique patterns in the test set when the Huffman tree is skewed. We show that jTDj must be exponential in m for the Huffman tree to be skewed. This property is often satisfied by deterministic test sets for sequential circuits with a large number of flip-flops but few primary inputs. Theorem 5. Let jTDj and m be the total number of patterns and the number of unique patterns in TD; res- pectively. If the Huffman tree for TD is skewed then denotes an asymptotic lower bound in the sense that f .m/ D !.g.m// implies limm!1 gf.m/ D1. Proof: Let f1 f2 fm be the numbers of Poccurrence of the unique patterns in TD. Then jTDjD imD1 fi. We know that fm1 fm 1, and fm2 fm 1. From (2), fm3 fm1 C fm 2. Sim- ilarly, fm4 3 and fm5 5. The lower bounds on thus form the FibPonacci series 3; 5;:::I therefore, jTDj1C imD1si, where si is the ith Fibonacci termp, given by si D p1p.'iC1 'O iC1/; where ' D 1 .1 C 5/ and 'O D 1 .1 5 5/ [21]. 'Om). For even m,1 from which it follows that jTDjD!.1:62m/. The proof for odd m is similar. Comma codes, being non-optimal, do not always yield better compression than equal-length codes. The following theorem establishes a sufficient condition under which Comma codes perform worse than equal-length codes. Theorem 6. Let p1 p2 pm be the probabilities of occurrence of the unique patterns in TD. 2dlog me If pm > m.mC2 1/ ; then lC > lE ; where lC .lE / is the average codeword length for Comma .equal-length/ coding. Proof: We know that lE Ddlog2 me and lC D p1 C 2p2 CCmpm. Since p1 p2 pm,lC if 12 pmm.m C 1/>dlog2 me, from which the theorem follows. For example, in the test set for s35932 obtained using Gentest, m D 86 and pm D 0:012, while 2dlog me=86.87/ D 0:0019. Hence, Comma codingperforms worse than equal-length coding for this test set. Note that Theorem 6 not provide a necessary condition for which lC > lE. In fact, it is easy to construct data sets for which lC > lE even though pm 2dlog me . The following theorem provides a tighter con- m.mC1/ dition under which Comma codes perform worse than equal-length codes. Theorem 7. Let p1 p2 pm be the probabilities of occurrence of the unique patterns in TD; and let lC .lE / be the average codeword length for Comma coding .equal-length coding/. Let fi D mini f ppiCi 1 me Proof: We first note that pi fipiC1 Pfi2 piC2 fimi pm,1i m. Since lC D imD1 ipi,it follows that lC fim1 pm C 2fim2 pm C 3fim3 pm C C.m1/fipm C mpm. Let E be defined as such that lC E. From (4), we get From (4) and (5), we obtain me fim1fi.mC21/Cm and the theorem follows. If m 1 then (3) can be simplified to pm fi.fim1/2mdlo.fig2m1e/ . In addition, if mini f ppiCi 1 g exceeds 2, i.e. Table 2. Huffman and Comma code words for the patterns in the test set of s444. Test Occur- Probability of Huffman Comma pattern rences occurrence codeword codeword every data pattern occurs twice as often the next most-frequent data pattern, then we can replace fi by2in (3) to obtain the following simpler sufficient condition under which Comma codes perform worse than equal-length codes. Corollary 1. Let p1 p2 pm be the probabilities of occurence of the unique patterns in TD; and let lC .lE / be the average codeword length for Comma coding .equal-length coding/.IffiDmini f ppiCi 1 g > 2 dlog me and pm > 2m12m2 ; then lC > lE . However, the skewing probability distribution property of Theorem 4 appears to be easy to satisfy in most cases. The probabilities of occurrence of patterns for a typical case (the s444 test set) are shown in Table 2 in Section 3. The decrease in compression resulting from the use of Comma codes, instead of optimal (Huffman) codes to compress such test sets, which is given by Deterministic Built-in Pattern Generation 103 lC lH D pm from Theorem 3, is extremely small in practice. For the s444 test set, pm D 0:0005;lH D 1:2121, and therefore lC D 1:2126, and the compression loss is only one bit. Therefore, both Huffman and Comma codes can efficiently encode sequential circuit test sets. 3. TGC Design In this section, we illustrate our methods for constructing employing statistical encoding of precomputed test sequences. We illustrate the steps involved in encoding and decoding with the test set for the s444 benchmark circuit as an example. Huffman Coding The first step in the encoding process is to identify the unique patterns in the test set. A codeword is then developed for each unique pattern using the Huffman code construction method outlined in Section 2. The Huffman tree used to construct codewords for the patterns of s444 is shown in Fig. 4. The unique test patterns and the corresponding codewords for s444 are listed in Table 2. The original (unencoded) test set TD, which contains 1881 test patterns of 3 bits each, requires bits of memory for storage. On the other hand, the encoded test set has only 1.2121 bits per codeword, and hence requires only 2280 bits of memory. Therefore, Huffman encoding of TD leads to 59.59% saving in storage, while both the order as well as the contiguity of test patterns are preserved. Once the encoded test set TE is determined by applying the Huffman encoding procedure to TD,itis Fig. 4. Huffman tree for the test set of s444. 104 Iyengar, Chakrabarty and Murray Fig. 5. Illustration of the proposed test application technique. stored on-chip and read out one bit at a time during test application. The sequence generator SG of Fig. 1 is therefore a ROM that stores TE. The test patterns in TD can be obtained by decoding using a simple finite-state machine (FSM) [20]. Table-lookup based methods that are typically used for software implementations of Huffman decoding are inefficient for on-chip, hardware-implemented decoding. The decoder DC is therefore a sequential circuit, unlike for combinational and full-scan circuits where a combinational decoder can be used [12, 22]. Figure 5 outlines the proposed test application scheme. We exploit the prefix-free property of the Huffman code; thus patterns can be decoded immediately as the bits in the compressed data stream are encountered. We next describe the state diagram of the FSM decoder DC using the s444 example. Figure 6 shows the state transition diagram of DC. The number of states is equal to the number of non-leaf nodes in the corresponding Huffman tree. For example, the Huffman tree of Fig. 4 has seven non-leaf nodes, hence the corresponding FSM of Fig. 6 has seven states-S1; S2;:::;S7. The FSM receives a single-bit input from SG, and produces n-bit-wide test patterns, as well as a single-bit control output TEST VEC. The control output is enabled only when a valid test pattern for the CUT is generated by the decoder-this happens whenever a transition is made to state S1. The use of the TEST VEC signal ensures that the test sequence TD is preserved and no additional test patterns are applied to the CUT. Hence Huffman codes provide an efficient encoding of the test patterns Fig. 6. State transition diagram for the FSM decoder of s444. and a straightforward decoding procedure can then be used during test application. The trade-off involved is the increase in test application time t since the decoder examines only one bit of in each clock cycle. Fortunately, the increase in t is directly related to the amount of test set compression achieved-the higher the degree of compression, the lesser is the impact on t. Theorem 8. The test application time t increases by a lH is the average length of a Huffman codeword. Proof: The state transition diagram of Fig. 6 shows that wi clock cycles are required to apply a test pattern Xi which is mapped to a codeword of wi bits. Hence the test application time (number of clock cycles) is given by t D iD1 wi , where jTDj is the total number of patterns in the test set TD. TPhejTtDejst application time therefore increases by a factor iD1 wi =jTDjDlH, the average length of a codeword. Experimental results on test set compression in Section 4 show that the average length of a Huffman codeword for typical test sets is less than 2. This implies that the increase in test application time rarely exceeds 100%. Since test patterns are applied in a BIST environment, this increase in testing time is acceptable, and it has little impact on testing cost or test quality. Figure 7 shows the netlist of the decoder circuit for s444. This circuit was generated for a test set obtained using Gentest. The design is simplified considerably by the presence of a large number of don't-cares in the decoder specification, which a design automation tool can exploit for optimization. The cost of the on-chip decoder DC can be reduced by noting that it is possible to share the same decoder on a chip among multiple CUTs. The encoding problem is now reformulated to encode the test sets of the CUTs together. We do this by combining these test sets to obtain a composite test set T 0 and applying the encoding procedure to T 0 to obtain an encoded test set Figure 8 illustrates a single sequence generator SG0 and pattern decoder DC0 used to apply test sets to multiple CUTs that have the same number of primary inputs. Note that such sharing of the pattern decoder is also possible if the CUTs have an unequal number of primary inputs. The sharing is, however, more effi- cient if the difference in the number of primary inputs is small. The slight increase in the size of T 0 (com- pared to TE) is offset by the hardware saving obtained Deterministic Built-in Pattern Generation 105 Fig. 7. Gate-level netlist of the FSM decoder for s444. by decoder sharing. We next present upper and lower bounds on the Huffman codeword length for two test sets that are encoded jointly. Theorem 9. Let TD1 and TD2 be test sets for two CUTs with the same number of primary inputs and let TD0 be obtained by combining TD1 and TD2. Let m1; N1;l1; and m2; N2;l2 be the number of unique pat- terns; total number of patterns; and average Huffman codeword length for TD1 and TD2; respectively. Let l0 be the average Huffman codeword length for T 0 and let m0 and N0 be defined as: m0 D maxfm1; m2g and In addition; let pi .qi /; 1 i m0; be the probability of occurrence of the ith unique pattern Fig. 8. A BIST sequence generator and decoder circuit used to test multiple CUTs. 106 Iyengar, Chakrabarty and Murray in TD1.TD2/. Then log flmin 1 l N0 log2 fimax C where .i/fimax is the largest value of fi such that N1 pi C is the smallest value of fl such that N1 pi C N2qi N0flpi and N1 pi C N2qi Proof:hskip10ptWe use the fact that H.TD1/ l1 H.TD1/ and H.TD2/ are the entropies of TD1 and TD2, respec- tively. The probability of occurrence of the ith unique pattern in TD0 is .N1 pi C N2qi /=N0. The entropy of TD0 is therefore given by H.T / D log It follows from the theorem statement that Therefore, N0 .l1 log2 fimax/ C N0 .l2 log2 fimax/ D log fimax Therefore, l0 [.N1l1 C N2l2/= Next we prove the lower bound. Once again from the theorem statement, flmin Note that the lower bound is meaningful only if pi D requiresthat TD1 and TD2 have the same set of unique patterns and therefore m0 D m1 D m2. Therefore, Now, H.TD1/ l11 and H.TD2/ l21. Therefore, l0 H.TD0 / [.N1l1 C N2l2/=N0] log2 flmin 1. A tighter lower bound on l0 is given by the following corollary to Theorem 9. Corollary 2. Let l1 D H.TD1/ C -1 and l2 D H.TD2/ C -2; and let flmin be defined as in Theorem 9; where As a special case, if N1 D N2 then 12 .l1 Cl2/ log2 For example, let TD1 and TD2 be test sets for two different CUTs with five primary inputs each. Suppose they contain the unique patterns shown in Fig. 9, with N1 D N2. The probabilities of occurrencePof patterns in TD1 and TD2 satisfy (2), therefore l1 D i4D1 ipi C 4p5 D 1:25. Similarly, l2 D 1:31. From Theorem 9, fimax D 0:58, and flmin D 3:5: Since N1 D N2, the bounds on l0 are given by 12 .l1 C l2/ log2 flmin 1 Therefore, 1:03 Fig. 9. Unique patterns and their probabilities of occurrence for the example illustrating Theorem 9. l0 3:55. Now, the patterns in T 0 also satisfy (2), and therefore l0 D 4 ip0 C4p0 D 1:33, which clearly lies between the calculated bounds, where p0 is the probability of occurrence of pattern Xi in TD0 . Experimental results on test set encoding and decoder overhead in Section 4 show that it is indeed possible to achieve high levels of compression while reducing decoder overhead significantly if test sets for two different CUTs with the same number of primary inputs are jointly encoded and a single decoder DC0 is shared among them. Comma Coding We next describe test set compression and test application using Comma encoding. Once again, we illustrate the encoding and decoding scheme using the s444 example. The unique patterns in the test set are first identified, and sorted in decreasing order of probability of occur- rence. Codewords are then generated for the patterns according to the Comma code construction procedure described in Section 2. Comma codewords generated for the unique patterns in the s444 test set are listed in Table 2. The probabilities of occurrence of test patterns, shown in Table 2 clearly satisfy (2) in Theorem 3, and therefore the encoding is near-optimal. The Comma encoded test set has 1.2126 bits per codeword, and requires 2281 bits for storage, an increase of only one bit from the optimally (Huffman) encoded test set described in Section 3. Hence the reduction in test set compression arising from the use of Comma codes instead of Huffman codes for this example is only 0.02%. The slight decrease in test set compression due to the use of the Comma code is offset by the reduced complexity of the pattern decoder DC. Figure 10 illustrates the pattern decoder for the s444 circuit test set. The decoder is constructed using a binary counter and combinational logic that maps the counter states to the test patterns. The test application scheme is the same as that in Fig. 5 for the Huffman decoder. The inverted input bit is used to generate the TEST VEC signal which ensures that the CUT is clocked only when a 0 is received. TEST VEC is also gated with the clock to the CUT and used to reset the the counter on the falling edge of the clock. Bits with value 1 received from SG therefore result in the flip- flops of the counter being clocked to the next state, while 0s (the terminating commas present at the end Deterministic Built-in Pattern Generation 107 Fig. 10. The Comma pattern decoder for the s444 test set. state after half a clock cyle. The test pattern can thus be latched by the CUT before the counter is reset. Comma decoders are simpler to implement than Huffman de- coders, and binary counters already present for normal operation can be used to reduce overhead. As in the case of Huffman coding (Theorem 8), the increase in testing time due to Comma coding equals the average length of a codeword. Run-Length Encoding Finally, we describe run-length encoding of the statistically encoded test sequence TE to achieve further compression. We exploit the fact that sequences of identical test patterns (runs) are common in test sets for sequential circuits having a high ratio of flip-flops to primary inputs. For example in the test set for s444, runs of the pattern 000 occur with lengths of up to 70. Huffman and Comma encoding exploit the large number of repetitions of patterns in the test sequence without directly making use of the fact that there are many contiguous, identical patterns. Run-length encoding exploits this property of the test sequences-it therefore complements statistical encoding. Huffman and comma encoding transform the sequence of test patterns to a compressed serial bit stream, and in the test set, each occurrence of the test pattern 000 is replaced bya0(Table 2). Therefore, long runs of 0s are present in the statistically compressed bit stream, 108 Iyengar, Chakrabarty and Murray Table 3. Distribution of runs in the Huffman encoded test set for the s444 circuit. No. of runs No. of runs Run- Run-length 0s 1s length 0s 1s which can be further compressed using run-length coding. Run-length coding is a data compression technique that replaces a sequence of identical symbols with a copy of the repeating symbol and the length of the se- quence. For example, a run of 5 0s (00000) can be encoded as (0,5) or (0,101). Run-length encoding has been used recently to reduce the time to download test sets to ATE across a network [23, 24]. We improve upon the basic run-length encoding scheme by considering only those runs that have a substantial probability of occurrence in the statistically encoded bit stream. A unique symbol representing a run of a particular length (and the corresponding bit) is then stored. The value of the repeating bit is generated from the bits representing the length of the run during decoding. We therefore obviate the need to store a copy of the repeating bit. We describe our run-length encoding process using the s444 example. An analysis of its Huffman encoded test set yields the distribution of runs shown in Table 3. Encoding all runs would obviously be expensive (4 bits would be required for each run) since very few instances of (0,4), (0,5) and (1,3), and no instances of (1,5), (1,6), (1,7) or (1,8) exist. We therefore use combinations of 3 bits (000; 001;:::;111) to encode the 8 most frequently occurring runs-(0,1), (0,2), (0,3), (0,7), (0,8), (1,1), (1,2) and (1,4). The less- frequently occurring runs (0,4), (0,5), (0,6) and (1,3) are divided into smaller consecutive runs for encoding. For example (0,5) is encoded as (0,3) followed by (0,2). Figure encoding applied to a portion of the Huffman encoded s444 test set. The encoded runs are stored in a ROM and output to a run-length decoder. The run-length decoder provides a single bit in every clock cycle to the Huffman (or Comma) decoder for test application. The run-length decoder consists of a binary down counter, and a small amount of combinational logic. Figure 12 illustrates the run-length decoder for the test set. The bits used to encode a run (e.g., 011 for (0,7)-Fig. 11(a)) are first mapped to the run-length bits). The run-length is loaded into the counter which outputs the first bit of the run. The counter then counts down from the preset value 110 to 000, sending a bit (0, for this example) to the Huffman decoderineveryclockcycle. Whenthecounterreaches 000, the NOR gate output becomes 1, enabling the ROM to output the bits representing the next run. Since one bit is received by the Huffman decoder in every clock cycle, run-length decoding does not add to testing time. 4. Experimental Results In this section, we present experimental results on test set encoding for several ISCAS 89 benchmark circuits to demonstrate the saving in on-chip storage achieved using Huffman, Comma and run-length encoding. We Fig. 11. Run-length encoding applied to a portion of the Huffman encoded s444 test set: (a) 3-bit encoding for 8 types of runs, (b) bit stream to be encoded, and (c) run-length encoded data. Fig. 12. Run-length decoder for the s444 test set. consider circuits in which the number of flip-flops f is considerably greater than the number of primary inputs n; we denote the ratio f=n by . Table 4 lists the values of for the ISCAS 89 circuits, with circuits having a high value of shown in bold. Such circuits are especially hard to test because of the relatively large number of internal states and few primary inputs. From Table 4 Table 4. The ratio of the number of flip-flops to the number of primary inputs , and jTDj, the length of the HITEC test sequences for the ISCAS 89 circuits. CUT jTDj CUT jTDj Deterministic Built-in Pattern Generation 109 we see that these circuits typically require longer sequences of test patterns. On the other hand, they are excellent candidates for our encoding approach. Several other ISCAS 89 benchmark circuits do not have a high value of , and are therefore more suitable for scan-based testing, than for the proposed approach of encoding non-scan test sets. We do not present results for these circuits, however, statistical encoding of full-scan test sets for these circuits, on the lines of the proposed approach, has recently been shown to be effective in reducing the amount of memory required for test storage [25]. We performed experiments on test sets for single- stuck line (SSL) faults obtained from the Gentest ATPG program, as well as the HITEC, GATEST, and test sets from the University of Illinois [26]. We measured the fault coverage of these test sets using the PROOFS fault simulator [27] and ensured that the coverage is comparable to the best-known fault coverage for these circuits. We next present results on the compression achieved using Huffman and Comma coding for all four test sets. Table 5 compares the number of bits required to store the encoded test set that required to store the corresponding unencoded test set TD. The number of bits required by our scheme is moderate, substantially less than that required to store unencoded test sets, and reduces significantly when the same test set can be shared among multiple CUTs of the same type included on a chip, as in core-based DSP circuits [14]. The saving in SG memory presented in Table 5 is substantial, and in most cases, the difference in compression due to the use of Comma coding instead of Huffman coding is very small. In Table 6, we show that further compression is achieved by applying run-length coding to TE. We present results on run-length encoding for the s382 and s444 circuits using the Gentest test set. The test application time required is considerably less than that required for pseudorandom testing, even though the number of clock cycles C is greater than the number of patterns in TD (C D lHjTDj for Huffman coding and C D lCjTDj for Comma coding). Table 7 compares the number of test patterns applied, the number of clock cycles required, and the fault coverage obtained for our method, with the corresponding figures reported recently for two pseudorandom testing schemes [5, 6]. The test application time required by our method is much less than for the pseudorandom testing method of [5]. We also achieve higher fault coverage for all circuits. Iyengar, Chakrabarty and Murray Table 5. Experimental results on test set compression for ISCAS 89 circuits with a high value of . Average Percentage codeword length compression ISCAS circuit nmjTDjTbits lH lC Hbits Cbits HE CE Gentest GATEST n: No. of primary inputs; m: No. of unique test patterns; Tbits: Total no. of bits in TD; Hbits: No. of bits in after Huffman encoding; Cbits: No. of bits in after comma encoding; aComma coding is not applicable for the test set of s35932, because the probabilities of occurrence of the test patterns do not satisfy (2) given in Theorem 4. Table 6. Percentage compression achieved by run-length coding after applying Huffman and Comma encoding to TD. Number of bits in Percentage compression ISCAS circuit Tbits Hbits Cbits HRbits CRbits HE CE HRE CRE HRbits: No. of bits in the encoded test set after Huffman and run-length encoding; CRbits: No. of bits in the encoded test set after Comma and run-length encoding. Deterministic Built-in Pattern Generation 111 Table 7. Number of clock cycles C required and fault coverage obtained using pseudorandom testing compared with the corresponding figures using precomputed deterministic test sets. Number of patterns jTDj Number of clock cycles C Fault coverage (%) ISCAS circuit [5]a [6]a Deta [5] [6] Det [5] [6] Detb a[5, 6]: Recently proposed pseudorandom BIST methods; Det: Deterministic testing using precomputed test sets. bThe best fault coverage achieved by precomputed deterministic testing. cResults for these circuits were not reported in [5, 6]. Table 8. Literal counts of the Huffman and Comma decoders for the four test sets. Decoder cost in literals Huffman decoders Comma decoders ISCAS circuit Gen HIT GAT STRAT Gen HIT GAT STRAT 28 43 s400 44 33 33 37 26 27 27 27 s526 43 47 28 27 Gen: Gentest; HIT: HITEC; GAT: GATEST; STRAT: STRATEGATE. We next present experimental results on the Huffman and Comma decoder implementations. We designed and synthesized the FSM decoders using the Epoch CAD tool from Cascade Design Automation [28]. The low to moderate decoder costs in Table 8 show that the decoding algorithm can be easily implemented as a BIST scheme. Note that the largest benchmark circuit s35932 requires an extremely small overhead (synthe- sized ROM area is 0.53% of CUT area, and decoder area is 6.18% of CUT area) to store the encoded test set and decoder, thus demonstrating that the proposed approach is scalable and it is feasible to incorporate the encoded test set on-chip for larger circuits. Note that, while Huffman and Comma encoding reduce the number of bits to be stored, the serialization of the ROM may increase the hardware requirements for ROM address generation. In a conventional fixed-length encoding scheme, the size of the counter required for ROM address generation is dlog2 jTDje, while an encoded ROM requires a dlog2.jTDjl/e-bit counter for address generation, where l is the average codeword length. However, since l is small, this logarithmic increase in counter size is also small, e.g., the size of the counter does not change for s444, while it increases from 7 to 10 for s35932. The hardware overhead figures in Table 8 do not include this small increase in counter size. It may be argued that a special-purpose, minimal- state FSM may be used to produce a precomputed sequence. However, we have seen that the overhead of such FSMs is prohibitive, especially for long test sequences. In addition, such a special-purpose FSM wouldbespecifictoasingleCUT;ontheotherhand, the decoder DC for the proposed scheme is shared among multiple CUTs, thereby reducing overall TGC overhead Table 9 compares the overhead of the proposed deterministic BIST scheme with the overhead of a pseudorandom BIST method [6] for several circuits. The overhead for the pseudorandom method was obtained 112 Iyengar, Chakrabarty and Murray Table 9. Literal counts for the proposed technique compared with pseudorandom testing. Deterministic TGC cost Pseudorandom Number of ISCAS 89 Decoder Total TGC cost test points circuit cost cost [6] [6] s526 by mapping the gate count figures from [6] to the literal counts of standard cells in the Epoch library. While the deterministic TGC requires greater area than the pseudorandom TGCs, the difference is quite small, and thus may be acceptable if higher fault coverage and shorter test times are required. Note also that the pseudorandom method requires the addition of a large number of observability test points. These require a gate-level model of the CUT, as well as additional primary outputs and routing. Moreover, they may also increase the size of the response monitor at the CUT outputs. The proposed TGCs require no circuit modification, thus making them more applicable to testing core-based designs using precomputed test sequences. Finally, we present experimental results for test set compression and decoder overhead, using a single decoder to test several CUTs on a chip. Table 10 shows that the levels of compression obtained for combined test sets are comparable to those obtained for the individual test sets. In fact, in several cases the overall compression is higher than that obtained for one of the individual test sets. The percentage area overhead required for the decoder reduces significantly, because a single decoder can now be shared among several CUTs. Note that in the case of the Comma decoders, a major part of the overhead is contributed by the binary coun- ters. For example, in the Comma decoder for the pair head, while the combinational logic represents only overhead. If the counter is also used for normal operation of the system, then the BIST overhead will reduce further. The test application technique is therefore clearly scalable with increasing circuit com- plexity. The decoder overhead also tends to decrease with an increase in . This clearly demonstrates that the proposed test technique is well suited to circuits for which is high. 5. Conclusion We have presented a novel technique for deterministic built-in pattern generation for sequential circuits. This approach is especially suited to sequential circuits that have a large number of flip-flops and relatively few Table 10. Percentage compression for test sets encoded jointly. Percentage Huffman compression Percentage Comma compression Circuit Gen HIT GAT STRAT Gen HIT GAT STRAT Table 11. Decoder cost in literals and percentage decoder overhead for a single decoder shared among several CUTs. Decoder cost in literals Percentage decoder overhead Circuit Gen HIT GAT STRAT Gen HIT GAT STRAT fs382; s444g 48 52 44 44 6.72 7.29 6.18 6.21 28 Deterministic Built-in Pattern Generation 113 primary inputs, and for circuits such as embedded 9. M.S. Hsiao, E.M. Rudnick, and J.H. Patel, Alternating Strate- cores, for which gate-level models are not available. We gies for Sequential Circuit ATPG, Proc. European Design and have shown that statistical encoding of precomputed Test Conf., 1996, pp. 368-374. 10. T.M.NiermannandJ.H.Patel, HITEC:ATestGenerationPack- test sequences leads to effective compression, thereby age for Sequential Circuits, Proc. European Design Automation allowingon-chipstorageofencodedtestsequences.We Conf., 1991, pp. 214-218. have also shown that the average codeword length for 11. D.G. Saab, Y.G. Saab, and J.A. Abraham, Automatic Test the non-optimal Comma code is nearly equal to the av- Vector Cultivation for Sequential VLSI Circuits Using Genetic erage codeword length for the optimal Huffman code if Algorithms, IEEE Trans. on Computer Aided Design, Vol. 15, pp. 1278-1285, Oct. 1996. the test sequence satisfies certain proeprties. These are 12. K. Chakrabarty, B.T. Murray, J. Liu, and M. Zhu, Test generally satisfied by test sequences for typical sequen- Compression for Built-in Self Testing, Proc. Int. Test Conf., tial circuits, therefore Comma coding is near-optimal 1997, pp. 328-337. in practice. 13. F. Brglez, D. Bryan, and K. Kozminski, Combinational Profiles Our results show that Huffman and Comma encod- of Sequential Benchmark Circuits, Proc. Int. Symp. on Circuits and Systems, 1989, pp. 1929-1934. ing of test sequences, followed by run-length encoding, 14. M.S.B. Romdhane, V.K. Madisetti, and J.W. Hines, Quick- can greatly reduce the memory required for test stor- Turnaround ASIC Design in VHDL: Core-Based Behavioral age. The small increase in testing time is offset by Synthesis, Kluwer Academic Publishers, Boston, MA, 1996. the high degree of test set compression achieved. Fur- 15. J.P. Hayes, Computer Architecture and Organization, 3rd ed., thermore, testing time is considerably less than that for McGraw-Hill, New York, NY, 1998. 16. G. Held, Data Compression Techniques and Applications: pseudorandom methods. We have developed efficient HardwareandSoftwareConsiderations, JohnWiley, Chichester, low-overhead pattern decoding methods for applying West Sussex, 1991. the test patterns to the CUT. We have also shown that 17. M. Jakobssen, Huffman Coding in Bit-Vector Compression, the overhead can be reduced further by using a sin- Information Processing Letters, Vol. 7, No. 6, pp. 304-307, Oct. gle decoder to test multiple CUTs on the same chip. 1978. 18. T.M. Cover and J.A. Thomas, Elements of Information Theory, The proposed technique thus offers a promising BIST John Wiley, New York, NY, 1991. methodology for complex non-scan and partial-scan 19. M. Mansuripur, Introduction to Information Theory, Prentice- circuits for which precomputed test sets are readily Hall, Inc., Englewood Cliffs, NJ, 1987. available. 20. V. Iyengar and K. Chakrabarty, An Efficient Finite-State Machine Implementation of Huffman Decoders, Information Processing Letters, Vol. 64, No. 6, pp. 271-275, Jan. 1998. 21. D.H. Greene and D.E. Knuth, Mathematics for the Analysis of --R and the M. Reliable Computing Laboratory of the Department of Electrical and Computer Engineering. University of Illinois at Urbana-Champaign are in computer-aided design of VLSI circuits and systems verification and built-in self test Krishnendu Chakrabarty received the B. Indian Institute of Technology and Ph. Engineering at Duke University. Professor of Electrical and Computer Engineering at Boston projects are in embedded core testing archival journals and referred conference of IEEE and Sigma Xi Activities in IEEE's Test Technology Technical Council (TTTC). Murray received the A. from Albion College in in Electrical Engineering from Duke University in of Michigan in of the technical staff at General Motors Research and Development where he led projects in testing systems and computer architecture. Adjunct Lecturer at the University of Michigan since is currently Project Manager of Dependable Embedded Systems in tools for system safety engineering. He currently serves on the editorial board of the Journal of Electronic Testing: Theory and Applications. --TR --CTR A. Chandra , K. Chakrabarty, Efficient test data compression and decompression for system-on-a-chip using internal scan chains and Golomb coding, Proceedings of the conference on Design, automation and test in Europe, p.145-149, March 2001, Munich, Germany Anshuman Chandra , Krishnendu Chakrabarty, Test Data Compression and Test Resource Partitioning for System-on-a-Chip Using Frequency-Directed Run-Length (FDR) Codes, IEEE Transactions on Computers, v.52 n.8, p.1076-1088, August Anshuman Chandra , Krishnendu Chakrabarty, Test Resource Partitioning for SOCs, IEEE Design & Test, v.18 n.5, p.80-91, September 2001 Michael J. Knieser , Francis G. Wolff , Chris A. Papachristou , Daniel J. Weyer , David R. McIntyre, A Technique for High Ratio LZW Compression, Proceedings of the conference on Design, Automation and Test in Europe, p.10116, March 03-07, Ismet Bayraktaroglu , Alex Orailoglu, Concurrent Application of Compaction and Compression for Test Time and Data Volume Reduction in Scan Designs, IEEE Transactions on Computers, v.52 n.11, p.1480-1489, November A. Touba, Test data compression using dictionaries with selective entries and fixed-length indices, ACM Transactions on Design Automation of Electronic Systems (TODAES), v.8 n.4, p.470-490, October Anshuman Chandra , Krishnendu Chakrabarty, Analysis of Test Application Time for Test Data Compression Methods Based on Compression Codes, Journal of Electronic Testing: Theory and Applications, v.20 n.2, p.199-212, April 2004

sequential circuit testing;statistical encoding;run-length encoding;embedded-core testing;BIST;pattern decoding;huffman coding;comma coding

608855

Structural Fault Testing of Embedded Cores Using Pipelining.

The purpose of this paper is to develop a global design for test methodology for testing a core-based system in its entirety. This is achieved by introducing a bypass mode for each core by which the data can be transferred from a core input port to the output port without interfering the core circuitry itself. The interconnections are thoroughly tested because they are used to propagate test data (patterns or signatures) in the system. The system is modeled as a directed weighted graph in which the accessibility (of the core input and output ports) is solved as a shortest path problem. Finally, a pipelined test schedule is made to overlap accessing input ports (to send test patterns) and output ports (to observe the signatures). The experimental results show higher fault coverage and shorter test time.

Introduction Progress in deep submicron VLSI technology enables the integration of large predefined macros or cores together with user defined logic (UDL) into a single chip. This leads to a design paradigm shift from single ASIC design to system-on-chip design with full scale leverage of third-party intellectual property (IP)[BhGV96][De97]. There are several advantages of core based design such as reduction in the overall system design time, productivity increase, accelerating time-to-market, and increasing competitive superiority. Design and test of core-based systems is a very important and challenging problem facing the semiconductor industry for the next several years. A major difficulty concerns accessibility of embedded cores from the I/O terminals of the system. This entails mapping of the stand-alone test requirements, provided by the core vendor, into the embedded cores. The basic technique suggested by many R&D groups is to access each embedded core for testing in isolation from the others [KoWa97, Vsi97, BhGV96]. However, there are a number of disadvantages with the isolation approach. Isolation testing does not test the system-on-chip as a whole, for example it does not address the testing of interconnects and interfaces between cores. It does not consider the interaction of cores for testing such as the effect of testing one core on surrounding cores, or vice versa (the surrounding cores also may affect the core under test). Implementing protection safeguards will be very costly increasing the test overhead of the isolation method even further [ToPo97]. 1.1 Background Loosely speaking, a core is a highly complex logic block which is fully defined, in terms of its behavior, also predictable and reusable [ChPa96]. Cores are distinguished into several categories in terms of design flexibility, IP protection, test development, programmability and other characteristics. Soft cores are reusable blocks in terms of synthesizable RTL description. Firm Cores are reusable blocks supplied in netlist of library cells and for a range of technologies. Finally, Hard cores are reusable blocks optimized in terms of power and performance, and supplied in layout form for a specific technology. Clearly, soft cores are the most flexible type but hard cores provide most IP protection. From test development viewpoint, cores can be mergeable or non-mergeable. Mergeable cores use an expandable test scheme and thus they can be merged with other mergeable cores so that the composite structure is tested as a whole. In terms of programmability, cores can be characterized in fully programmable (e.g. microprocessor cores), partially programmable (e.g. Application-Specific Integrated Processors - ASIP), and little or non-programmable (e.g. ASIC cores). There are a number of test methods that apply to ASIC cores based on Design for Test (DFT) techniques, e.g. Scan, BIST, ScanBIST, Test points, or without DFT using precomputed testing. To facilitate test integration there are a number of preliminary proposals from the industry, e.g using test sockets, Macro test or core-level boundary scan [KoWa97]. We remark that the above issues are not decoupled of each other, in fact they are related and can have significant effect on system testing. Also, the above issues apply to both soft and hard cores. However, hard cores limit the flexibility of the designer significantly for system-level test solutions. The basic testing strategy suggested by industrial groups for system-on-chip is to test each embedded core one by one rather than the system as a whole. This strategy requires accessibility, i.e. controllability and observability of each core I/Os from the system I/Os. There are number of core isolation techniques proposed [ImRa90] to ensure accessibility and some of them may provide good match to a particular core's internal test method: MUX-based isolation of each core mapping precomputed tests, ffl A boundary scan type of approach accessing the cores within the system-on-chip - this can apply to cores with embedded Scan or BIST DFT. ffl A test wrapper or collar DFT hardware inserted for isolation, In isolation method, a global BIST controller is usually employed to test schedule cores with embedded BIST structures to shorten the test time. A test bus has also been proposed to affect accessibility. All of these techniques proposed have a number of disadvantages. They may incur significant overhead of isolation-related test structures. Some performance and possibly power consumption penalty is also incurred due to these structures. Moreover, the isolation techniques do not address testing the system as a whole. Specifically, the faults in the interfaces (interconnects, and user logic) between cores remain undetected. The above shortcomings of isolation techniques motivate a coordinated approach to testing system-on-chip. The basic goal is to test the system as a whole, this means testing the cores themselves as well as testing their interface. Any type of cores in the system (e.g. soft, hard, etc.) for which a test data set is available (predefined or deterministically/empirically computable) can be handled by our method. The main contribution of our work is twofold. First we define the "bypass" for each core by which the data can be transferred from a core input port to the output port without interfering with the core circuitry itself. Since the test data travel in the existing interconnections, the core interface are thoroughly tested. Second, we model the core-port accessibility problem as finding the shortest path in a directed weighted graph and minimize the testing time by overlapping the time consumed to access paths. Conceptually, our method is a generalization of the scan approach at the system level by allowing the use of system interconnects, with various bit widths, for test data distribution and signature collection. This paper is organized as follows. Section 2 presents the bypass mode and test overhead cost. Section 3 models the accessibility problem as finding the shortest path in a directed graph. Section 4 discusses the algorithm to overlap execution of those paths to minimize the test time and details a design example step by step. The experimental results are in Section 5. Finally, the concluding remarks are summarized in section 6. Using Bypass Mode 2.1 Core Environment We distinguish faults with respect to the core environment. Given Core A, by definition, the environment of Core A is all the input/output connections from/to primary inputs/outputs and other cores in the system. Using ticker lines, we have shown the environment of Core 1 in the system pictured in Figure 1. System Primary Inputs System Primary Outputs Figure 1: Environment of Core 1. We also differentiate between isolated core and isolated core environment. Isolated core refers to the core itself (shaded core in Figure 1 for example), without any other components of the system, which is isolated from the system by a mechanism, such as multiplexors or tri-state buffers and so on. On the other hand, as Figure 1 shows for Core 1, the isolated core environment includes the core and all connections to/from it, all isolated by an appropriate mechanism. This distinction plays an important role in our discussion since for example a fault in the interconnect of a core can not be captured in isolated core testing while it may be caught when testing the isolated core environment. 2.2 Input/Output Test The overall objective is to use the existing wires and topology of the system to establish a path to carry test data between two test points in the system, called source and sink. In core testing, there are two types of test paths, that is input test path to access core input ports from system primary inputs and output test path to access core output ports from the system primary outputs. Figure 2(a) shows a core under test (Core these two paths and their corresponding source and sink. It also shows a general view to establish a route between two test points using the existing wires. In Figure 2(b) the blowup picture of a core in input/output test path (Core k) is shown. We symbolically showed that the inputs are bypassed to the output without interfering the core circuitry which are used in the normal mode. We will shortly elaborate on matching the bit widths (packetization) and the real implementation of bypass circuitry. The basic idea of having the bypass mode for each core is to have an independent route around a core to carry test data (e.g. predefined test patterns or core signatures) between port i (m i k bit wide) and port j (n j k bit wide) of that core (Core k). Our goal is to establish the shortest path (fastest route) to carry the packets of test data between source and sink. Note that by this formulation, accessibility of the core inputs from system primary inputs and of the core outputs from system primary outputs, are similar problems, i.e. identifying the shortest path between source and sink. (Under Test) Source Source Sink Sink Input Test Path Output Test Path b_in b_out Bit length of test data System Primary Output System Primary Input (a) (b) Bit-Match port j port i i_k j_k Figure 2: Test paths: (a) Input/output test paths; (b) A typical core in test paths. The important benefit of the bypass is to take advantage of the existing connections among the cores and the existing wires to transfer multi-bit data from source to sink. This, in the worst case, will be equivalent to the conventional scan which transfers the patterns serially using a separate routing in the system. Considering the multi-bit interconnections among the cores and the fact that we do not use a separate scan chain, we expect the average case to be quite superior in terms of time and comparable in terms of hardware overhead. The "bypass" mode that we use here is different from identity mode (I-mode) introduced by Abadir and Breuer [AbBr85] in many ways. In [AbBr85] the authors define the I-mode and I-path for RTL components (e.g. ALU's, MUXes and Registers) to transfer data unaltered from one port to another. For example, for an adder having one data value as zero creates the I-mode. In our approach, we have physical bypass routes through which the data is transferred from one point to another. Additionally, the I-mode and many other mode definitions, including transfer mode (T-mode) and sensitized mode (S-mode), all control components to efficiently realize one form of partial scan test. These modes and I-paths are used for transmitting data from scan registers to the input ports of a block under test and transmitting signatures to scan registers to be shifted out. Our approach does not use scan registers at all. It offers totally different test methodology by allowing the use of system interconnects, with various bit widths, for test data distribution and signature collection. We would like also to point out that many core providers, such as Philips [MABD98][Mari98], have already devised the bypass mode as a part of P1500 core test standardization. Such standard is well justified by providing considerable flexibility to the designer with a reasonable cost. Additionally, the bypass mode can be easily incorporated within soft and hard cores by core providers [MABD98]. If a core does not come with bypass mode, this mode and the required bit-match circuitry (explained in this section) should be added externally by a designer for the benefit of testing. 2.3 Packetizing Technique for Data Transfer Figure 2(a) clearly shows that the bit width of inputs/outputs of cores change in a path between two test points. This requires sort of bit matching. Let's assume that we need to transfer a b bit pattern between source and sink. In general, to transfer b bit test data from port i (m i k bit) to port j (n j k bit) of Core k we need to packetize data (to match the available bit width) and send it in several iterations. For example, a core with a bit input port test data in 4 iterations. We assume that data packetization and transfer is synchronized with the system clock. Note also that from Core k point of view it does not matter how b-bit data gets to its inputs. If they all come at once (e.g. or through many packets (e.g. b ? number of iterations (cycles) that Core k consumes to bypass data will not change. However, it will affect the n/m Stages S/P circuit (m <= n) m-bit m-bit m-bit Figure 3: Serial-to-Parallel (S/P) bit match circuit. scheduling of bypass activities. Before we discuss this we present the cost (hardware and time overhead) due to bypass and bit matching circuit. For simplicity, we have not shown i k and j k subscripts in the bit width variables. 2.3.1 Case 1: Input - Output (m - n) - Serial-to-Parallel Figure 3 shows the first type of the bit-match circuit required to assemble a larger pattern from different packets of data. The circuit consists of d n e stages of cascaded m-bit registers (e.g. D flip flops) controlled by the same clock. The circuit, which is like a shift register bank of m bits, has a serial-to-parallel (S/P) behavior whose worst case (in terms of time) corresponds to equivalent to the traditional scan-in discipline. Note that the d n e packets of m bit data will be parallelized to n bit in d n cycles. For a b-bit data we need to iterate d b times (overall d n Briefly, based on this implementation, to bypass b-bit data from an m-bit input port to an n-bit output port of a core we have the following cost values: Time Cost: d b Area (1) Where DFF denotes a D-type flip flop which is one possible implementation of 1-bit register. These equations clearly shows that depending on b the participating core may spend less or more time in bypassing data and some of the output wires (out of n) may not be needed. For example, suppose the data is transferred in 1 cycle using two out of eight available output wires. If the data is transferred in 8 cycles using all eight wires. 2.3.2 Case 2: Input ? Output (m ? n) - Parallel-to-Serial Figure 4 shows the second type of the bit-match circuit required to disassemble a large pattern to different packets of data. The circuit consists of n stages of parallel d m multiplexors and 1-bit registers (e.g. D flip flops) controlled by the same clock. The circuit has a parallel-to-serial (P/S) behavior whose worst case (in terms of time) corresponds to equivalent to the traditional scan-out discipline. Multiplexors need dlog e)e bit for select lines. Since the actual data to be transferred depends on b we assumed a self-starting counter (controlled by the test controller) controls the number of iterations actually needed. Note that the d m e packets of n bit data will be produced in d m clock cycles. For a b-bit data we need to iterate d b Briefly, based on this implementation, to bypass b-bit data from an m-bit input port to an n-bit output port of a core we have the following cost values: m/n m/n m/n Log m/n Stages P/S circuit (m > n) 1-bit 1-bit 1-bit Counter Self-starting Figure 4: Parallel-to-Serial (P/S) bit match circuit. Time Cost: d b Area (2) Again depending on b the participating core may spend less or more time in bypassing data and some of the input wires or MUX inputs (out of m) may not be needed. For example, suppose 2. If the data is transferred in 1 cycle using two out of eight available input wires. If the data is transferred in 8 cycles using all eight wires. Equations 2 and 1 can be joined together as follows: Time Cost: d b minfm;ng e cycles Area 2.3.3 Complete Bypass Circuitry In addition to the S/P or P/S bit match circuit, the complete bypass circuitry includes tri-state buffers and few additional logic gates to control the core activities as shown in Figure 5. Tri-state buffers are needed to protect the core when it bypasses data. This is a safeguard mechanism to ensure that in bypass mode the core does not receive any new data and so does not change the state. Note that many input ports can be bypassed to a single output port but one at a time. Logically speaking we have 1. Although in Figure 5 we showed the bypass circuitry for bypassing port i to port j only, "\Delta \Delta \Delta" shows that we may have additional bypass routes. This will be decided by the shortest path algorithm that we will explain in section 3. In summary, any new routes to bypass an input port to port j adds, one AND gate, one buffer and one entry to the NOR gate. The introduction of the glue logic among cores in order to bypass test data may slightly degrade the timing characteristics of the core due to its additional delay. In our experiments, using 0.8-micron CMOS library in the COMPASS Design Automation tool [Comp93], this additional delay was less than 3.5 nano-second. (S/P or P/S) Bit Match (Port port 3-State Buffer Enable Core k with port i to port bypass circuit BUS (Port i) Input stop_k Figure 5: Core bypass circuit (only port i to port j bypass is shown). Mode stopk bypassk Function of Core k Run Bypass Table 1: Different modes of each modified core. When bypass the core accepts the inputs and forwards the output to the output port. The core performs its functionality in this mode, thus we call it "normal mode". When core is under test (even in still it has to perform its normal functionality. bypass on the other hand, disconnects the input to the core and bypasses data to the output port based on the selected route determined by bypass ij k . The test controller has to make sure (by there will not be any conflict of bypassing different input ports to one output port simultaneously. Signal allows the global "clock" to reach the core and if stop masks the clock and so the core does not leave its present state. This signal can be used in interactive testing to freeze the system temporarily to read out the core outputs. This issue is not pursued in this paper. Note that, for the purpose of the normal or bypassing operations we could have considered bypass . However, we intentionally separated them for each core to provide the capability of bypassing data even when the core is under test. By doing so, we will be able to bypass data through Core k to test other cores even when Core k itself is under test. This will reduce the overall test time specifically when the application test time for a core is too long. We will clarify this later in section 4. Table 1 summarizes the operation modes. 2.4 Cost Calculation for Source-Sink Paths In the previous section we presented two cost functions for time and area (see Equation 3) of the bypass circuitry pertaining to single cores. In this work we decided to focus only on the test time optimization. Other heuristics can be proposed based on the area cost to look at the system testing from another angle. Based on Equation 3, when core k participates in a test path (by bypassing data between and test data between source and sink its time overhead will be proportional Source (1st test point) (2nd test point) ij_k b: bitwidth of test data (input pattern OR output signature) Cycles i_k j_k Figure The pipeline-like structure for each source-sink path. Cycles Cores (a) Char. Interface Cost: High Cycles Cores (c) Char. Interface Cost: Low Cycles Cores (b) C2 . Char. Function: 2(2C1 Interface Cost: Medium Source (1st test point) (2nd test point) Figure 7: An example of bypass scheduling. e cycles. However, the distribution of its -cycle activities is another issue. The whole path, as shown in Figure 6 is similar to a pipeline system with N stages in which each stage requires cycles. The difference, however, with conventional pipelines is in the scheduling method. In conventional pipelining [Ston90], we define the pipeline clock period to be equal to the slowest stage delay and then schedule the activities accordingly. In our problem, we don't want to devise too many registers in the interface between cores to pile up all data packets. Instead, we have to implement an innovative mechanism by which the bypassing is performed as soon as a packet of the appropriate size is ready. To make this point clear let's consider the example of Figure 7 in which the sink is an input port of a core under test and requires a test data. There are three cores in the source-sink path with time cost of 4, 4 and 2, respectively. These cost values correspond to the time overhead required to packetize (serial to parallel or parallel to serial) the test data. For example, 16-bit test data would be dis-assembled into four packets (of 4-bit each) in 4 cycles to transfer through Core 1. Three bypass scheduling choices are shown. We used space-time table similar to the reservation table [Ston90] in pipelining. Each row corresponds to a core and each column corresponds to a time step. An entry (C1, C2 or C3) in the table shows that the corresponding core is bypassing a packet of data in that cycle. For example, in all three schedules shown in this figure Core 1 bypasses a packet of 4-bit data in the first four cycles. In Figure 7(a), we are not taking advantage of the pipeline-like structure at all. Activity of one core starts when the previous one is finished. More importantly, we need an expensive hardware interface between cores to pile up all packets (at most four packets of 4-bit data) of data before sending it to the next core in the path, obviously a bad choice. Figure 7(b) pictures a case that we need cheaper interface (at most two packets of 4-bit data) and it consumes 7 cycles. Figure 7(c) pictures the superior choice which minimizes the cost (e.g. registers used anyway in bit match circuits) and the data transfer time into 6 cycles. We will shortly show that finding such optimized schedule is possible by constructing the characteristic function of the path and factorizing it as much as possible starting from the outside without changing the order of core variables. These functions are shown also in Figure 7. From above argument and example, it's clear that for a given shortest path the bounds for total bypass scheduling time for a path consists of N cores are: Upper Bound: e cycles Lower where i and j are assumed to be single specific input and output ports of Core k, respectively, through which data is bypassed. 2.4.1 Path Characteristic Function The bounds presented in equation 4 can be used as data transfer time evaluation heuristics to identify the shortest path between two test points. Although, the upper and lower bound may suggest different solutions. For example, suppose there are two paths between a source and a sink. The first path has four cores with 's equal to 4, 4, 3 and 3. The second path has three cores with 's equal to 6, 1 and 1. The upper bound selects the second path as shortest (cost of 6 compared to 4 while the lower bound heuristic chooses the first path as the shortest (cost of 4 compared to 6 Briefly speaking, the overall time cost of bypass scheduling depends on the time distribution of activities among cores. This example shows that we need a mechanism to evaluate the actual time needed for bypass scheduling when the data is packetized based on the available bit widths and transferred from one core to another. This is needed not only for overall time evaluation but also to have the complete bypass schedule for the test controller in test session to tell cores how to behave. To do this, we defined the path characteristic function (pcf). The pcf is written by starting from then factorizing the coefficients starting from the outmost possible factor and continuing inside. Any other type of factorizing would lead to different sub-optimal schedule. In the example of Figure 7 we start from equation 4C1 This corresponds to the schedule of Figure 7 (a). We interpret each "+" in the pcf as sign of sequential activity. So, the function here means, first Core 1 has to bypass data for 4 cycles, then Core 2 bypasses data for 4 cycles, and finally data for 2 cycles. If we factorize 2 out to become 2(2C1+2C2+C3) the pcf corresponds to the schedule of Figure 7 (b). Note that the factor out of the parenthesis reflects the number of repetitions of that sequencing, each starts p cycles after another where p is the largest coefficient inside the parentheses (2 for this pcf). Finally, if we continue factorizing the terms to: 2(2(C1 +C2)+C3), the pcf corresponds to Figure 7 (c) which consumes less time (6 cycles) and requires cheaper interface. According to 2(2(C1 suggests a schedule in which Core 2 bypasses data after Core 1. This sequence is repeated twice followed by bypassing data by Core 3. Finally, the whole thing is repeated twice. Briefly, the general formula for a pcf function is: I Note that in this function superscript r denotes the level factorized in the function. R r is the coefficient outside of the parenthesis after factoring it out and f I are the factorized terms inside the rth f r Table 2: Example of pcf recursive formula. level of parenthesis. The form of the inner-most terms will be as f 0 similar to 2C1 or C3 terms appeared in example of Figure 7. The following recursive formula is a simple way of computing the overall scheduling time (T (f r )) based on the pcf function: I As an example, consider the pcf corresponding to the example of Figure 7 (c), which is 2(2(C1+C2)+C3). Table shows how we get T (f r as the overall scheduling time: Graph Modeling and Shortest Path Problem Our objective is to model the port accessibility of cores within a system as a directed weighted graph in which the shortest path between any two points (called source and sink) reflects the fastest route to transfer packets of data between those two points. From testing perspective, with such model we can find the fastest route to transfer test data (predefined or random pattern) from the system primary inputs to any of the core input ports. Similarly, we can find the fastest route to transfer test data (signature) from any of the core output ports to the system primary output. Equation 3 defines the cost associated with bypassing data from P ort i to P ort j of a core. So, in our graph modeling, a node corresponds to a port and an edge corresponds to the interconnects between ports or the bypass possibilities. Those edges reflecting the bypass choices form a bipartite subgraph for each core whose cost (weight) will be determined based on Equation 3. The cost of an edge corresponding to the bypass delay shows the time required to transfer the packetized data from one point to another. The time cost of the existing interconnections between cores is assumed to be zero since no additional circuit/delay for packetizing or transfer-control is needed. Figure 8(a) shows a system, made of four cores with different ports and bit widths, under test. For consistency, we showed the test pattern generator (TPGR) and signature analyzer (MISR) also as cores. The system has four cores, two primary inputs (going to Core 1 and Core 4) and three primary outputs (from Core 2, Core 3 and Core 4). Figure 8(b) shows the corresponding Core Bypass Graph (CBG). Recall from Equation 3 that the time cost depends on the bit width of test data (b) that we desire to transfer. So, depending on the bit width of test data (b) different cost values should be used in finding the shortest paths. As an example, in Figure 8(b) near each edge we have shown two cost values. The cost values outside parenthesis are t ij e showing the time overhead to transfer 8-bit test data. Similarly, the cost values inside parenthesis are t ij TPGR MISR1616816816168164 Sink (a) Global Global Source (b) TPGR MISR Time Cost: Global Global Source Figure 8: A core-based system: (a) Bit widths; (b) The CBG graph. for repeat f Select an unmarked vertex vq such that sq is minimal; Mark vq foreach(unmarked vertex until (all vertices are marked); Figure 9: Dijkstra Algorithm d e reflecting the time overhead to transfer 16-bit test data. All edges without a cost value correspond to the existing interconnect between cores and are assumed to have time cost of zero because if we ignore resistance and capacitance of wires, any packet of data will be transferred almost immediately. As for the shortest path algorithm itself, there are many well-known solutions proposed in the literature, graph theory and operation research texts. Some of them are, Bellman, Dijkstra Bellman-Ford and Liao-Wong algorithms [DeMi94] [BoMu76] whose running time is between O(n 2 ) to O(n 3 ), where n is the number of nodes or edges, and their capability is between finding single shortest path to finding the shortest path between all pairs. In our work, we have used Dijkstra algorithm which is a greedy algorithm providing an exact solution with computational complexity of O(jEj+jV jlogjV and jEj are total number of nodes and edges in the graph, respectively. This is one of the fastest among such algorithms. Cycles are also considered in the Dijkstra Algorithm. The pseudocode of the algorithm is shown in Figure 9. The algorithm keeps a list of tentative shortest paths, which are then iteratively refined. Initially, all vertices are unmarked and fs Thus, the path weights to each vertex are either the weights on the edges from the source or infinity. Then, the algorithm iterates the following steps until all vertices are marked. It selects and marks the vertex that is head of a path from the source whose weight is minimal among those paths whose heads are unmarked. The corresponding tentative path is declared final. It updates the other (tentative) path weights by computing the minimum between the previous (tentative) path weights and the sum of the (final) path weight to the newly marked vertex plus the weights on the edges from that vertex. Details can be found in many graph theory and synthesis books [BoMu76] [DeMi94]. Our shortest test path algorithm is summarized in Figure 10. The algorithm first constructs the CBG graph and finds the time cost values corresponding to the edges in CBG. Then, it uses the Dijkstra algorithm Construct the Core Bypass Graph (CBG); Compute the time cost values, i.e. e Apply DIJKSTRA algorithm to find all input test shortest paths (1 - i - Nin k ); Apply DIJKSTRA algorithm to find all output test shortest paths (1 Figure 10: The shortest path algorithm applied to CBG. TPGR MISR Global Global Source Time Cost: Input Test Shortest Paths: Output Test Shortest Paths: Figure 11: Four input/output shortest paths for Core 1 using Dijkstra Algorithm. to find all the input/output test shortest paths. To show the process, we continue our running example by applying the Dijkstra algorithm to the CBG graph of Figure 8(b) for Core 1 only. Note carefully that Core 1 has two 8-bit and two 16-bit ports. So, we consider the appropriate cost of edges accordingly. That is the cost values outside parenthesis when a test point is the core 8-bit port and the cost values inside parenthesis when a test point is the core 16-bit port. The result is summarized in Figure 11 that shows the shortest paths between global source and core input ports, with thick solid and broken lines. Figure 11 also shows the shortest paths between two output ports of 1 and the global sink using the two different dotted lines. 4 The Structural Testing After we find all the shortest paths for all core input/output ports, we have to apply the bypass scheduling method explained in section 2 for each core. Finally, using some path scheduling method we have to combine these path schedules to overlap the test activities and minimize the test time as much as possible. The different steps of structural test process are summarized in the pseudocode of Figure 12. Briefly, the algorithm finds the best schedule of each input test path and schedules it according to the factorized form of the characteristic function. For the highest concurrency (test time minimization) the scheduling of each test path starts from the first possible time step, that is "1" for the input test paths transferring the test patterns to the core under test and "the time step that the core output signature becomes ready" for the test output paths. In this work, we have employed a simple As Soon As Possible scheduling method and consider scheduling the cores in order. Each path is scheduled in a pipeline fashion as explained in section 4 while the ASAP strategy is used to overlap the path execution times. STRUCTURAL TEST (Input/Output Test Shortest Construct and factorize the characteristic functions for input test paths; Construct and factorize the characteristic function for output test paths; for (all Core schedule ASAP the input test paths (1 - i - Nin k ) based on the available bypass/connections; test pathsg; Ttest application ; for (all Core schedule ASAP the output test paths (1 based on the available bypass/connections; Figure 12: Structural Test Algorithm 4.1 Possible Extensions We have considered only unidirectional interconnects and buses in this work. In our graph modeling an n-bit bidirectional bus is modeled as two p-bit and q-bit (p buses. Although such extension of our graph model is straight forward, the shortest-path formulation needs to be changed such that it can also decide for the best value of p and q in order to minimize the transfer time and bit-match overhead. At the moment, we can take the bidirectional buses into account only if the values for p and q are fixed (e.g. n=2). We don't address the general case in this paper but intend to consider it in near future. Not pursued in this paper, more sophisticated scheduling methods can be implemented for higher perfor- mance. For example, we can relax the above two assumptions, i.e. ASAP and fixed ordering. Moreover, by selecting a set of k disjoint (not necessarily the shortest) paths is selected to bypass test data for k different pairs of points concurrently, the test time will be reduced even further. For systems using few busses (e.g. VSI alliance bus wrapper design in which mainly three busses are responsible for connecting all cores in the system) our test scheme shows lower efficiency but can be still used to find the best order of bus access to optimize the test time. 4.2 Completing the Running Example Figure 13 shows all the input and output test shortest paths to all four cores in our example. The time cost is shown within the shaded circles attached to each core in the path. Figures 14 (a) through (d) show the complete schedules for the shortest paths of Core 1 through Core 4, respectively. We assumed that Core 1, 2, 3 and 4 require 9, 9, 11 and 25 cycles, respectively as test application time (the shaded area in Figures 14). Note also that to clarify the role of signals introduced before in section 2 we listed all signals. A "0" in this table means that signal is not active (low logic level). All other entries ("1", "2", "3" and "4") show an active signal (high logic level) is applied in the corresponding time step. We used different symbols (1 through 4) to refer to the core under test. For example, entries "4" show the signals to be active when Core 4 is under test. All empty locations in this table correspond to "\Theta" (don't care situation) and can be used for logic optimization in implementation of this test controller. Note that some of bypass ij k signals are defined "0" to guarantee data conflict on port j . Figure 15 shows the final schedule according to the scheduling method explained before. This scheduling is po1 po1 po1 po1 po2 po2 po2 MISR TPGR TPGR TPGR TPGR TPGR TPGR MISR MISR MISR MISR MISR MISR (a) (b) (c) (d) Infinity Infinity Infinity Infinity Infinity Infinity Infinity Infinity Infinity Infinity Infinity 44 Infinity Infinity TPGR Infinity 4Core 4 Figure 13: All the input/output shortest paths. basically obtained by overlapping the four schedules of Figure 14 and some time adjustment when some port or bypassing route is not available. This schedule shows that in the structural test session the input data packets (test patterns) and the output data packets (signatures) are somehow scrambled to overlap the activities and test data transfer in the test session. The importance of core capability to bypass when it executes a test pattern is clear in this figure. More specifically, when a core is under test, the bypass routes can transfer test data for other cores. This kind of overlapping reduces the test time dramatically. Note also in Figure 15 bypass 12 1 , bypass 22 2 and bypass 12 3 are omitted because the corresponding bypass routes are never needed. This removals will reduce the hardware overhead of the bypass circuitry. 5 Experimental Results In this section, we demonstrate our approach using a system made of four cores. The circuits have been synthesized from high level descriptions using the SYNTEST synthesis system [HPCN92]. Logic level synthesis is done using the ASIC Synthesizer from the COMPASS Design Automation suite of tools with a 0.8-micron CMOS library [Comp93]. Fault coverage curves are found for the resulting logic level circuits using AT&T's GENTEST fault simulator [Gent93]. The probability of aliasing within the MISRs is neglected, as are faults within the TPGRs and other test circuitry. Since standard core benchmarks have not been introduced yet we decided to design ourselves four example circuits as cores, all with eight bit wide datapaths. The first core (Core 2) evaluates a third degree polynomial. Our other examples implement three high level synthesis benchmarks: a differential equation solver (Core TPGR MISR stop_4 stop_3 stop_2 stop_1 send_port_1 read_port_1 read_port_2 read_port_3 send_port_2 Cores Signal Name 17 Under Test Time Steps TPGR MISR stop_4 stop_3 stop_2 stop_1 send_port_1 read_port_1 read_port_2 read_port_3 send_port_2 Cores Signal Name 17 Under Test22 Time Steps (a) Bypass schedule for Core 1 (b) Bypass schedule for Core 2 TPGR MISR stop_4 stop_3 stop_2 stop_1 send_port_1 read_port_1 read_port_2 read_port_3 send_port_2 Cores Signal Name 17 Under Test Time Steps TPGR MISR stop_4 stop_3 stop_2 stop_1 send_port_1 read_port_1 read_port_2 read_port_3 send_port_2 Cores Signal Name 17 Under Time Steps (c) Bypass schedule for Core 3 (d) Bypass schedule for Core 4 Figure 14: Bypass schedules for four cores. TPGR MISR stop_4 stop_3 stop_2 stop_1 send_port_1 read_port_1 read_port_2 read_port_3 4 44 Cores Signal Name Time Steps222 Figure 15: Complete bypass schedule (pipelined fashion) for all four cores. Core Circuit Normal Transistor Total Faults Fault Test Name Schedule Count Faults Detected Coverage Time Poly 9 8684 899 804 89.43 932 Table 3: Statistics for four cores when tested separately. Full Scan Test Structural Test Parameter Value Parameter Value Test Overhead Output 4484 Bit Match 3548 Test Controller 1045 Test Controller 6062 Test Per Pattern Core 2 101 Core 2 (Overlapped) 15 One Iteration 501 One Iteration 34 Fault Coverage 88.44% Fault Coverage 91.43% Test Test Statistics [Transistor] [Transistor] Test Time 57363 Test Time 9180 Table 4: Comparison between scan and our proposed structural test. and the Facet example (Core 1) [GDWL92], and a fifth order elliptical filter (Core 4) [KuWK85]. Note that each core consists of the datapath and controller both are made testable using BIST applied by SYNTEST in integrated fashion [HPCN92][NoCP97]. Also, to fit the cores in the port specification as Figure 8(a) shows we grouped or replicated the actual inputs/outputs of the circuit. Table 3 summarizes the transistor count, fault coverage and the test time for each core when tested as a non-embedded circuit separately. Note that the test time has been expressed in terms of "clock cycle". We execute the corresponding schedule of the core until the fault coverage curve is saturated. For example, for Diffeq core, the schedule consists of 11 cycles and the fault simulation process requires 70 iterations (each for a set of random patterns) until the curve is saturated. We then put these fours together and obtained the fault simulation of the whole system based on "scan" and our "structural" test. Table 4 summarizes the fault coverage statistics for these two methods. In scan testing, the order of cores in the scan-in and scan-out chain affects the test time. In our experiments we assumed a single scan-in chain with the order 4. In the single scan-out chain the order is assumed to be Core 1. The lower fault coverage of scan is expected because that method can not capture the interconnect faults. Our proposed structural test approach requires slightly more test circuitry (about 5.4%) than full scan mainly due to its test controller. Overall, the structural test overhead is 23.18% of the system cost in this example. Note that the test methods for individual cores (datapath and controller) achieve fault coverage in the range of of 87.97% to 92.81% (see Table 3). Thus, regardless of method employed for system testing achieving very high fault coverage in this system, without redesigning the cores, would not be possible. Using our structural Multiple-chain Structural Test Test Controller [Trans.] 4810 6062 Test Time [Cycle] 15390 9180 Tester Pin Table 5: Comparison between multiple-chain scan and our method. test approach, the fault coverage has increased by almost 3% mainly due to detecting the interconnect faults. The real advantage of our structural method is the test time since by bypass scheduling we overlap transferring test data between test points using the existing interconnections. This resulted in almost 84% test time reduction compared to scan. Note that in Table 4 we compared our method with single-chain scan. Using multiple-chain scan, obviously, will improve the test time with the expense of more overhead for control and wiring. To be more specific, the result of using the maximum number of scan chains, four in this example - one chain for each core - is summarized in Table 5. The basic idea of using multiple chains is to apply scan to different partitions in parallel. Independent multiple chains require dedicated control which increases the test control overhead by a factor of almost 4. The overall test overhead circuitry remain quite high. More importantly, independent chains require separate scan-in, scan-out, core-select and test=normal which increases the tester pins to 4 for four chains compared to one (test=normal) in our method. Test time reduction depends on the bottleneck core; the one with the largest test time which itself depends on inputs/outputs bit width, core execution time and number of test patterns required. In fact, in our example the bottleneck core is Core 1 with 48 bits inputs/outputs and requires about 270 test patterns. This resulted in test time of 15390 cycles which is still 67.5% higher than our proposed structural core testing method. 6 Summary We have proposed a test methodology for testing a core-based system in its entirety. A "bypass" mode circuitry is added to each core and is used to transfer test data from a source (data generation point) to a sink (data consumption point) through the existing interconnections. The system is modeled as a directed weighted graph in which the accessibility (of the core input and output ports) is solved as a shortest path problem. The test data distribution and collection of signatures are scrambled in a pipelined fashion to minimize test time. The experimental results are promising in terms of test time and quality testing of the interconnections among cores. --R "A Knowledge Based System for Designing Testable VLSI Chips," "Finding Defects with Fault Models," Testability Concepts for Digital ICs "A Unifying Methodology for Intellectual Property and Custom Logic Testing," Graph Theory With Applications "Test methodology for embedded cores which protects intellectual property," "Testability Analysis and Insertion for RTL Circuits Based on Pseudorandom BIST," "Testing Systems on a Chip," "User Manuals for COMPASS VLSI V8R4.4," Synthesis and optimization of digital circuits "User Manuals for GENTEST S 2.0," Introduction to Chip and System Design "SYNTEST: An Environment for System-Level Design for Test," "Direct Access Test Scheme - Design of Block and Core Cells for Embedded ASICs," "TestSockets: A Framework for System-On-Chip Design," VLSI and Modern Signal Processing "A Structured and Scalable Mechanism for Test Access to Embedded Reusable Cores," "A Structured and Scalable Mechanism for Test Access to Embedded Reusable Cores," "A Scheme for Integrated Controller-Datapath Fault Testing," "Test Synthesis in the Behavioral Domain," "Test Responses Compaction in Accumulators with Rotate Carry Adders," Application Specific Integrated Circuits High Performance Computer Architecture "Testing Embedded Cores Using Partial Isolation Rings," "0.8-Micron CMOS VSC450 Portable Library," Version 1.0 --TR --CTR Tomokazu Yoneda , Hideo Fujiwara, Design for Consecutive Testability of System-on-a-Chip with Built-In Self Testable Cores, Journal of Electronic Testing: Theory and Applications, v.18 n.4-5, p.487-501, August-October 2002 Tomokazu Yoneda , Masahiro Imanishi , Hideo Fujiwara, Interactive presentation: An SoC test scheduling algorithm using reconfigurable union wrappers, Proceedings of the conference on Design, automation and test in Europe, April 16-20, 2007, Nice, France Tomokazu Yoneda , Kimihiko Masuda , Hideo Fujiwara, Power-constrained test scheduling for multi-clock domain SoCs, Proceedings of the conference on Design, automation and test in Europe: Proceedings, March 06-10, 2006, Munich, Germany Mohammad Hosseinabady , Abbas Banaiyan , Mahdi Nazm Bojnordi , Zainalabedin Navabi, A concurrent testing method for NoC switches, Proceedings of the conference on Design, automation and test in Europe: Proceedings, March 06-10, 2006, Munich, Germany Rainer Dorsch , Hans-Joachim Wunderlich, Reusing Scan Chains for Test Pattern Decompression, Journal of Electronic Testing: Theory and Applications, v.18 n.2, p.231-240, April 2002 rika Cota , Luigi Carro , Marcelo Lubaszewski, Reusing an on-chip network for the test of core-based systems, ACM Transactions on Design Automation of Electronic Systems (TODAES), v.9 n.4, p.471-499, October 2004 rika Cota , Luigi Carro , Marcelo Lubaszewski , Alex Orailolu, Searching for Global Test Costs Optimization in Core-Based Systems, Journal of Electronic Testing: Theory and Applications, v.20 n.4, p.357-373, August 2004 Qiang Xu , Nicola Nicolici, Modular and rapid testing of SOCs with unwrapped logic blocks, IEEE Transactions on Very Large Scale Integration (VLSI) Systems, v.13 n.11, p.1275-1285, November 2005

shortest path;structural testing;pipelined test schedule;at-speed testing;bypass mode

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Test Wrapper and Test Access Mechanism Co-Optimization for System-on-Chip.

Test access mechanisms (TAMs) and test wrappers are integral parts of a system-on-chip (SOC) test architecture. Prior research has concentrated on only one aspect of the TAM/wrapper design problem at a time, i.e., either optimizing the TAMs for a set of pre-designed wrappers, or optimizing the wrapper for a given TAM width. In this paper, we address a more general problem, that of carrying out TAM design and wrapper optimization in conjunction. We present an efficient algorithm to construct wrappers that reduce the testing time for cores. Our wrapper design algorithm improves on earlier approaches by also reducing the TAM width required to achieve these lower testing times. We present new mathematical models for TAM optimization that use the core testing time values calculated by our wrapper design algorithm. We further present a new enumerative method for TAM optimization that reduces execution time significantly when the number of TAMs being designed is small. Experimental results are presented for an academic SOC as well as an industrial SOC.

Introduction System-on-chip (SOC) integrated circuits composed of proces- sors, memories, and peripheral interface devices in the form of embedded cores, are now commonplace. Nevertheless, there remain several roadblocks to rapid and efficient system integration. Test development is now seen as a major bottleneck in SOC design, and test challenges are a major contributor to the widening gap between design and manufacturing capability [23]. The 1999 International Technology Roadmap for Semiconductors [12] clearly identifies test access for SOC cores as one of the challenges for the near future. Test access mechanisms (TAMs) and test wrappers have been proposed as important components of an SOC test access architecture [23]. TAMs deliver pre-computed test sequences to cores on the SOC, while test wrappers translate these test sequences into patterns that can be applied directly to the cores. Test wrapper and TAM design is of critical importance in SOC system integration since it directly impacts the vector memory depth required on the ATE, as well as testing time, and thereby affects test cost. A TAM and wrapper design that minimizes the idle time spent by TAMs and wrappers during test directly reduces the number of don't-care bits in vectors stored on the tester, thereby reducing vector memory depth. The design of efficient test access architectures has become an important focus of research in core test integration [1, 3, 4, 5, 6, 14, 17, 19]. This is especially timely and relevant, This research was supported in part by the National Science Foundation under grant number CCR-9875324. since the proposed IEEE P1500 standard provides a lot of freedom in optimizing its standardized, but scalable wrapper, and leaves TAM optimization entirely to the system integrator [10, 18]. The general problem of SOC test integration includes the design of TAM architectures, optimization of the core wrappers, and test scheduling. The goal is to minimize the testing time, area costs, and power consumption during testing. The wrapper/TAM co-optimization problem that we address in this paper is as follows. Given the test set parameters for the cores on the SOC, as well as the total TAM width, determine an optimal number of TAMs for the SOC, an optimal partition of the total TAM width among the TAMs, an optimal assignment of cores to each TAM, and an optimal wrapper design for each core, such that the overall system testing time is minimized. In order to solve this problem, we examine a progression of three incremental problems structured in order of increasing com- plexity, such that they serve as stepping-stones to the more general problem of wrapper/TAM design. The first problem PW is related to test wrapper design. The next two problems PAW and PPAW are related to wrapper/TAM co-optimization. a wrapper for a given core, such that (a) the core testing time is minimized, and (b) the TAM width required for the core is minimized. 2. PAW : Determine (i) an assignment of cores to TAMs of given widths and (ii) a wrapper design for each core, such that SOC testing time is minimized. (Item (ii) equals PW .) 3. a partition of the total TAM width among the given number of TAMs, (ii) an assignment of cores to the TAMs, and (iii) a wrapper design for each core, such that SOC testing time is minimized. (Items (ii) and (iii) together equal PAW .) These three problems lead up to PNPAW , the more general problem of wrapper/TAM co-optimization described as follows. the number of TAMs for the SOC, (ii) a partition of the total TAM width among the TAMs, (iii) an assignment of cores to TAMs, and (iv) a wrapper design for each core, such that testing time is minimized. (Items (ii), (iii) and (iv) together equal In the remainder of this paper, we formally define and analyze these problems, and propose solutions for each. In this paper, we assume the "test bus" model for TAMs. We assume that the TAMs on the SOC operate independently of each however, the cores on a single TAM are tested sequentially. This can be implemented either by (i) multiplexing all the cores assigned to a TAM, or (ii) by testing one of the cores on the TAM, while the other cores on the TAM are bypassed. Furthermore, in this paper, we are addressing the problem of core test only; hence, we do not discuss issues related to test wrapper bypass and interconnect test. The organization of this paper is as follows. In Section 2, we discuss prior work in the area of TAM and test wrapper design. In Section 3, we present two SOCs that we use as running examples throughout the paper. In Section 4, we address Problem PW . In Section 5, we develop improved integer linear programming (ILP) models for optimizing core assignment to TAMs (Problem PAW ). In Section 6, we present new ILP models for TAM width partitioning (Problem PPAW ). In Section 7, we present an enumerative method that can often reduce the execution time required to solve PPAW when the number of TAMs is small. Finally, in Section 8, we examine PNPAW , the general problem of wrapper/TAM co-optimization. Section 9 concludes the paper. Prior work Test wrappers provide a variety of operation modes, including normal operation, core test, interconnect test, and (optional) by-pass [16]. In addition, test wrappers need to be able to perform test width adaptation if the width of the TAM is not equal to the number of core terminals. The IEEE P1500 standard addresses the design of a flexible, scalable wrapper to allow modular testing [10, 18]. This wrapper is flexible and can be optimized to suit the type of TAM and test requirements for the core. A "test collar" was proposed in [21] to be used as a test wrapper for cores. However, test width adaptation and interconnect test were not addressed. The issue of efficient de-serialization of test data by the use of balanced wrapper scan chains was discussed in [6]. Balanced wrapper scan chains, consisting of chains of core I/Os and internal scan chains, are desirable because they minimize the time required to scan in test patterns from the TAM. However, no mention was made of the method to be used to arrive at a balanced assignment of core I/Os and internal scan chains to TAM lines. The TESTSHELL proposed in [16] has provisions for the IEEE P1500 required modes of operation. Furthermore, heuristics for designing balanced wrapper scan chains, based on approximation algorithms for the well-known Bin Design problem [7], were presented in [17]. However the issue of reducing the TAM width required for a wrapper was not addressed. A number of TAM designs have been proposed in the literature. These include multiplexed access [11], partial isolation rings [20], core transparency [8], dedicated test bus [21], reuse of the existing system bus [9], and a scalable bus-based architecture called RAIL [16]. Bus-based TAMs, being flexible and scalable, appear to be the most promising. However, their design has largely been ad hoc and previous methods have seldom addressed the problem of minimizing testing time under TAM width constraints. While [1] presents several novel TAM architectures (i.e., multiplexing, daisy chaining and distribution), it does not directly address the problem of optimal sizing of TAMs in the SOC. In particular, only internal scan chains are considered in [1], while wrappers and functional I/Os are ignored. Moreover, the lengths of the internal scan chains are not considered fixed, and therefore [1] does not directly address the problem of designing test architectures for hard cores. More recently, integrated TAM design and test scheduling has been attempted in [14, 19]. However, in [14, 19], the problem of optimizing test bus widths and arbitrating contention among cores for test width was not addressed. In [19], the cost estimation for TAMs was based on the number of bridges and multiplexers used; the number of TAM wires was not taken into consideration. Fur- thermore, in [14] the impact of TAM widths on testing time was not included in the cost function. The relationship between testing time and TAM widths using ILP was examined in [3, 5], and TAM width optimization under power and routing constraints was studied in [4]. However, the problem of effective test width adaptation in the wrapper was not addressed. This led to an overestimation of testing time and TAM width. Improved wrapper designs and new ILP models for TAM design are therefore necessary. In this paper, we present a new wrapper/TAM co-optimization methodology that overcomes the limitations of previous TAM design approaches that have addressed TAM optimization and wrapper design as independent problems. The new wrapper design algorithm that we present improves upon previous approaches by minimizing the core testing time, as well as reducing the TAM width required for the core. We propose an approach based on ILP to solve the problems of determining an optimal partition of the total TAM width and determining an optimal core assignment to the TAMs. We also address a new problem, that of determining the optimal number of TAMs for an SOC. This problem gains importance with increasing SOC size. This paper, to the best of our knowledge, is the first in which a wrapper/TAM co-optimization methodology has been applied to an industrial SOC. 3 Example SOCs In order to illustrate the proposed wrapper/TAM co-optimization methods presented in this paper, and to demonstrate their effective- ness, we use two representative SOCs as running examples through-out this paper. The first one is an academic SOC named d695 (de- scribed as system S2 in [5]), and the second one is an industrial SOC from Philips, named p93791. The number (e.g., 93791) in each SOC name is a measure of its test complexity. This number is calculated by considering the numbers of functional inputs n i , functional outputs chains sc i , internal scan chain lengths l i;1 ; l test patterns p i for each Core i, as well as the total number of cores N in the SOC, all of which contribute to the complexity of the wrapper/TAM co-optimization prob- lem. We calculate the SOC test complexity number using the formula l i;r ). The letters "d" and "p" in d695 and p93791 refer to Duke University and Philips, respectively. SOC d695 consists of two combinational ISCAS'85 and eight sequential benchmark circuits. Table 1 presents the test data for each core in d695. We assume that the ISCAS'89 circuits contain well-balanced internal scan chains. The proposed wrapper/TAM co-optimization methodology is also applicable to SOCs containing non-scanned sequential cores, since these cores can be treated as combinational (having zero-length internal scan chains) for the purpose of testing time calculation. SOC p93791 contains cores. Of these, are memory cores embedded within hierarchical logic cores. Table 2 presents the data for the 14 logic cores and embedded memories in SOC p93791. We do not describe each core in p93791 individually due to insufficient space. Test data for each individual core in p93791 is presented in [13]. The experimental results presented in this paper were obtained using a Sun Ultra 80 with a 450 MHz processor and 4096 MB memory Number of Scan chain Circuit Test Functional Scan lengths patterns I/Os chains Min Max Table 1. Test data for the cores in d695 [5]. Number range Scan chain Circuit Test Functional Scan lengths patterns I/Os chains Min Max Logic cores 11-6127 109-813 11-46 1 521 Memory cores 42-3085 21-396 - Table 2. Ranges in test data for the 32 cores in p93791. 4 Test wrapper design A standardized, but scalable test wrapper is an integral part of the IEEE P1500 working group proposal [10]. A test wrapper is a layer of DfT logic that connects a TAM to a core for the purpose of test [23]. Test wrappers have four main modes of operation. These are (i) Normal operation, (ii) Intest: core-internal testing, (iii) Extest: core-external testing, i.e., interconnect test, and (iv) Bypass mode. Wrappers may need to perform test width adaptation when the TAM width is not equal to the number of core terminals. This will often be required in practice, since large cores typically have hundreds of core terminals, while the total TAM width is limited by the number of SOC pins. In this paper, we address the problem of TAM design for Intest, and therefore we do not discuss issues related to Bypass and Extest. The problem of designing an effective width adaptation mechanism for Intest can be broken down into three problems [17]: (i) partitioning the set of wrapper scan chain elements (internal scan chains and wrapper cells) into several wrapper scan chains, which are equal in number to the number of TAM lines, (ii) ordering the scan elements on each wrapper chain, and (iii) providing optional bypass paths across the core. The problems of ordering scan elements on wrapper scan chains and providing bypass paths were shown to be simple in [17], while that of partitioning wrapper scan chain elements was shown to be NP-hard. Therefore, in this sec- tion, we address only the problem of effectively partitioning wrapper scan chain elements into wrapper scan chains. Recent research on wrapper design has stressed the need for balanced wrapper scan chains [6, 17]. Balanced wrapper scan chains are those that are as equal in length to each other as possible. Balanced wrapper scan chains are important because the number of clock cycles to scan in (out) a test pattern to (from) a core is a function of the length of the longest wrapper scan-in (scan-out) chain. be the length of the longest wrapper scan-in (scan-out) chain for a core. The time required to apply the entire test set to the core is then given by where p is the number of test patterns. This time T decreases as both s i and so are reduced, i.e., as the wrapper scan-in (and scan- out) chains become more equal in length. Figure 1 illustrates the difference between balanced and unbalanced wrapper scan chains; Bypass and Extest mechanisms are not shown. In Figure 1(a), wrapper scan chain 1 consists of two input cells and two output cells, while wrapper scan chain 2 consists of three internal scan chains that contain 14 flip-flops in total. This results in unbalanced wrapper scan-in/out chains and a scan-in and scan-out time per test pattern of 14 clock cycles each. On the other hand, with the same elements and TAM width, the wrapper scan chains in Figure 1(b) are balanced. The scan-in and scan-out time per test pattern is now 8 clock cycles. Wrapper scan chain 1 Wrapper scan chain 2 Wrapper scan chain 2 Wrapper Wrapper scan chain 1 Wrapper (a) (b) Figure 1. Wrapper chains: (a) unbalanced, (b) balanced. The problem of partitioning wrapper scan chain elements into balanced wrapper scan chains was shown to be equivalent to the well-known Multiprocessor Scheduling and Bin Design problems in [17]. In this paper, the authors presented two heuristic algorithms for the Bin Design problem to solve the wrapper scan chain element partitioning problem. Given k TAM lines and sc internal scan chains, the authors assigned the scan elements to m wrapper scan chains, such that maxfs i ; sog was minimized. This approach is effective if the goal is to minimize only maxfs sog. However, we are addressing the wrapper design problem as part of the more general problem of wrapper/TAM co-optimization, and therefore we would also like to minimize the number of wrapper chains created. This can be explained as follows. Consider a core that has four internal scan chains of lengths 32, 8, 8, and 8, respectively, 4 functional inputs, and 2 functional outputs. Let the number of TAM lines provided be 4. The algorithms in [17] will partition the scan elements among four wrapper scan chains as shown in Figure 2(a), giving However, the scan elements may also be assigned to only 2 wrapper scan chains as shown in Figure 2(b), which also gives maxfs 32. The second assignment, however, is clearly more efficient in terms of TAM width utilization, and therefore would be more useful for a wrapper/TAM co-optimization strategy Consider Core 6, the largest logic core from p93791. Core 6 has 417 functional inputs, 324 functional outputs, 72 bidirectional I/Os, and 46 internal scan chains of lengths: 7 scan chains 500 bits, scan chains 520 bits, and 9 scan chains 521 bits, re- spectively. The Combine algorithm [17] was used to create wrapper configurations for Core 6, for values of k between 1 and 64 bits. Since the functional inputs in Core 6 outnumber the functional out- puts, . The value of s i obtained for each value of k is illustrated in Figure 3. From the graph, we observe that as increases, s i decreases in a series of distinct steps. This is be- Scan chain - Scan chain - 8 FF Scan chain - 8 FF I O I I Scan chain - 8 FF O I (a) Scan chain - O I I I I Scan chain - 8 FF Scan chain - 8 FF Scan chain - 8 FF O (b) Figure 2. Wrapper design example using (a) four wrapper scan chains, and (b) two wrapper scan chains. cause as k increases, the core internal scan chains are redistributed among a larger number of wrapper scan-in chains; thus s i decreases only when the increase in k is sufficient to remove an internal scan chain from the longest wrapper scan-in chain. For example, when the internal scan chains in Core 6 are distributed among 24 wrapper scan-in chains, s long. The value of s i remains at 1040 until k reaches 39, when s i drops to 1020. Hence, for 24 k 38, only 24 wrapper scan-in chains need be designed. Our wrapper design strategy is therefore to (i) minimize testing time by minimizing identify the maximum number k 0 of wrapper scan chains that actually need to be created to minimize testing time, when k TAM lines are provided to the wrapper. The set of values of corresponding to the values of 1 k 1 is known as the set of pareto-optimal points for the graph. TAM width (bits) 28 Longest wrapper scan-in chain Figure 3. Decrease in s i with increasing k for Core 6. PW , the two-priority wrapper optimization problem that this section addresses can now be formally stated as follows. Given a core with n functional inputs, m functional out- puts, sc internal scan chains of lengths l 1 tively, and TAM width k, assign the n +m+ sc wrapper scan chain elements to k 0 k wrapper scan chains such that (i) is the length of the longest wrapper scan-in (scan-out) chain, and (ii) k 0 is minimum subject to priority (i). Priority (ii) of PW is based on the earlier observation that can be minimized even when the number of wrapper scan chains designed is less than k. This reduces the width of the TAM required to connect to the wrapper. Problem PW is therefore analogous to the problem of Bin Design (minimizing the size of the bins), with the secondary priority of Bin Packing (minimiz- ing the number of bins). If the value of kmax in Problem PW is always fixed at k, then Problem PW reduces to the Partitioning of Scan Chains (PSC) Problem [17], and is therefore clearly NP-hard, since Problem PSC was shown to be NP-hard in [17]. We have developed an approximation algorithm based on the Best Fit Decreasing (BFD) heuristic [7] to solve PW efficiently. The algorithm has three main parts, similar to [17]: (i) partition the internal scan chains among a minimal number of wrapper scan chains to minimize the longest wrapper scan chain length, (ii) assign the functional inputs to the wrapper scan chains created in part (i), and (iii) assign the functional outputs to the wrapper scan chains created in part (i). To solve part (i), the internal scan chains are sorted in descending order. Each internal scan chain is then successively assigned to the wrapper scan chain, whose length after this assignment is closest to, but not exceeding the length of the current longest wrapper scan chain. Intuitively, each internal scan chain is assigned to the wrapper scan chain in which it achieves the best fit. If there is no such wrapper scan chain available, then the internal scan chain is assigned to the current shortest wrapper scan chain. Next the process is repeated for part (ii) and part (iii), considering the functional inputs and outputs as internal scan chains of length 1. The pseudocode for our algorithm Design wrapper is illustrated in Figure 4. Procedure Design wrapper Part (i) 1. Sort the internal scan chains in descending order of length 2. For each internal scan chain l 3. Find wrapper scan chain Smax with current maximum length 4. Find wrapper scan chain Smin with current minimum length 5. Assign l to wrapper scan chain S, such that 6. If there is no such wrapper scan chain S then 7. Assign l to Smin Part (ii) 8. Add the functional inputs to the wrapper chains created in part (i) Part (iii) 9. Add the functional outputs to the wrapper chains created in part (i) Figure 4. Algorithm for wrapper design that minimizes testing time and number of wrapper scan chains. We base our algorithm on the BFD heuristic mainly because BFD utilizes a more sophisticated partitioning rule than First Fit Decreasing since each scan element is assigned to the wrapper scan chain in which it achieves the best fit [7]. FFD was used as a subroutine to the wrapper design algorithm in [17]. In our algorithm, a new wrapper scan chain is created only when it is not possible to fit an internal scan chain into one of the existing wrapper scan chains without exceeding the length of the current longest wrapper scan chain. Thus, while the algorithms presented in [17] always use k wrapper scan chains, Design wrapper uses as few wrapper scan chains as possible, without compromising test application time. The worst-case complexity of the Design wrapper algorithm is O(sc log sc + sc k), where sc is the number of internal scan chains and k is the limit on the number of wrapper scan chains. From Figure 3, we further observe that as k is increased beyond 47, there is no further decrease in testing time, since the longest internal scan chain of the core has been assigned to a dedicated wrapper scan chain, and there is no other wrapper scan chain longer than the longest internal scan chain. We next derive an expression for this maximum value of TAM width kmax required to minimize testing time for a core. Theorem 1 If a core has n functional inputs, m functional outputs, and sc internal scan chains of lengths l 1 respectively, an upper bound kmax on the TAM width required to minimize testing time is given by Proof: The test application time for a core is given by is the length of the longest wrapper scan-in (scan-out) chain, and p is the number of test patterns in the test set. A lower bound on T , for any TAM width k, is therefore given by (1 sogg. The lowest value that maxfs i ; sog and can attain, is given by the length of the core's longest internal scan chain g. Therefore, g. Let the upper bound on k at which minfTg is reached be denoted as kmax . At this value of k, the number of flip-flops assigned to each wrapper scan chain (either scan-in or scan-out, whichever has more flip-flops) is at most max i g. Therefore kmax is the smallest integer, such that kmax is at least the sum of all the flip-flops on the wrapper scan chains, l i . Thus, kmax is the smallest integer, such that kmax maxfn;mg+ . Therefore, l i . 2 The value of kmax for each core can further be used to determine an upper bound on the TAM width for any TAM on the SOC. In Section 6, we show how kmax can be used to bound the TAM widths, when obtaining an optimal partition of total test width among TAMs on the SOC. Table 3 presents results on the savings in TAM width obtained using Design wrapper for Core 6. For larger values of k, the number of TAM lines actually used is far less than the number of available TAM lines; thus, with respect to TAM width utilization, Design wrapper is considerably more efficient than the wrapper design algorithms proposed in [17]. Available TAM Longest TAM width wrapper width utilized scan chain 9 9 3081 Available TAM Longest TAM width wrapper width utilized scan chain 22 22 1521 43-45 43 1000 Table 3. Design wrapper results for Core 6. In the next section, we address PAW , the second problem of our TAM/wrapper co-optimization framework-determining an assignment of cores to TAMs of given widths and optimizing the wrapper design for each core. 5 Optimal core assignment to TAMs In this paper, we assume the "test bus" model for TAM design. We assume that each of the B TAMs on the SOC are independent; however, the cores on each TAM are tested in sequential order. This can be implemented either by (i) multiplexing all the cores assigned to a TAM as in Figure 5(a), or (ii) by testing one of the cores on the TAM, while the other cores on the TAM are in Bypass mode as in Figure 5(b). Furthermore, the core bypass may either be an internal bypass within the wrapper or an external bypass. This paper does not directly the address the design of hierarchical TAMs. The SOC hierarchy is flattened for the purpose of TAM design and hierarchical cores are considered as being at the same level in test mode. Multiplexed cores Core A Core A Internal bypass External bypass (a) (b) Figure 5. Test bus model of TAM design: (a) multiplexed cores, (b) cores with bypass on a test bus. The problem that we examine in this section, that of minimizing the system testing time by assigning cores to TAMs when TAM widths are known, can be stated as follows. Given N cores and B TAMs of test widths determine an assignment of cores to the TAMs and a wrapper design for each core, such that the testing time is minimized. This problem can be shown to be NP-hard using the techniques presented in [5]. However, for realistic SOCs the sizes of the problem instances were found to be small and could be solved exactly using an ILP formulation in execution times less than a second. To model this problem, consider an SOC consisting of N cores and B TAMs of widths w1 ; assigned to TAM j, let the time taken to test Core i be given by T i (w j ) clock cycles. The testing time T i (w j ) is calculated as T i (w is the number of test patterns for Core i and s i (so ) is the length of the longest wrapper scan-in (scan-out) chain obtained from Design wrapper. We introduce binary variables x ij (where 1 i N and 1 j B), which are used to determine the assignment of cores to TAMs in the SOC. Let x ij be a 0-1 variable defined as follows: is assigned to TAM j The time needed to test all cores on TAM j is given by Since all the TAMs can be used simultaneously for testing, the system testing time equals A mathematical programming model for this problem can be formulated as follows. Objective: Min. subject to every core is connected to exactly one TAM. Before we describe how a solution to PAW can be obtained, we briefly describe ILP, and then present the ILP formulation based on the above mathematical programming model to solve PAW . The goal of ILP is to minimize a linear objective function on a set of integer variables, while satisfying a set of linear constraints [22]. A typical ILP model can described as follows: minimize: Ax subject to: Bx C, such that x 0, where A is an objective vector, B is a constraint matrix, C is a column vector of constants, and x is a vector of integer variables. Efficient ILP solvers are now readily available, both commercially and in the public domain [2]. The minmax objective function of the mathematical programming model for PAW can be easily linearized to obtain the following ILP model. Objective: Minimize T , subject to 1. T is the maximum testing time on any TAM 2. every core is assigned to exactly one TAM We solved this simple ILP model to determine optimal assignments of cores to TAMs for the SOCs introduced in Section 3. The number of variables and constraints for this model (a measure of the complexity of the problem) is given by NB, and N respectively. The user time was less than a second in all cases. The optimal assignments of cores to TAMs of given widths for SOC d695 are shown in Table 4. Note that the testing times shown are optimal only for the given TAM widths; lower testing times can be achieved if an optimal TAM width partition is chosen. For example, Table 5 in Section 6 shows that a testing time lower than 29451 cycles can be achieved using two TAMs, if an optimal TAM width partition is cho- sen. In Sections 6 and 7, we will address the problem of determining optimal width partitions. TAM TAM Testing time widths assignment (clock cycles) Table 4. Core assignment to TAMs for SOC d695. Lower bounds on system testing time. For an SOC with N cores and B TAMs of widths w1 ; respectively, a lower bound on the total testing time T is given by g. The testing time for Core i depends on the width of the test bus to which it is assigned. Clearly, the testing time for Core i is at least min j fT i (w j )g. Since the overall system testing time is constrained by the core that has the largest test time, therefore g. Intuitively, this value is the time needed to test the core that has the largest testing time when assigned to the widest TAM. For SOC d695 with two TAMs of bits and 16 bits, respectively, the lower bound on the testing time is 6215 cycles. This corresponds to the testing time needed for Core 5 if it is assigned to TAM 1. A lower bound on system testing time that does not depend on the given TAM widths can further be determined. This bound is related to the length of the longest internal scan chain of each core. The lower bound becomes tighter as we increase the number of TAMs. From Theorem 1, we know that for a Core i, where Core i has sc i internal scan chains of lengths l i;1 ; l respectively, and the test set for Core i has p i test patterns, a lower bound on the testing time is given by (1+maxrfl i;r g) p i +maxrfl i;r g. Therefore, for an SOC with N cores, a lower bound on the system testing time is given by gg. Intuitively, this means that the system testing time is lower bounded by the time required to test the core with the largest testing time. 6 Optimal partitioning of TAM width In this section, we address PPAW : the problem of determining (i) optimal widths of TAMs, and (ii) optimal assignments of cores to TAMs, in conjunction with wrapper design. This is a generalization of the core assignment problem PAW . We describe how testing times computed using the Design wrapper algorithm in Section 4 are used to design the TAM architecture. We assume that the total system TAM width can be at most W . From Theorem 1 in Section 4, we know that the width of each TAM need not exceed the maximum value of upper bound kmax for any core on the SOC. We denote this upper bound on the width of an individual TAM as wmax . For TAMs wider than wmax , there is no further decrease in testing time. Problem PPAW of minimizing testing time by optimal allocation of width among the TAMs can now be stated as follows. Given an SOC having N cores and B TAMs of total width W , determine a partition of W among the B TAMs, an assignment of cores to TAMs, and a wrapper design for each core, such that the total testing time is minimized. This problem can be shown to be NP-hard using the techniques presented in [5]. However, for many realistic SOCs, including p93791, the problem instance size is reasonable and can be solved exactly using an ILP formulation. This is also because the complexity of our solutions is related to the number of cores on the SOC and the numbers of their I/O ports and scan chains, and not the number of transistors or nets on the chip. A mathematical programming model for PPAW is shown below. Objective: Min. subject to 1. every core is connected to exactly one TAM 2. i.e., the sum of all TAM widths is 3. w j wmax , 1 j B, i.e., each TAM is at most wmax bits wide The objective function and constraints of this mathematical programming model must now be linearized in order to express them in the form of an ILP model that can be solved by an ILP solver. We first express T i (w j ) as a sum: T i (w adding new binary indicator variables - jk (where 1 j B; 1 to the mathematical programming model, such that: 1, if TAM j is k bits wide 0, otherwise In addition, the following constraints are included in the model: 1. a TAM can have values of width between 1 and wmax 2. a TAM can have only one width Intuitively, for every TAM j there is exactly one value of k for which therefore, the new indicator variables determine the width w j of each TAM. The objective function now becomes g. The testing time T i (k) for various values of TAM width k can be efficiently calculated using the Design Wrapper algorithm as shown in Section 4, and stored in the form of a look-up table for reference by the ILP solver. Finally, the non-linear term - jk x ij in the objective function can be linearized by replacing it with the variable y ijk and the following two constraints: 2. This is explained as follows. Consider first the case when x From Constraints 1 and 2, we have y ijk +1 - jk and 2y ijk - jk ; since - jk 1, therefore, y ijk must equal The new variables and constraints yield the following ILP formulation Objective: Minimize subject to 2. 3. 4. 5. 6. The number of variables and constraints for these ILP models is given by B (3N respectively. We solved the ILP model for PPAW for several values of W and B. Table 5 and Figure 6 present the values of testing time for SOC d695 obtained with two TAMs. The total TAM width partition among the two TAMs is shown and we also compare the testing times obtained with the testing times obtained in [5]. The testing time using the new wrapper design is at least an order of magnitude less than the time required in [5] for all cases. This was to be expected since an inefficient de-serialization model was used in [5]. The reductions in testing time diminish with increasing W . A pragmatic choice of W for the system might therefore be the point where the system testing time begins to level off. In Figure 6, this occurs at Total Results in [5] Current wrapper/TAM co-optimizaion TAM Partition Testing Partition Testing Execution 28 36 4+32 2174501 16+20 22246 11.0 44 12+32 2123437 10+34 20094 13.0 48 reported in [5] Table 5. Testing time for d695 for 2. Total TAM width (bits) 28 6420406080100120140160Testing time (1000 clock cycles) Figure 6. Testing time for d695 for 2. Table 6 presents the values of testing time obtained with three TAMs. The testing times for are lower than the values obtained in general. However, for W 14, the testing time more than that for included in Table 6). This is because for small values of W; a larger number of TAMs makes the widths of individual TAMs very small. Once again, the testing time begins to level off, this time at hence this is a good choice for trading off TAM width with testing time. Total TAM width Execution TAM partition Core Testing time 28 4+8+16 (2,1,2,1,3,2,3,1,1,3) 24021 52.3 36 4+16+16 (2,2,2,1,2,3,2,1,2,3) 19573 85.0 44 4+18+22 (1,1,1,1,2,3,2,1,2,3) 16975 48 4+18+26 (1,1,1,1,2,3,2,1,2,3) 16975 lpsolve was halted after 180 minutes. Table 6. Testing time for d695 obtained for 3. Table 7 presents the system testing times for SOC p93791 obtained using two TAMs. We halted the ILP solver after 1 hour for each value of W and tabulated the best results obtained. This was done to determine whether an efficient partition of TAM width and the corresponding testing time can be obtained using the ILP model within a reasonable execution time. In the next section, we present the optimal testing times obtained for p93791 using a new enumerative methodology, and show that the testing times obtained in Table 6 and Table 7 using the ILP model for PPAW are indeed either optimal or close to optimal. Testing 28 9+19 1119160 36 9+27 924909 Testing 44 9+35 873276 48 9+39 835526 52 9+43 807909 Table 7. Testing time for p93791 obtained with 2. 7 Enumerative TAM sizing In Section 6, we showed that TAM optimization can be carried out using an ILP model for the PPAW problem. However, ILP is in itself an NP-hard problem, and execution times can get high for large SOCs. A faster algorithm for TAM optimization that produces optimal results in short execution times is clearly needed. In Section 5, it was observed that the execution time of the model for Problem PAW was less than 1 second in all cases. We next demonstrate how the short execution time of this ILP model can be exploited to construct a series of PAW models that are solved to address the PPAW problem. The pseudocode for an enumerative algorithm for PPAW that explicitly enumerates the unique partitions of W among the individual TAMs is presented in Figure 7. Procedure PPAW enumerate() 1. Let 2. number of TAMs 3. While all unique partitions of W have not been enumerated 4. For TAM 5. For TAM width w (j 1) 6. Create an ILP model for PA for the TAM widths using Design wrapper 7. Determine core assignment and testing time 8. Record the TAM design for the minimum testing time Figure 7. Algorithm for enumerative TAM design. The ILP models generated for each value of W in line 6 of PPAW enumerate are solved and the TAM width partition and core assignment delivering the best testing time are recorded. The solution obtained using PPAW enumerate is always optimal, because we generate all unique TAM width partitions, and then choose the solution with the lowest cost. Since lines 6 and 7 each take less than a second to execute, the execution time for PPAW enumerate is at most 2 pB (W ) seconds, where pB (W ) is the number of partitions of W among B TAMs enumerated in line 3. The problem of determining the number of partitions pB (W ) for a given choice of B TAMs can be addressed using partition theory in combinatorial mathematics [15]. In [15], pB (W ) is shown to be approximately B!(B 1)! for W 7! 1. For W , since there are only W unique ways of dividing an integer W into two smaller integers w1 and w2 . Thus PPAW enumerate obtains the optimal solution for the PPAW problem for than 64 seconds. For 3, the number of partitions work out to p3(W W (3i+1) . From the above formula, the value of C for is found to be 341. There- fore, the execution time of PPAW enumerate for bounded by 682 seconds or 11.6 minutes. This execution time is clearly reasonable, even for large W . In our experiments, we used a Sun Ultra 80, which solved the PA models in well under a second of execution time. The time taken for PPAW enumerate was therefore significantly lower than the upper bound of 2 pB (W ) seconds even for large values of W . We used PPAW enumerate to obtain the optimal TAM width assignment and minimal testing time for d695 and p93791. Results for SOC d695 are presented in Table 8. While the testing time for 3 is always less than the testing time the difference between widens for larger W . This can be explained as follows. For smaller values of W , each individual TAM for the testing time on each individual TAM increases sharply, as was observed earlier in Figure 3. Exec. Exec. time 28 44 10+34 20094 2 4+18+22 16975 48 Table 8. Results for SOC d695 using PPAW enumerate. Table 9 presents optimal results for enumerative TAM optimization for the p93791 SOC. Comparing these results with those presented in Table 7, we note that the results in Table 7 are indeed close to optimal. For example, for the testing time presented in Table 7 is only 5% higher than optimal. Note that for both SOCs, the execution time for PAW is under 1 second. Hence similar execution times for PPAW enumerate are obtained for SOCs d695 and p93791. These execution times are significantly lower than those in Tables 5 and 6. Exec. Exec. time 28 5+23 1031200 1 2+3+23 1030920 13 44 21+23 711256 2 5+16+23 659856 33 48 23+25 634488 2 9+16+23 602613 42 Table 9. Results for SOC p93791 using PPAW enumerate. We compared the optimal testing times presented in Tables 8 and 9 with the testing times obtained using an equal partition of W among the B TAMs. The testing time using an optimal partition of W was significantly lower than that obtained using an equal partition for all values of W . For example, for a partition of (w1 ; testing time of clock cycles, which is an increase of 28.6% over the testing time of 475598 clock cycles obtained using an optimal partition of The execution time of PPAW enumerate is smaller than that of the ILP model in Section 6 because the number of enumerations for two and three TAMs is reasonable. However, when TAM optimization is carried out for a larger number of TAMs that have a larger number of partitions of W , the ILP model for PPAW is likely to be more efficient in terms of execution time. In addition, the ILP model presented in Section 6 is likely to be more efficient when constraints arising from place-and-route and power issues are included in TAM optimization [4]. 8 General problem of wrapper/TAM co- optimization In the previous sections, we presented a series of problems in test wrapper and TAM design, each of which was a generalized version of the problem preceding it. In this section, we present PNPAW , the more general problem of wrapper/TAM optimization that the problems of the preceding sections lead up to. We also show how solutions to the previous problems can be used to formulate a solution for this general problem. The general problem can be stated as follows. Given an SOC having N cores and a total TAM width W , determine the number of TAMs, a partition of W among the TAMs, an assignment of cores to TAMs, and a wrapper design for each core, such that the total testing time is minimized. We use the method of restriction to prove that PNPAW is NP- hard. We first define a new Problem PNPAW 1 , which consists of only those instances of PNPAW for which (i) cores on the SOC have a single internal scan chain and no functional terminals. Hence, each core will have the same testing time on a 1-bit TAM as on a 2-bit TAM. An optimal solution to PNPAW 1 will therefore always result in two TAMs of width one bit each. Problem reduces to that of partitioning the set C of cores on the SOC into two subsets C1 and C C1 , such that each subset is assigned to a separate 1-bit TAM, and the difference between the sum of the testing times of the cores (on the first 1-bit TAM) and the sum of the testing times of the cores (on the 2nd 1-bit TAM) is minimized. Formally, the optimization cost function for PNPAW 1 can be written as: Objective: Minimize is the testing time of core c on a 1-bit TAM. Next, consider the Partition problem [7], whose optimization variant can be stated as follows. Partition: Given a finite set A and a size s(a) 2 Z + for each element a 2 A, determine a partition of A into two subsets A1 and A A1 , such that s(a) s(a) is minimized. That Problem PNPAW 1 is equivalent to Partition can be established by the following four mappings: (i) C A, (ii) C1 A1 , s(a). Since Partition is known to be NP-hard [7], PNPAW 1 and PNPAW must also be NP-hard. To solve PNPAW , we enumerate solutions for PPAW over several values of B. For each value of W , the optimal number of TAMs, TAM width partition, core assignment, and wrapper designs for the cores are obtained. The solutions to PPAW for d695 for values of B ranging from 2 to 8 are illustrated in Figures 8 (a) and 8 (b) for W values of 12, and 16 bits, respectively. In each Figure, we observe that as B is increased from 2, the testing time decreases until a minimum value is reached at a particular value of B, after which the testing time stops decreasing and starts increasing as B is increased further. This is because for larger B, the width per TAM is small and testing time on each TAM increases significantly. Number of TAMs Testing time (1000 clock cycles) Number of TAMs 842.547.552.557.5Testing time (1000 clock cycles) (a) Figure 8. Testing time for d695 obtained with increasing values of B. We next present a conjecture that formalizes the observation made in Figures 8(a) and 8(b). Conjecture 1 Let T (S; W;B) denote the optimal testing time for and a total TAM width of W . If We conjecture that during the execution of PNPAW enumerate, if at a certain value of B, the testing time is greater than or equal to the testing time at the previous value of B for the same total TAM width W , then the enumeration procedure can be halted and the optimal value of B recorded. Therefore, Conjecture 1 can be used to prune the search space for the optimal wrapper/TAM design. Since the execution time of PNPAW enumerate is particularly high for large values of B, we can achieve significant speed-ups in TAM optimization by halting the enumeration as soon as the minimum value of T is reached. Based on Conjecture 1, we executed PNPAW enumerate for several values of W . In Table 10, we present the best testing times obtained for d695 for the values of W . For each value of W , the number of TAMs, width partition, testing time, and core assignment providing the minimum testing time is shown. TAM Optimum Optimal Optimal Optimum width number width core testing W of TAMs partition asignment time 28 5 1+2+8+8+9 (4,2,4,2,3,5,4,1,5,4) 24197 Table results obtained for d695 for several values of W . 9 Conclusion We have investigated the problem of test wrapper/TAM co-optimization for SOCs, based on the test bus model of TAM design. In particular, we have formally defined the problem of determining the number of TAMs, a partition of the total TAM width among the TAMs, an assignment of cores to TAMs, and a wrapper design for each core, such that SOC testing time is minimized. To address this problem, we have formulated three incremental problems in test wrapper and TAM optimization that serve as stepping-stones to the more general problem stated above. We have proposed an efficient heuristic algorithm based on BFD for the wrapper design problem PW that minimizes testing time and TAM width. For PAW , the problem of determining core assignments and wrapper designs, we have formulated an ILP model that results in optimal solutions in short execution times. We have formulated an ILP model to solve PPAW , the problem of TAM width partioning that PAW leads up to. This ILP model was solved to obtain optimal TAM designs for reasonably-sized problem instances. We have also presented a new enumerative approach for PPAW that offers significant reductions in the execution time. Finally, we have defined a new wrap- per/TAM design problem, PNPAW , in which the number of TAMs to be designed must be determined. PNPAW is the final step in our progression of incremental wrapper/TAM design problems, and it includes PW , PAW and PPAW . An enumerative algorithm to solve PNPAW has been proposed, in which the search space can be pruned significantly when no further improvement to testing time would result. We have applied our TAM optimization models to a realistic example SOC as well as to an industrial SOC; the experimental results demonstrate the feasibility of the proposed techniques. To the best of our knowledge, this is the first reported attempt at integrated wrapper/TAM co-optimization that has been applied to an industrial SOC. In future work, we intend to extend our TAM optimization models to include several other TAM configurations, including daisy-chained cores on TAMs [1] and "forked and merged" TAMs [5]. We intend to extend our models, such that multiple wrappers on the same TAM are active in the test data transfer mode at the same time; this will allow us to address the problems of both testing hierarchical cores, as well as Extest. While ILP is a useful optimization tool for reasonably-sized problem instances, execution times can increase significantly for complex SOCs and large values of B. This is also true of our enumerative approach to Problems PPAW and PNPAW . We are in the process of designing heuristic algorithms for each of the problems formulated in this paper that can efficiently address wrapper/TAM co-optimization for large TAM widths as well as large numbers of TAMs. Furthermore, we plan to add constraints related to power dissipation, routing complexity and layout area to our TAM optimization models. Acknowledgements The authors thank Harry van Herten and Erwin Waterlander for their help with providing data for the Philips SOC p93791, Henk Hollmann and Wil Schilders for their help with partition theory, and Jan Korst for his help with the NP-hardness proof for PNPAW . We also thank Sandeep Koranne and Harald Vranken for their constructive review comments on earlier versions of this paper. --R Scan chain design for test time reduction in core-based ICs lpsolve 3.0 Design of system-on-a-chip test access architectures using integer linear programming Design of system-on-a-chip test access architectures under place-and-route and power constraints Optimal test access architectures for system-on-a- chip Computers and Intractability: A Guide to the Theory of NP-Completeness A fast and low cost testing technique for core-based system-on-chip Testing re-usable IP: A case study IEEE P1500 Standard for Embedded Core Test. Direct access test scheme - Design of block and core cells for embedded ASICs Test wrapper and test access mechanism co-optimization for system-on-chip An integrated system-on-chip test framework A course in combinatorics A structured and scalable mechanism for test access to embedded reusable cores. Wrapper design for embedded core test. On using IEEE P1500 SECT for test plug-n-play An ILP formulation to optimize test access mechanism in system-on-chip testing Using partial isolation rings to test core-based designs A structured test re-use methodology for core-based system chips Model Building in Mathematical Programming. Testing embedded-core-based system chips --TR --CTR Feng Jianhua , Long Jieyi , Xu Wenhua , Ye Hongfei, An improved test access mechanism structure and optimization technique in system-on-chip, Proceedings of the 2005 conference on Asia South Pacific design automation, January 18-21, 2005, Shanghai, China Qiang Xu , Nicola Nicolici , Krishnendu Chakrabarty, Multi-frequency wrapper design and optimization for embedded cores under average power constraints, Proceedings of the 42nd annual conference on Design automation, June 13-17, 2005, San Diego, California, USA Tomokazu Yoneda , Kimihiko Masuda , Hideo Fujiwara, Power-constrained test scheduling for multi-clock domain SoCs, Proceedings of the conference on Design, automation and test in Europe: Proceedings, March 06-10, 2006, Munich, Germany Tomokazu Yoneda , Masahiro Imanishi , Hideo Fujiwara, Interactive presentation: An SoC test scheduling algorithm using reconfigurable union wrappers, Proceedings of the conference on Design, automation and test in Europe, April 16-20, 2007, Nice, France test scheduling with reconfigurable core wrappers, IEEE Transactions on Very Large Scale Integration (VLSI) Systems, v.14 n.3, p.305-309, March 2006 Julien Pouget , Erik Larsson , Zebo Peng, Multiple-constraint driven system-on-chip test time optimization, Journal of Electronic Testing: Theory and Applications, v.21 n.6, p.599-611, December 2005 Anuja Sehgal , Sandeep Kumar Goel , Erik Jan Marinissen , Krishnendu Chakrabarty, Hierarchy-aware and area-efficient test infrastructure design for core-based system chips, Proceedings of the conference on Design, automation and test in Europe: Proceedings, March 06-10, 2006, Munich, Germany Sudarshan Bahukudumbi , Krishnendu Chakrabarty, Wafer-level modular testing of core-based SoCs, IEEE Transactions on Very Large Scale Integration (VLSI) Systems, v.15 n.10, p.1144-1154, October 2007 Anuja Sehgal , Krishnendu Chakrabarty, Efficient Modular Testing of SOCs Using Dual-Speed TAM Architectures, Proceedings of the conference on Design, automation and test in Europe, p.10422, February 16-20, 2004 Anuja Sehgal , Vikram Iyengar , Mark D. Krasniewski , Krishnendu Chakrabarty, Test cost reduction for SOCs using virtual TAMs and lagrange multipliers, Proceedings of the 40th conference on Design automation, June 02-06, 2003, Anaheim, CA, USA Zhanglei Wang , Krishnendu Chakrabarty , Seongmoon Wang, SoC testing using LFSR reseeding, and scan-slice-based TAM optimization and test scheduling, Proceedings of the conference on Design, automation and test in Europe, April 16-20, 2007, Nice, France Sandeep Kumar Goel , Erik Jan Marinissen, Layout-Driven SOC Test Architecture Design for Test Time and Wire Length Minimization, Proceedings of the conference on Design, Automation and Test in Europe, p.10738, March 03-07, Sandeep Kumar Goel , Kuoshu Chiu , Erik Jan Marinissen , Toan Nguyen , Steven Oostdijk, Test Infrastructure Design for the Nexperia" Home Platform PNX8550 System Chip, Proceedings of the conference on Design, automation and test in Europe, p.30108, February 16-20, 2004 Jan Marinissen , Rohit Kapur , Maurice Lousberg , Teresa McLaurin , Mike Ricchetti , Yervant Zorian, On IEEE P1500's Standard for Embedded Core Test, Journal of Electronic Testing: Theory and Applications, v.18 n.4-5, p.365-383, August-October 2002 Anuja Sehgal , Sule Ozev , Krishnendu Chakrabarty, TAM Optimization for Mixed-Signal SOCs using Analog Test Wrappers, Proceedings of the IEEE/ACM international conference on Computer-aided design, p.95, November 09-13, Sandeep Kumar Goel , Erik Jan Marinissen, A Test Time Reduction Algorithm for Test Architecture Design for Core-Based System Chips, Journal of Electronic Testing: Theory and Applications, v.19 n.4, p.425-435, August Qiang Xu , Nicola Nicolici, Delay Fault Testing of Core-Based Systems-on-a-Chip, Proceedings of the conference on Design, Automation and Test in Europe, p.10744, March 03-07, Sandeep Kumar Goel , Erik Jan Marinissen, On-Chip Test Infrastructure Design for Optimal Multi-Site Testing of System Chips, Proceedings of the conference on Design, Automation and Test in Europe, p.44-49, March 07-11, 2005 Vikram Iyengar , Krishnendu Chakrabarty , Erik Jan Marinissen, Wrapper/TAM co-optimization, constraint-driven test scheduling, and tester data volume reduction for SOCs, Proceedings of the 39th conference on Design automation, June 10-14, 2002, New Orleans, Louisiana, USA A. Sehgal , K. Chakrabarty, Test planning for the effective utilization of port-scalable testers for heterogeneous core-based SOCs, Proceedings of the 2005 IEEE/ACM International conference on Computer-aided design, p.88-93, November 06-10, 2005, San Jose, CA Qiang Xu , Nicola Nicolici, Modular and rapid testing of SOCs with unwrapped logic blocks, IEEE Transactions on Very Large Scale Integration (VLSI) Systems, v.13 n.11, p.1275-1285, November 2005 Vikram Iyengar , Krishnendu Chakrabarty , Erik Jan Marinissen, Test Access Mechanism Optimization, Test Scheduling, and Tester Data Volume Reduction for System-on-Chip, IEEE Transactions on Computers, v.52 n.12, p.1619-1632, December Sandeep Kumar Goel , Erik Jan Marinissen, SOC test architecture design for efficient utilization of test bandwidth, ACM Transactions on Design Automation of Electronic Systems (TODAES), v.8 n.4, p.399-429, October Anuja Sehgal , Fang Liu , Sule Ozev , Krishnendu Chakrabarty, Test Planning for Mixed-Signal SOCs with Wrapped Analog Cores, Proceedings of the conference on Design, Automation and Test in Europe, p.50-55, March 07-11, 2005 Matthew W. Heath , Wayne P. Burleson , Ian G. Harris, Synchro-Tokens: Eliminating Nondeterminism to Enable Chip-Level Test of Globally-Asynchronous Locally-Synchronous SoC's, Proceedings of the conference on Design, automation and test in Europe, p.10410, February 16-20, 2004 Zahra S. Ebadi , Alireza N. Avanaki , Resve Saleh , Andre Ivanov, Design and implementation of reconfigurable and flexible test access mechanism for system-on-chip, Integration, the VLSI Journal, v.40 n.2, p.149-160, February, 2007 Jan Marinissen, The Role of Test Protocols in Automated Test Generation for Embedded-Core-Based System ICs, Journal of Electronic Testing: Theory and Applications, v.18 n.4-5, p.435-454, August-October 2002 Anuja Sehgal , Sule Ozev , Krishnendu Chakrabarty, Test infrastructure design for mixed-signal SOCs with wrapped analog cores, IEEE Transactions on Very Large Scale Integration (VLSI) Systems, v.14 n.3, p.292-304, March 2006

embedded core testing;testing time;integer linear programming;test access mechanism TAM;test wrapper

609176

Retraction Approach to CPS Transform.

We study the continuation passing style (CPS) transform and its generalization, the computational transform, in which the notion of computation is generalized from continuation passing to an arbitrary one. To establish a relation between direct style and continuation passing style interpretation of sequential call-by-value programs, we prove the Retraction Theorem which says that a lambda term can be recovered from its CPS form via a -definable retraction. The Retraction Theorem is proved in the logic of computational lambda calculus for the simply typable terms.

Introduction The notions of a continuation and a continuation passing style (CPS) transform have been introduced by a number of authors (see [Rey93] for a historical overview). The main motivation for the independent developments of these concepts seemed to be twofold: explaining the behavior of imperative features in functional languages, and compilation of programs with higher order procedures. Further research led to development of CPS denotational semantics [SW74] (see also [Sto77]), and later categorical semantics of computations [Mog89], as well as compilers based on the CPS transform [Ste78] (see also [App92]). In both kinds of applications one of the central goals of This research was supported in part by: "Types and Algorithms", Office of Naval Research (N00014-93-1-1015), "Computational efficiency of optimal reduction in lambda calculus", National Science Foundation (CDA-9504288), and "Logic, Complexity, and Programming Languages", National Science Foundation (CCR-9216185). the research has been to establish a relationship between original terms and their images under the transform. In this work we view the CPS transform as a formalization of the (contin- uation passing style) denotational semantics of a call-by-value programming language in the fij lambda calculus - n . To model call-by-value evaluation in a programming language we choose Moggi's [Mog88] computational lambda calculus, - c , for the two reasons: 1) the logic of - c is sound for call-by-value reasoning, and 2) the logic of - c is complete for the class of models (computational lambda models [Mog88]) in which most commonly used computational effects can be expressed. One way of asserting the correctness of the CPS transform, as an interpretation of - c in - n , is the equational correspondence result due to Sabry and Felleisen [SF92]. Theorem 1.1 (Sabry-Felleisen). For any two lambda terms M and N , The left-to-right implication in this theorem says that the CPS transform preserves equality, and the right-to-left implication says that the transform also preserves distinctions. Thus, the transform gives an accurate picture of -equivalence in - n . To formalize the problem we are trying to solve we observe that the left-to-right implication of the theorem also says that the CPS transform defines a function, T , mapping the - c -equivalence classes of lambda terms to the - n -equivalence classes of lambda terms. The right-to-left implication can be understood as saying that T is injective, and therefore has a left We ask the question whether functions T , or its inverse T are definable. More precisely, ffl is there a lambda term P , such that - n ' (P ffl is there a lambda term R, such that - c ' (R An elementary argument given in [MR] shows that the answer to the first question must be "no". In this work we give an affirmative answer to the second question. More precisely we prove the following theorem: Theorem 1.2 (Retraction, for the CPS transform). For any simple type oe there is a lambda term R oe , such that for all closed lambda terms M of type oe, (R oe A version of the above Retraction Theorem was proven by Meyer and Wand [MW85], where the conclusion of the theorem holds in the logic of - n . However, as the authors themselves have pointed out to us, their result is misleading. We are interested in behavior of a term M under a call-by-value evaluation, and - n is not sound for call-by-value reasoning in presence of any computational effects. An interesting point about the CPS transform, viewed as an interpretation of a call-by-value programs, is that not only it can interpret pure functional programs, but can also be extended to interpret programs with control operators such as call/cc and abort. Different extensions of a functional language with "impure" features can also be given denotational semantics using a similar transform. For example, an interpretation of programs in a language with mutable store can be given using the state passing style (SPS) transform. As shown by Moggi [Mog88], a number of such computational effects can be described by the notion of a monad, and the CPS and the SPS transforms can be generalized to, what we call, the computational transform [Wad90, SW96]. The equational correspondence for the computational transform holds as well [SW96], and it is natural to ask whether the Retraction Theorem (Theorem 1.2) generalizes. However, the computational transform maps lambda terms to the terms of the "monadic metalanguage", - ml [Mog91]. The language of - ml is extended with new constructs that the logic of the computational lambda calculus has no axioms for, so the question of whether there is a lambda term, R, (even in the language of - ml ) such that denotes the computational transform of M , is ill formed. In order to study the Retraction Theorem in an abstract setting that can be applied to other transforms, as well as to the CPS transform, we define a modified computational transform, T \Pi , mapping lambda terms to lambda terms extended with two constants E and R that satisfy axiom (R The modified computational transform satisfies the equational correspondence result for the closed terms, and we prove the Retraction Theorem in the logic of - c extended with the axiom (r-e). Theorem 1.3 (Retraction, for T \Pi ). For any simple type oe there is a term R oe , such that for all closed lambda terms M of type oe, (R oe M \Pi where M \Pi stands for the modified computational transform of M . The proof of the above theorem consists of defining interpretations of types and terms, as well as an "-relation between the interpretations, and proving that if a term M has a type oe, then the meaning of M and the meaning of oe are related by the "-relation. This framework is in many ways similar to, and was inspired by, the type inference models developed in [Mit88]. Even though the transforms of interests, namely the CPS transform and the SPS transform, are not special cases of the modified computational trans- form, we benefit from studying the abstract transform in that we obtain a proof that does not depend on details of a particular transform, and can be applied, by modifying definitions appropriately, to either the CPS or the SPS transform. Hence, we obtain the Retraction Theorems for the CPS and the SPS transforms. The fore mentioned results apply only to simply typed closed terms. To extend the applicability of these results we can proceed in two directions: we can extend the computational lambda calculus with new language constructs and axioms that define equational behavior of the new terms, and we can extend the type system so that our results apply to a larger class of terms. Our results can be easily extended to a calculus, extending - c , with a datatype such as natural numbers and primitive operators on natural numbers. A more important class of extensions consists of those extensions which introduce a computational effect to - c . We have been able to extend the Retraction Theorem to - c extended with a divergent element. However, we stop short of proving the Retraction Theorem for - c terms extended with recursion. In an attempt to prove the Retraction Theorem for all (untyped) closed terms, we extend the type system with recursive types and prove the following result. Call a term F a total function if for any value V , F (V ) is c -equivalent to a value. Theorem 1.4. Assume terms e and r exist such that e and r are total functions satisfying (in - c +(r-e)) (R (y (e x)))) Then for every term M , (R M \Pi Analogous theorems also hold for the CPS and the SPS transforms. The assumption of the above theorem is quite strong, and it remains to be seen whether such terms e and r exist. One should also investigate whether such elements exist in models that would allow interesting applications of the theorem. We assume the reader is familiar with elementary concepts of a lambda calculus. For details one is referred to [Bar84]. In this section we will provide concise definitions in order to disambiguate our notation. 2.1 Lambda calculus Lambda terms are terms formed over an infinite set of variables by lambda abstraction and application. We will use a number of standard conventions when writing lambda terms, such as that application associates to the left, and in general, use parentheses freely to make terms easier to read. We will (let x=M in N) to abbreviate the term ((-x:N) M ), and M ffi N for the term -x:(M (N x)). For most part (but not exclusively) we use letters M , N , P , etc. to range over arbitrary lambda terms, and letters U , V and W to range over values, that is lambda terms that are either variables, lambda abstractions or constants. Lower-case letters, x, y, z etc. will be used for variables. We study provable equality between untyped lambda terms. If - is a set of axions we if the equation can be derived using the rules of lambda congruence from the axioms in -. (let x=M in Table 1: Axioms of - c . For most part of this paper we consider equalities provable in Moggi's Table 1), possibly extended with additional axioms for constants. In particular we will use constants E and R satisfying axiom (R 2.2 Typing system We consider a type system for assigning types to untyped lambda terms. Simple types are defined over a base type -, i.e. - is a type and oe!- is a type whenever oe and - are types. The type inference system consists of set of rules (given in Table 2) for deriving sequents of the form \Gamma . M : oe, where oe is called a typing assertion and \Gamma is a typing hypothesis, i.e. a set of typing assertions of the form x where we always assume no variable x i occurs more than once in \Gamma. \Gamma; x: oe . x: oe (var) . (M N (app) \Gamma; x: oe . (abs) Table 2: Type inference rules for simple types. 2.3 Transforms In this work we concentrate on three transforms mapping lambda terms to lambda terms. First we study the modified computational transform, T \Pi , mapping pure lambda terms to lambda terms extended with two constants E and R (see Table 3). The transform T \Pi can be viewed as an abstract transform which captures, for our purposes, important properties of the CPS and the SPS transforms. Namely, in the transform M \Pi of a term M , the order of evaluation (left-to-right and call-by-value) is made explicit. However, the additional structure that makes, in particular, the CPS transform attractive to compiler designers, is not reflected in the definition of T \Pi . Table 3: The modified computational transform, T \Pi . The next transform we study is the call-by-value version of the Fischer- Reynolds CPS transform. The definition we use (as well as the overline is taken from [Plo75] (see Table 4). Analogous to the CPS transform, used to give a denotational semantics of programs with mutable store, instead of control operators, is the state passing style transform (SPS). In the definition, given in Table 5, we used the pairing constructs as abbreviations. Namely, (let hx 1 i=M in N) Table 4: The CPS transform. abbreviates the expression (let x=M in (let x 1 =- 1 (x) in (let x 2 =- 2 (x) in N))): x Table 5: The SPS transform. Even though we give untyped definitions of the transforms, we believe the transforms should be understood in the context of a typed language. This view is supported by the monadic framework, developed by Moggi [Mog88], in which programs are interpreted as "computations". We leave out the details of Moggi's monadic interpretation, as well as the definitions of the typed transforms, since they are not central to our development, but rather sketch the intuitive picture to help motivate some of our definitions. We think of a transform mapping terms of type oe (intuitively programs) to terms of type T (oe 0 ) (intuitively computations), where T is a unary type constructor (that depends on the particular transform), and oe 0 is defined inductively using T to be - 0). This can be made precise by defining the typed version of the transform mapping typing sequents to typing sequents. 3 Retraction Theorem In this section we state and prove the Retraction Theorem for the abstract transform, T \Pi , as well as, for the CPS and the SPS transforms. While one might not find the transform T \Pi interesting in itself, for us it serves the purpose. More precisely, we develop a framework which enables us to prove the Retraction Theorem for the transform T \Pi , and is free of the details specific to the CPS and the SPS transforms. Nevertheless, the framework can be effortlessly modified to prove the Retraction Theorem for each of these transforms. Thus we believe that focusing on the more abstract transform, improves the clarity of our presentation. 3.1 Retractions The Retraction Theorem asserts the definability of the inverse of the CPS transform (as well as other transforms of interest). We now give a construction of the type-indexed family of lambda terms that define the retraction. Note that we give the following definitions using terms E and R, but the definitions should be understood as parameterized by these terms. That is, the inverse of the modified computational transform will be defined using constants E and R, and in the definition of the inverse of the CPS transform, these constants will be replaced by the terms E K and R K . Likewise for the SPS transform, terms E S and R S will be used. If we think of transforms in the context of a typed language, mapping terms (of type oe) to terms representing computations (of type T (oe 0 )), in- tuitively, we can understand the pairs of terms defined below as retraction- embedding pairs between types oe and T (oe 0 ). One can also formally define a notion of a type oe being a retract of a type - , and in the sense of such definition, given below, we exhibit terms R oe and E oe that form a retraction- embedding pair between types oe and T (oe 0 ). Moreover, we will show that (suitable versions of) terms R oe define the required inverses of the transforms we study. Definition 3.1. A type oe is said to be a retract of a type - if there is a pair of lambda terms R oe;- and E oe;- of types -!oe and oe!- respectively, such that (R oe;- (E oe;- Definition 3.2. For each simple type, oe, define terms e oe and r oe inductively on the structure of oe as follows: and at higher types we have r (R (f (e Lemma 3.3. - c +(r-e) ' (r oe (e oe Proof: Easy by induction on oe. Note: Above lemma holds whenever we replace constants E and R with any values E 0 and R 0 such that - c ' (R 0 Finally we define terms R (R x)) and E Later we will show that R oe is an inverse of the transform T \Pi . To define an inverse of the CPS transform we first define terms R K def -xk:(k x). It is easy to show (in - c ) that (R K (E K the retraction-embedding pair (R K oe using terms R K and E K instead of constants R and E in the above definitions. Similarly for the SPS transform we define terms represents some initial state of the store, and define terms R S oe and E S oe using these terms instead. 3.2 Interpretations of Terms and Types The framework we develop to prove the Retraction Theorem is closely related to Mitchell's type inference models [Mit88]. As in the definition of a type inference model we define interpretations of terms and types, as well as a relation between the two. However, the definition of a type inference model assumes one works with the full fij-equality, and we need to relax the definitions to accommodate reasoning in the weaker logic of - c . We will first sketch the definitions for the general framework and then fill in the details that apply to each particular transform we study. Interpretation of terms: Assume (D; App) is an applicative structure, that is, D is a set and App is a binary operation on D. Assume also that there is a distinguished subset, V(D) ' D, of values. Given an environment, ae, mapping variables to V(D) we define an interpretation (relative to ae) that to each term M assigns an element JMKae of D. In addition we assume that In particular we will define D to be the set of equivalence classes of terms and the set of values to be the equivalence classes of terms that are values. (Note: below, we may be informal and identify terms with their equivalence classes.) The interpretation function will then be defined by the transform in consideration. Interpretation of types: The types will be interpreted as certain subsets of D called type sets. In particular we will chose type sets to contain only the equivalence classes of terms of the form . The interpretation of types will be defined (using the "-relation defined below) inductively: set of all terms of the form set of all M (of the form (E V )) such that 8N 2 JoeK: (R - (R oe N)) Relating the interpretations: To relate meanings of terms and types, instead of using the simple set-membership relation, we define an extended 2. This relation will in general depend on the structure of the transform in consideration, and intuitively, will serve the purpose of separating out "important" part of the transformed term. Truth and validity: Having defined the above notions, we say that a typing assertion M : oe is true (with respect to ae), written ae JoeK. The notions of satisfaction and validity are defined in the standard way relative to the definition of truth. Namely, ae satisfies a typing hypothesis \Gamma, written ae every typing assertion in \Gamma is true with respect to ae, and a typing sequent \Gamma . M : oe is valid, is true for every ae that satisfies \Gamma. Our aim is to prove a soundness lemma for the type inference system that would imply the Retraction Theorem. Before we state the lemma, we single out two conditions that are necessary for the lemma to hold. Namely we require that for all M " Joe!-K and N " JoeK: (R (R oe N)); (z) where R oe , etc. are defined in Section 3.1 and equality is provable equality in - c +(r-e). The main reason why these two conditions are singled out is that, having proved the Retraction Theorem for the modified computational transform, we modify the definition of the interpretations of terms and types, as well as the "-relation, to reflect properties of the CPS transform, and only the conditions (y) and (z) need to be proved again to show that the Retraction Theorem holds for the CPS transform. Similarly for the SPS transform. 3.3 Modified Computational Transform In the previous section we outlined the definitions of our framework, and we can now fill in the details. The following definitions are given, in particular, to prove the Retraction Theorem for the modified computational transform, but we will also indicate how these definitions need to be changed in the subsequent sections to prove the Retraction Theorem for the CPS and the SPS transforms. We write "=" to denote an equality provable in - c +(r-e). Definition 3.4. Let ae be a substitution mapping variables to values. Define the interpretation function J \DeltaK to be JMKae Definition 3.5. Type sets are sets of terms of the form a value. Note: this definition is to be understood as parameterized by the term E , that is, when we consider, say, the CPS transform we will use the term E K instead. Definition 3.6. If M and N are terms, we define the application in the codomain of the transform as follows: It is easy to see that J(M Definition 3.7. Let S be a type set and let M be a term. We write M " S if there is a value V and terms P 1 and Definition 3.8. The interpretation of types is defined inductively on the structure of type expressions. Namely, set of all terms of the form set of all M , of the form (E V ), such that 8N 2 JoeK, (R - (R oe N)) Recall that R Note: We should understand this definition as parameterized by E , R, App, and ". When we consider the CPS or the SPS transform, the appropriate definitions will be used instead. Equipped with the definitions we can prove the Soundness Lemma, but first we need some auxiliary results. Lemma 3.9. For each type oe and any value V , Proof: Easy by induction on oe. Lemma 3.10. For any two terms M and N such that M " Joe!-K and and (R (R oe N)): Proof: First observe that the statement of this lemma is stronger than what is given in Definition 3.8 above. The definition only requires that the two statements hold only for N 2 JoeK, and in the lemma we show that these two statements hold for all N " JoeK. Both conditions can be easily proved using the definitions and axioms of - c +(r-e). The Soundness Lemma consists of two parts, (S.1) and (S.2). The first part asserts the soundness of our interpretation with respect to typing rules of the simply typed lambda calculus, and the second part is, in fact, the statement of the Retraction Theorem. Lemma 3.11 (Soundness). Let \Gamma be a typing hypothesis and let ae be a substitution that satisfies \Gamma. Let ae 0 be a substitution such that for each and (R oe ae(M \Pi Proof: We prove the lemma by induction on the derivation of \Gamma . M : oe. The (var) case follows by assumptions, and the (app) case follows directly by Lemma 3.10 and induction hypotheses. The (abs) case is slightly more involved. Assume M j -x:N and oe j , and that \Gamma . -x:N was derived from \Gamma; x: - 1 using the (abs) rule. To show (S.1), first observe that ae((-x:N) \Pi so we need to show that ae((-x:N) \Pi any term in J- 1 K. Then and therefore since aefV=xg satisfies \Gamma; x: - 1 , by induction hypothesis (S.1), K. Moreover we can compute (R -2 App(ae((-x:N) \Pi (R using the definition of the retraction-embedding pairs, and both induction hypotheses (S.1) and (S.2). This establishes (S.1). To show (S.2) simply compute (R using properties of retraction-embedding pairs, Lemma 3.9 and induction hypothesis (S.2). The Retraction Theorem follows directly from this lemma. Theorem 3.12 (Retraction, for T \Pi ). For any closed term M of (simple) type oe, (R oe M \Pi 3.4 The CPS Transform To prove the the Retraction Theorem for the CPS transform, as indicated be- fore, we will use exactly the same framework, but will modify the definitions using appropriate definitions of application in the codomain of the transform and "-relation. First recall that we define the retraction-embedding (R to be It is easy to see that - c ' (R K (E K which is the only abstract property of E and R we use. We define the interpretation of terms (relative to a substitution ae) using the CPS transform, that is, JMKae The application in the codomain of the transform is defined to be so that (M Finally, we define the extended membership relation, " K , to be for some terms P i , a value V such that and a fresh variable k. It should be understood that all the definitions used in the preceding section are now defined using E K , R K , App K and " K , instead of E , R, App and ". With the new definitions, we prove the following lemma, analogous to Lemma 3.10, asserting that conditions (y) and (z) hold. Lemma 3.13. For any two terms M and N such that M " K Joe!-K and App K (M; and (R K oe!- M) (R K oe Proof: The proof is straightforward using the definitions and axioms of - c . Having shown the above lemma, the rest of the proof of the Soundness Lemma for the CPS transform is exactly the same as in the case of the modified computational transform, and as a corollary we obtain the retraction result. Theorem 3.14 (Retraction, for the CPS transform). For any closed term M of (simple) type oe, (R K 3.5 The SPS Transform To adopt our framework to the SPS transform we define E S , R S , App S and " S in place of E , R, App and ", and prove that the conditions (y) and (z) still hold. Recall that the terms E S and R S are defined to be where init is some initial state of the store. The interpretation of terms is defined using the SPS transform. Namely, JMKae The application in the codomain of the transform is define to be so that (M N) Finally we define the extended membership for some terms P i , some value V such that Furthermore, we interpret the definitions of type sets, interpretation of types, and retraction-embedding pairs defined earlier, as if given using in place of E , R, App and ". With the new definitions we can show that conditions (y) and (z) are still satisfied. This yields the Retraction Theorem for the SPS transform. Theorem 3.15 (Retraction, for the SPS transform). For any closed term M of (simple) type oe, (R S oe M 3.6 Extensions Thus far we have only proved the retraction results for the pure simply typed terms. In order to make these results more applicable we would like to extend the theorems to a larger class of terms. We have essentially two directions in which we can proceed. We can extend the class of terms by adding constants or term constructors (including possibly new axioms that define functional behavior of the new terms), and secondly, we can extend the type system to one that can type a larger class of terms. Extending the Retraction Theorem to an extension of - c with constants of a base type and primitive operators such as numerals is quite straight- forward. However, adding arbitrary constants of higher order types may be more difficult. The difficulty lies in ensuring the closure conditions imposed on type sets by the addition of such constants are satisfied. For example if a constant c of type oe!- is added to - c , we need to make sure that if M " oe then App(c \Pi ; M) " - . While such closure conditions are determined based on the type of new constants, the proof they are satisfied will, in general, depend on the functional behavior of the new constants. Divergence: The difference between call-by-name and call-by-value evaluation strategies becomes apparent only in presence of actual computational effects. So far we have only considered pure simply typed terms. In this setting every closed term is equivalent to a value in both logics of - c and - n . Therefore, if we were to stop here, it would be unjustified to claim significant improvement over the original Meyer-Wand Retraction Theorem. The simplest computational effect we can add to the language is diver- gence. In presence of divergence - n reasoning is no longer sound for call- by-value languages, so for any applications in - c extended with divergence, we really need the stronger version of the Retraction Theorem provable in the weaker logic of - c . While extension of the Retraction Theorem to a language with divergence, which we now present, is quite straightforward, it is important since it illustrates the difference between Meyer and Wand's and our formulation of the Retraction Theorem. Divergence is represented by the divergent element, \Omega\Gamma that is added to the language of - c as a constant, but it is not considered a value. The axioms for\Omega specify that an application diverges if either the operators or the operand diverges. Moreover, these axioms identify all divergent terms. The axioms are:(\Omega M) =\Omega and (M \Omega\Gamma One can verify that the resulting equational logic is consistent and that it cannot value V . The type system is extended with the axiom . \Omega\Gamma oe which says that\Omega has every type. The modified computational transform is defined on\Omega to be has every type, to prove the Retraction Theorem for - c+\Omega we need to extend the Soundness Lemma for the case of typing In other words we need to show (S.1): and (S.2): that (R =\Omega for every type oe. The second condition follows trivially from the definition of\Omega \Pi and the axioms To prove the first condition observe that x=\Omega in (E (e oe x))); and by Lemma 3.9, (E (e oe x)) 2 JoeK for every oe. The very same reasoning can be applied to extend the Retraction Theorem for the CPS transform to - c+\Omega\Gamma where the CPS transform is defined on\Omega to -k:\Omega k: Recursive types: It is well known that all terms can be typed using the recursive type system. In order to extend the Retraction Theorem to all closed terms we study the recursive types. The recursive type discipline introduces types of the form -t:oe (where we use t to denote a type variable). In order to extend our results to - c extended with recursive types we need to define retraction-embedding pairs oe ) at new types. In particular how does one define e -t:oe and r -t:oe , or even e t and r t ? To motivate a solution, consider the following example. Let -t:t!t. Then in the recursive type discipline one can type .(-x:x x): - . Assume we have defined terms e - and r - , and we try to compute (R (-x:x x) \Pi (R (R ((e - x) (e - x)))) To complete this derivation, one would like to have (e - so we can continue (R What we see from this example is that the two occurrences of x in -x:x x "act" as having types - and - , respectively. Similarly, we would like the two occurrences of e - in -x:(r - (R ((e - x) (e - x)))) to "act" as e - and e - . A solution to our problem is to find a uniform definition for e's and r's at all types. Namely, we want a retraction-embedding pair (r; e) that satisfies the following definition. Definition 3.16. A term F is called a total function if F is a value and, for any value V , provably equal to a value. A pair of total functions (r; e) is a uniform retraction-embedding pair if e and r satisfy system of equations r While it remains open whether there is a pair of terms satisfying the above conditions, we will assume we are given such a pair of functions and, under this assumption, show how the Retraction Theorem can be extended to recursive types. Moreover, since the recursive type system can type all terms, as a corollary we obtain the following theorem. Theorem 3.17. Assume total functions e and r exist that satisfy equations (?). Then, for any closed lambda term M , (R M \Pi Of course, the analogous theorems hold for the other transforms as well. Here we only sketch the main idea in the proof of the above theorem. A detailed proof can be found in [Ku-c97]. The recursive type system extends the simple types by adding type variables and type expressions of the form -t:oe. The new inference rules are (- I) E) One can understand these rules by considering the type -t:oe as the type - , satisfying equation oef-=tg. Thus we need to define the interpretation, J-K-, of - such that it satisfies the equation In other words J-K- should be a fixed point of the function -S:JoeK-fS=tg: (We assume the interpretation will satisfy JoeK-fJ-K-=tg.) The difficulty lies in showing that for any oe and -, the function -S:JoeK-fS=tg always has a fixed point. To do so, we define a metric on the space of all type sets so that the resulting metric space is complete. Then we show that each function -S:JoeK-fS=tg is a contraction, and thus, by Banach's Fixed-point Theorem, has a unique fixed point. Mac Queen et al. [MPS86] have developed such a framework, of which our development can be viewed as a special case. Namely, our domain consists only of finite elements (typ- ing sequents) ordered under discrete order, thus greatly simplifying general purpose structures used in [MPS86]. Concluding Remarks In this work, we have established a relation between direct style and CPS terms using definable retraction functions. The Retraction Theorem shows that a term can be recovered, up to - c -equivalence, from its image under the CPS transform. Therefore, the retraction approach, in fact, only provides a relation between equivalence classes of terms. To contrast our results with others that provide, perhaps even stronger relation between lambda terms and their CPS forms (e.g. [SW96]), we should emphasize that the inverse of the CPS transform we obtain is definable. Another important point is that the conclusion of our version of the Retraction Theorem is an equation provable in the logic of - c , which is a call-by-value logic, unlike the results in [MW85] and [Fil94] which give similar equalities, but in a call-by-name logic. As a consequence, our results are applicable even where call-by-name reasoning is not sound. Some open questions: In all practical applications, functional programming languages are equipped with some form of recursion. Therefore, to make the retraction approach applicable in practice, we need to extend our results to a language with recursion. This can be done in two ways: By extending the type system so that the fixed-point operator is definable in the pure language, or by adding a language construct such as constant Y , letrec, etc. The first approach, with some partial results, has been discussed One difficulty in adding fixed-point operator Y , or a similar language construct, is that additional closure conditions are needed in the definition of type sets, and we haven't been able to construct type sets satisfying these conditions. The other difficulty is determining the correct axiomatization of a fixed-point operator. It appears that the axiom does not suffice. In models of - c , fixed-point operator can be defined using the so called fixpoint object. Crole and Pitts [CP92] define such an object in models of - c , and discuss a logical system for reasoning about fixpoint computations, which may hold the answer to above questions. Another class of extensions is motivated by the application of the Retraction Theorem developed by Riecke and Viswanathan [RV95], where they show how one can isolate effects of an extension of a language with assignment or control from interfering with pure functional code. A natural question arises, whether it is possible to extend this approach to isolate one computational effect from interfering with code possibly containing a different computational effect. For instance, if M is a program in, say call- by-value PCF with assignment, can we define an operator, call it encap, so that, in an extension of call-by-value PCF with both assignment and control, (encap M) will behave the same as M behaves in the extension of call-by- value PCF with assignment. We believe that an appropriate extension of the Retraction Theorem to a programming language with imperative features may give us such results. --R Compiling with Continuations. The Lambda Calculus: Its Syntax and Se- mantics New foundations for fixpoint computations: FIX-hyperdoctrines and the FIX-logic Representing monads. "Free Theorems" type inference and containment. Computational lambda-caluclus and monads Computational lambda-caluclus and monads Notions of computation and monads. An ideal model for recursive polymorphic types. Continuations may be unrea- sonable Continuation semantics in typed lambda-calculi (summary) The discoveries of continutions. Isolating side effects in sequential languages. Reasoning about programs in continuation-passing style A compiler for Scheme. Denotational Semantics: The Scott-Strachey Approach to Programming Language Theory A mathematical semantics for handling full jumps. A reflection on call-by-value Comprehending monads. --TR --CTR Andrzej Filinski, On the relations between monadic semantics, Theoretical Computer Science, v.375 n.1-3, p.41-75, May, 2007

continuation passing style;continuations;CPS transform;monads;retractions

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A Syntactic Theory of Dynamic Binding.

Dynamic binding, which traditionally has always been associated with Lisp, is still semantically obscure to many. Even though most programming languages favour lexical scope, not only does dynamic binding remain an interesting and expressive programming technique in specialised circumstances, but also it is a key notion in formal semantics. This article presents a syntactic theory that enables the programmer to perform equational reasoning on programs using dynamic binding. The theory is proved to be sound and complete with respect to derivations allowed on programs in dynamic-environment passing style. From this theory, we derive a sequential evaluation function in a context-rewriting system. Then, we further refine the evaluation function in two popular implementation strategies: deep binding and shallow binding with value cells. Afterwards, following the saying that deep binding is suitable for parallel evaluation, we present the parallel evaluation function of a future-based functional language extended with constructs for dynamic binding. Finally, we exhibit the power and usefulness of dynamic binding in two different ways. First, we prove that dynamic binding adds expressiveness to a purely functional language. Second, we show that dynamic binding is an essential notion in semantics that can be used to define exceptions.

Introduction Dynamic binding has traditionally been associated with Lisp dialects. It appeared in McCarthy's Lisp 1.0 [24] as a bug and became a feature in all succeeding implemen- tations, like for instance MacLisp 2 [28], Gnu Emacs Lisp [23]. Even modern dialects of the language which favour lexical scoping provide some form of dynamic variables, with special declarations in Common Lisp [43], or even simulate dynamic binding by lexically-scoped variables as in MITScheme's fluid-let [18]. Lexical scope has now become the norm, not only in imperative languages, but also in functional languages such as Scheme [39], Common Lisp [43], Standard ML [26], or Haskell [21]. The scope of a name binding is the text where occurrences of this name refer to the binding. Lexical scoping imposes that a variable in an expression refers to the innermost lexically-enclosing construct declaring that variable. This rule implies that nested declarations follow a block structure organisation. On the contrary, the scope of a name is said to be indefinite [43] if references to it may occur anywhere in the program. On the other hand, dynamic binding refers to a notion of dynamic extent. The dynamic extent of an expression is the lifetime of this expression, starting and ending when control enters and exits this expression. A dynamic binding is a binding which exists and can only be used during the dynamic extent of an expression. A dynamic variable refers to the latest active dynamic binding that exists for that variable [1]. The expression dynamic scope is convenient to refer to the indefinite scope of a variable with a dynamic extent [43]. Dynamic binding was initially defined by a meta-circular evaluator [24] and was later formalised by a denotational semantics by Gordon [15, 16]. It is also part of the This research was supported in part by EPSRC grant GR/K30773. Author's address: Department of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ. United Kingdom. E-mail: L.Moreau@ecs.soton.ac.uk. At least, the interpreted mode. folklore that there exists a translation, the dynamic-environment passing translation, which translates programs using dynamic binding into programs using lexical binding only [36, p. 180]. Like the continuation-passing transform [35], the dynamic-passing translation adds an extra argument to each function, its dynamic environment, and every reference to a dynamic variable is translated into a lookup in the current dynamic environment. The late eighties saw the apparition of "syntactic theories", a new semantic frame-work which allows equational reasoning on programs using non-functional features like first-class continuations and state [10, 11, 12, 44]. Those frameworks were later extended to take into account parallel evaluation [9, 14, 29, 30]. The purpose of this paper is to present a syntactic theory that allows the user to perform equational reasoning on programs using dynamic binding. Our contribution is fivefold. First, from the dynamic-environment passing translation, we construct an inverse translation. Using Sabry and Felleisen's technique [40, 41], we derive a set of axioms and define a calculus, which we prove to be sound and complete with respect to the derivations accepted in dynamic-environment passing style (Section 3). Second, we devise a sequential evaluation function, i.e. an algorithm, which we prove to return a value whenever the calculus does so. The evaluation function, which relies on a context-rewriting technique [11], is presented in Section 4. Third, in order to strengthen our claim that dynamic binding is an expressive programming technique and a useful notion in semantics, we give a formal proof of its expressiveness and use it in the definition of exceptions. In Section 5, we define a relation of observational equivalence using the evaluation function, and we prove that dynamic binding adds expressiveness [8] to a purely functional programming language, by establishing that dynamic binding cannot be macro-expressed in the call-by-value lambda-calculus. In Section 6, we use dynamic binding as a semantic primitive to formalise two different models of exceptions: non-resumable exceptions as in ML [26] and resumable ones as in Common Lisp [43, 34]. Fourth, we refine our evaluation function in the strategy called deep binding , which facilitates the creation and restoration of dynamic environments (Section 7). Fifth, we extend our framework to parallel evaluation, based on the future construct [14, 17, 30]. In Section 8, we define a parallel evaluation function which also relies on the deep binding technique. Before deriving our calculus, we further motivate our work by describing three broad categories of use of dynamic binding: conciseness, control delimiters, and distributed computing. Let us insist here and now that our purpose is not to denigrate the qualities of lexical binding, which is the essence of abstraction by its block structure organisation, but to present a syntactic theory that allows equational reasoning on dynamic binding, to claim that dynamic binding is an expressive programming technique if used in a sensible manner, and to show that dynamic binding can elegantly be used to define semantics of other constructs. Let us note that dynamic binding is found not only in Lisp but also in T E X [22], Perl [45], and Unix TM shells. Practical Uses of Dynamic Binding 2.1 Conciseness A typical use of dynamic binding is a printing routine print-number which requires the basis in which the numbers should be displayed. One solution would be to pass an explicit argument to each call to print-number. However, repeating such a programming pattern across the whole program is the source of programming mistakes. In addition, this solution is not scalable, because if later we require the print-number routine to take an additional parameter indicating in which font numbers should be displayed, we would have to modify the whole program. Scheme I/O functions take an optional input/output port. The procedures with- input-from-file and with-output-to-file [39] simulate dynamic binding for these parameters. Gnu Emacs [23] is an example of large program using dynamic variables. It contains dynamic variables for the current buffer, the current window, the current cursor position, which avoid to pass these parameters to all the functions that refer to them. These examples illustrate Felleisen's conciseness conjecture [8], according to which sensible use of expressive programming constructs can reduce programming patterns in programs. In order to strengthen this observation, we prove that dynamic binding actually adds expressiveness to a purely functional language in Section 5. 2.2 Control Delimiters Even though Standard ML [26] is a lexically-scoped language, raised exceptions are caught by the latest active handler. Usually, programmers install exception handlers for the duration of an expression, i.e. the handler is dynamically bound during the extent of the expression. MacLisp [28] and Common Lisp [43] catch and throw, Eulisp let/cc [34] are other examples of exception-like control operators with a dynamic extent. More generally, control delimiters are used to create partial continuations, whose different semantics tolerate various degrees of dynamicness [5, 20, 31, 38, 42]. 2.3 Parallelism and Distribution Parallelism and distribution are usually considered as a possible mean of increasing the speed of programs execution. However, another motivation for distribution, exacerbated by the ubiquitous WWW, is the quest for new resources: a computation has to migrate from a site s 1 to a another site s 2 , because s 2 holds a resource that is not accessible from s 1 . For our explanatory purpose, we consider a simple resource which is the name of a computer. There are several solutions to model the name of the running host in a language; the last one only is entirely satisfactory. (i) A lexical variable hostname could be bound to the name of the computer whenever a process is created. Unfortunately, this variable, which may be closed in a closure, will always return the same value, even though it is evaluated on a different site. (ii) A primitive (hostname), defined as a function of its arguments only (by ffi in [35]), cannot return different values in different contexts, unless it is defined as a non-deterministic function, which would prevent equational reasoning. (iii) A special form (hostname) could satisfy our goal, but it is in contradiction with the minimalist philosophy of Scheme, which avoids adding unnecessary special forms. Furthermore, as we would have to define such a special form for every resource, it would be natural to abstract them into a unique special form, parameterised by the resource name: this introduces a new name space, which is exactly what dynamic binding offers. (iv) Our solution is to dynamically bind a variable hostname with the name of the computer at process-creation time. Every occurrence of such a variable would refer to the latest active binding for the variable. Besides, control of tasks in a parallel/distributed setting usually relies on a notion of dynamic extent: sponsors [33, 37] allow the programmer to control hierarchies of tasks. 3 A Calculus of Dynamic Binding Figure 1 displays the syntax of u , the language accessible to the end user. Let us observe that the purpose of u is to capture the essence of dynamic variables and not to propose a new syntax for them.The language u is based on two disjoint sets of variables: the dynamic and static (or lexical) variables. As a consequence, the programmer can choose between lexical abstractions -x s :M which lexically bind their parameter when applied, or dynamic abstractions -x d :M , which dynamically bind their parameter. The former represent regular abstractions of the -calculus [3], while the latter model constructs like Common Lisp abstractions with special variables [43], or dynamic-scope [6]. Fig. 1. The User Language u It is of paramount importance to clearly state the naming conventions that we adopt for such a language. Following Barendregt [3], we consider terms that are equal up to the renaming of their bound static variables as equivalent. On the contrary, two terms that differ by their dynamic variables are not considered as equivalent. E) D[[(dlet Fig. 2. Dynamic-Environment Passing Transform D In Figure 2, the dynamic-environment passing translation, which we call D, is a program transformation that maps programs of u into the target language deps( d ), an extended call-by-value -calculus based on lexical variables only (Figure 3). Intuitively, each abstraction (static or dynamic) of u is translated by D into an abstraction taking an extra dynamic environment in argument; the target language contains a variable e which denotes an unknown environment. As a result, the application protocol in the target language is changed accordingly: operator values are applied to pairs. In the translation of the application, the dynamic environment E is used in the translations of the operator and operand, and is also passed in argument to the operator. Dynamic abstractions are translated into abstractions which extend the dynamic environment. Each dynamic variable is translated into a lookup for the corresponding constant in the current dynamic environment. The source language of D extends u with a dlet construct, (dlet ((x d1 which stands for "dynamic let". Such a construct, inaccessible to the programmer, is used internally by the system to model the bindings of dynamic variables x di to values . The syntax of the input language, called d , appears in Figure 5. Binding lists are defined with the concatenation operator x, satisfying the following property. Vn Vn Evaluation in the target language is based on the set of axioms displayed in the second part of Figure 3. Applications of binary abstractions require a double fi v -reduction as modelled by rule (fi \Theta environment lookup is implemented by (lk 1 ) and (lk 2 ). Following Sabry and Felleisen, our purpose in the rest of this Section is to derive the set of axioms that can perform on terms of d the reductions allowed on terms of The Language deps( d E) j (-y:P )P (Term) e (Unknown Env. Variable) Axioms: (lookup xd (extend E xd W (lookup xd (extend E xd1 W E) if xd1 6= xd (lk 2 ) (j \Theta Fig. 3. Syntax and Axioms of the deps(-d )-calculus Fig. 4. The Inverse Dynamic-Environment Passing Transform D \Gamma1 deps( d ). More precisely, we want to define a calculus on d that equationally corresponds to the calculus on deps( d ). The following definition of equational correspondence is taken verbatim from [40]. Definition 1 (Equational Correspondence) Let R and G be two languages with calculi -XR and -XG . Also let f G be a translation from R to G, and h be a translation from G to R. Finally let G. Then the calculus -XR equationally corresponds to the calculus -XG if the following four conditions hold: 1. 2. -XG ' Figure 4 contains an inverse dynamic-environment passing transform mapping terms of deps( d ) into terms of d . The first case is worth explaining: a term (W 1 hE; W 2 i) represents the application of an operator value W 1 on a pair dynamic environment E and operand value its inverse translation is the application of the inverse translations of W 1 and W 2 , in the scope of a dlet with the inverse translation of E. For the following cases, the inverse translation removes the environment argument added to abstractions, and translates any occurrence of a dynamic environment into a dlet-expression. State Space: (binding list) Primary Axioms: (dlet (dlet (dlet (dlet (dlet Derived Axioms: (dlet Compatibility (dlet Fig. 5. Syntax and Axioms of the -d -calculus If we apply the dynamic-environment passing transform D to a term of d , and immediately translate the result back to d by D \Gamma1 , we find the first six primary axioms of Figure 5. For explanatory purpose, we prefer to present the derived axioms (dlet intro 0 ) and (dlet propagate 0 ). The axiom (dlet intro 0 ) is the counterpart of (fi v ) for dynamic abstraction: applying a dynamic abstraction on a value V creates a dlet- construct that dynamically binds the parameter to the argument V and that has the same body as the abstraction. Rule (dlet propagate 0 ), rewritten below using the syntactic sugar let, tells us how to transform an application appearing inside the scope of a dlet. (dlet The operator and the operand can each separately be evaluated inside the scope of the same dynamic environment, and the application of the operator value on the operand value also appears inside the scope of the same dynamic environment. The interpretation of (dlet merge), (dlet elim 1 ), (dlet We can establish the following properties concerning the composition of D and D Lemma 2 For any term M 2 d , any value V 2 V alue d , any list of bindings Bind d , for any environment E 2 deps( d ), let For any term P 2 deps( d ), any value W 2 deps(V alue d ), any dynamic environments by applying the inverse translation D \Gamma1 to each axiom of deps(- d ), we obtain the four last primary axioms of Figure 5. Rules (lookup 1 ) and (lookup 2 ) are the immediate correspondent of (lk 1 ) and (lk 2 ) in deps(- d ), while (fi 0\Omega ) and (j v ) were axioms discovered by Sabry and Felleisen in applying the same technique to calculi for continuations and assignments [40]. The intuition of the set of axioms of - d can be explained as follows. In the absence of dynamic abstractions, - d behaves as the call-by-value -calculus. Whenever a dynamic abstraction is applied, a dlet construct is created. Rule (dlet propagate 0 ) propagates the dlet to the leaves of the syntax tree, and replaces each occurrence of a dynamic variable by its value in the dynamic environment by (lookup 1 ) and (lookup 2 ). Rule (dlet propagate 0 ) also guarantees that the dynamic binding remains accessible during the extent of the application of the dynamic abstraction, i.e. until it is deleted by (dlet us also observe here and now that parallel evaluation is possible because the dynamic environment is duplicated for the operator and the operand, and both can be reduced independently. This property will be used in Section 8 to define a parallel evaluation function. We obtain the following soundness and completeness results: Lemma 4 (Soundness) For any terms M for any E 2 deps( d ), we have that: deps(- d Lemma 5 (Completeness) For any terms P The following Theorem is a consequence of Lemmas 2 to 5. Theorem 1 The calculus - d equationally corresponds to the calculus deps(- d ). 2 Within the calculus, we can define a partial evaluation relation: the value of a program M is V if we can prove that M equals V in the calculus. Definition 6 (eval c ) For any program M 2 0 This definition does not give us an algorithm, but it states the specification that must be satisfied by any evaluation procedure. The purpose of the next Section is to define such a procedure. 4 Sequential Evaluation The sequential evaluation function is defined in Figure 6. It relies on a notion of evaluation context [11]: an evaluation context E is a term with a "hole", [ ], in place of the next subterm to evaluate. We use the notation E [M ] to denote the term obtained by placing M inside the hole of the context E . Four transition rules only are necessary: (dlet intro) and (dlet elim) are derived from the - d -calculus. Rule (lookup) is a replacement for (dlet propagate), (dlet merge), (dlet lookup 1 ), and (dlet lookup 2 ) of the - d -calculus. State Space: alued ::= xs j (-xs:M) j (-xd:M) (Value) E) j E) (Evaluation Context) Transition Rules: Evaluation Function: For any program M 2 0 error if M 7! Dynamically Bound Variables: Stuck Terms: Fig. 6. Sequential Evaluation Function Intuitively, the value of a dynamic variable is given by the latest active binding for this variable. In this framework, the latest active binding corresponds to the innermost dlet that binds this variable. The dynamic extent of a dlet construct is the period of time between its apparition by (dlet intro) and its elimination by (dlet elim). The evaluation algorithm introduces the concept of stuck term, which is defined by the occurrence of a dynamic variable in an evaluation context that does not contain a binding for it. The evaluation function is then defined as a total function returning a value when evaluation terminates, ? when evaluation diverges, or error when a stuck term is reached. The correctness of the evaluation function is established by the following Theorem, which relates eval c and eval d . Let us observe that eval c may return a value V 0 that differs from the value V returned by eval d because the calculus can perform reductions inside abstractions. Theorem 2 For any program M 2 0 d , eval c If we were to implement (lookup), we would start from the dynamic variable to be evaluated, and search for the innermost enclosing dlet. If it contained a binding for the variable, we would return the associated value. Otherwise, we would proceed with the next enclosing dlet. This behaviour exactly corresponds to the search of a value in an associative list (assoc in Scheme). Such a strategy is usually referred to as deep binding . In Section 7, we further refine the sequential evaluation function by making this associative list explicit. But, beforehand, we show that dynamic binding adds expressiveness to a functional language. 5 Expressiveness In Section 2.1, we stated that dynamic binding was an expressive programming technique that, when used in a sensible manner, could reduce programming patterns in programs. In this Section, we give a formal justification to this statement, by proving that dynamic binding adds expressiveness [8] to a purely functional language. First, we define the notion of observational equivalence. Definition 7 (Observational Equivalence) Given a programming language L and an evaluation function eval L , two terms are observationally equivalent , context C 2 L, such that C[M 1 ] and C[M 1 are both programs of L, eval L (M 1 ) is defined and equal to V if and only if eval L (M 2 ) is defined and equal to V . 2 We shall denote the observational equivalences for the call-by-value -calculus and for the - d -calculus by - =v and - =d , respectively. In order to prove that dynamic binding adds expressiveness [8] to a purely functional language, let us consider the following lambda terms, assuming the existence of a primitive cons to construct pairs. (cons v (f (-d:v)))) The terms are observationnally equivalent in the - v -calculus, i.e. M 1 - =v M 2 , but we have that M 1 then This example shows that dynamic binding enables us to distinguish terms that the call-by-value -calculus cannot distinguish. As a result, - =v 6ae - =d , and using Felleisen's definition of expressiveness [8, Thm 3.14], we conclude that: Proposition 1. v cannot macro-express dynamic binding relative to d . 6 Semantics of Exceptions First-class continuations and state can simulate exceptions [13]. We show here that exceptions can be defined in terms of first-class continuations and dynamic binding. In the semantics of ML [26], a raised exception returns an exceptional value, distinct from a normal value, which has the effect to prune its evaluation context until a handler is able to deal with the exception. By merging the mechanism that aborts the computation and the mechanism that fetches the handler for the exception, the handler can no longer be executed in the dynamic environment in which the exception was raised. As a result, such an approach cannot be used to give a semantics to other kinds of exceptions, like resumable ones [43]. In order to model the abortive effect, we extend the sequential evaluation function of Figure 6 with Felleisen and Friedman's abort operator A [11]. For the sake of simplicity, we assume that there exists only one exception type (discrimination on the kind of exception can be performed in the handler). We also assume the existence of a distinguished dynamic variable x ed . In Figure 7, we give the semantics of ML-style exceptions. When an exception is raised, the latest active handler is called, escapes, and then applies f in the same dynamic environment as handle, and not in the dynamic environment where the exception was raised 3 . On the other hand, there exist other kinds of exceptions, like resumable exceptions, e.g. Common Lisp resumable errors [43], or Eulisp resumable conditions [34]. They essentially offer the opportunity to resume the computation at the point where the exception was raised. In the sequel, we present a variant of Queinnec's monitors [36, 3 The usage of a first-class continuation appears here as the rule for handle duplicates the evaluation context E. Let us also observe that the continuation is only used in a downward way, which amounts to popping frames from the stack only. E[(-xed :M) (-v:A E[(f v)])] Fig. 7. ML-style exceptions p. 255], which give the essence of resumable exceptions. The primitives monitor/signal play the role that handler/raise had for ML-style exceptions. Let us note that signal is a binary function, which takes not only a value, but also a boolean r indicating whether the exception should be raised as resumable. E[(monitor f M)] 7!d E [ (-xed :M) (let (old xed (if r x E[(signal r V )] 7!d E[(xed r V )] Fig. 8. Resumable exceptions Like handle, monitor installs an exception handler for the duration of a computation. If an exception is signalled, the latest active handler is called in the dynamic environment of the signalled exception. If an exception is signalled by the handler itself, it will be handled by the handler that existed before monitor was called: this is why x ed is shadowed for the duration of the execution of the handler f , but will be again accessible if the "normal" computation resumes. If the exception was signalled as resumable, i.e. if the first argument of signal is true, the value returned by the handler is returned by signal, and computation continues in exactly the same dynamic environment 4 . This approach to define the semantics of exception has two advantages, at least. First, as we model each effect by the appropriate primitive (abortion by A and handler installation by dynamic binding), we have the ability to model different kinds of semantics for exceptions. Second, defining the semantics of exceptions with assignments weakens the theory [12] because assignments break some equivalences that would hold in the presence of exceptions: so, our definition provides a more precise characterisation of a theory of exceptions. Refinement We refine the evaluation function by representing the dynamic environment explicitly by an associative list. By separating the evaluation context from the dynamic environment, we facilitate the design of a parallel evaluation function of Section 8. Figure 9 displays the state space and transition rules of the deep binding strategy. The dynamic environment is represented in a new dlet construct which can only appear at the outermost level of a configuration, called state. The list of bindings ffi can be regarded as a global stack, initially empty when evaluation starts. A binding is pushed on the binding list, every time a dynamic abstraction is applied, and popped at the end of the dynamic extent of the application. In Section 4, the dlet construct was also modelling the dynamic extent of a dynamic-abstraction application; now that the dlet construct no longer appears inside terms, we introduce a (pop M) term playing the same role: it is created when a dynamic abstraction is applied and is destroyed at the end of the dynamic extent, after popping the top binding of the binding list. Theorem 3 establishes the correctness of the deep binding strategy. 4 Such a semantics assumes that there exists an initial handler in which evaluation can proceed. State Space: (Binding list) E) j E) (Evaluation Context) Transition Rules: (dlet (dlet (dlet (dlet ffix((x d V Evaluation Function: db (dlet error if (dlet () M) 7! Stuck State: S 2 Stuck( db Fig. 9. Deep Binding Theorem 3 eval The deep binding technique is simple to implement: bindings are pushed on the binding list ffi at application time of dynamic abstractions and popped at the end of their extent. However, the lookup operation is inefficient because it requires searching the dynamic list, which is an operation linear in its length. There exist some techniques to improve the lookup operation. The shallow binding technique consists in indexing the dynamic environment by the variable names [1]. A further optimisation, called shallow binding with value cell is to associate each dynamic variable with a fixed location which contains the correct binding for that variable: the lookup operation then simply requires to read the content of that location. 8 Parallel Evaluation In Section 3, we observed that the axiom (dlet propagate 0 ) was particularly suitable for parallel evaluation because it allowed the independent evaluation of the operator and operand by duplicating the dynamic environment. It is well-known that the deep binding strategy is adapted to parallel evaluation because the associative list representing the dynamic environment can be shared between different tasks. As in our previous work [30], we follow the "parallelism by annotation" approach, where the programmer uses an annotation future [17] to indicate which expressions may be evaluated in parallel. The semantics of future has been described in the purely functional framework [14] and in the presence of first-class continuations and assignments [30]. In Figure 10, we present the semantics of future in the presence of dynamic binding. As in [14, 30], the set of terms is augmented with a future construct, and we add to the set of values a placeholder variable, "which represents the result of a computation that is in progress". In addition, a new construct (f-let (p M) S) has a double goal: first as a let, it binds p to the value of M in S; second, it models the potential evaluation of S in parallel with M . The component M is the mandatory term because it is the first that would be evaluated if evaluation was sequential, while S is speculative because its value is not known to be needed before M terminates. State Space: Transition Rules: (dlet ae (dlet (dlet (dlet (dlet ae (dlet (dlet (dlet ffix((x d V (dlet error (error) (dlet (dlet (dlet (dlet (dlet Evaluation Function: For any program M 2 0 (dlet IN such that (dlet error if (dlet () M) 7! error Fig. 10. Parallel Evaluation (differences with Figure It is important to observe that (future [ ]) is not a valid evaluation context. Otherwise, if evaluation was allowed to proceed inside the future body, it could possibly change the dynamic environment, which would make (fork) unsound. Instead, rule (ltc), which stands for lazy task creation [27, 7], replaces a (future M) expression by (fmark which should be interpreted as a mark indicating that a task may be created. If the runtime elects to create a new task, (fork) creates a f-let expression, whose mandatory component is the argument of fmark, i.e. the future argument, and whose speculative component is a new state evaluating the context of fmark filled with the placeholder variable, in the scope of the duplicated dynamic environment ffi 1 . If the run-time does not elect to spawn a new task, evaluation can proceed in the fmark argument. Rules (ltc) and (future id) specify the sequential behaviour of future: the value of future is the value of fmark, which is the value of its argument. When the evaluation of the mandatory component terminates, rule (join) substitutes the value of the placeholder in the speculative state. Rule (speculative) indicates that speculative transitions are allowed in the f-let body. Following [14], Figure 10 defines a relation S 1 7! n;m meaning that n steps are involved in the reduction from S 1 to S 2 , among which m are mandatory. The correctness of the evaluation function follows from a modified diamond property and by the observation that the number of pop terms in a state is always smaller or equal to the length of the dynamic environment. Theorem 4 eval As far as implementation is concerned, rule (ltc) seems to indicate that the dynamic environment should be duplicated. A further refinement of the system indicates that it suffices to duplicate a pointer to the associative list, as long as the list remains accessible in a shared store. Rule (ltc) adds an overhead to every use of future, by duplicating the dynamic environment even if dynamic variables are not used. Feeley [7] describes an implementation that avoids this cost by lazily recreating a dynamic environment when a task is stolen. Due to the orthogonality between assignments and dynamic binding, our previous results [30] with assignments can be merged within this framework. Adding assignments permits the definition of mutable dynamic variables (with a construct like dynamic-set! [34]). Due to the purely dynamic nature of the semantics, the presence of mutable dynamic variables offers less parallelism as observed in [30]. The interaction of dynamic binding and continuations is however beyond the scope of this paper [19]. 9 Related Work In the conference on the History of Programming Languages, McCarthy [25] relates that they observed the behaviour of dynamic binding on a program with higher-order functions. The bug was fixed by introducing the funarg device and the function con- struct[32]. Cartwright [4] presents an equational theory of dynamic binding, but his language is extended with explicit substitutions and assumes a call-by-name parameter passing technique. The motivation of his work fundamentally differs from ours: his goal is to derive a homomorphic model of functional languages by considering - as a combinator. His axioms are derived from the -oe-calculus axioms, while ours are constructed during the proof of equational correspondence of the calculus. The authors of [6] discuss the issue of tail-recursion in the presence of dynamic binding. They observe that simple implementations of fluid-let [18] are not tail-recursive because they restore the previous dynamic environment after evaluating the fluid-let body. Therefore, they propose an implementation strategy, which in essence is a dynamic-environment passing style solution. Programs in dynamic-environment passing style are characterised by the fact that they do not require a growth of the control state for dynamic binding; however, they require a growth of the heap space. An analogy is the continuation-passing translation, which generates a program where all function calls are in terminal position although it does not mean that all cps-programs are iterative. Feeley [7] and Queinnec [36] observe that programs in dynamic-environ- ment passing style reserve a special register for the current dynamic environment. Since every non-terminal call saves and then restores this register, such a strategy penalises programs that do not use dynamic binding, especially in byte-code interpreters where the marginal cost of an extra register is very high. Both of them prefer a solution that does not penalise all programs, at the price of a growth of the control state for every dynamic binding. Consequently, we believe that implementors have to decide whether dynamic binding should or not increase the control state; in any case, it will result in a non-iterative behaviour. In the tradition of the syntactic theories for continuations and assignments, we present a syntactic theory of dynamic binding. This theory helps us in deriving a sequential evaluation function and a refined implementation like deep binding. We also integrate dynamic-binding constructs into our framework for parallel evaluation of future-based programs. Besides, we prove that dynamic binding adds expressiveness to purely functional language and we show that dynamic binding is a suitable tool to define the semantics of exceptions-like notions. Furthermore, we believe that a single framework integrating continuations, side-effects, and dynamic binding would help us in proving implementation strategies of fluid-let in the presence of continuations [19]. Acknowledgement Many thanks to Daniel Ribbens, Christian Queinnec, and the anonymous referees for their helpful comments. --R Anatomy of Lisp. Shallow binding in lisp 1.5. The Lambda Calculus: Its Syntax and Semantics Lambda: the Ultimate Combinator. Abstracting Control. Dynamic Identifiers Can Be Neat. An Efficient and General Implementation of Futures on Large Scale Shared-Memory Multiprocessors On the Expressive Power of Programming Languages. A Reduction Semantics for Imperative Higher-Order Languages A Syntactic Theory of Sequential State. A Syntactic Theory of Sequential Control. The Revised Report on the Syntactic Theories of Sequential Control and State. Controlling Effects. The Semantics of Future and Its Use in Program Optimization. Operational Reasoning and Denotational Semantics. Towards a Semantic Theory of Dynamic Binding. MIT Scheme Reference Manual. Embedding Continuations in Procedural Objects. Continuations and Concurrency. Report on the Programming Language Haskell. The GNU Emacs Lisp Reference Manual Recursive Functions of Symbolic Expressions and Their Computation by Machine History of Lisp. The Definition of Standard ML. Lazy Task Creation Maclisp reference manual. Sound Evaluation of Parallel Functional Programs with First-Class Contin- uations The Semantics of Scheme with Future. Partial Continuations as the Difference of Continu- ations The function of function in lisp or why the funarg problem should be called the environment problem. Speculative Computation in Multilisp. The Eulisp Definition Lisp in Small Pieces. Design of a Concurrent and Distributed Lan- guage A Dynamic Extent Control Operator for Partial Continuations. Revised 4 Report on the Algorithmic Language Scheme. The Formal Relationship between Direct and Continuation-Passing Style Optimizing Compilers: a Synthesis of Two Paradigms Reasoning about Programs in Continuation-Passing Style Control Delimiters and Their Hierarchies. The Language. Rum : an Intensional Theory of Function and Control Abstractions. Programming Perl. --TR --CTR Christian Queinnec, The influence of browsers on evaluators or, continuations to program web servers, ACM SIGPLAN Notices, v.35 n.9, p.23-33, Sept. 2000 Matthias Neubauer , Michael Sperber, Down with Emacs Lisp: dynamic scope analysis, ACM SIGPLAN Notices, v.36 n.10, October 2001 Gavin Bierman , Michael Hicks , Peter Sewell , Gareth Stoyle , Keith Wansbrough, Dynamic rebinding for marshalling and update, with destruct-time ?, ACM SIGPLAN Notices, v.38 n.9, p.99-110, September Zena M. Ariola , Hugo Herbelin , Amr Sabry, A type-theoretic foundation of continuations and prompts, ACM SIGPLAN Notices, v.39 n.9, September 2004 Oleg Kiselyov , Chung-chieh Shan , Amr Sabry, Delimited dynamic binding, ACM SIGPLAN Notices, v.41 n.9, September 2006 Magorzata Biernacka , Olivier Danvy, A syntactic correspondence between context-sensitive calculi and abstract machines, Theoretical Computer Science, v.375 n.1-3, p.76-108, May, 2007

dynamic binding and extent;parallelism;functional programming;syntactic theories

609199

Continuation-Based Multiprocessing.

Any multiprocessing facility must include three features: elementary exclusion, data protection, and process saving. While elementary exclusion must rest on some hardware facility (e.g., a test-and-set instruction), the other two requirements are fulfilled by features already present in applicative languages. Data protection may be obtained through the use of procedures (closures or funargs), and process saving may be obtained through the use of the catch operator. The use of catch, in particular, allows an elegant treatment of process saving.We demonstrate these techniques by writing the kernel and some modules for a multiprocessing system. The kernel is very small. Many functions which one would normally expect to find inside the kernel are completely decentralized. We consider the implementation of other schedulers, interrupts, and the implications of these ideas for language design.

Introduction In the past few years, researchers have made progress in understanding the mechanisms needed for a well-structured multi-processing facility. There seems to be universal agreement that the following three features are needed: 1. Elementary exclusion 2. Process saving 3. Data protection By elementary exclusion, we mean some device to prevent processors from interfering with each other's access to shared resources. Typically, such an elementary exclusion may be programmed using a test and set instruction to create a critical region. Such critical regions, however, are not by themselves adequate to describe the kinds of sharing which one wants for controlling more complex resources such as disks or regions in highly structured data bases. In these cases, one uses an elementary exclusion to control access to a resource manager (e.g., a monitor [11] or serializer [1]), which in turn regulates access to the resource. Unfortunately, access to the manager may then become a system bottleneck. The standard way to alleviate this is to have the manager save the state of processes which it wishes to delay. The manager then acts by taking a request, considering the state of the resource, and either allowing the requesting program to continue * Research reported herein was supported in part by the National Science Foundation under grant numbers MCS75-06678A01 and MCS79-04183. This paper originally appeared in J. Allen, editor, Conference Record of the 1980 LISP Conference, pages 19-28, Palo Alto, CA, 1980. The Company, Republished by ACM. * Current address: College of Computer Science, Northeastern University, 360 Huntington Av- enue, 161CN Boston, MA 02115, USA. or delaying it on some queue. In this picture, the manager itself does very little computing, and so becomes less of a bottleneck. To implement this kind of manager, one needs some kind of mechanism for saving the state of the process making a request. The basic observation of this paper is that such a mechanism already exists in the literature of applicative languages: the catch operator [14], [15], [21], [25]. This operator allows us to write code for process-saving procedures with little or no fuss. This leaves the third problem: protecting private data. It would do no good to monitors if a user could bypass the manager and blithely get to the resource. The standard solution is to introduce a class mechanism to implement protected data. In an applicative language, data may be protected by making it local to a procedure (closure). This idea was exploited in [19], but has been unjustly neglected. We revive it and show how it gives an elegant solution to this problem. We will demonstrate our solution by writing the kernel and some modules for a multiprocessing system. The kernel is very small. Many functions which one would normally expect to find inside the kernel, such as semaphore management [2], may be completely decentralized because of the use of catch. Our system thus answers one of the questions of [3] by providing a way to drastically decrease the size of the kernel. We have implemented the system presented here (in slightly different form) using the Indiana SCHEME 3.1 system [26]. The remainder of this paper proceeds as follows: In Section 2, we discuss our assumptions about the system under which our code will run. Sections 3 and 4 show how we implement classes and process-saving, respectively. In Section 5 we bring these ideas together to write the kernel of a multiprocessing system. In Sections 6 and 7 we utilize this kernel to write some scheduling modules for our system In Section 8, we show how to treat interrupts. Last, in Section 9, we consider the implications of this work for applicative languages. 2. The Model of Computation Our fundamental model is that of a multiprocessor, multiprocess system using shared memory. That is, we have many segments of code, called processes, which reside in a single shared random-access memory. The extent to which processes actually share memory is to be controlled by software. We have several active units called processors which can execute processes. Several processors may be executing the same process simultaneously. We make the usual assumption that memory access marks the finest grain of interleaving; that is, two processors may not access (read or write) the sane word in memory at the same time. This elementary memory exclusion is enforced by the memory hardware. At the interface between the processes and the processors is a distinguished process called the kernel. The kernel's job is to assign processes which are ready to run to processors which are idle. In a conventional system, e.g., [22], this entails keeping track of many things. We shall see that the kernel need only keep track of ready processes. It may be worthwhile to discuss the author's SCHEME 3.1 system, which provided the context for this work. SCHEME is an applicative-order, lexically-scoped, full-funarg dialect of LISP [25]. The SCHEME 3.1 system at Indiana University translates input Scheme code into the code for a suitable multistack machine. The machine is implemented in LISP. Thus, we were under the constraint that we could no LISP code, since such an addition would constitute a modification to the machine. The system simulates a multiprocessor system by means of interrupts, using a protocol to be discussed in Section 8. However, all primitive operations, including the application of LISP functions, are uninterruptible. This allows us to write an uninterruptible test-and-set operation, such as (de test-and-set-car (x) (prog2 nil (car x) (rplaca x nil))) which returns the car of its argument and sets the car to nil. Two other features of SCHEME are worth mentioning. First, SCHEME uses call-by-value to pass parameters. That means that after an actual parameter is evaluated, a new cons-cell is allocated, and in this new cell a pointer to the evaluated actual parameter is planted. (In the usual association list implementation, the pointer is in the cdr field; in the rib-cage implementation [24], it is in the car.) This pointer may be changed by the use of the asetq procedure. Thus, if we write (define scheme-demo-1 (x) (asetq x any call to SCHEME-DEMO-1 always returns 3, since the second asetq changes a different cell from the first one. (This feature of applicative languages has always been rather obscure. See [17] Sec. 1.8.5 for an illuminating discussion.) The second property on which we depend is that the "stack" is actually allocated from the LISP heap using cons and is reclaimed using the garbage collector. This allows us to be quite our coding techniques. We shall have more to say about this assumption in our conclusions. 3. Implementing classes For us, the primary purpose of the class construct is to provide a locus for the retention of private information. In Simula [6], a class instance is an activation record which can survive its caller. In an applicative language, such a record may be constructed in the environment (association list) of a closure (funarg). This idea is stated clearly in [23]; we discuss it briefly here for completeness. For example, a simple cons-cell may be modelled by: (define cons-cell (x y) (lambda (msg) (cond (lambda (val) (asetq x val))) (lambda (val) (asetq y val))) (define car (x) (x 'car)) (define cdr (x) (x 'cdr)) (define rplaca (x v) ((x 'rplaca) v)) (define rplacd (x v) ((x 'rplacd) v)) Here a cons cell is a function which expects a single argument; depending on the type of argument received, the cell returns or changes either of its components. (We have arbitrarily chosen one of the several ways to do this). Such behaviorally defined data structures are discussed in [10], [20], [23]; at least one similar object was known to Church [5], cited in [24]. Another example, important for our purposes, is (define busy-wait () (let ((x (cons t nil))) (labels ((self (lambda (msg) (cond (if (test-and-set-car x) (car (rplaca x t))) busy-wait is a function of no arguments, which, when called, creates a new locus of busy-waiting. It does this by creating a function with a new private variable x. (This x is guaranteed new because of the use of call-by-value). This returned function (here denoted self) expects a single argument, either P or V. Calling it with P sends it into a test-and-set loop, and calling it with V resets the car of x to t, thus releasing the semaphore. There is no way to access the variable x except through calls on this function. Note that we are not advocating busy-waiting (except perhaps in certain very special circumstances). Any use of busy-wait in the rest of this paper may be safely replaced with any hardware-supported elementary exclusion device which the reader may prefer. Our concern is how to build complex schedulers from these elementary exclusions. In particular, we shall consider better ways to build a semaphore in Section 6. 4. Process saving with catch catch is an old addition to applicative languages. The oldest version known to the author is Landin's, who called it either ``pp'' (for ``program point'') [15] or "J-lambda" [14]. 1 Reynolds [21] called it "escape." A somewhat restricted form of catch exists in LISP 1.5, as errset [16]; another version is found in MACLISP, as the pair catch and throw. The form we have adopted is Steele and Sussman's [25], which is similar to Reynolds'. In SCHEME, catch is a binding operator. Evaluation of the expression (catch id causes the identifier id to be bound (using call-by-value) to a "continuation object" which will be described shortly. The expression expr is then evaluated in this extended environment. The continuation object is a function of one argument which, when invoked, returns control to the caller of the catch expression. Control then proceeds as if the catch expression had returned with the supplied argument as its value. This corresponds to the notion of an "expression continuation" in denotational semantics. To understand the use of catch, we may consider some examples. (catch m (cons (m returns 3; when the (m 3) is evaluated, it is as if the entire catch expression returned 3. The form in which we usually will use catch is similar. In (define foo evaluation of (m -junk-) causes the function foo to return to its caller with the value of -junk-. The power of catch arises when we store the value of m and invoke it from some other point in the program. In that case, the caller of foo is restarted with m's argu- ment. The portion of the program which called m is lost, unless it has been preserved with a strategically placed catch. A small instance of this phenomenon happened even in our first example-there, m's caller was the cons which was abandoned. Calling a continuation function is thus much like jumping into hyperspace-one loses track entirely of one's current context, only to re-emerge in the context that set the continuation. There will actually be very few occurrences of catch in the code we write. For the remainder of this section and the next, we shall consider what things we can do with continuations which have already been created by catch. When we get to Section 6, we shall start to use catch in our code. We shall use continuations to represent processes. A process is a self-contained computation. We may represent a process as a pair consisting of a continuation and an argument to be sent to that continuation. This corresponds to the notion of "command continuation" in denotational semantics. (define cons-process (cont arg) (lambda (msg) (cond Here we have defined a process as a class instance with two components, a continuation and an argument, and a single operation, run-it, which causes the continuation to be applied to the argument, thus starting the process. Because cont is a contin- uation, applying it causes control to revert to the place to which it refers, and the caller of (x 'run-it) is lost. This is not so terrible, since the caller of (x 'run-it) may have been saved as a continuation someplace else. 5. The Kernel We now have enough machinery to write the kernel of our operating system. The kernel's job is to keep track of those processes which are ready to run, and to assign a process to any processor which asks for one. The kernel is therefore a class instance which keeps a queue of processes and has two operations: one to add a process to the ready queue and one to assign a process to a processor (thereby deleting it from the ready queue). We shall need to do some queue manipulation. We therefore assume that we have a function (create-queue) which creates an empty queue, a function (addq q x) which has the side effect of adding the value of x to the queue q, and the function (deleteq q), which returns the top element of the queue q, with the side-effect of deleting it from q. We may now write the code for the kernel: (define gen-kernel () (let ((ready-queue (create-queue)) (mutex (busy-wait))) (lambda (msg) (cond (lambda (cont arg) (block (mutex 'P) (addq ready-queue (cons-process cont arg)) (mutex (mutex 'P) (let ((next-process (deleteq ready-queue))) (block (mutex 'V) (next-process 'run-it)))))))) (asetq kernel (gen-kernel)) (define made-ready (cont arg) ((kernel 'make-ready) cont arg)) (define dispatch () (kernel 'dispatch)) We have now defined the two basic functions, make-ready and dispatch. The call (make-ready cont arg) puts a process, built from cont and arg, on the ready queue. To do this, it must get past a short busy-wait. (This busy-wait is always short because the kernel is never tied up for very long. This construction is also in keeping with the idea of building complex exclusion mechanisms from very simple ones.) It then puts the process on the queue, releases the kernel's exclusion, and exits. (Given the code for busy-wait above, it always returns t. The value returned must not be a pointer to any private data). dispatch is subtler. A processor will execute (dispatch) whenever it decides it has nothing better to do. Normally, a call to dispatch would be preceded by a call to make-ready, but this need not be the case. After passing through the semaphore, the next waiting process is deleted from the ready queue and assigned to next-process. A (mutex 'V) is executed, and the next-process is started by sending it a run-it signal. The subtlety is in the order of these last two operations. They cannot be reversed, since once next-process is started, there would be no way to reset the semaphore. The given order is safe however, because of the use of call-by-value. Every call on (dispatch) uses a different memory word for next-process. Therefore, the call (next-process 'run-it) uses no shared data and may be executed outside the critical region. few explanatory words on the code itself are in order. First, note that (kernel 'make-ready) returns a function which takes two arguments and performs the required actions. (kernel 'dispatch), however, performs its actions directly. We could have made (kernel 'dispatch) return a function of no arguments, but we judged that to be more confusing than the asymmetry. Second, block is SCHEME's sequencing construct, analogous to progn. Also, cond uses the so-called "generalized cond," with an implicit block (or progn) on the right-hand-side of each alternative.) 6. Two Better Semaphores Our function busy-wait would be an adequate implementation of a binary semaphore if one was sure that the semaphore was never closed for very long. In this section, we shall write code for two better implementations of semaphores. For our first implementation, we use the kernel to provide an alternative to the test-and-set loop. If the test-and-set fails, we throw the remainder of the current process on the ready queue, and execute a DISPATCH. This is sometimes called a "spin lock." (define spin-lock-semaphore () (let ((x (cons t nil))) (labels ((self (lambda (msg) (cond (cond ((test-and-set-car (car (rplaca x t))) (define give-up-and-try-later () (catch caller (block (make-ready caller t) Here, the key function is give-up-and-try-later. It puts on the ready-queue a process consisting of its caller and the argument t. It then calls dispatch, which switches the processor executing it to some ready process. When the enqueued process is restarted (by some processor executing a dispatch), it will appear that give-up-and-try-later has quietly returned t. The effect is to execute a delay of unknown duration, depending on the state of the ready queue. Thus a process executing a P on this semaphore will knock on the test-and-set cell once; if it is closed, the process will go to sleep for a while and try again later. While this example illustrates the use of catch and make-ready, it is probably not a very good implementation of a semaphore. A better implementation (closer to the standard one) would maintain a queue of processes waiting on each semaphore. A process which needs to be delayed when it tries a P will be stored on this queue. When a V is executed, a waiting process may be restarted, or, more precisely, placed on the ready queue. We code this as follows: (define semaphore () (let ((q (create-queue)) ; a queue for waiting processes (count (mutex (busy-wait))) (lambda (msg) (cond (mutex 'P) (catch caller (block (cond ((greaterp count (asetq count (sub1 count)) (mutex 'V) (mutex 'V) (mutex 'P) (if (emptyq q) (asetq count (add1 count)) (make-ready (deleteq q) t)) (mutex 'V) Executing (semaphore) creates a class instance with a queue q, used to hold processes waiting on this semaphore, an integer count, which is the traditional "value" of the semaphore, and a busy-wait locus mutex. mutex is used to control access to the scheduling code, and is always reopened after a process passes through the semaphore. As was suggested in the introduction, this use of a small busy-wait to control entrance to a more sophisticated scheduler is typical. When a P is executed, the calling process first must get past mutex into the critical region. In the critical region, the count is checked. If it is greater than 0, it is decremented, the mutex exclusion is released, and the semaphore returns a value of T to its caller. If the count is zero, the continuation corresponding to the caller of the semaphore is stored on the queue. mutex is released, and the processor executes a (dispatch) to find some other process to work on. When a V is executed, the calling process first gets past mutex into the semaphore's critical region. The queue is checked to see if there are any processes waiting on this semaphore. If there are none, the count is incremented. If there is at least one, it is deleted from the queue by (deleteq q), and put on the kernel's ready queue with argument t. When it is restarted by the kernel, it will think it has just completed its call on P. (Since a P always returns t, the second argument to make-ready must likewise be a t). After this bookkeeping is accomplished, mutex is released and the call on V returns t. All of this is just what a typical implementation of semaphores (e.g., [2]) does. The difference is that our semaphore is an independent object which lies outside the kernel. It is in no way privileged code. We have also written code to implement more complex schedulers. The most complex scheduler for which we have actually written code is for Brinch Hansen's "process" [4]. We have written this as a SCHEME syntactic macro. The code is only about a page long. 7. Doing more than one thing at once We now turn to the important issue of process creation. Although the semaphores in the previous section used catch to save the state of the current process, they did not provide any means to increase the number of processes in the system. We may do this with the function create-process. create-process takes one argument, which is a function of no arguments, and creates a process which will execute this function in "parallel" with the caller of create-process. (define create-process (fn) (catch caller (block (catch process (block (make-ready process t) (caller When create-process is called with fn, it first creates a continuation containing its caller and calls it caller. It enters the block, and creates a continuation called process, which, when started, will continue execution of the block with (fn). This continuation process is then put on the kernel's ready queue (with argument t, which will be ignored when process is restarted). Then (caller t) is executed, which causes create-process to return to its caller with value t. Thus, the process which called create-process continues in control of its pro- cessor, but process is put onto the ready queue. When the kernel decides to run process, (fn) will be executed. The processor which runs process will then do a (dispatch) to find something else to do. (The reader who finds this code tricky may take some comfort in our opinion that this is the trickiest piece of code in this paper. The difficulty lies in the fact that its execution sequence is almost exactly reversed from its lexical sequence [8].) We can use create-process to implement a fork-join. The function fork takes two functions of no arguments. Its result is to be the cons of their values. The execution of the two functions is to proceed as two independent processes, and the process which called fork is to be delayed until they both return. (define fork (fn1 fn2) (catch caller (let ((one-done? nil) (ans1 nil) (ans2 nil) (mutex (busy-wait))) (let ((check-done (lambda (dummy) (block (mutex 'P) (if one-done? (make-ready caller (cons ans1 ans2)) (asetq one-done? t)) (mutex 'V) )))) (block (create-process (lambda (check-done (asetq ans1 (fn1))))) (create-process (lambda (check-done (asetq ans2 (fn2))))) fork sets up four locals: one for each of the two answers, a flag called one-done?, and a semaphore to control access to the flag. It creates the two daughter processes and then dispatches, having saved its caller in the continuation caller. Each of the two processes computes its answer, deposits it in the appropriate local variable, and calls check-done. check-done uses mutex to obtain access to the flag one-done?, which is initially nil. If its value is nil, then it is set to T. If its value is T, signifying that the current call to check-done is the second one, then caller is moved to the ready queue with argument (cons ans1 ans2). 8. Interrupts What we have written so far is quite adequate for a non-preemptive scheduling system [2]. If we wish to use a pre-emptive scheduling system (as we must if we wish to use a single processor), then we must consider the handling of interrupts. We shall consider only the problem of pre-emption of processes through timing interrupts as non-preempting interrupts can be handled through methods analogous to those in [3], [27]. We model a timing interrupt as follows: When a processor detects a timing interrupt, the next identifier encountered in the course of its computation (say X) will be executed as if it had been replaced by (preempt X). preempt is the name of the interrupt-handling routine. If we believe, with [23], that a function application is just a GO-TO with binding, then this model is quite close to the conventional model, in which an interrupt causes control to pass to a predefined value of the program counter. A very similar treatment of interrupts was developed independently for use in the MIT/Xerox PARC SCHEME chip [12]. The simplest interrupt handler is: (define preempt (x) (catch caller (block (make-ready caller x) With this interrupt handler, the process which the processor is executing is thrown back on the ready queue, and the processor executes a dispatch to find something else to do. A complication that arises with pre-emptive scheduling is that interrupts must be inhibited inside the kernel. This may be accomplished by changing the busy-wait in the kernel to kernel-exclusion: (define kernel-exclusion () (let ((sem (busy-wait))) (lambda (msg) (cond (sem 'P) (sem 'V) Note the order of the operations for V. The reverse order is wrong; an interrupt might occur after the enable-preemption but before the (sem 'V), causing instant deadlock. (We discovered this the hard way!) Now, for the first time, we have introduced some operations which probably should be privileged: disable-preemption and enable-preemption. 2 We can make those privileged without changing the architecture of the machine by introducing a read-loop like: (define user-read-loop () (let ((disable-preemption (lambda () (error 'protection-error))) (enable-preemption (lambda () (error 'protection-error)))) (labels (lambda (dummy) (loop (print (eval (read))))))) (loop This is intended to suggest the user's input is evaluated in an environment in which disable-preemption and enable-preemption are bound to error-creating func- tions. This is not actually the way the code is written in SCHEME, but we have written it in this way to avoid dealing with the complications of SCHEME's version of eval. 9. Conclusions and Issues In this paper, we have shown how many of the most troublesome portions of the "back end" of operating systems may be written simply using an applicative language with catch. In the course of doing so, we have drawn some conclusions In three categories: operating system kernel design, applicative languages, and language design in general. For operating systems, this work answers in part Brinch Hansen's call to simplify the kernel [3]. Because all of the scheduling apparatus except the ready queue has been moved out of the kernel, the kernel becomes smaller, is called less often, and therefore becomes less of a bottleneck. By passing messages to class instances (functions) instead of passing them between processes, we avoid the need for individuation of processes, and thereby avoid the need to maintain process tables, etc., further reducing the size of the kernel. This is not meant to imply that we have solved all the problems associated with system kernels. Problems of storage allocation and performance are not addressed. In the areas of process saving and protection, however, the approach discussed here seems to offer considerable advantages. In the area of applicative languages, our work seems to address the issue of "state." A module is said to have "state" if different calls on that module with identical arguments may give different results at different times in the computation. Another way of describing this phenomenon is that the model is "history-dependent." (This is not to be confused with issues of non-determinism). If an object does not have state, then it should never matter whether two processes are dealing with the same object or with two copies of it. For processes to communicate, however, they must be talking to the same module, not just to two copies of it. For instance, all modules must communicate with the same kernel, not just with two or more modules produced by calls on gen-kernel. Therefore, the kernel and similar modules must have state-they must have uses of asetq in their code. This seems to us to be an important observation. It means that we must come to grips with the concept of the state if we are to deal with the semantics of parallelism. This observation could not have been made in the context of imperative languages, where every module has state. Only in an applicative context, where we can distinguish true state from binding (or internal state), could we make this distinction. 3 A related issue is the use of call-by-value. A detailed semantics of SCHEME, incorporating the Algol call-by-value mechanism, would give an unambiguous account of when two modules were the "same," and thus also give an account of when two modules share the same state. Such an account is necessary to explain the use of asetq in our programs and to determine which data is private and which is shared (as in the last lines of (kernel 'dispatch)). In such a description, we would find that restarting a continuation restores the environment (which is a map from identifiers to L-values), but does not undo changes in the global state (the map from L-values to R-values) which is altered by asetq. Nonetheless, we find this account unsatisfying, because its systematic introduction of a global state at every procedure call seems quite at odds with the usual state-free picture of an applicative program. We find it unpleasant to say that we pass parameters by worth (i.e., without copying), except when we need to think harder about the program. In this regard, we commend to applicative meta-programmers a closer study of denotational semantics. Descriptive denotational semantics, as expounded in Chapter 1 of [17] or in [9], provides the tools to give an accurate description of what actually happens when a parameter is passed. There are, however, some measures which would help alleviate the confusion. For example, we could use a primitive cell operation in place of the unrestricted use of asetq. Then all values could be passed by worth (R-value); L-values would arise only as denotations of cells, and explicit dereferencing would be required. Such an approach is taken, in various degrees, in PLASMA [10], FORTH [13], and BLISS [28]. 4 Also, John Reynolds and one of his students are investigating semantics which do not rely on a single global state [personal communication]. 5 Last, we essay some ideas about the language design process. Our choice to work in the area of applicative languages was motivated in part by Minsky's call for the separation of syntax from semantics in programming [18]. We have attempted to home in on the essential semantic ideas in multiprogramming. By "semantic" we do not simply mean those ideas which are expressible in denotational semantics, though surely the use of denotational semantics has exposed and simplified the basic ideas in programming in general. We add to these ideas some basic operational knowledge about how one goes from semantics to implementations (e.g., [21]) and some additional operational knowledge not expressed in the "formal semantics" at all, e.g., our treatment of interrupts. Only after we have a firm grasp on these informal semantic ideas should we begin to consider syntax. Some syntax is for human engineering-replacing parentheses and positional structure with grammars and keywords. Other syntax may be introduced to restrict the class of run-time structures which are needed to support the language. The design of RUSSELL [7] is a good example of this paradigm. One spectacular success which may be claimed for this approach is that of PASCAL, which took the well-understood semantics of ALGOL and introduced syntactic restrictions which considerably simplified the run-time structure. In our case, we should consider syntactic restrictions which will allow the use of sequential structures to avoid spending all one's time garbage-collecting the stack. Other clever data structures for the run-time stack should also be considered. Another syntactic restriction which might be desirable is one which would prevent a continuation from being restarted more than once. Any language or language proposal must embody a trade-off between generality (sometimes called "functionality") and efficiency. By considering complete generality first, we may more readily see where the trade-offs may occur, and what is lost thereby. Unfortunately, the more typical approach to language design is to start with a given run-time structure (or, worse yet, a syntactic proposal). When the authors realize that some functionality is lacking, they add it by introducing a patch. By introducing the generality and cleanness first, and then compromising for efficiency, one seems more likely to produce clean, small, understandable, and even efficient languages. Notes 1. Though catch and call/cc are clearly interdefinable, J and call/cc differ importantly in details; see Hayo Thielecke, "An Introduction to Landin's 'A Generalization of Jumps and Labels','' Higher-Order and Symbolic Computation, 11(2), pp. 117-123, December 1998. 2. These were additional primitives that were added to the Scheme 3.1 interpreter. 3. This paragraph grew out of conversations I had had with Carl Hewitt over the nature of object identity. I had objected that Hewitt's notion of object identity in a distributed system required some notion of global state (C. Hewitt and H. G. Baker, Actors and Continuous Functionals, in E. J. Neuhold (ed.) Formal Descriptions of Programming Concepts, pages 367-390. North Holland, Amsterdam, 1978; at page 388). This is an issue that remains of interest in the generation of globally-unique identifiers for use in large distributed systems such as IP, DCOM or the World-Wide Web. 4. This approach was of course adopted in ML. At the time, changing Scheme in this way was at least conceivable, and we seriously considered it for the Indiana Scheme 84 implementation. After the Revised 3 Report in 1984, such a radical change became impossible. Sussman and Steele now list this as among the mistakes in the design of Scheme (G.J. Sussman and G.L. Steele Jr., The First Report on Scheme Revisited, Higher-Order and Symbolic Computation 11(2), pp. 5. I am not sure to what this refers. My best guess is that it refers to his work with Oles on stack semantics (J. C. Reynolds, "The Essence of Algol," in J. W. deBakker and J. C. van Vliet, eds., Algorithmic Languages, pages 345-372. North Holland, Amsterdam, 1981). --R Synchronization in actor systems. Operating Systems Principles. The Architecture of Concurrent Programs. Distributed processes: A concurrent programming concept. The Calculi of Lambda-Conversion Hierarchical program structures. Data types Go to statement considered harmful. The Denotational Description of Programming Languages. Viewing control structures as patterns of passing messages. Monitors: An operating system structuring concept. The SCHEME-79 chip FORTH for microcomputers. A correspondence between ALGOL 60 and Church's lambda-notation: Part I The next 700 programming languages. A Theory of Programming Language Semantics. Form and content in computer science. Protection in programming languages. Definitional interpreters for higher-order programming languages The Logical Design of Operating Systems. LAMBDA: The ultimate declarative. The art of the interpreter The revised report on SCHEME. SCHEME version 3.1 reference manual. Modula: a language for modular multiprogramming. BLISS: A language for systems program- ming --TR --CTR Edoardo Biagioni , Robert Harper , Peter Lee, A Network Protocol Stack in Standard ML, Higher-Order and Symbolic Computation, v.14 n.4, p.309-356, December 2001 Manuel Serrano , Frdric Boussinot , Bernard Serpette, Scheme fair threads, Proceedings of the 6th ACM SIGPLAN international conference on Principles and practice of declarative programming, p.203-214, August 24-26, 2004, Verona, Italy Zena M. Ariola , Hugo Herbelin , Amr Sabry, A type-theoretic foundation of continuations and prompts, ACM SIGPLAN Notices, v.39 n.9, September 2004

continuations;operating sytems;multiprocessing;scheme;language design

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Combining Program and Data Specialization.

Program and data specialization have always been studied separately, although they are both aimed at processing early computations. Program specialization encodes the result of early computations into a new program&semi; while data specialization encodes the result of early computations into data structures.In this paper, we present an extension of the Tempo specializer, which performs both program and data specialization. We show how these two strategies can be integrated in a single specializer. This new kind of specializer provides the programmer with complementary strategies which widen the scope of specialization. We illustrate the benefits and limitations of these strategies and their combination on a variety of programs.

Introduction Program and data specialization are both aimed at performing computations which depend on early values. However, they dioeer in the way the result of early computations are encoded: on the one hand, program specialization encodes these results in a residual program, and on the other hand, data specialization encodes these results in data structures. More precisely, program specialization performs a computation when it only relies on early data, and inserts the textual representation of its result in the residual program when it is surrounded by computations depending on late values. In essence, it is because a new program is being constructed that early computations can be encoded in it. Furthermore, because a new program is being constructed it can be pruned, that is, the residual program only corresponds to the control AEow which could not be resolved given the available data. As a consequence, program specialization optimizes the control AEow since fewer control decisions need to be taken. However, because it requires a new program to be constructed, program specialization can lead to code explosion if the size of the specialization values is large. For example, this situation can occur when a loop needs to be unrolled and the number of iterations is high. Not only does code explosion cause code size problems, but it also degrades the execution time of the specialized program dramatically because of instruction cache misses. The dual notion to specializing programs is specializing data. This strategy consists of splitting the execution of program into two phases. The -rst phase, called the loader, performs the early computations and stores their results in a data structure called a cache. Instead of generating a program which contains the textual representation of values, data specialization generates a program to perform the second phase: it only consists of the late computations and is parameterized with respect to the result of early computations, that is, the cache. The corresponding program is named the reader. Because the reader is parameterized with respect to the cache, it is shared by all specializations. This strategy fundamentally contrasts with program specialization because it decouples the result of early computations and the program which exploits it. As a consequence, as the size of the specialization problem increases, only the cache parameter increases, not the program. In practice, data specialization can handle problem sizes which are far beyond the reach of program specialization, and thus opens up new opportunities as demonstrated by Knoblock and Ruf for graphics applications [7, 4]. However, data specialization, by de-nition, does not optimize the control AEow: it is limited to performing the early computations which are expensive enough to be worth caching. Because the reader is valid for any cache it is passed, an early control decision leading to a costly early computation needs to be part of the loader as well as the reader: in the loader, it decides whether the costly computation much be cached; in the reader, the control decision determines whether the cache needs to be looked up. In fact, data specialization does not apply to programs whose bottlenecks are limited to control decisions. A typical example of this situation is interpreters for low-level languages: the instruction dispatch is the main target of specialization. For such programs, data specialization can be completely ineoeective. Perhaps the apparent dioeerence in the nature of the opportunities addressed by program and data specialization has led researchers to study these strategies in isolation. As a consequence, no attempt has ever been made to integrate both strategies in a specializer; further, there exist no experimental data to assess the bene-ts and limitations of these specialization strategies. In this paper, we study the relationship between program and data specialization with respect to their underlying concepts, their implementation techniques and their applicability. More precisely, we study program and data specialization when they are applied separately, as well as when they are combined (Section 2). Furthermore, we describe how a specializer can integrate both program and data specialization: what components are common to both strategies and what components dioeer. In practice, we have achieved this integration by extending a program special- izer, named Tempo, with the phases needed to perform data specialization (Section 3). Finally, we assess the bene-ts and limitations of program and data specialization based on experimental data collected by specializing a variety of programs exposing various features (Section 4). 2 Concepts of Program and Data Specialization In this section, the basic concepts of both program and data specialization are presented. The limitations of each strategy are identi-ed and illustrated by an example. Finally, the combination of program and data specialization is introduced. 2.1 Program Specialization The partial evaluation community has mainly been focusing on specialization of programs. That is, given some inputs of a program, partial evaluation generates a residual program which encodes the result of the early computations which depend on the known inputs. Although program specialization has successfully been used for a variety of applications (e.g., operating systems [10, 11], scienti-c programs [8, 12], and compiler generation [2, 6]), it has shown some limitations. One of the most fundamental limitations is code explosion which occurs when the size of the specialization problem is large. Let us illustrate this limitation using the procedure displayed on the left-hand side of Figure 1. In this example, stat is considered static, whereas dyn and d are dynamic. Static constructs are printed in boldface. Assuming the specialization process unrolls the loop, variable i becomes static and thus the gi procedures (i.e., g1, g2 and g3) can be fully evaluated. Even if the gi procedures correspond to non-expensive computations, program specialization still optimizes procedure f in that it simpli-es its control AEow: the loop and one of the conditionals are eliminated. A possible specialization of procedure f is presented on the right-hand side of Figure 1. However, beyond some number of iterations, the unrolling of a loop, and the computations it enables, do not pay for the size of the resulting specialized program; this number depends on the processor features. In fact, as will be shown later, the specialized program can even get slower than the unspecialized program. The larger the size of the residual loop body, the earlier this phenomenon happens. void f (int stat, int dyn, int d[ ]) void f-1(int dyn, int d[ ]) int j; for if if (E-dyn) d[j] += (a) Source program (b) Specialized program Figure 1: Program specialization For domains like graphics and scienti-c computing, some applications are beyond the reach of program specialization because the specialization opportunities rely on very large data or iteration bounds which would cause code explosion if loops traversing these data were unrolled. In this situation, data specialization may apply. 2.2 Data Specialization In late eighties, an alternative to program specialization, called data specialization, was introduced by Barzdins and Bulyonkov [1] and further explored by Malmkj#r [9]. Later, Knoblock and Ruf studied data specialization for a subset of C and applied it to a graphics application [7]. Data specialization is aimed at encoding the results of early computations in data structures, not in the residual program. The execution of a program is divided into two stages: a loader -rst executes the early computations and saves their result in a cache. Then, a reader performs the remaining computations using the result of the early computations contained in the cache. Let us illustrate this process by an example displayed in Figure 2. On the left-hand side of this -gure, a procedure f is repeatedly invoked in a loop with a -rst argument (c) which does not vary (and is thus considered its second argument, the loop index (k) varies at each iteration. Procedure f is also passed a dioeerent vector at each iteration, which is assumed to be late. Because this procedure is called repeatedly with the same -rst argument, data specialization can be used to perform the computations which depend on it. In this context, many computations can be performed, namely the loop test, E-stat and the invocation of the gi procedures. Of course, caching an expression assumes that its execution cost exceeds the cost of a cache reference. Measurements have shown that caching expressions which are too simple (e.g. a variable occurrence or simple comparisons) actually cause the resulting program to slow down. In our example, let us assume that, like the loop test, the cost of expression E-stat is not expensive enough to be cached. If, however, the gi procedures are assumed to consist of expensive computations their invocations need to be examined as potential candidate for caching. Since the -rst conditional test E-stat is early, it can be put in the loader so that whenever it evaluates to true the invocation of procedure g1 can be cached; similarly, in the reader, the cache is looked up only if the conditional test evaluates to true. However, the invocation of procedure g2 cannot be cached according to Knoblock and Ruf's strategy, since it is under dynamic control and thus caching its result would amount to performing speculative evaluation [7]. Finally, the invocation of procedure g3 needs to be cached since it is unconditionally executed and its argument is early. The resulting loader and reader for procedure f are presented on the right-hand side of Figure 2, as well as their invocations. extern int w[N][M]; extern int w[N][M]; struct data-cache - int val1; . int val3;- cache[MAX]; f-load (c, cache); f (c, k, w[k]); f-read (c, k, w[k], cache); . void f-load (stat,cache[ ]) int stat; void f (int stat, int dyn, int d[ ]) struct data-cache cache[ ]; int j; int j; if if (E-dyn) d[j] += int stat, dyn, d[ ]; struct data-cache cache[ ]; int j; if if (E-dyn) d[j] += (a) Source program (b) Specialized program Figure 2: Data specialization To study the limitations of data specialization consider a program where computations to be cached are not expensive enough to amortize the cost of memory reference. In our example, assume the gi procedures correspond to such computations. Then, only the control AEow of procedure f remains a target for specialization. 2.3 Combining Program and Data Specialization We have shown the bene-ts and limitations of both program and data specialization. The main parameters to determine which strategy -ts the specialization opportunities are the cost of the early computations and the size of the specialization problem. Obviously, within the same program (or even a procedure), some fragments may require program specialization and others data specialization. As a simple example consider a procedure which consists of two nested loops. The innermost loop may require few iterations and thus allow program specialization to be applied. Whereas, the outermost loop may iterate over a vector whose size is very large; this may prevent program specialization from being applied, but not data specialization from exploiting some opportunities. Concretely performing both program and data specialization can be done in a simple way. One approach consists of doing data specialization -rst, and then applying the program specializer on either the loader or the reader, or both. The idea is that code explosion may not be an issue in one of these components; as a result, program specialization can further optimize the loader or the reader by simplifying its control AEow or performing speculative specialization. For example, a reader may consist of a loop whose body is small; this situation may thus allow the loop to be unrolled without causing the residual program to be too large. Applying a program specializer to both the reader and the loader may be possible if the fragments of the program, which may cause code explosion, are made dynamic. Alternatively, program specialization can be performed prior to data specialization. This combination requires program specialization to be applied selectively so that only fragments which do not cause code explosion are specialized. Then, the other fragments ooeering specialization opportunities can be processed by data specialization. As is shown in Section 4, in practice combining both program and data specialization allows better performance than pure data specialization and prevents the performance gain from dropping as quickly as in the case of program specialization as the problem size increases. Integrating Program and Data Specialization We now present how Tempo is extended to perform data specialization. To do so let us brieAEy describe its features which are relevant to both data specialization and the experiments presented in the next section. 3.1 Tempo is an ooe-line program specializer for C programs. As such, specialization is preceded by a preprocessing phase. This phase is aimed at computing information to guide the specialization process. The main analyses of Tempo's preprocessing phase are an alias analysis, a side-eoeect analyses, a binding-time analysis and an action analysis. The -rst two analyses are needed because of the imperative nature of the C language, whereas the binding-time analysis is typical of any ooe-line specializer. The action analysis is more unusual: it computes the specialization actions (i.e., the program transformations) to be performed by the specialization phase. The output of the preprocessing phase is a program annotated with specialization actions. Given some specialization values, this annotated program can be used by the specialization phase to produce a residual program at compile time, as is traditionally done by partial evaluators. In addition, Tempo can specialize a program at run time. Tempo's run-time specializer is based on templates which are eOEciently compiled by standard C compilers [3, 12]. Tempo has been successfully used for a variety of applications ranging from operating systems [10, 11] to scienti-c programs [8, 12]. 3.2 Extending Tempo with Data Specialization Tempo includes a binding-time analysis which propagates binding times forward and backward. The forward analysis aims at determining the static computations; it propagates binding times from the de-nitions to the uses of variables. The backward analysis performs the same propagation in the opposite direction; when uses of a variable are both static and dynamic, its de-nition is annotated static&dynamic. This annotation indicates that the de-nition should be evaluated both at specialization time and run time. This process, introduced by Hornof et al., allows a binding-time analysis to be more accurate; such an analysis is said to be use sensitive [5]. When a de-nition is static&dynamic and occurs in a control construct (e.g., while), this control construct becomes static&dynamic as well. The specialized program is the code where constructs and expressions annoted static are evaluated at specialization time and its result are introduced in the residual code and where constructs and expressions annoted dynamic or static&dynamic are rebuilt in the residual code. To perform data specialization an analysis is inserted between the forward analysis and backward analysis. In essence, this new phase identi-es the frontier terms, that is, static terms occurring in a dynamic (or static&dynamic) context. If the cost of the frontier term is below a given threshold (de-ned as a parameter of the data specializer), it is forced to dynamic (or static&dynamic). Furthermore, because data specialization does not perform speculative evaluation, static computations which are under dynamic control are made dynamic. Once these adjustments are done, the backward phase of the binding-time analysis then determines the -nal binding times of the program. Later in the process, the static computations are included in the loader and the dynamic computations in the reader; the frontier terms are cached. The rest of our data specializer is the same as Knoblock and Ruf's. Performance Evaluation In this section, we compare the performance obtained by applying dioeerent specialization strategies on a set of programs. This set includes several scienti-c programs and a system program. 4.1 Overview Machine and Compiler. The measurements presented in this paper were obtained using a Sun Model 170 with 448 mega bytes of main memory, running Sun-OS version 5.5.1. Times were measured using the Unix system call getrusage and include both iuserj and isystemj times. Figure 3 displays the speedups and the size increases of compiled code obtained for dioeerent specialization strategies. For each benchmark, we give the program invariant used for specialization and an approximation of its time complexity. The code sources are included in the appendices. All the programs were compiled with gcc -O2. Higher degrees of optimization did not make a dioeerence for the programs used in this experiment. Specialization strategies. We evaluate the performance of -ve dioeerent specialization meth- ods. The speedup is the ratio between the execution times of the specialized program and the original one. The size increases is the ratio between the size of the specialized program and the original one. The data displayed in Figure 3 correspond to the behavior of the following specialization strategies: ffl PS-CT: the program is program specialized at compile time. ffl PS-RT: the program is program specialized at run time. ffl DS: the program is data specialized. PS-CT: the program is data specialized and program specialized at compile time. The loops which manipulate the cache (for data specialization) are kept dynamic to avoid code explosion. PS-RT: the program is data specialized and program specialized at run time. As in the previous strategy, the loops which manipulate the cache are kept dynamic to avoid code explosion. Source programs. We consider a variety of source programs: a one-dimensional fast Fourier transformation (FFT), a Chebyshev approximation, a Romberg integration, a Smirnov integration, a cubic spline interpolation and a Berkeley packet -lter (BPF). Given the specialization strategies available, these programs can be classi-ed as follows. Control AEow intensive. A program which mainly exposes control AEow computations; data AEow computations are inexpensive. In this case, program specialization can improve performance whereas data specialization does not because there is no expensive calculations to cache. Data AEow intensive. A program which is only based on expensive data AEow computations. As a result, program specialization at compile time as well as data specialization can improve the performance of such program. Control and data AEow intensive. A program which contains both control AEow computations and expensive data AEow computations. Such program is a good candidate for program specialization at compile time when applied to small values, and well-suited for data specialization when applied to large values. We now analyze the performance of -ve specialization methods in turn on the benchmark programs. 4.2 Results Data specialization can be executed at compile time or at run time. At run time, the loader of the cache is executed before the execution of the specialized program, while at compile time, the cache is constructed before the compilation. The cache is then used by the specialized program during the execution. For all programs, data specialization yields a greater speedup than program specialization at run time. The combination of these two specialization strategies does not make a better result. In this section, we characterize dioeerent opportunities of specialization to illustrate our method in the three categories of program. 4.2.1 Program Specialization We analyze two programs where performance is better with program specialization: the Berkeley packet -lter (BPF), which interprets a packet with respect to an interpreter program, and the cubic spline interpolation, which approximates a function using a third degree polynomial equation. Characteristics: For the BPF, the program consists exclusively of conditionals whose tests and branches contain inexpensive expressions. For the cubic spline interpolation, the program consists of small loops whose small body can be evaluated in part. Concretely, a program which mainly depends on the control AEow graph and whose leaves contain few calculations but partially reducible, is a good candidate for program specialization. By program specialization, the control AEow graph is reduced and some calculations are eliminated. Since there is no static calculation expensive enough to be eOEciently cached by data specialization, the specialized program is mostly the same as the original one. For this kind of programs, only program specialization gives signi-cant improvements: it reduces the control AEow graph and it produces a small specialized program. Applications: The BPF (Appendix F) is specialized with respect to a program (of size n). It mainly consists of the conditionals; its time complexity is linear in the size of the program and it does not contain expensive data computations. As the program does not contain any loop, the size of the specialized program is mostly the same as the original one. In Figure 3-F, program specialization at compile time and at run time yields a good speedup, whereas data specialization only improves performance marginally. The combination of program and data specialization does not improve the performance further. The cubic spline interpolation (Appendix E) is specialized with respect to the number of points (n) and their x-coordinates. It contains three singly nested loops; its time complexity is O(n). In the -rst two loops, more than half of the computations of each body can be completely evaluated or cached by specialization, including real multiplications and divisions. Nevertheless, there is no expensive calculation to cache, and data specialization does not improve performance signi-cantly. The unrolled loop does not really increase the code size because of the small complexity of the program and the small body of the loop. As a consequence, for each number of points n, the Figure 3: Program, data and combined specializations speedup of each specialization barely changes. In Figure 3-E, program specialization at compile time produces a good speedup, whereas program specialization at run time does not improve performance. Data specialization obtains a minor speedup because the cached calculations are not expensive. 4.2.2 Program Specialization or Data Specialization We now analyze two programs where performance is identical to program specialization or data specialization: the polynomial Chebyshev, which approximates a continuous function in a known interval, and the Smirnov integration, which approximates the integral of a function on an interval using estimations. Characteristics: These two programs only contain loops and expensive calculations in doubly nested loops. As for the cubic spline interpolation (Section 4.2.1), more than half of the computations of each body loop can be completely evaluated or cached by specialization. In contrast with cubic spline interpolation, the static calculations in Chebyshev and Smirnov are very expensive and allow data specialization to yield major improvements. For the combined specialization, data specialization is applied to the innermost loop and program specialization is applied to the rest of the program. For this kind of programs, program and data specialization both give signi-cant improvements. However, for the same speedup, the code size of the program produced by program specialization is a hundred times larger than the specialized program using data specialization. Applications: The Chebyshev approximation (Appendix C) is specialized with respect to the degree (n) of the generated polynomial. This program contains two calls to the trigonometric function cos: one of them in a singly nested loop and the other call in a doubly nested loop. Since this program mainly consists of data AEow computations, program specialization and data specialization obtain similar speedups (see Figure 3-C). The Smirnov integration (Appendix D) is specialized with respect to the number of iterations (n, m). The program contains a call to the function fabs which returns the absolute value of its parameter. This function is contained in a doubly nested loop and the time complexity of this program is O(m n ). As in the case of Chebyshev, program and data specialization produce similar speedups (see Figure 3-D). 4.2.3 Combining Program Specialization and Data Specialization Finally, we analyze two programs where performance improves using program specialization when values are small, and data specialization when values are large: the FFT and the Romberg in- tegration. The FFT converts data from the time domain to a frequency domain. The Romberg integration approximates the integral of a function on an interval using estimations. Characteristics: These two programs contain several loops and expensive data AEow computations in doubly nested loops; however more than half of the computations of each loop body cannot be evaluated. Beyond some number of iterations, when the program specialization unrolls these loops, it increases the code size of the specialized program and then degrades performance. The specialized program becomes slower because of its code size. Furthermore, beyond some problem size, the specialization process cannot produce the program because of its size. In contrast, data specialization only caches the expensive calculations, does not unroll loops, and improves perfor- mance. The result is that the code size of the program produced by program specialization is a hundred times larger than the specialized program using data specialization, for a speedup gain of 20%. The combined specialization delays the occurrence of code explosion. Data specialization is applied to the innermost loop, which contains the cache computations, and program specialization is applied to the rest of the program. Applications: The FFT (Appendix A) is specialized with respect to the number of data points (N ). It contains ten loops with several degrees of nesting. One of these loops, with complexity contains four calls to trigonometric functions, which can be evaluated by program specialization or cached by data specialization. Due to the elimination of these expensive library calls, program specialization and data specialization produce signi-cant speedups (see Figure 3-A). However, in the case of program specialization, code unrolling degrade performance. In contrast, data specialization produces a stable speedup regardless of the number of data points. When N is smaller than 512, data specialization does not obtain a better result in comparison to program specialization. However, when N is greater than 512, program specialization becomes impossible to apply because of the specialization time and the size of the residual code. In this situation, data specialization still gives better performance than the unspecialized program. Because this program also contains some conditionals, combined specialization, where the innermost loop is not unrolled, improves performance better than data specialization alone. The Romberg integration (Appendix B) is specialized with respect to the number of iterations (M) used in the approximation. The Romberg integration contains two calls to the costly function intpow. It is called twice: once in a singly nested loop and another time in a doubly nested loop. Because both specialization strategies eliminate these expensive library calls, the speedup is consequently good. As for FFT, loop unrolling causes the program specialization speedup to decrease, whereas the data specialization speedup still remains the same, even when M increases Figure 3-B). 5 Conclusion We have integrated program and data specialization in a specializer named Tempo. Importantly, data specialization can re-use most of the phases of an ooe-line program specializer. Because Tempo now ooeers both program and data specialization, we have experimentally compared both strategies and their combination. This evaluation shows that, on the one hand program specialization typically gives better speed-up than data specialization for small problem size. However, as the problem size increases, the residual program may become very large and often slower than the unspecialized program. On the other hand, data specialization can handle large problem size without much performance degradation. This strategy can, however, be ineoeective if the program to be specialized mainly consists of control AEow computations. The combination of both program and data specialization is promising: it can produce a residual program more eOEcient than with data specialization alone, without dropping in performance as dramatically as program specialization, as the problem size increases. Acknowledgments We thank Renaud Marlet for thoughtful comments on earlier versions of this paper, as well as the Compose group for stimulating discussions. A substantial amount of the research reported in this paper builds on work done by the authors with Scott Thibault on Berkeley packet -lter and Julia Lawall on Fast Fourier Transformation. --R Mixed computation and translation: Linearisation and decomposition of compilers. Tutorial notes on partial evaluation. Specializing shaders. Partial Evaluation and Automatic Program Genera- tion Data specialization. Faster Fourier transforms via automatic program specialization. Program and data specialization: Principles Fast, optimized Sun RPC using automatic program specialization. Scaling up partial evaluation for optimizing the Sun commercial RPC protocol. --TR --CTR Jung Gyu Park , Myong-Soon Park, Using indexed data structures for program specialization, Proceedings of the ASIAN symposium on Partial evaluation and semantics-based program manipulation, p.61-69, September 12-14, 2002, Aizu, Japan Vytautas tuikys , Robertas Damaeviius, Metaprogramming techniques for designing embedded components for ambient intelligence, Ambient intelligence: impact on embedded system design, Kluwer Academic Publishers, Norwell, MA, Mads Sig Ager , Olivier Danvy , Henning Korsholm Rohde, On obtaining Knuth, Morris, and Pratt's string matcher by partial evaluation, Proceedings of the ASIAN symposium on Partial evaluation and semantics-based program manipulation, p.32-46, September 12-14, 2002, Aizu, Japan Charles Consel , Julia L. Lawall , Anne-Franoise Le Meur, A tour of tempo: a program specializer for the C language, Science of Computer Programming, v.52 n.1-3, p.341-370, August 2004 Torben Amtoft , Charles Consel , Olivier Danvy , Karoline Malmkjr, The abstraction and instantiation of string-matching programs, The essence of computation: complexity, analysis, transformation, Springer-Verlag New York, Inc., New York, NY, 2002

program specialization;data specialization;partial evaluation;program transformation;combining program

609203

Certifying Compilation and Run-Time Code Generation.

A certifying compiler takes a source language program and produces object code, as well as a certificate that can be used to verify that the object code satisfies desirable properties, such as type safety and memory safety. Certifying compilation helps to increase both compiler robustness and program safety. Compiler robustness is improved since some compiler errors can be caught by checking the object code against the certificate immediately after compilation. Program safety is improved because the object code and certificate alone are sufficient to establish safety: even if the object code and certificate are produced on an unknown machine by an unknown compiler and sent over an untrusted network, safe execution is guaranteed as long as the code and certificate pass the verifier.Existing work in certifying compilation has addressed statically generated code. In this paper, we extend this to code generated at run time. Our goal is to combine certifying compilation with run-time code generation to produce programs that are both fast and verifiably safe. To achieve this goal, we present two new languages with explicit run-time code generation constructs: Cyclone, a type safe dialect of C, and TAL/T, a type safe assembly language. We have designed and implemented a system that translates a safe C program into Cyclone, which is then compiled to TAL/T, and finally assembled into executable object code. This paper focuses on our overall approach and the front end of our system&semi; details about TAL/T will appear in a subsequent paper.

Introduction 1.1 Run-time specialization Specialization is a program transformation that optimizes a program with respect to invariants. This technique has been shown to give dramatic speedups on a wide range of appli- cations, including aircraft crew planning programs, image shaders, and operating systems [4, 11, 17]. Run-time specialization exploits invariants that become available during the execution of a program, generating optimized code on the fly. Opportunities for run-time specialization occur when dynamically changing values remain invariant for a period of time. For example, networking software can be specialized to a particular TCP connection or multicast tree. Run-time code generation is tricky. It is hard to correctly write and reason about code that generates code; it is not obvious how to optimize or debug a program that has yet to be generated. Early examples of run-time code generation include self-modifying code, and ad hoc code generators written by hand with a specific function in mind. These approaches proved complicated and error prone [14]. More recent work has applied advanced programming language techniques to the problem. New source languages have been designed to facilitate run-time code generation by providing the programmer with high-level constructs and having the compiler implement the low-level details [15, 21, 22]. Program transformations based on static analyses are now capable of automatically translating a normal program into a run-time code generating program [6, 10, 12]. And type systems can check run-time code generating programs at compile time, ensuring that certain bugs will not occur at run time (provided the compiler is correct) [22, 25]. These techniques make it easier for programmers to use run-time code generation, but they do not address the concerns of the compiler writer or end user. The compiler writer still needs to implement a correct compiler-not easy even for a language without run-time code generation. The end user would like some assurance that executables will not crash their machine, even if the programs generate code and jump to it-behavior that usually provokes suspicion in security-concious users. We will address both of these concerns through another programming language technique, certifying compilation. 1.2 Certifying compilation A certifying compiler takes a source language program and produces object code and a "certificate" that may help to show that the object code satisfies certain desirable properties [16, 18]. A separate component called the verifier examines the object code and certificate and determines whether the object code actually satisfies the properties. A wide range of properties can be verified, including memory safety (unallocated portions of memory are not accessed), control safety (code is entered only at valid entry points), and various security properties (e.g., highly classified data does appear on low security channels). Often, these properties are corollaries of type safety in an appropriate type system for the object code. In this paper we will describe a certifying compiler for Cyclone, a high-level language that supports run-time code generation. Cyclone is compiled into TAL/T, an assembly language that supports run-time code generation. Cyclone and TAL/T are both type safe; the certificates of our system are the type annotations of the TAL/T output, and the verifier is the TAL/T type checker. As compiler writers, we were motivated to implement Cyclone as a certifying compiler because we believe the approach enhances compiler correctness. For example, we were forced to develop a type system and operational semantics for TAL/T. This provides a formal framework for reasoning about object code that generates object code at run time. Eventually, we hope to prove that the compiler transforms type correct source programs into type correct object pro- grams, an important step towards proving correctness for the compiler. In the meantime, we use the verifier to type check the output of the compiler, so that we get immediate feedback when our compiler introduces type errors. As others have noted [23, 24], this helps to identify and correct compiler bugs quickly. We also wanted a certifying compiler to address the safety concerns of end users. In our system, type safety only depends on the certificate and the object code, and not on the method by which they are produced. Thus the end user does not have to rely on the programmer or the Cyclone compiler to ensure safety. This makes our system usable as the basis of security-critical applications like active networks and mobile code systems. 1.3 The Cyclone compiler The Cyclone compiler is built on two existing systems, the specializer [19] and the Popcorn certifying compiler [16]. It has three phases, shown in Fig. 1. The first phase transforms a type safe C program into a Cyclone program that uses run-time code generation. It starts by applying the static analyses of the Tempo system to a C program and context information that specifies which function arguments are invariant. The Tempo front end produces an action-annotated program. We added an additional pass to translate the action-annotated program into a Cyclone run-time specializer. The second phase verifies that the Cyclone program is type safe, and then compiles it into TAL/T. To do this, we modified the Popcorn compiler of Morrisett et al.; Popcorn compiles a type safe dialect of C into TAL, a typed assembly language. We extended the front end of Popcorn to handle Cyclone programs, and modified its back end so that it outputs TAL/T. TAL/T is TAL extended with instructions for manipulating templates, code fragments parameterized by holes, and their corresponding types. This compilation phase not only transforms high-level Cyclone constructs into low-level assembly instructions, but also transforms Cyclone types into TAL/T types. The third phase first verifies the type safety of the TAL/T program. The type system of TAL/T ensures that the templates are combined correctly and that holes are filled in correctly. This paper describes our overall approach and the front end in detail, but the details of TAL/T will appear in a subsequent paper. Finally, the TAL/T program is assembled and linked into an executable. This three phase design offers a very flexible user interface since it allows programs to be written in C, Cyclone, or TAL/T. In the simplest case, the user can simply write a C program (or reuse an existing program) and allow the system to handle the rest. If the user desires more explicit control over the code generation process, he may write (or modify) a Cyclone program. If very fine-grain control is de- sired, the user can fine-tune a TAL/T program produced by Cyclone, or can write one by hand. Note that, since verification is performed at the TAL/T level, the same program Cyclone verify & compile Cyclone verify, assemble, link executable translate action-annotated program to Cyclone Figure 1: Overview of the Cyclone compiler safety properties are guaranteed in all three of these cases. 1.4 Example We now present an example that illustrates run-time code generation and the phases of our Cyclone compiler. Fig. 2 shows a modular exponentiation function, mexp, written in standard C. Its arguments are a base value, an exponent, and a modulus. Modular exponentiation is often used in cryptography; when the same key is used to encrypt or decrypt several messages, the function is called repeatedly with the same exponent and modulus. Thus mexp can benefit from specialization. To specialize the function with respect to a given exponent and modulus, the user indicates that the two arguments are invariant : the function will be called repeatedly with the same values for the invariant arguments. In Fig. 2, invariant arguments are shown in italics. A static analysis propagates this information throughout the program, producing an action-annotated program. Actions describe how each language construct will be treated during specializa- tion. Constructs that depend only on invariants can be evaluated during specialization; these constructs are displayed in italics in the second part of the figure. To understand how run-time specialization works, it is C code (invariant arguments in italics) int mexp(int base, int exp, int mod) f int s, t, u; while if ((u&1) != Action-annotated code (italicized constructs can be evaluated) int mexp(int base, int exp, int mod) f int s, t, u; while Specialized source code int mexp sp(int base) f int s, t; Figure 2: Specialization at the source level helpful to first consider how specialization could be achieved entirely within the source language. In our example, the specialized function mexp sp of Fig. 2 is obtained from the action-annotated mexp when the exponent is 10 and the modulus is 1234. Italicized constructs of mexp, like the while loop, can be evaluated (note that the loop test depends only on the known arguments). Non-italicized constructs of mexp show up in the source code of mexp sp. These constructs can only be evaluated when mexp sp is called, because they depend on the unknown argument. We can think of mexp sp as being constructed by cutting and pasting together fragments of the source code of mexp. These fragments, or templates, are a central idea we used in designing Cyclone. Cyclone is a type safe dialect of C extended with four constructs that manipulate templates: codegen, cut, splice, and fill. Using these constructs, it is possible to write a Cyclone function that generates a specialized version of mexp at run time. int (int) mexp gen(int exp, int mod) f int u; return codegen( int mexp sp(int base) f int s, t; cut while splice splice Figure 3: A run-time specializer written in Cyclone In fact, our system can automatically generate a Cyclone run-time specializer from an action-annotated pro- gram. Fig. 3 shows the Cyclone specializer produced from the action-annotated modular exponentiation function of Fig. 2. The function mexp gen takes the two invariant arguments of the original mexp function and returns the function mexp sp, a version of mexp specialized to those arguments. In the figure we have italicized code that will be evaluated when mexp gen is called. Non-italicized code is template code that will be manipulated by mexp gen to produce the specialized function. The template code will only be evaluated when the specialized function is itself called. In our example, the codegen expression begins the code generation process by allocating a region in memory for the new function mexp sp, and copying the first template into the region. This template includes the declarations of the function, its argument base, and local variables s and t, and also the initial assignments to s and t. Recall that this template code is not evaluated during the code generation process, but merely manipulated. The cut statement marks the end of the template and introduces code (italicized) that will be evaluated during code generation: namely, the while loop. The while test and body, including the conditional statement, splice state- ments, and shift/assignment statement, will all be evalu- ated. After the while loop finishes, the template following the cut statement (containing return(s)) will be added to the code generation region. Evaluating a splice statement causes a template to be appended to the code generation region. In our example, each time the first splice statement is executed, an assignment to s is appended. Similarly, each time the second splice statement is executed, an assignment to t is ap- pended. The effect of the while loop is thus to add some number of assignment statements to the code of mexp sp; exactly how many, and which ones, is determined by the arguments of mexp gen. A fill expression can be used within a template, and it marks a hole in the template. When fill(e) is encountered in a template, e is evaluated at code generation time to a value, which is then used to fill the hole in the template. In our example, fill is used to insert the known modulus value into the assignment statements. After code generation is complete, the newly generated function mexp sp is returned as the result of codegen. It takes the one remaining argument of mexp to compute its result. Cyclone programs can be evaluated symbolically to produce specialized source programs, like the one in Fig. 2; this is the basis of the formal operational semantics we give in the appendix. In our implementation, however, we compile Cyclone source code to object code, and we compile source templates into object templates. The Cyclone object code then manipulates object templates directly. Our object code, TAL/T, is an extension of TAL with instructions for manipulating object templates. Most of the TAL/T instructions are x86 machine instructions; the new template instructions are CGSTART, CGDUMP, CGFILL, CGHOLE, START, and TEMPLATE END. For example, the Cyclone program in Fig. 3 is compiled into the TAL/T program shown in Fig. 4. (We omitted some instructions to save space, and added source code fragments in comments to aid readability.) The beginning of mexp gen contains x86 instructions for adding the local variable u to the stack and assigning it the value of the argument exp. Next, CGSTART is used to dynamically allocate a code generation region, and the first template is dumped (copied) into the region with the CGDUMP instruction. Next, the body of the loop is unrolled. Each Cyclone splice statement is compiled into a CGDUMP instruc- tion, followed by instructions for computing hole values and a CGFILL instruction for filling in the hole. At the end of the mexp gen function, a final CGDUMP instruction outputs code for the last template. Next comes the code for each of the four templates. The first template allocates stack space for local variables s and t and assigns values to them. The second and third templates come from the statements contained within the Cyclone splice instructions, i.e., the multiplications, mods, and assignments. The final template contains the code for return(s). Each CGHOLE instruction introduces a place-holder inside a template, filled in during specialization as described above. Summary We designed a system for performing type safe run-time code generation. It has the following parts: ffl C to action-annotated program translation Action-annotated program to Cyclone translation ffl Cyclone language design ffl Cyclone verifier ffl Cyclone to TAL/T compiler ffl TAL/T language design ffl TAL/T verifier -mexp-gen: MOV [ESP+0],EAX 1st template) ifend$24: MOV EAX,[ESP+0] MOV [ESP+0],EAX whileend$22: CGEND EAX RETN (1st template) MOV [ESP+0],EAX TEMPLATE-START splc-beg$25,splc-end$26 TEMPLATE-END splc-end$26 TEMPLATE-END splc-end$32 TEMPLATE-START cut-beg$36,cut-end$37 ADD ESP,8 RETN TEMPLATE-END cut-end$37 Figure 4: TAL/T code ffl TAL/T to assembly translation ffl Assembler/Linker For some parts, we were able to reuse existing software. Specifically, we used Tempo for action-annotated program generation, Microsoft MASM for assembling, and Microsoft Visual C++ for linking. Other parts extend existing work. This was the case for the Cyclone language, type system, verifier, compiler, and the TAL/T language. Some components needed to be written from scratch, including the translation from an action-annotated program into a Cyclone pro- gram, and the definition of the new TAL/T instructions in terms of x86 instructions. We've organized the rest of the paper as follows. In Section 2, we present the Cyclone language and its type sys- tem. In Section 3, we give a brief description of TAL/T; due to limited space we defer a full description to a later pa- per. We give implementation details and initial impressions about performance in Section 4. We discuss related work in Section 5, and future work in Section 6. Our final remarks are in Section 7. 2.1 Design decisions Cyclone's codegen, cut, splice, and fill constructs were designed to express a template-based style of run-time code generation cleanly and concisely. We made some other design decisions based on Cyclone's relationship to the C programming language, and on implementation concerns. First, because a run-time specializer is a function that returns a function as its result, we need higher order types in Cyclone. In C, higher order types can be written using pointer types, but Cyclone does not have pointers. There- fore, we introduce new notation for higher order types in Cyclone. For example: int (float,int) f(int x) - This is a Cyclone function f that takes an int argument x, and returns a function taking a float and an int and returning an int. When f is declared and not defined, we use int (int) (float,int) f; Note that the type of the first argument appears to the left of the remaining arguments. This is consistent with the order the arguments would appear in C, using pointer types. A second design decision concerns the extent to which we should support nested codegen's. Consider the following example. int (float) (int) f(int x) - return(codegen( int (int) g(float y) - return(codegen( int h(int z) - . body of h . Here f is a function that generates a function g using codegen when called at run time. In turn, g will generate a function h each time it is called. Nested codegen's are thus used to generate code that generates code. The first version of Tempo did not support code that generates code (though it has recently been extended to do so), and some other sys- tems, such as 'C [20, 21], also prohibit it. We decided to permit it in Cyclone, because it adds little complication to our type system or implementation. Nested codegen's are not generated automatically in Cyclone, because of the version of Tempo that we use, but the programmer can always them explicitly. A final design decision concerns the extent to which Cyclone should support lexically scoped bindings. In the last example, the function h is nested inside of two other func- tions, f and g. In a language with true lexical scoping, the arguments and local variables of these outer functions would be visible within the inner function: f, x, g, and y could be used in the body of h. We decided that we would not support full lexical scoping in Cyclone. Our scoping rule is that in the body of a function, only the function itself, its arguments and local variables, and top-level variables are visible. This is in keeping with C's character as a low-level, machine- and systems-oriented language: the operators in the language are close to those provided by the machine, and the cost of executing a program is not hidden by high-level abstractions. We felt that closures and lambda lifting, the standard techniques for supporting lexical scoping, would stray too far from this. If lexical scoping is desired, the programmer can introduce explicit closures. Or, lexical scoping can be achieved using the Cyclone features, for example, if y is needed in the body of h, it can be accessed using fill(y). 2.2 Syntax and typing rules Now we formalize a core calculus of Cyclone. Full Cyclone has, in addition, structures, unions, arrays, void, break and continue, and for and do loops. We use x to range over variables, c to range over con- stants, and b to range over base types. There is an implicit signature assigning types to constants, so that we can speak of "the type of c." Figure 5 gives the grammars for programs modifiers m, types t, declarations d, sequences D of declarations, function definitions F , statements s, and expressions e. We write t ffl m for the type of a function from m to t: D is defined to be the modifier so that a function definition t x(D) s declares x to be of type t ffl e D. We sometimes consider a sequence of declarations to be a finite function from variables to types: This assumes that the x i are distinct; we achieve this by alpha conversion when neces- sary, and by imposing some standard syntactic restrictions on Cyclone programs (the names of a function and its formal parameters must be distinct, and global variables have distinct names). We define type environments E to support Cyclone's scoping rules: local ); D global local local Informally, a type environment is a sequence of hidden and visible frames, followed by an outermost frame that gives vis outermost(t Figure Cyclone environment functions Programs Modifiers m ::= Types Declarations d ::= t x Decl. sequences D ::= Function defns. F ::= t x(D) s Statements s ::= e; return e; splice s Expressions e ::= x Figure 5: The grammar of core Cyclone the type of a top level function, the types of its local vari- ables, and the types of global variables. The non-outermost frames contain the type of a function that will be generated at run time, and types for the parameters and local variables of the function. If E is a type environment, we write E vis for the visible declarations of E; E vis is defined in Figure 6. Informally, the definition says that the declarations of the first non-hidden frame and the global declarations are vis- ible, and all other declarations are not visible. Note that vis is a sequence of declarations, so we may write E vis (x) for the type of x in E. Figure 6 also defines two other important operations on environments: rtype(E) is the return type for the function of the first non-hidden frame, and E + d is the environment obtained by adding declaration d to the local declarations of the first non-hidden frame. The typing rules of Cyclone are given in Figure 7. The interesting rules are those for codegen, cut, splice, and fill. A codegen expression starts the process of run time code generation. To type codegen(t x(D) s) in an environment E, we type the body s of the function in an environment This makes the function x and its parameters D visible in the body, while any enclosing func- tion, parameters, and local variables will be hidden. An expression fill(e) should only appear within a tem- plate. Our typing rule ensures this by looking at the envi- ronment: it must have the form frame(t x(D); D 0 If so, the expression fill(e) is typed if e is typed in the environment That is, the function being generated with codegen, as well as its parameters and local variables, are hidden when computing the value that will fill the hole. This is necessary because the parameters and local variables will not become available until the function is called; they will not be available when the hole is filled. The rules for cut and splice are similar. Like fill, cut can only be invoked within a template, and it changes frame to hidden for the same reason as fill. Splice is the dual of cut; it changes a frame hidden by cut back into a visible frame. Thus splice introduces a template, and cut interrupts a template. (p is a well-formed program) well-formed statement) is the type of the constant c Figure 7: Typing rules of Cyclone An operational semantics for Cyclone and safety theorem are given in an appendix. The output of the Cyclone compiler is a program in TAL/T, an extension of the Typed Assembly Language (TAL) of Morrisett et al. [16]. In designing TAL/T, our primary concern was to retain the low-level, assembly language character of TAL. Most TAL instructions are x86 machine in- structions, possibly annotated with type information. The exceptions are a few macros, such as malloc, that would be difficult to type in their expanded form; each macro expands to a short sequence of x86 instructions. Since each instruction is simple, the trusted components of the system-the typing rules, the verifier, and the macros-are also simple. This gives us a high degree of confidence in the correctness and safety of the system. TAL already has instructions that are powerful enough to generate code at run time: malloc and move are sufficient. The problem with this approach is in the types. If we malloc a region for code, what is its type? Clearly, by the end of the code generation process, it should have the type of TAL code that can be jumped to. But at the start of code generation, when it is not safe to jump to, it must have a different type. Moreover, the type of the region should change as we move instructions into it. The TAL type system is not powerful enough to show that a sequence of malloc and move instructions results in a TAL program that can safely be jumped to. Our solution, TAL/T, is an extension of TAL with some types and macros for manipulating templates. Since this paper focuses on Cyclone and the front end of system, we will only sketch the ideas of TAL/T here. Full details will appear in a subsequent paper. In TAL, a procedure is just the label or address of a sequence of TAL instructions. A procedure is called by jumping to the label or address. The type of a procedure is a precondition that says that on entry, the x86 registers should contain values of particular types. For example, if a procedure is to return it will have a precondition saying that a return address should be accessible through the stack pointer when it is jumped to. In TAL/T, a template is also the label of a sequence of instructions. Unlike a TAL procedure, however, a template is not meant to be jumped to. For example, it might need to be concatenated with another template to form a TAL procedure. Thus the type of a template includes a postcondition as well as a precondition. Our typing rules for the template instructions of TAL/T will ensure that before a template is dumped into a code generation region, its precondition matches the postcondition of the previous template dumped. Also, a template may have holes that need to be filled; the types of these holes are also given in the type of the template. The type of a code generation region is very similar to that of a template: it includes types for the holes that remain to be filled in the region, the precondition of the first template that was dumped, and the postcondition of the last template that was dumped. When all holes have been filled and a template with no postcondition is dumped, the region will have a type consisting of just a precondition, i.e., the type of a TAL procedure. At this point code generation is finished and the result can be jumped to. int f(int x) - return(codegen( int g(int y) - return int h(int x)(int) - return(codegen( int k(int y) - Figure 8: An example showing that two codegen expressions can be executing at once. When called, h starts generating k, but stops in the middle to call f which generates g. Now we give a brief description of the new TAL/T macros. This is intended to be an informal description showing that each macro does not go beyond what is already in TAL-the macros are low level, and remain close to machine code. The macros manipulate an implicit stack of code generation regions. Each region in the stack is used for a function being generated by a codegen. The stack is needed because it is possible to have two codegen expressions executing at once (for an example, see Figure 8). ffl cgstart initiates run-time code generation by allocating a new code generation region. This new region is pushed onto the stack of code generation regions and becomes the "current" region. The cgstart macro is about as complicated as malloc. copies the template at label L into the current code generation region. After execution, the register r points to the copy of the template, and can be used to fill holes in the copy. Cgdump is our most complicated macro: its core is a simple string-copy loop, but it must also check that the current code generation region has enough room for a copy of the template. If there is not enough room, cgdump allocates a new region twice the size of the old region, copies the contents of the old region plus the new template to the new region, and replaces the old region with the new on the region stack. This is the most complex TAL/T instruction, consisting of roughly twenty x86 instructions. ffl cghole r, L template , L hole is a move instruction containing a hole. It should be used in a template with label L template , and declares the hole L hole . ffl cgfill r1, L template , L hole , r2 fills the hole of a tem- plate; it is a simple move instruction. Register r1 should point to a copy of the template at label L template , which should have a hole with label L hole . Register r2 contains the value to put in the hole. ffl cgfillrel fills the hole of a template with a pointer into a second template; like cgfill it expands to a simple move instruction. It is needed for jumps between templates int f() - return(codegen( int g(int i) - cut - return 4; - Figure 9: An example that shows the need for cgabort. When called, the function f starts generating function g but aborts in the middle (it returns 4). ffl cgabort aborts a code generation; it pops the top region off the region stack. It is needed when the run-time code generation of a function stops in the middle, as in the example of Figure 9. cgend r finalizes the code generation process: the current region is popped off the region stack and put into register r. TAL can then jump to location r. Implementation Status We now describe some key aspects of our implementation. As previously mentioned, some components were written from scratch, while others were realized by modifying existing software. 4.1 Action-annotated program to Cyclone We translate Tempo action-annotated programs into run-time specializers written in Cyclone. Using the Tempo front end, this lets us automatically generate a Cyclone program from a C program. An action-annotated program distinguishes two kinds of code: normal code that will be executed during specializa- tion, indicated in italics in Fig. 2; and template code that will emitted during specialization (non-italicized code). The annotated C program is translated into a Cyclone program that uses codegen, cut, splice, and fill. Since italicized constructs will be executed during code generation, they will occur outside codegen, or within a cut statement or a fill expression. Non-italicized constructs will be placed within a codegen expression or splice statement. Our algorithm operates in two modes: "normal" mode translates constructs that should be executed at code generation time and "template" mode translates constructs that will be part of a template. The algorithm performs a recursive descent of the action-annotated abstract syntax, keeping track of which mode it is in. It starts off in "normal" mode and produces Cyclone code for the beginning of the run-time specializer: its arguments (the invariants) and any local variables and initial statements that are annotated with italics. When the first non-italic construct is encoun- tered, a codegen expression is issued, putting the translation into "template" mode. The rest of the program is translated as follows. An italic statement or expression must be translated in "normal" mode. Therefore, if the translation is in "tem- plate" mode, we insert cut (if we are processing a statement) or fill (if we are processing an expression) and switch into "normal" mode. Similarly, a non-italic statement should be translated in "template" mode; here we insert splice and switch modes if necessary. It isn't possible to encounter a non-italic expression within an italic expression. Another step needs to be taken during this translation since specialization is speculative, i.e., both branches of a conditional statement can be optimistically specialized when the conditional test itself cannot be evaluated. This means that during specialization, the store needs to be saved prior to specializing one branch and restored before specializing the other branch. Therefore, we must introduce Cyclone statements to save and restore the store when translating such a conditional statement. This is the same solution used by Tempo [6]. 4.2 Cyclone to TAL/T To compile Cyclone to TAL/T, we extended an existing com- piler, the Popcorn compiler of Morrisett et al. Popcorn is written in Caml, and it compiles a type safe dialect of C into TAL, a typed assembly language [16]. Currently, Popcorn is a very simple, stack based compiler, though it is being extended with register allocation and more sophisticated optimizations. The Popcorn compiler works by performing a traversal of the abstract syntax tree, emitting TAL code as it goes. It uses an environment data structure of the following form: args-on-stack: int - The environment maintains the execution state of each function as it is compiled. The field local env contains each variable identifier and its corresponding stack offset. Arguments are pushed onto the stack prior to entry to the function body; the field args on stack records the number of arguments, so they can be popped off the stack upon exiting the function. To compile Cyclone we needed to extend the environment datatype: first, because Cyclone switches between generating normal code and template code, and second, because Cyclone has nested functions. Therefore, we use environments with the same structure as the environments used in Cyclone's typing rules: Outermost of env * (id list) - Frame of env * cyclone-env - Hidden of env * cyclone-env That is, environments are sequences of type frames for func- tions. A frame can either be outermost, normal, or hidden. Once we have this type of environment, visible bindings are defined as they are for E vis in Section 2. An Outermost frame contains the local environment for a top-level function as well as the global identifiers. A Frame is used when compiling template code. A new Frame environment is created each time codegen is encountered. A Frame becomes Hidden to switch back to "normal" mode when a cut or fill is encountered. Popcorn programs are compiled by traversing the abstract syntax tree and translating each Popcorn construct into the appropriate TAL instructions; the resulting sequence of TAL instructions is the compiled program. Compiling a Cyclone program, however, is more complicated; it is performed in two phases. The first phase alternates between generating normal and template TAL/T instructions and a second phase rearranges the instructions to put them in their proper place. In order for the instructions to be rearranged in the second phase, the first phase interleaves special markers with the TAL/T instructions: M-TemplateBeg of id * id - M-Fill of id * exp These markers are used to indicate which instructions are normal, which belong within a template, and which are used to fill holes. M TemplateBeg takes two arguments, the beginning and ending label of a template, and is issued at the beginning of a template (when codegen or splice is encountered, or cut ends). Similarly, M TemplateEnd is issued at the end of a template (at the end of a codegen or splice, or the beginning of a cut). Note that between corresponding M TemplateBeg and M TemplateEnd markers, other templates may begin and end. Therefore, these markers can be nested. When a hole is encountered, a M Fill marker is issued. The first argument of M Fill is a label for the hole inside the template. The second argument is the Cyclone source code expression that should fill the hole. The following example shows how the cut statement is compiled. match stmt of Cut s -? match cyclone-env with Outermost -? raise Error cg-fill-holes (Hidden(env,cyclone-env')); emit-mark(M-TemplateBeg(id-new "a", id-new "b")); Hidden -? raise Error The function compile stmt takes a Cyclone statement and an environment, and emits TAL/T instructions as a side-effect. The first thing to notice is that a cut can only occur when the compiler is in "template" mode, in which case the environment begins with Frame. A cut statement ends a template. Therefore, cg fill holes is called, which emits a M TemplateEnd marker, and emits TAL/T code to dump the template and fill its holes. Filling holes must be done using a "normal" environment, and therefore the first frame becomes Hidden. Next, compile stmt is called recursively to compile the statement s within the cut. Since the statement s should also be compiled in normal mode, it also keeps the first frame Hidden. Finally, a M TemplateBeg marker is emitted so that the compilation of any constructs following the cut will occur within a new template. The second phase of the code generation uses the markers to rearrange the code. The TAL/T instructions issued between a M TemplateBeg and a M TemplateEnd marker are extracted and made into a template. The remaining, normal instructions are concatenated to make one function; hole filling instructions are inserted after the instruction which dumps the template that contains the hole. The example in Fig. 4 shows a TAL/T program after the second phase is completed; the normal code includes instructions to dump templates and fill holes, and is followed by the templates. 4.3 TAL/T to executable TAL is translated into assembly code by expanding each TAL macro into a sequence of x86 instructions. Similarly, the new TAL/T macros expand into a sequence of x86 and TAL instructions. A description of each TAL/T macro is given in Section 3. The resulting x86 assembly language program is assembled with Microsoft MASM and linked with the Microsoft Visual C++ linker. 4.4 Initial Impressions We have implemented our system and have started testing it on programs to assess its strengths and weaknesses. Since there is currently a lot of interest in specializing interpreters, we decided to explore this type of application program. A state-of-the-art program specializer such as Tempo typically achieves a speedup between 2 and 20, depending on the interpreter and program interpreted. To see how our system compares, we took a bytecode interpreter available in the Tempo distribution and ran it through our system. Preliminary results show that Cyclone achieves a speedup of over 3. This is encouraging, since this is roughly the speedup Tempo achieves on similar programs. A more precise comparison of the two systems still needs to be done, however. On the other hand, in our initial implementation, the cost of generating code is higher than in Tempo. One possible reason is that for safety, we allocate our code generation regions at run time, and perform bounds checks as we dump templates. The approach taken by Tempo, choosing a maximum buffer size at compile time and allocating a buffer of that size, is faster but not safe. 5 Related Work Propagating types through all stages of a compiler, from the front end to the back end, has been shown to aid robust compiler construction: checking type safety after each stage quickly identifies compiler bugs [23, 24]. Additionally, Necula and Lee have shown that proving properties at the assembly language level is useful for safe execution of untrusted mobile code [18]. So far, this approach has been taken only for statically generated code. Our system is intended to achieve these same goals for dynamically generated code. Many of the ideas in Cyclone were derived from the Tempo run-time specializer [7, 12, 13]. We designed Cyclone and TAL/T with a template-based approach in mind, and we use the Tempo front end for automatic template identi- fication. Another run-time specializer, DyC, shares some of the same features, such as static analyses and a template-like back end [5, 9, 10]. There are, however, some important differences between Cyclone and these systems. We have tried to make our compiler more robust than Tempo and DyC, by making Cyclone type safe, and by using types to verify the safety of compiled code. Like Tempo and DyC, Cyclone can automatically construct specializers, but in ad- dition, Cyclone also gives the programmer explicit control over run-time code generation, via the codegen, cut, splice, and fill constructs. It is even possible for us to hand-tweak the specializers produced by the Tempo front end with complete type safety. Like DyC, we can perform optimizations such as inter-template code motion, since we are writing our own compiler. Tempo's strategy of using an unmodified, existing compiler limits the optimizations that it can perform. ML-box, Meta-ML, and 'C are all systems that add explicit code generation constructs to existing languages. ML- box and Meta-ML are type safe dialects of ML [15, 25, 22], while 'C is an unsafe dialect of C [20, 21]. All three systems have features for combining code fragments that go beyond what we provide in Cyclone. For example, in 'C it is possible to generate functions that have n arguments, where n is a value computed at run time; this is not possible in Cyclone, ML-box, or Meta-ML. On the other hand, 'C cannot generate a function that generates a function; this can be done in Cyclone (using nested codegens), and also in ML-box and Meta-ML. An advantage we gain from not having sophisticated features for manipulating code fragments is simplicity: for example, the Cyclone type system does not need a new type for code fragments. The most fundamental difference, however, is that the overall system we present will provide type safety not only at the source level, but also at the object level. This makes our system more robust and makes it usable in a proof carrying code system. 6 Future Work In this paper we presented a framework for performing safe and robust run-time code generation. Our compiler is based on a simple, stack-based, certifying compiler written by Morrisett et al. They are extending the compiler with register allocation and other standard optimizations, and we expect to merge Cyclone with their improvements. We are interested in studying template-specific optimiza- tions. For example, because templates appear explicitly in TAL/T, we plan to study inter-template optimizations, such as code motion between templates. Performing inter- template optimizations is more difficult in a system, like Tempo, based on an existing compiler that is not aware of templates. We are also interested in analyses that could statically bound the size of the dynamic code generation region. This would let us allocate exactly the right amount of space when we begin generating code for a function, and would let us eliminate bounds checks during template dumps. We would like to extend the front end of Tempo so that it takes Cyclone, and not just C, as input. This would mean extending the analyses of Tempo to handle Cyclone, which is an n-level language like ML-box. Additionally, we may implement the analysis of Gl-uck and J-rgensen [8] to produce n-level Cyclone from C or Cyclone. 7 Conclusion We have designed a programming language and compiler that combines dynamic code generation with certified com- pilation. Our system, Cyclone, has the following features. Robust dynamic code generation Existing dynamic code generation systems only prove safety at the source level. Our approach extends this to object code. This means that bugs in the compiler that produce unsafe run-time specializers can be caught at compile time, before the specializer itself is run. This is extremely helpful because of the complexity of the analyses and transformations involved in dynamic code generation. Flexibility and Safety Cyclone produces dynamic code generators that exploit run-time invariants to produce optimized programs. The user interface is flexible, since the final executable can be generated from a C program, a Cyclone program, or TAL/T assembly code. Type safety is statically verified in all three cases. Safe execution of untrusted, dynamic, mobile code generators This approach can be used to extend a proof-carrying code system to include dynamic code generation. Since verification occurs prior to run time, there is no run-time cost incurred for the safety guarantees. Sophisticated optimization techniques can be employed in the certifying compiler. The resulting system could produce mobile code that is not only safe, but potentially extremely fast. Acknowledgements . We were able to implement Cyclone quickly because we worked from the existing Tempo and TAL implementations. We'd like to thank Charles Consel and the Tempo group, and Greg Morrisett and the TAL group, for making this possible. The paper was improved by feedback from Julia Lawall and the anonymous referees. --R Partial evaluation in aircraft crew plan- ning A general approach for run-time specialization and its application to C Fast binding-time analysis for multi-level specialization DyC: An expressive annotation-directed dynamic compiler for C Specializing shaders. Static Analyses for the Effective Specialization of Realistic Applications. Accurate binding-time analysis for imperative languages: Flow A case for runtime code generation. Lightweight run-time code gener- ation From System F to typed assembly language. Fast, optimized Sun RPC using automatic program specialization. The design and implementation of a certifying compiler. tcc: A system for fast Design and Implementation of Code Optimizations for a Type-Directed Compiler for Standard ML --TR --CTR George C. Necula , Peter Lee, The design and implementation of a certifying compiler, ACM SIGPLAN Notices, v.39 n.4, April 2004 Christopher Colby , Peter Lee , George C. Necula , Fred Blau , Mark Plesko , Kenneth Cline, A certifying compiler for Java, ACM SIGPLAN Notices, v.35 n.5, p.95-107, May 2000 Ingo Strmer, Integration Of The Code Generation Approach In The Model-Based Development Process By Means Of Tool Certification, Journal of Integrated Design & Process Science, v.8 n.2, p.1-11, April 2004 Hongxu Cai , Zhong Shao , Alexander Vaynberg, Certified self-modifying code, ACM SIGPLAN Notices, v.42 n.6, June 2007 Scott Thibault , Charles Consel , Julia L. Lawall , Renaud Marlet , Gilles Muller, Static and Dynamic Program Compilation by Interpreter Specialization, Higher-Order and Symbolic Computation, v.13 n.3, p.161-178, Sept. 2000 Cristiano Calcagno , Walid Taha , Liwen Huang , Xavier Leroy, Implementing multi-stage languages using ASTs, Gensym, and reflection, Proceedings of the second international conference on Generative programming and component engineering, p.57-76, September 22-25, 2003, Erfurt, Germany Simon Helsen, Bisimilarity for the Region Calculus, Higher-Order and Symbolic Computation, v.17 n.4, p.347-394, December 2004

program specialization;run-time code generation;partial evaluation;certifying compilation;typed assembly language;proof-carrying code

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A Generic Reification Technique for Object-Oriented Reflective Languages.

Computational reflection is gaining interest in practical applications as witnessed by the use of reflection in the Java programming environment and recent work on reflective middleware. Reflective systems offer many different reflection programming interfaces, the so-called Meta-Object Protocols (MOPs). Their design is subject to a number of constraints relating to, among others, expressive power, efficiency and security properties. Since these constraints are different from one application to another, it would be desirable to easily provide specially-tailored MOPs.In this paper, we present a generic reification technique based on program transformation. It enables the selective reification of arbitrary parts of object-oriented meta-circular interpreters. The reification process is of fine granularity: individual objects of the run-time system can be reified independently. Furthermore, the program transformation can be applied to different interpreter definitions. Each resulting reflective implementation provides a different MOP directly derived from the original interpreter definition.

Introduction Computational reflection, that is, the possibility of a software system to inspect and modify itself at runtime, is gaining interest in practical applications: modern software frequently requires strong adaptability conditions to be met in order to fit a heterogenous and evolving computing environment. Reflection allows, for instance, host services to be determined dynamically and enables the modification of interaction protocols at runtime. Concretely, the JAVA programming environment [java] relies heavily on the use of reflection for the implementation of the JAVABEANS component model and its remote method invocation mechanism. Furthermore, adaptability is a prime requirement of middle-ware systems and several groups are therefore doing research on reflective middleware [coi99][bc00]. Reflective systems offer many different reflection programming interfaces, the so-called Meta-Object Protocols (MOPs) 1 . The design of such a MOP is subject to a number of constraints relating to, among others, expressive power, efficiency and security properties. For instance, using reflection Extended version. c 2001 Kluwer Academic Publishers. "Higher-Order and Symbolic Computation", 14(1), 2001, to appear. We use the term "MOP" in the sense of Kiczales et al. [kic91] (page 1): "Metaobject protocols are interfaces to the language that give users the ability to incrementally modify the language's behavior and implementation, as well as the ability to write programs within the language." for debugging purposes may require the MOP to provide access to the execution stack. However, because of security concerns stack access must frequently be restricted: in JAVA, for example, it is not allowed to modify the (untyped) stack because security properties essentially rely on type information. Since these constraints are different from one application to another, we should be able to provide a specially-tailored MOP for a particular set of constraints. Moreover, the constraints may change during the overall software life cycle. Hence, the development of such specially-tailored MOPs should be a lightweight process. Traditional approaches to the development of MOPs do not meet this goal instead each of them only provides a specific MOP which can hardly be modified (see the discussion of related work in Section 9). Consider, for instance, a single-processor application which is to be distributed. In this case, distinct tasks have to be performed on the message sending side and the receiving side: for example, on the sender side local calls are replaced by remote ones (instead of relying on proxies) and on the receiver side incoming messages can be synchronized. Many existing MOPs do not allow the behavior of message senders to be modified. Hence, such a distribution strategy cannot be implemented using reflection in these systems. Some systems (see, for instance CODA [aff95]) provide access to senders right from the start. Therefore, they can introduce an overhead for local applications. In this paper, we present a reification mechanism for object-oriented interpreters based on program transformation techniques. We use a generic transformation which can be applied at compile time to any class of a non-reflective interpreter definition. This mechanism can be used to transform different subsets of a metacircular interpreter in order to generate increasingly reflective interpreters. It can also be applied to different interpreter definitions in order to automatically get different reflective interpreters. Each resulting reflective implementation provides a different MOP directly derived from the original interpreter definition. The paper is structured as follows: in Section 2, we briefly introduce Smith's seminal reflective towers upon which our work is based and we sketch the architecture of our transformational system. Section 3 provides an overview of a metacircular interpreter for JAVA. Our generic reification technique is formally defined and its application to the non-reflective interpreter is exemplified in Section 4. Section 5 is devoted to reflective programming: it details our reification technique at work by presenting several applications. Section 6 complements Section 4 by presenting a few technicalities postponed for the sake of readability. Section 7 discusses the correctness of the transformation and sketches a formal correctness proof. Section 8 illustrates how a refined definition of the non-reflective interpreter produces a more expressive reflective interpreter. Section 9 discusses related work. Fi- nally, Section 10 concludes and discusses future work. Code occuring in the paper refers to a freely available prototype implementation, called METAJ [metaj], which enables execution of the reflective programming examples we present and provides a platform for experimentation with our technique. 2 Overview of the reification process In our opinion, Smith' definition of reflection [smi84] remains a key reference because of its clean semantic foundation and generality. This paper proposes one method to transpose his technique into the domain of object-oriented languages. In this section, we first introduce Smith-like reflection before presenting the architecture of our reification method. Level 0 Level 1 Level 2 Level 3 Interpreter Program Program Interpreter Interpreter Interpreter Program Interpreter Program Figure 1: Smith-like reflective towers 2.1 Smith-like Reflection Smith's seminal work on reflective 3-Lisp defines reflection with the notion of reflective towers. In Figure 1, the left hand side tower shows a user-written (i.e. level 0) Program in a double-square box and its Interpreter, which defines its operational semantics. A simple classic example of reflective programming deals with the introduction of debugging traces. Trace generation requires the interpreter to be modified, that is, two steps have to be performed at runtime: provide an accessible representation of the current interpreter and change this representation. Such a computation creates an extra interpretation layer by means of a reification operator "reify," so that the level 1 Interpreter becomes now part of the program: in the illustration, it is included in the double square box. We get a second tower with three levels. The Program can now modify the standard semantics of the language defined by the level 1 Interpreter to get Interpreter' which generates traces during execution (see the third tower). Finally, when a non-standard semantics Interpreter" of Interpreter' is required, a further extra interpretation level can be introduced as illustrated by the fourth tower. The fourth tower would be required, for example, to trace Interpreter'. On a more abstract level, Smith's reflection model - as well as our reification technique - has two essential properties: there is a potentially infinite tower of reflective interpreters and the interpreter at level n interprets the actual code of the interpreter at level n 1. 2.2 Making object-oriented interpreters reflective In order to get a first intuition of our reification technique, consider the following simple example of how we intend reflection to be used: color information (represented by the class Color) should be added to pairs at runtime. Using reflection, we could dynamically modify the inheritance graph such that Pair inherits from Color. This can be achieved by // Pair extends Object // Pair extends Color where 4 is a reification operator. The application of the reification operator to an expression yields an accessible representation of the value denoted by the expression. In this example, the expression Pair denotes the corresponding Class object, say c, in the interpreter's memory (see Figure 7). 4Pair returns an instance (say i), i.e. an object of type Instance, in the interpreter's memory which represents c and which can be inspected and modified. The default superclass of Pair is replaced by Color by assigning the field extendsLink. From now on, newly instantiated pairs contain color information. It is crucial to our approach that the reified representation i is based on the definition of c. This Parser Java.jjt Runtime System ExpAssign.java Reflective Interpreter Reflective_Prog.java Java2ExpVisitor.java Parser Java.jjt Runtime System ExpAssign.java Non-Reflective Interpreter Prog.java Java2ExpVisitor.java Instance.java BaseClass.java Instance.java generation of reflective interpreter program transformation Figure 2: System architecture is achieved by the system architecture shown in Figure 2. Our non-reflective JAVA interpreter (repre- sented by the box at the top) takes a non-reflective program Prog.java as input. This program is parsed into a syntax tree and evaluated. According to the required reflective capabilities, the language designer 2 transforms a subset of the classes of the non-reflective interpreter. Basically, this transformation generates two classes for each original class. In our example, the file Class.java, which represents classes in the non-reflective interpreter, becomes BaseClass.java and a different version of Class.java in the reflective one. The reflective interpreter relies on the non-reflective interpreter in order to build levels of the reflective tower. This is the core issue of our approach: the tower levels shown in Figure 1 are effectively built at runtime on the basis of the (verbatim) definition of the non-reflective interpreter as in Smith's model. This is why the original definition of Class.java is an input of the reflective interpreter in Figure 2. So, the behavior of the reflective interpreter is derived from the non-reflective one. Furthermore, our approach is selective and complete because the transformation is applicable to any class of the non-reflective interpreter definition. implements one version of this system architecture. Its parser has been implemented by means of JAVACC and JJTREE (versions 0.8pre2 and 0.3pre6, respectively). METAJ itself is operational with the JDK versions 1.1.6 and 1.2. 3 A simple non-reflective interpreter We have implemented a non-reflective metacircular interpreter for a subset of JAVA, which provides support for all essential object-oriented and imperative features, such as classes, objects, fields, meth- ods, local variables and assignment statements. (We did not implement features such as some primitive 2 Note that building a reflective interpreter by transformation and writing reflective programs are two different tasks: the former is performed by language designers and the latter by application programmers. class ExpId extends Exp { private String id; ExpId(String id) { Data eval(Environment localE) { return localE.lookup(this.id); Figure 3: Class ExpId class ExpAssign extends Exp { private Exp lhs; private Exp rhs; ExpAssign(Exp lhs, Exp rhs) { Data eval(Environment localE) { Data Data return d2; Figure 4: Class ExpAssign types or loop constructs; all of these could be integrated and reified similarly.) JAVA programs are represented as abstract syntax trees the nodes of which denote JAVA's syntactic constructs and are implemented by corresponding classes. For example, variables, assignment statement, method call, and class instantiation expressions are respectively encoded by the classes ExpId, ExpAssign, ExpMethod and ExpNew. All of these classes define an evaluation method Data eval(Environment localE) that takes the values of local variables in localE and returns the value of the expression (wrapped in a Data object). In particular, ExpId (see Figure 3) holds the name of a variable and its evaluation method yields the value currently associated to the variable in the local environment. An ExpAssign node (see Figure stores the two subexpressions of an assignment. Its evaluation method evaluates the location of the right-hand side expression, followed by the value represented by the left-hand side expression and finally performs the assignment. ExpMethod (see Figure 5) represents a method call with a receiver expression (exp), a method name (methodId) and its argument expressions (args). Method call evaluation proceeds by evaluating the receiver, constructing an environment from the argument values, looking up the method definition and applying it. ExpNew (see Figure 6) encodes the class name (classId) and constructor argument expressions. Its evaluation fetches the class definition from the global environment, instantiates it and possibly calls the constructor. As suggested before, the interpreter defines a few other classes to provide a runtime system and implement an operational semantics. For example, the class Class (see Figure 7) represents classes by a reference to a superclass (extendsLink), a list of fields (dataList) and a list of methods (methodList). It provides methods for instantiating the class (instantiate()), accessing the list of methods including those in super classes (methodList()), etc. Methods are represented by class ExpMethod extends Exp { private Exp exp; // receiver private String methodId; // method name private ExpList args; // arguments ExpMethod(Exp exp, String methodId, ExpList args) { Data eval(Environment localE) { // evaluate the lhs (receiver) Instance // evaluate the arguments to get a new local environment Environment // lookup and apply method return m.apply(argsE, i); Figure 5: Class ExpMethod class ExpNew extends Exp { private String classId; // class name private ExpList args; // constructor arguments ExpNew(String classId, ExpList args) { Data eval(Environment localE) { // get the Class and create an Instance Instance // call non default constructor if it exists if (i.getInstanceLink().methodList().member(this.classId).booleanValue()) { Environment // lookup and apply method return new Figure class Class { Class extendsLink; // superclass DataList dataList; // field list MethodList methodList; Class(Class eL, DataList dL, MethodList mL) { Class getExtendsLink() { return this.extendsLink; } // implementation of Java's 'new' operator Instance instantiate() { . } // compute complete method list (incl. superclasses) MethodList methodList() { . } Figure 7: Class Class class Method { private StringList args; // parameter names private Exp body; // method body Method(StringList args, Exp body) { Data apply(Environment argsE, Instance i) { // name each argument argsE.add("this", new // eval the body definition of the method return this.body.eval(argsE); Figure 8: Class Method the class Method (see Figure 8) by means of a list of argument names (args) and a body expression (body). Its method apply() binds argument names to values including this and evaluates the body. Other classes include Instance (contains a reference instanceLink to its class and a list of field values; provides field lookup and method lookup), MethodList, Data (implements mutable memory cells such as fields), DataList, Environment (maps identifiers to values), etc. The architecture of the interpreter follows the standard design for object-oriented interpreters as presented in Gamma et al. [ghjv95] by the Interpreter design pattern. Instantiating this design pattern, the following correspondences hold: their 'Client' is our interpreter's main() method, their methods 'Interpret(Context)' is our eval(Environment). The reification technique described in this paper is applicable to other interpreters having such an architecture. Note that these interpreters may implement many different runtime systems. 4 Generic reification by code transformation In this section, we give an overview of our generic reification scheme for the class Class and formally define the underlying program transformation. (For the sake of readability, we postpone the discussion of a few technicalities to Section 6.) Then, we apply it in detail to the class Instance. 4.1 Overview of the generic reification scheme Reification of an object should not change the semantics of that object but change its representation and provide access to the changed representation. For example, it is not possible to modify the superclass of a class at runtime in our non-reflective interpreter (although a reference representing the inheritance relation exists in the memory of the underlying implementation). The reified representation of a class provides access to this reference. Once the internal representation has been exposed, access to this structure allows the semantics of the program to be changed (e.g. by means of dynamic class changes). Note that this form of structural reification of the interpreter memory subsumes the traditional notions of structural and behavioral reflection. For illustration purposes, consider a class Pair with two fields fst and snd which is implemented in the interpreter memory by a Class (from here on, C denotes an instance of the class C). In order to reify Pair, we choose Class to be reifiable. Basically, a reifiable entity can have two different representations as exemplified in Figure 9: either a base representation or a reified represen- tation. Since reification of any object does not change its behavior, the object should provide the same method interface in both representations. This common interface is implemented using a dispatch object 3 : the Class denoted by Pair. The dispatch object points to the currently active representation: either the base representation (BaseClass in Figure 9a) or the reified representation (Instance denoted by 4Pair in Figure 9b). The dispatch object provides a method reify() (triggered by 4) to switch from the base representation to the reified one: a call to reify() creates a new tower level. The dispatch object executes incoming method calls according to the active representation: when the base representation is active, the dispatch simply delegates incoming method calls to it. When the reified representation is active, the dispatch object interprets the method call. Whether an object is accessed through its dispatch object or through its reified representation is irrelevant, that is, modification of the object through the access path Pair is visible through the other access path 4Pair. (This property is commonly referred to as the causal connection between levels.) 3 The dispatch technique is close to the bridge and state patterns introduced in Gamma et al. [ghjv95]. denotes methodList dispatch object Class Instance dataList methodList, extendLink active representation instance of a) before D b) after D Pair Pair Pair Pair Pair different representations Figure 9: Before and after reification of the class Pair Obviously, the two paths provide different interfaces. Consider, for example, the problem of keeping track of the number of Pair instances using a static field countInstances: this field could be accessed either by Pair.countInstances or by (4(4Pair).staticDataList).lookup 4 . In the last expression, the outermost reification operation is necessary in order to call lookup() on a data list object (cf. the fourth item below). In order to conclude this overview, we briefly mention other important properties of our reification scheme: Since reflection provides objects representing internal structure for use in user-level programs, every reification operation returns an Instance (e.g. the one in Figure 9b). This implies that reification of reified entities requires that Instances are reifiable. 4Exp yields an accessible representation of the value denoted by Exp (i.e. an object in the interpreter's memory, such as Class, Instance, Method) 5 . References from dispatch objects to their active representations cannot be accessed by user programs. Only a call to the reification operator may modify these references. This ensures that the tower structure cannot be messed up by user programs. The scope of the reification process is limited to individual objects in the interpreter's memory. For example, the reification of a class does not reify its list of methods methodList nor its superclass. So, three categories of objects coexist at runtime: reified objects, non-reified (but 4 METAJ does not allow static fields but could be extended easily to deal with such examples. 5 4 is a strict operator. A syntax extension would be necessary to reify the expression (e.g. the AST representing 1+4) rather than the value denoted by the expression (e.g. the integer 5). class Name { Type f1 field f1 ; Type fn field fn ; Name(Type f1 arg f1 , ., Type fn arg fn ) { body } Type m1 method m1 (Type m11 arg m11 , ., Type m 1k arg m 1k ) { body m1 } Figure 10: Original class definition reifiable) ones and non-reifiable ones. If a program accesses an object o through a reified one, the use of restricted exactly as in the non-reflective case. 4Pair.extendsLink, for example, references a Class representing the superclass of Pair. Therefore, the only valid operations on this reference are new (4Pair.extendsLink)() 6 as well as accesses to static fields and members of this class. If the structure or behavior of the superclass is to be changed, it must be reified first. This implies that accesses to non-reifiable objects through reified ones are safe. 4.2 Formal definition of the generic reification scheme Based on the implementation technique outlined above, our generic reification scheme is an automatic program transformation which can be applied to an arbitrary class, called Name in the following definition, of the original interpreter. As shown in Figure 10, such classes consist of a number of fields and methods and must have a constructor with arguments for all of their fields. The transformation of a set of classes has time and space complexity linear in the number of classes. The transformation consists of two main steps: 1. Introduce the class BaseName (see Figure 11) which defines the base representation of the original class Name. This class is very similar to the original class Name. 2. Redefine the class Name (see Figure 12) such that it implements the corresponding dispatch object. This class provides the same method interface as the original class Name and implements a method reify() which creates the reified representation and switches from the base representation to the reified one. Figure 11 shows the generated base class. (In the figures of this section, we use different style conventions for verbatim text, schema variables and [text substitutions].) Ba- sically, the original class is renamed and a field referent is added. Remember that a reifiable entity is implemented by a dispatch object that points to the current representation. The referent field, which is initialized in the constructor and points back from the representation to the dispatch object, is mandatory to distinguish the dispatch object and the representation: if this is not used to access 6 The current parser of METAJ does not allow such an expression: new requires a class identifier. However, the parser could be easily extended to deal with such expressions and we allow this notation in this paper. class BaseName { Type f1 field f1 ; Type fn field fn ; Name referent; BaseName(Type f1 arg f1 , ., Type fn arg fn , Name referent) { body Type m1 method m1 (Type m11 arg m11 , ., Type m 1k arg m 1k ) { Figure Generated base class fields or methods in the base class it should denote the dispatch object 7 . In the transformation, this is implemented by substituting this(~.) (matching the keyword this followed by anything but a dot) by this.referent. The generated dispatch class, shown in Figure 12, has two fields: representation that points to either the base representation or the reified representation, and a boolean field isReified that discriminates the active representation. Its constructor creates a base representation for the object. The methods method m i have the same signature as their original version. When the base representation is active (i.e. isReified is false), the method call is delegated to the base representation. When the reified representation is active (i.e. isReified is true), the method call is interpreted: the corresponding call expression is parsed (Parser.java2Exp()), a local environment is built (argsE.add()) from the method arguments and the field representation of the dispatch object and the method call is evaluated (eval()). Note that for the sake of clarity, this code is intentionally naive. The actual implemented version could be optimized: for example, the call to the parser could be replaced by the corresponding syntax tree. The method reify() builds a reified representation of the base representation by evaluating a new-expression. The corresponding class is cloned in order to build a new tower level. So, every reified object has its own copy of a Class. This way, the behavior of each reified object can be specialized independently. If sharing is required the application programmer can achieve it by explicitly manipulating references. Finally, the reified representation is installed as the current representation and a reference to it is returned. A series of experiments led us to this sharing strategy. A previous version of the transformation did not clone the class. This sharing led to cycling dependency relationships and reflective overlap after reification: in particular, reification of the class Class introduced 7 This is a typical problem of wrapper-based techniques that introduce two different identities for an object. class Name { Object representation; boolean isReified; Name(Type f1 arg f1 , ., Type fn arg fn ) { new BaseName(arg f1 , ., arg fn , this); Type m1 method m1 (Type m11 arg m11 , ., Type m 1k arg m 1k ) { if (this.isReified) { "reifiedRep.method m1 (arg m11 ,.,arg m 1k )"); Environment argsE.add("reifiedRep", this.representation); argsE.add("arg m11 ", arg m11 ); Data return result.read(); else return (BaseName) this.representation.method m1 (arg m11 , . , arg m 1k ); Instance reify() { if (!this.isReified) { baseRep_field fn , Environment argsE.add("baseRep_field f1 ", this.representation.field f1 ); argsE.add("baseRep_field fn ", this.representation.field fn ); argsE.add("aClass", aClass); return (Instance)this.representation; Figure 12: Generated dispatch class non-termination. Alternatively, we experimented with one copy of each class per level but in this case the reification (without modification) of an object could already change its behavior. This generic reification technique is based on only two assumptions: 1. Each syntactic construct is represented by an appropriate expression during interpreter execu- tion. We assume that all of these expressions can be evaluated using the method eval(argsE) where argsE contains the current environment, i.e. the values of the free variables in the current expression. 2. We assume that the textual definitions of all reifiable classes have been parsed at interpreter creation time and that they are stored as Class objects in the global environment Main.globalE. These objects have to be cloneable. This way, reify() creates an extra interpreter layer based on the actual interpreter definition. Note that these simple assumptions and the formal definition enable the transformation to be performed automatically. In Java, the operator new returns an object (i.e. an Instance). Therefore, in order to let the user build other runtime entities than Instances, such as Classes and Methods, we provide a family of deification 8 operators, one for each of these entities. These operators are the inverse of the generic reification operator. For example, in the reflective program (where r Class denotes the deification operator for classes): the right-hand side expression returns a Class dispatch object in front of the Instance created by new. Note that the deification operators - while functionally inverting the reification operation - do not change the representation of an object "back" to its unreified structure (e.g. to a BaseClass in the case of classes). The dispatch objects engender the structure of the reflective tower; their implementation is not accessible to the user. In particular, the reification operator and the deification operators encapsulate the fields representation and isReified of dispatch objects as well as the field referent from the base class. So, user programs cannot arbitrarily change the tower structure. However, the user or a type system to be developed should avoid the creation of meaningless structures, such as r Class (new Method(. 4.3 Example: making the class Instance reifiable To illustrate the definition of the transformation, we apply it to the class Instance (see Figure 14), which is used in the examples of reflective programming in the next section. This class implements objects in the interpreter. For example, a pair object with two fields fst and snd is implemented by an Instance the field dataList of which contains two memory cells labelled fst and snd. Its field instanceLink points to a Class containing the methods of the class Pair. The method lookupData() is called whenever a field of pair is accessed. (For the sake of conciseness, we did not show the other methods of Instance, such as lookupMethod().) The application of the transformation defined above to Instance yields the two classes Ba- seInstance (see Figure 15) and the dispatch class Instance (see Figure 16). Now, pair is implemented by a dispatching Instance as shown in Figure 13. Its default unreified representation is a BaseInstance (say b 1 whose dataList field contains the fields labelled fst and snd (see 8 We prefer the term 'deification' [iyl95] to the equivalent terms `reflection' [wf88] and 'absorption' [meu98]. dispatch object Instance dispatch object Instance denotes active representation instance of a) before b) after BaseInstance dataList instanceLink Instance different representations dispatch object Instance Figure 13: Before and after reification of the object pair class Instance { public Class instanceLink; // ref. to Class public DataList dataList; // field list Instance(Class instanceLink, DataList dataList) { // field access Data lookupData(String name) { return this.dataList.lookup(name); Figure 14: Original class Instance class BaseInstance { Class instanceLink; DataList dataList; Instance referent; BaseInstance (Class instanceLink, DataList dataList, Instance referent) { Data lookupData(String name) { return this.dataList.lookup(name); Figure 15: Class BaseInstance Figure 13a). Once pair has been reified (see Figure 13b), it is represented by an Instance which points to a BaseInstance (say b 2 ). Note that in contrast to the reification of classes shown in Figure 9, the reified representation of an instance is reifiable (because it is an instance itself; hence, the second dispatching Instance in Figure 13b). Since the reification is based on the actual definition of the original Instance, the dataList of b 2 contains the three fields instanceLink,dataList (itself containing fst and snd) and referent. The definition of the method lookupData() in the dispatch object calls the method lookupData()of b 1 as long as pair is not reified. Once it is reified, the definition of lookupData() of Instance is interpreted. In order to prove the feasibility of our approach, we applied this reification technique to different classes defining object-oriented features of our JAVA interpreter resulting in the prototype METAJ. The imperative features of the non-reflective interpreter can be tackled analogously. This way we could, for example, redefine the sequentialization operator ';' in order to count the number of execution steps in a given method (say m). One way to achieve this is by reification of occurrences of ExpS in reified m and dynamically changing their classes by a class performing profiling within the eval() method. Another solution would be to replace ExpS nodes in reified m by nodes including profiling. Reflective Programming In this section, we express several classic examples of reflective programming in our framework. These detailed examples of our reflective interpreter at work should help the reader's understanding of the system's working. The examples highlight an important feature of our design: since our reification scheme relies on the original interpreter definition, the meta-object protocol of the corresponding reflective interpreter (i.e. the interface of a reflective system) is quite easy to apprehend. It consists of a few classes which are reifiable in METAJ, the reification operator 4 and the deification operators r . In Figure 17 the class Pair is defined, and in main() a new instance pair is created. In the interpreter, the object pair is represented by an Instance (see Figure 13a). Our generic reification method provides access to a representation of this Instance which we name metaPair (denoted by 4pair in Figure 13b). The most basic use of reflection in object-oriented languages consists in class Instance { Object representation; boolean isReified; Instance(Class instanceLink, DataList dataList) { new BaseInstance(instanceLink, dataList, this); Data lookupData(String name) { if (this.isReified) { // interpret lookup method call // pass already evaluated values Environment argsE.add("name", name); argsE.add("reifiedRep", this.representation); Data // unpack result return (Data)result.read(); else return ((BaseInstance)this.representation).lookupData(name); Data reify() { if (!this.isReified) { // copy the base class BaseInstance // create and initialize new representation Environment argsE.add("baseRep_instanceLink", this.representation.instanceLink); argsE.add("baseRep_dataList", this.representation.dataList); argsE.add("aClass", aClass); return new Figure Dispatch class Instance class Pair { String fst; String snd; Pair(String fst, String snd) { class PrintablePair extends Pair { String toString() { return "(" this.fst class InstanceWithTrace extends Instance { Method lookupMethod(String name) { // trace method-called System.out.println("method return this.instanceLink.methodList().lookup(name); class Main { void main() { Pair new Pair("1", "2"); invariance under reification Instance test existence of a super class if (metaClass.getExtendsLink() == null) System.out.println("Class Pair has no superclass"); class change method-call semantics Instance metaMetaPair.setInstanceLink(InstanceWithTrace); instance and class deification System.out.println((r InstancemetaPair).fst); Figure 17: Examples of Reflective Programming reifying an object: changing the internal representation without modifying its behavior (see Example 1). Another simple use is introspection. Let us consider the problem of testing the existence of a super class of a given class. In Example 2, the class Pair (represented by a Class in the interpreter) is reified which enables its method getExtendsLink() to be called. In METAJ, reflective programming is not limited to introspection, but the internal state of the interpreter can also be modified (aka intercession). The third example in main() shows how the behavior of an instance can be modified by changing its class dynamically. Imagine that we would like to print pairs using a method called toString(). We define a class PrintablePair which extends the original class Pair and implements a method toString(). A pair can then be made printable by dynamically changing its class from Pair to PrintablePair (remember that the field instanceLink of Instance holds the class of the represented instance, see Figure 15). Afterwards the object pair understands the method toString(). The fourth example deals with method call tracing for debugging purposes. The class Instance of the interpreter defines the method Method lookupMethod(String name) that returns the effective method to be called within the inheritance hierarchy. In our interpreter each lookup- is followed by an apply(). Thus, method call tracing can be introduced by defining a class InstanceWithTrace which specializes the class Instance of the interpreter such that its method lookupMethod() prints the name of its parameter. In order to install the tracing of method calls of the instance pair, its standard behavior defined in the interpreter by the class Instance (note that this class can be accessed because the interpreter definition is an integral part of the reflective system built on top of the reflective interpreter) is replaced by InstanceWithTrace. Reification of pair provides access to an Instance whose field instanceLink denotes the class Pair. A sequence of two reification operations on pair provides access to an Instance whose instanceLink denotes the class Instance. This link can then be set to the class Instance- WithTrace. A method call of the object pair then prints the name of the method. Therefore, "toString" is printed by our third example. Finally, note that our tower-based reflection scheme makes it easy to trace the tracing code if required because any number of levels may be created by a sequence of calls to 4. The fifth (rather artificial) example illustrates deification by deifying metaPair and meta-Class in order to create an instance and a class at the base level. After deification of the reified representation metaPair we show that base-level operations can be performed on the resulting ob- ject. In the case of class deification, we restore the original class of pair. More advanced examples that illustrate our approach rely on the capacity to reify arbitrary parts of the underlying interpreter. As discussed in Section 4.3, the reification of ExpS allows the behavior of the sequence operator ';' to be changed. This way, we could, for instance, stop program execution at every statement for debugging purposes or handle numeric overflow exceptions by re-executing the current statement block with higher-precision data representations. Furthermore, reification of the control stack would allow Java's try/catch-mecanism for exception handling to be extended by a retry variant. 6 The nuts and bolts of generic reification Section 4 presents the essential parts of the generic reification mechanism. However, the actual implementation of a full-fledged reflective system requires several intricacies to be handled. In the current section, we motivate the problems which must be handled and sketch the solution we developed. For an in-depth understanding of these technicalities we refer the reader to the METAJ source code. class ExpId extends Exp { // same fields and constructor as in Data eval_original(Environment localE) { // same definition as eval in Data eval(Environment localE) { if (!localE.member("#meta_level").booleanValue()) return this.eval_original(localE); else { if (this.id.equals("this")) return new Data(((Instance) localE.lookup("this").read()).referent); else return eval_original(localE); Figure class ExpMethod extends Exp { Data eval(Environment localE) { Instance if (!localE.member("#meta_level").booleanValue()) return this.eval_original(localE); else { // evaluate the lhs (object part) Object if (o instanceof Reifiable && ((Reifiable) o).getIsReified()) { // evaluate the receiver // evaluate the arguments to get a new local environment Environment new Environment(null, null, null); argsE.add("#meta_level", new // lookup the method and apply it return m.apply(argsE, i); } else { if ((o instanceof DataList) && this.methodId.equals(lookup)) { Environment return new Data(((DataList) o) } else . // other delegation cases Figure 19: Class ExpMethod First, in the reflective interpreter a reified object is represented by a dispatch object and a reified representation. So, basically a reified object has two different identities. With our technique, this is bound to the representation rather than the dispatch object by parsing the expression "reifiedRep. method m1 (arg m11 ,\dots,arg m 1k )" in the dispatch object (see Figure 12). However, if a statement return this is to be interpreted, this should denote the dispatch object. Otherwise, user-level programs could expose the reified representations. The interpreter class ExpId is in charge of identifier evaluation (including this) and has therefore to be modified to account for this be- havior. In Figure 18 the method eval() distinguishes two cases by means of the environment-tag #meta_level. 9 First, interpretation has been initiated by the interpreter's entry point and non-reflective evaluation is necessary. Second, interpretation has been initiated by a dispatch object and reflective interpretation is required. In the first case eval_original() is called: this method has the same definition as eval() in the non-reflective interpreter. In the second case if the identifier is this, the dispatch object of the current representation is returned. Remember that the field referent points back from the base representation to the dispatch object, the same mechanism is used to link the reified representation to the dispatch object. This field must be set by the methods reify(), so the class Instance has to provide such a field 10 . Second, remember that the scope of reification is limited to a single object in the interpreter memory. This means interpretation involves reified and non-reified objects. For example, the reification of an Instance does not reify neither its field list dataList nor its class denoted by in- stanceLink. In particular, once an Instance has been reified, the interpretation of its method lookupData (repeated from Figure 14): Data lookupData(String name){return this.dataList.lookup(name);} requires this.dataList to be interpreted and the call lookup(name) to be delegated because this.dataList denotes a non-reifiable object. In abstract terms, a dispatch object introduces an interpretation layer (a call to eval()) and this layer has to be eliminated when the scope of the current (reified) object is left. This scheme is implemented in ExpMethod.eval() (see Figure 19). Because of these two problems, the methods ExpData.eval() and ExpNew.eval() have to be modified similarly. This means that our reification scheme cannot be applied to the four classes ExpId, ExpMethod, ExpData, ExpNew 11 . However, our method provides much expressive power: these restrictions fix the relationship between certain syntactic constructs and the runtime system, but the runtime mechanisms themselves can still be modified as exemplified in Section 5. In order to weaken this restriction, we designed and implemented a variant 12 of our reification scheme that does not require ExpId and ExpData to be modified. Unfortunately, this advantage comes at a price: the field referent can be exposed and modified by reification in this case. 7 Discussion of the correctness of the transformation A complete treatment of the correctness of our technique is beyond the scope of this paper. However, in this section we discuss very briefly work related to semantics of reflective systems and sketch a few essential properties constituting a skeleton for a formal correctness proof of our technique. 9 The dispatch objects insert this tag into the local environment. 10 For the sake of simplicity, the code shown in Figures 12 and 16 does not mention the field referent. 11 The restriction that all parts of a reflective system cannot be reified seem to be inherent to reflection [wf88]. This variant is also bundled in the METAJ distribution. Semantics of reflective programming systems is a complex research domain. Almost all of the existing body of research work in this domain is about reflection in functional programming languages [wf88][dm88][mul92][mf93]. Even in this context, foundational problems still exist. For example, it seems impossible to give a clean semantics which avoids introducing non-reifiable components [wf88] and logics of programming languages must be considerably weakened in order to obtain a consistent theory of reification [mul92]. One of the very few formal studies of reflection in a non-functional setting has been done by Malenfant et al. [mdc96]. This work deals with reflection in prototype-based languages and focuses on the lookup() ; apply() MOP formalized by means of rewriting systems. This approach is thus too restricted to serve as a basis for our correctness concerns. In general, semantic accounts of imperative languages are more difficult to define than in the functional case. In particular, the transposition of the results obtained in the functional case to our approach requires further work. We anticipate that this should be simpler in a transformational setting such as ours than for arbitrary reflective imperative systems. In order to prove the correctness of our scheme, the basic property to satisfy would be equivalence between a non-reflective interpreter I nr and a reflective interpreter generated by applying our transformation to I nr , i.e. Since the transformation Tr is operating on individual classes, this property can be tackled by establishing an equivalence between an arbitrary class (say c) of the non-reflective interpreter and its transformed counterpart. Essentially, the transformation introduces an extra interpretation layer into the evaluation of the methods of c. Programs and their interpretations introduced by transformation satisfy the property This property can be proven by induction on the structure of the AST representation of p. (Note that the formulation of this property is intentionally simplistic and should be parameterized with contextual information, such as a global environment and a store.) It can be applied to the dispatch classes (see Figure 12) to fold interpreting code into delegating code. When the then-branches of dispatching methods are rewritten using the property from left to right, the then-branches equal the corresponding else-branches. Henceforth, the conditionals become useless and the dispatch objects become simple indirections that can be suppressed. In the case of the method reify(), the rewriting leads to the expression new Name(.) that creates a copy of the non-reified representation. Finally, we strongly believe our transformation is type-safe (although we did not formally prove every well-typed interpreter is transformed into a well-typed reflective interpreter. Obviously, wrongly-typed user programs may crash the non-reflective interpreter. In the same way, some reflective programs may crash the reflective interpreter, for instance by confusing reflective levels or trying to access a field which has been previously suppressed using intercession. Specialized type systems and static analysis methods for safe reflective programming should be developed. Generating alternative metaobject protocols We have already mentioned that each set of reified classes along with their definitions determines a MOP of its own. We think that this is a key property of our approach because it provides a basis for the systematic development of specially-tailored MOPs. In this section, we modify the message-sending part of the non-reflective interpreter in order to provide a finer-grained MOP which distinguishes the sender and the receiver of a message. class Instance { // add two new methods Data send(Msg msg) { return msg.to.receive(msg); Data receive(Msg msg) { return msg.to.lookupMethod(msg.methodId) class ExpMethod extends Exp { Data eval(Environment localE) { // as before evaluate receiver and arguments: o, argsE // new code: determine sender, build and send message Instance new Msg(self, o, this.methodId, argsE); return self.send(msg); Figure 20: Alternative original interpreter class InstanceWithSenderTrace extends Instance { Data send(Msg msg) { System.out.println("method called return Figure 21: (User-defined) extension of Instance In the original interpreter, ExpMethod.eval() evaluates a method call by implementing the composition lookupMethod();apply(). So, the behavior of the receiver of a method call can be modified easily by changing the definition of lookupMethod() (as illustrated by trace insertion in the Section 5). However, a modification concerning the sender of the method call (see CODA [aff95] for a motivation of making the sender explicit in the context of distributed programming) is much more difficult to implement. Such a change would require the modification of all instances of ExpMethod in the abstract syntax tree, i.e. all occurrences of the operator '. Indeed, we have to check whether the object this in such contexts has a non-standard behavior. A solution to this problem is to modify the non-reflective interpreter, such that its reflective version provides a MOP enabling explicit access to the sender in a method call. Intuitively, we split message sending in two parts: the sender side and the receiver side. First, we introduce a new class Msg which is a four-tuple. For each method call, it contains the sender from, the receiver to, the method name methodId and the corresponding argument values argsE. Then, two methods dealing with messages are added to the definition of Instance in the original interpreter: send() and receive() (see Figure 20). Finally, ExpMethod.eval() is redefined such that it creates and sends a message to the receiver. This new version of the non-reflective interpreter is made reflective by applying our program transformation. Then, the user can, for example, introduce tracing for message senders (see Figure 21), the same way traces have been introduced in the previous section. This example highlights three advantages of our approach: MOPs are precisely defined, application programmers are provided with the minimal MOPs tailored to their needs and language designers can extend MOPs at compile time without anticipation of these changes. 9 Related work A comparison between reflective systems is inherently difficult because of the wide variety and the conceptual complexity of reflective models and implementations. For example, the detailed definition of the CLOS MOP requires a book [kic91] and a thorough comparison between CLOS and already fills a book chapter [coi93]. Consequently, we restrict our comparison to the three basic properties our reflection model obeys (the first and second characterizing Smith-like approaches, the third being fundamental to our goal of the construction of specially-tailored MOPs): 1. (tower) There is a potentially infinite tower of reflective interpreters. 2. (interpreter) The interpreter at level n interprets the code of the interpreter at level n 1. 3. (selectivity & completeness) Any part of the runtime system and almost all of the syntax tree (see Section 6) of an interpreter at level n can be reified and has an accessible representation at level First, most reflective systems are based on some notion of reflective towers and provide a potentially infinite number of levels. A notable exception to this are OPEN-C++ [chi95] and IGUANA [gc96] whose MOPs only provide one metalevel. Second, our approach is semantics-based following Smith's seminal work on reflective 3-LISP [smi84] for functional languages. This is also the case for the prototype-based languages 3-KRS [mae87] and AGORA [meu98]. The other object-oriented approaches to reflection (including OBJ- VLISP [coi87], SMALLTALK [bri89] [riv96], CLASSTALK [bri89], CLOS [kic91], MetaXa [gol97]) are not semantics-based (in the sense of the second property cited above) because they do not feed higher-level interpreters with the code of lower-level interpreters. Instead, different levels are represented by appropriate pointer structures. This proceeding allows more efficient implementations but has no semantic foundation. Moreover, these reflective languages are monolithic entities while our modular approach consists of three simple parts: a non-reflective interpreter, the operator 4 and the operators r . Third, our approach enables language designers to precisely select which mechanisms of the language are reflective. With the exception of IGUANA and OPEN-C++, all the reflective systems cited above do not have this characteristic. Finally, note that our approach shares a general notion of completeness with 3-LISP, 3-KRS and AGORA: the programming model is defined by the interpreter and almost all of its features can be made reifiable ("up" and "down" are primitives in 3-LISP and cannot be reified, for instance). Asai et al. [amy96] also starts from such a complete model but this interesting approach to reflection in functional languages restricts reifiable entities in order to allow optimization by partial evaluation. In contrast, the remaining reflective systems described above do not base reflection on features of an underlying interpreter but implement an ad hoc MOP. The notion of completeness therefore does not make sense for them. Conclusion and future work In this paper we have presented a program transformation technique to generate reflective object oriented interpreters from non-reflective ones. This technique allows specially-tailored MOPs to be produced quickly. New MOPs can be developed from scratch or by refinement from existing ones as exemplified in Section 8. Compared to general MOPs, specially-tailored ones could be tuned, for instance, towards better efficiency and security properties. To the best of our knowledge, the resulting framework for reflective object-oriented languages is the first one satisfying the three basic properties mentioned in Section 9. Consequently, our approach cleanly distinguishes between reifiable and non-reifiable entities, thus helping the understanding of reflective programs. A prototype implementation, called METAJ [metaj], is available. Future work. We presented a generic reification technique for object-oriented reflective languages, which provides a basis for the exploration of the metaprogramming design space, optimization techniques and the formalization of reflective systems. First, at the system level the design space of MOPs should be explored by defining and refining different non-reflective interpreters as exemplified in Section 8, yielding a taxonomy of reflective mechanisms. At the user level, the proliferation of reflective dialects requires appropriate design and programming tools, including libraries of user-friendly reflective operators, program analyses and type systems. Second, reflection is deeply related to interpretation. Each dispatch object introduces a new interpretation layer by calling the method eval(). So, specialization techniques like partial evaluation [bn00] are prime candidates for efficiency improvements. Furthermore, user-written reflective programs may not use all reflective capabilities provided by a reflective interpreter (e.g. only make use of a bound number of reflective levels). In this case, optimization techniques such as that presented by Asai et al. [amy96] could be used to merge interpretation levels. Third, since reflective programming is a rather complex task, it should be based on a formal semantics, e.g. to define and ensure security properties. We believe that our transformation could be used to generate specially-tailored reflective semantics from a non-reflective one. Finally, we firmly believe that our reification technique can also be applied to (parts of) applications instead of an interpreter in order to make them reflective (preliminary results can be found in a related paper by the authors [ds00]). Acknowledgements . We thank the anonymous referees for their numerous constructive comments and the editor Olivier Danvy. The work reported here has also benefited from remarks by Kris de Volder, Shigeru Chiba and Jan Vitek. It has been improved through many discussions with our colleagues Noury Bouraqadi, Mathias Braux and Thomas Ledoux. --R Duplication and Partial Evaluation - For a Better Understanding of Reflective Languages Programming with Explicit Metaclasses in SMALLTALK. A Metaobject Protocol for C are First Class Objects: the OBJVLISP Model. "Object-Oriented Programming: The CLOS perspectives?" Intensions and Extensions in a Reflective Tower. On the lightweight and selective introduction of reflective capabilities in applications. Design Patterns. Design and Implementation of a Meta Architecture for Java. Using Meta-Objects to Support Optimisation in the Apertos Operating System Sun Microsystems The Art of the Metaobject Protocol. Concepts and Experiments in Computational Reflection. A Semantics of Introspection in a Reflective Prototype-Based Language Towards a Theory of Reflective Programming Languages. http://www. "Prototype-based Programming" M-LISP: A Representation-Independant Dialect of LISP with Reduction Seman- tics SMALLTALK: a Reflective Language. Reflection and Semantics in LISP. The Mystery of the Tower Revealed: A Non-Reflective Description of the Reflective Tower --TR --CTR Gregory T. Sullivan, Aspect-oriented programming using reflection and metaobject protocols, Communications of the ACM, v.44 n.10, p.95-97, Oct. 2001 Manuel Clavel , Jos Meseguer , Miguel Palomino, Reflection in membership equational logic, many-sorted equational logic, Horn logic with equality, and rewriting logic, Theoretical Computer Science, v.373 n.1-2, p.70-91, March, 2007

reflection;language implementation;OO languages;program transformation

609228

A Per Model of Secure Information Flow in Sequential Programs.

This paper proposes an extensional semantics-based formal specification of secure information-flow properties in sequential programs based on representing degrees of security by partial equivalence relations (pers). The specification clarifies and unifies a number of specific correctness arguments in the literature and connections to other forms of program analysis. The approach is inspired by (and in the deterministic case equivalent to) the use of partial equivalence relations in specifying binding-time analysis, and is thus able to specify security properties of higher-order functions and partially confidential data. We also show how the per approach can handle nondeterminism for a first-order language, by using powerdomain semantics and show how probabilistic security properties can be formalised by using probabilistic powerdomain semantics. We illustrate the usefulness of the compositional nature of the security specifications by presenting a straightforward correctness proof for a simple type-based security analysis.

Introduction 1.1 Motivation You have received a program from an untrusted source. Let us call it company M. M promises to help you to optimise your personal financial invest- ments, information about which you have stored in a database on your home computer. The software limited time), under the condition that you permit a log-file containing a summary of your usage of the program to be automatically emailed back to the developers of the program (who claim they wish to determine the most commonly used features of their tool). Is such a program safe to use? The program must be allowed access to your personal investment information, and is allowed to send information, via the log-file, back to M. But how can you be sure that M is not obtaining your sensitive private financial information by cunningly encoding it in the contents of the innocent-looking log-file? This is an example of the problem of determining that the program has secure information flow. Information about your sensitive "high-security" data should not be able to propagate to the "low-security" output (the log- file). Traditional methods of access control are of limited use here since the program has legitimate access to the database. This paper proposes an extensional semantics-based formal specification of secure information-flow properties in sequential programs based on representing degrees of security by partial equiv- Department of Computer Science, Chalmers University of Technology and the University of G-oteborg, fandrei,daveg@cs.chalmers.se alence relations 1 . The specification clarifies and unifies a number of specific correctness arguments in the literature, and connections to other forms of program analysis. The approach is inspired by (and equivalent to) the use of partial equivalence relations in specifying binding-time analysis [HS91], and is thus able to specify security properties of higher order functions and "partially confidential data" (e.g. one's financial database could be deemed to be partially confidential if the number of entries is not deemed to be confidential even though the entries themselves are). We show how the approach can be extended to handle nondeter- minism, and illustrate how the various choices of powerdomain semantics affects the kinds of security properties that can be expressed, ranging from termination-insensitive properties (corresponding to the use of the Hoare (partial correctness) pow- erdomain) to probabilistic security properties, obtained when one uses a probabilistic powerdomain. 1.2 Background The study of information flow in the context of systems with multiple levels of confidentiality was pioneered by Denning [Den76, DD77] in an extension of Bell and LaPadula's early work [BL76]. Den- ning's approach is to apply a static analysis suitable for inclusion into a compiler. The basic idea is that security levels are represented as a lattice (for example the two point lattice PublicDomain - TopSecret ). The aim of the static analysis is to ensure that information from inputs, variables or processes of a given security level only flows to out- Equivalence relation is symmetric and transitive but not necessarily reflexive puts, variables or processes which have been assigned a higher or equal security level. Semantic Foundations of Information Flow Analysis In order to verify a program analysis or a specific proof a program's security one must have a formal specification of what constitutes secure information flow. The value of a semantics-based specification for secure information flow is that it contributes significantly to the reliability of and the confidence in such activities, and can be used in the systematic design of such analyses. Many approaches to Denning-style analyses (including the original articles) contain a fair degree of formalism but arguably are lacking a rigorous soundness proof. Volpano et al [VSI96] claim to give the first satisfactory treatment of soundness of Den- ning's analysis. Such a claim rests on the dissatisfaction with soundness arguments based on an instrumented operational e.g., [-rb95] or denotational semantics e.g., [MS92], or on "axiomatic" approaches which define security in terms of a program logic [AR80] without any models to relate the logic to the semantics of the programming lan- guage. The problem here is that an "instrumented semantics" or a "security logic" is just a definition, not subject to any further mathematical justifica- tion. McLean points out [McL90] in a related discussion about the (non language-specific) Bell and LaPadula model: One problem is that LaPadula security properties] constitute a possible implementation of security, rather than an abstract specification of what all secure systems must satisfy. By concerning themselves with particular controls over files inside the computer, rather than limiting themselves to the relation between input and output, they make it harder to reason about the re- This criticism points to more abstract, extensional notions of soundness, based on, for example, the idea of noninterference introduced in [GM82]. Semantics-based models of Information Flow The problem of secure information flow, or "non- interference" is now quite mature, and very many specifications exist in the literature - see [McL94] for a tutorial overview. Many approaches have been phrased in terms of abstract, and sometimes rather ad hoc models of computation. Only more recently have attempts been made to rephrase and compare various security conditions in terms of well-known semantic models, e.g. the use of labelled transition systems and bisimulation semantics in [FG45]. In this paper we consider the problem of information-flow properties of sequential systems, and use the framework of denotational semantics as our formal model of compu- tation. Along the way we consider some relations to specific static analyses, such as the Security Lambda Calculus [HR98] and an alternative semantic condition for secure information flow proposed by Leino and Joshi [LJ98]. 1.3 Overview The rest of the paper is organised as follows. Section 2 shows how the per-based condition for soundness of binding times analysis is also a model of secure information flow. We show how this provides insight into the treatment of higher-order functions and structured data. Section 3 shows how the approach can be adapted to the setting of a nondeterministic imperative language by appropriate use of a powerdomain-based semantics. We show how the choice of powerdomain (upper, lower or convex) affects the nature of the security condition Section 4 focuses on an alternative semantic specification due to Leino and Joshi. Modulo some technicalities we show that Leino's condition - and a family of similar conditions - are in agreement with, and can be represented using our form of specification. Section 5 considers the problem of preventing unwanted probabilistic information flows in programs. We show how this can be solved in the same framework by utilising a probabilistic semantics based on the probabilistic powerdomain [JP89]. Per Model of Information Flow In this section we introduce the way that partial equivalence relations (pers) can be used to model dependencies in programs. The basic idea comes from Hunts use of pers to model and construct abstract interpretations for strictness properties in higher-order functional programs [Hun90, Hun91], and in particular its use to model dependencies in binding-time analysis [HS91]. Related ideas already occur in the denotational formulation of live- variable analysis [Nie90]. 2.1 Binding Time Analysis as Dependency Analysis Given a description of the parameters in a program that will be known at partial evaluation time (called the static arguments), a binding-time analysis must determine which parts of the program are dependent solely on these known parts (and therefore also known at partial evaluation time). The safety condition for binding time analysis must ensure that there is no dependency between the dynamic (i.e., non-static) arguments and the parts of the program that are deemed to be static. Viewed in this way, binding time analysis is purely an analysis of dependencies. 2 Dependencies in Security In the security field, the property of absence of unwanted dependencies is often called noninterference, after [GM82]. Many problems in security come down to forms of dependency analysis. For example, in the case of confidentiality, the aim is to show that the outputs of a program which are deemed to be of low confidentiality do not have any dependence 2 Unfortunately, from the perspective of a partial evalua- tor, BTA is not purely a matter of dependencies; in [HS95] it was shown that the pure dependency models of [Lau89] and [HS91] are not adequate to ensure the safety of partial evaluation. on inputs of a higher degree of confidentiality. In the case of integrity (trust ), one must ensure that the value of some trusted data does not depend on some untrusted source. Some intuitions about information flow Let us consider a program modelled as a function from some input domain to an output domain. Now consider the following simple functions mapping inputs to outputs: snd : D \Theta E ! E for some sets (or domains) D and E, and shift and test, functions in N \Theta N ! N, defined by Now suppose that (h; l) is a pair where h is some high security information, and l is low, "public do- main", information. Without knowing about what the actual values h and l might be, we know about the result of applying function snd will be a low value, and, in the case that we have a pair of num- bers, the result of applying shift will be a pair with a high second component and a low first component Note that the function test does not enjoy the same security property that snd does, since although it produces a value which is constructed from purely low-security components, the actual value is dependent on the first component of the input. This is what is known as an indirect information flow [Den76]. It is rather natural to think of these properties as "security types": high \Theta low ! low high \Theta low ! high \Theta low test : high \Theta low ! high But what notion of "type", and what interpretation of "high" and "low" can formalise these more intuitive type statements? Interpreting types as sets of values is not adequate to model "high" and "low". To track degrees of dependence between inputs and outputs we need a more dynamic view of a type as a degree of variation. We must vary (parts of) the input and observe which (parts of) the output vary. For the application to confidentiality we want to determine if there is possible information leakage from a high level input to the parts of an output which are intended to be visible to a low security observer. We can detect this by observing whether the "low" parts of the output vary in any way as we vary the high input. The simple properties of the functions snd and shift described above can be be captured formally by the following formulae: Indeed, this kind of formula forms the core of the correctness arguments for the security analyses proposed by Volpano and Smith et al [VSI96, SV98], and also for the extensional correctness proofs in core of the Slam-calculus [HR98]. High and Low as Equivalence Relations We show how we can interpret "security types" in general as partial equivalence relations. We will interpret high(for values in D) as the equivalence relation All D , and low as the relation all x All D x 0 (3) x For a function f binary relations iff For binary relations P , Q we define the relation Now the security property of snd described by (1) can be captured by and (2) is given by 2.2 From Equivalence Relations to Pers We have seen how the equivalence relations All and may be used to describe security "properties" high and low . It turns out that these are exactly the same as the interpretations given to the notions "dynamic" and "static" given in [HS91]. This means that the binding-time analysis for a higher-order functional language can also be read as a security information-flow analysis. This connection between security and binding time analysis is already e.g. [Thi97] for a comparison of a particular security type system and a particular binding-time analysis, and [DRH95] which shows how the incorporation of indirect information flows from Dennings security analysis can improve binding time analyses). It is worth highlighting a few of the pertinent ideas from [HS91]. Beginning with the equivalence relations All and Id to describe high and low respectively, there are two important extensions to the basic idea in order to handle structured data types and higher-order functions. Both of these ideas are handled by the analysis of [HS91] which rather straightforwardly extends Launchbury's projection-based binding-time analysis [Lau89] to higher types. To some extent [HS91] anticipates the treatment of partially-secure data types in the SLam calculus [HR98], and the use of logical relations in their proof of noninterference. For structured data it is useful to have more refined notions of security than just high and low ; we would like to be able to model various degrees of security. For example, we may have a list of records containing name-password pairs. Assuming passwords are considered high , we might like to express the fact that although the whole list cannot be considered low , it can be considered as a (low \Theta high)list. Constructing equivalence relations which represent such properties is straightforward see [HS91] for examples (which are adapted directly from Launchbury's work), and [Hun91] for a more general treatment of finite lattices of "bind- ing times" for recursive types. To represent security properties of higher-order functions we use a less restricted class of relations than the equivalence relations. A partial equivalence relation (per) on a set D is a binary relation on D which is symmetric and transitive. If P is such a per let jP j denote the domain of P , given by Note that the domain and range of a per P are both equal to jP j (so for any x; y 2 D, if x P y then x P x and y P y), and that the restriction of P to jP j is an equivalence relation. Clearly, an equivalence relation is just a per which is reflexive equivalence relations over various applicative structures have been used to construct models of the polymorphic lambda calculus (see, for example, [AP90]). As far as we are aware, the first use of pers in static program analysis is that presented in [Hun90]. For a given set D let Per(D) denote the partial equivalence relations over D. Per(D) is a meet semi-lattice, with meets given by set-intersection, and top element All . Given pers P 2 Per(D) and Q 2 Per(E), we may construct a new per (D - E) 2 Per(D ! E) defined by: If P is a per, we will write x : P to mean x 2 jP j. This notation and the above definition of P - Q are consistent with the notation used previously, since now Note that even if P and Q are both total (i.e., equivalence relations), P - Q may be partial. A simple example is All - we know that given a high input, f returns a low output. A constant function -x:42 has this prop- erty, but clearly not all functions satisfy this. 2.3 Observations on Strictness and Termination Properties We are interested in the security properties of functions which are the denotations of programs (in a Scott-style denotational semantics), and so there are some termination issues which should address. The formulation of security properties given above is sensitive to termination. Consider, for example, the following Clearly, if the argument is high then the result must be high. Now consider the security properties of the function g ffi f where g the constant function We might like to consider that g has low . However, if function application is considered to be strict (as in ML) then g is not in Hence the function g ffi f does not have security type high ! low (in our semantic interpretation). This is correct, since on termination of an application of this function, the low observer will have learned that the value of the high argument was non-zero. The specific security analysis of e.g. Smith and Volpano [SV98] is termination sensitive - and this is enforced by a rather sweeping measure: no branching condition may involve a high variable. On the other hand, the type system of the SLam calculus [HR98] is not termination sensitive in gen- eral. This is due to the fact that it is based on a call-by-value semantics, and indeed the composition could be considered to have a security type corresponding to "high ! low ". The correctness proof for noninterference carefully avoids saying anything about nonterminating executions. What is perhaps worth noting here is that had they chosen a non-strict semantics for application then the same type-system would yield termination sensitive security properties! So we might say that lazy programs are intrinsically more secure than strict ones. This phenomenon is closely related to properties of parametrically polymorphic functions [Rey83] 3 . From the type of a polymorphic function one can predict certain properties about its behaviour - the so-called "free theorems" of the type [Wad89]. However, in a strict language one must add an additional condition in order that the theorems hold: the functions must be bottom-reflecting 3 Not forgetting that the use of Pers in static analysis was inspired, in part, by Abadi and Plotkin's Per model of polymorphic types [AP90] ?). The same side condition can be added to make the e.g. the type system of the Slam-calculus termination-sensitive. To make this observation precise we introduce one further constructor for pers. If R 2 Per(D) then we will also let R denote the corresponding per on D? without explicit injection of elements from D into elements in D? . We will write R? to denote the relation in Per(D? ) which naturally extends R by ? R ?. Now we can be more precise about the properties of g under a strict (call-by-value) interpreta- which expresses that g is a constant function, modulo strictness. More informatively we can say that that which expresses that g is a non-bottom constant function. It is straightforward to express per properties in a subtype system of compositional rules (although we don't claim that such a a system would be in any sense complete). Pleasantly, all the expected subtyping rules are sound when types are interpreted as pers and the subtyping relation is interpreted as subset inclusion of relations. For the abstract interpretation presented in [HS91] this has already been undertaken by e.g. Jensen [Jen92] and Hankin and Le M'etayer [HL94]. 3 Nondeterministic Information Flow In this section we show how the per model of security can be extended to describe nondeterministic computations. We see nondeterminism as an important feature as it arises naturally when considering the semantics of a concurrent language (al- though the treatment of a concurrent language remains outside the scope of the present paper.) In order to focus on the essence of the problem we consider a very simplified setting - the analysis of commands in some simple imperative language containing a nondeterministic choice operator. We assume that there is some discrete (i.e., unordered) domain St of states (which might be viewed as finite maps from variables to discrete values, or simply just a tuple of values). 3.1 Secure Commands in a Deterministic Setting In the deterministic setting we can take the denotation of a command C, written JCK, to be a function in [St ? we mean the set of strict and continuous maps between domains D? and E? . Note that we could equally well take the set of all functions in St ! St ? , which is isomorphic Now suppose that the state is just a simple partition into a high-security half and a low-security half, so the set of states is the product St high \Theta St low . Then we might define a command C to be secure if no information from the high part of the state can leak into the low part: C is secure Which is equivalent to saying that JCK : (All \Theta since we only consider strict functions. Note that this does not imply that JCK terminates, but what it does imply is that the termination behaviour is not influenced by the values of the high part of the state. It is easy to see that the sequential composition of secure commands is a secure command, since firstly, the denotation of the sequential composition of commands is just the function-composition of denotations, and sec- ondly, in general for functions and R 2 Per(F ) it is easy to verify the soundness of the inference rule: 3.2 Powerdomain Semantics for Nonde- A standard approach to giving meaning to a non-deterministic language - for example Dijkstra's guarded command language - is to interpret a command as a mapping which yields a set of re- sults. However, when defining an ordering on the results in order to obtain a domain, there is a tension between the internal order of State ? and the subset order of the powerset. This is resolved by considering a suitable powerdomain structure [Plo76, Smy78]. The idea is to define a preoreder on the finitely generated subsets 4 of S? in terms of the order on their elements. By quotienting "equiv- alent sets" one obtains a partial ordering, each depending on a different view of what sets of values should be considered equivalent. Consider the following three programs (an example from [Plo81]) In the "Hoare" or partial correctness interpretation the first two programs are considered to be equal since, ignoring nontermination, they yield the same sets of outcomes. This view motivates the definition of the Hoare or lower powerdomain, P L [St ? ]. In the "Smyth" or total correctness interpreta- tion, programs (2) and (3) are considered equal (equally bad!) because neither of them can guarantee an outcome. In the general case this view motivates the Smyth or upper powerdomain, P U [St ? [Smy78]. In the "Egli-Milner" interpretation (leading to the convex or Plotkin powerdomain in the general case) all three programs are considered to have distinct denotations. The three powerdomains are built from a domain D by starting with the finitely generated (f.g.) subsets of D? (those non-empty subsets which are either finite, or contain ?), and a preorder on these sets. Quotienting the f.g. sets using the associated equivalence relation yields the corresponding do- main. We give each construction in turn, and give an idea about the corresponding discrete powerdo- main P[St ? ]. ffl Upper powerdomain The upper ordering on f.g. sets u, v, is given by In this case the induced discrete powerdomain is isomorphic to the set of finite non-empty subsets of St together with St ? itself, ordered by superset inclusion. y. Here the induced discrete powerdomain P L [St ? ] is isomorphic to 4 If a set is infinite then it must contain ?. the powerset of St ordered by subset inclusion. This means that the domain [St is isomorphic to all subsets of St \Theta St - i.e. the relational semantics. and u - L v. This is also known as the Egli- Milner ordering. The resulting powerdomain is isomorphic to the f.g. subsets of A few basic properties and definitions on pow- erdomains will be needed. For each powerdo- main constructor P[\Gamma] define the order-preserving which takes each element a 2 D into (the powerdomain equivalence class of) the singleton set fag. For each function there exits a unique extension of f , denoted f where f which is the unique mapping such that In the particular setting of the denotations of commands, it is worth noting that JC 1 K would be given by: K: 3.3 Pers on Powerdomains Give one of the discrete powerdomains, P[St ? ], we will need a "logical" way to lift a per P to a per in Per(P[St ? ]). Definition 1 For each R 2 Per(D? ) and each choice of power domain P[\Gamma], let P[R] denote the relation on P[D? ] given by It is easy to check that P[R] is a per, and in particular that P[Id D? Henceforth we shall restrict our attention to the semantics of simple commands, and hence the three discrete powerdomains P[St ? ]. Proposition 1 For any f 2 [St any R, S 2 Per(St ? ), From this it easily follows that the following inference rule is sound: 3.4 The Security Condition We will investigate the implications of the security condition under each of the powerdomain in- terpretations. Let us suppose that, as before the state is partitioned into a high part and a low part: high \Theta St low . With respect to a particular choice of powerdomain let the security "type" high \Theta low ! high \Theta low denote the property In this case we say that C is secure. Now we explore the implications of this definition on each of the possible choices of powerdomain: 1. In the lower powerdomain, the security condition describes in a weak sense termination- insensitive information flow. For example, the program (h is the high part of the state) is considered secure under this interpretation but the termination behaviours is influenced by h (it can fail to terminate only when 2. In the upper powerdomain nontermination is considered catastrophic. This interpretation seems completely unsuitable for security unless one only considers programs which are "totally correct" - i.e. which must terminate on their intended domain. Otherwise, a possible nonterminating computation path will mask any other insecure behaviours a term might exhibit. This means that for any program C, the program C 8 loop is secure! 3. The convex powerdomain gives the appropriate generalisation of the deterministic case in the sense that it is termination sensitive, and does not have the shortcomings of the upper powerdomain interpretation. 4 Relation to an Equational Characterisation In this section we relate the Per-based security condition to a proposal by Leino and Joshi [LJ98]. Following their approach, assume for simplicity we have programs with just two variables: h and l of high and low secrecy respectively. Assume that the state is simple a pair, where h refers to the first projection and l is the second projection. In [LJ98] the security condition for a program C is defined by where "=" stands for semantic equality (the style of semantic specification is left unfixed), and HH is the program that "assigns to h arbitrary values" - aka "Havoc on H". We will refer to this equation as the equational security condition. Intuitively, the equation says that we cannot learn anything about the initial values of the high variables by variation of the low security variables. The postfix occurrences of HH on each side mean that we are only interested in the final value of l. The prefix HH on the left-hand side means that the two programs are equal if the final value of l does not depend on the initial value of h. In relating the equational security condition to pers we must first decide upon the denotation of HH . Here we run into some potential problems since it is necessary in [LJ98] that HH always terminates, but nevertheless exhibits unbounded nondeterminism. Although this appears to pose no problems in [LJ98] (in fact it goes without mention), to handle this we would need to work with non-continuous semantics, and powerdo- mains for unbounded nondeterminism. Instead, we side-step the issue by assuming that the domain of h, St high , is finite. Secondly we must find common ground for our semantic interpretation. It is not the style of semantic definition that is important (viz. operational vs denotational vs axiomatic), but rather the interpretation of nondeterminism itself. Leino and Joshi consider two styles of interpretation with different treatments of nondeterminism: the relational interpretation (corresponding to the choice of the lower powerdomain) and the wlp/wp seman- tics, which corresponds to the convex powerdomain interpretation. Leino and Joshi claim that considering a relational semantics, the security condition is equivalent to a notion used elsewhere in the lit- erature. As we shall see, the relational semantics interpretation of the security condition allows programs to leak information via their termination be- haviour, so this observation is not entirely correct. 4.1 Equational Security and Projection Analysis A first observation is that the the equational security condition is strikingly similar to the well-known form of static analysis for functional programs known as projection analysis [WH87]. Given a function f , a projection analysis aims to find projections (continuous lower closure operators on the domain) ff and fi such that For (generalised) strictness analysis and dead- variable analysis, one is given fi, and ff is to be determined; for binding time analysis [Lau89] it is a forwards analysis problem: given ff one must determine some fi. For strict functions (e.g., the denotations of commands) projection analysis is not so readily ap- plicable. However, in the convex powerdomain HH is rather projection-like, since it effectively hides all information about the high variable; in fact it is an embedding (an upper closure operator) so the connection is rather close. 4.2 The equational security condition is subsumed by the per security conditio Hunt [Hun90] showed that projection properties of the form fi could be expressed naturally as a per property of the for equivalence relations derived from ff and fi by relating elements which get mapped to the same point by the corresponding projection. Using the same idea we can show that the per- based security condition subsumes the equation specification in a similar manner. We will establish the following: Theorem 1. For any command C iff high \Theta low ! high \Theta low : The idea will be to associate an equivalence relation to the function HH . More generally, for any command C let ker(C), the kernel of C, denote the relation on P[St ? Define the extension of ker(C) by A ker (C) B () JCK Recall the per interpretation of the type signature of C. high \Theta low ! high \Theta low Observe that (All \Theta Id) since for any The proof of the theorem is based on this observation and on the following two facts: Let us first prove the latter fact by proving a more general statement similar to Proposition 3.1.5 from [Hun91] (the correspondence between projections and per-analysis). Note that we do not use the specifics of the convex powerdomain semantics here, so the proof is valid for any of the three choices of powerdomain. Theorem 2. Let us say that a command B is idempotent iff JB; JBK. For any commands C and D, and any idempotent command B Proof. and JB; C; DKs 1 . Thus s 1 implies that , since JBKs 1 , and by idempotence, which implies Corollary. Since JHH K is idempotent we can conclude that It remains to establish the first fact. Theorem 3. P[All \Theta Proof. Suppose A P[All \Theta need to show JHH K bottom-reflecting ((): For the other direction assume JHH K JHH K B. Thus, the equational and per security conditions in this simple case are equivalent. 5 A Probabilistic Security Conditio There are still some weaknesses in the security condition when interpreted in the convex powerdo- main when it comes to the consideration of non-deterministic programs. In the usual terminology of information flow, we have considered possibilistic information flows. The probabilistic nature of an implementation may allow probabilistic information flows for "secure" programs. Consider the program This program is secure in the convex powerdomain interpretation since regardless of the value of h, the value of l can be any value in the range 99g. But with a reasonably fair implementation of the nondeterministic choice and of the randomised as- signment, it is clear that a few runs of the program, for a fixed input value of h, could yield a rather clear indiction of its value by observing only the possible final values of l: - from which we might reasonably conclude that the value of h was 2. To counter this problem we consider probabilistic powerdomains [JP89] which allows the probabilistic nature of choice to be reflected in the semantics of programs, and hence enables us to capture the fact that varying the value of h causes a change in the probability distribution of values of l. In the "possibilistic" setting we had the denotation of a command C to be a function in [St ? ! In the probabilistic case, given an input to C not only we keep track of possible outputs, but also of probabilities at which they appear. Thus, we need to consider a domain E [St ? ] of distributions or evaluations over St ? . The denotation of C is going to be a function in [St Let us first present the general construction of probabilistic powerdomain. If D is an inductively complete partial order (ipo for short, it has lubs of directed subsets, and it is countable), then the probabilistic powerdomain of evaluations E [D] is built as follows. An evaluation on D, -, is a function from D to [0; 1] such that E [D] to be the set of evaluations on D partially ordered by - iff 8d 6= ?: -d -d. Define the point-mass evaluation j D (x) for an ae 0; otherwise. A series of theorems from [JP89] proves that is an ipo with directed lubs defined pointwise, and with a least element To lift a function f : we define the extension of f by The structure (E [D]; j D (x); ) is a Kleisli triple. and thus we have a canonical way of composing the probabilistic semantics of any two given programs. are such. Then the lifted composition (g ffi f) can be computed by one of the Kleisli triple laws as g ffif . The next step towards the security condition is to define how pers work on probabilistic powerdo- mains. Recall the definition for pers on powerdo- mains introduced in section 3. If R 2 Rel(D) then for To lift pers to E [D] we need to consider a definition which takes into consideration the whole of each R-equivalence class in one go. Define the per relation E [R] on E [D] for - 2 where [d] R , as usual, stands for the R\Gammaequivalence class which contains d. Naturally, - As an example, consider E[(All \Theta Id) ? ] Two distributions - and - in (All \Theta Id) ? ! [0; 1] are equal if the probability of any given low value l in the left-hand distribution, given by h -(h; l), is equal to the probability in the right hand distribution, namely To make sure we have built a stronger security model, let us prove that the Egly-Milner power- domain security condition follows from the probabilistic powerdomain one. In other words, Theorem 4. Suppose R and S are equivalence relations on D. For any command C we have R -E [S] implies JCK C Proof. For some a; b 2 D let deduce a R b =) -e. What we need to prove is a R b =) JCK C a P C [S] JCK C b. So, assume a R b and let us show that Take any x 2 JCK C a. Observe, that -x ? 0, since if x is a possible output of program C run on data a, then the probability of getting this output must be greater then 0. Therefore, -e. There must exist a y 2 [x] S such that -y ? 0. Thus, y is a possible output of program C run on data b, and y 2 JCK C b. y 2 [x] S implies Let us derive the probabilistic powerdomain security condition for the case of two variables h and l and domain . C is secure iff l l -e low :-? & h 02St high So, a command C is secure iff and h2St high h2St high for any i l h and o l . Intuitively the equation means that if you vary i h the distribution of low variables (the sums provide "forgetting" the highs) does not change. Let us introduce probabilistic powerdomain semantics definitions for some language constructs. Here we omit the E-subscripts to mean the probabilistic semantics. Given two programs such that JC 1 the composition of two program semantics is defined by The semantics of the uniformly distributed nondeterministic choice is defined by Consult [JP89] to get a full account on how to define the semantics of other language constructs. Example. Recall the program Let us check the security condition on it. Take The left-hand side is whereas the right-hand side is So, the security condition does not hold and the program must be rejected. We recently became aware of probabilistic security type-system due to Volpano and Smith [VS98] with a soundness proof based on a probabilistic operational semantics. Although the security condition that they use in their correctness argument is not not directly comparable - due to the fact that they consider parallel deterministic threads and a non-compositional semantics - we can easily turn their examples into nondeterministic sequential programs with the same probabilistic behaviours, and it seems (although we have not checked all of the details) that their examples can all be verified using our security condition. 6 Conclusions There are many possible extensions to the ideas we have sketched, and also many limitations. We consider a few: Multi-level security There is no problem with handling lattices of security levels rather than the simple high-low distinction. But one cannot expect to assign any intrinsic semantic meaning to such lattices of security levels, since they represent a "social phenomenon" which is external to the programming language semantics. In the presence of multiple security levels one must simply formulate conditions for security by considering information flows between levels in a pairwise fashion (although of course a specific static analysis is able to do something much more efficient). Downgrading and Trusting There are operations which are natural to consider but which cannot be modelled in an obvious way in an extensional framework. One such operation is the downgrading of information from high to low without losing information - for example representing the secure encryption of high level information. This seems impossible since an encryption operation does not lose information about a value and yet should have type high ! low - but the only functions of type high ! low are the constant func- tions. An analogous problem arises with -rbaek and Palsberg's trust primitive if we try to use pers to model their integrity analysis [-P97]. Operational Semantics We are not particularly married to the denotational perspective on programming language semantics. There are also interesting operational formulations of pers on a higher-order language, based on partial- bisimulations. We hope to investigate these further Constructing Program Analyses Although the model seems useful to compare other formali- further work is needed to show that it can assist in the systematic design of program analyses. Concurrency Handling nondeterminism can be viewed as the main stepping stone to formulating a language-based security condition for concurrent languages, but this remains a topic for further work. --R A per model of polymorphism and recursive types. An axiomatic approach to information flow in programs. Secure Computer Systems: Unified Exposition and Multics Interpretation. A lattice model of secure information flow. Semantic foundations of binding-time analysis for imperative programs A classification of security properties for process algebra. Security policies and security models. The SLam calculus: Programming with secrecy and integrity. Binding Time Analysis: A New PERspec- tive A semantic model of binding times for safe partial evaluation. PERs generalise projections for strictness analysis. Abstract Interpretation of Functional Languages: From Theory to Practice. Abstract Interpretation in Logical Form. A probabilistic powerdomain of evaluations. Projection Factorisations in Partial Evaluation. A semantic approach to secure information flow. The specification and modeling of computer security. models. A security flow control algorithm and its denotational semantics correctness proof. A powerdomain con- struction "Pisa Notes" Types, abstraction and parametric polymorphism. Journal of Computer and Systems Sciences Secure information flow in a multi-threaded imperative language University of Nottingham (submitted for publication) Probabilistic noninterference in a concurrent language. A sound type system for secure flow anal- ysis Theorems for free. Projections for strictness analysis. --TR --CTR Kyung Goo Doh , Seung Cheol Shin, Detection of information leak by data flow analysis, ACM SIGPLAN Notices, v.37 n.8, August 2002 Pablo Giambiagi , Mads Dam, On the secure implementation of security protocols, Science of Computer Programming, v.50 n.1-3, p.73-99, March 2004 Stephen Tse , Steve Zdancewic, Translating dependency into parametricity, ACM SIGPLAN Notices, v.39 n.9, September 2004 Sebastian Hunt , David Sands, On flow-sensitive security types, ACM SIGPLAN Notices, v.41 n.1, p.79-90, January 2006 Nick Benton , Peter Buchlovsky, Semantics of an effect analysis for exceptions, Proceedings of the 2007 ACM SIGPLAN international workshop on Types in languages design and implementation, January 16-16, 2007, Nice, Nice, France Aslan Askarov , Andrei Sabelfeld, Localized delimited release: combining the what and where dimensions of information release, Proceedings of the 2007 workshop on Programming languages and analysis for security, June 14-14, 2007, San Diego, California, USA Anindya Banerjee , Roberto Giacobazzi , Isabella Mastroeni, What You Lose is What You Leak: Information Leakage in Declassification Policies, Electronic Notes in Theoretical Computer Science (ENTCS), 173, p.47-66, April, 2007 Gilles Barthe , Leonor Prensa Nieto, Formally verifying information flow type systems for concurrent and thread systems, Proceedings of the 2004 ACM workshop on Formal methods in security engineering, October 29-29, 2004, Washington DC, USA Nick Benton, Simple relational correctness proofs for static analyses and program transformations, ACM SIGPLAN Notices, v.39 n.1, p.14-25, January 2004 Roberto Giacobazzi , Isabella Mastroeni, Abstract non-interference: parameterizing non-interference by abstract interpretation, ACM SIGPLAN Notices, v.39 n.1, p.186-197, January 2004 Mads Dam, Decidability and proof systems for language-based noninterference relations, ACM SIGPLAN Notices, v.41 n.1, p.67-78, January 2006 Torben Amtoft , Anindya Banerjee, A logic for information flow analysis with an application to forward slicing of simple imperative programs, Science of Computer Programming, v.64 n.1, p.3-28, January, 2007 Steve Zdancewic , Andrew C. Myers, Secure Information Flow via Linear Continuations, Higher-Order and Symbolic Computation, v.15 n.2-3, p.209-234, September 2002 Heiko Mantel , Andrei Sabelfeld, A unifying approach to the security of distributed and multi-threaded programs, Journal of Computer Security, v.11 n.4, p.615-676, 01/01/2004 Andrew C. Myers , Andrei Sabelfeld , Steve Zdancewic, Enforcing robust declassification and qualified robustness, Journal of Computer Security, v.14 n.2, p.157-196, January 2006 Anindya Banerjee , David A. Naumann, Stack-based access control and secure information flow, Journal of Functional Programming, v.15 n.2, p.131-177, March 2005

powerdomains;semantics;noninterference;confidentiality;partial equivalence relations;security;probabilistic covert channels

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Formalization and Analysis of Class Loading in Java.

Since Java security relies on the type-safety of the JVM, many formal approaches have been taken in order to prove the soundness of the JVM. This paper presents a new formalization of the JVM and proves its soundness. It is the first model to employ dynamic linking and bytecode verification to analyze the loading constraint scheme of Java2. The key concept required for proving the soundness of the new model is augmented value typing, which is defined from ordinary value typing combined with the loading constraint scheme. In proving the soundness of the model, it is shown that there are some problems inside the current reference implementation of the JVM with respect to our model. We also analyze the findClass scheme, newly introduced in Java2. The same analysis also shows why applets cannot exploit the type-spoofing vulnerability reported by Saraswat, which led to the introduction of the loading constraint scheme.

Introduction Unlike its predecessor, C++, Java supports platform-independent bytecodes which are compiled from source programs written in Java, sent over the network as mobile codes, and executed by the Java Virtual Machine running within a local application such as a Web browser. The JVM links bytecodes sent over the network in a type-safe manner, whose meaning is as follows. It is guaranteed that linking bytecodes, with their type information consistent with themselves, does not destroy the consistency of the JVM state, which has its own type information. The following requirement should also be satised. If the current JVM state is consistent with its own type information, then at its next execution step, it is still consistent. Java is in this way a type-safe language, if these two requirements are satised. Note that if the above consistency is broken, then the JVM incorrectly interprets the contents pointed to by its inner pointer references. The type safety of the JVM guarantees the memory safety of the JVM, and thus it plays the primary role in the Java Security. In order to show the type safety property of the JVM, Akihiko Tozawa and Masami Hagiya are at the Graduate School of Science, University of Tokyo, Japan. Any suggestions or comments to this article are appreciated. Please send email: fmiles, hagiyag@is.s.u-tokyo.ac.jp a number of studies have been made so far. They are brie y summaried in the next subsection. This paper gives another formalization of the JVM and proves its soundness. Bytecodes running inside the JVM are structures into classes, each of which is separatedly loaded and linked by the JVM. Objects called class loaders have the responsibility of loading and linking a class. By supporting a variety of class loaders, the JVM achieves the exibility of class loading. However, this exibility has been causing problems with respect to the above mentioned type-safety of Java. The rst contribution of our work is that of developing the new model of the JVM. It is described in Section 2 of this paper in detail. Our model has several improvements over those dened in the previous studies. First of all, it includes class loaders. Java class loaders are instances of a user-dened class, whose primary function is a map from class names to class objects. They are, however, closely related to how the JVM internally builds its class environments. In other words, the class environments of the JVM are not statically given, but they are lazily built by dynamic evaluations of class loaders. Our model gives dynamic environments which represent internal heaps of the JVM. It seems to be the best approach to model the lazy linking semantics of the JVM by its class loaders. Since we do not consider any static language from which these environments can be constructed, our model is a practical one that faithfully re ects JVM implementations. In 1998, Sun Microsystems released the version 1.2 of Java Development Kit (JDK1.2), which is based on the newly proposed design principle called Java2. This comes with the rewritten specication of the JVM, The Java Virtual Machine Specication (2nd Edition) [9]. The most important feature introduced to the new specication is the loading constraint scheme, which is originally introduced by Liang and Bracha [8]. The scheme is the x of Saraswat's problem [13] related to the unique design of Java class loaders. Our formalization also includes this scheme. The second contribution of our work is that we have found three problems inside the current o-cial implementation of the JVM with respect to this scheme. These problems are not trivial ones because they require a careful analysis of the scheme, which is done through our work on the formalization. They are described in Section 3. The third contribution of our work is that of proving the soundness of our model, given in Section 2. The key notion required for the soundness proof is the augmented value typing, which is dened from the ordinary value typing combined with the loading constraint scheme. This new typing is shown to be consistent with the subtype relation under the existence of the loading constraint scheme. Note that this consistency is crucial for the soundness of the model, and the problems we found inside the current o-cial implementation of the JVM are due to its violation against the consistency. Another new feature of Java2 is the findClass implementation of the class loading. Technically speaking, both the loading constraint scheme and the findClass scheme give constraints to the dynamic class loading of Java. Such constraints restrict what will happen in the future, so that they further complicate the lazy linking semantics of Java. For example, the problem we found in the loading constraint scheme is sensitive to the timing of introducing con- straints. This fact re ects the subtleties of the semantics of such constraints. The analysis of the findClass scheme in Section 4 is the fourth contribution. By the analysis, we can also answer the old question why applets cannot cause Saraswat's problem [13]. 1.1 Related Work Since the Java security deeply relies on the JVM and its type-safety, giving formal models to the JVM is recently one of the major research issues in network security. Stata and Abadi [15] gave the JVM model including its bytecode verication. They try to grasp the JVM as a type system, and its bytecode verication as typing rules, whereby the correctness of the JVM is proved in the form of a soundness theorem. Our work lays its concept for modeling the bytecode verier on Stata and Abadi's work. In other words, we extended their model to cope with class loaders. Freund and Mitchell worked on a specic problem related to object initialization [4]. Their work is also based on Stata and Abadi's. Qian rst succeeded in modeling a large part of the JVM [12]. Formalizations covering a wider range of the JVM (but without proofs) were given by Goldberg [5] and Jensen, et al. [6]. In particular, Jensen, et al. dealt with Saraswat's bug by modeling both class loading and operational semantics. Saraswat [13] himself gave formal explanations to his problem. Dean also gave a formal model of class loading [2]. However, it cannot explain the type spoong problems. The lazy semantics of class loaders has not yet been fully modeled as far as we know. We think that it has something to do with the study of modularity as in the work by Cardelli [1]. Machine verications are also being applied to prove the type soundness of Java [10][11]. 1.2 Organization The rest of the paper is structured as follows. Section 2 gives a basic formal model of the JVM and also a soundness proof of the model. Section 3 mainly explains problems with an implementation of the JVM. In Section 4, we give further discussions related to the findClass scheme. We formally answer the old question why applets cannot cause Saraswat's problem [13] in Section 4. We also have discovered a new method to implement the loading constraint scheme e-ciently. Model and its Soundness The formal model of the JVM presented in this paper has the following new features compared to those in previous studies. Modeling lazy class loading of the JVM. Dening loading constraints. Modifying the value typing statement. Giving a rigorous denition of an environment, which is enough to model all the above features. In particular, lazy class loading of the JVM was rst modeled in this study. Note also that for the last feature, it is necessary to dene the well-formedness type type option := None Some of type Env := f !(StringString listT Class ) cl (iface : String list) class I : Instruction list lvars : TValue list list type VerifyRecord := f lvars : String list stack : String list invokevirtual of String String String areturn :::: Figure 1: Type denitions of an environment, which in turn is dened in terms of the well-formedness of its various components. In this section, we rst explain several basic denitions. Section 2.4 gives the denition of the loading constraint scheme. Our main soundness theorem is described in Section 2.5. 2.1 Environments 2.1.1 Denition of Environments An environment, which represents an internal heap of the JVM, is the most basic data structure in our formalization. Of course, every component of such a heap cannot be modeled. Instead we give a well-dened set of components to which some mathematical examinations are possible. It is represented by type Env. See the denition of types in Figure 1. An environment, i.e., an element of Env, is a record consisting of four subsets, T Class , T Loader , TValue and T Method , of the set, Loc, of locations, and ve maps, C, W , R. The meaning of each subset should be clear from its index. As we will note at the end of this section, type Env can be considered as a dependent type, since each map and R) has an arrow type, which is constructed by using the subsets (T Class , T Loader , TValue and T Method ) as types. Members M, C and V map locations to appropriate denitions, i.e., they return denitions stored in E from their references. Member W represents class loading, while R represents method resolution. These two maps are explained later. We use the notation, E:m, to extract member m from environment E. Figure 2: Abbreviations inside judgments For any type , option denotes the type composed of constructor None and values of wrapped with constructor Some. For example, the value of E:W(l)(n) is either None or of the form Some c. For any type , list denotes the type of nite lists of . For list, the i-th element of x is denotes by x[i], and its length by len(x). Note that in Classdef(T ), T denotes the tuple consisting of T Class , T Loader , TValue and T Method . We dene T Class and Classdef(T ) separately, so that there are two kinds of identity between classes. The same situation happens for values and methods. We usually identify an object by its reference. A state, i.e., an element of State(T ), is either exeption , which denotes an error state, or a record consisting of four elements, pc, method, lvars and stack. Elements of VerifyRecord give types to the local variables and the stack of a state. They are used in the bytecode verication. The type of instructions, Instruction, is not dened completely. For the sake of this study, we only assume the invokevirtual instruction and the areturn instruction. Type theories use a -type for representing such types as Env. Tuple T is considered as an element of the index set, Index, dened as follows. We use a reference, i.e., an element of Loc, to access a component of an en- vironment. Of course, there are dierent ways to dene and identify these components. In some studies a class is identied by a tuple of a class name and a class loader, because there should not be two classes with the same name and loader inside a single JVM. Our model gives a low level abstraction which re ects practical JVM im- plementations. In any such implementation, the class identity property is only one of the various properties implicitly satised by the internal structure of the heap. In practice, there is no two classes with the same name and loader only when class loaders are synchronized. If two threads start loading the same class by the same loader at the same time, the property would easily be violated. Here we are not going into how to implement the JVM which always satises such properties. Instead we have extracted a set of properties, which are essential for the soundness of the model, called the well-formedness of the en- vironment, and given an assumption that any modication or update of the environment must preserve this well-formedness. The well-formedness of the environment, denoted by wf (E), is composed of the following properties. The property related to the classes inside an environment. The property related to the methods inside an environment. The property that states that all methods are statically veriable, i.e., have gone through the bytecode verication. The Bridge Safety property related to the loading constraint scheme. The above properties are denoted by wf class(E), wf method (E), veried(E) and bridge safe(E), and are described in Sections 2.2.4, 2.3.5, 2.5.1 and 2.4.3, respectively. 2.2 Objects, Classes and Loaders This section gives basic denitions of objects, classes and class loaders. Each class of Java is actually an object instance of java.lang.Class and each class loader is of java.lang.ClassLoader, but unlike other objects, they play essential roles in the JVM architecture, which refers to them automatically and implicitly. Therefore, it is appropriate to dene objects, classes and class loaders separately. 2.2.1 Classes Let us rst explain the denition of classes in Figure 1. A class denition, i.e., an element of Classdef(T ), is a tuple consisting of a class name, a direct super class name and a class loader. As we have noted, if we assume c is a class reference, then the components of c's denition are simply written inside a judgement as c:name, c:super and c:cl, respectively. The two other components of c, the method dispatch table, c:mt, and the implemented interfaces of the class, c:iface, will be discussed later. 2.2.2 Class Loaders Each Java object instantiating java.lang.ClassLoader has a private member declared as a hashtable which maps names of classes to class objects themselves. We model this mapping with E:W as follows. Denition 2.1 (Subtyping) Denition 2.2 (Widening) cl Denition 2.3 (Normal Value Typing) Denition 2.4 (Well-Formedness) Figure 3: Predicates related to classes The class name resolution n l (= E:W(l)(n)) results in Some c if name n is associated with a certain class c by class loader l, and None if not. We also say that n is resolved into c by l. In the above sense, each class loader has its own name space. Therefore, a class loader is also called a context. Each class c has its dening class loader c:cl, whereby every class name resolution related to c is done. In this sense, c:cl is the context of c. Throughout this paper, we use the word context and class loader as synonyms. As Figure shows, each value uniquely refers to a class, and thus to a class loader. We say that value v is created in context v:class:cl or method m is executed in context m:cc:cl. 2.2.3 Subtyping Subtyping between classes is dened in Figure 3. In our model, predicate sub represents the relation. For class c, if its parent class name, c:super, is resolved into c 0 in context c:cl, then c is a direct subtype of c 0 . The (indirect) subtyping is dened as the re exive and transitive closure of the direct subtyping. 2.2.4 Well-Formedness of Classes An actual class loader of the JVM resolve classes in one of the following two ways. When a class loader, l, denes class c whose name is n, it sets c:cl to l, and resolves n into c. When a class loader, l, delegates class loading to another class loader, it resolves n into c whose c:cl is not equal to l. Actual class loaders therefore satisfy the following statement, the well-formedness of classes dened in Figure 1, since each class should be once dened by the above rst process which sets n l to that class. For each c in E:T class , c:name is resolved into c itself in context c:cl. Note that this statement immediately implies the class identity property described in Section 2.1.3. 2.2.5 Object Values and States If a JVM state, : State(E:T ), is not an error state, exception , it has the following components. :method denotes the method that the JVM is now processing. :pc, the program counter, points to a certain instruction of :method:I. Both :lvars and :stack are represented by a list of values which stores contents of the local variables and the local stack, respectively. While both the state and the environment represent internal data structures of the JVM, their transitions, ! and , respectively, are distinct and independent In our model, all values found inside :lvars or :stack are objects. An object value, v, is a record with just one member, v:class, which represents a type (class) of that value 1 . In practice, a Java object value may also have a reference to an array of values, i.e., its eld values. There is no di-culty to extend our model to represent elds, since it already has a value heap. We can just add v:eld, which represents a list of value references. Our model omits elds of Java, because the get=puteld instructions are in large part similar to invokevirtual and also easier to handle. 2.2.6 Predicates with Respect to Subtyping and Value Typing We introduce two predicates with respect to subtyping, one of which represents the widening conversion, and the other represents the typing judgment of values. Their formalizations are given in Figure 3. The widening conversion denes a quasi-order 2 between class names. The order between two names is dened only if both of them are resolved in cl , except for the case that two names are equal. The ordering itself follows the subclass relation between the classes into which names are resolved. The normal value typing is a typing judgment which denotes that the type of value v is a subtype of n in context cl . It is also dened only if n is already resolved in cl . Note that both predicates depend on a context, cl . 2.3 Methods This section simply describes the specication of invokevirtual [9], while the practical method invocation of the JVM involves further di-culties and problems. Such remaining problems will be explained in Section 3. 2.3.1 Instructions As dened in Figure 1, our JVM model includes only two instructions, invokevirtual and areturn. Instruction invokevirtual is the most fundamental but the most non- 1 Java has a null value which represents an uninitialized object value. We do not model this for simplicity. 2 According to our denition, both sub and cl satisfy transitivity but not anti-symmetry. Of course, they should do so in practice. trivial instruction of the JVM. It is used with three arguments as follows. The above judgement means that the current JVM state, , is now about to process invokevirtual, which calls a certain instance method. String name is the name of the method, and string list desc is the descriptor representing the type signature of the method, where are argument types (i.e., class names) of the method, and desc[0] is its return value type. String classname is the name of the symbolically referenced class [9]. Instruction areturn is one of the return instructions of the JVM, which returns to the caller, holding one object value as a return value. 2.3.2 Method Invocation Execution of invokevirtual consists of the following three processes. The method resolution looks up a method according to the three arguments of invokevirtual. The method found by this process is called the symbolically referenced method and denoted by SRmethod. The method selection looks up a method accessible from the class of the object on top of the local stack. The method invocation calls the method found by the method selection process. Assume that invokevirtual is inside a method whose class is c and whose context is l(= c:cl). The method resolution process is modeled as follows. We just look up the method, SRmethod, which is already registered to the en- vironment, E:R (cf. Section 2.1), by the given name and descriptor, name and desc. SRclass, the symbolically referenced class, is the result of resolving classname by l. Remember that (name; desc; SRclass) l abbreviates E:R(l)(name; desc; SRclass). Its further denition is given at the end of this section and also in Section 3. Practically, the JVM remembers whether a symbolic reference is already re-solved or not, so that it never resolves the same reference twice 3 . Our model re ects this behavior. The method selection process will be dened in Section 3. In this section, we just assume that the process results in a unique value, denoted by method. When we write a judgement having E and in its left hand side, we assume that SRmethod and method inside the right hand side denote the symbolically referenced method and the selected method, respectively. ' SRmethod method Although the dention of the value, method , is left blank until we discuss the implementation later, we can examine the soundness of the JVM, i.e., the 3 This is implemented by the internal class representation which attaches a ag to each resolvable constant pool entry, which remembers whether the entry is resolved or not. It also remembers the result of resolution. correctness of the new loading constraint scheme (cf. Section 2.4), provided that we assume Lemmas 1, 1' and 1", described in Sections 2.3.4 and 2.3.5. The value, SRmethod , is identical to the value of the same variable, appearing in the predicate, invv OK (E; ), dened below. Denition 2.5 (invokevirtual OK) where Some The predicate means that the JVM is now about to execute invokevirtual, and the method resolution has normally succeeded, i.e., it has not raised exceptions. Unlike what is specied about the method resolution, once invv OK (E; ) holds, i.e., the method resolution succeeds, our JVM model ensures that the method selection and method invocation will also succeed. This is because method is always dened in such a case. Formally, name, desc, classname, SRclass, SRmethod and objectref in the denition of invv OK (E; ) should be existentially quantied. However, in a situation where we assume invv OK (E; ), we refer to them in a judgement as if they leak out from the denition of invv OK (E; ), i.e., those variables are assumed to satisfy invv OK (E; ). 2.3.3 Operational Semantics Denition 2.6 describes the state transition semantics of invokevirtual, which is dened in accordance with the specication of the three processes of invokevir- tual. (Note that @ denotes list concatenation.) It rst tries to resolve its method. If it cannot be resolved, If the method can be resolved, it generates a new state, in , whose method is the selected one. Its local variables should store the proper value of values of arguments, each of which is originally stored in :stack. There must be a transition of the form where out should point at instruction areturn, the return instruction of the JVM. This means that the invoked method can be safely executed. At last, out :stack has a return value on its top, which will be pushed onto denotes the transitive closure of the one-step transition, !. Transitions for other instructions are not given in this paper (See [15] or [7]). exception . Denition 2.6 (Operational Semantics of invokevirtual) exeception in :lvars = [objectref ]@arg ; in in out :method:I[ out [retval [retval]@rest ; State < references to method :cc Figure 4: Subtyping relations related to invokevirtual 2.3.4 Invocation Correctness With respect to SRmethod and method , the following lemma should be con- sidered, while the lemma itself will be examined in Section 3. In the following several sections, the lemma will be assumed to hold. (Correctness of invokevirtual) Let E be an environment, and be a JVM state. method overrides SRmethod In other words, Lemma 1 (Correctness of invokevirtual) guarantees that if the JVM is about to execute invokevirtual, then if the JVM is in a safe state, a method whose dening class is a superclass of the class of objectref should be invoked. Furthermore, if the JVM is in a safe state, then a method that overrides the symbolically referenced method should be invoked. The second premise of the lemma, the subtyping relation between objectref :class and SRclass , is a property that should be invariantly satised by a safe execution of the JVM (cf. Section 2.4.1). The rst consequence of the lemma, the subtyping relation between objectref :class and method :cc, is derived from the method selection algorithm described in Section 3.1.1. See Figure 4. The method overriding relation, overrides, has not yet been dened for certain reasons. One reason is that although the term, override, is often used in the specication of the JVM, its accurate meaning is not dened, so that its interpretation depends on implementations. In this paper, we will dene predicate overrides in Section 3.1.1 in conjunction with predicate select. The predicate, overrides, should satisfy the following conditions. LEMMA 1' The implementation of Sun's JDK1.2 is inconsistent with Lemma 1' and exposes a new aw, as we will explain in Section 3. The relation, , between environments is dened in Section 2.5.3. 2.3.5 Well-Formedness of Methods The following well-formedness property of methods is required mainly for the discussions in Section 4. selects SRmethod Predicate selects models the method selection process and its denition can be found in Denition 3.1 in Section 3.1.1. class RT { . RR new RR(); . } Figure 5: Saraswat's bug code With predicate selects, the equivalence of descriptors should also hold. LEMMA 1" 2.4 Bridge Safety Bridge safety of the JVM is a notion originally introduced by Saraswat in his report on type-spoong in JDK1.1 [13]. He insisted that applet loaders never suers from his bug because they never break this property. Sheng and Bracha have devised a x of Saraswat's bug, implemented in JDK1.2 [8][9], which forces the JVM to check the bridge safety at runtime. 2.4.1 Type-Spoong in JDK1.1 See the source code in Figure 5, which revealed the bug of type-spoong in JDK1.1. The code itself has nothing suspicious, but if there exist two contexts l 1 and l 2 (the code itself runs under l 1 ) and delegation of class loading RR is dened, the invocation of r.speakup() would result in a serious violation of the type system of the JVM. Expression r.speakUp() is complied into the following instruction of the JVM. The above ()V is dierent from our notation of method descriptors. It represents a method which takes no argument and returns nothing. With respect to this invokevirtual, its objectref , the value on top of the local stack, is equal to the value of r, which comes from context l 2 via the invocation of rr.getR(). l 2 The symbolically referenced class, SRclass, is the class into which the current context l 1 resolves R (the third string of invokevirtual). l 1 The code causes a problem when R l 1 is dierent from R l 2 . For any invokevirtual to be correctly executed, the subtyping relation Denition 2.7 (Loading Constraints) (method overriding) (method resolution) cl m:desc[i] m:cc:cl (re exivity) cl n cl cl n cl cl 00 cl n cl 00 cl n cl 0 cl 0 n cl is absolutely necessary (of course, objectref :class = SRclass su-ces). Recall that this subtyping relation is the second premise of Lemma 1. In other words, method in Section 2.3 is completely unrelated to SRmethod . More accurately, the bug results in the incompatibility of the dispatch tables of two classes, by applying a method index obtained from SRclass onto a completely unrelated method table of objectref :class. JDK1.1 would thus either invoke method with argument values of incompatible types or just core- dump. The method dispatch table, which is excluded from Sun's specication and therefore from our model, is an implementation technique of the method selection algorithm (cf. Section 3.1.3). 2.4.2 Loading Constraints Go back to the example of Section 2.4.1. Suppose that the JVM has already noticed at the invocation of rr.getR() that the method brought a value of type R from context l 2 to l 1 . In this case we can check beforehand if this ow of the value is acceptable or not. We may simply try to check it as follows. l 2 But this attempt should fail, since before the method resolution of speakUp(), R l 1 cannot be evaluated, i.e., under the environment, E ' R we must consider an alternative, i.e., the loading constraint scheme. In this case, the new scheme introduces the following loading constraint. R If the JVM remembers the above constraint, R l 1 is not allowed to be resolved into a dierent class from R l 2 in the method resolution process. How the JVM can notice the possibility of a value ow is not easy to un- derstand. It is related to Lemma 3 (Existence of Constraints) described later in this section. Here, we only give rules to introduce the loading constraints. For any method m, which overrides m 0 , the following constraint is introduced with respect to each class name n appearing in their descriptor. For any method resolution at which nds SRmethod , the following constraint is introduced (for each class name n appearing in the descriptor of SRmethod ). SRmethod :cl Relation n is dened to be transitive, re exive and symmetric, so that it is an equivalence relation. It should also be the minimal relation among all satisfying the above conditions. Denition 2.7 formally describes the predicate by inductive denition. As shown later in Section 2.5, the relation is -invariant, i.e., relation between the environments subsumes n . This enables the JVM to incrementally construct the relation, n , as it resolves a method reference of invokevirtual, or it links a class whose method overrides another method. This is the reason why the loading constraint scheme is light-weight, and why it has been actually adopted among many other solutions. The history of Saraswat's problem and its solutions are described in [8][13]. 2.4.3 Bridge Safety Predicate Denition 2.8 denes predicate bridge safe(E) as follows. For environment E, any class loader in E has made no resolutions of classes which contradict with loading constraints existing inside E. This predicate forbids any environment modication, i.e., class loading and method linking, which violates constraints. It is the same as what JDK1.2 is doing. 2.4.4 Augmented Value Typing The augmented value typing, cl n, is dened as follows. There is another context cl 0 , in which v has type n in the normal sense, cl 0 n, and there also exists the following constraint. cl n cl 0 The intent of the denition may not obvious. Roughly, it means that value v itself has been created, i.e., instantiated from n, in context cl 0 , and it has been transfered to the current context, cl. This predicate was called dynamic conformity in our previous paper [16]. As we noted in the introduction, it is crucial for proving the type soundness of the loading constraint scheme, and its denition is one of the contributions of this study. With respect to this predicate, the following lemma is important. cl -invariance of Augmented Typing) Denition 2.8 (Bridge Safety) bridge safe(E) def cl cl n cl cl cl 0 cl 0 Denition 2.9 (Augmented Value Typing) cl cl n cl cl 0 n list cl (ns : String list) cl ns[i] Let E be an environment, n and n 0 be class names, and cl be any class loader, bridge cl n 0 cl cl Proof The following fact is proved by examining Def.2.3 Def.2.9 and Def.2.8. bridge cl n =) (n cl 6= None =) cl n) Assume the rst line of the lemma. From Def.2.2 Def.2.3 and Def.2.1, we have cl n =) v :: cl By applying E ' cl n cl to Def.2.9, the lemma is proved. The lemma states that the augmented typing is invariantly satised against the widening conversion described in Section 2.2.6. This is an important statement which relates runtime typing to static typing, i.e., the bytecode verication. The proof of the lemma requires that the denition of the widening conversion, cl n 0 , force additional loadings of n cl and n 0cl in case of n 6= n 0 . We have found two inconsistencies between our denition of cl and the bytecode verication of JDK1.2. By exploiting each of these, we can still entirely escape additional checks newly imposed. See Section 3 or [16] for detail. Note that it is relatively easy to show that the augmented typing is also -invariant, which will be used later in the soundness proof. 2.4.5 Constraint Existence Lemma 3 (Existence of Constraints) states that if the JVM is in a safe state and about to execute invokevirtual, there already exist loading constraints for each class name appearing in the descriptor of invokevirtual between the current context and the context of the invoked method. We now assume Lemma 1 (Correctness of invokevirtual). Note that E ' cl desc[] cl 0 abbreviates cl desc[i] cl 0 . Denition 2.10 (Verication Rules) (rule of (rule of invokevirtual)6 6 6 6 6 6 4 (i):stack[0] m:cl cname (rule of areturn)4 LEMMA 3 (Existence of Constraints) Let E be an environment, and be a JVM state. method:desc[] method :cl Assume the rst line of the lemma. From Def.2.7(method overriding and symmetry) and Lemma 1, we have ' SRmethod :cl method:desc[] method :cl: From Def.2.7(method rosolution) and Def.2.5, we have Applying Lemma 1' and Def.2.7(transitivity) yields the lemma. 2.5 Soundness This section follows the framework of the soundness proof by Stata and Abadi [15]. One dierence is in the introduction of environments. Another is in the treatment of invokevirtual, the instruction that does not exist in their model. 2.5.1 Bytecode Verication One uniqueness underlying the language design of Java is found in its bytecode verication. The idea is to guarantee the runtime well-typedness by the static verication, which allows minimum type checks at runtime. See Denition 2.10. Type VerifyRecord represents an imaginary store for the static class name information about local variables and local stacks. A map, , stores an element of VerifyRecord for each instruction of :method. The bytecode verication is a problem of nding such that is consistent with the veried method, m. Denition 2.11 (k-transition) exeception out kn Denition 2.12 (well-typedness) veries :method :cl (:pc):lvars :cl (:pc):stack It should be veried that contents of satisfy all m:cl relations imposed by the verication rules. In our model, the consistency of is represented by predicate veries. Predicate veried(E) states that each method in E has already been veried. The well-typedness predicate, E ' wt(), means that state is safe inside environment E. Both :lvars and :stack should have their values typed by the class names recorded in which veries :method. State exception representing an error state is always well-typed. Note that :lvars :: :cl (:pc):lvars abbreviates 8i :cl (:pc):lvars[i]. The interesting work by Yelland [17] implements the bytecode verier based on the type inference of Haskell. 2.5.2 Soundness Theorem Here is our main theorem. THEOREM (Soundness) be environments, and and 0 be states. This theorem states that any execution step of the JVM preserves the well-typedness While the intuitive meaning of the invariant is not so trivial, it is su-cient to guarantee the correctness of invokevirtual. If then we have which is the second premise of Lemma 1. Any instruction that is not modeled in this paper also preserves this invariant, whereby the correct behavior of the instruction should be guaranteed. The theorem can be divided into two lemmas in the following sections. Denition 2.13 (Sub-Environment relation) String list E:T Class : 2.5.3 Soundness of Environment Updates Denition 2.13 describes how environments can be updated as the JVM dynamically links objects, classes, etc. It denes the sub-environment relation, between two environments. Following is the rst lemma needed to prove the main soundness theorem. This lemma states that any modication of environments preserves the well- typedness, and its proof is as follows. Remember that we are assuming Lemmas 1, 1' and 1". We rst assume guarantees that any components of E are compatible with those of E 0 . Therefore, from Def.2.13 and Def.2.1, we have We also have, from Def.2.13 and Def.2.3, cl cl n: We say that relations sub and :: cl are -invariant. Furthermore, Lemma 1' and Def.2.13 guarantee the following facts, respectively. They imply that relation n is also -invariant, because it is minimal among those that satisfy Denition 2.7. Therefore, the augmented cl , is also -invariant, because it is dened in Def.2.9 as follows. cl cl n cl 0 From Def.2.12, the lemma is proved. 2.5.4 Soundness of State Transitions Following is the second lemma for the soundness theorem. This lemma states that every state transition under a xed environment, E, preserves well-typedness. Before examining the lemma, we redene state transitions as described in which adds depth k to each transition. Obviously, we have Note that 0 denotes a state transition to exception or a transition by an instruction other than invokevirtual, which we do not dene in this paper. The proof of the lemma is as follows: We rst note that the following fact holds. cl cl n cl 0 cl 0 It expresses the n -invariance of the augmented typing and can be easily proved by examining Def.2.9 and Def.2.7(transitivity). By the transitivity of ! , the following fact is a su-cient condition of the lemma. This is proved by induction on k. The base case of the induction is as follows. Its proof is omitted here. See [15] or [7] for detail. The remaining subgoal is to show the case of k+1 from the induction hypothesis. Assume the rst line of [iii]. From the only state transition rule that denes where in and out are dened as in Def.2.6. From the induction hypothesis, we have holds here, there exists that satises the following condition. :cl ([:pc]):stack Applying Def.2.10(rule of invokevirtual) and Lemma 2, we obtain where x[0; ::; n 1] denotes a sublist, [x[0]; ::; x[n 1]], and n is the length of desc. Assuming invv OK (E; ), we have Lemma 1 can be used here. It implies method:cl method :cc:name together with wf class(E). On the other hand, Lemmas 1, 1' and 1" imply together with wf method(E). We can then use Lemma 3 and [i] to obtain method:cl method :desc[1; ::; n 1]: Therefore, Lemma 2 and Def.2.10(rule of imply method:cl 0 (0):lvars for some 0 that satises 0 veries method(= in :method) and Finally, Def.2.6 implies in :lvars = :stack[0; ::; n and therefore From the induction hypothesis, [iv], we have of areturn) and Lemma 2 lead to method:cl method :desc[0]: Similarly as above, we can use Lemma 3 and [i] to obtain Finally, Def.2.10(rule of invokevirtual), Lemma 2, Def.2.6 and Def.2.12 imply Denition 3.1 (Predicate selects) 3 Analysis of Implementations Another main topic of this paper is the analysis of Sun's JVM implementation. The latter half of this section describes several aws that we have found in JDK1.2 with respect to Saraswat's bug and the loading constraint scheme. Before describing the aws, let us examine Lemma 1 (Correctness of invokevirtual). 3.1 (Correctness of invokevirtual) 3.1.1 Method Selection In Section 2.3, we dened the method invocation processes only partially. In this section, Denition 3.1 formalizes the recursive procedure employed in method resolution and selection. We dene it as predicate selects. A pair of a key and a class, (key ; c), selects method m whose key is equal to key , if m is found by a lookup over the subtype tree from c to its superclasses. In the denition, key denotes the pair, (name; desc), and the key of m, m:key, denotes a pair (m:name; m:desc). If ( ; desc; c) selects m, then we have the following The latter condition is what Lemma 1" requires. The predicate is also - invariant from its denition. In the denition of selects, the selection process terminates once a method, m, is found inside c, no matter whether another method, m 0 , can be found inside some superclass of c or not. However, in such a case, i.e., if m makes m 0 invisible from c, we say that For any class c and method m such that (m:key; c) selects m, if a direct superclass of c selects a dierent method, m 0 , with its key the same as m:key, we dene Predicate overrides is then dened as the re exive and transitive closure of predicate overrides - . If we faithfully follow Sun's specication, SRmethod and method are simply dened as follows. The method resolution process will nd SRmethod which satises the following selects SRmethod The method selection process will nd method which satises the following. selects method As we see below, the above two denitions derive Lemma 1 (Correctness of invokevirtual) by simply applying the premise of the lemma, objectref :class sub SRclass, to the dention of overrides. By the denitions and Lemma 1", method :desc holds no matter whether a method overriding exists or not. This is because the specication explicitly uses desc to select a method. However, the implementation diers from the speciciation in that it employs the method dispatch table of a class to select a method. As long as the above descriptor equivalence holds, the JVM never falls into an error state or coredumps, even though state may be badly-typed. This is the reason why it is di-cult for the specication to explain the type-spoong problem. Figure 6 describes the problem graphically. 3.1.2 Proof of Lemma 1 In this paper, we only show Lemma 1 with respect to the above specication, though it must be and can be proved for existing implementations. We rst show the existence of method , since in our model the method selection process should always succeed. Assume invv OK (E; ), which implies E; ' (SRmethod :key; SRclass) selects SRmethod . Also assume from which Denition 3.1 implies the existence of method that sat- ises As we already noted, relation select holds between objectref :class and method :cc, so we obtain the rst consequence of the lemma. The second consequence is the following. method overrides SRmethod This can be proved by induction on sub , because overrides is the reexive and transitive closure of overrides - . If objectref :class sub SRclass holds, then either method = SRmethod or method overrides - SRmethod . 3.1.3 Method Dispatch Table A method dispatch table is a list of methods which satises the following conditions A method is selected by a class, c, i the method is inside the method dispatch table of the class, c:mt. An overriding method has the same index as the overridden one, i.e., for any class c and any superclass c 0 of c, and any index i less than the length of c 0 :mt. The JVM can incrementally build such a table structure for each class that satises the above conditions, by referring to the table of its direct superclass, which has already been built. Here, suppose that we already have a method dispatch table for each class. Since SRclass:mt is a collection of methods selected by SRclass , the method resolution process of JDK1.2 searches inside SRclass :mt. In the method resolution, threfore, this dierence between the specication and the implementation by method dispatch tables cannot be seen from outside. On the other hand, as to the method selection process, the implementation remarkably diers from the specication. It is an O(1)-time procedure rather than a recursive procedure represented by predicate selects. A method selection process is as follows. There is index i which satises because SRclass selects SRmethod . The selected method, method , is The above selection process is also sound, as it also satises Lemma 1 (Correct- ness of invokevirtual). For the proof of the lemma, the sub relation, should necessarily be used. In the implementation, the equivalence of the descriptors, desc = method :desc, and even the existence of method , depend on this sub relation. Without such a relation, index i of SRclass :mt has no meaning inside method :mt. By exploiting the inconsistency between desc and method :desc, one can falsify an integer value as an object value, and vice vasa. If method does not exist, the JVM coredumps [13]. 3.2 Bytecode Verier and Loading Constraints 3.2.1 Problem with Respect to the Widening Conversion Now we go back to Saraswat's bug code (Figure 5) in Section 2.4. Suppose that a modication described below is applied to the original bug code. original code modied code6 4 new R'() R is not load by L 1 R L1 inside method resolution loaded JDK1:2 . :r :: L1 R R L 2 not loaded VIOLATION VIOLATION JDK1:1 # spec: . & Executing r.speakUp() Some In any case where method even if method is unexpected, execution continues. It is posible that method :desc 6= desc, or in a worse case, method may not exist. Type confusion or coredump. Figure Type spoong chart class RR { public R getR() { return new R'(); // originally ``new R()'' Assume that L 2 loads class RR, and L 1 loads class RT, which invokes rr.getR() and also r.speakUp(). It is also assumed that L 1 resolves R dierently from L 2 . The method invocation, r = rr.getR(), which calls a method inside class RR, returns a value not of type R but of type R'. Inside context L 2 , R' is a subtype of R. Recall that to check this widening conversion, R L2 should have already been resolved. Therefore, the constraint, R will be checked at the invocation of r.speakUp(), when R L1 is resolved. Assume, conversely, that R L2 has not been resolved yet. Even though class R L1 has been resolved, and though the above constraint indeed exists, the constraint will never be checked. Figure 6 describes what happens in such a case. In fact, JDK1.2 sometimes does not resolve R L2 , although its resolution is a role of the bytecode verier. 3.2.2 Two Inconsistencies We have found two inconsistencies in the bytecode verier of JDK1.2 against our model, each of which still enables the type spoong. These inconsistencies are as follows. Some widening conversion, n cl n 0 , is not correctly checked and n 0 is not resolved. System classes are not veried at runtime. These bugs are brie y explained in [16] with example codes. Let us emphasize the signicance of our work. The problem is concerned with the augmented typing, which is an alternative of the naturally dened typing. Since the bytecode verication seemed unrelated with Saraswat's bug, the designer of JDK1.2 did not modify the bytecode verication of JDK1.1. However, the JVM requires Lemma 2 ( cl -invariance) as well, which relates the well-typedness invariant to the bytecode verication based on the widening conversion. It is our model that makes all of these points clear and visible. 3.3 Interfaces and Loading Constraints In addition to the above problem with its bytecode verier, one more aw inside the JDK1.2 implementation was found during the analysis of the invokeinterface instruction, which has been excluded from our model so far. 3.3.1 The invokeinterface Instruction In order to discuss the problem of invokeinterface, we extend our model to be able to deal with the interfaces of Java. The only thing to do is to allow a class to have multiple parent classes. Throughout this section, a class, c, has its list of the names of implemented interfaces, c:iface. We redene the subtyping relation, c sub c 0 , i.e., c is a direct subtype of We introduce a new predicate, is class(c), which is true i c is a pure class, i.e., c is not an interface. Though any class has no more than a single direct supertype in the previous sections, this fact was not used throughout our soundness proof, which thus requires no further changes. The invokeinterface instruction, which similarly resolves and selects a method, will have exactly the same semantics as invokevirtual. The only modication must be considered with predicate selects. If pure class c itself does not declare a method with the required key, the predicate should select a method not inside implementing interfaces of c, but inside some pure superclass of c. Therefore, we should redene the second rule of Denition 3.1 as follows. class(c) =) is class(c 3.3.2 Problem with Respect to Constraint Existence A question may be raised about the code in Figure 7. Is D.getR overriding I.getR, or not? If we accept our denition of overrides, the answer is yes, even though there is no subtyping relation between the declaring classes of two methods. Since class D selects D.getR for the key representing R getR() and one of its direct supertype, I, also selects I.getR, we conclude that D.getR overrides I.getR from the denition of override, In fact, the following code invokes a method inside class D safely. interface I { R getR(); class D { R getR() { . } class C extends D implements I {} Figure 7: Problem of invokeinterface I Since JDK1.2 fails to recognize such complex method overriding relations, the code in Figure 7 is what brings another problem of the loading constraint scheme. The overriding relation between D.getR and I.getR is not recognized, so that there may be no constraint between the loaders that dene D and I with respect to the name of the return type, R. 4 The findClass Scheme 4.1 Formalization of the findClass Scheme There is also one more new feature in Java2, i.e., the implementation of class loadings by findClass. Java2 recommends to implement class loaders by findClass rather than the old loadClass method, though loadClass is also accepted for backward compatibility. The findClass scheme denes a tree structure among class loaders. The delegation of class loadings should follow this tree structure. In the old version of Java, Applet loaders are implemented in a similar manner as the findClass scheme. It is also known that applets never cause the Saraswat's problem (though it has never been proved completely). This leads to the following question. Can the findClass scheme replace the loading constraint scheme? Our last theorem, Theorem (Trusted Environments), which will be proved in Section 4.2, gives a negative answer. Even if we follow this findClass scheme, class loadings may violate constraints unless no delegations are allowed other than those to system loaders As the above theorem states, and also as Saraswat has correctly mentioned, applet loaders are safe since they only delegate to system loaders. However, as [8] describes in its rst half, class loaders are recently increasing their variety of applications. Consider an applet loader which delegates to another applet loader. Although such loaders seem to be safe at a glance, the theorem correctly states that they possibly violate constraints. 4.1.1 Parent Loaders The following denitions incorporate the findClass scheme into our model. Denition 4.1 (Parent Loader) true l vP :Parentdef l l vP l 0 The direct parent loader of each loader. If l does not have a parent loader, we assume P l. The inductively dened predicate, l vP l 0 , denotes that l is one of the (indirect) parents of l 0 . Denition 4.2 (Correct Delegation to Parent Loaders) The denition formalizes the following delegation strategy of Java2. If a loader, l, has its direct parent loader, l 0 , any class loading by l is rst delegated to l 0 . The class is loaded by l itself only if l 0 cannot resolve the class name. Otherwise, l will return the same class as l 0 returns. In the above denition, =) expresses that there is no loading delegation that does not follow the parent loader relation. (= expresses that two loaders in the parent loader relation correctly delegate class loadings according to the above strategy. The second condition implies that if c:cl vP l, then l has already resolved c:name. Therefore, if wf parent(E; P ) holds, each loader in E is considered to have resolved all the classes it can do so. In other words, E represents such an environment that all possible class loadings have been completed. Ordinary environments are thus considered as sub-environments of such environments. We therefore use dient font, such as E , for those environments. Note that wf parent(E ; P ) implies wf class(E). 4.1.2 Parent Environment We dene a relation between envirnoments that represents an extension of environments through delegations to parent loaders. Denition 4.3 (Parent Environment) The second line of the denition of E that a method loaded by a loader in E should already exist in E . The third line states that delegations are allowed only to a direct parent loader in E . 4.2 Trusted Environments Theorem (Trusted Environments) states that even if we follow the ndClass scheme, the JVM never violates constraints only if all parent loaders are system loaders. We st dene system environments. Denition 4.4 (System Environment) The above predicate denes a condition that system environments should satisfy. That is, when all system classes are loaded, any class name appearing in any method descriptor should have been resolved. THEOREM (Trusted Environments) The following proposition is satised if and only if i 1, called a trusted environment if it satises the consequence of the theorem. The theorem states that E 1 is a trusted environment and E 2 is not. For example, applet loaders are inside a certain E 1 and applets never violate constraints. Prior to the proof of the theorem, we introduce an additional relation. The l 0 , means there is a one-step constraint between l and l 0 . We allow (method resolution), (method overriding), and (symmetry) in Deni- tion 2.7. If we ignore the symmetry, there are two cases in which E ' l n l 0 holds. Assume wf method(E) in Section 2.3.5 for now. There is a method resolution where l Remember that l should be the context which resolves classname. This implies that if wf parent(E; P ), then SRclass:cl vP l. Note that also holds from wf method (E) and Def.3.1. Generally, if the one-step subtyping relation, holds, then for any P such that wf parent(E; P ), we have from Def.2.1 and Def.4.2. Consequently, the subtyping relation, sub , also implies vP . Therefore, if wf parent(E; P ), then holds. Otherwise, we have the overrides relation, In this case, according to Section 3, there are a class, c, which selects m, and a superclass of c, which selects m 0 . The fact implies the following one. Therefore, From Def.4.1, if a loader has two dierent (indirect or direct) parent load- ers, then one is a (direct) parent of the other, so we have In both cases, we have the following result with respect to n l 0 =) l vP l 0 _ l 0 vP l ::[i] Furthermore, we can easily show the following in both cases. Note that l <P l 0 abbreviates l vP l From these results, Theorem (Trusted Environment) can be proved as follows: (proof) Assume that there are given E Assume that l and l 0 l 0 and also l <P l 0 . Def.4.3 implies that l should be a system loader. Therefore [ii] implies: Note that a method dened by some system loader should be a system method (cf. Def.4.3). From Def.4.4, we have Our next goal is to prove the following fact. This is proved by induction on the derivation of the loading con- straint, l n l 0 . Suppose that the following one-step constraint is appended to above l n l 0 to generate l n l 00 by the transitivity. l 00 We assume by the induction hypothesis and show l 0 vP l 00 _ l 00 The left case of the disjunction, l 0 vP l 00 , is easy since vP is transi- tive. Therefore, assume l 00 <P l 0 . In this case, [iii] says that there exists c 0 in E 1 :T Class which satises Together with Def.4.2, this leads to the induction hypothesis implies implies Some c; so l 00 is derived from E 1 ' c 0 :cl vP l 00 . Lemma [iv] naturally leads to bridge It is easy to show that bridge safe(E) holds if We can make a counter example which violates con- straints. For example, E 1 is as follows. loader 0. { Loader 0 denes class C. Class C has method M with signature [X]. does not satisfy system(E 1 ), we assume that class X is never resolved by loader 0, i.e., where m is the method M resolved by loader 0 (= m:cl). Additionally, loader 1. { Loader 1 denes class D. Class D calls method M of class C with signature [X]. { Loader 1 denes class X. { Loader 1 delegates to loader 0 for other classes. loader 2. { Loader 2 denes class D. Class D extends C. Class D has method M with signature [X]. { Loader 2 denes class X. { Loader 2 delegates to loader 0 for other classes. We can assume that where c is the class, C, dened by loader Obviously, M in loader 2 overrides m. Therefore, from the denition of - , we have 2: However, X in loader 1 and X in 1 are dierent. Therefore, bridge does not hold. 5 Conclusion We have presented a new model of the JVM, which explains various unique features of the JVM, and also species several conditions on its implementations. In particular, the model includes the loading constraint scheme and the ndclass scheme, both of which are new features of JDK1.2, Through the formalization, we could analyze the extremely subtle relationship between the loading constraint scheme and the bytecode verication. We believe that such an analysis is possible only through a rigorous formalization and soundness proofs. However, our model excludes many features of the JVM: its primitive types, eld members, array types, member modiers, threads, most of its instructions, etc. We have several ideas to incorporate them into our model. For example, our model can easily express the object and class nalization of the JVM. The soundness theorem in Section 2.5 states that when an environment is updated into a larger environment, the well-typedness invariant is preserved. There- fore, if we can introduce a reduced environment which also preserves the same invariant, then the soundness of the nalization is guaranteed. As for the ndclass scheme, we showed that it should work with the loading constraint scheme. However, we have also obtained a method which allows some loading constraints to be omitted under the cooperation with the ndclass scheme. This result is not included in this paper, as we do not think that it is the best solution, and we expect that both schemes should be improved in the future. --R Linking and Moduralization Formal Aspects of Mobile Code Security Web Browers and Beyond. A Type System for Object Initialization in the Java Bytecode Language A speci Security and Dynamic Class Loading in Java: A Formalisation On a New Method for Data ow Analysis of Java Virtual Machine Subroutines Dynamic Class Loading in the Java Virtual Ma- chine The Java Virtual Machine Speci Java light is type-safe - de nitely Proving the Soundness of a Java Bytecode Veri A Formal Speci Java is not type-safe Nicht veri A Type System for Java Bytecode Subroutines Careful Analysis of Type Spoo A compositional account of the Java virtual machine --TR --CTR Modeling multiple class loaders by a calculus for dynamic linking, Proceedings of the 2004 ACM symposium on Applied computing, March 14-17, 2004, Nicosia, Cyprus

class loading;security

609232

Dependent Types for Program Termination Verification.

Program termination verification is a challenging research subject of significant practical importance. While there is already a rich body of literature on this subject, it is still undeniably a difficult task to design a termination checker for a realistic programming language that supports general recursion. In this paper, we present an approach to program termination verification that makes use of a form of dependent types developed in Dependent ML (DML), demonstrating a novel application of such dependent types to establishing a liveness property. We design a type system that enables the programmer to supply metrics for verifying program termination and prove that every well-typed program in this type system is terminating. We also provide realistic examples, which are all verified in a prototype implementation, to support the effectiveness of our approach to program termination verification as well as its unobtrusiveness to programming. The main contribution of the paper lies in the design of an approach to program termination verification that smoothly combines types with metrics, yielding a type system capable of guaranteeing program termination that supports a general form of recursion (including mutual recursion), higher-order functions, algebraic datatypes, and polymorphism.

Introduction Programming is notoriously error-prone. As a conse- quence, a great number of approaches have been developed to facilitate program error detection. In practice, the programmer often knows certain program properties that must hold in a correct implementation; it is therefore an indication of program errors if the actual implementation violates some of these properties. For instance, various type systems have been designed to detect program errors that cause violations of the supported type disciplines. It is common in practice that the programmer often knows for some reasons that a particular program should terminate if implemented correctly. This immediately implies that a termination checker can be of great value for detecting program errors that cause nonterminating program ex- Partially supported by NSF grant no. CCR-0092703 ecution. However, termination checking in a realistic programming language that supports general recursion is often prohibitively expensive given that (a) program termination in such a language is in general undecidable, (b) termination checking often requires interactive theorem proving that can be too involved for the programmer, (c) a minor change in a program can readily demand a renewed effort in termination checking, and (d) a large number of changes are likely to be made in a program development cycle. In order to design a termination checker for practical use, these issues must be properly addressed. There is already a rich literature on termination verifica- tion. Most approaches to automated termination proofs for either programs or term rewriting systems (TRSs) use various heuristics, some of which can be highly involved, to synthesize well-founded orderings (e.g., various path orderings [3], polynomial interpretation [1], etc. While these approaches are mainly developed for first-order languages, the work in higher-order settings can also be found (e.g., [7]). When a program, which should be terminating if implemented correctly, cannot be proven terminating, it is often difficult for the programmer to determine whether this is caused by a program error or by the limitation of the heuristics involved. Therefore, such automated approaches are likely to offer little help in detecting program errors that cause nonterminating program execution. In addition, automated approaches often have difficulty handling realistic (not necessarily large) programs. The programmer can also prove program termination in various (interactive) theorem proving systems such as NuPrl [2], Coq [4], Isabelle [8] and PVS [9]. This is a viable practice and various successes have been reported. However, the main problem with this practice is that the programmer may often need to spend so much time on proving the termination of a program compared with the time spent on simply implementing the program. In addition, a renewed effort may be required each time when some changes, which are likely in a program development cycle, are made to the program. Therefore, the programmer can often feel hesitant to adopt (interactive) theorem proving for detecting program errors in general programming. We are primarily interested in finding a middle ground. In particular, we are interested in forming a mechanism in a programming language that allows the programmer to provide information needed for establishing program termination else if withtype {i:nat,j:nat} <i,j> => int(i) -> int(j) -> [k:nat] int(k) Figure 1. An implementation of Ackerman function and then automatically verifies that the provided information indeed suffices. An analogy would be like allowing the user to provide induction hypotheses in inductive theorem proving and then proving theorems with the provided induction hypotheses. Clearly, the challenging question is how such information for establishing program termination can be formalized and then expressed. The main contribution of this paper lies in our attempt to address the question by presenting a design that allows the programmer to provide through dependent types such key information in a (relatively) simple and clean way. It is common in practice to prove the termination of recursive functions with metrics. Roughly speaking, we attach a metric in a well-founded ordering to a recursive function and verify that the metric is always decreasing when a recursive function call is made. In this paper, we present an approach that uses the dependent types developed in DML [18, 14] to carry metrics for proving program termination. We form a type system in which metrics can be encoded into types and prove that every well-typed program is terminating. It should be emphasized that we are not here advocating the design of a programming language in which only terminating programs can be written. Instead, we are interested in designing a mechanism in a programming language, which, if the programmer chooses to use it, can facilitate program termination verification. This is to be manifested in that the type system we form can be smoothly embedded into the type system of DML. We now illustrate the basic idea with a concrete example before going into further details. In Figure 1, an implementation of Ackerman function is given. The withtype clause is a type annotation, which states that for natural numbers i and j, this function takes an argument of type int(i) and another argument of type and returns a natural number as a result. Note that we have refined the usual integer type int into infinitely many singleton types int(a) for such that int(a) is precisely the type for integer expressions with value equal to a. We write fi:nat,j:natg for universally quantifying over index variables i and j of nat , that is, the sort for index expressions with values being natural numbers. Also, we write [k:nat] int(k) which represents the sum of all types :. The novelty here is the pair hi; ji in the type annotation, which indicates that this is the metric to be used for termination checking. We now informally explain how termination checking is performed in this case; assume that i and j are two natural numbers and m and n have types int(i) and int(j), respectively, and attach the metric hi; ji to ack m n; note that there are three recursive function calls to ack in the body of ack; we attach the metric hi 1; 1i to the first ack since m 1 and 1 have types int(i 1) and int(1), respectively; similarly, we attach the metric hi 1; ki to the second ack, where k is assumed to be some natural number, and the metric hi; j 1i to the third ack; it is obvious that hi and hi; j 1i < hi; ji hold, where < is the usual lexicographic ordering on pairs of natural numbers; we thus claim that the function ack is terminating (by a theorem proven in this paper). Note that although this is a simple example, its termination cannot be proven with (lexicographical) structural ordering (as the semantic meaning of both addition and subtraction is needed). 1 More realistic examples are to be presented in Section 5, involving dependent datatypes [15], mutual recursion, higher-order functions and polymorphism. The reader may read some of these examples before studying the sections on technical development so as to get a feel as to what can actually be handled by our approach. Combining metrics with the dependent types in DML poses a number of theoretical and pragmatic questions. We briefly outline our results and design choices. The first question that arises is to decide what metrics we should support. Clearly, the variety of metrics for establishing program termination is endless in practice. In this pa- per, we only consider metrics that are tuples of index expressions of sort nat and use the usual lexicographic ordering to compare metrics. The main reasons for this decision are that (a) such metrics are commonly used in practice to establish termination proofs for a large variety of programs and (b) constraints generated from comparing such metrics can be readily handled by the constraint solver already built for type-checking DML programs. Note that the usual structural ordering on first-order terms can be obtained by attaching to the term the number of constructors in the term, which can be readily accomplished by using the dependent datatype mechanism in DML. However, we are currently unable to capture structural ordering on higher-order terms. The second question is about establishing the soundness of our approach, that is, proving every well-typed program in the type system we design is terminating. Though the idea mentioned in the example of Ackerman function seems intu- itive, this task is far from being trivial because of the presence of higher-order functions. The reader may take a look at the higher-order example in Section 5 to understand this. We seek a method that can be readily adapted to handle various common programming features when they are added, 1 There is an implementation of Ackerman function that involves only primitive recursion and can thus be easily proven terminating, but the point we drive here is that this particular implementation can be proven terminating with our approach. including mutual recursion, datatypes, polymorphism, etc. This naturally leads us to the reducibility method [12]. We are to form a notion of reducibility for the dependent types extended with metrics, in which the novelty lies in the treatment of general recursion. This formation, which is novel to our knowledge, constitutes the main technical contribution of the paper. The third question is about integrating our termination checking mechanism with DML. In practice, it is common to encounter a case where the termination of a function f depends on the termination of another function g, which, unfor- tunately, is not proven for various reasons, e.g., it is beyond the reach of the adopted mechanism for termination checking or the programmer is simply unwilling to spend the effort proving it. Our approach is designed in a way that allows the programmer to provide a metric in this case for verifying the termination of f conditional on the termination of g, which can still be useful for detecting program errors. The presented work builds upon our previous work on the use of dependent types in practical programming [18, 14]. While the work has its roots in DML, it is largely unclear, a priori, how dependent types in DML can be used for establishing program termination. We thus believe that it is a significant effort to actually design a type system that combines types with metrics and then prove that the type system guarantees program termination. This effort is further strengthened with a prototype implementation and a variety of verified examples. The rest of the paper is organized as follows. We form a language ML ; 0 in Section 2, which essentially extends the simply typed call-by-value -calculus with a form of dependent types, developed in DML, and recursion. We then extend ML ;to ML ; in Section 3, combining metrics with types, and prove that every program in ML ; 0; is termi- nating. In Section 4, we enrich ML ; with some significant programming features such as datatypes, mutual recursion and polymorphism. We present some examples in Section 5, illustrating how our approach to program termination verification is applied in practice. We then mention some related work and conclude. There is a full paper available on-line [16] in which the reader can find details omitted here. start with a language ML ; 0 , which essentially extends the simply typed call-by-value -calculus with a form of dependent types and (general) recursion. The syntax for ML ;is given in Figure 2. 2.1 Syntax We fix an integer domain and restrict type index expres- sions, namely, the expressions that can be used to index a type, to this domain. This is a sorted domain and subset sorts can be formed. For instance, we use nat for the subset sort 0g. We use (~) for a base type indexed with a sequence of index expressions~, which may be empty. For instance, bool(0) and bool(1) are types for boolean values false and true, respectively; for each integer i, int(i) is the singleton type for integer expressions with value equal to i. We use satisfaction relation, which means P holds under , that is, the formula ()P , defined below, is satisfied in the domain of integers. For instance, the satisfaction relation holds since the following formula is true in the integer domain Note that the decidability of the satisfaction relation depends on the constraint domain. For the integer constraint domain we use here, the satisfaction relation is decidable (as we do not accept nonlinear integer constraints). We use a : : for the usual dependent function and sum types, respectively. A type of form : is essentially equivalent to a where we use ~a : ~ for n . 2 We also introduce -variables and -variables in ML ;and use x and f for them, respectively. A lambda-abstraction can only be formed over a -variable while recursion (via fixed point op- erator) must be formed over a -variable. A -variable is a value but a -variable is not. We use for abstracting over index variables, lam for abstracting over variables, and fun for forming recursive func- tions. Note that the body after either or fun must be a value. We use hi j ei for packing an index i with an expression e to form an expression of a dependent sum type, and open for unpacking an expression of a dependent sum type. 2.2 Static Semantics We write ' : to mean that is a legally formed type under and omit the standard rules for such judgments. index substitutions I ::= [] j I [a 7! i] substitutions ::= [] j [x 7! e] j [f 7! e] A substitution is a finite mapping and [] represents an empty mapping. We use I for a substitution mapping index variables to index expressions and dom( I ) for the domain of I . Similar notations are used for substitutions on variables. We write [ I ] ([]) for the result from applying I () to , where can be a type, an expression, etc. The standard In practice, we also have types of form ~a : ~ : , which we omit here for simplifying the presentation. index constants c I ::= index expressions i ::= a j c I j index propositions P index sorts index variable contexts ::= index constraints ::= types ::= (~) contexts constants c ::= true expressions e ::= c j x values Figure 2. The syntax for ML ;; Figure 3. Typing Rules for ML ;definition is omitted. The following rules are for judgments of form ' I : 0 , which roughly means that I has "type" We write dom() for the domain of , that is, the set of variables declared in . Given substitutions I and , we say We write for the congruent extension of index expressions to types, determined by the following rules. It is the application of these rules that generates constraints during type-checking. We present the typing rules for ML ;in Figure 3. Some of these rules have obvious side conditions, which are omit- ted. For instance, in the rule (type-ilam), ~a cannot have free occurrences in . The following lemma plays a pivotal r"ole in proving the subject reduction theorem for ML ;, whose standard proof is available in [14]. Lemma 2.1 Assume ; derivable and holds. Then we can derive ; ' 2.3 Dynamic Semantics We present the dynamic semantics of ML ;through the use of evaluation contexts defined below. Certainly, there are other possibilities for this purpose, which we do not explore here. 3 evaluation contexts E ::= We write E[e] for the expression resulting from replacing the hole [] in E with e. Note that this replacement can never result in capturing free variables. Definition 2.2 A redex is defined below. are redexes for false , which reduce to e 1 and e 2 , respectively. (lam x : :e)(v) is a redex, which reduces to e[x 7! v]. Let e be fun f [~a : ~ e is a redex, which reduces to ~a : ~ :v[f 7! e]. :v)[~] is a redex, which reduces to v[~a 7!~]. open hi j vi as ha j xi in e is a redex, which reduces to e[a 7! i][x 7! v]. We use r for a redex and write r ,! e if r reduces to e. If e, we write e 1 ,! e 2 and say reduces to e 2 in one step. Let ,! be the reflexive and transitive closure of ,!. We say reduces to e 2 (in many steps) if e 1 ,! e 2 . We omit the standard proof for the following subject reduction theorem, which uses Lemma 2.1. Theorem 2.3 (Subject Reduction) Assume ; derivable in ML ; derivable in ML ;2.4 Erasure We can simply transform ML ;into a language ML 0 by erasing all syntax related to type index expressions in . Then ML 0 basically extends simply typed - calculus with recursion. Let jej be the erasure of expression e. We have e 1 reducing to e 2 in ML ; reducing to je 2 j in ML 0 . Therefore, if e is terminating in ML ;then jej is terminating in ML 0 . This is a crucial point since the evaluation of a program in ML ; 0 is (most likely) done through the evaluation of its erasure in ML 0 . Please find more details on this issue in [18, 14]. 3 For instance, it is suggested that one present the dynamic semantics in the style of natural semantics and then later form the notion of reducibility for evaluation rules. We combine metrics with the dependent types in ML ;, forming a language ML ; . We then prove that every well-typed program in ML ; is terminating, which is the main technical contribution of the paper. 3.1 We use for the usual lexicographic ordering on tuples of natural numbers and < for the strict part of . Given two tuples of natural numbers hi holds if . Evi- dently, < is a well-founded. We stress that (in theory) there is no difficulty supporting various other well-founded orderings on natural numbers such as the usual multiset ordering. We fix an ordering solely for easing the presentation. Definition 3.1 (Metric) Let be a tuple of index expressions and be an index variable context. We say is a metric under if ' are derivable for to mean is a metric under . A decorated type in ML ; 0; is of form ~a : ~ the following rule is for forming such types. The syntax of ML ; is the same as that of ML ;except that a context in ML ; maps every -variable f in its domain to a decorated type and a recursive function in ML ; is of form fun f [~a : ~ v. The process of translating a source program into an expression in ML ; is what we call elaboration, which is thoroughly explained in [18, 14]. Our approach to program termination verification is to be applied to elaborated programs. 3.2 Dynamic and Static Semantics The dynamic semantics of ML ; is formed in precisely the same manner as that of ML ; 0 and we thus omit all the details. The difference between ML ; and ML ;lies in static semantics. There are two kinds of typing judgments in ML ;, which are of forms 0 . We call the latter a metric typing judgment, for which we give some explanation. Suppose and roughly speaking, for each free occurrence of f in e, f is followed by a sequence of index expressions [~] such that [~a 7! ~], which we call the label of this occurrence of f , is less than 0 under . Now suppose we have a well-typed closed recursive function 0; and~ are of sorts ~ then f [~][f 7! holds; by the rule (type-fun), we know that all labels of f in v are less than [~a 7!~], which is the label of f in f [~]; since labels cannot decrease forever, this yields some basic intuition on why all recursive functions in ML ; are terminating. However, this intuitive argument is difficult to be formalized directly in the presence of high-order functions. The typing rules in ML ; for a judgment of form ; ' are essentially the same as those in ML ;except the following ones. We present the rules for deriving metric typing judgments in Figure 4. Given means that for some 1 k < are satisfied for all 1 j < k is also satisfied. Lemma 3.2 We have the following. 1. Assume ; holds. Then we can derive ; ' 2. Assume ; derivable and dom(). Then we can derive Proof (1) and (2) are proven simultaneously by structural induction on derivations of ; Theorem 3.3 (Subject Reduction) Assume ; derivable in ML ; 0; . If e ,! e 0 , then ; ' e derivable in ML ; Obviously, we have the following. Proposition 3.4 Assume that D is a derivation ; f 0 . Then then there is a derivation of ; with the same height 4 as D. 3.3 Reducibility We define the notion of reducibility for well-typed closed expressions. Definition 3.5 (Reducibility) Suppose that e is a closed expression of type and e ,! v holds for some value v. The reducibility of e is defined by induction on the complexity of . 4 For a minor technicality reason, we count neither of the rules (type-var) and (-var) when calculating the height of a derivation. 1. is a base type. Then e is reducible. 2. are reducible for all reducible values v 1 of type . 3. reducible if e[~] are reducible 4. some i and v 1 such that v 1 is a reducible value of type Note that reducibility is only defined for closed expressions that reduce to values. Proposition 3.6 Assume that e is a closed expression of type and e ,! e 0 holds. Then e is reducible if and only if e 0 is reducible. Proof By induction on the complexity of . The following is a key notion for handling recursion, which, though natural, requires some technical insights. Definition 3.7 (-Reducibility). Let e be a well-typed closed recursive function fun f [~a : ~ be a closed metric. e is 0 -reducible if e[~] are reducible for all satisfying [~a 7!~] < 0 . Definition 3.8 Let be a substitution that maps variables to expressions; for every x 2 dom(), is x-reducible if (x) is reducible; for every f 2 dom(), is (f; f )-reducible if (f) is f -reducible. In some sense, the following lemma verifies whether the notion of reducibility is formed correctly, where the difficulty probably lies in its formulation rather than in its proof. Lemma 3.9 (Main Lemma) Assume that ; ' e : and are derivable. Also assume that is x-reducible for every x 2 dom() and for every f 2 derivable and is (f; f )-reducible. Then e[ I ][] is reducible. Proof Let D be a derivation of ; ' e : and we proceed by induction on the height of D. We present the most interesting case below. All other cases can be found in [16]. Assume that the following rule (type-fun) is last applied in D, where we have and Suppose that e I ][] is not reducible. Then by definition there exist ~ 1 such that e [~ 0 ] is not reducible but e [~] are reducible for all satisfying ~ In other words, e is f1 - reducible for that we can derive Figure 4. Metric Typing Rules for ML ; . By Proposition 3.4, there is a derivation D 1 of such that the height of D 1 is less than that of D. By induction hypothesis, we have that v Note that e [~ 0 ] ,! v and thus e [~ 0 ] is reducible, contradicting the definition of~ 0 . Therefore, e is reducible. The following is the main result of the paper. Corollary 3.10 If ; ' e : is derivable in ML ; in ML ; is reducible and thus reduces to a value. Proof The corollary follows from Lemma 3.9. Extensions We can extend ML ; with some significant programming features such as mutual recursion, datatypes and poly- morphism, defining the notion of reducibility for each extension and thus making it clear that Lemma 3.9 still holds after the extension. We present in this section the treatment of mutual recursion and currying, leaving the details in [16]. 4.1 Mutual Recursion The treatment of mutual recursion is slightly different from the standard one. The syntax and typing rules for handling mutual recursion are given in Figure 5. We use the type of an expression representing n mutually recursive functions of types respectively, which should not be confused with the product of types . Also, the n in e:n must be a positive (constant) integer. Let v be the following expression. funs Then for every 1 k n, v:k is a redex, which reduces to and we form a metric typing judgment ; ' e ~ f 0 for verifying that all labels of f in e are less than 0 under . The rules for deriving such a judgment are essentially the same as those in Figure 4 except (-lab), which is given below. f in ~ f The rule (-funs) for handling mutual recursion is straight-forward and thus omitted. Definition 4.1 (Reducibility) Let e be a closed expression of reduces to v. e is reducible if e:k are reducible for 4.2 Currying A decorated type must so far be of form ~a : ~ and this restriction has a rather unpleasant consequence. For types ::= j expressions e ::= j e:n j funs f 1 values v ::= j funs f 1 ~ f f (type-funs) Figure 5. The Syntax and Typing Rules for Mutual Recursion instance, we may want to assign the following type to the implementation of Ackerman function in Figure 1: fi:natg int(i) -> fj:natg int(j) -> int; which is formally written as If we decorate with a metric , then can only involve the index variable a 1 , making it impossible to verify that the implementation is terminating. We generalize the form of decorated types to the following so as to address the problem. Also, we introduce the following form of expression e for representing a recursive function. We require that e 0 be a value if In the following, we only deal with the case 1. For n > 1, the treatment is similar. For have e ,! :e 0 and the following typing rule , and the following metric typing rule Definition 4.2 (-reducibility) Let e be a closed recursive function a closed metric. e is 0 -reducible if e[~ 1 ](v)[~] are reducible for all reducible values 1 and 5 Practice We have implemented a type-checker for ML ; in a prototype implementation of DML and experimented with various examples, some of which are presented below. We also address the practicality issue at the end of this section. 5.1 Examples We demonstrate how various programming features are handled in practice by our approach to program termination verification. Primitive Recursion The following is an implementation of the primitive recursion operator R in Godel's T , which is clearly typable in ML ; . Note that Z and S are assigned the types Nat(0) and respectively. datatype Nat with Z(0) | {n:nat} S(n+1) of Nat(n) u | R (S n) u (R n u v) withtype {n:nat} => (* Nat is for [n:nat] Nat(n) in a type *) By Corollary 3.10, it is clear that every term in T is terminating (or weakly normalizing). This is the only example in this paper that can be proven terminating with a structural ordering. The point we make is that though it seems "evident" that the use of R cannot cause non-termination, it is not trivial at all to prove every term in T is terminating. Notice that such a proof cannot be obtained in Peano arithmetic. The notion of reducibility is precisely invented for overcoming the difficulty [12]. Actually, every term in T is strongly normalizing, but this obviously is untrue in 0; . Nested Recursive Function Call The program in Figure 6 involving a nested recursive function call implements Mc- Carthy's ``91'' function. The withtype clause indicates that for every integer x, f91(x) returns integer 91 if x 100 We informally explain why the metric in the type annotation suffices to establish the termination of f91; for the inner call to f91, we need to prove that for which is obvious; for the outer call to f91, we need to verify that 1 and max(0; 101 assumed in ). Clearly, this example can not be handled with a structural ordering. Mutual Recursion The program in Figure 7 implements quicksort on a list, where the functions qs and par are defined mutually recursively. We informally explain why this program is typable in ML ; 0; and thus qs is a terminating function by Corollary 3.10. For the call to par in the body of qs, the label is (0 1), where a is the length of xs 0 . So we need to verify that is satisfied for obvious. For the two calls to qs in the body of par, we need to verify that of which hold since This also indicates why we need r of r in the metric for par. For the two calls to par in the body of par, we need to verify that and both of which hold since this example can not be handled with a structural ordering. Higher-order Function The program in Figure 8 implements a function accept that takes a pattern p and a string s and checks whether s matches p, where the meaning of a pattern is given in the comments. The auxiliary function acc is implemented in continuation passing style, which takes a pattern p, a list of characters cs and a continuation k and matches a prefix of cs against p and call k on the rest of characters. Note that k is given a type that allows k to be applied only to a character list not longer than cs. The metric used for proving the termination of acc is hn; ii, where n is the size of p, that is the number constructors in p (excluding Empty) and i is the length of cs. Notice the call acc p cs 0 k in the last pattern matching clause; the label attached to this call is is the length of cs 0 ; we have i 0 i since the continuation has the type a where must be false when this call hap- pens; therefore we have It is straightforward to see that the labels attached to other calls to acc are less than hn; ii. By Corollary 3.10, acc is termi- nating, which implies that accept is terminating (assuming explode is terminating). In every aspect, this is a non-trivial example even for interactive theorem proving systems. Notice that the test length(cs 0 in the body of acc can be time-consuming. This can be resolved by using a continuation that accepts as its arguments both a character list and its length. In [5], there is an elegant implementation of accept that does some processing on the pattern to be matched and then eliminates the test. Run-time Check There are also realistic cases where termination depends on a program invariant that cannot (or is difficult to) be captured in the type system of DML. For instance, the following example is adopted from an implementation of bit reversing, which is a part of an implementation of fast Fourier transform (FFT). fun loop (j, if (k<j) then loop (j-k, k/2) else j+k withtype {a:nat,b:nat} int(a) * int(b) -> int Obviously, loop(1; 0) is not terminating. However, we may know for some reason that the second argument of loop can never be 0 during execution. This leads to the following im- plementation, in which we need to check that k > 1 holds before calling loop(j k; k=2) so as to guarantee that k=2 is a positive integer. fun loop (j, else raise Impossible withtype {a:nat,b:pos} <max(0, a-b)> => int(a) * int(b) -> int It can now be readily verified that loop is a terminating func- tion. This example indicates that we can insert run-time checks to verify program termination, sometimes, approximating a liveness property with a safety property. 5.2 Practicality There are two separate issues concerning the practicality of our approach to program termination verification, which are (a) the practicality of the termination verification process and (b) the applicability of the approach to realistic programs 5 Note that length(cs 0 ) and length(cs) have the types int(i 0 ) and int(i), respectively, and thus length(cs has the type depending on whether i 0 equals i. Thus, can be inferred in the type system. withtype Figure 6. An implementation of McCarthy's ``91'' function case xs of [] => [] | x :: xs' => par cmp (x, [], [], xs') withtype ('a * 'a -> bool) -> {n:nat} <n,0> => 'a list(n) -> 'a list(n) and('a) par cmp (x, l, r, case xs of | x' :: xs' => if cmp(x', x) then par cmp (x, x' :: l, r, xs') else par cmp (x, l, x' :: r, xs') withtype ('a * 'a -> bool) -> {p:nat,q:nat,r:nat} <p+q+r,r+1> => 'a * 'a list(p) * 'a list(q) * 'a list(r) -> 'a list(p+q+r+1) Figure 7. An implementation of quicksort on a list It is easy to observe that the complexity of type-checking in ML ; is basically the same as in ML ;since the only added work is to verify that metrics (provided by the pro- are decreasing, which requires solving some extra constraints. The number of extra constraints generated from type-checking a function is proportional to the number of recursive calls in the body of the function and therefore is likely small. Based on our experience with DML, we thus feel that type-checking in ML ; is suitable for practical use. As for the applicability of our approach to realistic pro- grams, we use the type system of the programming language C as an example to illustrate a design decision. Obviously, the type system of C is unsound because of (unsafe) type casts, which are often needed in C for typing programs that would otherwise not be possible. In spite of this practice, the type system of C is still of great help for capturing program errors. Clearly, a similar design is to allow the programmer to assert the termination of a function in DML if it cannot be verified, which we may call termination cast. Combining termination verification, run-time checks and termination cast, we feel that our approach is promising to be put into practice. 6 Related Work The amount of research work related to program termination is simply vast. In this section, we mainly mention some related work with which our work shares some similarity either in design or in technique. Most approaches to automated termination proofs for either programs or term rewriting systems (TRSs) use various heuristics to synthesize well-founded orderings. Such ap- proaches, however, often have difficulty reporting comprehensible information when a program cannot be proven ter- minating. Following [13], there is also a large amount of work on proving termination of logic programs. In [11], it is reported that the Mercury compiler can perform automated termination checking on realistic logic programs. However, we address a different question here. We are interested in checking whether a given metric suffices to establish the termination of a program and not in synthesizing such a metric. This design is essentially the same as the one adopted in [10], where it checks whether a given structural ordering (possibly on high-order terms) is decreasing in an inductive proof or a logic program. Clearly, approaches based on checking complements those based on synthesis. Our approach also relates to the semantic labelling approach [19] designed to prove termination for term rewriting systems (TRSs). The essential idea is to differentiate function calls with labels and show that labels are always decreasing when a function call unfolds. The semantic labelling approach requires constructing a model for a TRS to verify whether labelling is done correctly while our approach does this by type-checking. The notion of sized types is introduced in [6] for proving the correctness of reactive systems. There, the type system is capable of guaranteeing the termination of well-typed programs. The language presented in [6], which is designed for embedded functional programming, contains a significant restriction as it only supports (a minor variant) of primitive recursion, which can cause inconvenience in programming. For instance, it seems difficult to implement quicksort by using only primitive recursion. From our experience, general recursion is really a major programming feature that greatly complicates program termination verification. Also, the notion of existential dependent types, which we deem indispensable in practical programming, does not exist in [6]. When compared to various (interactive) theorem proving datatype pattern with string matches Empty *) | Char(1) of char (* "c" matches Char (c) *) | {i:nat,j:nat} Plus(i+j+1) of pattern(i) * pattern(j) (* cs matches Plus(p1, p2) if cs matches either p1 or p2 *) | {i:nat,j:nat} Times(i+j+1) of pattern(i) * pattern(j) (* cs matches Times(p1, p2) if a prefix of cs matches p1 and the rest matches p2 *) | {i:nat} Star(i+1) of pattern(i) (* cs matches Star(p) if cs matches some, possibly 0, copies of p *) (* 'length' computes the length of a list *) length | len withtype in len (xs, withtype {i:nat} <> => 'a list(i) -> int(i) (* empty tuple <> is used since 'length' is not recursive *) case p of Empty => k (cs) | Char(c) => (case cs of | c' :: cs' => if | Plus(p1, p2) => (* in this case, k is used for backtracking *) if acc p1 cs k then true else acc p2 cs k | Times(p1, p2) => acc p1 cs (fn cs' => acc p2 cs' | if k (cs) then true else acc p0 cs (fn cs' => else acc p cs' withtype {n:nat} pattern(n) -> {i:nat} <n, i> => char list(i) -> ({i':nat | i' <= i} char list(i') -> bool) -> bool (* 'explode' turns a string into a list of characters *) withtype <> => pattern -> string -> bool Figure 8. An implementation of pattern matching on strings systems such as NuPrl [2], Coq [4], Isabelle [8] and PVS [9], our approach to program termination is weaker (in the sense that [many] fewer programs can be verified terminating) but more automatic and less obtrusive to programming. We have essentially designed a mechanism for program termination verification with a language interface that is to be used during program development cycle. We consider this as the main contribution of the paper. When applied, the designed mechanism intends to facilitate program error detection, leading to the construction of more robust programs. 7 Conclusion and Future Work We have presented an approach based on dependent types in DML that allows the programmer to supply metrics for verifying program termination and proven its correctness. We have also applied this approach to various examples that involve significant programming features such as a general form of recursion (including mutual recursion), higher-order functions, algebraic datatypes and polymorphism, supporting its usefulness in practice. A program property is often classified as either a safety property or a liveness property. That a program never performs out-of-bounds array subscripting at run-time is a safety property. It is demonstrated in [17] that dependent types in DML can guarantee that every well-typed program in DML possesses such a safety property, effectively facilitating run-time array bound check elimination. It is, however, unclear (a priori) whether dependent types in DML can also be used for establishing liveness properties. In this paper, we have formally addressed the question, demonstrating that dependent types in DML can be combined with metrics to establish program termination, one of the most significant liveness properties. Termination checking is also useful for compiler opti- mization. For instance, if one decides to change the execution order of two programs, it may be required to prove that the first program always terminates. Also, it seems feasible to use metrics for estimating the time complexity of programs. In lazy function programming, such information may allow a compiler to decide whether a thunk should be formed. In future, we expect to explore along these lines of research. Although we have presented many interesting examples that cannot be proven terminating with structural orderings, we emphasize that structural orderings are often effective in practice for establishing program termination. Therefore, it seems fruitful to study a combination of our approach with structural orderings that handles simple cases with either automatically synthesized or manually provided structural orderings and verifies more difficult cases with metrics supplied by the programmer. --R Termination of rewriting systems by polynomial interpretations and its implementation. Implementing Mathematics with the NuPrl Proof Development System. Orderings for term rewriting systems. Proving the correctness of reactive systems using sized types. The higher-order recursive path ordering A Generic Theorem Prover. PVS: Combining specification Termination and Reduction Checking in the Logical Framework. Termination Analysis for Mercury. Intensional Interpretations of Functionals of Finite Type I. Efficient tests for top-down termination of logic rules Dependent Types in Practical Programming. Dependently Typed Data Structures. Dependent Types for Program Termination Verifica- tion Eliminating array bound checking through dependent types. Dependent types in practical program- ming Termination of term rewriting by semantic la- belling --TR --CTR Kevin Donnelly , Hongwei Xi, A Formalization of Strong Normalization for Simply-Typed Lambda-Calculus and System F, Electronic Notes in Theoretical Computer Science (ENTCS), v.174 n.5, p.109-125, June, 2007 Chiyan Chen , Hongwei Xi, Combining programming with theorem proving, ACM SIGPLAN Notices, v.40 n.9, September 2005 Kevin Donnelly , Hongwei Xi, Combining higher-order abstract syntax with first-order abstract syntax in ATS, Proceedings of the 3rd ACM SIGPLAN workshop on Mechanized reasoning about languages with variable binding, p.58-63, September 30-30, 2005, Tallinn, Estonia Amir M. Ben-Amram , Chin Soon Lee, Program termination analysis in polynomial time, ACM Transactions on Programming Languages and Systems (TOPLAS), v.29 n.1, p.5-es, January 2007 Arne John Glenstrup , Neil D. Jones, Termination analysis and specialization-point insertion in offline partial evaluation, ACM Transactions on Programming Languages and Systems (TOPLAS), v.27 n.6, p.1147-1215, November 2005

dependent types;termination

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Axioms for Recursion in Call-by-Value.

We propose an axiomatization of fixpoint operators in typed call-by-value programming languages, and give its justifications in two ways. First, it is shown to be sound and complete for the notion of uniform T-fixpoint operators of Simpson and Plotkin. Second, the axioms precisely account for Filinski's fixpoint operator derived from an iterator (infinite loop constructor) in the presence of first-class continuations, provided that we define the uniformity principle on such an iterator via a notion of effect-freeness (centrality). We then explain how these two results are related in terms of the underlying categorical structures.

Introduction While the equational theories of xpoint operators in call-by-name programming languages and in domain theory have been extensively studied and now there are some canonical axiomatizations (including the iteration theories [1] and Conway theories, equivalently traced cartesian categories [12] { see [27] for the latest account), there seems no such widely-accepted result in the context of call-by-value (cbv) programming languages, possibly with side eects. Although the implementation of recursion in \impure" programming language has been well-known, it seems that the underlying semantic nature of recursive computation in the presence of side-eects has not been studied at a suciently general level. Regarding the widespread use of call-by-value programming languages and the importance of recursion in real life programming, it is desirable to have theoretically motivated and justied principles for reasoning about recursive computation in a call-by-value setting. In this paper we propose a candidate of such an axiomatization, which consists of three simple axioms, including a uniformity principle analogous to that in the call-by-name setting. Our axiomatization, of stable uniform call-by-value xpoint operators to be introduced below, is justied by the following two main results: y An extended abstract of this work appeared in Proc. Foundations of Software Science and Computation Structures (FoSSaCS 2001), Springer LNCS Vol. 2030. Hasegawa and Kakutani 1. The c -calculus (computational lambda calculus) [18] with a stable uniform cbv xpoint operator is sound and complete for the models based on the notion of uniform T -xpoint operators of Simpson and Plotkin [27]. 2. In the call-by-value -calculus [25] (= the c -calculus plus rst- class continuations) there is a bijective correspondence between stable uniform cbv xpoint operators and uniform iterators, via Filinski's construction of recursion from iteration [5]. The notion of uniform T-xpoint operators arose from the context of Axiomatic Domain Theory [7, 26]. By letting T be a lifting monad on a category of predomains, a uniform T-xpoint operator amounts to a uniform xpoint operator on domains (the least xpoint operator in the standard order-theoretic setting). In general, T can be any strong monad on a category with nite products, thus a uniform T-xpoint operator makes sense for any model of the computational lambda calculus in terms of strong monads [18], and Simpson and Plotkin [27] suggest the possibility of using uniform T-xpoint operators for modelling call-by-value recursion. This line of considerations leads us to our rst main result. In fact, we distill our axioms from the uniform T-xpoint operators. A surprise is the second one, in that the axioms precisely account for Filinski's cbv xpoint operator derived from an iterator (innite loop constructor) and rst-class continuations, provided that we rene Filin- ski's notion of uniformity, for which the distinction between values and eect-insensitive programs (characterised by the notion of centrality) [22, 28, 10] is essential. Using our axioms, we establish the bijectivity result between xpoint operators and iterators. Therefore here is an interesting coincidence of a category-theoretic axiomatics (of Simpson and Plotkin) with a program construction (of Filinski). However, we also show that, after sorting out the underlying categorical semantics, Filinski's construction combined with the Continuation-Passing Style (CPS) transformation can be understood within the abstract setting of Simpson and Plotkin. The story is summarised as follows. As noted by Filinski, the CPS-transform of an iterator is a usual (call-by-name) xpoint operator on the types of the form R A in the target -calculus, where R is the answer type. If we let T be the continuation monad R R ( , then the uniform T-xpoint operator precisely amounts to the uniform xpoint operator on the types R A . Since our rst main result is that the stable uniform cbv xpoint operator is sound and complete for such uniform T-xpoint operators, it turns out that Filinski's construction combined with the CPS transformation can be regarded as a consequence of the general categorical axiomatics; by Axioms for Recursion in Call-by-Value 3 specialising it to the setting with a continuation monad, we obtain a semantic version of the second main result. Construction of this paper In Section 2 we recall the c -calculus and the call-by-value -calculus, which will be used as our working languages in this paper. In Section 3 we introduce our axioms for xpoint operators in these calculi (De- nition and give basic syntactic results. Section 4 demonstrates how our axioms are used for establishing Filinski's correspondence between recursion and iteration (which in fact gives a syntactic proof of the second main result). Up to this section, all results are presented in an entirely syntactic manner. In Section 5 we start to look at the semantic counterpart of our axiomatization, by recalling the categorical models of the c -calculus and the call-by-value -calculus. We then recall the notion of uniform T-xpoint operators on these models in Section 6, and explain how our axioms are distilled from the uniform T-xpoint operators (Theorem 2, the rst main result). In Section 7, we specialise the result in the previous section to the models of the call-by-value - calculus, and give a semantic proof of the second main result (Theorem 4). Section 8 gives some concluding remarks. 2. The Call-by-Value Calculi The c -calculus (computational lambda calculus) [18], an improvement of the call-by-value -calculus [21], is sound and complete for 1. categorical models based on strong monads (Moggi [18]) 2. Continuation-Passing Style transformation into the -calculus (Sabry and Felleisen [23]) and has proved useful for reasoning about call-by-value programs. In particular, it can be seen as the theoretical backbone of (the typed version of) the theory of A-normal forms [8], which enables us to optimise call-by-value programs directly without performing the CPS transformation. For these reasons, we take the c -calculus as a basic calculus for typed call-by-value programming languages. We also use an extension of the c -calculus with rst-class continuations, called the call-by- value -calculus, for which the soundness and completeness results mentioned above have been extended by Selinger [25]. 4 Hasegawa and Kakutani 2.1. The c -calculus The syntax, typing rules and axioms on the well-typed terms of the c -calculus are summarised in Figure 1. The types, terms and typing judgements are those of the standard simply typed lambda calculus (including the unit > and binary products ). 1 c ranges over the constants of type . As an abbreviation, we write let x be M in N for denotes the set of free variables in M . (As long as there is no confusion, we may use italic small letters for both variables and values. Capital letters usually range over terms, though we may also use some capital letters like F , G, H for higher-order functional values.) The crucial point is that we have the notion of values, and the axioms are designed so that the above-mentioned completeness results hold. Below we may call a term a value if it is provably equal to a value dened by the grammar. We write g f for the composition x:g (f x) of values f and g, and id for the identity function x :x. In the sequel, we are concerned not just about the pure c -calculus but also about its extensions with additional constructs and axioms. A c -theory is a typed equational theory on the well-typed expressions of the c -calculus (possibly with additional constructs) which is a congruence on all term constructions and contains the axioms of the c -calculus. A c -theory can be typically specied by the additional axioms (as the congruence generated from them), or as the equational theory induced by a model in the sense of Section 5, i.e. ' Centre and focus In call-by-value languages, we often regard values as representing eect- (nished or suspended) computations. While this intuition is valid, the converse may not always be justied; in fact, the answer depends on the computational eects under consideration. DEFINITION 1 (centre, focus). In a c -theory, we say that a term central if it commutes with any other computational eect, that is, let x be M in let y be N in be N in let x be M in L : holds for any N : and L : , where x and y are not free in M and N . In addition, we say that it is central and moreover copyable and discardable, i.e., let x be M in hx; and let x be M in 1 We do not include the \computation types" T and associated constructs, as they can be dened by Axioms for Recursion in Call-by-Value 5 Types ranges over base types Terms Typing Rules: Axioms: let x be V in let x be M in let y be (let x be L in M) in be L in let y be M in N be M in let x be N in f x be M in let y be N in hx; yi where let x be M in N stands for Figure 1. The c-calculus It is worth emphasising that a value is always focal, but the converse is not true (see Section 7.3). A detailed analysis of these concepts in several c -theories is found in [10]; see also discussions in Section 5. 2.2. The call-by-value -calculus Our call-by-value -calculus, summarised in Figure 2, is the version due to Selinger [25]. We regard it as an extension of the c -calculus with rst-class continuations and sum types (the empty type ? and binary sums +). We write : for the type The typing judgements take the form ' is a sequence of names (ranged over by , ,. ) with their types. A judgement represents a well-typed term M with at most m free We write FN(M) for 6 Hasegawa and Kakutani Types Terms Additional Typing Rules: Additional Axioms: be M in [; ]x Figure 2. The call-by-value -calculus the set of free names in M . In this judgement, M can be thought of as a proof of the sequent or the proposition in the classical propositional logic. Among the additional axioms, the rst one involves the mixed substitution M [C( )=[]( )] for a term M , a context C( ) and a name , which is the result of recursively replacing any subterm of the form []N by C(N) and any subterm of the form [ or further details on these syntactic conventions. Remark 1. We have chosen the cbv -calculus as our working language rstly because we intend the results in this paper to be compatible with the duality result of the second author [15] (see Section 7) which is based on Selinger's work on the -calculus [25], and secondly because it has a well-established categorical semantics, again thanks to Selinger. However our results are not specic to the -calculus; they apply also to any other language with similar semantics { for example, we could have used Hofmann's axiomatization of control operators [13]. Also, strictly speaking, the inclusion of sum types (coproducts) is not Axioms for Recursion in Call-by-Value 7 necessary in the main development of this paper, though they enable us to describe iterators more naturally (as general feedback operators, see Remark 3 in Section 4) and are also used in some principles on iterators like diagonal property (see Section 8), and crucially needed for the duality result in [25, 15]. Example 1. As an example, we can dene terms for the \double- negation elimination" and the \initial map": One can check that these combinators satisfy Hofmann's axioms in [13]. Also, we can use C (as will be done in the programming example in SML/NJ in Section 4). Centre and focus In the presence of rst-class continuations, central and focal terms coincide [28, 25], and enjoy a simple characterisation (thunkability [28]). LEMMA 1. In a cbv -theory, the following conditions on a term are equivalent. 1. M is central. 2. M is focal. 3. (thunkability) let x be M in u > We also note that central terms and values agree at function types [25]. LEMMA 2. In a cbv -theory, a term M : ! is central if and only if it is a value, i.e., (with x not free in M) holds. 3. Axioms for Recursion Throughout this section, we work in a c -theory. 3.1. Rigid functionals The key in our axiomatization of call-by-value xpoint operators is the notion of uniformity. In the call-by-name setting, we dene the 8 Hasegawa and Kakutani uniformity for xpoint operators with respect to the strict maps, i.e., those that preserve the bottom element (divergence). In a call-by-value setting, however, we cannot dene uniformity via this particular notion of strict maps, simply because everything is strict { if an input does not terminate, the whole program cannot terminate. Instead we propose to dene the uniformity principle with respect to a class of functionals that use their argument functions in a constrained way. called rigid if H (x:M holds for any M : ! , where x and y are not free in H and M . The word \rigid" was coined by Filinski in [5] (see discussions in Section 7.3). Intuitively, a rigid functional uses its argument exactly once, and it does not matter whether the argument is evaluated beforehand or evaluated at its actual use. LEMMA 3. are rigid, so is H 0 LEMMA 4. If H holds (where f and y are not free in H). Example 2. The reader may want to see rigid functionals in more concrete ways. In the case of settings with rst-class continuations, we have such a characterisation of rigid functionals, see Section 7.3. In general cases, a rigid functional typically takes the following form: :let x be f (h y) in N the following property: h v is central for any { later, such an h will be called \total" (Denition 4). N can be any term, possibly with side eects. It is easily seen that such an H y in any c -theory. On the other hand, in the presence of side eects, many purely functional terms fail to be rigid { e.g., constant functionals, as well as functionals like f:f f . 3.2. Axioms for recursion Now we are ready to state the main denition of this paper: our axiomatization of the call-by-value xpoint operators. Axioms for Recursion in Call-by-Value 9 DEFINITION 3 (stable uniform call-by-value xpoint operator). A type-indexed family of closed values x v is called a stable uniform call-by-value xpoint operator if the following conditions are satised: 1. (cbv xpoint) For any value F (where x is not free in F ) 2. (stability) For any value F (where f , x are not free in F ) 3. (uniformity) For values F holds, then H F G The rst axiom is known as the call-by-value xpoint equation; the eta-expansion in the right-hand-side means that x v F is equal to a value. The second axiom says that, though the functionals F and may behave dierently, their xpoints, when applied to values, satisfy the same xpoint equation and cannot be distinguished. The last axiom is a call-by-value variant of Plotkin's uniformity prin- ciple; here the rigid functionals play the r^ole of strict functions in the uniformity principle for the call-by-name xpoint operators. Our uniformity axiom can be justied by the fact that H (x v same xpoint equation as x v holds: xpoint equation for x v The following consideration conrms that the rigidness assumption cannot be dropped from the uniformity axiom. Let H be any value so that holds. Take any term M of type Hasegawa and Kakutani by be M in y:H f y (with f , g not free in M ). Then we have H H, and the uniformity would ask G to be hold; it is easily seen that x v and x v y, hence H must be rigid. Remark 2. As easily seen, the uniformity implies that any rigid functional preserves the xpoints of the identity maps: H . It is tempting to dene a notion of \call-by-value strictness" as this preservation of the xpoints of identities. In the pure functional settings (like the call-by- value PCF) where the divergence is the only eect, this call-by-value strictness actually coincides with the rigidness (this can be veried by inspecting the standard domain-theoretic model; see also Section 6). However, in the presence of other eects (in particular the case with rst-class continuations which we will study in the Section 4), rigidness is a much stronger requirement than this call-by-value strictness; for instance, the constant functional f:x v id as well as the \twice" functional are call-by-value strict (hence rigid in a pure functional setting), but they are not rigid in many c -theories and cannot be used in the uniformity principle. 3.3. On the axiomatizations of uniformity There are some alternative ways of presenting the axioms of stable uniform cbv xpoint operators. In particular, in [5] Filinski proposed a single uniformity axiom which amounts to our stability and uniformity axioms. LEMMA 5. For values F and g, y are not free in G). F G Proof. Axioms for Recursion in Call-by-Value 11 PROPOSITION 1. The stability axiom and uniformity axiom are equivalent to the following Filinski's uniformity axiom [5]: For (with x, f not free in F ), then H G. G Proof. Stability and uniformity imply Filinski's uniformity, because, stability Conversely, Filinski's uniformity implies stability and uniformity. First, for a value F is rigid, by Filinski's uniformity we have the stability x v For uniformity, suppose that we have values F holds. Then, by Lemma 5, we have H (f:x:F f (g:y:G g y) H. By applying Filinski's uniformity axiom, we obtain Since we have already seen that stability follows from Filinski's unifor- mity, it follows that x v we have H 4. Recursion from Iteration For grasping the r^ole of our axioms, it is best to look at the actual construction in the second main result: the correspondence of recursors and iterators in the presence of rst-class continuations due to Filinski [5]. So we shall describe this syntactic development before going into the semantic investigation which is the main issue of this paper. In this section we work in a call-by-value -theory, unless otherwise stated. 12 Hasegawa and Kakutani 4.1. Axioms for iteration As the case of recursion, we introduce a class of functions for determining the uniformity principle for an iterator. DEFINITION 4 (total value). In a c -theory, a value called is central for any value v : . The word \total" is due to Filinski [5], though in his original denition h v is asked to be a value rather than a central term. 3 DEFINITION 5 (uniform iterator). A type-indexed family of closed values called a uniform iterator if the following conditions are satised: 1. (iteration) For any value f (i.e. loop 2. (uniformity) For values is total and h f h, we have (loop So it is natural to expect that (loop g) h behaves in the same way as loop f for \well-behaved" h. The uniformity axiom claims that this is the case when h is total. It seems that this totality assumption is necessary. For example, let always-jumping function which is not total); then (loop g) performs the jump to the label , while loop f just diverges. 3 We shall warn that there is yet another use of the word \total" by Filinski [4] where a term is called total when it is discardable (in the sense of Denition 1); see [29] for a detailed analysis on this concept. Another possible source of confusion is that our notion of totality does not correspond to the standard notions of \total relations" or \total maps (in domain theory)". However, in this paper we put our priority on the compatibility with Filinski's development in [5]. Axioms for Recursion in Call-by-Value 13 Remark 3. Despite of its very limited form, the expressive power of an iterator is not so weak, as we can derive a general feedback operator from an iterator using sums and rst-class continuations, which satises (with a syntax sugar for sums) case f a of (in 1 x for 4.2. The construction Surprisingly, in the presence of rst-class continuations, there is a bijective correspondence between the stable uniform cbv xpoint operators and the uniform iterators. We recall the construction which is essentially the same as that in [5]. The construction is divided into two parts. For the rst part, we introduce a pair of contravariant constructions: Note that here we need rst-class continuations to implement step ; (it has \classical" type). One can easily verify that LEMMA 6. holds. pets holds. LEMMA 7. For values is rigid or F is total. pets The following observation implies that the two notions of uniformity for recursors and iterators are intimately related by this contravariant correspondence. 14 Hasegawa and Kakutani LEMMA 8. step bijective correspondence between rigid functionals of total functions of ! . Proof. The only non-trivial part is that step ; ( ) sends a rigid functional to a total function (the other direction and the bijectivity follow immediately from Lemma 4 and Lemma 6). Suppose that H : rigid. We show that step central. This can be veried as follows: let u be :H (y:[]y) x in let v be M in N be M in N))x be M in []N) x let v be M in let u be :H (y:[]y) x in N let v be M in :H (u:[]N) x :(let v be M in H (u:[]N) x) be M in u:[]N) x Since H is rigid, it follows that H (u:let v be M in []N) be M in u:[]N) x:We are then able to see that, if loop is a uniform iterator, the composition loop step yields a stable uniform xpoint operator restricted on the negative types :. The cbv xpoint axiom is veried as (by noting the equation (loop step ; The stability axiom holds as step (f:x:F f . The uniformity axiom follows from Lemma 7 and Lemma 8. If H :!: F G :!: H :!: and H is rigid (hence total by Lemma 4), the rst half of Lemma 7 implies (step Since step ; H is total by Lemma 8, by the uniformity of loop we have loop ; (step Axioms for Recursion in Call-by-Value 15 Conversely, if x v is a stable uniform xpoint operator, gives a uniform iterator: Again, the uniformity is a consequence of Lemma 7 and Lemma 8. One direction of the bijectivity of these constructions is guaranteed by the stability axiom (while the other direction follows from step ; pets So we have established PROPOSITION 2. There is a bijective correspondence between uniform iterators and stable uniform cbv xpoint operators restricted on negative types. The second part is to reduce xpoints on an arrow type ! to those on a negative type This is possible because we can implement a pair of isomorphisms between these types (again using rst-class continuations): switch 1 It is routinely seen that both switch 1 switch hold. It is also easy to verify (by direct calculation or by applying Proposition 8 in Section 7) that LEMMA 9. switch ; and switch 1 are rigid. By applying the uniformity axiom to the trivial equation switch ; (switch 1 PROPOSITION 3. There is a bijective correspondence between stable uniform cbv xpoint operators restricted on negative types and those on general function types. Hasegawa and Kakutani Proof. From a stable uniform cbv xpoint operator restricted on negative types, one can dene that on general function types by taking the equation above as denition; because of the uniformity, this in fact is the unique possibility of extending the operator to that on all function types. The only nontrivial point is that the uniformity axiom on this dened xpoint operator on general function typed can be derived from the uniformity axiom on the xpoint operator on negative types, which we shall spell out below. Suppose that we have values F such that H holds. Since rigid functionals are closed under composition (Lemma 3) and switch and switch 1 are rigid (Lemma 9), switch 1 H switch is also rigid. By applying the uniformity axiom (on negative types) to the equation (switch 1 (switch 1 we obtain switch 1 which implies (by applying switch to both sides of the equation) In summary, we conclude that, in the presence of rst-class continu- ations, stable uniform cbv xpoint operators are precisely those derived from uniform iterators, and vice versa: THEOREM 1. There is a bijective correspondence between uniform iterators and stable uniform cbv xpoint operators. switch loop code written in SML/NJ [17, 11] is found in Figure 3. Axioms for Recursion in Call-by-Value 17 (* an empty type "bot" with an initial map A : bot -> 'a *) datatype fun A (VOID (* the C operator, C : (('a -> bot) -> bot) -> 'a *) (* basic combinators *) fun step F fun switch l (* an iterator, loop : ('a -> 'a) -> 'a -> bot *) (* recursion from iteration *) Figure 3. Coding in SML/NJ (versions based on SML '97 [17]) 5. Categorical Semantics The rest of this paper is devoted to investigating the semantic counterpart of our stable uniform cbv xpoint operators and for giving our two main results in a coherent way. In this section we recall some preliminaries on the underlying categorical structures which will be used in our semantic development. 5.1. Models of the c -calculus Let C be a category with nite products and a strong monad and are the unit and multiplication of the monad T , and is the tensorial strength with respect to the nite products of C (see e.g. [18, 19] for these category-theoretic concepts). We write C T for the Kleisli category of T , and for the associated left adjoint explicitly, J is the identity on objects and sends f 2 C (X; Y ) to Y f 2 C T (X; Y We assume that C has Kleisli exponentials, i.e., for every X in C the functor J(( Hasegawa and Kakutani has a right adjoint X This gives the structure for modelling computational lambda calculus [18]. Specically, we x an object for each base type b and dene the interpretation of types as well-typed interpreted inductively as a morphism of once we x the interpretations of constants; see Appendix A for a summary. Following Moggi, we call such a structure a computational model. PROPOSITION 4. [18] The computational models provide a sound and complete class of models of the computational lambda calculus. In fact, we can use the c -calculus as an internal language of a computational model { up to the choice of the base category C (which may correspond to either syntactically dened values, or more semantic values like thunkable terms, or even something between them; see [10] for a detailed consideration on this issue) { in a similar sense that the simply typed lambda calculus is used as an internal language of a cartesian closed category [16]. 5.2. Models of the call-by-value -calculus Let C be a distributive category, i.e., a category with nite products and coproducts so that preserves nite coproducts for each A. We call an object R a response object if there exists an exponential R A for each A, i.e., C ( A; R) ' C ( ; R A ) holds. Given such a structure, we can model the cbv -calculus in the Kleisli category C T of the strong monad [25]. A term ' M : j is interpreted as a morphism of C T ([[]]; for The interpretation is in fact a typed version of the call-by-value CPS transformation [21, 25], as sketched in Appendix B. Following Selinger, we call C a response category and the Kleisli category C T a category of continuations and write R C for C T (though in [25] a category of continuations means the opposite of R C ). PROPOSITION 5. [25] The categories of continuations provide a sound and complete class of models of the cbv -calculus. As the case of the c -calculus, we can use the cbv -calculus as an internal language of a category of continuations [25]. Axioms for Recursion in Call-by-Value 19 5.3. Centre and focus We have already seen the notion of centre and focus in the c -calculus and the cbv -calculus in a syntactic form (Denition 1). However, these concepts originally arose from the analysis on the category-theoretic models given as above. Following the discovery of the premonoidal structure on the Kleisli category part C T (R C ) of these models [22], Thielecke [28] proposed a direct axiomatization of R C not depending on the base category C (which may be seen as a chosen category of \values") but on the subcategory of \eect-free" morphisms of R C , which is the focus (equivalently centre) of R C . Fuhrmann [10] carries out further study on models of the c -calculus along this line. DEFINITION 6 (centre, semantic denition). Given a computational model with the base category C and the strong monad T , an arrow called central if, for any g : compositions (Y g) (f Note that the products are not necessarily bifunctorial on C T ; they form premonoidal products in the sense of [22] (the reader familiar with this notion might prefer to use instead of for indicating that they are not cartesian products). This notion of centrality amounts to the semantic version of centrality in Denition 1. In this paper we do not go into the further details of these semantic analyses. However, we will soon see that these concepts naturally arise in our analysis of the uniformity principles for recursors and iterators. In particular, a total value (equivalently the term x : ' precisely corresponds to the central morphisms in the semantic models. In the case of the models of the cbv -calculus, the centre can be characterised in terms of the category of algebras, for which our uniformity principles are dened; that is, we have PROPOSITION 6. f central if and only if its counterpart in C is an algebra morphism from the algebra (R B to (R A ; R A ). We discuss more about this in Section 7; there this observation turns out to be essential in relating the uniformity principles for recursion and iteration in the cbv -theories. We note that this result has been observed in various forms in [28, 25, 10]. 4 In terms of C , f 2 C T (X; Y holds for any g 2 C T (X are the left-rst and right-rst pairings (Appendix A). 20 Hasegawa and Kakutani 6. Uniform T -Fixpoint Operators In this section we shall consider a computational model with the base category C and a strong monad T . 6.1. Uniform T-Fixpoint Operators We rst recall the notion of uniform T -xpoint operator of Simpson and Plotkin [27], which arose from considerations on xpoint operators in Axiomatic Domain Theory (ADT) [7, 26]. In ADT, we typically start with a category C of predomains, for example the category of !-complete partial orders (possibly without bottom) and continuous functions. Then we consider the lifting monad T on C , which adds a bottom element to !-cpo's. Then objects of the form TX are pointed cpo's (!-cpo's with bottom), on which we have the least xpoint op- erator. It is also easily checked that such a pointed cpo has a unique T -algebra structure (in fact any T -algebra arises in this way in this setting, though we will soon see that this is not the case if we take a continuation monad as T ), and an algebra morphism is precisely the bottom-preserving maps, i.e., the strict ones. As is well known, the least xpoint operator enjoys the uniformity principle with respect to such strict maps. By abstracting this situation we have: DEFINITION 7 (uniform T-xpoint operator [27]). A T-xpoint operator on C is a family of functions such that, for any f : holds. It is called uniform if, for any f : imply TTY TY TY TY f Thus a T-xpoint operator is given as a xpoint operator restricted on the objects of the form TX. One may easily check that, in the domain-theoretic example sketched as above, the condition h Th says that h is a strict map. Axioms for Recursion in Call-by-Value 21 This limited form of xpoint operators, however, turns out to be sucient to model a call-by-value xpoint operator. To see this, suppose that we are given an object A with a T -algebra structure : (that is, we ask we have ( f Therefore we can extend a T-xpoint operator ( ) to be a xpoint objects with T -algebra structure by dening f Moreover, given a uniform T-xpoint operator ( ) , it is easy to see that this extended xpoint operator ( ) on T -algebras is uniform in the following sense: for T -algebras h is a T -algebra morphism) and g f Furthermore, such a uniform extension is unique: given a uniform xpoint operator ( ) on objects with T -alpgebra structure, by applying this uniformity to ( f completely determined by its restriction on free algebras (TX;X ), i.e., a uniform T-xpoint operator. In particular, Kleisli exponentials X ) Y t in this scheme, where the T -algebra structure given as the adjoint mate (currying) of (see Appendix A for notations). Since we interpret a function type as a Kleisli exponential, this fact enables us to use a uniform T-xpoint operator for dealing with a xpoint operator on function types. We note that corresponds to an eta-expansion in the c -calculus. That is, if a term ' represents an arrow f 22 Hasegawa and Kakutani LEMMA 10. For any ' M : ! , holds. Proof. This observation is frequently used in distilling the axioms of the stable uniform cbv xpoint operators below. 6.2. Axiomatization in the c -calculus Using the c -calculus as an internal language of C T , the equation f f f on X ) Y can be represented as The side condition means that F corresponds to an arrow in C (X the operator ( can be equivalently axiomatized by a slightly dierent operator subject to f z = X;Y f f z , with an additional condition f X;Y f) z . In fact, we can dene such a ( ) z as ( X;Y ( )) and conversely and it is easy to see that these are in bijective correspondence. The condition f equivalently z , is axiomatized in the c -calculus as (by recalling that X;Y ( ) gives an eta-expansion) which is precisely the cbv xpoint axiom. The additional condition f) z is axiomatized as F is a value This is no other than the stability axiom. We thus obtain the rst two axioms of our stable uniform cbv xpoint operators, which are precisely modelled by T-xpoint operators. Axioms for Recursion in Call-by-Value 23 6.3. Uniformity axiom Next, we shall see how the uniformity condition on T-xpoint operators can be represented in the c -calculus. Following the previous discus- sions, we consider H it is an algebra morphism from (X ) Y; X;Y ) to (X 0 Spelling out this condition, we ask H to satisfy H equivalently In terms of the c -calculus, this means that an eta-expansion commutes with the application of H; therefore, in the c -calculus, we to be a value such that holds for any M : We have called such an H rigid, and dened the uniformity condition with respect to such rigid functionals. Remark 4. Actually the uniformity condition obtained by the argument above is as follows, which is slightly weaker than stated in Denition 3: For G. However, thanks to Lemma 5, we can justify the uniformity axiom in Denition 3. 6.4. Soundness and completeness Now we give one of the main result of this paper. THEOREM 2. The computational models with a uniform T -xpoint operator provide a sound and complete class of models of the computational lambda calculus with a stable uniform call-by-value xpoint operator. 5 A characterisation of rigid functions (on computation types) in the same spirit is given in Filinski's thesis [6] (Section 2.2.2) though unrelated to the uniformity of Hasegawa and Kakutani This extends Proposition 4 with the stable uniform call-by-value xpoint operator and uniform T-xpoint operators. Most part of soundness follows from a routine calculation. However, the interpretation of the stable uniform call-by-value xpoint operator and the verication of the axioms do require some care: we need to consider a parameterized xpoint operator (with parameterized uniformity) for interpreting the free variables. Thus we have to parameterize the considerations in Section 6.1. This can be done along the line of Simpson's work [26]. Below we outline the constructions and results needed for our purpose. PROPOSITION 7. A uniform T -xpoint operator uniquely extends to a family of functions where X ranges over objects of C and such that 1. (parameterized xpoint) For f holds 2. (parameterized uniformity) For Th X;A and h h X;A imply g ThX;A A Here we only give the construction of f A and omit the proof (which largely consists of lengthy diagram chasings and we shall leave it for interested readers { see also [26]). Let (recall that is the T -algebra structure on Using , we dene dfe Axioms for Recursion in Call-by-Value 25 Finally we have f A. By trivialising the parameterization and by considering just the free algebras, one can recover the original uniform T-xpoint operator. The uniqueness of the extension follows from the uniformity (essentially in the same way as described in Section 6.1). Using this parametrically uniform parameterized xpoint operator, now it is not hard to interpret a stable uniform call-by-value xpoint operator in a computational model with a uniform T-xpoint operator and check that all axioms are validated. Completeness is shown by constructing a term model, for which there is no diculty. Since the uniform T-xpoint operator on this term model is directly dened by the stable uniform call-by-value x- point operator on the types also because we have already observed that rigid functionals are characterized as the algebra morphisms in this model, this part is truly routine. 7. Recursion from Iteration Revisited 7.1. Iteration in the category of continuations Let C be a response category with a response object R. An iterator on the category of continuations R C is a family of functions ( R C Spelling out this denition in C , to give an iterator on R C is to give a family of functions ( (R A ; R A holds for f 2 C (R A ; R A ). Thus an iterator on R C (hence in the cbv - calculus) is no other than a xpoint operator on C (hence the target call-by-name calculus) restricted on objects of the form R A (\negative objects"). Example 3. We give a simple-minded model of the cbv -calculus with an iterator. Let C be the category of !-cpo's (possibly without bottom) and continuous maps, and let R be an !-cpo with bottom. Since C is a cartesian closed category with nite coproducts, it serves as a response category with the response object R. Moreover there is a least xpoint operator on the negative objects R A because R A has a bottom element, thus we have an iterator on R C (which in fact is a unique uniform iterator in the sense below). Remark 5. A careful reader may notice that we actually need a parameterized version of the iterator for interpreting free variables as well as free names: should be dened as a function from R C (X A+Y ) to R C (X A; Y ). However, this parameterization, including 26 Hasegawa and Kakutani that on uniformity discussed below, can be done in the same way as in the previous section (and is much easier); a uniform iterator uniquely extends to a parametrically uniform parameterized iterator { we leave the detail to the interested reader. 7.2. Relation to uniform T-fixpoint operators For any object A, the negative object R A canonically has a T -algebra structure :x A :m (f R A for the monad . Thus the consideration on the uniform T - xpoint operators applies to this setting: if this computational model has a uniform T-xpoint operator, then we have a xpoint operator on negative objects, hence we can model an iterator of the cbv -calculus in the category of continuations. Conversely, if we have an iterator on R C , then it corresponds to a xpoint operator on negative objects in C , which of course include objects of the form . Therefore we obtain a T-xpoint operator. It is then natural to expect that (along the consideration in Section 6.1), if the iterator satises a suitable uniformity condition, then it bijectively corresponds to a uniform T-xpoint operator. This uniformity condition on an iterator must be determined again with respect to algebra morphisms. So we regard h 2 R C as \strict" when its counterpart in C (R B ; R A ) is an algebra morphism from (R to (R A ; A ), i.e., h holds in C . We say that an iterator ( ) on R C is uniform if f holds for THEOREM 3. Given a response category C with a response object R, to give a uniform R R ( -xpoint operator on C is to give a uniform iterator on R C . Proof. Immediate, since a uniform R R ( -xpoint operator uniquely extends to a uniform xpoint operator on negative objects (hence a uniform iterator) { the uniqueness of the extension follows from the uniformity (by the same argument as given in Section 6.1). 2 Axioms for Recursion in Call-by-Value 27 Fortunately, the condition to be an algebra morphism is naturally represented in a cbv -theory. A value h : A!B represents an algebra morphism if and only if holds { in fact, the CPS transformation (see Appendix B) of this equation is no other than the equation h . By Lemma 1, in a cbv -theory, this requirement is equivalent to saying that hx is a central term for each value x (this also implies Proposition 6 in Section 5), hence h is total. Therefore we obtain the uniformity condition for an iterator in Section 4. This is remarkable, as it says that the idea of dening the uniformity principle of xpoint operators with respect to algebra morphisms (from ADT) and the idea of dening the uniformity principle of iterators with respect to eect-free morphisms (from Filin- ski's work) coincide in the presence of rst-class continuations, despite their very dierent origins; technically, this is the substance of the left- to-right implication of Proposition 6. In summary, we have semantically shown Theorem 1: THEOREM 4 (Theorem 1 restated). In a cbv -theory, there is a bijective correspondence between the stable uniform cbv xpoint operators and the uniform iterators. In a sense, the syntactic proof in Section 4 gives an example of direct style reasoning, whereas this semantic proof provides a continuation-passing style reasoning on the same result. We can choose either stable uniform xpoint operators (in syntactic, direct style) or uniform T - xpoint operators (in semantic, monadic or continuation-passing style) as the tool for reasoning about recursion in call-by-value setting; they are as good as the other (thanks to Theorem 2). 7.3. On Filinski's uniformity In [5] Filinski introduced uniformity principles for both cbv xpoint operators and iterators, for establishing a bijective correspondence between them. While his denitions turn out to be sucient for his purpose, in retrospect they seem to be somewhat ad hoc and are strictly weaker than our uniformity principles. Here we give a brief comparison. First, Filinski calls a value is a value for each value v : . However, while a value is always central, the converse is not true. Note that, while the notion of centre is uniquely determined for each cbv -theory (and category of continuations), the notion of value is not canonically determined (a category of continuations 28 Hasegawa and Kakutani can arise from dierent response categories [25]). Since the uniformity principle is determined not in terms of the base category C but in terms of the category of algebras, it seems natural that it corresponds to the notion of centre which is determined not by C but by C T . Second, Filinski calls a value H there are total such that holds (cf. Example 2). It is easily checked that if H is rigid in the sense of Filinski, it is also rigid in our sense { but the converse does not hold, even if we change the notion of total values to ours (for instance, switch ; in Section 4 is not rigid in the sense of Filinski). By closely inspecting the correspondence of rigid functionals and total functions via the step/pets and switch constructions, we can strengthen Filinski's formulation to match ours: PROPOSITION 8. In a cbv if and only if there are total such that holds. Proof. By pre- and post-composing switch and switch 1 , rigid functionals of are in bijective correspondence with those of :( are, by Lemma 8, in bijective correspondence with the total functions of the step/pets construction. A total function of ( is equal to hy; ki:hh 2 hy; ki; h 1 total functions We note that total functions of are in bijective correspondence with those of we take hy; ki:x:k (g y In summary, for any rigid functional H have total such that holds. By simplifying the right hand side of this equation, we obtain the result. 2 This subsumes Filinski's rigid functionals as special cases where h 2 does not use the second argument. Axioms for Recursion in Call-by-Value 29 8. Conclusion and Further Work We have proposed an axiomatization of xpoint operators in typed call-by-value programming languages, and have shown that it can be justied in two dierent ways: as a sound and complete axiomatization for uniform T-xpoint operators of Simpson and Plotkin [27], and also by Filinski's bijective correspondence between recursion and iteration in the presence of rst-class continuations [5]. We also have shown that these results are closely related, by inspecting the semantic structure behind Filinski's construction, which turns out to be a special case of the uniform T-xpoint operators. We think that our axioms are reasonably simple, and we expect they can be a practical tool for direct-style reasoning about call-by-value programs involving recursion, just in the same way as the equational theory of the computational lambda calculus is the theoretical basis of the theory of A-normal forms [23, 8]. 8.1. Further principles for call-by-value recursion It is an interesting challenge to strengthen the axioms in some systematic ways. Below we give some results and perspectives. Dinaturality, diagonal property, and Iteration Theories By adding other natural axioms on an iterator in the presence of rst- class continuations, one may derive the corresponding axioms on the cbv xpoint operator. In particular, we note that the dinaturality loop (g on an iterator loop precisely amounts to the axiom on the corresponding cbv xpoint operator x v (note that this axiom implies both the cbv xpoint axiom and the stability axiom). Similarly, the diagonal property on the iterator loop (x::[; ](f corresponds to that on the xpoint operator These can be seen axiomatizing the call-by-value counterpart of Conway theories [1, 12]. In [27], Simpson and Plotkin have shown that the Hasegawa and Kakutani equational theory induced by a uniform Conway operator (provided it is consistent) is the smallest iteration theory of Bloom and Esik [1], which enjoys very general completeness theorem. Regarding this fact, we conjecture that our axioms for stable uniform cbv xpoint operators together with the dinaturality and diagonal property capture all the valid identities on the cbv xpoint operators, at least in the presence of rst-class continuations. Mutual recursion and extensions to product types One may further consider the call-by-value version of the Bekic property (another equivalent axiomatization of dinatural and diagonal properties [12]) along this line, which could be used for reasoning about mutual recursion. For this purpose it is natural to extend the denition of xpoint operators on product types of function types, and also extend the notion of rigid functionals to those with multiple parameters. These extensions are syntactically straightforward and semantically natural (as the category of algebras is closed under nite products). Spelling this out, for , we can (uniquely) extend the xpoint operator on 0 by (using the idea of Section property is stated as, for . For example, from Bekic property and uniformity, we can show equations like x v Fixpoint objects Another promising direction is the approach based on xpoint objects [2], as a uniform T-xpoint operator is canonically derived from a x- point object whose universal property implies strong proof principles. For instance, in Example 3, a uniform iterator is unique because the monad R R ( has a xpoint object. For the setting with rst-class con- tinuations, it might be fruitful to study the implications of the existence of a xpoint object of continuation monads. Graphical axioms Jerey [14] argues the possibility of partial traces as a foundation of graphical reasoning on recursion in call-by-value languages. Schweimeier and Jerey [24] demonstrate that such graphical axioms can be used to Axioms for Recursion in Call-by-Value 31 verify the closure conversion phase of a compiler. Similar consideration is found in Fuhrmann's thesis [10]. It follows that most of the equalities proposed in these approaches can be derived (up to some syntactic dierences) from the axioms for stable uniform call-by-value xpoint operators with dinatural and diagonal (or Bekic) property; detailed comparisons, however, are left as a future work. In a related but dierent direction, Erkok and Launchbury [3] propose graphical axioms for reasoning about recursion with monadic eects in lazy functional programming languages. Friedman and Sabry [9] also discuss about recursion in such settings (\unfolding recursion" versus \updating recursion") and propose an implementation of \up- dating recursion" via a monadic eect. Although these approaches have the common underlying semantic structure as the present work, the problems cosidered are rather of dierent nature and it is not clear how they can be compared with our work. 8.2. Relating recursion in call-by-name and call-by-value The results reported here can be nicely combined with Filinski's duality [4] between call-by-value and call-by-name languages with rst-class control primitives. In his MSc thesis [15], the second author demonstrates that recursion in the call-by-name -calculus [20] exactly corresponds to iteration in the call-by-value -calculus via this duality, by extending Selinger's work [25]. Together with the results in this paper, we obtain a bijective correspondence between call-by-name recursion and call-by-value recursion (both subject to suitable uniformity principles) Recursion in cbn -calculus , Iteration in cbv -calculus , Recursion in cbv -calculus which seems to open a way to relate the reasoning principles on recursive computations under these two calling strategies. Acknowledgements We thank Shin-ya Katsumata and Carsten Fuhrmann for helpful discussions and their interests on this work, and the anonymous reviewers of FoSSaCS 2001 and this submission for numerous insightful sugges- tions. Part of this work was done while the rst author was visiting Laboratory for Foundations of Computer Science, University of Edinburgh. Hasegawa and Kakutani --R Iteration Theories. New foundations for Declarative continuations: an investigation of duality in programming language semantics. Recursion from iteration. Controlling Axiomatic Domain Theory in Categories of Partical Maps. The essence of compiling with continuations. Recursion is a computational e Models of Sharing Graphs: A Categorical Semantics of let and letrec. Sound and complete axiomatisations of call-by-value control operators Duality between Call-by-Name Recursion and Call-by- Value Iteration Introduction to Higher Order Categorical Logic. Computational lambda-calculus and monads Notions of computation and monads. Premonoidal categories and notions of computation. Reasoning about programs in continuation-passing style Control categories and duality: on the categorical semantics of the lambda-mu calculus Recursive types in Kleisli categories. Complete axioms for categorical Categorical Structure of Continuation Passing Style. Using a continuation twice and its implications for the expressive power of call/cc. --TR --CTR Yoshihiko Kakutani , Masahito Hasegawa, Parameterizations and Fixed-Point Operators on Control Categories, Fundamenta Informaticae, v.65 n.1-2, p.153-172, January 2005 Atsushi Ohori , Isao Sasano, Lightweight fusion by fixed point promotion, ACM SIGPLAN Notices, v.42 n.1, January 2007 Carsten Fhrmann , Hayo Thielecke, On the call-by-value CPS transform and its semantics, Information and Computation, v.188 n.2, p.241-283, 29 January 2004 Martin Hyland , Paul Blain Levy , Gordon Plotkin , John Power, Combining algebraic effects with continuations, Theoretical Computer Science, v.375 n.1-3, p.20-40, May, 2007

continuations;iteration;categorical semantics;recursion;call-by-value

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Challenges and Solutions to Adaptive Computing and Seamless Mobility over Heterogeneous Wireless Networks.

Recent years have witnessed the rapid evolution of commercially available mobile computing environments. This has given rise to the presence of several viable, but non-interoperable wireless networking technologies each targeting a niche mobility environment and providing a distinct quality of service. The lack of a uniform set of standards, the heterogeneity in the quality of service, and the diversity in the networking approaches makes it difficult for a mobile computing environment to provide seamless mobility across different wireless networks. Besides, inter-network mobility will typically be accompanied by a change in the quality of service. The application and the environment need to collaboratively adapt their communication and data management strategies in order to gracefully react to the dynamic operating conditions.This paper presents the important challenges in building a mobile computing environment which provides seamless mobility and adaptive computing over commercially available wireless networks. It suggests possible solutions to the challenges, and describes an ongoing research effort to build such a mobile computing environment.

Introduction Recent years have witnessed explosive growth in the field of mobile computing, resulting in the development of mobile computing environments which can provide a very respectable level of computing and communications capabilities to a mobile user. Unfortunately, one consequence of this rapid evolution has been the emergence of several viable, but non-interoperable wireless networking technologies, each targeting a specific operation environment and providing a distinct quality of service (QoS). The lack of a single set of standards, the heterogeneity in QoS, and the diversity in networking approaches makes it difficult for a mobile computing environment to provide seamless mobility across different wireless networks. Since inter-network migration may result in significant changes in bandwidth or other QoS parameters, adaptive computing is necessary in a mobile computing environment which provides seamless mobility over multiple wireless networks. A graceful reaction to sudden QoS changes can only happen when both the environment and the applications collaboratively adapt to the dynamic operating conditions. Seamless mobility and adaptive computing are thus related and complementary goals. The ideal scenario in mobile computing is the following: a mobile user carries a portable computer with one or more built-in wireless network interfaces (ranging from O(Kbps) CDPD WMAN to O(Mbps) 2.4 GHz WLAN), and moves around both indoors and outdoors while executing applications in a uniform working environment (oblivious of the dynamics of the underlying inter-network migration). In order to achieve this scenario, the mobile computing environment needs to provide at least the following functionality: ffl seamless mobility across different wireless networks. ffl graceful adaptation to dynamic QoS variations. ffl a uniform framework for executing applications across diverse underlying environments. ffl backbone support for mobile applications. ffl seamless recovery in the presence of failures. In summary, the goal of a mobile computing environment is to provide the illusion of a uniform working environment (with possibly dynamically varying QoS) on top of underlying heterogeneous technologies. Inherent limitations of wireless and imposed limitations of the state-of-the-art technology currently restrict the realizability of the provisions above (example of the former: the bandwidth of a wireless MAN is one to two orders of magnitude lower than a wireless LAN; example of the latter: switching between RAM and CDPD may require the portable computer to reboot). The focus of this paper is to distinguish the inherent scientific limitations from the state-of-the-art limitations, and identify the challenges which need to be solved in order to approximate the ideal scenario. In contemporary research, there is no working implementation of a mobile computing environment which provides seamless mobility on top of heterogeneous commercial networks. Several critical research issues in both seamless mobility and adaptive computing have not been addressed, or even identified. In order to understand the challenges in this field, and also experiment with some preliminary solutions, we started building the PRAYER 1 mobile computing environment, which provides a platform for seamless mobility and adaptive computing on top of diverse commercially available wireless networks. We achieve seamless mobility by providing the 'software glue' to hold together diverse commercial networks, and build an adaptive computing framework on top of the seamless network. Three important components of PRAYER relevant to this paper are connection management, adaptive file system support, and OS/language support for application-level adaptation. The connection manager enables seamless mobility over diverse networks and provides coarse estimates of available network QoS. The file system supports application-directed adaptation for partially connected opera- tion. PRAYER provides simple OS/language support to structure applications in an adaptive environment. Preliminary results indicate that PRAYER offers an effective environment for adaptive computing and seamless mobility over commercial networks. Our system is named PRAYER after the most popular type of wireless uplink transmissions to an infinite server. server application home computer virtual point-to-point link computer portable wired backbone wireless network1 wireless network2 Figure 1: Mobile Computing Model The rest of the paper is organized as follows. Section 2 describes the target mobile computing model. Section 3 identifies the research goals. Section 4 explores the research challenges and evaluates different approaches. Section 5 describes the PRAYER mobile computing envi- ronment. Section 6 summarizes related work, and Section 7 concludes the paper. Mobile Computing Model Our mobile computing model is fairly standard, as shown in Figure 1. Five components are of relevance: (a) portable computer, equipped with multiple wireless network interfaces (b) home computer, the repository of data and computing resources for a mobile user on the backbone network (c) application servers, e.g. database server, file server etc. (d) wireless networks, typically autonomously owned and managed, and (e) backbone network. In our environment, the following points are distinctive. 1. A wireless network may be autonomously owned and operated. It may use proprietary network protocols. Its base stations will not be accessible for software changes to support seamless mobility. Therefore, it is important to operate on top of off-the-shelf commercial networks. 2. The wireless network may vary in its offered quality of service (QoS) (Figure 2) and can be parametrized by the following: bandwidth, latency, channel error, range, access protocol, connection failure probability, connection cost, pricing structure and security. 3. The portable computer may vary in sophistication from a PDA to a high-end notebook, and can be parametrized by the following: compute power, memory, available network interfaces, disk space, available compression techniques, battery power and native OS. 4. The dynamics of the underlying environment are primarily influenced by user mobility. A portable computer will see a variable QoS networking environment depending on its current connections (disconnection is the extreme case). In the PRAYER model, the home computer plays an important role. It runs the application stubs and file system clients on behalf of the portable. Essentially, it performs the role of a client with respect to the application servers, and the server with respect to the portable. In this Technology Range Bandwidth Access Latency Cost setup country 10ms TDMA 2.4 GHz radio Ethernet 500ft 1Mbps 5ms CSMA/CA 1.5Mbps 10Mbps km 100us CSMA/CD setup $1000/mth (AT&T) CDPD 500ms AMPS CSMA/CA between $80/MB (Motorola) Cellular mdm MAN MAN 50ms TDMA 40c/min Radio mdm MAN 100ms CSMA/CA $100/MB $10000/mth FDMA 100ms 100Kbps country Satellite 30ft 1Mbps 1ms setup point point-to Infrared 10Kbps 10Kbps 10Kbps Figure 2: Comparison Chart of Wired/Wireless Networking Technologies (numbers represent model, the goal of seamless mobility is to establish a 'virtual point-to-point' connection between the portable computer and the home computer over multiple wireless networks. Likewise, the adaptive computing solutions focus on the consistency management issues between the home and the portable. Though restrictive, this model simplifies the systems architecture while still allowing us to study the issues in seamless mobility and adaptive computing. 3 Research Goals The fundamental goal is to provide a mobile computing environment which enables a portable computer with multiple wireless networking interfaces to seamlessly move between the different proprietary networks without disturbing the application, and to provide systems support for applications to adapt to dynamic QoS variations gracefully. The technical goals fall into two classes: service goals - what services need to be provided to the applications, and system goals what systems issues need to be solved to provide the services. 3.1 Service Goals Three major types of activity from the portable computer include computation, communication, and information access. The mobile computing environment should provide services for each of the above within a uniform framework independent of the dynamics of the underlying system. Computation: Delegating compute intensive and non-interactive communication-intensive tasks to a backbone computer saves critical wireless bandwidth, and also portable compute cycles. Remote agent invocation at the backbone is a general and flexible mechanism to provide backbone computation services [19]. Communication: Efficient, medium-transparent, secure communication is a primary service requirement. Since a portable computer may have multiple wireless connections, the efficient choice for communication depends on the application and network characteristics (e.g. CDPD for short bursts of data, cellular modem for periodic data traffic). Medium-transparent communication requires seamless migration [4] (with possible notification of QoS change to appli- cations) between autonomously managed and potentially non-interoperable networks. Information access: Filtering information destined for the portable at the backbone network saves wireless bandwidth. This is traditionally achieved by application-level filtering [40]. An alternative approach, which we pursue, is to have applications use the file system as a filtering mechanism by imposing different semantics-driven consistency policies. Uniform framework: User mobility, migration between wireless networks, and network fail- ures/partitions cause the underlying environment to change dynamically. The mobile computing environment should provide a uniform framework for the abstraction of the current capabilities of the environment, and a simple mechanism for applications to be notified upon change in the environment. 3.2 System Goals This section identifies the systems goals needed to support the services listed in Section 3.1 within the constraints imposed by the mobile computing environment in Section 2. Seamless mobility: Medium-transparent data transport in the presence of network connec- tions/disconnections requires seamless migration between the cells of a wireless network (com- mon case), and between autonomously managed wireless networks with possibly very different QoS (general case). Multiple connections: A portable computer may have simultaneous access to multiple wireless networks. Depending on the type of application data traffic, required QoS, and priority of transmission, the mobile computing environment should be able to make an appropriate choice of wireless network for data transmission. The portable should thus be provided with the ability to use different wireless networks for different types of application traffic. Caching and consistency mechanism: Since the wireless medium is a scarce resource, caching data at the portable may significantly reduce access time and communication traffic, thereby improving performance and reducing cost. However, caching also introduces data consistency issues. While contemporary research has typically concentrated on caching and consistency issues for disconnected operation, partial connection (with varying degrees of QoS) will soon become a common mode of operation. Caching and consistency policies should adapt with the network QoS. We argue for application-directed consistency policies and simple APIs for applications to interact with the mobile computing environment in order to enforce these policies. Seamless recovery: When network connections fail, it may be possible to use alternative redundant network connections in order to recover from primary link failure. Such recovery should be seamless, but may still change the caching/consistency policies due to a change in QoS. Previously cached data needs to be reintegrated according to the revised consistency poli- cies. Recovery of essential state and the mechanism for migration between different consistency policies should be transparent to the application. Backbone support: Providing agent invocation and application-directed caching/consistency policies requires backbone support. We suggest the use of a dedicated home computer in order to provide the backbone support for executing agents, application stubs, and file system clients. Although this approach may induce greater overhead, it reduces the consistency management problem to just keeping the portable consistent with its home, and permits supporting different consistency policies on the backbone. Application support: In order to effectively adapt to QoS changes in the network, applications need to make 'aware' decisions [31, 39]. The trade-off between providing applications the flexibility to adapt based on context-awareness, and introducing complexity in the application logic in order to handle such adaptation, is a delicate one. Ideally, the applications will control the policies for consistency management of their data, while the system will provide the mechanisms to realize these policies. The split between policies and mechanisms will enable the applications to adapt to the dynamic QoS while not being concerned with the actual mechanisms of imposing consistency. Research Challenges and Solutions The mobile computing environment consists of five entities: portable computers, home com- puters, application servers, wireless networks, and the backbone wired network. The wireless network is typically proprietary, and the base station is thus inaccessible for customization to support seamless mobility. The service goals in Section 3.1 all require some form of backbone support. The home computer is the natural entity to provide this support, since we cannot provide systems software support on the base stations or mobile service stations 2 . The system architecture at a very high level is thus fairly obvious: the computing entities are the portable computer (one per user), home (one per user) and application servers (one per application) 3 . The home and the portable computer are connected by a dynamically changing set of network connections with diverse QoS - each with a wireless and wired component. The home and application servers are connected via the backbone network, and application servers are oblivious of user mobility. The home interacts with the application server on behalf of the portable. The home and the portable together provide the functionality required by the goals in Section 3. Within the above framework, a number of research challenges need to be addressed. We classify them into three broad areas: ffl application structure ffl seamless mobility and management of multiple connections 2 Henceforth, the term base station generically refers to the backbone agent which supports mobility in a network. 3 In practice, an application server may serve several applications, and a home computer may be a dedicated computer from which mobile users can lease home service. ffl adaptive computing and consistency management 4.1 Application Structure Applications need to adapt to the dynamic network conditions because the mobile computing environment has no knowledge of the application semantics. While the environment can perform some application-independent adaptation (such as compression, batching writes, etc.), semantics-dependent adaptation - such as deciding which part of the data is critical and needs to be kept consistent over a low QoS network - needs to be performed by the application. The application structure thus depends on the type of adaptation support provided by the environment to applications. In particular, two issues are of interest: (a) whether the environment notifies the applications of QoS changes, and (b) whether the environment provides support for notification handling and dynamic QoS negotiation. 4.1.1 Transparency versus Notification The major source of problems in our mobile computing environment is the dynamics of the network connections. Due to user mobility, connections may be set-up/torn-down, typically accompanied with QoS changes. Even within the same network, mobility may cause a user to move from uncongested cell to a congested cell. The issue is whether, and how, to provide a framework for applications to react to the dynamic network QoS. There are three broad approaches to this problem: 1. Applications are provided with a seamless mobile computing environment without notifications of QoS change. This is consistent with a purely networking solution to seamless cross-network migration, and does not work well when QoS changes by orders of magnitude (e.g. indoor to outdoor mobility). 2. Applications are allowed to dynamically (re)negotiate QoS with the network. If the pre-negotiated QoS is violated by the network, the application is notified. The onus of QoS negotiation and reaction to notifications is on the applications. Several emerging adaptive computing solutions fit into this approach[31, 39]. 3. Applications specify a sequence of acceptable QoS classes, the procedure to execute if a QoS class is granted, and the exception handling policy to execute if the network violates its QoS contract. The mobile computing environment handles the dynamic QoS (re)negotiation and reaction to notification. If the network notifies the application of a failure to deliver a pre-negotiated QoS class, the system executes the exception procedure and then renegotiates a lower acceptable QoS class from the list. PRAYER follows this approach, as described in Section 5. 4.1.2 Adaptation Support The application structure will depend on the level of adaptation support provided by the mobile computing environment. In case the QoS changes are transparent to the application, no special support needs to be provided for application-level adaptation. However, 'aware' applications will adapt better to the dynamics of the environment [39]. Support for QoS-awareness in mobile computing environments is a very complex task. While there are systems which provide application-level reaction to measured coarse-grain QoS, we are not aware of a working system which supports dynamic QoS (re)negotiation by the application. Even application-level support for reaction to QoS-notification (upon migration of networks, for example) is a challenging task. There are three broad approaches: 1. The mobile computing environment provides the measured QoS in a structure which can be retrieved by applications. Applications essentially poll the environment periodically in order to retrieve current QoS value and then adapt accordingly. 2. The mobile computing environment provides the measured QoS in a structure which can be retrieved by applications. In addition, an application registers with the environment and specifies acceptable QoS bounds. If the QoS bounds are violated, the application is notified, and may then adapt accordingly [31]. 3. The application program is split into ranges separated by QoS system calls. In each call, the application specifies a sequence of acceptable QoS classes, the procedure to execute if each class is satisfied, and the exception handling if the measured QoS class changes. In this case, notification is handled by the exception handling procedure, which may pursue one of four actions: block, abort, rollback, or continue. Section 5 describes this approach further. 4.2 Seamless Mobility and Management of Multiple Connections At any time, a portable computer may have access to multiple wireless networks. The appropriate choice for the wireless network for data transport depends on the application traffic characteristics, wireless network characteristics, priority of transmission, and cost (making this trade-off is in itself a challenging task). Migration between networks may be induced by a variety of reasons, such as user mobility, network failure/partition, or application-dependent trade-offs. Seamless mobility requires that when connections are broken or established, the process of switching between networks should be done transparently (with possible QoS notifi- cations). An important distinction between the considerations of seamless mobility in this paper and related work [4, 26] is that we address the issue of providing seamless mobility over autonomously owned and operated, possibly non-interoperable wireless networks. This imposes the constraint of not being able to access or modify base station software. There are four levels of mobility, as described below. 1. Handoff within the organization and the network: a mobile user moves between two cells of the same service-provider within a network (e.g. handoff between two RAM cells). 2. Handoff between organizations but within the same network: a mobile user switches between service-providers on the same network (e.g. switches carriers while using the same cellular phone line). TCP/IP handoff Backbone Backbone TCP/IP agent MH FH network2 network1 agent mobile IP' TCP/IP drivers Backbone agent agent connection manager transport transport connection manager MH home network2 network1 (c) (a) (b) Figure 3: Network Support for Seamless Mobility 3. Migration within the organization but between networks: a mobile user switches between networks of the same service-provider (e.g. switches between cellular phone and CDPD of same carrier). 4. Migration between organizations and networks: a mobile user switches both networks and service-provider (e.g. switches between locally owned WLAN to RAM). Handoff between the cells of the same network and organization is the common case, and has been solved at the network layer [25]. Handoff between cells of different organizations introduces the problem of authentication and accounting [9]. Handoff across the cells of different networks and organizations is the general case, and is addressed in this paper. In the general case, the following constraints apply: (a) a mobile user may migrate between different networks which are owned and operated autonomously, (b) the networks may use different protocol stacks (typically the lower 3 layers of the stack), (c) mobile service stations for the different networks will not communicate with each other directly for mobility support, and (d) each network will require a unique address for the computer. Several important issues arise in this case: (a) how to structure the network, (b) when to migrate between networks and (c) how to make application-related trade-offs. Authentication, accounting and security in inter-network mobility are other major issues, but are not discussed in this paper. A discussion of these issues, and a preliminary solution are proposed in [9]. 4.2.1 Network Structure Figure 3 shows alternative network structures for handling mobility. Figure 3.a shows the standard Mobile-IP structure which supports handoffs between two cells of the same network. Agents provide backbone support for mobility. Performance enhancements may include multicasting between the old and new base station in order to reduce handoff latency [10], and snoop caches in base stations in order to eliminate transport layer retransmissions [8]. This network structure uses a single network address and only considers handoff between cells of the same network. Figure 3.b shows a network structure which supports inter-network mobility at the network layer by enhancements through the Mobile-IP protocol. It uses different drivers for the different networks, and provides a network level solution to the mobility problem [5]. It is still possible to provide some notion of QoS, by having the network layer measure the QoS over each net-work and then propagate it to higher layers. Likewise, application-level trade-offs and choice of the wireless network can be achieved by informing the higher layers of the wireless networks currently accessible to the portable computer. We are not, however, aware of any implementation which provides adaptation support over a network level solution for seamless mobility across different wireless networks. While network level solutions may be the eventual goal for seamless mobility, they do involve making changes at the base station and cooperation between the different networks. In the current scenario, providing seamless mobility over autonomously owned and operated wireless networks is not possible using this approach. Figure 3.c shows a network structure which handles inter-network mobility at the transport layer and connection manager in the context of the PRAYER model. The connection manager maintains individual connections for each network, and can make the choice of network for data transport. The connection manager can also measure the end-to-end QoS parameters for each network between the portable and the home. The advantage of this architecture is that the choice of the wireless network for data transport is made outside the network. Thus seamless mobility is provided without the necessity of software support at the base stations. There are two advantages in not requiring base stations to different networks to co-operate in order to provide seamless mobility the networks: (a) many autonomous networks follow their own protocol standards, and (b) authentication and trust between the base stations of different organizations is avoided. The disadvantage is performance degradation. PRAYER uses a similar solution, but it merges the transport layer and connection manager. 4.2.2 Choice of Wireless Network for Data Transport networking resources being scarce and expensive, the choice of wireless network for data transport can significantly affect the performance and cost of an application. Typical network related trade-offs include bandwidth, delay, medium access patterns, security, and pricing structure. Primarily due to bandwidth considerations, the choice of networks is typically wire, indoor wireless and outdoor wireless in descending order of preference. The interesting trade-offs happen in the outdoor wireless networks, where access patterns and pricing structure play important roles. For example, CDPD and RAM support packetized data transport and are susceptible to much larger bursts of throughput, delay and jitter than cellular modems, which have periodic medium access patterns. If charged by data size, short packet bursts can cost up to an order of magnitude more in cellular modems as compared to RAM, while large data transmissions can cost an order of magnitude more in RAM than cellular modems. However, several packet data wireless providers also support monthly rates, in which case the pricing trade-off is irrelevant. Security is another important issue. For example, cellular modems are insecure while CDPD offers 6 levels of security. We are not aware of any good solution to the problem of selecting a medium for transport based on an application-directed tradeoff. In PRAYER, the current approach is to pre-specify a descending order of network choices based on raw data rates, and try to satisfy the application connection request through the 'best' available network. Clearly, this is inadequate, because the dynamically available data rate may be significantly lower than the raw data rate. Besides, bandwidth is an important, but by no means only criterion for network selection. 4.3 Adaptive Computing and Consistency Management mobility across different wireless networks is typically accompanied with a change in QoS. For example, mobility from indoor to outdoor wireless networks may result in a bandwidth decrease by two orders of magnitude. In order to provide a graceful degradation of the operating environment, mechanisms for system level and application level adaptation are necessary. This section explores the issues in caching and consistency management for adaptive mobile computing environments. Caching data and meta-data at the portable computer reduces access time and offered wireless traffic, but introduces a related problem of consistency when multiple copies of shared data are maintained. In mobile computing environments, disconnection is always a possibility. Thus most approaches to caching 'hoard' data aggressively, and allow the mobile user to manipulate the cached copy at the portable when disconnected. The modified data is reintegrated with the server copy upon reconnection, and update conflicts are typically reconciled by human intervention in the worst case [27]. This approach is suitable for disconnected operation on mostly private data. Given the increasing availability of wide area wireless connectivity and also the potential for collaborative applications in mobile computing, there arise four scenarios for caching and consistency management of data. 1. Disconnected operation on private data 2. Disconnected operation on shared data 3. Partially connected operation on private data 4. Partially connected operation on shared data In the above classification, indoor and outdoor wireless network connectivity are termed as 'partially connected' because (a) disconnections are possible at any time, (b) network errors are orders of magnitude higher than on wire, and (c) the significant cost associated with data transmission over the scarce resource may induce voluntary intermittent connectivity on the part of the mobile user. Distributed file systems which support disconnected operation typically assume that most of the data is private and unshared. The general approach in this case is to hoard data while in connected mode [27, 42]. Just prior to voluntary disconnection by the user, explicit user-directed hoarding is allowed. Once disconnected, the user is allowed to access and update the hoarded local copy, and all update operations are logged. Upon reconnection, the hoarded files are checked into the server. Update conflicts are resolved by log replay and in the worst case, user intervention [27, 34]. While file systems assume that most of the files are private (user data) or shared read-only (program binaries), this is not true of other data repositories such as databases. In such cases, update conflicts upon reconnection cannot be assumed to happen rarely, and automatic mechanisms for update conflict resolution need to be provided [15]. The possibility of conflict resolution voiding a previously concluded transaction also gives rise to the notion of provisional and committed transactions. Thus distributed databases which support disconnected operation must also support multiple levels of reads and writes. Related work has often assumed the two extreme modes of operation - connection or dis- connection. The emergence of wide area wireless networks provides intermediate modes of connection - where communication is expensive, but possible. Partial connection is particularly useful in two situations in the context of data management: (a) when files which were not hoarded in the connected mode are required, and (b) when certain parts of the application data need to be kept consistent with the data on the backbone server. In an environment which supports seamless mobility over heterogeneous wireless networks, the QoS may change dynam- ically. Thus the caching and consistency policies need to adapt to change in the network QoS in order to efficiently exploit the benefits of partial connectivity without incurring a significant communication cost. In our model, the home retains the 'true' copy of the cached data at any time. Since the home is in a connected mode on the backbone, it can execute any application dependent consistency policy on the backbone. Essentially, the onus of keeping its cached data consistent with the home is on the portable computer. This approach is at variance with contemporary schemes such as Coda [27] or Bayou [15], which do not have an intermediate home computer. The advantage of having a home that is known to maintain the true version of the data is twofold: (a) the consistency management in the mobile computing environment is now restricted to two known endpoints connected by a variable QoS network, and (b) distributed applications which support different types of consistency policies on the backbone can be supported, since the applications on the backbone need only care about keeping their data consistent with the home computer. The disadvantage of this model is poor availability - as far as the portable is concerned, the home is the only server for its cached data. Within this framework, a number of caching and consistency issues arise: (a) what data should be hoarded, (b) how consistency will be maintained, and (c) how applications will interact with the mobile computing environment in order to adapt the consistency management policies upon dynamic QoS changes. These issues are discussed below in the context of file system support for partially connected operation. 4.3.1 Hoarding Policy What to hoard is a non-trivial question in mobile computing. The factors involved are: (a) the nature of the data - ownership, mutability, and level of consistency, (b) currently available network QoS, (c) portable computer characteristics - available disk and battery power, and (d) predicted future connection or disconnection. Privately owned data or read-only data can be cached without involving any communication overhead for consistency management. Cached shared read/write data may involve high communication overhead for consistency management, depending on the type of consistency guarantees provided on shared data. Data which is loosely consistent or data which is not modified often can be cached with low overhead. Based on the these observations, a user who voluntarily plans on initiating a disconnection or migration to a low QoS network (e.g. indoor to outdoor) may choose to hoard private and read-only files, and flush the dirty cached data for shared read/write files. Most of the caching decisions mentioned above are highly application dependent. Ideally, the mobile computing environment will be smart enough to provide some assistance in predictive caching [28, 42]. For example, a request for caching an application binary will also cache the files it has frequently accessed during its previous executions (e.g. resource files) [42]. The current approach in PRAYER is simplistic - caching files which have been accessed in the recent past, and allowing the user to explicitly select files for caching. The fact that partial connectedness (as opposed to disconnectedness) is the common mode of operation reduces the negative impact of such a simple predictive caching approach. 4.3.2 Caching Granularity The tradeoff in caching granularity is that a large grain size may induce false sharing while a small grain size may induce higher processing overhead at the portable and the home. There are three broad alternatives for the grain size of caching: 1. Whole file caching: The whole file is cached upon file open. In most current distributed file systems which support disconnected operation [20, 23, 27], whole files are cached during connection. 2. Block caching: Caching is at the granularity of file system blocks. A variation of block-size caching is to have applications vary the block size, depending on the available network QoS. 3. Semantic record caching: An application imposes a semantic structure on its files. The semantic structure is contained in a pre-defined template, which specifies record and field formats in terms of regular expressions. The application is allowed to specify (by means of a file system interface) the cache size to be per-record level or per-field level. The advantage of this scheme is that it allows an 'aware' application to adapt its caching granularity to both the network QoS and application-semantics. The disadvantage is the increased complexity in caching/consistency management. The PRAYER caching approach implements a variant of this approach. An important point to note in both application-defined block level caching and semantic record caching is that the caching/consistency is being done here between the home and the portable. In the absence of a home, different portable clients may have different cache granularities, which will make providing consistency management incredibly hard for the distributed file system. In our model, the distributed file system provides whatever consistency policy it may choose to, with respect to the home. The block and semantic record caching schemes refer to the ways the portable keeps itself consistent with the home. 4.3.3 Consistency Management One of the advantages of partial connectedness is the option of providing a variable level of consistency on a whole file or parts of it. File systems which support disconnected operation must inevitably provide some form of session semantics, wherein a disconnection period is treated as a session. Since disconnection is a special case of partially connected operation, the consistency management for partially connected operation must reduce to session semantics in the event of disconnection. PRAYER supports semantic record caching and consistency. An aware application opens a file and imposes a template structure on it. The open file is thus treated as a sequence of (possibly multi-level) records. An application can specify certain fields in all records, or certain records in the file to be kept consistent with the home. For each cached data element, there are two possible consistency options: reintegrate and invalidate. Reintegration keeps data con- sistent, but requires communication between the home and the portable in order to propagate updates. Invalidation tolerates inconsistencies, but requires no communication. Depending on the available QoS, the application can dynamically choose to reintegrate or invalidate each record or field. Note, that invalidate still allows the application to access the local copy, but with no guarantees on consistency. In connected mode, the whole file is kept in the reintegration mode (between the portable and the home). In disconnected mode, the whole file is kept in the invalidation mode. In partially connected mode, the application has the flexibility to keep critical fields or records consistent while accepting inconsistencies for the rest of the file. In addition to maintaining consistency on certain parts of the file, there also needs to be support for explicit consistent reads and writes. PRAYER supports two types of reads: local read and consistent read. A local read is the default operation, and reads the local copy. A consistent read checks the consistency between the portable and home, and retrieves the copy from the home if the two copies are inconsistent. PRAYER supports three types of writes: local write, deferred write and consistent write. A local write updates only the local copy. A deferred batches write updates. A consistent write flushes the write to the home. In disconnected mode, consistent read and write return errors. Two simple examples illustrate the operation of the application-directed adaptive consistency management. ffl Calendar: If a user goes on a trip and maintains a distributed calendar with the secretary, the user would keep the 'time' and `place' fields of appointments in reintegration mode, but the 'content' field in invalidation mode. This will enable the user and the secretary to prevent scheduling conflicts, though the content fields of the appointments may not be consistent. ffl Email: If a user goes on a trip, the email application could keep the 'sender' and `subject' fields in reintegration mode, but the 'content' field in invalidation mode. If the user wants to read a particular email, an explicit 'consistent read' will be issued in order to access the contents. 5 The PRAYER Mobile Computing Environment mobility across diverse indoor and outdoor wireless networks is a very desirable goal, since it will enable a user to operate in a uniform mobile computing environment anytime, anywhere. However, lack of inter-operability standards pose a significant challenge in building such an environment. Even if seamless mobility is provided across different networks, the wide variation in bandwidth and other QoS parameters imply that the systems software and application will have to collaboratively adapt to the dynamic operating conditions in order to gracefully react to inter-network migration. Simple, yet effective mechanisms need to be provided to applications for adaptive computing. In order to explore the challenges in building a uniform operating environment which provides adaptive computing and seamless mobility on top of commercially available wireless net- works, we are building the PRAYER mobile computing environment. A preliminary PRAYER prototype has been operational for six months, and has served as a platform to test some of our solutions. While the focus of this paper has been to identify the major challenges and discuss possible solutions for providing seamless mobility and adaptive computing, a brief discussion of the PRAYER environment will serve to provide an overall context for our approach. There are three important components in PRAYER: connection management, data manage- ment, and adaptation management. Connection management provides seamless mobility over multiple wireless networks, and provides the abstraction of a virtual point-to-point connection between the portable computer and the home computer. Data management provides filtered information access on top of a file system which implements application-directed caching and consistency policies. Adaptation management provides language and systems support to applications for dynamic QoS (re)negotiation and reaction to notifications by the network (though at this point, we do not perform end-to-end QoS negotiation in the network ). 5.0.4 Seamless Mobility across Multiple Wireless Networks We adapt TCP/IP in order to provide support for a virtual point-to-point connection over multiple wireless networks between the portable computer and the home computer. The portable computer may have different IP addresses, corresponding to the different networks. Each TCP virtual connection is identified by a 4-tuple consisting of a logical IP address for the portable, a port at the portable, the IP address of the home, and a port at the home. The logical IP address for the portable for a TCP connection is the IP address of the portable corresponding to the wireless network over which the connection is first set up. Thus, the logical IP address, which serves to identify the connection, can be different from the IP address of the wireless network over which the portable actually transmits the packets. All TCP connections to or from the portable pass through the home if seamless mobility over multiple wireless networks is desired. The home is bypassed if a TCP connection over a single wireless network is desired. In order to establish a virtual connection between the home and the portable, the application pre-specifies the sequence of acceptable wireless networks in a descending order of priority. When a connection request is initiated at the home or the portable, the networks are polled for access in the descending order of priority. We define a socket interface to applications for setting up virtual connections over multiple wireless networks. The details of the implementation and the programming interface are described in [18]. 5.0.5 Caching and Consistency Management in the File System The file system uses the home as the server and the portable as the client, and provides strong consistency on application-specified portions of files cached at the client (note that the home itself may be an NFS client). The key feature of the file system is the support for application- directed caching and consistency policies. When an application opens a file at the portable, it imposes a template on the open file. Basically, a template specifies the semantic structure of the file. For example, a mailbox is a sequence of mail records, where a mail record has some pre-defined fields such as sender, subject, content, etc. It is possible for a template to specify records with variable length fields, optional fields, or fields appearing in different orders (all of which occur in the mailbox template). Once the application imposes a semantic structure for a file, the file system at the home and the portable create a sequence of objects for the file, each object representing a record. The application may then specify a subset of the fields of every record, or a subset of the records of the file to be kept consistent with the home through a pconsistency() system call, which causes these fields/records to be marked at both the portable and the home. Whenever a marked data element is updated at either the home or the portable, it is propagated to the other entity. In addition to requiring consistency on parts of the file, the application may also perform explicit consistent reads and writes (through the pread() and pwrite() system calls, which basically read/write through the home if the data at the portable is not consistent. The goal of the PRAYER file system is to facilitate adaptive application-directed consistency policies, while shielding the application from the mechanisms of keeping parts of the file consistent. When the portable is fully connected, the whole file may be kept consistent with the home (i.e. reintegration mode). When the portable disconnects, the whole file now operates in cached-only (invalidation) mode. When the portable has connectivity through wide area wireless networks and communication is expensive, only the critical parts of the file are kept consistent, and the rest of the file is accessed in a cached-only mode. Supporting this level of adaptation has a definite penalty in terms of file system performance, though we are yet to perform a quantitative evaluation of the overhead. A detailed description of the file system design and implementation is available in [16]. 5.0.6 Application Support for Adaptation We provide simple language and OS support for modifying existing applications or building new applications in our adaptive mobile computing environment. We classify QoS requests into commonly used QoS classes. A program is divided into regions, and may explicitly initiate QoS re-negotiation between the regions. Within a region, the application expects the network to provide a fixed QoS class. If the network is unable to do so, it notifies the application, which causes pre-specified exception handling procedures to be executed (as described below). Exceptions are handled in one of four ways: best effort, block, abort or rollback. 'Best effort' ignores the notification and continues with the task. 'Block' suspends the application till the desired QoS class is available. 'Abort' aborts the rest of the task within the region and moves to the next region. 'Rollback' aborts the rest of the task, and reinitiates QoS negotiation within the same region. Ideally, we would like rollback to also undo the actions taken thus far in the region before re-negotiation. The QoS negotiation is performed by a system call getQoS(), which takes in a sequence of options. Each option is a 3-tuple, consisting of the desired QoS class, the procedure to execute if that class is granted, and the exception handling policy. At the start of a region, the getQoS() call measures the network QoS, and returns the highest desired QoS class it could satisfy. When a QoS class is granted, the corresponding procedure is executed. If during the execution of the procedure, the network is unable to sustain the QoS class, then the exception handling routine is invoked with the application-defined policy. In this framework, both QoS negotiation and notification handling are supported by the system; the application need only specify the policy, and not bother about the mechanism in order to achieve adaptation. We expect applications to use the adaptation support and consistency management policy in concert. We expect that an application will initially open a file and impose a template structure on it. Then, depending on the QoS class granted through the getQoS() call, the application can change the fields/records it keeps consistent through the pconsistency() call. While we have not yet written a large application in this environment, we expect that the mechanisms for adaptation are simple, yet sufficiently powerful to support adaptive computing. 6 Related Work Mobile computing has witnessed a rapid evolution in the recent past, both in industry and academia. However, related work on seamless mobility and adaptive computing support for applications has just begun to emerge. In this section, we provide an overview of contemporary work in seamless mobility, adaptive computing, consistency management, and disconnected operation. We also identify key work in related areas which motivated several of the design decisions in PRAYER. Most of the projects which provide seamless mobility across heterogeneous wireless networks provide a network level solution. While this is a desirable goal, it will require mobile service stations of different networks to interact and trust each other. MosquitoNet [4, 5] and Barwan [26] address seamless mobility issues as described above. data consistency in variable QoS environments. For high-QoS, caching/consistency is as in backbone networks. In low-QoS, nothing is cached. In variable QoS, there is a two level caching scheme: on the backbone between homes, and on the wireless network between the home and the portable. Bayou [15, 43, 44] provides a replicated weakly-consistent distributed database to support shared data-driven mobile applications and supports per-application consistency. Bayou deals with small-to-medium database applications (calendars, etc) since it assumes that a large part of the database may be cached at the mobile. Odyssey [31, 36] provides a framework for adaptive applications to react to QoS changes. The applications can specify QoS bounds to the network. If the network is unable to satisfy these bounds, it notifies the applications, which can then adapt to the dynamic QoS change. Coda [27] provides disconnected access of file systems. When the portable is in connected mode, it hoards files by periodic 'hoard walks'. Upon disconnection, the portable accesses the cached files, and logs the updates. Upon reconnection, it checks the mutated cached files for potential conflict, which is then resolved by the user. Disconnected AFS [23] preserves the AFS semantics for disconnected operation. A user explicitly disconnects from the network, upon which the callbacks are retrieved by the server. Seer [28] and MFS [42] propose sophisticated hoarding mechanisms for disconnected operation. PRAYER uses several ideas from current and past related work, which are mentioned below. I-TCP [6] (intermediate host), Daedelus [8] (snoop cache) and [11] provide approaches for efficient wireless TCP. NFS [35], Sprite [30] and Andrew [21] provided distributed file systems caching approaches, which may be used for backbone consistency. Distributed databases establish consistency among replicated data by clustering [33], tokens [24] and partitioning [14]. [48] discusses fundamental issues in ubiquitous and mobile computing. [45] highlights OS issues in mobile computing. Conclusions The ideal scenario in mobile computing is when a user equipped with a portable computer with multiple wireless interfaces roams around between different indoor and outdoor networks while operating in a seamless computing environment which gracefully adapts to the dynamically changing quality of service. In order to achieve this scenario, at least two important components need to be satisfied: seamless mobility, and adaptive computing. The emergence of several viable wireless networking technologies with different standards, networking architectures and protocol stacks makes the problem of seamless mobility across different wireless networks a very challenging task. Contemporary research typically proposes network level solutions to this problem. Such a solution, though scientifically desirable, poses some serious problems because the commercial networks cannot inter-operate. This paper identifies the challenges in providing seamless mobility on top of commercial networks. mobility across wireless networks is typically accompanied with a change in QoS. In order to react gracefully to the change in QoS, both the mobile computing environment and the application need to collaboratively adapt to the dynamic operating conditions. Adaptation to QoS changes introduces challenges in data management and caching/consistency. It is the application which can best make semantics-based decisions on adaptation. However, burdening the application with dynamic QoS negotiation and reaction to network notifications will complicate application logic, and lead to an unviable environment to develop real-world programs. This paper identifies the challenges in language and operating systems support for applications to interact with the underlying file system. With the increasing popularity for mobile computing, the importance of a uniform mobile computing environment providing seamless mobility across commercial wireless networks and graceful adaptation to dynamic operating conditions cannot be over emphasized. However, there are significant challenges which need to be overcome before such an environment can be effectively deployed. This paper explores the issues and offers some preliminary solutions, which are being implemented in the PRAYER mobile computing environment. Acknowledgements I am grateful to Dane Dwyer for providing multiple reviews of this paper and for implementing the PRAYER file system. --R Structuring Distributed Algorithms for Mobile Hosts Changing Communication Environmeents in MosquitoNet Supporting Mobility in MosquitoNet I-TCP: Indirect TCP for Mobile Hosts Handoff and System Support for Indirect TCP/IP Improving TCP/IP Performance over Wireless Networks A Protocol for Authentication The Effects of Mobility on Reliable Transport Protocols Improving the Performance of Reliable Transport Protocols in Mobile Computing Environments Experiences with a Wireless Network in MosquitoNet Network Access for Personal Communications Consistency in Patitioned Networks The Bayou Architecture: Support for Data Sharing Among Mobile Users Mobile File Systems for Partially Connected Operation The Challenges of Mobile Computing Mobility across Commercial Wireless Networks Networks Primarily Disconnected Operation: Experience with Ficus Scale and Performance in a Distributed File System Data Replication for Mobile Computers Disconnected Operation for AFS Data Management for Mobile Computing The Bay Area Research Wireless Access Network(BARWAN) Disconnected Operating in the Coda File System The Design of the SEER Predictive Caching System Large Granularity Cache Coherence for Intermittent Connectivity Caching in the Sprite Network Filesystem A Programming Interface for Application-Aware Adaptation in Mobile Computing An Empirical Study of a Highly Available File System Maintaining Consistency of Data in Mobile Distributed En- vironments Resolving File Conflicts in the Ficus File System for a Distributed Workstation Enviroonment Design and Implementation of the Sun Network Filesystem Experience with Disconnected Operation in a Mobile Computing Environment Customizing Mobile Applications Context Aware Computing Applications Information Organization using Rufus Service Interface and Replica Management Algorithm for Mobile File System Clients Intelligent File Hoarding for Mobile Com- puters Managing Update Conflicts in Bayou Session Guarantees for Weakly Consistent Replicated Data Operating System Issues for PDA's Effective Wireless Communication through Application Partitioning Application Design for Wireless Computing Some Computer Science Issues in Ubitquitous Computing --TR --CTR Dane Dwyer , Vaduvur Bharghavan, A mobility-aware file system for partially connected operation, ACM SIGOPS Operating Systems Review, v.31 n.1, p.24-30, Jan. 1997 S. K. S. Gupta , P. K. Srimani, Adaptive Core Selection and Migration Method for Multicast Routing in Mobile Ad Hoc Networks, IEEE Transactions on Parallel and Distributed Systems, v.14 n.1, p.27-38, January Ahmad Rahmati , Lin Zhong, Context-for-wireless: context-sensitive energy-efficient wireless data transfer, Proceedings of the 5th international conference on Mobile systems, applications and services, June 11-13, 2007, San Juan, Puerto Rico

seamless mobility;adaptive computing

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On Multirate DS-CDMA Schemes with Interference Cancellation.

This paper investigates interference cancellation (IC) in direct-sequence code-division multiple access (DS-CDMA) systems that support multiple data rates. Two methods for implementing multiple data rates are considered. One is the use of mixed modulation and the other is the use of multicodes. We introduce and analyze a new approach that combines these multiple data rate systems with IC. The cancellation in the receiver is performed successively on each user, starting with the user received with the highest power. This procedure can in turn be iterated, forming a multistage scheme, with the number of iterations set as a design parameter. Our analysis employs a Gaussian approximation for the distribution of the interference, and it includes both the AWGN and the flat Rayleigh fading channel. The systems are also evaluated via computer simulations. Our analysis and simulations indicate that the IC schemes used in mixed modulation or multicode systems yield a performance close to the single BPSK user bound and, consequently, give a prospect of a considerable improvement in performance compared to systems employing matched filter detectors.

Introduction An important feature of future mobile communication systems is the ability to handle other services besides speech, e.g., fax, Hi-Fi audio and transmission of images, services that are not readily available today. To achieve this, it is essential to have a flexible multiple access method that maintains both high system capacity and the ability to handle variable data rates. Direct-sequence code-division multiple access (DS-CDMA) is believed to be a multiple access method able to fulfill these requirements [1]. There are two main factors that limit the capacity of a multiuser DS-CDMA system and make the signal detection more difficult. They are signal interference between users, referred to as multiple access interference (MAI), and possibly large variations in the power of the received signals from different users, which is known as the near-far effect. The power variations that causes the near-far effect is due to the difference in distance between the mobile terminals and the base station as well as fading and shadowing. One way to counteract the near-far effect is to use stringent power control [2]. Another approach would be to use more sophisticated receivers which are near-far resistant [3]. Because of the MAI's contribution to the near-far problem and its limiting of the total system capacity, much attention has been given to the subject of multiuser detectors that have the prospect of both mitigating the near-far problem and cancelling the MAI. Research in this area was initiated by Verd'u [4, 5], who related the multiple access channel to a periodically time varying, single-user, intersymbol interference (ISI) channel and who derived the optimal multiuser detector. Unfortunately, the complexity of this detector increases exponentially with the number of users. This has motivated further research in the area of suboptimal detectors with lower complexity [6-21]. The objective of our work is to propose and evaluate an efficient detector for a multiuser and multirate DS-CDMA system. The proposed multiuser detectors in [6-18] are all designed for single rate systems. In [22], however, a dual rate scheme based on multi-processing gains is considered for the decorrelating detector. In this paper we consider two other multirate schemes together with a single- and multistage non-decision directed interference canceller (NDDIC) [19-21]. The IC 1 schemes are generalizations and extensions of the single-stage SIC scheme for BPSK derived by Patel and Holtzman in [14, 15]. The operation of the IC scheme is as follows. The receiver is composed of a bank of filters matched to the I and Q spreading sequences of each user. Initially the users are ranked in decreasing order of their received signal power. Then the output of the matched filter of the strongest user is used to estimate that user's baseband signal, which is subsequently cancelled from the composite signal. In other words, the projection of the received signal in the direction of the spreading sequence of the strongest user is subtracted from the composite signal. This is how we attempt to cancel the interference that affects the remaining users. Since we consider the uplink, that is, communication from the mobile terminal to the base station, we are interested in detection of all received signals and, thus, we continue by cancelling the second strongest user successively followed by all the other users. This scheme may be extended to an iterative multistage IC scheme by repetition of the IC one or more times. In the multistage IC scheme the estimated signal from the previous stage is added to the resulting composite signal and the output of the matched filter is used to obtain a new estimate of the signal, which in turn is cancelled. Hence, in this manner the interference can be further reduced and the signal estimates improved. The IC scheme is generalized to apply to the two multiple data rate schemes, mixed modulation and multicodes. Mixed modulation refers to the use of different modulation formats to change the information rate. That is, given a specific symbol rate, each user chooses a modulation format, for example, BPSK, QPSK or any M-ary QAM format, depending on the required data rate [23]. Multicodes is the second approach for implementing multiple data rates. It allows the user to transmit over one or several parallel channels according to the requirements [23]. Hence, the user transmits the information synchronously employing several signature sequences. To simplify notation IC is used both for interference cancellation and interference canceller. This approach can, of course, also be used in combination with different modulation formats. For our analysis and simulations we consider coherent demodulation, known time delays and two types of channels: a stationary AWGN channel and a channel with frequency-nonselective Rayleigh fading. Perfect power ranking is assumed in the performance analysis and in most of the simulations, which implies knowledge of the channel gain for each signal. This knowledge is, however, not used in the IC scheme itself. The paper is organized as follows. In Section 2 we present the system model and the decoder structure for rectangular M-ary QAM. A single- and multistage IC are then presented for this model in Section 3. The performance of a single-rate system with IC in AWGN and in flat Rayleigh fading is analysed in Section 4 and Section 5. Thereafter, the performance of mixed modulation systems with IC is analysed in Section 6 and the corresponding analysis for multicodes is given in Section 7. Numerical results are presented in Section 8 and in Section 9 we discuss performance improvements for high-rate users in mixed modulation systems. Finally, the conclusions and future considerations are discussed in Section 10. System Model and Decoder Structure We consider a model for a system with square lattice QAM, where the received signal for K users is modelled as ae r d I r oe (1) which is the sum of all transmitted signals embedded in AWGN. d I=Q k (t) is a sequence of rectangular pulses of duration T with amplitude A I=Q k;l , where I=Q denotes in-phase (I) or quadrature branch. T is the inverse of the symbol rate, which is assumed to be equal for all users. The amplitudes of the quadrature carriers for the k th user's l th symbol element, A I k;l and A Q k;l , generate together M equiprobable and independent symbols. They take the discrete values A I=Q since amplitude levels are required for the I and Q components to form a signal constellation for M-ary QAM. The energy of the signal with lowest amplitude is then 2E 0 . The k th user's signature sequence that is used for spreading the signal in the I or Q branch is denoted c I=Q k (t). It consists of a sequence of antipodal, unit-amplitude, rectangular pulses of duration T c . The period of all the users' signature sequences is hence there is one period per data symbol 2 . - k is the time delay and OE k is the phase of the k th user. These are, in the asynchronous case, i.i.d. uniform random variables in [0; T ) and [0; 2-). Both parameters are assumed to be known in the analysis and in the simulations with known channel parameters. However, if complex spreading 2 In this paper N is also referred to as processing gain. and despreading is used, OE k is only needed for the coherent detection and not for the NDDIC scheme [18]. Furthermore, ! c represents the common centre frequency, ff k represents the channel gain, which could be constant or Rayleigh distributed, and n(t) is the AWGN with two sided power spectral density of N 0 =2. Figure 1 shows the structure of the k th user's receiver when detecting the l th symbol. The receiver is the standard coherent matched filter detector for M-ary QAM, from which we obtain two decision variables, S I k;l and S Q k;l , which are the sufficient statistics for the I and Q components. The low pass filter removes the double frequency components and for the I branch we get ae r d I r sin OE koe l s I where s I k;l (t) is the baseband signal for the l th symbol of the k th user. A similar expression can be derived for is the baseband equivalent of n(t). PSfrag replacements I (t) Z II k;l Z IQ k;l Z QI k;l Z QQ k;l I k;l R dtT R dtT R dtT R dt c I c I sin Figure 1: M-ary QAM receiver for DS-CDMA systems. The I branch as well as the Q branch is correlated with both the I and Q signature sequences of the k th user to form four different correlator outputs, which are the outputs at integer multiples of T . These outputs contain all information about the amplitudes and they are used to form the decision variables, S I k;l and S Q k;l . Let us consider detection of the first user's zeroth symbol. Then, Z II 1;0 , from the first correlator, is determined as Z II r ae I II oe where A I A I , A Q A Q and the noise component is given by I (t)c I The sum of I II k;1 terms in (4) represents the interference due to the remaining users and each term can be expressed as I II A I cos A Q sin is a unit-amplitude, rectangular pulse of length T and - . The delay - k is assumed, without loss of generality, to be shorter than - 1 . This is discussed more in the next section. All the other Z 1;0 terms are derived in the same manner as above and we get the decision variables I r r where N I=Q 1 is the noise term of the decision variable including both Gaussian noise and noise caused by multiuser interference. The noise term depends actually on the symbol but for convenience and reasons explained later we write N I=Q 1 instead of N I=Q 1;0 . It can be shown with the help of trigonometric functions that the noise terms are given by I r I II r I QQ where OE k;1 is the function given in (6), where OE k has been replaced by OE k;1 . I QQ k;1 is given by a similar expression with the appropriate changes in indices. The four noise components are assumed to be uncorrelated Gaussian random variables. The two pairs of noise components, 1 and n IQ 1 and n QQ 1 , are uncorrelated only if the signature sequences, c I are orthogonal, which is a mild restriction. We assume, however, that the correlation is zero also for non-orthogonal signature sequences with large processing gain to enable analytical evaluation of the system performance. 3 Non-Decision Directed Interference Cancellation 3.1 Single-Stage Interference Cancellation The receiver for M-ary QAM is composed of a bank of filters matched to the I and Q signature sequences of each user according to Figure 1. From the filter outputs we obtain the decision variables, which are used both for determining which of the users is the strongest and in the cancellation of that user's signal. The users are then decoded and cancelled in decreasing order of their power. The detector is a coherent demodulator and we assume decision boundaries according to minimum Euclidean distance. A block diagram of a receiver for M-ary QAM with IC is shown in Figure 2. Without loss of generality, we assume that ff 1 ? Hence, if the users have the same average transmitted power, the first user is the strongest. The strongest user is cancelled first, since this user is likely to cause most interference and also the one less affected by the interference from the other users. The decision variables of the strongest user is used to estimate its baseband signal, which is subsequently cancelled from the composite signal. In other words, the projection of the received signal in the direction of the spreading sequence, is subtracted from the composite signal. The scheme continues with cancellation of the second strongest user and thereafter all the users in order of their received power. Select Decode MF K PSfrag replacements I (t) I 1;l 1;l I 2;l 2;l I K;l K;l I k;l I k;l k;l k;l c I c I sin Figure 2: M-ary QAM receiver with interference cancellation. For the sake of notational simplicity we assume that that all the interfering symbols of the stronger users have been cancelled before the considered symbol is decoded. That is, all the symbols prior to the zeroth have already been decoded and cancelled and we can consider the detection of the zeroth symbol for all users. This is not a restriction, since it does not affect the results of the analysis and it is not used in the simulations. It only simplifies the expressions, thus we avoid considering different symbols for different users. We use the decision variables in (7) to estimate the baseband signal of the first user's zeroth symbol and subsequently cancel it from the composite signal. The cancellations are however not perfect: besides the desired signal, the filter output also contains Gaussian noise and noise caused by MAI, and, for each cancellation, noise is projected in the directions of the other users in the system. Nevertheless, the scheme has the advantage of being simple. We proceed by cancelling the users successively and after cancellations the decision variable for the h th user is given by r After cancellation of the h th user, the resulting baseband signal of the I branch is expressed as I where there are h cancelled and K \Gamma h remaining zeroth symbols. When h is equal to 1 the term I 0;0 (t) corresponds to the remaining baseband signal after cancellation of all symbols prior to the zeroth, and consequently, we get ffi I 1;0 (t) after cancelling the first user's zeroth symbol. We rewrite the expression in the following way I s I I ae r A I r A Q oe \Theta N I where the first sum is the remaining baseband signal after cancellation of all symbols prior to the zeroth. The second sum is the additional noise caused by imperfect cancellation of these symbols and it is defined as I \Theta N I k;l c I k;l c Q where we have given the noise term N I=Q k;l an additional subscript, l, specifying the symbol. This index indicates that the noise term do vary over time. The third term in (11) is the in-phase Gaussian noise and the subsequent sum is the cancelled baseband signals corresponding to the zeroth symbol of the h strongest users in the system. Finally, we have the additional noise components caused by imperfect cancellation of these h users' zeroth symbols. The omission of a second subscript in the noise term in the last line of (11) is explained below. The total noise component in (9) for the h th user in the I branch is I r I II k;h \Theta I where the first sum consists of noise caused by the remaining interfering users, the second term is Gaussian noise and the last sum is the resulting noise caused by imperfect cancellations. N Q is given by a similar expression. The correlation terms, J II j;h and J QI j;h , are given by J II where the correlation is over the noise caused by imperfect cancellation of the symbols -1 and 0 of the j th user, since we assume This is illustrated in Figure 3, where shaded lines indicate cancelled symbols. However, since we consider a slowly fading channel, which implicates that the channel changes slowly and the interference-power can be regarded as equal for two subsequent symbols, we do not distinguish between, e.g., the noise terms N I I j;0 . This is the reason for simply using N I j and N Q in (11) and (13). -T PSfrag replacements Figure 3: Cross-correlation between users in an asynchronous system. 3.2 Multistage Interference Cancellation The derived single-stage scheme may be iterated to form a multistage scheme. The motivation for this is that the users received with high power have the advantage of being strong but they are still exposed to interference from the weaker users. Hence, if better estimates of the strong users' signals can be achieved the estimates and the cancellations of the weak users' signals would be improved. Therefore, iterating the IC scheme can improve the performance of the whole system. We still have to keep in mind that in our simple IC scheme we make non-decision directed or 'soft' cancellations using the matched filter outputs. The effect is that the Gaussian noise is not removed through hard decisions and for each cancellation a small amount of noise is projected in the directions of the other users in the system. Hence, the performance of a system can be improved through cancellation of the MAI by employing a limited number of IC stages, but after the optimum number of stages the performance will degrade. However, simulated results in [18] and analytical results in [24] show that the multistage NDDIC in a synchronous system performs better than the decorrelator [6] after a limited number of stages and that they are asymptotically equivalent. To simplify the notations when describing the multistage IC, we drop the subscript for the symbol and replace it with a subscript that represent the stage. That is, the first subscript of a variable defines the user and the second subscript defines the stage and the assumption of detection of symbol zero is implicit. To describe the multistage IC scheme we use an interference cancellation unit (ICU), which is illustrated in Figure 4 using a simplified block diagram. First we add the estimated baseband signal from the previous stage (denoted s k;i\Gamma1 in Figure 4) to the resulting composite signal. Then we use the output of the matched filter to obtain a new estimate of the signal, which in turn is cancelled. The variable r k;i defines the composite baseband signal after cancellation of user denotes the I and Q signature sequences, which are used to regenerate the estimated baseband signal s k;i of the k th user. The scheme is repeated for all the users in the system for the desired number of stages. This is shown in Figure 5, where each block, IC k (i) , is the k th user's ICU at the i th stage. PSfrag replacements r i;k+1 r i;k s i\Gamma1;k s i;k y i;k Figure 4: Linear non-decision directed interference cancellation unit. The corresponding expression to (10) for the resulting baseband signal in multistage IC is determined as I h;i a I h;i a Q I a I where a I . a Q h is given by a similar expression using the sine-function. The decision variable for the h th user at the i th stage is given by r PSfrag replacements r 1;1 r 1;2 r 1;K r 1;K+1 r 2;1 r 2;2 r 2;K s 1;1 s 1;2 r 2;K+1 r i;1 r i;2 r i;K . ICU K (1) ICU K (2) ICU K (i) Figure 5: Multistage successive interference cancellation. where N I=Q h;i contains only Gaussian noise and noise caused by imperfect cancellations. For the I branch the noise term is given by I \Theta N I \Theta N I where the first term is Gaussian noise, the first sum is the noise caused by imperfect cancellation at the i th stage and the second sum is the noise caused by imperfect cancellation at the (i \Gamma 1) th stage. J II j;h and J QI j;h are defined in (14) and N Q h;i is given by an expression similar to (17). 3.3 Ranking of the Users In this paper we do not consider algorithms for ranking the users. We assume perfect ranking in the analysis and in most of the simulations. In simulations where we estimate the channel, power ranking is performed before the IC using pilot symbols for initial channel estimates. In mixed modulation systems, the QAM users are scaled with their average power giving a ranking according to the channel gain. Discussions about ranking is found in [15, 19]. Performance Analysis of Systems in AWGN In this section we analyse the performance of a single-rate system in an AWGN environment. To analyse the scheme, we let the noise components caused by MAI be modelled as independent Gaussian noise [25, 26]. We have chosen to use a Gaussian approximation partly since it is commonly used and partly because it yields a practical way to evaluate the performance of an asynchronous system. When a Gaussian approximation is used, an increase in noise and interference variance immediately leads to an increase in error probability, which is likely to occur also for the true distribution. Absolute performance, however, is likely to be too optimistic [26]. In this section we consider an AWGN channel where all the users are received with equal power, which corresponds to perfect power control. This will make the ranking of the users completely random and the order of cancellation will change continuously. Thus, the average probability of symbol error for each user is obtained taking the average of the symbol error rates (SER) for all the users. 4.1 Single-Stage Interference Cancellation First we calculate the variance of the decision variable of the I branch conditioned on ff, i.e., I \Theta N I where N I h is defined in (13) and ff includes all ff k . ff k is constant for stationary AWGN channels but we write the variance conditioned on ff to enable the use of (18) in the analysis for Rayleigh fading channels. With the assumption that all the Gaussian noise terms are uncorrelated (which is true if c I k and c Q are orthogonal), it can be shown that all the random variables in N I=Q are independent and with zero mean. Consequently, we model N I=Q h as an independent Gaussian random variable with zero mean and variance j I=Q h . Rewriting (18) we get I h =Var I II k;h I J II where the first term is the variance of the Gaussian noise, the first sum results from the MAI and the last sum is due to imperfect cancellations. For deterministic signature sequences and rectangular chip pulses, the variance in (19) is I k;h I 1)=3 is the normalized average transmitted power in each branch, k;h is the average interference [25] between the signature sequences in the I branches of user k and h. We have also assumed that - j;h and OE j;h are uniformly distributed over [0; T ) and [0; 2-). For random sequences the variance is I I where we have used j I j and that the average interference is 2N 2 [25]. It is straightforward to obtain the probability of error from the theory of single transmission of QAM signals over an AWGN channel [27] when the distribution of the MAI is approximated to be Gaussian. We use the variance given in (20) or (21) and define a signal-to-noise ratio, ae I for the h th user in the ideal coherent case as ae I r I The probability of error for transmission over the I branch is then [27] I Q(ae I where the Q-function defines the complementary Gaussian error function 3 . P Q e h is obtained in a similar manner and together they give the SER Finally, the average probability of symbol error is obtained taking the average of all the users' SERs. 4.2 Multistage Interference Cancellation The multistage scheme is analysed in the same manner as the single-stage scheme. The expression for N I h;i in (17) is used in (18) to obtain the variance of the decision variable. The variance for deterministic sequences is given by I I k;h I while for random sequences we get I I I j;i is used. The variance in (25) or (26) is used in (22) to obtain the signal-to-noise ratio, ae I h;i , which in turn is used together with (23) and (24) to calculate the average probability of symbol error for the multistage scheme. Performance Analysis of Systems in Flat Rayleigh Fading In this section we analyse single-rate systems in flat Rayleigh fading. That is, systems where the users' signals are received through independent, frequency-nonselective, slowly fading channels. This model is suitable in areas with small delay spread and for mobiles with slow speed (small Doppler frequency). These conditions also make estimation of OE k and ff k feasible, which is needed for coherent detection and to obtain decision boundaries for M-ary QAM [27]. The expressions for noise variances and error probabilities, that were derived in the previous sections, are all conditioned on the channel gain. We will use them as in [14, 15] to derive the error probabilities for flat Rayleigh fading channels. 5.1 Single-Stage Interference Cancellation The users' amplitudes are assumed to be Rayleigh distributed with unit mean square value. That is, the average power of the received signals at the base station is equal, assuming perfect power control for shadowing and distance attenuation. To obtain the unconditional probability of error, I , we average the conditional probability of error over the fading as follows I Z 1P I where f ff h (x) is the pdf of the h th ordered amplitude, which is obtained using order statistics [28] and stated here for convenience, i.e., We define a conditional signal-to-noise ratio, ae I h , for the h th user in the I branch according to (22). The only difference is that j I h is replaced by E ff [j I h ], which is the expected value of the conditional variance with respect to ff. The expected value is taken with respect to all ff When using a Gaussian approximation we calculate the second moment of the MAI and add it to the variance of the Gaussian noise. The expected value is therefore determined as \Theta I \Theta \Theta I \Theta ff 2 is the mean square value of the ordered amplitude, ff k , given by \Theta ff 2 It should be noted that this integral, as well as the integral in (27), is calculated numerically in the analysis. Taking the average of the different users' SERs, yields the average probability of symbol error. This is the proper measure of performance, since the order of cancellation will change with the fading and the average of all users will be the same as the time average for each user. 5.2 Multistage Interference Cancellation The multistage scheme is analysed in the manner described above using order statistics. The expected value, with respect to ff, of the variance of the h th user's decision variable at the i th stage is given by \Theta j I \Theta j I \Theta j I when using random spreading sequences. The variance in (31) is used in (22) to form ae I h;i , which in turn is used in Eqns. (23), (24) and (27) to calculate the unconditional SER. 6 Mixed Modulation Systems with IC Mixed modulation is one possible scheme that can be used when handling multirate systems [23]. In this scheme, the information rate is determined by the modulation format, which can be BPSK, QPSK or any M-ary QAM format. Accordingly, if a user transmits with a specific data rate using BPSK modulation, the user would change to QPSK modulation when a twice as high information rate was required. In the following paragraphs we evaluate the performance of a system where the users employ different modulation formats, in this case, a combination of BPSK, QPSK and 16-QAM. 6.1 Mixed Modulation Systems with Single-Stage NDDIC We consider a system where we have K 1 BPSK, K 2 QPSK and K 3 16-QAM users. To compare different forms of modulation, we let the transmitted bit energy, E b , be equal for all users independent of modulation format. Rewriting the energy E 0 as a function of E b yields which is valid for M-ary QAM. For BPSK modulation, . The expression for the variance of the decision variable for a BPSK user is given in [15], and it is reproduced here for convenience together with the expression for a M-ary QAM user, i.e., I If we define M (16-QAM), we can express the signal-to-noise value conditioned on ff, for the h th QPSK user as ae I s I is the variance of the decision variable of the h th QPSK user in the mixed modulation system. is the number of cancelled BPSK, QPSK and 16-QAM users, respectively. Rewriting (34), using (33) to derive j I , we get ae I (ae I K3 (ae I K1 ae \Gamma2 where the noise variance caused by interference from QPSK users is found on the first line, from 16-QAM users on the second line and from BPSK users on the last line. For BPSK and 16-QAM users we get similar expressions. 6.2 Mixed Modulation Systems with Multistage NDDIC The variance of the noise in mixed modulation systems with multistage IC is derived in a similar manner as for the single-stage case. The signal-to-noise ratio, ae I h;i , for the h th QPSK user is formed using (34) together with the for multistage modified expressions in (33). We then obtain ae I (ae I (ae I (ae I (ae I ae \Gamma2 ae \Gamma2 where the first line results from imperfect cancellation of QPSK users, the second line from 16- QAM users and the last line from BPSK users. The signal-to-noise ratios for BPSK and 16-QAM users are given by similar expressions. 6.3 Performance Analysis for Stationary Channels To evaluate the performance of a mixed modulation system, we first calculate the bit error rate (BER) for each user before the BER for the whole system can be derived. For BPSK users the BER is determined by \Theta - To compare the results obtained for QAM users with BPSK users we assume a Gray encoded version of M-ary QAM. The log 2 M-bit Gray codes differ only in one bit position for neighbouring symbols, and when the probability of symbol error is sufficiently small, the probability of mistaking a symbol for the adjacent one vertically or horizontally is much greater than any other possible symbol error. The SER is easily derived using (23) in (24) and then the BER is obtained through To get the BER for the whole system, each user's BER is weighted together as are the data rates for the BPSK, QPSK and 16-QAM users, respectively, and the P b terms are the individual BERs. 6.4 Performance Analysis for Rayleigh Fading Channels In mixed modulation systems, the average transmitted energy per bit is equal for all users independent of modulation format. The average power of the QAM users is therefore log 2 M times higher than for the BPSK users. For the mixed modulation scheme, we have ordered the users according to the channel gain and not the received power. This will improve the performance of the high-rate users, which are more sensitive to noise, and, consequently, it improves the performance of the whole system. However, the improvement is mainly noticed for single-stage IC, since the effect of ranking is less important when the interference in the system is due only to imperfect cancellations. The expected values, with respect to ff, of the variances in (33) are used in (34) to obtain the corresponding expressions of the conditional signal-to-noise ratio. The signal-to-noise ratio is then used to derive the error probability for each user according to (23) or (37), depending on the user's modulation format, and the unconditional error probability is obtained using (27). Finally, the weighted BER is obtained using Eqns. (37) - (39). The corresponding signal-to-noise ratio, ae I h;i , in (36), after taking the expected value of the variances with respect to ff, is used to calculate the performance of the multistage IC. The procedure for obtaining the BER of the whole system is then the same as described above. 7 Multicode Systems with IC Multicodes is the second of the two considered multirate schemes [23]. In this scheme we let each user transmit information simultaneously over as many parallel channels as required for a specific data rate. Thus, a user employs several spreading codes and the information is transmitted synchronously at a given base rate. If there are users with very high rates in the system, there can be a large number of interfering signals. However, this affects the high-rate user itself very little, since sequences with low cross-correlation or orthogonal sequences can be used, in which case the synchronous signals interfere little or nothing with each other. 7.1 Synchronous Systems with Single-Stage NDDIC We consider first a system with only synchronous transmission. We have \Delta parallel channels with identical channel parameters, since we assume that the signals are transmitted simultaneously from the same location. In other words, the relative time delay and phase between the channels are equal to zero. The variance of the noise component in (13) when both - k;h and OE k;h is zero is given by I I k;h (0) is the periodic cross-correlation function [25]. The expression for random sequences follows using E[(' II 7.2 Multicode Systems with Single-Stage NDDIC We consider a system with K users, where each user, k, transmits over \Delta k channels. The total number of information bearing channels is then equal to the sum of all \Delta k and there are both synchronous and asynchronous interferers. When deterministic sequences are used, all, or the major part, of the interference comes from the asynchronous users, depending on if the sequences are orthogonal or not. Therefore, cancellation of parallel signals is excluded and instead we consider a receiver where each user's parallel signals are decoded and cancelled simultaneously. Combining (20) and (40) we can write the variance of the decision variable of the h th user's th signal in a QAM system with both asynchronous and synchronous transmission as I r II I r II r QI where r II denotes the average interference between the in-phase signals, and k j denotes the th channel of the k th user and h g the g th channel of the h th user. The variance in (41) is used together with Eqns. (22) - (24) to obtain the probability of error in stationary AWGN channels. 7.3 Multicode Systems with Multistage NDDIC The use of orthogonal spreading sequences does not improve the performance considerably for single-stage IC, since the interference caused by the remaining asynchronous users in each step of the cancellation scheme determines the performance. However, when employing a multistage scheme it is preferable to use orthogonal spreading sequences, because after the first complete stage of IC the MAI is partly rendered for noise caused by imperfect cancellation. Hence, there are not any remaining users that dominates the interference. The corresponding variance of the decision variable for multistage IC and orthogonal spreading sequences is given by a similar expression to (25), i.e., I hg ;i =6N 3 I I where g is one of the signals belonging to user h. The average error probability for stationary AWGN channels is then obtained using (42) in Eqns. (22) - (24) as above. 7.4 Performance Analysis of Multicode Systems in Fading The performance of a multicode system in fading is analysed using order statistics as described in Section 5. The K users are ordered, each one with \Delta k parallel channels, according to their total received power. The same pdf, f ff k (x), and mean square value, E ff [ff 2 k ], are assigned to all the \Delta k channels of user k and, for the single-stage IC, (41) is used to obtain the expected value of the variance with respect to ff. The error probability is then derived using Eqns. (22) - (24) and (27). The performance of the multistage scheme is evaluated for orthogonal spreading sequences. Thus, the remaining noise consists only of the noise caused by imperfect cancellation of the asynchronous users and Gaussian noise. The expected value with respect to ff of the variance in (42) is used to obtain the error probability using order statistics as described above. 8 Numerical Results 8.1 Simulations All presented simulations are for asynchronous systems. We consider both stationary AWGN channels and slow, frequency-nonselective Rayleigh fading channels. The stationary AWGN channel corresponds to a system with perfect power control and, for the Rayleigh fading channel, we assume average power control for distance and shadow fading. The average received power is assumed to be equal for all users in single-rate systems and it is equal for all channels in multicode Table 1: Parameter settings for the simulations. Simulation Parameters Single/Mixed Mod. Multicodes Channel Stationary AWGN/ Rayleigh Fading Stationary AWGN/ Rayleigh Fading Detection Coherent Coherent Modulation BPSK, QPSK, Channel Estimation Known Channel/ Pilot Symbols Known Channel/ Pilot Symbols Ranking Perfect Ranking/ MF Outputs Perfect Ranking/ MF Outputs Signature Sequences Random Codes Orth. Gold Codes Time Delays Perfect Estimates Perfect Estimates Processing Gain 127 128 Block Length systems. In mixed modulation systems, the E b =N 0 value is the same for all users, which makes the M-ary QAM users log 2 M times stronger in average power than the BPSK users. The IC scheme is performed block-wise on the data and it is assumed that the channel does not change during the transmission of a block, which corresponds to slow vehicle speed. We also assume that pilot symbols are added between the data blocks in those cases where we consider estimation of channel parameters. The estimate is then obtained from an average of the pilot symbols in the beginning as well as at the end of each block of data. For QPSK modulation, the k th user's channel estimate is obtained from Y I where Y I k;p and Y Q k;p are the matched filter outputs when the received baseband signal is despread with c I k and c Q k respectively. Moreover, I p denotes the indices for the P complex pilot symbols, which are defined as 1 + j. For QAM we get a similar expression. Furthermore, in simulations using known channel parameters we assumed perfect ranking of the users. On the other hand, in simulations with estimated parameters, ranking is performed using the pilot symbols to obtain initial channel estimates. Note that ff k is only used to determine the decision boundaries of the QAM users and not in the IC scheme. All the resulting parameter settings for the simulations are given in Table 1. All the simulated systems are chip-rate sampled, which limits the possible time lags between users to multiples of chip times. It should also be pointed out that, as a consequence of reduced simulation time, the confidence level is not sufficiently high to give completely accurate results for BER values below 10 \Gamma3 . However, we have considered a higher confidence level in the simulations of single-rate and mixed modulation systems in Figures 8, 9 and 10. 8.2 Stationary AWGN Channels 8.2.1 Multicode Systems The performance of a multicode system in AWGN is shown in Figure 6. There are 15 asyn- Average Bit Probability Processing Orth. Gold Sequences MF Rec. Simulation Analysis Single BPSK Figure Performance of a multicode system with 15 QPSK users and two parallel channels per user. The graph shows analytical and simulation results for one, two and five stages of IC and simulation results for an MF receiver. chronous QPSK users in the system. Each user transmits over two parallel channels and orthogonal Gold codes of length 128 are used. The graph shows analytical and simulation results for one, two and five stages of IC and we can see that a Gaussian approximation is too optimistic for the single-stage IC, but the analytical and simulation results for two and five stages of IC agree well. A Gaussian approximation is probably good for evaluating the performance of the users cancelled first in the single-stage IC scheme, however, as the scheme proceeds, the Gaussian approximation of the MAI becomes less accurate, especially for high E b =N 0 values. This is due to the relatively strong interference, compared to the Gaussian noise, from a small group of users. That is, the central limit theorem does not apply. For multistage IC a Gaussian approximation is good, since then the interference is due only to imperfect cancellation of the users. As can be seen in the graph, after the second stage of IC, most of the performance gain is acquired and the performance is close to the single-user bound. 8.2.2 Mixed Modulation Systems Simulations as well as analytical results of a mixed modulation system is shown in Figure 7. There are 20 BPSK, 10 QPSK and 5 16-QAM users in the system and random sequences of length 127 are used. The analytical results are obtained from an average of 100 rankings, where s Mix (20/10/5) MF Rec. Simulation Analysis Single BPSK Figure 7: Performance of a mixed modulation system with users. The graph shows analytical and simulation results for one, two and five stages of IC and simulation results for an MF receiver. each ranking gives a different ordering of the users according to the modulation format. That is, when the users in the mixed modulation system are ranked according to the channel gain, the order according to modulation format is completely random. As noted from the results, a Gaussian approximation is not reliable for single-stage IC, and using higher modulation formats increases the average BER compared to multicodes. The effect is noticed especially for high values of E b =N 0 . However, after five stages of IC the analytical and simulation results agree well. Studying the average BER of the BPSK, QPSK and 16-QAM users respectively we see that the BER of the 16-QAM users clearly dominates the performance and causes the relatively high average BER in Figure 7. Nonetheless, for the two- and five-stage IC we get a considerable reduction in average BER compared to the single-stage IC and the MF receiver. 8.3 Flat Rayleigh Fading Channels 8.3.1 Single-Rate Systems The average BER of a single-rate system with 20 QPSK users in Rayleigh fading is shown in Figure 8. The length of the random sequences is 127. The results from one, two and five stages of IC are compared with the single-user bound for BPSK users and the results from the corresponding system employing a conventional detector. The graph shows that a Gaussian approximation is too optimistic for a single-stage IC, but it works well for E b =N 0 values up to 20 dB. For multistage IC, the performance is close to the single-user bound and a Gaussian approximation works better, even though the results do not agree perfectly. Nevertheless, in [15] Patel and Holtzman show simulation results for BPSK users and single-stage IC that support the analytical results, which are obtained employing a Gaussian approximation, surprisingly well. We have, however, not been able to reproduce their results. The results for BPSK and 16-QAM s users Simulation Analysis MF Rec. Single BPSK Figure 8: Performance of a QPSK system with 20 users in Rayleigh fading. Analytical and simulation results for an MF receiver and an IC with one, two and five stages are shown. users are similar to the results presented in Figure 8 for QPSK users. The performance of a multistage IC is in both cases close to the respective single-user bound. 8.3.2 Mixed Modulation Systems Average analytical performance and simulation results of a mixed modulation system with 20 users and random sequences of length 127 are shown in Figure 9. The results show that the analysis is too optimistic for the single-stage IC but the accuracy improves with an increasing number of stages. After five stages of IC the analytical performance is 1 dB from the single-user bound. The simulation results for the same mixed modulation system are shown in Figure 10, where average system performance is presented together with average BER for each modulation format. The figure shows great improvement in performance for each additional stage of IC and after five stages the average BER of the different users is close to their respective single-user bound. 8.3.3 Multicode Systems Figure depicts analytical and simulation results for a multicode system with 15 QPSK users, two parallel channels per user and orthogonal Gold codes of length 128. The correspondence between the curves is relatively good for single-stage IC, but for multistage IC, the results agree well for E b =N 0 values up to 20 dB. In this region, the multicode system with multistage IC has a performance within 1 dB of the single-user bound. In Figure 12 we compare the performance of a multicode system (15 QPSK users and two parallel channels per user), with the performance of two single-rate systems (30 QPSK users and is equal for all users in the systems. The simulation results s Mix (20/10/5) MF Rec. Simulation Analysis Single BPSK Figure 9: Performance of a mixed modulation system with users in Rayleigh fading. Both analytical and simulation results are shown. for one, two and five stages of IC show that QPSK modulation together with two parallel channels are preferable to 16-QAM. However, it should be noted that the 16-QAM system outperforms the other two systems for a two-stage IC in the high E b =N 0 region, where the interference is limiting instead of the Gaussian noise. The performance of the 16-QAM system is then close to its single-user bound. The other two systems, the asynchronous QPSK system and the multicode system, have almost the same performance. They perform well and both systems are within 1 dB of the single-user bound for a five-stage IC. The difference in performance for high E b =N 0 values is presumably mainly due to inaccuracy in the simulation results. 8.3.4 Systems with Parameter Estimation In simulations with channel estimation, the channel parameters were estimated using pilot symbols according to (43). In this case we did not assume perfect ranking. The order of the users was instead determined from initial channel estimates as described in Section 3.3. In Figure 13, we compare the simulated performance of a multicode system using estimated channel parameters to a system where the channel parameters are assumed to be known. The degradation due to estimated channel parameters is several dB for a single-stage IC but for two- and five-stage ICs the degradation is only about 1 dB. It can also be noted that the degradation in systems employing a conventional detector is very large. Note that E b is the energy per bit on the channel. That is, there is no compensation for the energy used for the pilot symbols. Random Sequences Mix (20/10/5) Ave. QPSK Ave. 16-QAM MF Rec. Single BPSK Figure 10: Performance of a mixed modulation system with users in Rayleigh fading. Simulation results show average system performance and average BER for each modulation format. 9 Performance Improvements for M-ary QAM Users in Mixed Modulation Systems A disadvantage of using mixed modulation when handling multiple data rates is that high-rate users (16-QAM users) have a higher average BER than low-rate users (BPSK and QPSK users) for the same E b , as indicated in Figure 10. A possible way to reduce the BER for the 16-QAM users is to increase their transmitted power such that they are received with higher E b than the BPSK and QPSK users. The average BER for a mixed modulation system where the users have unequal energy per bit is shown in Figure 14. The system has users and the length of the signature sequences is 127. The E b =N 0 value for the 16-QAM users is increased in steps of 2 dB relative to the E b =N 0 for the other users, which is kept constant to value for the E b =N 0 was chosen since it seemed to give a relevant bit error probability for the system. We have not evaluated the performance for other fixed E b =N 0 values because of the time consuming simulations. The graph depicts that the average BER of the 16-QAM users may be decreased, by increasing the E b of these users, with almost no degradation of the performance of the BPSK and QPSK users. For the single-stage IC a minor degradation can be noticed for the BPSK and QPSK users but for the five-stage IC the performance of the BPSK and QPSK users is unchanged. That is, after five stages of IC, most of the interference is removed and the signals are separated in the signal space independently of their power. Accordingly, increasing the power of the 16-QAM users with an amount corresponding to an increase of E b =N 0 by approximately 2 dB, the average BER for the 16-QAM users is the same as the average BER for the BPSK and QPSK users for both one and five stages of IC. Orth. Gold Sequences QPSK, 15 users, P=2 MF Rec. Simulation Analysis Single BPSK Figure 11: Performance of a multicode system with 15 QPSK users and two parallel channels per user. Simulation and analytical results for an IC with one, two and five stages and simulation results for an MF receiver are shown in the graph. Conclusions The development in mobile communications makes it essential to evolve an efficient system capable of supporting both multiuser detection and variable data rates for the users. The optimum detector is too complex to be implemented in a practical system and the conventional matched filter detector does not perform well without stringent power control. Suboptimal multiuser detectors that has less computational complexity than the optimal detector, but performs better than the conventional detector, are therefore required. In this paper we have demonstrated the use of M-ary rectangular QAM with multistage non-decision directed interference cancellation (NDDIC), which has computational complexity that is linear in the number of users and stages. The two multiple data rate schemes, mixed modulation and multicodes, were analysed for both stationary AWGN channels and flat Rayleigh fading channels, and analytical performance estimates using a Gaussian approximation of the MAI were presented. The analytical results for flat Rayleigh fading channels agreed well with the results from computer simulations for E b =N 0 values up to 20 dB and the correspondence between the results improved with increasing number of IC stages. The performance of the multistage IC, even for systems with many users, was then close to the single-user bound. Consequently, the multistage IC scheme yields a considerable increase in performance compared to the conventional matched filter detector. Considering a mixed modulation system, we found that the users have different average BER depending on their modulation format. That is, the BPSK and QPSK users have lower BER than the 16-QAM users, like in ordinary single-user transmission. However, a small increase in received energy per bit for the 16-QAM users (relative to the BPSK and QPSK users) decreases QPSK, 15 users, P=2 QPSK, users users MF Rec. Single BPSK Figure 12: Performance of three different systems in Rayleigh fading. Simulation results for a multicode system (15 QPSK users, compared with two single-rate systems (30 QPSK users and 15 16-QAM users). their BER without great effect on the other users in the system. On the other hand, if we consider a multicode system, the users' average performance is equal when all users have the same number of parallel channels. To take advantage of the synchronous signalling between a user's parallel channels, orthogonal signature sequences can be used, which improves the overall performance and makes the high-rate users perform better than the low-rate users. To conclude, comparing the performance of the two multirate schemes for the same number of IC stages, multicodes is the preferable scheme. Although, the greatest system flexibility is obtained if the two schemes are combined in such a way that for each new user that is added to the system a decision is made in favour of a number of parallel channels and/or a certain modulation format. Future work within this project will be to study multistage IC schemes together with multi-path Rayleigh fading channels. The inclusion of channel coding and channel estimation will also be investigated. Some of this work has been carried out since this paper was first submitted. It can be found in [24, 29, 30]. Acknowledgment The authors would like to acknowledge Karim Jamal at Ericsson Radio Systems for his initial assistance in obtaining the simulation results. This work was supported by the Swedish National Board of Industrial and Technical Development project 9303363-5. Orth. Gold Sequences QPSK, 15 users, P=2 Known Channel Est. Channel MF Rec. Single BPSK Figure 13: Performance of a multicode system with 15 QPSK users and two parallel channels per user. Simulation results for known and estimated channel parameters for an IC with one, two and five stages and an MF receiver are shown in the graph. --R "Design study for a CDMA-based third-generation mobile radio system," "On the capacity of a cellular CDMA system," "Near-far resistance of multiuser detectors in asynchronous chan- nels," Optimum multi-user signal detection "Minimum probability of error for asynchronous Gaussian multiple-access chan- nels," "Linear multiuser detectors for synchronous code-division multiple-access channels," "Multistage detection in asynchronous code-division multiple access communications," "A family of suboptimum detectors for coherent multi-user communications," "Multiuser detection for CDMA systems," "A spread-spectrum multiaccess system with cochannel interference cancellation for multipath fading channels," "MMSE interference suppression for direct-sequence spread spectrum CDMA," "Decorrelating decision-feedback multi-user detector for synchronous code-division multiple access channel," "Analytic limits on performance of adaptive multistage interference cancellation for CDMA," "Analysis of successive interference cancellation in M-ary orthogonal DS-CDMA system with single path rayleigh fading," "Analysis of a simple successive interference cancellation scheme in a DS/CDMA system," "CDMA with interference can- cellation: A technique for high capacity wireless systems," "Pilot symbol-assisted coherent multi-stage interference canceller for DS-CDMA mobile radio," "Multi-stage serial interference cancellation for DS-CDMA," Interference cancellation for DS/CDMA systems in flat fading chan- nels "Successive interference cancellation schemes in multi-rate DS/CDMA systems," "Multistage interference cancellation in multi-rate DS/CDMA systems," "Decorrelating detectors for dual rate synchronous DS/CDMA systems," "On schemes for multirate support in DS-CDMA systems," "Convergence of linear successive interference cancellation in CDMA," "Performance evaluation for phase-coded spread-spectrum multiple-access communication-Part I: System analysis," "Performance of binary and quaternary direct-sequence spread-spectrum multiple-access systems with random signature sequences," "Multistage interference cancellation in multirate DS/CDMA on a mobile radio channel," "Joint interference cancellation and Viterbi decoding in DS-CDMA," --TR Minimum probability of error for asynchronous Gaussian multiple-access channels On Schemes for Multriate Support in DS-CDMA Systems Analysis of Successive Interference Cancellation in M-ary Orthogonal DS-CDMA System with Single Path Rayleigh Fading

multiple data rates;multi-user detection;multicodes;direct-sequence code-division multiple access DS-CDMA;mixed modulation;interference cancellation

609397

A Channel Sharing Scheme for Cellular Mobile Communications.

This paper presents a channel sharing scheme, Neighbor Cell Channel Sharing (NCCS) , based on region partitioning of cell coverage for wireless cellular networks. Each cell is divided into an inner-cell region and an outer-cell region. Cochannel interference is suppressed by limiting the usage of sharing channels in the inner-cell region. The channel sharing scheme achieves a traffic-adaptive channel assignment and does not require any channel locking. Performance analysis shows that using the NCCS scheme leads to a lower call blocking probability and a better channel utilization as compared with other previously proposed channel assignment schemes.

Introduction One of the major design objectives of wireless cellular communication systems is high network capacity and flexibility, while taking into account time-varying teletraffic loads and radio link quality. The limited radio frequency spectrum requires cellular systems to use efficient methods to handle the increasing service demands and to adapt system resources to various teletraffic (referred to as traffic) in different cells. Many current cellular systems use the conventional radio channel management, fixed channel assignment (FCA), where a set of nominal channels is permanently allocated to each cell for its exclusive use according to traffic load estimation, cochannel and adjacent channel interference constraints [1]. Due to the mobility of users, the traffic information is difficult to accurately predict in any case. As a result, the FCA scheme is not frequency efficient in the sense that the channel assignment cannot adapt to the dynamically changing distribution of mobile terminals in the coverage area. In order to overcome the deficiency of FCA, various traffic-adaptive channel assignment schemes have been proposed, such as dynamic channel assignment (DCA) [2]-[4] and hybrid channel assignment (HCA) [5]. In centralized DCA schemes, all channels are grouped into a pool managed by a central controller. For each call connection request, the associated base station will ask the controller for a channel. After a call is completed, the channel is returned to the channel pool. In distributed DCA schemes, a channel is either selected by the local base station of the cell where the call is initiated, or selected autonomously by the mobile station. A channel is eligible for use in any cell provided that signal interference constraints are satisfied. Since more than one channel may be available in the channel pool to be assigned to a call when required, some strategy must be applied to select the assigned channel. Although the DCA schemes can adapt channel assignment to dynamic traffic loads, it can also significantly increase network complexity due to cochannel cell locking and other channel management, because it is a call-by-call based assignment. In order to keep both cochannel interference and adjacent channel interference under a certain threshold, cells within the required minimum channel reuse distance from a cell that borrows a channel from the central pool cannot use the same channel. DCA also requires fast real-time signal processing and associated channel database updating. A compromise between the radio spectrum efficiency and channel management complexity is HCA, which combines FCA with DCA. In HCA, all available channels are divided into two groups, FCA group and DCA group, with an optimal ratio. It has been shown that both DCA and HCA can achieve a better utilization of radio channel resources than FCA in a light traffic load situation, due to the fact that both schemes can adapt to traffic load dynamics. However, they may perform less satisfactorily than FCA in a heavy traffic load situation due to the necessary channel locking [2]-[5]. Another approach to adaptive channel assignment is channel borrowing, in which the channel resources are divided into borrowable and non-borrowable channel groups [6]-[7]. The non-borrowable group is assigned to a cell in the same way as FCA. When all of its fixed channels are occupied, a cell borrows channels from its neighbor cells which have a light traffic load. More recently, a channel borrowing scheme called channel borrowing without locking (CBWL) is proposed [8], where the C channels of each base station are divided into seven distinct groups. The C 0 channels of group 0 are reserved for exclusive use of the given cell. The (C of the other six groups can be borrowed by the six adjacent cells respectively, one group by one adjacent cell. Each borrowing channel is used with a limited power level. That is, the borrowed channel is directionally limited as well as power limited. Therefore, the channel locking for cochannel cells is not necessary. In this paper, we propose a channel sharing scheme based on a channel sharing pool strategy. The scheme can adapt to traffic dynamics so that a higher network capacity can be achieved. The method partitions cell coverage region to eliminate the cochannel interference due to the dynamic channel sharing; therefore, it does not need any channel locking. In addition, because the borrowable channels are a portion of total available channels and are shared only among adjacent cells, the channel sharing management is relatively simple as compared with that of DCA and HCA. Compared with the CBWL borrowing scheme, the advantage of the newly proposed scheme is the relaxed constraint on directional borrowing, which results in a higher degree of traffic adaptation and a lower call blocking probability. This paper is organized as follows. In Section 2, after studying the cochannel interference issue, we calculate the cochannel interference spatial margin for cell region partitioning, and then propose the Neighbor Cell Channel Sharing (NCCS) scheme for traffic adaptive channel assignment. The adjacent channel interference using the NCCS scheme is also discussed. In Section 3, the call blocking probability using the NCCS scheme is derived. Numerical analysis results are presented in Section 4, which demonstrate the performance improvement of the NCCS over that of previously proposed schemes including FCA, HCA, and CBWL. The conclusions of this work are given in Section 5. 2 The Neighbor Cell Channel Sharing (NCCS) Scheme A. Cochannel Interference A cellular network employs distance separation to suppress cochannel interference. Fig. 1 shows the frequency reuse strategy for a cellular system with frequency reuse factor equal to 7, where the shadowed cells are the cochannel cells of Cell (17) using the same frequency channels (as an example). We assume that the received carrier-to-interference ratio (CIR) at a mobile station (e.g., in Cell (17)) caused by the base stations in the cochannel cells is, on the average, the same as the CIR at the base station of Cell (17) caused by the mobile stations in the cochannel cells. The CIR can be calculated by [9] R (1) where R is the radius of each cell, D i is the distant between the interested cell and its ith cochannel cell, q and fl is a propagation path-loss slope determined by the actual terrain environment (usually fl is assumed to be 4 for cellular radio systems). In a fully equipped hexagonal cellular system, there are always six cochannel cells in the first tier. It can be shown that the interference caused by cochannel cells in the second tier and all other higher-order tiers is negligible as compared with that caused by the first tier cochannel cells [10]. As a result, if we consider the cochannel interference only from the cells in the first tier, in equation (1), where q is a constant. In order to achieve a probability of at least 90% that any user can achieve satisfactory radio link quality for voice service, it requires that the CIR value be or higher, which corresponds to 4. If a channel of Cell (17) is lent to any of its six neighbor cells, then the cochannel interference to and from any of the six cochannel cells in the first tier may increase. For example, if a channel of Cell (17) is lent to Cell (24), the distances between cochannel cells will be D 3:5R. The shortest one is 3R. The channel borrowing of Cell (24) from Cell (17) reduces the CIR value of the channel in Cell (24) from dB to 16 dB, if R is kept unchanged. A similar degradation on radio link quality also happens in Cell (38), where the CIR value is reduced to 17.4 dB. The decrease of the CIR value is due to the decrease of the q i values. B. Cell Region Partition From the above discussion, we conclude that any reduction of the q i value due to a channel borrowing will degrade the radio link quality, because a channel borrowing will result in a decrease of some D i values. One way to keep the q i value unchanged even with channel borrowing is to reduce the value of R accordingly when D i is reduced, which can be implemented by reducing the transmission power. By reducing R to R r borrowed channel can be used only inside the circle (called the inner-cell region) centered at the base station with radius equal to R r in all cochannel cells. In other words, each cell is divided into two regions, the inner-cell region and the rest (called the outer-cell region). For example, if Cell (24) borrows a channel from Cell (17), with D should be 0.652 in order to ensure that q i - 4:6. Since the overall CIR value will be greater than Correspondingly, all the C channels of each base station are also divided into two groups: one consists of N nominal channels to be used exclusively in the cell (in both the inner-cell and outer-cell regions), and the other consists of the rest S (= sharing channels to be used in the inner-cell regions of the given cell and its six neighbor cells. For each cell, there is a pool of sharing channels to be used in its inner-cell region. The sharing pool consists of all the available sharing channels of the cell and the neighbor cells. C. The Proposed Channel Sharing Scheme In the following, it is assumed that: i) all base stations work in the same condition: omnidirection antennas are used, and transceivers are available at a given carrier frequency; ii) only cochannel interference and adjacent channel interference are considered, and all other kinds of noise and interferences are neglected; and iii) neighbor cells can communicate with each other. Without loss of generality, Fig. 2 shows the flowchart of the channel assignment for a two-cell network, where "CH" stands for "channel". Using NCCS, the channel resources in each cell consist of N nominal channels to serve the users in the whole cell as conventional FCA and S sharing channels to serve users in the inner-cell regions. When a call connection is requested, a nominal channel will be assigned to it. In the case that all the nominal channels of the cell are occupied, if the mobile is in the inner-cell region, then a channel from the sharing pool will be used; otherwise, if the mobile is in the outer-cell region and there is a mobile in the inner-cell region using a nominal channel, then an event of channel swapping occurs: the inner-cell mobile switches to a channel from the sharing pool and gives up its original nominal channel to the new call from the outer-cell region. The purpose of the channel swapping is to make room for new calls so that the system channel resources can be fully deployed. Note that when there is no channel available in the sharing pool, channel swapping may be carried out in a neighbor cell to allow for channel borrowing. The call will be blocked if i) all the nominal channels are occupied and no channel swapping is possible when the mobile is in the outer-cell region; ii) all the nominal channels and sharing channels in the pool are occupied when the mobile is in the inner-cell region. When a connected user moves from the outer-cell region into the inner-cell region, the transmitters of both mobile terminal and base station will reduce the transmitting power automatically since the mobile terminal gets closer to the base station. When an inner-cell mobile terminal using a sharing channel moves into the outer-cell region, not only the associated power control occurs, but also an intra-cell handoff happens because the sharing channel cannot be used in the outer-cell region. If there is no nominal channel available for the intra-cell handoff, the link will be forced to drop. In implementing the NCCS, each borrowable channel has an "on/off" register in the associated channel sharing pools to indicate whether the channel is available. During the channel borrowing from a neighbor cell, the borrowed channel register is turned to "off" in the sharing pools. For example, when Cell (24) borrows a channel from Cell (17), the channel register is turned "off" in the sharing pools of Cell (17), Cell (10), Cell (11), Cell (16), Cell (18), and Cell (25). When the borrowed channel is returned, the register is then turned back to "on" in the pools. One aspect that should be taken into account is the potential borrowing conflict, that is, two adjacent cells try to borrow the same channel from the different cells at the same time. This will violate the channel reuse distance limitation. To prevent the borrowing conflict, some selective borrowing algorithms should be introduced such as borrowing with ordering [11], borrowing from the richest [12]. The cell selectivity for borrowing can achieve higher capacity at the expense of higher complexity. One simple approach is to use directional borrowing restriction. For example, in Fig. 1, when Cell (24) borrows a channel from Cell (17), it is required that Cell (23) not borrow the same channel from Cell (22), and Cell (31) and Cell (32) not borrow the same channel from Cell (38). All other neighbor cells of the six cochannel cells are allowed to borrow the same channel. In order words, the borrowing restriction is limited only to those cells affected by the borrowing of Cell (24) from Cell (17). The restriction can be implemented by turning "off" the register of the channel in the sharing pools of Cell (23), Cell (31) and Cell (32). It should be mentioned that in the CBWL scheme [8], directional lending is used to avoid the borrowing conflict, where each cell can borrow up to one sixth of the borrowable channels from its neighbor cells. As a result, using the NCCS scheme each cell has a much larger channel sharing pool than that using the CBWL scheme under the same condition of the channel resource arrangement. It is expected that the NCCS scheme can adapt channel assignment to traffic dynamics to a larger extent as compared with the CBWL scheme, leading to a lower call blocking probability. D. Adjacent Channel Interference Adjacent channel interference is a result of the splatter of modulated RF signals. Because the mobility of network users, the distance between a mobile terminal and its base station changes with time. At each moment, some mobile terminals are close to the base station and others are not. Considering the receiver at the base station, the adjacent channel interference may not be a problem if the signals from the desired channel and both its adjacent channels are received with the same power level. The bandpass filter of the receiver should provide adequate rejection to the interference from the adjacent channels. However, the problems may arise if two users communicate to the same base station at significantly different transmitting power levels using two adjacent channels. Signal from the adjacent channel can be stronger than that from the desired channel to such a degree that the desired signal is dominated by the signal carried by the adjacent channel. This situation is referred to as "near-far" effect in wireless mobile communication systems. The larger the difference between the near-far distances, the worse the adjacent channel interference in radio links. Severe adjacent channel interference may occur when the difference in the received power levels exceeds the base station receiver's band rejection ratio. Therefore, channel separations are required, which is primarily determined by the distance ratio, the path-loss slope fl, and the receiver filter characteristics. The required channel separation, in terms of channel bandwidth W , is 2 (d a =d b ) fl , L in dB is the falloff slope outside the passband of the receiver bandpass filter, d a is the distance between the base station and mobile terminal M a using the desired channel, and d b is the distance between the base station and mobile terminal M b using one of the adjacent channels. In order to overcome the adjacent channel interference, FCA achieves channel separation by channel interleaving in such a way that there is sufficient channel guard band between any two channels assigned to a base station. For a call connection, the user can just randomly choose any channel with the strongest signal from all the available channels, without violating the adjacent channel interference constraint. With channel borrowing, if a cell has traffic congestion, it will borrow channels from its neighbor cell(s). A borrowed channel may be located in the channel guard band, which can introduce excessive interference to the desired signal when the difference between d a and d b is large. Therefore, two aspects need to be taken into account with channel borrowing: one is the cochannel interference issue as to whether channel borrowing is allowed; the other is the adjacent channel interference issue as to whether the borrowed channel can provide satisfactory link quality. For the NCCS, it has been shown that with dynamic power control, if d a =d b - 16, no extra channel separation is required between any two channels assigned to the same base station in order to overcome the adjacent channel interference [10]. If we consider the users, M a and M b , both in the inner-cell region, then the requirement d a =d b - 16 is eqivalent to that the radius of the is the minimal distance between a mobile terminal and the base station. Under the condition, no channel spacing is needed among the sharing channels used in the inner-cell regions. With the cell radius R adjacent channel interference does not affect the channel interleaving and the target radio link quality when channel borrowing happens in the NCCS operation as long as R 0 - 0:04075R. Compared with other channel assignment schemes, the NCCS scheme offers the following advantages: i) it ensures satisfactory link quality (taking into account both cochannel interference and adjacent channel interference) for both nominal channels and borrowable channels, which cannot be achieved using directed retry and its enhanced schemes [6]; ii) it does not need global information and management of channel assignment which is required when using DCA schemes, resulting in simplicity of implementation; iii) each base station is required to operation on its nominal channels and the borrowable channels of its sharing pool, which is a much smaller set as compared with that when using DCA schemes; and iv) no channel locking is necessary, which leads to a better utilization of the channel resources and a simpler management for channel assignment as compared with DCA and HCA. Performance Analysis In the following performance analysis of the NCCS scheme, we consider that the network operates on a blocked call cleared (BCC) basis, which means once a call is blocked it leaves the system. Under the assumption that the number of users is much larger than the number of channels assigned to a base station, each call arrival is independent of the channel occupancy at the base station [13]. Handoff calls are viewed as new calls. A. Basic Modeling In general, requests for radio channels from mobile users can be modeled as a Poisson arrival process. The occupancy of radio channels at a base station conventionally is considered as a "birth and death" process with states f0; is the number of total channels assigned to the base station. A new call arrival enters the system with a mean arrival rate - and leaves the system with a mean departure rate -. Defining the traffic density A = -, it can be derived that the probability of the channel occupancy being at state j is [13] From equation (2), the probability of radio channel resource congestion (i.e, the call blocking probability) is which provides a fundamental measure of the mobile cellular network performance. Equation (3) is usually referred to as B. Call Blocking Probability with Channel Sharing In order to formulate the call blocking probability for the NCCS scheme, we consider the channel sharing between two adjacent cells without loss of generality. Both base stations are equipped with the same numbers of nominal channels and borrowable channels respectively. The uniform topology of this scenario is shown in Fig. 3. For Cell (i) (where the traffic density in the inner-cell region; y i the traffic density in the outer-cell region; u Sg the channel occupancy in the inner-cell; and v i 2 f0; 1; 2; :::; ; Ng the channel occupancy in the outer-cell. We assume that u i and v i are independent random variables. For each cell, the channel occupancy in the inner-cell region may be different from that in the outer- cell region, so is the call blocking probability. Let PB (C 1;in ) and PB (C 1;out ) denote the blocking probability of Cell (1) for inner-cell and outer-cell regions respectively. Due to the channel sharing, the blocking probabilities of Cell (1) depend on the channel occupancy of the neighbor cell, Cell (2), which is at one of the following two states: (I) The Cell (2) channel occupancy is within its C equipped channels. In this case, we say that Cell (2) is underflow with a 2 f0; unused and borrowable channels. The probability of Cell (2) being underflow is (II) The Cell (2) channel occupancy is over its assigned C channels. In this case, we say that Cell (2) is overflow with b 2 f1; :::Sg borrowed channels from Cell (1). The probability of Cell (2) being overflow is It can be verified that P (C For state (I), the blocking probability of the outer-cell region of Cell (1) (represented by outer cell (1) for simplicity) is The call blocking probability of the inner-cell region of Cell (1) (represented by inner-cell (1)), only conditioned upon Cell (2) channel occupancy, but also upon the situation of outer-cell (1). If there are n unused channels out of the N nominal channels in Cell (1), the inner-cell users can use them. Under the assumption that the channel occupancies in outer cell (1) and Cell (2) are independent, we have For state (II), the blocking probability of the outer cell (1) is and the blocking probability of the inner cell (1) is Using equation (2) to compute P 2, the four conditional probabilities of equations (6)-(9) can be obtained. Then, using equations (4)-(5) the blocking probabilities of both inner-cell and outer-cell regions of Cell (1) can be calculated according to the theorem on total probability C. Effect of User Mobility Taking user mobility into consideration, the number of mobile terminals in a cell at a given moment is a random variable. For the two-cell network, the overall traffic load is dynamically distributed over the two cells. The network is designed in such a way that each cell has a fair share of resources depending on its traffic load in a long term. However, the traffic load over each cell is a random process. Let A i ( 4 the total traffic density of Cell (i) for and 2, and ~ the overall traffic density of the cellular network. Given the number of subscribers, the traffic of the whole network, ~ A, is a constant. Let A A and A where ff 2 [0; 1] is a random variable referred to as a traffic load distributor whose value indicates the traffic load in Cell (1) and Cell (2). If Cell (1) and Cell (2) are identical (Fig. 3), the traffic load distributor should have a mean value 0:5. The following relations are considered for the blocking probability equations: x (referred to as an interior distributor inside Cell (i)) is a random variable with a mean value of coverage of inner-cell(i) coverage of whole 2: Using these two distributors, the blocking probabilities in equations (10)-(11) can be denoted as where g(\Delta) and h(\Delta) denote any measurable function. If we assume that mobile terminals are uniformly distributed in the coverage area and the cell sizes are the same, then ae Furthermore, the number of terminals in each cell or cell region follows a binomial distribution, from which we can obtain the joint probability distribution function p(ff; ae) of the distributors ff and ae. As a result, the blocking probabilities related to the overall traffic density ~ A and the design parameters N and S are PB;C 1;outer PB;C 1;inner In reality, cellular network service operators will try to achieve service fairness, that is, appropriate channel resources will be allocated to each base station in order to obtain the same call blocking probability over all the cells in the service area. Therefore, the system is designed to have PB;C 1;inner As a result, the blocking probabilities in terms of the traffic density ~ A and channel resources N and S is where p ff (ff i ) is the probability distribution function of ff. The analysis for the two-cell network can be extended to a multiple-cell network as shown in Fig 1, where for a cell under consideration all of its 6 neighbor cells can be equivalently modeled by a composite neighbor cell. 4 Numerical Results and Discussion The numerical analysis in this section is to provide a performance comparison between the NCCS and other channel assignment schemes. The following assumptions are made in the analysis: i) All the base stations are equipped with the same numbers of nominal channels and borrowable Each new call is initiated equally likely from any cell and is independent of any other calls. Except in the analysis of the bounds of the blocking probability, the following assumptions are also made: iii) Taking into account the possible borrowing conflict, the channel sharing pool for each cell consists of available borrowable channels of the cell and four of its neighbor cells; iv) The traffic loads in all the cells are statistically the same. Under the assumptions, given the total traffic loads in the network, the traffic load distributor for the cell under consideration follows a binomial distribution. Fig. 4 shows the call blocking probabilities of the FCA, HCA [5] and NCCS schemes. In FCA, each base station has 28 nominal channels; in HCA, each base station has 20 FCA channels and 8 DCA channels; and in NCCS, In Fig. 4 and all the following figures, A is the traffic density for each cell. The performance of the FCA scheme is calculated based on Equation (3), while the performance of the NCCS scheme is based on Equation (17). It is observed that: i) At a low traffic load, HCA has a much lower blocking rate than FCA; however, as the traffic load increases, the advantage of HCA over FCA disappears. In fact, HCA may have a higher blocking probability, due to the necessary DCA channel locking; ii) The NCCS scheme outperforms the HCA scheme because the NCCS scheme can adapt to traffic dynamics without channel locking; iii) The NCCS scheme performs much better than the FCA scheme, but the improvement is reduced as the traffic load increases. This is because with a large value of A, all the cells tend to be in a congestion state, so that the probability of having any sharing channel available for borrowing is greatly reduced. Fig. 5 shows the blocking probabilities of the FCA, CBWL with channel rearrangement [8] and NCCS schemes. In the CBWL scheme, each base station has 24 channels with C and 30% of call arrivals can use borrowed channels. In the NCCS scheme, corresponding to 25% of calls can use borrowed channels. It is observed that the NCCS scheme has a lower blocking probability than the CBWL scheme, due to the fact that the CBWL is limited to the directional lending, resulting in a channel sharing pool with much less borrowable channels as compared with that of the NCCS scheme. The call blocking probability of the NCCS scheme depends on the traffic load dynamics, which can be difficult to generalize. In the following, we consider two extreme cases which lead to the lower and upper bounds on the call blocking probability for the NCCS scheme with channel sharing among m (= 2; 3; 4; 5; 6; 7) neighbor cells. First, consider the situation where one cell is a traffic "hot spot" and its neighbor cells have many idle channels, which we refer to as a local burst situation. The heavily traffic loaded cell can borrow most or all of the sharing channels from its neighbor cells, resulting in a lower bound of blocking probability for the cell. The other situation is that all the cells are heavily loaded and no channel sharing is possible, which is referred to as a global busy situation. If the channel resources, C, in each cell is properly divided into the nominal channel group and sharing channel group, then the global busy situation results in the upper bound of the call blocking probability of the NCCS scheme, which is the same as the call blocking probability of the FCA scheme. Fig. 6 shows the lower and upper bounds of the call blocking probability of the NCCS scheme with channel sharing among m multiple cells. Each base station has 15 nominal and 5 sharing channels. It is observed that the lower bound decreases significantly as m increases, due to an increased number of sharing channels available in the sharing pool. However, when the traffic density increases, the performance improvement of the NCCS over FCA (the upper bound) is significantly reduced. Even with all the sharing channels from the m neighbor cells, it is still possible that the channel resources available to the cell are not enough to provide service to all the incoming calls in the hot spot. Conclusions In this paper, we have developed the neighbor cell channel sharing (NCCS) scheme for wireless cellular networks. Both cochannel interference and adjacent interference issues regarding the channel sharing have been discussed. It has been shown that the NCCS scheme achieves a lower call blocking probability for any traffic load and traffic dynamics as compared with other channel assignment schemes. The performance improvement is obtained at the expense of additional intra-cell handoffs. With more neighbor cells in channel sharing, the proposed scheme offers better traffic handling capacity. The advantages of the proposed scheme include i) that no channel locking is necessary, and ii) larger channel sharing pools are available due to less strict constraint on directional borrowing, which lead to both simpler channel resource management and lower call blocking probability. Acknowledgements The authors wish to thank ITRC (the Information Technology Research Center - Center of Excellence supported by Technology Ontario) for the research grant which supported this work. --R "Handover and channel assignment in mobile cellular networks" "Performance analysis of cellular mobile communication systems with dynamic channel assignment" "Distributed dynamic channel allocation algorithms with adjacent channel constraints" "Performance issues and algorithms for dynamic channel assignment" "A hybrid channel assignment scheme in large-scale cellular-structured mobile communication systems" "A cellular mobile telephone system with load sharing - an enhancement of directed retry" "Load sharing sector cells in cellular systems" "CBWL: a new channel assignment and sharing method for cellular communications systems" Mobile cellular telecommunication systems. "Channel assignment and sharing for wireless cellular networks" "A new frequency channel assignment algorithm in high capacity mobile communication systems" "Comparisons of channel assignment strategies in cellular mobile telephone systems" Queueing Systems. --TR Mobile Cellular Telecommunications Systems --CTR Jean Q.-J. Chak , Weihua Zhuang, Capacity Analysis for Connection Admission Control in IndoorMultimedia CDMA Wireless Communications, Wireless Personal Communications: An International Journal, v.12 n.3, p.269-282, March 2000

cellular mobile communications;neighbor cell channel sharing;channel assignment

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Adaptive Modulation over Nakagami Fading Channels.

We first study the capacity of Nakagami multipath fading (NMF) channels with an average power constraint for three power and rate adaptation policies. We obtain closed-form solutions for NMF channel capacity for each power and rate adaptation strategy. Results show that rate adaptation is the key to increasing link spectral efficiency. We then analyze the performance of practical constant-power variable-rate M-QAM schemes over NMF channels. We obtain closed-form expressions for the outage probability, spectral efficiency and average bit-error-rate (BER) assuming perfect channel estimation and negligible time delay between channel estimation and signal set adaptation. We also analyze the impact of time delay on the BER of adaptive M-QAM.

Introduction The radio spectrum available for wireless services is extremely scarce, while demand for these services is growing at a rapid pace [1]. Hence spectral efficiency is of primary concern in the design of future wireless data communications systems. In this paper we first investigate the theoretical spectral efficiency limits of adaptive transmission in Nakagami multipath fading (NMF) channels [2]. We then propose and study adaptive multi-level quadrature amplitude modulation (M-QAM) schemes which improve link spectral efficiency (R=W [Bits/Sec/Hz]), defined as the average transmitted data rate per unit band-width for a specified average transmit power and bit-error-rate (BER). We also evaluate the performance of these schemes relative to the theoretical spectral efficiency limit. Mobile radio links can exhibit severe multipath fading which leads to serious degradation in the link carrier-to-noise ratio (CNR) and consequently a higher BER. Fading compensation such as an increased link budget margin or interleaving with channel coding are typically required to improve link performance. However, these techniques are designed relative to the worst-case channel conditions, resulting in poor utilization of the full channel capacity a good percentage of the time (i.e., under negligible or shallow fading conditions). Adapting certain parameters of the transmitted signal to the channel fading leads to better utilization of the channel capacity. The basic concept of adaptive transmission is real-time balancing of the link budget through adaptive variation of the March 13, 1998 transmitted power level, symbol transmission rate, constellation size, coding rate/scheme, or any combination of these parameters [3], [4], [5], [6], [7]. Thus, without wasting power or sacrificing BER, these schemes provide a higher average link spectral efficiency by taking advantage of the time-varying nature of wireless channels: transmitting at high speeds under favorable channel conditions and responding to channel degradation through a smooth reduction of their data throughput. Good performance of these schemes requires accurate channel estimation at the receiver and a reliable feedback path between that estimator and the transmitter. Furthermore since outage probability of such schemes can be quite high, especially for channels with low average CNR, buffering of the input data may be required, and adaptive systems are therefore best suited to applications without stringent delay constraints. The Shannon capacity of a channel defines its maximum possible rate of data transmission for an arbitrarily small BER, without any delay or complexity constraints. Therefore the Shannon capacity represents an optimistic bound for practical communication schemes, and also serves as a bench-mark against which to compare the spectral efficiency of adaptive transmission schemes [8]. In [9] the capacity of a single-user flat-fading channel with perfect channel measurement information at the transmitter and receiver was derived for various adaptive transmission policies. In this paper we apply the general theory developed in [9] to obtain closed-form expressions for the capacity of NMF channels under different adaptive transmission schemes. In particular, we consider three adaptive policies: optimal simultaneous power and rate adaptation, constant power with optimal rate adaptation, and channel inversion with fixed rate. We then present numerical results showing that rate adaptation is the key to achieving high link spectral efficiency. Rate adaptation can be achieved through a variation of the symbol time duration [3] or constellation size [5]. The former method requires complicated hardware and results in a variable-bandwidth system, whereas the latter technique is better suited for hardware implementation, since it results in a variable-throughput system with a fixed bandwidth. Based on these advantages we analyze the performance of constant-power variable-rate M-QAM schemes for spectrally efficient data transmission over NMF channels. Similar analysis has been presented in [6] for a variable-power variable-rate M-QAM in Rayleigh fading and log-normal March 13, 1998 Feedback Path Channel Nakagami Channel Slowly Varying Modulator Input Transmitter Channel Estimator Constellation Size Selector Pilot Demodulator Receiver Data Output Data AGC Carrier Recovery Data Fig. 1. Adaptive communication system model. shadowing, and in [10] for constant-power variable-rate M-QAM in Rayleigh fading. We extend the results of [6], [10] to constant-power variable-rate M-QAM by analyzing the resulting spectral efficiency and BER for the more general NMF distribution. We also analyze the impact of time delay on the performance of adaptive M-QAM. The remainder of this paper is organized as follows. In Section II we outline the channel and communication system models. In Section III we derive the capacity of NMF channels for the optimal adaptive policy, constant power policy, and channel inversion policy, and we present some numerical examples comparing (i) the NMF channel capacity with the capacity of an additive white Gaussian noise (AWGN) channel, and (ii) the NMF channel capacity for the various adaptive policies. In Section IV we propose and evaluate the performance of an adaptive constant-power variable-rate M-QAM system assuming perfect channel estimation and negligible time delay. The BER degradation due to time delay is analyzed in Section V. A summary of our results is presented in Section VI. II. System and Channel Models A. Adaptive Communication System Model A block diagram of the adaptive communication system is shown in Fig. 1. A pilot tone continually sends a known "channel sounding" sequence so that the channel-induced envelope fluctuation ff and phase shift OE can be extracted at the channel estimation stage. Based on this channel gain estimate - ff, a decision device selects the rate and power to be transmitted, configures the demodulator accordingly, and informs the transmitter about that decision via the feed back path. The constellation size assignment for the proposed March 13, 1998 constant-power variable-rate M-QAM scheme will be discussed in more detail in Section IV-A. The transmission system keeps its configuration unchanged (i.e, no re-adaptation) for a duration - t [s]. Meanwhile the phase estimate - OE is used at the receiver for full compensation of the phase variation (i.e., ideal coherent phase detection), whereas the channel gain estimate - ff is used on a continuous basis by the automatic gain controller (AGC)/demodulator for symbol-by-symbol maximum-likelihood detection. For satisfactory operation the modulator and demodulator must be configurated at any instant for the same constellation size. Efficient error control schemes are therefore required to insure an error-free feedback path. However such schemes inevitably introduce a certain time delay - fb [s], which may include decoding/ARQ delay, and propagation time via the feedback path. Hence, even if perfect channel estimates are available at the receiver, the system will not be able to adapt to the actual channel fading but rather to at best a - fb delayed version of it. In practice, the choice of the power and/or constellation is based on a channel estimate at time t, but the data are sent over the channel at time t +- such that is the rate at which we change the constellation size and power. The goal is to operate with the smallest possible - fb to minimize the impact of feedback delay, and with the largest possible - t to minimize the rate of system reconfiguration. This issue will be further discussed in Section V. B. Channel Model and Fading Statistics We consider a slowly-varying flat-fading channel changing at a rate much slower than the symbol data rate, so the channel remains roughly constant over hundreds of symbols. The multipath fading environment can be characterized by different statistical models. For NMF channels the probability distribution function (PDF) of the channel gain ff is given by [2, (11)] \Omega \Gamma(m) exp \Gammam ff 2 \Omega is the average received power, m is the Nakagami fading parameter (m - 1=2), and \Gamma(:) is the gamma function [11]. The received CNR, fl, is then gamma March 13, 1998 distributed according to the PDF, p fl (fl), given by \Gamma(m) exp \Gammam where fl is the average received CNR. The phase OE of the Nakagami fading is uniformly distributed over [0,2-]. The Nakagami fading represents a wide range of multipath channels via the m fading parameter [2]. For instance, the Nakagami-m distribution includes the one-sided Gaussian distribution which corresponds to worst-case fading) and the Rayleigh distribution special cases. In addition, when m ? 1, a one-to-one mapping between the Rician factor and the Nakagami fading parameter allows the Nakagami-m distribution to closely approximate the Rice distribution [2]. Finally, and perhaps most importantly, the Nakagami-m distribution often gives the best fit to urban [12] and indoor [13] multipath propagation. III. Capacity of Nakagami Fading Channels A. Optimal Adaptation Given an average transmit power constraint, the channel capacity of a fading channel with received CNR distribution p fl (fl) and optimal power and rate adaptation (!C ? opra [Bit/Sec]) is given in [9] as log 2 where W [Hz] is the channel bandwidth and fl o is the optimal cutoff CNR level below which data transmission is suspended. This optimal cutoff must satisfy the equation To achieve the capacity (3), the channel fade level must be tracked at both the receiver and transmitter, and the transmitter has to adapt its power and rate accordingly, allocating high power levels and rates for good channel conditions (fl large), and lower power levels and rates for unfavorable channel conditions (fl small). Since no data is sent when the optimal policy suffers a probability of outage P out , equal to the probability of no March 13, 1998 transmission, given by Substituting (2) in (4) we find that fl must satisfy where \Gamma(:; :) is the complementary incomplete gamma function [11]. For the special case of the Rayleigh fading channel reduces to e \Gammafl is the exponential integral of first order [11]. Let Note that df(x) Moreover, from (8), lim x!0 there is a unique positive x o for which f(x equivalently, there is a unique fl which satisfies (6). An asymptotic expansion of (6) shows that as Our numerical results show that fl o increases as fl increases, so lies in the interval [0,1]. Substituting (2) in (3), and defining the integral J n (-) as we can rewrite the channel capacity !C? opra as Jm The evaluation of J n (-) for n a positive integer is derived in [14, Appendix A]. Using that result we obtain the NMF channel capacity per unit bandwidth [Bits/Sec/Hz] under the optimal power and rate adaptation policy as which can also be written as denotes the Poisson distribution defined by For the special case of the Rayleigh fading channel, using (7) in (12) for m=1, the optimal capacity per unit bandwidth reduces to the simple expression e \Gammafl Using (2) in the probability of outage equation (5) yields B. Constant Transmit Power With optimal rate adaptation to channel fading with a constant transmit power, the channel capacity !C? ora [Bits/Sec] becomes [9] !C? ora was previously introduced by Lee [15], [16] as the average channel capacity of a flat-fading channel, since it is obtained by averaging the capacity of an AWGN channel over the distribution of the received CNR. In fact, (16) represents the capacity of the fading channel without transmitter feedback (i.e. with the channel fade level known at the receiver only) [17], [18], [19]. Substituting (2) into (16) and defining the integral I n (-) as I Z +1t the channel capacity !C? ora of a NMF channel can be written as Im March 13, 1998 The evaluation of I n (-) for n a positive integer is derived in [14, Appendix B]. Using that result, we can rewrite !C? ora =W [Bits/Sec/Hz] as One may also express (20) in terms of the Poisson distribution as [16] Note that Yao and Sheikh [20] provided a closed-form expression for the capacity of NMF channels in terms of the complementary incomplete gamma function. However their derivation is different then ours and their resulting expression [20, (7)] contains m order deriva- tives. For the special case of the Rayleigh fading channel reduces to C. Channel Inversion with Fixed Rate The channel capacity when the transmitter adapts its power to maintain a constant CNR at the receiver (i.e., inverts the channel fading) was also investigated in [9]. This technique uses fixed-rate modulation and a fixed code design, since the channel after channel inversion appears as a time-invariant AWGN channel. As a result, channel inversion with fixed rate is the least complex technique to implement, assuming good channel estimates are available at the transmitter and receiver. The channel capacity with this technique (!C? cifr [Bits/Sec]) is derived from the capacity of an AWGN channel and is given in [9] as Channel inversion with fixed rate suffers a large capacity penalty relative to the other techniques, since a large amount of the transmitted power is required to compensate for the deep channel fades. Another approach is to use a modified inversion policy which inverts the channel fading only above a fixed cutoff fade depth fl . The capacity with this truncated channel inversion and fixed rate policy (!C ? tifr [Bits/Sec]) was derived in [9] to be R +1 where P out is given by (5). The cutoff level fl o can be selected to achieve a specified outage probability or, alternatively (as shown in Figures 2, 3, and 4), to maximize (24). By substituting the CNR distribution (2) in (23) we find that the capacity per unit bandwidth of a NMF channel with total channel inversion, !C ? cifr =W , is given for all Thus the capacity of a Rayleigh fading channel zero in this case. Note that the capacity of this policy for a NMF channel is the same as the capacity of an AWGN channels with equivalent CNR= With truncated channel inversion the capacity per unit bandwidth !C? tifr =W [Bits/Sec/Hz] can be expressed in terms of fl and fl o by substituting (2) into (24), which yields \Gamma(m; mfl For the special case of the Rayleigh fading channel 1), the capacity per unit band-width with truncated channel inversion reduces to e \Gammafl D. Numerical Results Figures 2, 3, and 4 show the capacity per unit bandwidth as a function of fl for a NMF channel under the three different adaptive policies for respectively. We see from these figures that the capacity of NMF channels is always smaller than the capacity of an AWGN channel for fl - 0 dB but converges to it as the m parameter increases or, equivalently, as the amount of fading decreases. We also see that optimal power and rate adaptation yields a small increase in capacity over just optimal rate adaptation, and this small increase in capacity diminishes as the average received CNR and/or fading parameter m increase. Note finally that fixed rate transmission with channel inversion suffers the largest capacity penalty. However, this penalty diminishes as the amount of fading decreases. Average Received CNR [dB] Capacity per Unit Bandwidth AWGN Channel Capacity Optimal Power and Rate Optimal Rate and Constant Power Truncated Channel Inversion Total Channel Inversion Fig. 2. Capacity per unit bandwidth for a Rayleigh fading channel (m=1) under different adaption policies. IV. Adaptive M-QAM Modulation A. Proposed Adaptive Schemes The BER of coherent M-QAM with two-dimensional Gray coding over an additive white Gaussian noise (AWGN) channel assuming perfect clock and carrier recovery can be well approximated by [6] Exact expressions for the BER of "square" M-QAM (when the number of bits per symbol n is even) are known [21, Chapter 5], and are plotted by the solid lines in Fig. 5. On the other hand, tight upper-bounds on the BER of "non-square" M-QAM (when the number of bits per symbol n is odd) are also available [22, p. 283], and are plotted by the cross/solid lines in Fig. 5. For comparison, the dash lines in this figure show the BER approximation for different values of M . Note that the approximate BER expression upper bounds the exact BER for M - 4 and for which is the BER range of interest. We will use this approximation when needed in our analysis since it is "invertible" in the sense that it provides a simple closed-form expression for the link spectral efficiency of M-QAM as a function of the CNR and the BER. In addition, (28) and its inverse are very simple functions which lead, as shown below, to closed-form analytical expressions and insights March 13, 1998 ALOUINI AND GOLDSMITH: ADAPTIVE MODULATION OVER NAKAGAMI FADING CHANNELS 11 Average Received CNR [dB] Capacity per Unit Bandwidth AWGN Channel Capacity Optimal Power and Rate Optimal Rate and Constant Power Truncated Channel Inversion Total Channel Inversion Fig. 3. Capacity per unit bandwidth for a Nakagami fading channel with m=2, and for different adaption policies. that are unattainable with more complicated BER expressions. Assuming ideal Nyquist pulses and given a fixed CNR (fl) and BER (BER 0 ) the spectral efficiency of continuous-rate M-QAM can be approximated by inverting (28), giving R where adaptive continuous rate (ACR) M-QAM scheme responds to the instantaneous channel CNR fluctuation by varying the number of bits per symbol according to (29). In the context of this paper, continuous-rate means that the number of bits per symbol is not restricted to integer values. While continuous-rate M-QAM is possible [23] it is more practical to study the performance of adaptive discrete rate (ADR) M-QAM, where the constellation size M n is restricted to 2 n for n a positive integer. In this case the scheme responds to the instantaneous channel CNR fluctuation by varying its constellation size as follows. The CNR range is divided into N fading regions, and the constellation size M n is assigned to the nth region When the received CNR is estimated to be in the nth region, the constellation size M n is transmitted. Suppose we set a target BER, BER 0 . The region boundaries (or switching thresholds) are then set to the CNR required to achieve the target BER 0 using M n -QAM over March 13, 1998 Average Received CNR [dB] Capacity per Unit Bandwidth AWGN Channel Capacity Optimal Power and Rate Optimal Rate and Constant Power Truncated Channel Inversion Total Channel Inversion Fig. 4. Capacity per unit bandwidth for a Nakagami fading channel with m=4, and for different adaption policies. Carrier-to-Noise-Ratio CNR g [dB] Bit Rate BER Approximation (3) Exact Upper Bound Fig. 5. BER for M-QAM versus CNR. Assignment of Constellation Size Relative to Received CNR for a Target BER Received Carrier-to-Noise-Ratio CNR g [dB] Constellation Size logM Continuous Rate Adaptive M-QAM (4) Discrete Rate Adaptive M-QAM Fig. 6. Number of bits per symbol versus CNR. an AWGN channel. Specifically denotes the inverse complementary error function. When the switching thresholds are chosen according to (30), the system will operate with a BER below the target BER, as will be confirmed in Section IV-D. Note in particular that all the are chosen according to (28). Since (28) is an upper-bound of the BER only for M - 4, fl 1 is chosen according to the exact BER performance of 2-QAM (BPSK). The thick line in Fig. 6 shows the number of bits per symbol as a function of the received CNR for ADR M-QAM with 8-regions, along with the corresponding switching thresholds. For comparison the thin line in this figure shows the bits per symbol of ACR M-QAM. B. Outage Probability Since no data is sent when the received CNR falls below fl 1 , the ADR M-QAM scheme suffers an outage probability, P out , of Figs. 7 shows the outage probability for various values of the Nakagami fading parameter and for target BERs of 10 \Gamma3 and 10 \Gamma6 , respectively. Probability of Outage Average Received Carrier-to-Noise-Ratio g- [dB] Probability of Outage out Target BER=10 -3 Target BER=10 -6 Fig. 7. Outage probability in Nakagami fading. C. Achievable Spectral Efficiency Integrating (29) over (2) and following the same steps of Section III-B which obtained (20), we find the average link spectral efficiency, !R? acr =W , of the ACR M-QAM over NMF channels as !R? acr March 13, 1998 Achievable Rates in Rayleigh Fading (m=1) for a Target BER Average Received Carrier-to-Noise-Ratio g- [dB] Spectral Efficiency Regions Regions Regions Capacity (Optimal Rate and Constant Power) Continuous Rate Adaptive M-QAM Discrete Rate Adaptive M-QAM Non Adaptive 2-QAM (B-PSK) Fig. 8. Achievable spectral efficiency for a target BER of 10 \Gamma3 and The average link spectral efficiency, of the ADR M-QAM over NMF channels is just the sum of the data rates (log 2 [M n associated with the individual weighted by the probability a that the CNR fl falls in the nth region: where the a n s can be expressed as a Figs. 8, show the average link spectral efficiency of ACR M-QAM (32) and ADR M-QAM (33) for a target BER tively. The Shannon capacity using constant-power and variable-rate (20) is also shown for comparison, along with the spectral efficiency of nonadaptive 2-QAM (BPSK). This latter efficiency is found by determining the value of the average received CNR for which the average BER of nonadaptive BPSK over Nakagami fading channel, as given by (38), equals the target BER. Note that the achievable spectral efficiency of ACR M-QAM comes within 5 dB of the Shannon capacity limit. ADR M-QAM suffers a minimum additional 1.2 dB penalty, whereas nonadaptive BPSK suffers a large spectral efficiency penalty. Achievable Rates in Nakagami Fading (m=2) for a Target BER Average Received Carrier-to-Noise-Ratio g- [dB] Average Spectral Efficiency Regions Regions Regions Capacity (Optimal Rate and Constant Power) Continuous Rate Adaptive M-QAM Discrete Rate Adaptive M-QAM Non Adaptive 2-QAM (B-PSK) Fig. 9. Achievable spectral efficiency for a target BER of 10 \Gamma3 and 2. 3013579Achievable Rates in Nakagami Fading (m=4) for a Target BER Average Received Carrier-to-Noise-Ratio g- [dB] Average Spectral Efficiency Regions Regions Regions Capacity (Optimal Rate and Constant Power) Continuous Rate Adaptive M-QAM Discrete Rate Adaptive M-QAM Non Adaptive 2-QAM (B-PSK) Fig. 10. Achievable spectral efficiency for a target BER of 10 \Gamma3 and D. Average Bit Error Rate ACR M-QAM always operates at the target BER. However, since the choice of M n in ADR M-QAM is done in a conservative fashion, this discrete technique operates at an average smaller than the target BER. This BER can be computed exactly as the ratio of the average number of bits in error over the total average number March 13, 1998 of transmitted bits where Using (2) and the approximation (28) in (36) BER n can be expressed in closed-form as (b where BER n can also be computed exactly by using the exact expressions for the BER(M n ; fl) as given in [21, Chapter 5] and [10]. Figs. 11, 12, and 13 show the average BER for ADR M-QAM for a target BER of 10 \Gamma3 and for respectively. The BER calculations based on the approximation (37) are plotted in solid lines whereas the exact average BERs are plotted by the star/solid lines. The average BER of nonadaptive BPSK over Nakagami fading channel is given by s denotes the Gauss' hypergeometric function [11]. We plot (38) in Figs. 11, 12, and 13 in dash lines for comparison with (35). In these figures we observe similar trends in the average BER for various m parameters. For instance we see that the average BER of ADR M-QAM is always below the 10 \Gamma3 target BER. Recall that the approximation (28) lower bounds the exact BER for M=2 and that ADR M-QAM often uses the 2-QAM constellation (B-PSK) at low average CNRs. This explains why the average BER based on the approximation (37) lower bounds the exact average BER for Conversely because of the fact that the approximation (28) upper bounds the exact BER for M ? 2 and because ADR M-QAM often uses the high constellation sizes at high average CNRs the closed-form approximate average BER for ADR M-QAM tightly upper-bounds the exact average BER for March 13, 1998 Average BER in Rayleigh Fading (m=1) for a Target BER Average Received Carrier-to-Noise-Ratio g- [dB] Average Bit Rate Regions Regions Regions ACR M-QAM ADR M-QAM (Approx.) ADR M-QAM (Exact) Non-Adaptive 2-QAM Fig. 11. Average BER for a target BER of 10 \Gamma3 and M-QAM uses the largest available constellation often when the average CNR is large, the average BER prediction as fl increases becomes dominated by the BER performance of that constellation. V. Impact of Time Delay Recall from Section II-A that the choice of the constellation size is based on a channel estimate at time t, whereas the data are sent over the channel at time t - such that . If a delay of - fb degrades BER significantly, then this adaptive technique will not work, since - fb is an inherent and unavoidable parameter of the system. However, if a delay of - fb has a small impact on the BER then we should choose - t as large as possible so that we meet the BER requirement while minimizing the rate of system reconfiguration. In this section we analyze the impact of time delay on the performance of adaptive M-QAM over NMF channels, assuming perfect channel estimates. Average BER in Nakagami Fading (m=2) for a Target BER Average Received Carrier-to-Noise-Ratio g- [dB] Average Bit Rate Regions Regions Regions ACR M-QAM ADR M-QAM (Approx.) ADR M-QAM (Exact) Non-Adaptive 2-QAM Fig. 12. Average BER for a target BER of 10 \Gamma3 and 2. A. Fading Correlation Investigating the impact of time delay requires the second-order statistics for the channel variation, which are known for Nakagami fading. Let ff and ff - denote the channel gains at a time t and t respectively. For a slowly-varying channel we can assume that the average received power remains constant over the time delay - Under these conditions the joint these two correlated Nakagami-m distributed channel gains is given by [2, (126)] \Omega I ae)\Omega exp ae)\Omega where I m\Gamma1 (.) is the (m \Gamma 1)th-order modified Bessel function of the first kind [11], and ae is the correlation factor between ff and ff - . Since Nakagami fading assumes isotropic scattering of the multipath components, ae can be expressed in terms of the time delay - , the mobile speed, v [m/s], and the wavelength of the carrier frequency - c [m] as where J 0 (:) is the zero-order Bessel function of the first kind [11], and f the maximum Doppler frequency shift [24, p. 31]. The PDF of ff - conditioned on ff, p ff - =ff (ff - =ff), is given by Average BER in Nakagami Fading (m=4) for a Target BER Average Received Carrier-to-Noise-Ratio g- [dB] Average Bit Rate Regions Regions Regions ACR M-QAM ADR M-QAM (Approx.) ADR M-QAM (Exact) Non-Adaptive 2-QAM Fig. 13. Average BER for a target BER of 10 \Gamma3 and Inserting (1) and (39) in (40) and expressing the result in terms of the CNRs fl and fl - yields ae I exp B. Analysis B.1 Adaptive Continuous Rate M-QAM For all delays - let the communication system be configured according to fl (CNR at such that M(fl) is given by The constellation size M(fl) is based on the value fl at time t, but that constellation is transmitted over the channel at time t when fl has changed to fl - . Since M does not depend on fl - (CNR at time - delay does not affect the link spectral efficiency as calculated in Section IV-C. However, delay affects the instantaneous BER, which becomes a function of the "mismatch" between fl - and fl: Integrating (43) over the conditional PDF (41) yields the average BER conditioned on fl, BER(fl), as Inserting (41) and (43) in (44), BER(fl) can be written in a closed-form with the help of the generalized Marcum Q-function of order m, Qm (:; :) [25, p. 299, (11.63)] exp Qm Using the recurrence relation [25, p. 299, (11.64)] x I m (2 p we get that for all x, Qm (x; which can be shown to equal 1. Therefore reduces to: exp Although this formula was derived for integer m it is also valid for all non-integer values of m - 1=2. Averaging (47) over the PDF of fl (2) yields the average BER, !BER? acr , as Finally, using (47) in (48) and making the substitution yields exp where Since this analysis assumes continuous rate adaptation and since M n (fl) - M(fl) for all fl, (49) represents an upper-bound on the average BER degradation for ADR M-QAM, as will be confirmed in the following sections. B.2 Adaptive Discrete Rate M-QAM The constellation size M n is chosen based on the value of fl according to the ADR M-QAM scheme described in Section IV-A. However the constellation is transmitted over the channel when fl has changed to fl - . As in Section V-B-1, we can easily see that the link spectral efficiency of ADR M-QAM is unaffected by time delay. However, delay affects !BER? adr , which can be computed as in (35) with BER n replaced by BER 0 Using again the generalized Marcum Q-functions it can be shown that (b where Note that as ae reduces to BER n (37), as expected. C. Numerical Results Figs. 14 and 15 show adr as a function of the normalized time delay f D - for different values of the Nakagami m parameter, for a target BER of respectively. It can be seen from Figs. 14 and 15 that a normalized time delay up to about 10 \Gamma2 can be tolerated without a noticeable degradation in the average BER. For example, for a 900 MHz carrier frequency and a target BER of 10 \Gamma3 , a time delay up to 3.33 ms can be tolerated for pedestrians with a speed of 1 m/s (3.6 km/hr), and a time delay up to 0.133 ms can be tolerated for mobile vehicles with a speed of 25 m/s (90 km/hr). Comparing Figs. 14 and 15 we see that systems with the lower BER requirements of 10 \Gamma6 are more sensitive to time delay, as they will suffer a higher "rate of increase" in BER. For example, in Rayleigh fading, systems with a 10 \Gamma3 BER requirement suffer about one order of magnitude degardation for f D systems with a 10 \Gamma6 BER requirement suffer about four order of magnitude degardation for the same range of f D - . However, in both cases these systems will be able to operate satisfactorily if the normalized delay is below the critical value of 10 \Gamma2 . Average BER Degradation due to Time Delay for a Target BER 0 =10 -3 Normalized Time Delay f D t Average Bit Rate Adaptive Continuous Rate M-QAM Adaptive Discrete Rate M-QAM Fig. 14. Average BER vs. normalized time delay for a BER 0 of 10 \Gamma3 , fl=20 dB, and 5 fading regions. VI. Conclusion We have studied the capacity of NMF channels with an average power constraint for three power and rate adaptation policies. We obtain closed-form solutions for NMF channel capacity for each power and rate adaptation strategy. Our results show that optimal power and rate adaptation yields a small increase in capacity over just optimal rate adaptation with constant power, and this small increase in capacity diminishes as the average received carrier-to-noise ratio, and/or the m parameter increases. Fixed rate transmission with channel inversion suffers the largest capacity penalty. However, this penalty diminishes as the amount of fading decreases. Based on these results we conclude that rate rather than power adaptation is the key to increasing link spectral efficiency. We therefore proposed and studied the performance of constant-power variable-rate M-QAM schemes over NMF channels assuming perfect channel estimation and negligible time delay. We determined their spectral efficiency performance and compared this to the theoretical maximum. Our March 13, 1998 Average BER Degradation due to Time Delay for a Target BER 0 =10 -6 Normalized Time Delay f D t Average Bit Rate Adaptive Continuous Rate M-QAM Adaptive Discrete Rate M-QAM Fig. 15. Average BER vs. normalized time delay for a BER 0 of 10 \Gamma6 , fl=20 dB, and 5 fading regions. results show that for a target BER of 10 \Gamma3 , the spectral efficiency of adaptive continuous rate M-QAM comes within 5 dB of the Shannon capacity limit and adaptive discrete rate M-QAM comes within 6.2 dB of this limit. We also analyzed the impact of time delay on the BER of adaptive M-QAM. Results show that systems with low BER requirements will be more sensitive to time delay but will still be able to operate satisfactorily if the normalized time delay is below the critical value of 10 \Gamma2 . --R "Wireless data communications," "The m-distribution- A general formula of intensity distribution of rapid fading," "Variable-rate transmission for Rayleigh fading channels," "Symbol rate and modulation level controlled adaptive modula- tion/TDMA/TDD for personal communication systems," "Variable rate QAM for mobile radio," "Variable-rate variable-power M-QAM for fading channels," "Adaptive modulation system with variable coding rate concatenated code for high quality multi-media communication systems," "Variable-rate coded M-QAM for fading channels," "Capacity of fading channels with channel side information," "Upper bound performance of adaptive modulation in a slow Rayleigh fading channel," Table of Integrals "A statistical model for urban multipath propagation," "Indoor mobile radio channel at 946 MHz: measurements and modeling," "Capacity of Rayleigh fading channels under different adaptive transmission and diversity techniques." "Estimate of channel capacity in Rayleigh fading environment," "Comment on " "Channels with block interference," "Information theoretic considerations for cellular mobile radio," "A Gaussian channel with slow fading," "Evaluation of channel capacity in a generalized fading channel," Modern Quadrature Amplitude Modulation. New York "Efficient modulation for band-limited channels," Special Functions- An Introduction to the Classical Functions of Mathematical Physics --TR --CTR Hong-Chuan Yang , Nesrine Belhaj , Mohamed-Slim Alouini, Performance analysis of joint adaptive modulation and diversity combining over fading channels, Proceeding of the 2006 international conference on Communications and mobile computing, July 03-06, 2006, Vancouver, British Columbia, Canada Andreas Mller , Joachim Speidel, Adaptive modulation for MIMO spatial multiplexing systems with zero-forcing receivers in semi-correlated Rayleigh fading channels, Proceeding of the 2006 international conference on Communications and mobile computing, July 03-06, 2006, Vancouver, British Columbia, Canada Qingwen Liu , Shengli Zhou , Georgios B. Giannakis, Cross-layer modeling of adaptive wireless links for QoS support in heterogeneous wired-wireless networks, Wireless Networks, v.12 n.4, p.427-437, July 2006 Dalei Wu , Song Ci, Cross-layer combination of hybrid ARQ and adaptive modulation and coding for QoS provisioning in wireless data networks, Proceedings of the 3rd international conference on Quality of service in heterogeneous wired/wireless networks, August 07-09, 2006, Waterloo, Ontario, Canada Chengzhi Li , Hao Che , Sanqi Li , Dapeng Wu, A New Wireless Channel Fade Duration Model for Exploiting Multi-User Diversity Gain and Its Applications, Proceedings of the 2006 International Symposium on on World of Wireless, Mobile and Multimedia Networks, p.377-383, June 26-29, 2006 Vegard Hassel , Mohamed-Slim Alouini , Geir E. ien , David Gesbert, Rate-optimal multiuser scheduling with reduced feedback load and analysis of delay effects, EURASIP Journal on Wireless Communications and Networking, v.2006 n.2, p.53-53, April 2006 Dalei Wu , Song Ci, Cross-layer design for combining adaptive modulation and coding with hybrid ARQ, Proceeding of the 2006 international conference on Communications and mobile computing, July 03-06, 2006, Vancouver, British Columbia, Canada

and Nakagami fading;adaptive modulation techniques;link spectral efficiency

609902

PAC learning with nasty noise.

We introduce a new model for learning in the presence of noise, which we call the Nasty Noise model. This model generalizes previously considered models of learning with noise. The learning process in this model, which is a variant of the PAC model, proceeds as follows: Suppose that the learning algorithm during its execution asks for m examples. The examples that the algorithm gets are generated by a nasty adversary that works according to the following steps. First, the adversary chooses m examples (independently) according to a fixed (but unknown to the learning algorithm) distribution D as in the PAC-model. Then the powerful adversary, upon seeing the specific m examples that were chosen (and using his knowledge of the target function, the distribution D and the learning algorithm), is allowed to remove a fraction of the examples at its choice, and replace these examples by the same number of arbitrary examples of its choice; the m modified examples are then given to the learning algorithm. The only restriction on the adversary is that the number of examples that the adversary is allowed to modify should be distributed according to a binomial distribution with parameters (the noise rate) and m.On the negative side, we prove that no algorithm can achieve accuracy of > 2 in learning any non-trivial class of functions. We also give some lower bounds on the sample complexity required to achieve accuracy . On the positive side, we show that a polynomial (in the usual parameters, and in 1/(-2)) number of examples suffice for learning any class of finite VC-dimension with accuracy > 2. This algorithm may not be efficient; however, we also show that a fairly wide family of concept classes can be efficiently learned in the presence of nasty noise.

Introduction Valiant's PAC model of learning [22] is one of the most important models for learning from examples. Although being an extremely elegant model, the PAC model has some drawbacks. In particular, it assumes that the learning algorithm has access to a perfect source of random examples. Namely, upon request, the learning algorithm can ask for random examples and in return gets pairs (x; c t (x)) where all the x's are points in the input space distributed identically and independently according to some fixed probability distribution D, and c t (x) is the correct classification of x according to the target function c t that the algorithm tries to learn. Since Valiant's seminal work, there were several attempts to relax these assumptions, by introducing models of noise. The first such noise model, called the Random Classification Noise model, was introduced in [2] and was extensively studied, e.g., in [1, 6, 9, 12, 13, 16]. In this model the adversary, before providing each example (x; c t (x)) to the learning algorithm tosses a biased coin; whenever the coin shows "H", which happens with probability j, the classification of the example is flipped and so the algorithm is provided with the, wrongly classified, example (stronger) model, called the Malicious Noise model, was introduced in [23], revisited in [17], and was further studied in [8, 10, 11, 20]. In this model the adversary, whenever the j-biased coin shows "H", can replace the example (x; c t (x)) by some arbitrary pair any point in the input space and b is a boolean value. (Note that this in particular gives the adversary the power to "distort" the distribution D.) In this work, we present a new model which we call the Nasty (Sample) Noise model. In this model, the adversary gets to see the whole sample of examples requested by the learning algorithm before giving it to the algorithm and then modify E of the examples, at its choice, where E is a random variable distributed by the binomial distribution with parameters j and m, where m is the size of the sample. This distribution makes the number of examples modified be the same as if it were determined by m independent tosses of an j-biased coin. However, we allow the adversary's choice to be dependent on the sample drawn. The modification applied by the adversary can be arbitrary (as in the Malicious Noise model). 1 Intuitively speaking, the new adversary is more powerful than the previous ones - it can examine the whole sample and then remove from it the most "informative" examples and replace them by less useful and even misleading examples (whereas in the Malicious Noise Model for instance, the adversary also may insert to the sample misleading examples but does not have the freedom to choose which examples to remove). The relationships between the various models are shown in Table 1. Random Noise-Location Adversarial Noise-Location Label Noise Only Random Classification Noise Nasty Classification Noise Point and Label Noise Malicious Noise Nasty Sample Noise Table 1: Summary of models for PAC-learning from noisy data We argue that the newly introduced model, not only generalizes the previous noise models, including variants such as Decatur's CAM model [11] and CPCN model [12], but also, that in many real-world situations, the assumptions previous models made about the noise seem unjustified. For example, when training data is the result of some physical experiment, noise may tend to be stronger in boundary areas rather than being uniformly distributed over all inputs. While special models were We also consider a weaker variant of this model, called the Nasty Classification Noise model, where the adversary may modify only the classification of the chosen points (as in the Random Classification Noise model). devised to describe this situation in the exact-learning setting (for example, the incomplete boundary query model of Blum et al., [5]), it may be regarded as a special case of Nasty Noise, where the adversary chooses to provide unreliable answers on sample points that are near the boundary of the target concept (or to remove such points from the sample). Another situation to which our model is related is the setting of Agnostic Learning. In this model, a concept class is not given. Instead, the learning algorithm needs to minimize the empirical error while using a hypothesis from a predefined hypotheses class (see, for example, [18] for a definition of the model). Assuming the best hypothesis classifies the input up to an j fraction, we may alternatively see the problem as that of learning the hypotheses class under nasty noise of rate j. However, we note that the success criterion in the agnostic learning literature is different from the one used in our PAC-based setting. We show two types of results. Sections 3 and 4 show information theoretic results, and Sect. 5 shows algorithmic results. The first result, presented in Section 3, is a lower bound on the quality of learning possible with a nasty adversary. This result shows that any learning algorithm cannot learn any non-trivial concept class with accuracy better than 2j when the sample contains nasty noise of rate j. We further show that learning a concept class of VC dimension d with accuracy requires examples. It is complemented by a matching positive result in Section 4 that shows that any class of finite VC-dimension can be learned by using a sample of polynomial size, with any accuracy ffl ? 2j. The size of the sample required is polynomial in the usual PAC parameters and in 1=\Delta where is the margin between the requested accuracy ffl and the above mentioned lower bound. The main, quite surprising, result (presented in Section 5) is another positive result showing that efficient learning algorithms are still possible in spite of the powerful adversary. More specifically, we present a composition theorem (analogous to [3, 8] but for the nasty-noise learning model) that shows that any concept class that is constructed by composing concept classes that are PAC-learnable from a hypothesis class of fixed VC-dimension, is efficiently learnable when using a sample subject to nasty noise. This includes, for instance, the class of all concepts formed by any boolean combination of half-spaces in a constant dimension Euclidean space. The complexity here is, again, polynomial in the usual parameters and in 1=\Delta. The algorithm used in the proof of this result is an adaptation to our model of the PAC algorithm presented in [8]. Our results may be compared to similar results available for the Malicious Noise model. For this model, Cesa-Bianchi et al. [10] show that the accuracy of learning with malicious noise is lower bounded by matching algorithm for learning classes similar to those presented here with malicious noise is presented in [8]. As for the Random Classification Noise model, learning with arbitrary small accuracy, even when the noise rate is close to a half, is possible. Again, the techniques presented in [8] may be used to learn the same type of classes we examine in this work with Random Classification Noise. Preliminaries In this section we provide basic definitions related to learning in the PAC model, with and without noise. A learning task is specified using a concept class, denoted C, of boolean concepts defined over an instance space, denoted X . A boolean concept c is a function c : X 7! f0; 1g: The concept class C is a set of boolean concepts: C ' f0; 1g X . Throughout this paper we sometimes treat a concept as a set of points instead of as a boolean function. The set that corresponds to a concept c is simply 1g. We use c to denote both the function and the corresponding set interchangeably. Specifically, when a probability distribution D is defined over X , we use the notation D(c) to refer to the probability that a point x drawn from X according to D will have 2.1 The Classical PAC Model The Probably Approximately Correct (PAC) model was originally presented by Valiant [22]. In this model, the learning algorithm has access to an oracle PAC that returns on each call a labeled example according to a fixed distribution D over X , unknown to the learning algorithm, and c t 2 C is the target function the learning algorithm should "learn". Definition 1: A class C of boolean functions is PAC-learnable using hypothesis class H in polynomial time if there exists an algorithm that, for any c t 2 C, any input parameters and any distribution D on X , when given access to the PAC oracle, runs in time polynomial in log jX j, 1=ffi, 1=ffl and with probability at least outputs a function h 2 H for Pr 2.2 Models for Learning in the Presence of Noise Next, we define the model of PAC-learning in the presence of Nasty Sample Noise (NSN for short). In this model, a learning algorithm for the concept class C is given access to an (adversarial) oracle NSN C;j (m). The learning algorithm is allowed to call this oracle once during a single run. The learning algorithm passes a single natural number m to the oracle, specifying the size of the sample it needs, and gets in return a labeled sample S 2 (X \Theta f0; 1g) m . (It is assumed, for simplicity, that the algorithm knows in advance the number of examples it needs; the extension of the model for scenarios where such a bound is not available in advance is given in Section 6.) The sample required by the learning algorithm is constructed as follows: As in the PAC model, a distribution D over the instance space X is defined, and a target concept c t 2 C is chosen. The adversary then draws a sample S g of m points from X according to the distribution D. Having full knowledge of the learning algorithm, the target function c t , the distribution D, and the sample drawn, the adversary chooses points from the sample, where E(S g ) is a random variable. The E points chosen by the adversary are removed from the sample and replaced by any other point-and-label pairs by the adversary. The not chosen by the adversary remain unchanged and are labeled by their correct labels according to c t . The modified sample of m points, denoted S, is then given to the learning algorithm. The only limitation that the adversary has on the number of examples that it may modify is that it should be distributed according to the binomial distribution with parameters m and j, namely: where the probability is taken by first choosing S g 2 D m and then choosing E according to the corresponding random variable E(S g ). Definition 2: An algorithm A is said to learn a class C with nasty sample noise of rate j - 0 with accuracy parameter ffl ? 0 and confidence parameter access to any oracle NSN C;j (m), for any distribution D and any target c t 2 C it outputs a hypothesis h : X 7! f0; 1g such that, with probability at least Pr We are also interested in a restriction of this model, which we call the Nasty Classification Noise learning model (NCN for short). The only difference between the NCN and NSN models is that the NCN adversary is only allowed to modify the labels of the E chosen sample-points, but it cannot modify the E points themselves. Previous models of learning in the presence of noise can also be readily shown to be restrictions of the Nasty Sample Noise model: The Malicious Noise model corresponds to the Nasty Noise model with the adversary restricted to introducing noise into points that are chosen uniformly at random, with probability j, from the original sample. The Random Classification Noise model corresponds to the Nasty Classification Noise model with the adversary restricted so that noise is introduced into points chosen uniformly at random, with probability j, from the original sample, and each point that is chosen gets its label flipped. 2.3 VC Theory Basics The VC-dimension [24], is widely used in learning theory to measure the complexity of concept classes. The VC-dimension of a class C, denoted VCdim(C), is the maximal integer d such that there exists a subset Y ' X of size d for which all 2 d possible behaviors are present in the class C, and if such a subset exists for any natural d. It is well known (e.g., [4]) that, for any two classes C and H (over X ), the class of negations fcjX n c 2 Cg has the same VC-dimension as the class C, and the class of unions fc [ hjc 2 C; h 2 Hg has VC-dimension at most VCdim(C)+VCdim(H)+1. Following [3] we define the dual of a concept class: Definition 3: The dual H ? ' f0; 1g H of a class H ' f0; 1g X is defined to be the set defined by x ? If we view a concept class H as a boolean matrix where each row represents a concept and each column a point from the instance space, X , then the matrix corresponding to H ? is the transpose of the matrix corresponding to H. The following claim, from [3], gives a tight bound on the VC dimension of the dual class: 1: For every class H, log In the following discussion we limit ourselves to instance spaces X of finite cardinality. The main use we make of the VC-dimension is in constructing ff-nets. The following definition and theorem are from [7]: Definition 4: A set of points Y ' X is an ff-net for the concept class H ' f0; 1g X under the distribution D over X , if for every h 2 H such that D(h) - ff, Y " h 6= ;. Theorem 1: For any class H ' f0; 1g X of VC-dimension d, any distribution D over X , and any ae 4 ff log 2 ff log 13 ff oe examples are drawn i.i.d. from X according to the distribution D, they constitute an ff-net for H with probability at least 1 \Gamma ffi. In [21], Talagrand proved a similar result: Definition 5: A set of points Y ' X is an ff-sample for the concept class H ' f0; 1g X under the distribution D over X , if it holds that for every h 2 H: Theorem 2: There is a constant c 1 , such that for any class H ' f0; 1g X of VC-dimension d, and distribution D over X , and any ff ? 0, examples are drawn i.i.d. from X according to the distribution D, they constitute an ff-sample for H with probability at least 2.4 Consistency Algorithms Let P and N be subsets of points from X . We say that a function h : X 7! f0; 1g is consistent on "positive point" x 2 P and "negative point" x 2 N . A consistency algorithm (see [8]) for a pair of classes (C; H) (both over the same instance space X , with C ' H), receives as input two subsets of the instance space, runs in time t(jP [ N j), and satisfies the following. If there is a function in C that is consistent with (P; N ), the algorithm outputs "YES" and some h 2 H that is consistent with (P; N ); the algorithm outputs "NO" if no consistent exist (there is no restriction on the output in the case that there is a consistent function in H but not in C). Given a subset of points of the instance space Q ' X , we will be interested in the set of all possible partitions of Q into positive and negative examples, such that there is a function h 2 H and a function c 2 C that are both consistent with this partition. This may be formulated as: CON is a consistency algorithm for (C; H). The following is based on Sauer's Lemma [19]: Lemma 1: For any set of points Q, Furthermore, an efficient algorithm for generating this set of partitions (along with the corresponding functions h presented, assuming that C is PAC-learnable from H of constant VC dimension. The algorithm's output is denoted h is consistent with Information Theoretic Lower Bound In this section we show that no learning algorithm (not even inefficient ones) can learn a "non- trivial" concept class with accuracy ffl better than 2j under the NSN model; in fact, we prove that this impossibility result holds even for the NCN model. We also give some results on the size of samples required to learn in the NSN model with accuracy ffl ? 2j. Definition class C over an instance space X is called non-trivial if there exist two points Theorem 3: Let C be a non-trivial concept class, j be a noise rate and ffl ! 2j be an accuracy parameter. Then, there is no algorithm that learns the concept class C with accuracy ffl under the NCN model (with rate j). Proof: We base our proof on the method of induced distributions introduced in [17, Theorem 1]. We show that there are two concepts distribution D such that and an adversary can force the labeled examples shown to the learning algorithm to be distributed identically both when c 1 is the target and when c 2 is the target. Let c 1 and c 2 be the two concepts whose existence is guaranteed by the fact that C is a non-trivial class, and let x be the two points that satisfy c 1 define the probability distribution D to be D(x 1 g. Clearly, we indeed have PrD [c 1 Now, we define the nasty adversary strategy (with respect to the above probability distribution D). Let m be the size of the sample asked by the learning algorithm. The adversary starts by drawing a sample S g of m points according to the above distribution. Then, for each occurrence of x 1 in the sample, the adversary labels it correctly according to c t , while for each occurrence of x 2 the adversary tosses a coin and with probability 1=2 it labels the point correctly (i.e., c t it flips the label (to )). The resulted sample of m examples is given by the adversary to the learning algorithm. First, we argue that the number of examples modified by the adversary is indeed distributed according to the binomial distribution with parameters j and m. For this, we view the above adversary as if it picks independently m points and for each of them decides (as above) whether to flip its label. Hence, it suffices to show that each example is labeled incorrectly with probability j independently of other examples. Indeed, for each example independently, its probability of being labeled incorrectly equals the probability of choosing x 2 according to D times the probability that the adversary chooses to flip the label on an x 2 example; i.e. 2j \Delta as needed. (We emphasize that the binomial distribution is obtained because D is known to the adversary.) Next observe that, no matter whether the target is c 1 or c 2 , the examples given to the learning algorithm (after being modified by the above nasty adversary) are distributed according to the following probability distribution: Therefore, according to the sample that the learning algorithm sees, it is impossible to differentiate between the case where the target function is c 1 and the case where the target function is c 2 . Note that in the above proof we indeed take advantage of the "nastiness" of the adversary. Unlike the malicious adversary, our adversary can focus all its "power" on just the point x 2 , causing it to suffer a relatively high error rate, while examples in which the point is x 1 do not suffer any noise. We also took advantage of the fact that E (the number of modified examples) is allowed to depend on the sample (in our case, it depends on the number of times x 2 appears in the original sample). This allows the adversary to further focus its destructive power on samples which were otherwise "good" for the learning algorithm. Finally, since any NCN adversary is also a NSN adversary, Theorem 3 implies the following: Corollary 4: Let C be a non-trivial concept class, j ? 0 be the noise rate, and ffl ! 2j be an accuracy parameter. There is no algorithm that learns the concept class C with accuracy ffl under the NSN model, with noise rate j. Once we have settled 2j as the lower bound on the accuracy possible with a nasty adversary with error rate j, we turn to the question of the number of examples that are necessary to learn a concept class with some accuracy Again, in this section we are only considering information-theoretic issues. These results are similar to those presented by Cesa-Bianchi et al. [10] for the Malicious Noise model. Note, however, that the definition of the margin \Delta used here is relative to a lower bound different than the one used in [10]. In the proofs of these results, we use the following claim (see [10]) that provides a lower bound on the probability that a random variable of binomial distribution deviates from its expectation by more than the standard deviation: 2: [10, Fact 3.2] Let SN;p be a random variable distributed by the binomial distribution with parameters N and p, and let p. For all N ? 37=(pq): Pr ki 19 (1) Pr ki 19 (2) Theorem 5: For any nontrivial concept class C, any noise rate j ? 0, confidence parameter the sample size needed for PAC learning C with accuracy and confidence tolerating nasty classification noise of rate j is at least =\Omega Proof: Let c 1 and c 2 be the two concepts whose existence is guaranteed by the fact that C is a non-trivial class, and let x be the two points that satisfy c 1 Let us define a distribution D that gives weight ffl to the point x 2 and weight 1 \Gamma ffl to x 1 , making by f the target function (either c 1 or c 2 ). The Nasty Classification Adversary will use the following strategy: for each pair of the form in the sample, with probability j=ffl reverse the label (i.e., present to the learning algorithm the pair instead). The rest of the sample (all the pairs of the form unmodified. Note that for each of the m examples the probability that its classification is changed is therefore exactly ffl \Delta j so the number of points that suffer noise is indeed distributed according to the binomial distribution with parameters j and m. The induced probability distribution on the sample that the learning algorithm sees is: For contradiction, let A be a (possibly randomized) algorithm that learns C with accuracy ffl using a sample generated by the above oracle and whose size is m by p A (m) the error of the hypothesis h that A outputs when using m examples. Let B be the Bayes strategy of outputting c 1 if the majority instances of x 2 are labeled c 1 Clearly, this strategy minimizes the probability of choosing the wrong hypothesis. This implies Define the following two events over runs of B on samples of size m: Let N denote the number of examples in m showing the point x 2 . BAD 1 is the event that at least dN=2e are corrupted, and BAD 2 is the event that N - 36j(j answer incorrectly, as there will be more examples showing x 2 with the wrong label than there will be examples showing x 2 with the correct label. Examine now the probability that BAD 2 will occur. Note that N is a random variable distributed by the binomial distribution with parameters m and ffl (and recall that \Delta). We are interested in: Since the probability that N is large is higher when m is larger, and m is upper bounded by (17j(1 \Gamma But this may be bounded, using Hoeffding's inequality to be at least We therefore have: On the other hand, if we assume that BAD 2 holds, namely that N - 36j(j+ additionally assume that N - 37(2j+ \Delta) 2 =(j(j+ \Delta)) then, by Claim 2 (with and using the following inequality $s it follows that Pr[BAD 1 To see that the inequality of Equation (3) indeed holds when BAD 2 holds, note that (3) is implied by: s which is, in turn, implied by the two conditions:2 s s It can be verified that these two conditions hold if we take N to be in the range we assume: 2 is an optimal strategy, and hence no worse than a strategy that ignores some of the sample points, its error can only decrease if more points are shown for x 2 . Therefore, the same results will hold if we remove the lower bound on N . We thus have that Pr[BAD 1 A second type of a lower bound on the number of required examples is based on the VC dimension of the class to be learned, and is similar to the results (and the proof techniques) of [7] for the standard PAC model: 2 By our conditions on \Delta there must be at least one integer in the range we assume for N . Theorem 6: For any concept class C with VC-dimension d - 3, and for any 0 ! ffl - 1=8, 1=12, the sample size required to learn C with accuracy ffl and confidence ffi when using samples generated by a nasty classification adversary with error rate \Delta) is greater than =\Omega be a set of d points shattered by C. Define a probability distribution D as follows: Assume for contradiction that at most (d \Gamma 2)=32\Delta examples are used by the learning algorithm. We let the nasty adversary behave as follows: it reverses the label on each example x d\Gamma1 with probability 1=2 (independent of any other sample points), making the labels of x d\Gamma1 appear as if they are just random noise (also note that the probability of each example to be corrupted by the adversary is exactly 2j \Delta j). Thus, with probability 1=2 the point x d\Gamma1 is misclassified by the learner's hypothesis. The rest of the sample is left unmodified. Denote by BAD 1 the event that at least half of the points x are not seen by the learning algorithm. Given BAD 1 , we denote UP the set of (d \Gamma 2)=2 unseen points with lowest indices, and define BAD 2 as the event that the algorithm's hypothesis misclassifies at least (d \Gamma 2)=8 points from UP. Finally, let BAD 3 denote the event that x d\Gamma1 is misclassified. It is easy to see that BAD 1 - BAD 2 - BAD 3 imply that the hypothesis has error of at least ffl, as it implies that the hypothesis errs on (d \Gamma 2)=8 points where each of these points has weight 8\Delta=(d \Gamma 2) and on the point x whose weight is 2j, making the total error at least \Delta Therefore, if an algorithm A can learn the class with confidence ffi , it must hold that Pr[BAD 1 - BAD 2 - BAD 3 As noted before, x d\Gamma1 appears to be labeled by random noise, and hence Pr[BAD 3 is independent of BAD 1 and BAD 2 , thus . As for the other events, since at most (d \Gamma 2)=32\Delta examples are seen, the expected number of points from x that the learning algorithm sees is at most (d \Gamma 2)=4. From the Markov inequality it follows that, with probability at least 1 2 , no more than (d \Gamma 2)=2 points are seen. Hence, 2 . Every unseen point will be misclassified by the learning algorithm with probability at least half (since for each such point, the adversary may set the target to label the point with the label that has lower probability to be given by the algorithm). Thus Pr[BAD 2 jBAD 1 ] is the probability that a fair coin flipped (d \Gamma 2)=2 times shows heads at least (d \Gamma 2)=8 times. Using [10, Fact 3.3] this probability can be shown to be at least 1=3. We thus have: This completes the proof. Since learning with Nasty Sample Noise is not easier than learning with Nasty Classification Noise, the results of Theorems 5 and 6 also hold for learning from a Nasty Sample Noise oracle. Information Theoretic Upper Bound In this section we provide a positive result that complements the negative result of Section 3. This result shows that, given a sufficiently large sample, any hypothesis that performs sufficiently well on the sample (even when this sample is subject to nasty noise) satisfies the PAC learning condition. Formally, we analyze the following generic algorithm for learning any class C of VC-dimension d, whose inputs are a certainty parameter ffi ? 0, the nasty error rate parameter 2 and the required Algorithm NastyConsistent: 1. Request a sample 2. Output any h 2 C such that (if no such h exists, choose any h 2 C arbitrarily). Theorem 7: Let C be any class of VC-dimension d. Then, (for some constant c) algorithm Nasty- Consistent is a PAC learning algorithm under nasty sample noise of rate j. In our proof of this theorem (as well as in the analysis of the algorithm in the next section), we use, for convenience, a slightly weaker definition of PAC learnability than the one used in Definition 1. We require the algorithm to output, with probability at least Pr (rather than a strict inequality). However, if we use the same algorithm but give it a slightly smaller accuracy parameter (e.g., ffl ffl), we will get an algorithm that learns using the original criterion of Definition 1. Proof: First, we argue that with "high probability" the number of sample points that are modified by the adversary is at most m(j \Delta=4). As the random variable E is distributed according to the binomial distribution with expectation jm, we may use Hoeffding's inequality [14] to get: Pr (by the choice of c), this event happens with probability of at most ffi=2. Now, we note that the target function c t , errs on at most E points of the sample shown to the learning algorithm (as it is completely accurate on the non-modified sample S g ). Thus, with probability at least NastyConsistent will be able to choose a function h 2 C that errs on no more that (j + \Delta=4)m points of the sample shown to it. However, in the worst case, these errors of the function h occur in points that were not modified by the adversary. In addition, h may be erroneous for all the points that the adversary did modify. Therefore, all we are guaranteed in this case, is that the hypothesis h errs on no more that 2E points of the original sample S g . By Theorem 2, there exists a constant c such that, with probability by taking S g to be of size at least c the resulting sample S g is a \Delta -sample for the class of symmetric differences between functions from C. By the union bound we therefore have that, with probability at least \Delta=4)m, meaning that jS \Delta=2)m, and that S g is a \Delta=2-sample for the class of symmetric differences, and so: Pr as required. 5 Composition Theorem for Learning with Nasty Noise Following [3] and [8], we define the notion of "composition class": Let C be a class of boolean functions . Define the class C ? to be the set of all boolean functions F (x) that can be represented as f(g 1 boolean function, and g i 2 C for We define the size of f(g to be k. Given a vector of hypotheses following [8], the set W(h to be the set of sub-domains W a ag for all possible vectors a 2 f0; 1g t . We now show a variation of the algorithm presented in [8] that can learn the class C ? with a nasty sample adversary, assuming that the class C is PAC-learnable from a class H of constant VC dimension d. Our algorithm builds on the fact that a consistency algorithm CON for (C; H) can be constructed, given an algorithm that PAC learns C from H [8]. This algorithm can learn the concept class C ? with any confidence parameter ffi and with accuracy ffl that is arbitrarily close to the lower bound of 2j, proved in the previous section. Its sample complexity and computational complexity are both polynomial in k, 1=ffi and 1=\Delta, where Our algorithm is based on the following idea: Request a large sample from the oracle. Randomly pick a smaller sub-sample from the sample retrieved. By randomly picking this sub-sample, the algorithm neutralize some of the power the adversary has, since the adversary cannot know which examples are the ones that will be most "informative" for us. Then use the consistency algorithm for (C; H) to find one representative from H for any possible behavior on the smaller sub-sample. These hypotheses from H now define a division of the instance space into "cells", where each cell is characterized by a specific behavior of all the hypotheses picked. The final hypotheses is simply based on taking a majority vote among the complete sample inside each such cell. To demonstrate the algorithm, let us consider (informally) the specific, relatively simple, case where the class to be learned is the class of k intervals on the straight line (see Figure 1). The algorithm, given a sample as input, proceeds as follows: 1. The algorithm uses a relatively small, random sub-sample to divide the line into sub-intervals. Each two adjacent points in the sub-sample define such a sub-interval. 2. For each such sub-interval the algorithm calculates a majority vote on the complete sample. The result is our hypothesis. The number of points (which in this specific case is the number of sub-intervals) that the algorithm chooses in the first step depends on k. Intuitively, we want the total weight of the sub-intervals containing the target's end-points to be relatively small (this is what is called the ``bad part'' in the formal analysis that follows). Naturally, there will be 2k such "bad" sub-intervals, so the larger k Target Concept: Sub-sample and intervals: "Bad" "Bad" "Bad" "Bad" Algorithm's hypothesis: Figure 1: Example of NastyLearn for intervals. is, the larger the sub-sample needed. Except for these "bad" sub-intervals, all other subintervals on which the algorithm errs have to have at least half of their points modified by the adversary. Thus the total error will be roughly 2j, plus the weight of the "bad" sub-intervals. Now, we proceed to a formal description of the learning algorithm. Given the constant d, the size k of the target function, the bound on the error rate j, the parameters ffi and \Delta, and two additional parameters M;N (to be specified below), the algorithm proceeds as follows: Algorithm NastyLearn: 1. Request a sample S of size N . 2. Choose uniformly at random a sub-sample R ' S of size M . 3. Use the consistency algorithm for (C; H) to compute 4. Output the hypotheses H(h computed as follows: For any W a 2 W(h is not empty, set H to be the majority of labels in S " W a . If W a is empty, set H to be 0 on any x 2 W a . Theorem 8: Let log 8 log 78k are constants. Then, Algorithm NastyLearn learns the class C ? with accuracy confidence ffi in time polynomial in k, 1 As in Theorem 7, this Theorem refers to the modified PAC criterion that require the algorithm to output, with probability at least 1 \Gamma ffi, a function h for Pr The same technique we mentioned for Algorithm NastyConsistent may be used to modify this algorithm to be a PAC learning algorithm in the sense of Definition 1. Before commencing with the actual proof, we present a technical lemma: Lemma 2: Assuming N is set as in the statement of Theorem 8, with probability at least 4 the number of points in which errors are introduced, E, is at most (j + \Delta=12)N . Proof of Lemma 2: Note that, by the definition of the model, E is distributed according to a binomial distribution with parameters j and N . Thus, E behaves as the number of successes in independent Bernoulli experiments, and the Hoeffding inequality [14] may be used to bound its value: Pr Therefore, if we take N ? 72 ln(4=ffi) we have, with probability at least 4 , that E is at most (j \Delta=12)N . Note that the value we have chosen for N in the statement of Theorem 8 is clearly large enough. We are now ready to present the proof of Theorem 8: Proof: To analyze the error made by the hypothesis that the algorithm generates, let us denote the adversary's strategy as follows: 1. Generate a sample of the requested size N according to the distribution D, and label it by the target concept F . Denote this sample by S g . 2. Choose a subset S out ' S g of size is a random variable (as defined in Section 2.2). 3. Choose (maliciously) some other set of points S in ' X \Theta f0; 1g of size E. 4. Hand to the learning algorithm the sample in . Assume the target function F is of the form the hypothesis that the algorithm have chosen in step 3 that exhibits the same behavior g i has over the points of R (from the definition of - SCON we are guaranteed that such a hypothesis exists). By definition, there are no points from R in h j i As the VC-dimension of both the class C of all g i 's and the class H of all h i 's is d, the class of all their possible symmetric differences also has VC-dimension O(d) (see Section 2.3). By applying Theorem 1, when viewing R as a sample taken from S according to the uniform distribution, and by choosing M to be as in the statement of the theorem, R will be an ff-net (with respect to the uniform distribution over S) for the class of symmetric differences, with at least 1 \Gamma ffi=4. Note that there may still be points in S which are in h j i 4g i . Hence, we let using (4) we get: with probability at least 1 \Gamma ffi=4, simultaneously for all i. For every sub-domain B 2 W(h N in B= jS in " Bj In words, NB and N in simply stand for the size of the restriction of the original (noise-free) sample S g and the noisy examples S in introduced by the adversary to the sub-domain B. As for the rest of the definitions, they are based on the distinction between the "good" part of B, where the g i s and the h j i s behave the same, and the "bad" part, which is present due to the fact that the g i s and the h j i s exhibit the same behavior only on the smaller sub-sample R, rather than on the complete sample S. We use N ff B to denote the number of sample points in the bad part of B, and N out,g B to denote the number of sample points that were removed by the adversary from the good and bad parts of B, respectively. Since our learning algorithm decides on the classification in each sub-domain by a majority vote, the hypothesis will err on the domain of B) if the number of examples left untouched in B is less than the number of examples in B that were modified by the adversary, plus those that were misclassified by the h j i s (with respect to the g i s). This may be formulated as the following condition: N in Therefore, the total error the algorithm may experience is at most: B: NB-N in We now calculate a bound for each of the two terms above separately. To bound the second term, note that by Theorem 2 our choice of N guarantees S g to be a \Delta -sample for our domain with probability at least 1 \Gamma ffi=4. Note that from the definition of W(h and from the Sauer Lemma [19] we have that jW(h Our choice of N indeed guarantees, with probability at least B: NB-N in B: NB-N in N in ?From the above choice of N , it follows that S g is also a \Delta -sample for the class of symmetric differences of the form h j i 4g i . Thus, with probability at least 1 \Gamma ffi=4, we have: The total error made by the hypothesis (assuming that none of the four bad events happen) is therefore bounded by: Pr N in as required. This bound holds with certainty at least 1 \Gamma ffi. 6 Conclusion We have presented the model of PAC learning with nasty noise, generalizing on previous models. We have proved a negative information-theoretic result, showing that there is no learning algorithm that can learn any non-trivial class with accuracy better than 2j, paired with a positive result showing this bound to be tight. We complemented these results with lower bounds on the sample size required for learning with accuracy 2j \Delta. We have also shown that for a wide variety of "interesting" concept classes, an efficient learning algorithm in this model exists. Our negative result can be generalized for the case where the learning algorithm uses randomized hypotheses, or coin rules (as defined in [10]); in such a case we get an information theoretic lower bound of j for the achievable accuracy, compared to a lower bound of j=2(1 proved in [10] for learning with Malicious Noise of rate j. While the partition into two separate variants: the NSN and the NCN models seem intuitive and well-motivated, it remains an open problem to come up with any results that actually separate the two models. Both the negative and positive results we presented in this work apply equally to both the NSN and the NCN models. Finally, note that the definition of the nasty noise model requires the learning algorithm to know in advance the sample size m (or an upper bound on it). The model however can be extended so as to deal with scenarios where no such bound is known to the learning algorithm. There are several scenarios of this kind. For example, the sample complexity may depend on certain parameters (such as the "size" of the target function) which are not known to the algorithm 3 . The adversary, who knows the learning algorithm and knows the target function (and in general can know all the parameters hidden from the learning algorithm) can thus "plan ahead" and draw in advance a sample S g of size which is sufficiently large to satisfy, with high probability, all the requests the learning algorithm will make. It then modifies S g as defined above and reorders the resulted sample S randomly 4 . Now, the learning algorithm simply asks for one example at a time (as in the PAC model) and the adversary supplies the next example in its (randomly-ordered) set S. If the sample is exhausted (which may happen in those cases where we have only an expected sample-complexity guarantee), we say that the learning algorithm has failed; however, when using a large enough sample (with respect to 1=ffi), this will happen with sufficiently small probability. --R "General Bounds on Statistical Query Learning and PAC Learning with Noise via Hypothesis Boosting" "Learning from Noisy Examples" "A Composition Theorem for Learning Algorithms with Applications to Geometric Concept Classes" "Combinatorial Variability of Vapnik-Chervonenkis Classes with Applications to Sample Compression Schemes" "Learning with Unreliable Boundary Queries" "Weakly Learning DNF and Characterizing Statistical Query Learning Using Fourier Analysis" "Learnability and the Vapnik-Chervonenkis dimension" "A New Composition Theorem for Learning Algorithms" "Noise-Tolerant Distribution-Free Learning of General Geometric Concepts" "Sample-efficient strategies for learning in the presence of noise" "Learning in Hybrid Noise Environments Using Statistical Queries" "PAC Learning with Constant-Partition Classification Noise and Applications to Decision Tree Induction" "On Learning from Noisy and Incomplete Examples" "Probability Inequalities for Sums of Bounded Random Variables" "Efficient Noise-Tolerant Learning from Statistical Queries" "Learning in the Presence of Malicious "Toward Efficient Agnostic Learning" "On the Density of Families of sets" "The Design and Analysis of Efficient Learning Algorithms" "Sharper Bounds for Gaussian and Empirical Processes" "A Theory of the Learnable" "Learning Disjunctions of Conjunctions" "On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities" --TR A theory of the learnable Learnability and the Vapnik-Chervonenkis dimension The design and analysis of efficient learning algorithms Learning in the presence of malicious errors Efficient noise-tolerant learning from statistical queries Weakly learning DNF and characterizing statistical query learning using Fourier analysis Toward Efficient Agnostic Learning Learning with unreliable boundary queries On learning from noisy and incomplete examples Noise-tolerant distribution-free learning of general geometric concepts A composition theorem for learning algorithms with applications to geometric concept classes A new composition theorem for learning algorithms Combinatorial variability of Vapnik-Chervonenkis classes with applications to sample compression schemes Sample-efficient strategies for learning in the presence of noise Learning From Noisy Examples --CTR Marco Barreno , Blaine Nelson , Russell Sears , Anthony D. Joseph , J. D. Tygar, Can machine learning be secure?, Proceedings of the 2006 ACM Symposium on Information, computer and communications security, March 21-24, 2006, Taipei, Taiwan

nasty noise;PAC learning;learning with noise

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Foundation of a computable solid modelling.

Solid modelling and computational geometry are based on classical topology and geometry in which the basic predicates and operations, such as membership, subset inclusion, union and intersection, are not continuous and therefore not computable. But a sound computational framework for solids and geometry can only be built in a framework with computable predicates and operations. In practice, correctness of algorithms in computational geometry is usually proved using the unrealistic Real RAM machine model of computation, which allows comparison of real numbers, with the undesirable result that correct algorithms, when implemented, turn into unreliable programs. Here, we use a domain-theoretic approach to recursive analysis to develop the basis of an effective and realistic framework for solid modelling. This framework is equipped with a well defined and realistic notion of computability which reflects the observable properties of real solids. The basic predicates and operations on solids are computable in this model which admits regular and non-regular sets and supports a design methodology for actual robust algorithms. Moreover, the model is able to capture the uncertainties of input data in actual CAD situations.

Introduction Correctness of algorithms in computational geometry are usually proved using the Real RAM machine [22] model of computation. Since this model is not realistic, correct algorithms, when implemented, turn into unreliable programs. In CAGD modeling operators, the effect of rounding errors on consistency and robustness of actual implementations is an open question, which is handled in industrial software by various unreliable and expensive "up to epsilon" heuristics that remain very unsatisfactory. The authors claim that a robust algorithm is one whose correctness is proved with the assumption of a realistic machine model [17, 18]. A branch of computer science, called recursive analysis, defines precisely what it means, in the context of the realistic Turing machine model of computation, to compute objects belonging to non-countable sets such as the real numbers. In this paper, we use a domain-theoretic approach to recursive analysis to develop the basis of an effective framework for solid modeling. The set-theoretic aspects of solid modeling is revis- ited, leading to a theoretically motivated model that shows some interesting similarities with the Requicha Solid Model [23, 24]. Within this model, some unavoidable limitations of solid modeling computations are proved and a sound framework to design specifications for feasible modeling operators is provided. Some consequences in computation with the boundary representation paradigm are sketched that can incorporate existing methods [13, 28, 16, 14, 15] into a general, mathematically well-founded theory. Moreover, the model is able to capture the uncertainties of input data [8, 19] in actual CAD situations. We need the following requirements for the mathematical model: (1) the notion of computability of solids has to be well defined, (2) the model has to reflect the observable properties of real solids, (3) the model has to be closed under the Boolean operations, (4) non-regular sets 1 have to be captured by the model as well, (5) it has to support a design methodology for actual robust algorithms. In Section 2, we outline some elements of recursive analysis and domain theory used in subsequent sections. Section 3 presents the solid domain, a mathematical model for computable rigid solids. In Section 4, we give an illustration, on a simple case, how one can design a robust algorithm in the light of our domain-theoretic approach. Recursive analysis and domain theory In this section, we briefly outline some elements of recursive analysis [26, 7, 5, 21] and domain theory [1, 29, 2] that we need in this paper. We first deal with N, the set of all non-negative integers. A function f : N ! N, is recursive if it is computable by a general purpose computer (e.g. a Turing machine or a C++ program); this means that there is a finite program written in some general language such that its output is f(n) whenever its input is any n 2 N. A recursively enumerable subset of N, or an r.e. set, is the image of a recursive function. A recursive set is an r.e. set whose complement is also r.e. There are r.e. sets which are not recursive, but their construction is non-trivial. Next, we consider the set Q of rational numbers. Since Q is countable, it is in one-to-one correspondence with N and we can write only if Therefore, computability over Q reduces to computability over N. The theory of computability over the set of real numbers R, which is uncountable, is more involved. Since the set of (finite) programs written in a general purpose computer is countable, it follows that the set of computable real numbers, i.e. those which are the output of a finite program, is also countable. These can be characterized in terms of recursive functions. A real number r is computable if there exists recursive functions f and g such that This means that r is the effective limit of a computable sequence of rational numbers. We say that a real number is lower (respectively, upper) semi- computable if it is the limit of an increasing (respectively, decreasing) computable sequence of rational numbers. It then follows that a real number is computable if and only if it is lower and upper semi-computable. Similarly, a function f : [a; b] ! R is computable if it is the effective limit (in the sup norm) of a computable sequence of rational polynomials. Intuitively, in a suitable representation such as the sign binary system, a real number is given by an infinite sequence of digits and a computable function is one which can compute any finite part of the output sequence by reading only a finite part of the input sequence. From this, it follows that a computable function is always continuous with respect to the Euclidean topology of R. Domain theory was originally introduced independently by Scott [27] as a mathematical theory of semantics of programming languages and by Ershov [12] for studying partial computable functionals of finite type. A domain is a structure for modeling a computational process or a data type with incomplete or uncertain specified information. It is a partially ordered set where the partial order corresponds to some notion of information. A simple example is the domain ftt; ff; ?g of the Boolean values tt and ff together with a least element ? below both. One thinks of ? here as the 1 A set is regular if it the closure of its interior. undefined Boolean value. A domain is also equipped with a notion of completion (as in Cauchy completeness for metric spaces) and a notion of approximation. There is a so-called Scott topology on a domain which is T 0 and is such that every open set is upward closed, i.e. whenever a Scott open set contains an element, it also contains any element above that element. See the appendix for precise definitions. A class of so-called !-continuous domains has in recent years been successfully used in modeling computation in a number of areas of analysis [9]. An !-continuous domain has a countable subset of basis elements such that every element of the domain can be completely specified by the set of basis elements which approximate it. One can use this countable basis to provide an effective structure for the domain and obtain the notions of a computable element of an effectively given domain and of a computable function between two effectively given !-continuous domains. We give two examples of useful continuous domains in this section which will motivate the idea of the solid domain introduced in the next section. The interval domain I[0; 1] n of the unit box [0; 1] n ae R n is the set of all non-empty n-dimensional sub-rectangles in [0; 1] n ordered by reverse inclusion. A basic Scott open set is given, for every open subset O of R n , by the collection of all rectangles contained in O. The map x is an embedding onto the set of maximal elements of I[0; 1] n . Every maximal element fxg can be obtained as the least upper bound (lub) of an increasing chain of elements, i.e. a shrinking, nested sequence of sub-rectangles, each containing fxg in its interior and thereby giving an approximation to fxg or equivalently to x. The set of sub-rectangles with rational coordinates provides a countable basis. One can similarly define, for example, the interval domain IR n of R n . For the interval domains I[\Gamma1; 1] and IR , where R is the one point compactification of R, have been used to develop a feasible framework for exact real arithmetic using linear fractional transformations [10, 20]. An important feature of domains, in the context of this paper, is that they can be used to obtain computable approximations to operations which are classically non-computable. For example, comparison of a real number with 0 is not computable. However, the function neg : I[\Gamma1; neg([a; b]) =! is the best computable approximation to this predicate. The upper space UX of a compact metric space X is the set of all non-empty compact subsets of X ordered by reverse inclusion. In fact, UX is a generalization of the interval domain and has similar properties; for example a basic Scott open set is given, for every open subset O ae X , by the collection of all non-empty compact subsets contained in O. As with the interval domain, the map x is an embedding onto the set of maximal elements of UX . The upper space gives rise to a computational model for fractals and for measure and integration theory [9]. The idea of the solid domain in the next section is closely linked with the upper space of [0; 1] n . 3 A domain-theoretic model In this section, we introduce the solid domain, a mathematical model for representing rigid solids. We focus here on the set-theoretic aspects of solid modeling as Requicha did in introducing the r-sets model [23]. Our model is motivated by requirements (1) to (5) given in the introduction. For any subset A of a topological space, A, A ffi , @A and A c denote respectively the closure, the interior, the boundary and the complement of A. The regularization of a subset A is defined, by Requicha [23, 24], as the subset A ffi . We say that a set is regular if it is equal to its regularization. 3.1 The solid domain The solid domain S[0; 1] n of the unit cube [0; 1] n ae R n is the set of ordered pairs (A; B) of compact subsets of [0; endowed with the information order: . The elements of S[0; 1] n are called partial solids. Proposition 3.1 (S[0; is a continuous domain and Proposition 3.2 For any (A; B) 2 S[0; 1] n , there exists a subset Y of [0; 1] n such One can take for example: We say that represents the subset Y . It follows that the partial order S[0; 1] n is isomorphic with the quotient of the power set of [0; 1] n under the equivalence relation with the ordering Given any subset X of [0; 1] n , the classical membership predicate 2: [0; continuous except on @X . It follows that the best continuous approximation of this predicate is where the value ? is taken on @X (recall that any open set containing ? contains the whole set ftt; ff; ?g). Then, two subsets are equivalent if and only if they have the same best continuous approximation of the membership predicate. By analogy with general set theory for which a set is completely defined by its membership predicate, the solid domain can be seen as the collection of subsets that can be distinguished by their continuous membership predicates. The definition of the solid domain is then consistent with requirement (1) since a computable membership predicate has to be continuous. Our definition is also consistent with requirement (2) in a closely related way. We consider the idealization of a machine used to measure mechanical parts. Two parts corresponding to equivalent subsets cannot be discriminated by such a machine. Moreover, partial solids, and, more generally, domain-theoretically defined data types (cf. Section allow us to capture partial, or uncertain input data [8, 19] encountered in realistic CAD situations. Starting with the continuous membership predicate, the natural definition for the complement would be to swap the values tt and ff. This means that the complement of (A; B) is (B; A), cf. requirement (3). As for requirement (4), the figure below represents a subset X of [0; 1] 2 that is not regular. Its regularization removes both the external and internal "dangling edge". This set can be captured in our framework but not in the Requicha model. Here and in subsequent figures, the two components A and B of the partial solid are depicted separately below each picture for clarity. Proposition 3.3 The maximal elements of S[0; 1] n are precisely those that represent regular sets. In other words, maximal elements are of the form (A; B) such that A and B are regular with Next we consider the Boolean operators. We first note that the regularized union [23, 24] of two adjacent three dimensional boxes (i.e. product of intervals) is not computable, since, to decide whether the adjacent faces are in contact or not, one would have to decide the equality of two real numbers which is not computable [21]. Requirements (1) and (3) entail the existence of Boolean operators which are computable with respect to a realistic machine model (e.g. the Turing machine). A In order to define Boolean operators on the solid domain, we obtain the truth table of logical Boolean operators on ftt; ff; ?g. Consider the logical Boolean operator "or", which, applied to the continuous membership predicates of two partial solids, would define their union. This is indeed the truth table for parallel or in domain theory; see [2, page 133]. One can likewise build the truth table for "and". Note the similarities with the (In,On,Out) points classifications used in some boundary representation based algorithms [25, 3]. From these truth tables follow the definition of Boolean operators on partial solids: Beware that, given two partial solids representing adjacent boxes, their union would not represent the set-theoretic union of the boxes, as illustrated in the figure below. A 1o/oo A 2 We have defined the continuous membership predicate for points of [0; 1] n . In order to be able to compute this predicate, we extend it to the interval domain I[0; 1] n by defining 2: I[0; A ff A A A A Proposition 3.4 The following maps are continuous: ffl 2: I[0; Similarly, one can define the continuous predicate ae: S[0; ?g. 3.2 Computability on the solid domain In order to endow S[0; 1] n with a computability structure, we introduce two different countable bases that lead to the same notion of computability, but correspond to different types of algorithms in use. A rational hyperplane is a subset of R n of the form: f(x i (0 are rational numbers such that at least one of them is non-zero. A rational polyhedron is a regular subset of [0; 1] n whose boundary is included in a finite union of rational hyper-planes. Notice that a rational polyhedron may not be connected and may also be the empty set. A dyadic number is a rational number whose denominator is a power of 2. A dyadic voxel set is a finite union of cubes, each the product of n intervals whose endpoints are dyadic numbers. Notice that every voxel set is a rational polyhedra. A partial rational polyhedron (PRP) is an element (A; B) 2 S[0; 1] n such that A and B are rational polyhedra. In the following, PRP stands for the set of PRP's. A partial dyadic voxel set (PDVS) is an element (A; B) 2 S[0; 1] n such that A and B are dyadic voxel sets. PDVS stands for the set of PVDS's. The set PRP is effectively enumerable, that is, each PRP can be represented by a finite set of integers (i.e. the rational coordinates of the vertices and the incidence graph) and there exists a program to compute a one to one correspondence between N and PRP so that we can write \Deltag. Proposition 3.5 PRP forms a countable basis for the solid domain S[0; 1] n . Moreover, the solid domain is effectively given with respect to the enumeration fR 0 \Deltag of this basis. Therefore: (i) every element of S[0; 1] n is the least upper bound of a sequence of PRP's approximating it, and (ii) the predicate R k - R j is r.e. in k; j. In fact, this predicate is recursive, that is, there exists a program able to decide, for any pair of integers k and j, whether or not R k - R j . From a more practical point of view, this implies that the Boolean operators on rational polyhedra are computable (see [6] for an efficient implementation), and that a subset is compact if and only if it is the intersection of a countable set of rational polyhedra. By the general notion of computability in domains (see the appendix), an element (A; B) 2 computable if the set fkjR k - (A; B)g is r.e. We obtain the same class of computable partial solids if we replace the PRP basis with the PDVS basis. Our notion of computability is somewhat weaker that one could expect. Consider a computable partial solid (A; B) and a computable point x 2 [0; 1] n n A. There exists a program to compute an increasing sequence converging to (A; B) and a program to compute an increasing sequence I k of rational intervals of I[0; 1] n converging to x. From these two programs, one can obtain a program to compute the increasing sequence of rational numbers representing the square of the minimum distance between A k and I k . It follows that the minimum distance between A and x is a lower semi-computable real number. However, this distance may not be computable. We introduce a stronger notion of computability, that will make the above distance computable. An element (A; B) 2 S[0; 1] n is recursive if the set fkjR k - (A; B)g is recursive. It can be shown [4] that (A; B) is recursive if and only if there exists a program to compute two nested sequences of rational polyhedra such that A and B are the effective limits of the sequences with respect to the Hausdorff metric. In [4], several related notions of computability for compact sets are given. In this setting, our notion of computable partial solid (A; B) means that A and B are co.r.e. and our notion of recursive partial solid means that A and B are recursive. We have now a positive and a negative result. Proposition 3.6 The Boolean operators over S[0; 1] n are computable. However, the intersection of two recursive partial solids may not be recursive as illustrated in the figure below. The two initial recursive partial solids represent regular sets. The details of construction of will be presented in the full version of the paper. The crucial property is that the left endpoint of the lower horizontal line segment is the limit of a computable, increasing, bounded sequence of rational numbers which is lower semi-computable but not computable. The intersection of A 1 and A 2 is therefore a horizontal segment whose left-end point is not computable. Therefore, requirement(3) prevents us to choose recursive partial solids for our model. A 1-A 2 However, we can choose the following notion which is stronger than computability but is neither weaker nor stronger than recursiveness. We say an element (A; B) 2 S[0; 1] n is Lebesgue computable if it is computable and if the Lebesgue measures of A and B are computable. Note that is the Lebesgue measure of C ae R n . Therefore, (A; B) is Lebesgue computable if and only if there exists a program to compute an increasing sequence and Proposition 3.7 The Boolean operators over S[0; 1] n are Lebesgue computable. In other words, there exists a program that, given two increasing sequences of PRP's defining two partial solids such that their Lebesgue measures are effectively converging, computes an increasing sequence of PRP's defining their intersection such that their Lebesgue measures is effectively converging. A Lebesgue computable partial solid (A; B), with can be manufactured with an error that can be made as small as we want in volume, assuming an idealized manufacturing device. The table below compares in general the three notions for computable solids. Partial solid Distance to a point Boolean operators Lebesgues measure computable semi-computable computable non-computable recursive computable non-computable non-computable Lebesgues computable semi-computable computable computable At this stage of our work, our model of choice would be the Lebesgue computable partial solids, since they are stable under Boolean operators. 4 Robustness Issues We illustrate, on a very rudimentary class of boundary represented solids, how our domain-theoretic approach matches requirement (5). Usually, robustness issues show up in two (related) way: (i) A numerical computation is not well-specified in case of discontinuities, as for example in the intersection of tangential, partially overlapping surfaces. (ii) The values of the logical predicates evaluated from numerical computations are inconsistent, resulting in an invalid output or the catastrophic failure of the algorithm. 4.1 The disk domain We consider d, the set of disks in the Euclidean plane. Each disk a of d is represented by the three real numbers giving the coordinates of the center and the radius: (x a ; y a ; R a ), with R a - 0. By an abuse of notation, such an element a denotes both the real triple defining it and the corresponding disk in the plane; the context always makes it clear which meaning we have in mind. We now define the domain D of interval disks. It is the set of interval triples with K K and add the bottom element ? to D and partially order it with reverse inclusion: K v L 3). The domain D is isomorphic with its maximal elements can be identified with the elements of d. An element is said to be rational if x \Gamma K and R K are rational numbers. ?From the general theory of computability in domains (see the appendix), K is computable if it is the least upper bound of an increasing computable sequence of rational interval disks, that is if there exists a program to compute such an increasing sequence. This definition is consistent with the solid domain introduced in Section 3 as we have the where the image f(K) of an interval disk K 2 D is the partial solid a It can be easily shown that f is monotonic, continuous and in fact computable with respect to the natural effective structure on D induced from I(R 2 \Theta R + ). When restricted to interval disks contained in [0; 1] 2 , f is in fact an embedding. 4.2 The domain of the relative position of disks We consider here the combinatorial part of the computation of Boolean operators over disks. For this purpose we consider the following map from d \Theta d to R 3 and the predicates 3 from d \Theta d to the domain f\Gamma; +; ?g defined, for 0: The domain topology on f\Gamma; +; ?g ensures that these predicates are continuous. Because of the inequalities, the range of made of 11 values, defining the relative position of the two disks. We denote this set of 11 values by F which is a subset of the domain f\Gamma; +; ?g 3 , whose order relation, induced by the order relation on f\Gamma; +; ?g, is represented in the figure below. 4.3 Extension to D and actual computation We define the Where inf denotes infimum or the greatest lower bound, which exists for every subset since F is a bounded complete domain (see the appendix). P is the best continuous extension of p. It is possible to compute the image P (K; L) of any pair (K; L) of rational interval disks, as this reduces to the evaluation of the sign of a few polynomials over Q (see [6]). Then, from two increasing sequences of rational interval disks (increasing with respect to v) defining a pair of interval disks, one can compute an increasing sequence in F defining their relative position. The actual image is computed after a finite time. However, when this image is not a maximal element of F, one never knows if the output will be refined by using a more accurate input (i.e. more terms from the two rational interval disk sequences). This behaviour is consistent with requirement (2): in the physical world, the statement "two disks are tangent", for example, means that there are tangent up to a relevant accuracy and a more accurate measuring may reveal that they actually intersect or are in fact disjoint. a b a b a b a b a b a b a b a A# A# A# A#A#A A#A #A# A#A A #A A a b A# A#A #A A# A#A# A# a a 5 Conclusion The solid domain described here satisfies the requirements of computability, having observable properties, closure under Boolean operations, admission of non-regular sets and the design of robust algorithms as stated in the introduction. The classical analysis framework, allowing discontinuous behaviour and exact real number comparisons, is neither realistic as a model of our interaction with the physical world (measuring, manufacturing), nor realistic as a basis for the design of algorithms implemented on realistic machines, which are only able to deal with finite data. The authors believe that the domain-theoretic approach, even at this initial stage of application to solid modeling and computational geometry, is a powerful mathematical framework both to model partial or uncertain data and to guide the design of robust software. The solid model can be defined for a more general class of topological spaces, in particular for locally compact Hausdorff spaces such as R n . It can also be represented equivalently in terms of pairs of open sets or equivalently in terms of continuous functions from the space to the Boolean domain ftt; ff; ?g. We will deal with these issues in a future paper. In our future work, we will use the domain-theoretic framework to capture more information on solids and geometric objects. In particular, we will deal more generally with the boundary representation and the differential properties of curves and surfaces (that is the C k or G k properties). We will also focus on actual computations, applying the methodology illustrated in Section 4 to more complex situations. Appendix In this section, we give the formal definitions of a number of notions in domain theory used in the paper. We think of a partially ordered set (poset) (P; v) as the set of output of some computation such that the partial order is an order of information: in other words, a v b indicates that a has less information than b. For example, the set f0; 1g 1 of all finite and infinite sequences of bits 0 and 1 with a v b if the sequence a is an initial segment of the sequence b is a poset and a v b simply means that b has more bits of information than a. A non-empty subset A ' P is directed if for any pair of elements there exists c 2 A such that a v c and b v c. A directed set is therefore a consistent set of output elements of a computation: for every pair of output a and b, there is some output c with more information than a and b. A directed complete partial order (dcpo) or a domain is a partial order in which every directed subset D ' P has a least upper bound (lub) denoted F A. It is easily seen that f0; 1g 1 is a dcpo. We say that a dcpo is pointed if it has a least element which is usually denoted by ? and is called bottom. For two elements a and b of a dcpo we say a is way-below or approximates b, denoted by a - b, if for every directed subset A with b v F A there exists c 2 A with a v c. The idea is that a is a finitary approximation to b: whenever the lub of a consistent set of output elements has more information than b, then already one of the input elements in the consistent set has more information than a. In f0; 1g 1 , we have a - b iff a v b and a is a finite sequence. The closed subsets of the Scott topology of a domain are those subsets C which are downward closed (i.e. closed under taking lub's of directed subsets (i.e. for every directed subset A ' C we have F A 2 C). A basis of a domain D is a subset B ' D such that for every element x 2 D of the domain the set B fy 2 Bjy - xg of elements in the basis way-below x is directed with F An (!)-continuous domain is a dcpo with a (countable) basis. In other words, every element of a continuous domain can be expressed as the lub of the directed set of basis elements which approximate it. A domain is bounded complete if every bounded subset has a lub; in such a domain every subset has an infimum or greatest lower bound. One can easily check that f0; 1g 1 is an !-continuous dcpo for which the set of finite sequences form a countable basis. It can be shown that a function f dcpo's is continuous with respect to the Scott topology if and only if it is monotone (i.e. a v b ) f(a) v f(b)) and preserves lub's of directed sets i.e. for any directed A ' D, we have f( F F a2A f(a). An !-continuous domain D with a least element ? is effectively given wrt an enumeration of a countable base \Deltag with b if the set f! m;n is r.e., where is the standard pairing function i.e. the isomorphism (x; y) 7! (x+y)(x+y+1) This means that there is a master program which generates all pairs of basis elements (b We say x 2 D is computable if the set fnjb n - xg is r.e. This is equivalent to say that there is a recursive function g such that (b g(n) ) n-0 is an increasing chain in D with F We say that a continuous effectively given !-continuous domains D (with basis computable if the set f! m;n ? jb m - f(an )g is r.e. This is equivalent to say that f maps computable elements to computable elements in an effective way. Every computable function can be shown to be a continuous function [30, Theorem 3.6.16]. It can be shown [11] that these notions of computability for the domain IR of intervals of R induce the same class of computable real numbers and computable real functions as in the classical theory [21] described in Section 2. Acknowledgements The first author has been supported by EPSRC and would like to thank the hospitality of the Institute for Studies in Theoretical Physics and Mathematics in Tehran where part of this work was done. --R Domain theory. Domains and Lambda-Calculi Toward a topology for computational geometry. Computability on subsets of Euclidean space I: Closed and compact subsets. Computing exact geometric predicates using modular arithmetic with single precision. An Introduction to Recursive Function Theory. Robustness of numerical methods in geometric computation when problem data is uncertain. Domains for computation in mathematics A new representation for exact real numbers. A domain theoretic approach to computability on the real line. Computable functionals of finite types. Epsilon Geometry Towards Robust Interval Solid Modeling of Curved Objects. Robust interval algorithm for curve intersections. Boundary Representation Modelling with local Tolerances. Repr'esentation b. Toward a data type for Solid Modeling based on Domain Theory. Algorithmic tolerances and semantics in data exchange. Efficient on-line computation of real functions using exact floating point Computability in Analysis and Physics. Computational Geometry: an introduction. Mathematical Foundations of Constructive Solid Geometry Representation for Rigid Solids Boolean Operations in Solid Modeling: Boundary Evaluation and Merging Algorithms. Outline of a mathematical theory of computation. Using tolerances to guarantee valid polyhedral modeling results. Mathematical Theory of Domains --TR Computational geometry: an introduction Computability Epsilon geometry: building robust algorithms from imprecise computations Using tolerances to guarantee valid polyhedral modeling results Dynamical systems, measures, and fractals via domain theory Boundary representation modelling with local tolerances Effective algebras Domain theory Towards robust interval solid modeling of curved objects Algorithmic tolerances and semantics in data exchange A domain-theoretic approach to computability on the real line Foundation of a computable solid modeling Domains and lambda-calculi Computability on subsets of Euclidean space I Computable banach spaces via domain theory Representations for Rigid Solids: Theory, Methods, and Systems Type Theory via Exact Categories On The Measure Of Two-Dimensional Regions With Polynomial-Time Computable Boundaries --CTR Martin Ziegler, Effectively open real functions, Journal of Complexity, v.22 n.6, p.827-849, December, 2006 Abbas Edalat , Andr Lieutier, Domain theory and differential calculus (functions of one variable), Mathematical Structures in Computer Science, v.14 n.6, p.771-802, December 2004

solid modelling;robustness;domain theory;turing computability;model of computation

610819

Real number computation through gray code embedding.

We propose an embedding G of the unit open interval to the set {0, 1},1 of infinite sequences of {0, 1} with at most one undefined element. This embedding is based on Gray code and it is a topological embedding with a natural topology on {0, 1},1. We also define a machine called an indeterministic multihead Type 2 machine which input/output sequences in {0, 1},1, and show that the computability notion induced on real functions through the embedding G is equivalent to the one induced by the signed digit representation and Type 2 machines. We also show that basic algorithms can be expressed naturally with respect to this embedding.

Introduction One of the ways of dening computability of a real function is by representing a real number x as an innite sequence called a name of x, and dening the computability of a function by the existence of a machine, called a Type-2 machine, which inputs and outputs the names one-way from left to right. This notion of computability dates back to Turing[Tur36], and is the basis of eective analysis [Wei85,Wei00]. This notion of computability depends on the choice of representation we use, and signed digit representation and equivalent ones such as the Cauchy representation and the shrinking interval representation are most commonly used; they have the property that every arbitrarily small rational interval including x can be obtained from a nite prex of a name of x, and therefore induces computability notion that a function f is computable if there is a machine which can output arbitrary good approximation information of f(x) as a rational interval when arbitrary good approximation information of x as a rational interval is given. The naturality of this computability notion is also justied by the fact that it coincides with those de- Preprint submitted to Elsevier Science 15 June 2000 ned through many other approaches such as Grzegorczyk's ([Grz57]), Pour-El and Richards ([PER89]), and domain theoretic approaches([ES98], [Gia99]). One of the properties of these representations is that they are not injective [Wei00]. More precisely, uncountably many real numbers have innitely many names with respect to representations equivalent to the signed digit representation [BH00]. This kind of redundancy is considered essential in many approaches to exact real arithmetic [BCRO86,EP97,Gia96,Gia97,Vui90]. Thus, computability of a real function is dened in two steps: rst the computability of functions over innite sequences is dened using Type-2 machines, and then it is connected with the computability of real functions by representations. The redundancy of representations means that we cannot dene the computability of a real function more directly by considering an embedding of real numbers into the set of innite sequences on which a Type-2 machine operates. In this paper, we consider such a direct denition by extending the notion of innite sequences and modifying the notion of computation on innite sequences. Our embedding, called the Gray code embedding, is based on the Gray code expan- sion, which is another binary expansion of real numbers. The target of this embedding is the set f0; 1g ! ?;1 of innite sequences of f0; 1g in which at most one ?, which means undenedness, is allowed. We dene the embbeding G of the unit open interval I, and then explain how it can be extended to the whole real line in the nal section. ?;1 has a natural topological structure as a subspace of f0; . We show that G is a topological embedding from I to the space f0; 1g ! ?;1 . Because of the existence of ?, a machine cannot have sequential access to inputs and outputs. However, because ? appears only at most once, we can deal with it by putting two heads on a tape and by allowing indeterministic behavior to a machine. We call such a machine Indeterministic Multihead Type 2 machine (IM2-machine for short). Here, indeterministic computation means that there are many computational paths which will produce valid results [She75,Bra98]. Thus, we dene computation over using IM2-machines, and consider the induced computational notion on I through the embedding G. We show that this computational notion is equivalent to the one induced by the signed digit representation and Type-2 machines. We also show how basic algorithms like addition can be expressed with this represen- tation. One remarkable thing about this representation is that it has three recursive structures though it is characterized by two recursive equations. This fact is used in composing basic recursive algorithms. We introduce Gray code embedding in Section 2 and an IM2 machine in Section 3. Then, we dene the Gray code computability of real functions in Section 4, and show that it is equivalent to the computability induced by the signed digit representation and Type 2 machines in Section 5. In Section 6, we study topological structure. number Binary code Gray code 9 1001 1101 Fig. 1. Binary code and Gray code of integers In Section 7 and 8, we consider basic algorithms with respect to this embedding. We will discuss how this embedding can be extended to R, give some experimental implementations, and give conclusion in Section 9. Notation: Let be an alphabet which does not include ?. We write for the set of nite sequences of , ! for the set of innite sequences of , and ! for the set of innite sequences of in which at most n instances of the undenedness character ? are allowed to exist. We from X to Y , and multi-valued function from X to Y , that is, F is a subset of X Y considered as a partial function from X to the power set of Y . We call a number of the form m 2 n for integers m and n a dyadic number. Gray Code Embedding Gray code is another binary encoding of natural numbers. Figure 1 shows the usual binary code and the Gray code of integers from 0 to 15. In this way, n-bit Gray code is composed by putting the n-th bit on and reversing the order of the coding up to (n- 1)-bits, instead of repeating the coding up to (n-1)-bits as we do in the usual binary code. The importance of this code lies in the fact that only one bit diers between the encoding of a number and that of its successor. This code is used in many areas of computer science such as image compression [ASD90] and nding minimal digital circuits [Dew93]. The conversion between these two encodings is easy. Gray code is obtained from the usual binary code by taking the bitwise xor of the sequence and its one-bit shift. Therefore, the function to convert from binary code to Gray code is written using the notation of a functional language Haskell [HJ92] as follows: This conv function has type [Int] -> [Int], where [Int] is the Haskell type of (possibly innite) list of integers. a:b means the list composed of a as the head and b as the tail, xor is the \exclusive or" dened as xor (0, xor (0, xor (1, xor (1, and zip is a function taking two lists (of length l and m) and returning a list of pairs (of length min(l; m)). This conversion is injective and the inverse is written as with [] the empty list. We will extend this coding to real numbers. Since the function conv is applicable to innite lists, we can obtain the Gray code expansion of a real number x by applying conv to the binary expansion of x. The Gray code expansion of real numbers in the unit interval I = (0; 1) is visualized in Figure 2. Here, a horizontal line means that the corresponding bit has value 1 on the line and value 0 otherwise. This gure has a ne fractal structure and shows symmetricity of bits greater than n at every dyadic number m 2 n . bit3 bit6Fig. 2. Gray code of real numbers In the usual binary expansion, we have two expansions for dyadic numbers. For example, can be expressed as 0:110000::: and also as 0:101111:::. This is also the case for the Gray code expansion. For example, by applying conv to these sequences, we have the two sequences 0:101000::: and 0:111000::: of However, one can nd that the two sequences dier only at one bit (in this case, the 2nd). This means that the information that this number is only by the remaining bits and the 2nd bit does not contribute to this fact. Therefore, it would be natural to introduce the character ? denoting undenedness and consider the sequence 0:1?1000::: as the unique representation of Note that the sequence after the bit where they dier is always 1000:::. Thus, we dene Gray code embedding of I as a modication of the Gray code expansion in that a dyadic number is represented as s?1000::: with Denition 1 The Gray code embedding of the unit open interval I is an injective function G from I to ! ?;1 which maps x to an innite sequence a 0 a as follows: a for an odd number m, a if the same holds for an even number m, and a for some integer m. We call G(x) the modied Gray code expansion of x, or simply Gray code of x. When or 0, according as x is bigger than, equal to, or less . The tail function which maps x to G 1 (a 1 a denotes the so-called tent It is in contrast to the binary expansion in that the tail function of the binary expansion denotes the function Note that Gray code expansion coincides with the itinerary by the tent map which is essential for symbolic dynamical systems [HY84]. 3 Indeterministic Multihead Type 2 Machine Consider calculating a real number x (0 < x < 1) as the limit of approximations and output the result as the modied Gray code expansion. More precisely, we consider a calculation which produces shrinking intervals (r successively so that lim n!1 s When we know that x < n), we can write 0 as the rst digit. And when we know that n), we can write 1. However, when neither will happen and we cannot ever write the rst digit. Even so, when we know that we can skip the rst and write 1 as the second digit, and when we know that as the third digit. Thus, when , we can continue producing the digits skipping the rst one and we can write the sequence from the second digit. In order to produce the Gray code of x as the result, we need to ll the rst cell with ?, which is impossible because we cannot obtain the information in a nite time. To solve this, we dene ? as the \blank character" of the output tape and consider that the output tape is lled with ? at the beginning. Thus, when a cell is skipped and is not lled eternally, it is left as ?. Suppose that we know we have written the second digit as 1 skipping the rst one. As the next output, we have two possibilities: to write the third digit as 0 because we know that or to write the rst digit because we obtain the information x < Therefore, when we consider a machine with Gray code output, the output tape is not written one-way from left to right. To present this behavior in a simple way, we consider two one-way heads H 1 (O) and H 2 (O) on an output tape O which move automatically after an output. At the beginning, H 1 (O) and H 2 (O) are located above the rst and the second cell, respectively. After an output from H 2 (O), H 2 (O) is moved to the next cell, and after an output from H 1 (O), H 1 (O) is moved to the position of H 2 (O) and H 2 (O) is moved to the next cell. Thus, in order to ll the output tape as Here, H(j) (H is H 1 (O) or H 2 (O) and means to output j from H . With this head movement rule, each cell is lled at most once and a cell is not lled eternally only when H 1 (O) is located on that cell and output is made solely from H 2 (O). are on the s-th and t-th cell of an output tape, the i-th cells (i < s; s < i < t) are already output and no longer accessible. Therefore, H 1 (O) and are always located at the rst and the second unlled cells and the machine treats the tape as if it were [O[s]; Next, we consider how to input a modied Gray code expansion of a real number. We dene our input mechanism so that nite input contains only approximation information. Therefore, our machine should not recognize that the cell under the head is ?, because the character ? with its preceding prex species the number exactly. This requirement is also supported by the way an input tape is lled when it is produced as an output of another machine; the character ? may be overwritten by 0 or 1 in the future and it is impossible to recognize that a particular cell is left eternally as ?. Therefore, our machine needs to have something other than the usual sequential access. To solve this, we consider multiple heads and consider that the machine waits for multiple cells to be lled. Since at most one cell is left unlled, two heads are su-cient for our purpose. Therefore, we consider two on an input tape I, which move in the same way as output heads when they input characters. Note that the character ? cannot be recognized by our machine, unlike the blank character used by a Turing machine. Thus, we dene a machine which has two heads on each input/output tape. Though we have explaind this idea based on the modied Gray code expansion, this machine can input/output sequences in ! ?;1 generally. In order to give the same computational power as a Turing machine, we consider a state machine controlled by a set of computational rules, which has some ordinary work tapes in addition to the input/output tapes. In order that the machine can continue working even when the cell under H 1 (I) or for an input tape I is ?, we need, at each time, a rule applicable only reading from H 1 (I) or H 2 (I). Therefore, the condition part of each rule should not include input from both H 1 (I) and H 2 (I). This also means that, if both head positions of an input tape are lled, we may have more than one applicable rules. Since a machine may execute both rules, both computational paths should produce valid results. To summarize, we have the following denition. Denition 2 Let be the input/output alphabet. Let be the work-tape alphabet which includes a blank character B. An indeterministic multihead Type 2 machine (IM2-machine in short) with k inputs is composed of the following: tapes named I 1 ; I one output tape named O. Each tape T has two heads (ii) several work tapes with one head, (iii) a nite set Q of states with one initial state q 0 2 Q, (iv) computational rules of the following form: Here, q and q 0 are states in Q, i j are heads of dierent input tapes, o is a head of the output tape, w , and w 00 are heads of work tapes, c j (j c are characters from , d j and d 0 are characters from , and M j (j are '+' or ' '. Each part of the rule is optional; there may be a rule without o(c), for example. The meaning of this rule is that if the state is q and the characters under the heads i are c j and d e , respectively, then change the state to q 0 , write the characters c and d 0 the heads , respectively, move the heads w 00 or backward depending on whether M move the heads of input/output tapes as follows. For each when it is a head moved to the position of H 2 (T ) and H 2 (T ) is moved to the next cell, and when it is H 2 (T ), the position of H 1 (T ) is left unchanged moved to the next cell. The machine starts with the output tape lled with ?, work tapes lled with B, the state set to q 0 , the heads of work tapes located on the rst cell, and the heads H 1 (T ) of an input/output tape T are located above the rst and the second cell, respectively. At each step, the machine chooses one applicable rule and applies it. When more than one rules is applicable, only one is selected in a nondeterministic way. Note 1. We can dene an indeterministic multihead Type 2 machine more generally in that each input/output tape may have input/output sequences in ! . We dene the head movements after an input/output operation as follows. If input/output is made from H l (T ) (l n) are moved to the position of H j+1 (T ) and H n+1 (T ) is moved to the next cell. If input/output is made from H n+1 (T ) then H n+1 (T ) is moved to the next cell. Note that when ?;0 is nothing but ! and a tape has only one head which moves to the next cell after an input/output. Note 2. Here, we acted as if the full contents of the input tapes were given at the beginning. However, an input is usually generated as an output of another machine, and given incrementally. In this case, the machine behaves like this: it repeats executing an applicable rule until no rule is applicable, and waits for input tapes to be lled so that one of the rules become applicable, and repeats this process indenitely. Note 3. A machine can have dierent input/output types on the tapes. The in- put/output types we consider are ?;n (n 0) and , where we may write ! for ?;0 . We extend an IM2-machine with a sequence (Y indicating that it has k input tapes with type Y one output tape of type Y 0 . When Y i is ! ?;n , the corresponding tape has the properties written in Note 1. When Y i is , the corresponding tape has the alphabet [ fBg and it has one head which moves to the next cell when it reads/writes a character. In this case, the blank cells are initialized with B. In addition, when Y 0 is , we consider that the machine has a halting state at which the machine stops execution. Gray Code Computability of Real Functions As we have seen, an IM2-machine has a nondeterministic behavior and thus it has many possible outputs to the same input. Therefore, we consider that an IM2-machine computes a multi-valued function. Note that multi-valued functions appear naturally when we consider computation over real numbers [Bra98]. Denition 3 An IM2-machine M with k inputs realizes a multi-valued function ?;1 if all the computational paths M have with the input tapes lled with (p outputs, and the set of outputs forms a subset of F (p We say that F is IM2-computable when it is realized by some IM2-machine. This denition can be generalized to a multi-valued function for the case Y i is or ! Note that our nondeterministic computation is dierent from nondeterminism used, for example, in a non-deterministic Turing machine; a non-deterministic Turing machine accepts a word when one of the computational paths accepts the word, whereas all the computational paths should produce valid results in our machine. To distin- guish, we use the word indeterminism instead of nondeterminism following [She75] and [Bra98]. Denition 4 A multi-valued function F : I I is realized by M if G(F ) is realized by M . We say that F is Gray-code-computable if G - F - G 1 is IM2- computable. Denition 5 A partial function f : I k ! I is Gray-code-computable if it is computable as a multi-valued function. 5 Equivalence to the Computability induced by the Signed Digit Repre- sentation Now, we prove that Gray code computability is equivalent to the computability induced by a Type-2 machine and the (restricted) signed digit representation. Denition 6 A Type-2 machine is an IM2-machine whose type includes only ! and , and whose computational rule is deterministic. This denition is equivalent to the one in [Wei00]. Proposition 1 Let Y i be ! or There is an IM2-machine which there is a deterministic IM2-machine which computes F . Proof: The if part is immediate. For the only if part, we need to construct a deterministic machine from an indeterministic machine for the case that the input/output tapes have only one head. Suppose that M is an IM2-machine which realizes F . Since the set of rules of M is nite, we give a numbering to them. We can determine whether or not each rule is applicable because the input tapes do not have the character ?. Therefore, we can modify M to construct a deterministic machine M 0 which chooses the rst applicable rule with respect to the numbering. The result of M 0 to uniquely determined and is in F (x). Denition 7 A representation of a set X is a surjective partial function from ! to X. If is a representation of X and a -name of x. Denition 8 I and - I be representations. We say that - is reducible to - 0 (- 0 ) when there is a computable function f that dom(-). We say that - and - 0 are equivalent ( - 0 ) when - 0 and - 0 -. Denition 9 1) The signed digit representation sd of I uses the alphabet denoting 1, and it is a partial function sd I dened on such that a j 6= 1 and a l 6= 1g and returns 1 a i 2 i to a 1 a 2) The restricted signed digit representation sdr of I is a restriction of sd to a smaller domain such that a j 6= 1 and a l 6= 1g without the rst character a 1 (= 1). By sd , 3=8 has innitely many names :. The domain of sdr means that we do not use a name which lasts as therefore 3=8 has only two sdr -names Proposition 2 sdr sd . Proof: It is an easy exercise to give an algorithm that converts a sd -name to a sdr -name. Denition I be a representation of I. A multi-valued function I is (-computable if there is a Type-2 machine M of type that if -(p) 2 dom(F ), then M with input p produces an innite sequence q such that A partial function is (-computable if it is computable as a multi-valued function. This denition can easily be extended to a function with several arguments. Equivalent representations induce the same computability notion on I. As we explained in the introduction, the equivalence class to which signed digit representation belongs induces a suitable notion of computability on real numbers. Proposition 3 Let M be an IM2-machine which realizes a multi-valued function be IM2-machines which realize multi-valued functions Suppose that Im(hG Then, there is an IM2-machine M -hN which realizes the multi-valued function Here, the composition of multi-valued functions F and G is dened to be y 2 Proof: First, we consider the case We write N for N 1 . We use the input tapes of N as those of M - N and the output tape of M as that of M - N . We use a work tape T with the alphabet [ fBg which connects the parts representing N and M , and work tapes to simulate the head movements of the input tape of M and the output tape of N . It is easy to change the rules of M and N so that M reads from T and N writes on T . We also need to modify the rules so that it rst looks for an applicable rule coming from M and if there is no such rule, then looks for a rule coming from N . It is possible because the former rules do not access to the input tapes and therefore a machine can determine whether a particular rule is applicable or not. When k > 1, we need to copy the input tapes onto work tapes so that they can be shared by the parts representing N . We dene that it executes rules coming from N i until it outputs a character, and then switch to the next part. As we will show in Section 7, we have the followings. Lemma 4 There is an IM2-machine of type (f1; 0; 1g ?;1 ) which converts a sdr -name of x to G(x) for all x 2 I. Lemma 5 There is an IM2-machine of type (f0; 1g ! G(x) to the sdr -name of x for all x 2 I. Now, we prove the equivalences. Theorem 6 A multi-valued function F : I k ! I is Gray-code-computable i it is Proof: Suppose that M is an IM2-machine which Gray code computes F . By composing it with the IM2-machines in Lemma 4 and Lemma 5, we can form, by Proposition 3, an IM2-machine of type ((f1; 0; outputs a sdr -name of a member of F -names of x i are given. Therefore, we have a desired Type-2 machine by Proposition 1. On the other hand, suppose that there is a Type-2 machine which (( sdr computes F . Since a Type-2 machine is a special case of an IM2-machine, again, by composing the IM2-machines in Lemma 4 and Lemma 5, we can form an IM2-machine which Gray code computes F . 6 Topological Properties 1g. In this section, we show that G from I to ! ?;1 is homeomorphic, and therefore is a topological embedding. Since the character ? may be overwritten by 0 or 1, it is not appropriate to consider Cantor topology on ! ?;1 . Instead, we dene the order structure ? < 0 and ? < 1 on our alphabet and consider the Scott topology on f0; We consider its product topology on f0; its subspace topology on ! ?;1 . Let " p denote the set fx j p xg. Then, the set f" (d? is a base of f0; From this, we have a base f" ?;1 . Note that P corresponds to the states of output tapes of IM2-machines after a nite time of execution, and " (d? ?;1 is the set of possible outputs of an IM2-machine after it outputs d 2 P . Thus, if q 2 O for an open set O ! ?;1 and for an output q of an IM2-machine, then this fact is available from a nite time of execution of the machine. In this sense, the observation that open sets are nitely observable properties in [Smy92] holds for our IM2-machine. We can prove the following fundamental theorem in just the same way as we do for Type-2 computability and Cantor topology on f0; 1g ! . Theorem 7 An IM2-computable function f ?;1 is continuous. Now, Im(G) is the set f0; 1g ! f0; 1g ?;1 . We also consider the subspace topology on Im(G), which has the base f" g. We consider the inverse image of this base by G. When d 2 f0; 1g range over open intervals of the form respectively, for m and i integers. Since these open intervals form a base of the unit open interval I, I and Im(G) become homeomorphic through the function G. Thus, we have the following: Theorem 8 The Gray code embedding G is a topological embedding of I into ! ?;1 . As a direct consequence, we have the following: Corollary 9 A Gray-code-computable function f : I k ! I is continuous. As an application of our representation, we give a simple proof of Theorem 4.2.6 of which says that there is no eective enumeration of computable real numbers. Here, we dene to be a computable sequence if there is an IM2-machine of outputs G(x i ) when a binary name of i is given. Theorem is a computable sequence, then a computable number x with x exists. be an IM2-machine which computes s i to the binary name of i. By Proposition 1, we can assume that M is deterministic. This means that, by selecting one machine, the order the output tape is lled is xed. Since s either s i [2i] or s i [2i + 1] is written in a nite time. When s i [2i] is written rst, we put rst, we put not is dened as not not Then, the resulting sequence t is computable and is in Im(G), but is not equal to s i for (i 2 !). Therefore, G 1 (t) is not equal to x i because of the injectivity of the representation. 7 Conversion with signed digit representation As an example of an IM2-machine, we consider conversions between the Gray code and the restricted signed digit representation. Recall that a sdr -name of x 2 I is given as a sequence 1 : xs with xs an innite sequence of f0; 1; 1g. In this section, we consider xs as the sdr -name of x. Since the intervals represented by nite prexes of both representation coincide, the conversions become simple automaton-like algorithms which do not use work tapes. Example 1 Conversion from the signed digit representation to Gray code. It has the type simply write the head of the input tape as I. It has four states (i; the initial state, and 12 rules: In order to express this more simply, we use the notation of the functional language Haskell as follows: ds ds Here, where produces bindings of c and ds to the head and the tail of stog0 (xs,0,1), respectively. It is clear that the behavior of an IM2-machine can be expressed using this notation with the state and the contents of the work tapes before and after the head positions passed as additional arguments. In the program stog0, the states are used to invert the output: the result of stog0(xs; 1; 0) is that of stog0(xs; 0; 0) with the rst character inverted, and the result of stog0(xs; 0; 1) is stog stog0 0 input tape output tape Fig. 3. The behavior of stog IM2-machine when it reads 0 that of stog0(xs; 0; 0) with the second character inverted. Therefore, we can simplify the above program as follows: ds where c : ds = stog xs Here, nh is the function to invert the rst element of an innite list. That is, not not not s:ds The behavior of stog with input 0:xs is given in Figure 1. Here, a small circle on an output head means to invert the output from that head before lling the tape. The program stog is a correct Haskell program and works on a Haskell system. However, if we evaluate stog([0,0.]), there will be no output because it tries to calculate the rst digit, which is ?. Of course, tail(stog([0,0.])) produces the answer [1,0,0,0,. Next, we consider the inverse conversion, which is an example of Gray code input. Example 2 Conversion from Gray code to signed digit representation. Now, we only show a Haskell program. It has the type (f0; 1g ! In this case, indeterminism occurs and yields many dierent valid results: the results are actually signed digit representations of the same number. This is also a correct Haskell program. However, it fails to calculate, for example, gtos(stog([0,0.])) because the program gtos, from the rst two rules, tries to pattern match the head of the argument and starts its non-terminating calculation. Therefore, it fails to use the third rule. This is a limitation of the use of an existing functional language. We will discuss how to implement an IM2-machine as a program in Section 9. These programs are based on the recursive structure of the Gray code and is not as di-cult to write such a program as one might imagine. One can see from Figure 2 the following three recursive equations: (1) Here, (p). The rst equation corresponds to the fact that on the interval with the rst bit 0, i.e. the left half of Figure 2, the remaining bits form a 1 / 2 reduction of Figure 2. The second equation corresponds to the fact that on the interval with the rst bit 1, i.e. the right half of Figure 2, the remaining bits with the rst bit inverted form a 1 / 2 reduction of Figure 2, and if we use the equation with parenthesis, we can also state that the remaining bits form the reversal of Figure 2. These two equations characterize Figure 2. One interesting fact about this representation is that we also have the third equation. It says that on the interval with the second bit 1, i.e. the middle half of Figure 2, the remaining bits with the second bit inverted form a 1 / 2 reduction of Figure 2. From Equations (1), we have the following recursive scheme. Here, g 1 is a function to calculate f(x) from f(2x) when 0 < x is a function to calculate f(x) from f(2x 1) when 3 is a function to calculate f(x) from derived immediately from this scheme. On the other hand, Equations (1) can be rewritten as follows: stog uses this scheme to calculate the gray code output. These recursive schemes are used to derive the algorithm for addition in the next section. 8 Some simple algorithms in Gray code We write some algorithms with respect to Gray code. Example 3 Multiplication and division by 2. They are simple shifting operations. (suppose that the input is 0 < x < 1/2) Example 4 The complement x 7! 1 x. It is a simple operation to invert the rst digit, i.e.,the nh function in Example 1. Note that with the usual binary representation and the signed digit representation, we need to invert all the bits to calculate 1 x and thus this operation needs to be dened recursively. We can also see that the complement operation (x 7! k=2 n x) with respect to a dyadic number k=2 n+1 for can be implemented as inverting one digit. Example 5 Shifting x 7! x Addition with a dyadic number is nothing but two continuous complement operations over dyadic numbers. In the case of the rst axis is and the second axis is Therefore, the function AddOneOfTwo operates as x 7! x Example 6 Addition We consider addition x+y with 0 < x; y < 1. Since the result is in (0; 2), we consider the average function pl (0:as) pl (1:as) pl (0:as) pl (1:as) pl (a:1:as) pl (a:1:0:as) pl (a:1:0:as) pl (a:1:0:as) pl (a:1:0:as) pl (0:0:as) pl (1:0:as) pl (0:a:1:as) pl (1:a:1:as) To calculate the sum with respect to the signed digit representation, we need to look ahead two characters. It is also the case with the Gray code representation. Since it does not have redundancy, we can reduce the number of rules from 25 to 13 compared with the program written in the same way with the signed digit representation. 9 Extension to the Whole Real Line, Implementation, and Conclusion We have dened an embedding G of I to f0; 1g ! ?;1 based on Gray code, and introduced an indeterministic multihead Type 2 machine as a machine which can input/output sequences in f0; 1g ! ?;1 . Since G is a topological embedding of I into f0; 1g ! ?;1 , our IM2- machines are operating on a topological space which includes I as a subspace. We hope that this computational model will propose a new perspective on real number computation. In this paper, we only treated the unit open interval I = (0; 1). We discuss here how this embedding can be extended to the whole real line R. First, by using the rst digit as the sign bit: 1 if positive, 0 if negative, and ? if the number is zero, we can extend it to the interval ( 1; 1). We can also extend it to ( by assuming that there is a decimal point after the k-th digit. However, there seems to be no direct extension to all of the real numbers without losing injectivity and without losing the simplicity of the algorithms in Section 7 and 8. One possibility is to use some computable embedding of R into ( 1; 1), such as the function arctan(x)=. It is known that this function is computable, and therefore, we have IM2-machines which convert between the signed digit representation of x 2 R and the Gray code of f(x) in ( 1; 1). Therefore, we can dene our new representation as G 0 R). It is clear that this representation embeds R into ! ?;1 , and all the properties we have shown in Section 4 to 6 hold if we replace I with R and G with G 0 . In particular, the computability notion on R induced by G 0 and IM2-machines is equivalent to the one induced by the signed digit representation and Type-2 machiens. However, we will lose the symmetricity of the Gray code expansion and simplicity of the algorithms in Section 7 and 8. Another possibility is to introduce the character \." indicating the decimal point into the sequence. In order to allow an expression starting with ? (i.e. integers of the need to consider an expression starting with 0 because it should be allowed to ll the ? with 0 or 1 afterwards. Thus, we lose the injectivity of the expansion because we have 1:xs. We also have the same kind of di-culty if we adopt the oating-point-like expression: a pair of a number indicating the decimal point and a Gray code on ( 1; 1). Although this expansion becomes redundant, the redundancy introduced here by preceding zeros is limited in that we only need at most one zero at the beginning of each representation and thus each number has at most two names. As is shown in [BH00], we need innitely many names to innitely many real numbers if we use representations equivalent to the signed binary representation. Therefore, the redundancy we need for this extension is essentially smaller than that of the signed binary representation. Finally, we show some experimental implementations we currently have. As we have noted, though we can express the behavior of an IM2-machine using the syntax of a functional language Haskell, the program comes to have dierent semantics under the usual lazy evaluation strategy. We have implemented this Gray code input/output mechanism using logic programming languages. We have written gtos, stog, and the addition function pl of Section 8 using KL1 [UC90], a concurrent logic programming language based on Guarded Horn Clauses. We have also implemented them using the coroutine facility of SICStus Prolog. We are also interested in extending lazy functional languages so that programs in Section 7 and 8 become executable. The details about these implementations are given in [Tsu00]. Acknowledgements The author thanks Andreas Knobel for many interesting and illuminating discus- sions. He also thanks Mariko Yasugi, Hiroyasu Kamo, and Izumi Takeuchi for many discussions. --R A data structure based on gray code encoding for graphics and image processing. Exact real arithmetic: A case study in higher order programming. Topological properties of real number representations. Recursive and Computable Operations over Topological Structures. The New Turing Omnibus. A new representation for exact real numbers. Real number computability and domain theory. A golden ratio notation for the real numbers. An abstract data type for real numbers. On the de Haskell report. The takagi function and its generalization. Computability in Analysis and Physics. Computation over abstract structures: serial and parallel procedures. Implementation of indeterministic multihead type 2 machines with ghc for real number computations. On computable real numbers Design of the kernel language for the parallel inference machine. Exact real computer arithmetic with continued fractions. Type 2 recursion theory. An introduction to computable analysis. --TR Exact Real Computer Arithmetic with Continued Fractions Design of the kernel language for the parallel inference machine Topology Real number computability and domain theory A domain-theoretic approach to computability on the real line An abstract data type for real numbers Exact real arithmetic: a case study in higher order programming Computable analysis Topological properties of real number representations A golden ratio notation for the real numbers --CTR Hideki Tsuiki, Compact metric spaces as minimal-limit sets in domains of bottomed sequences, Mathematical Structures in Computer Science, v.14 n.6, p.853-878, December 2004

multihead;gray code;indeterminism;IM2-machines;real number computation

611399

New and faster filters for multiple approximate string matching.

We present three new algorithms for on-line multiple string matching allowing errors. These are extensions of previous algorithms that search for a single pattern. The average running time achieved is in all cases linear in the text size for moderate error level, pattern length, and number of patterns. They adapt (with higher costs) to the other cases. However, the algorithms differ in speed and thresholds of usefulness. We theoretically analyze when each algorithm should be used, and show their performance experimentally. The only previous solution for this problem allows only one error. Our algorithms are the first to allow more errors, and are faster than previous work for a moderate number of patterns (e.g. less than 50-100 on English text, depending on the pattern length).

Introduction Approximate string matching is one of the main problems in classical string algorithms, with applications to text searching, computational biology, pattern recognition, etc. Given a text T 1::n of length n and a pattern P 1::m of length m (both sequences over an alphabet \Sigma of size oe), and a maximal number of errors allowed, m, we want to find all text positions where the pattern matches the text with up to k errors. Errors can be substituting, deleting or inserting a character. We use the term "error level" to refer to In this paper we are interested in the on-line problem (i.e. the text is not known in advance), where the classical solution for a single pattern is based on dynamic programming and has a running time of O(mn) [26]. In recent years several algorithms have improved the classical one [22]. Some improve the worst or average case by using the properties of the dynamic programming matrix [30, 11, 16, 31, 9]. Others filter the text to quickly eliminate uninteresting parts [29, 28, 10, 14, 24], some of them being "sublinear" on average for moderate ff (i.e. they do not inspect all the text characters). Yet other approaches use bit-parallelism [3] in a computer word of w bits to reduce the number of operations [33, 35, 34, 6, 19]. The problem of approximately searching a set of r patterns (i.e. the occurrences of anyone of them) has been considered only recently. This problem has many applications, for instance This work has been supported in part by FONDECYT grant 1990627. ffl Spelling: many incorrect words can be searched in the dictionary at a time, in order to find their most likely variants. Moreover, we may even search the dictionary of correct words in the "text" of misspelled words, hopefully at much less cost. ffl Information retrieval: when synonym or thesaurus expansion is done on a keyword and the text is error-prone, we may want to search all the variants allowing errors. Batched queries: if a system receives a number of queries to process, it may improve efficiency by searching all them in a single pass. ffl Single-pattern queries: some algorithms for a single pattern allowing errors (e.g. pattern partitioning [6]) reduce the problem to the search of many subpatterns allowing less errors, and they benefit from multipattern search algorithms. A trivial solution to the multipattern search problem is to perform r searches. As far as we know, the only previous attempt to improve the trivial solution is due to Muth & Manber [17], who use hashing to search many patterns with one error, being efficient even for one thousand patterns. In this work, we present three new algorithms that are extensions of previous ones to the case of multiple search. In Section 2 we explain some basic concepts necessary to understand the algorithms. Then we present the three new techniques. In Section 3 we present "automaton which extends a bit-parallel simulation of a nondeterministic finite automaton In Section 4 we present "exact partitioning", that extends a filter based on exact searching of pattern pieces [7, 6, 24]. In Section 5 we present "counting", based on counting pattern letters in a text window [14]. In Section 6 we analyze our algorithms and in Section 7 we compare them experimentally. Finally, in Section 8 we give our conclusions. Some detailed analyses are left for Appendices A and B. Although [17] allows searching for many patterns, it is limited to only one error. Ours are the first algorithms for multipattern matching allowing more than one error. Moreover, even for one error, we improve [17] when the number of patterns is not very large (say, less than 50-100 on English text, depending on the pattern length). Our multipattern extensions improve over their sequential counterparts (i.e. one separate search per pattern using the base algorithm) when the error level is not very high (about ff 0:4 on English text). The filter based on exact searching is the fastest for small error levels, while the bit-parallel simulation of the NFA adapts better to more errors on relatively short patterns. Previous partial and preliminary versions of this work appeared in [5, 20, 21]. 2 Basic Concepts We review in this section some basic concepts that are used in all the algorithms that follow. In the paper S i denotes the i-th character of string S (being S 1 the first character), and S i::j stands for the substring S i S i+1 :::S j . In particular, if ffl, the empty string. 2.1 Filtering Techiques All the multipattern search algorithms that we consider in this work are based in the concept of filtering, and therefore it is useful to define it here. Filtering is based on the fact that it is normally easier to tell that a text position does not match than to ensure that it matches. Therefore, a filter is a fast algorithm that checks for a simple necessary (though not sufficient) condition for an approximate match to occur. The text areas that do not satisfy the necessary condition can be safely discarded, and a more expensive algorithm has to be run on the text areas that passed the filter. Since the filters can be much faster than approximate searching algorithms, filtering algorithms can be very competitive (in fact, they dominate on a large range of parameters). The performance of filtering algorithms, however, is very sensitive to the error level ff. Most filters work very well on low error levels and very bad with more errors. This is related with the amount of text that the filter is able to discard. When evaluating filtering algorithms, it is important not only to consider time efficiency but also their tolerance to errors. A term normally used when referring to filters is "sublinearity". It is said that a filter is sublinear when it does not inspect all the characters of the text (like the Boyer-Moore [8] algorithms for exact searching, which can be at best O(n=m)). Throughout this work we make use of the two following lemmas to derive filtering conditions. with at most k errors, and concatenation of sub-patterns), then some substring of S matches at least one of the p i 's, with at most bk=jc errors. Proof: Otherwise, the best match of each p i inside S has at least bk=jc errors. An occurrence of P involves the occurrence of each of the p i 's, and the total number of errors in the occurrences is at least the sum of the errors of the pieces. But here, just summing up the errors of all the pieces we have more than errors and therefore a complete match is not possible. Notice that this does not even consider that the matches of the p i must be in the proper order, be disjoint, and that some deletions in S may be needed to connect them. In general, one can filter the search for a pattern of length m with k errors by the search of j subpatterns of length m=j with k=j errors. Only the text areas surrounding occurrences of pieces must be checked for complete matches. An important particular case of Lemma 1 arises when one considers since in this case some pattern piece appears unaltered (zero errors). Lemma 2: [32] If there are i j such that ed(T i::j ; P ) k, then T j \Gammam+1::j includes at least m \Gamma k characters of P . Proof: Suppose the opposite. If then we observe that there are less than characters of P in T i::j . Hence, more than k characters must be deleted from P to match the text. we observe that there are more than k characters in T i::j that are not in P , and hence we must insert more than k characters in P to match the text. A contradiction in both cases. Note that in case of repeated characters in the pattern, they must be counted as different occurrences. For example, if we search "aaaa" with one error in the text, the last four letters of each occurrence must include at least three a's. simplification of that in [32]) says essentially that we can design a filter for approximate searching based on finding enough characters of the pattern in a text window (without regarding their ordering). For instance, the pattern "survey" cannot appear with one error in the text window "surger" because there are not five letters of the pattern in the text. However, the filter cannot discard the possibility that the pattern appears in the text window "yevrus". 2.2 Bit-Parallelism Bit-parallelism is a technique of common use in string matching [3]. It was first proposed in [2, 4]. The technique consists in taking advantage of the intrinsic parallelism of the bit operations inside a computer word. By using cleverly this fact, the number of operations that an algorithm performs can be cut down by a factor of at most w, where w is the number of bits in the computer word. Since in current architectures w is 32 or 64, the speedup is very significant in practice (and improves with technological progress). In order to relate the behavior of bit-parallel algorithms to other works, it is normally assumed that dictated by the RAM model of computation. We prefer, however, to keep w as an independent value. Some notation we use for bit-parallel algorithms is in order. We denote as b ' :::b 1 the bits of a mask of length ', which is stored somewhere inside the computer word. We use C-like syntax for operations on the bits of computer words, e.g. "j" is the bitwise-or and "!!" moves the bits to the left and enters zeros from the right, e.g. b m b We can also perform arithmetic operations on the bits, such as addition and subtraction, which operates the bits as if they formed a number. For instance, b ' :::b x We explain now the first bit-parallel algorithm, since it is the basis of much of which follows in this work. The algorithm searches a pattern in a text (without errors) by parallelizing the operation of a non-deterministic finite automaton that looks for the pattern. Figure 1 illustrates this automaton. l a h a Figure 1: Nondeterministic automaton that searches "aloha" exactly. This automaton has m+ 1 states, and can be simulated in its non-deterministic form in O(mn) time. The Shift-Or algorithm achieves O(mn=w) worst-case time (i.e. optimal speedup). Notice that if we convert the non-deterministic automaton to a deterministic one to have O(n) search time, we get an improved version of the KMP algorithm [15]. However, KMP is twice as slow for The algorithm first builds a table B[ ] which for each character c stores a bit mask The mask B[c] has the bit b i in zero if and only if P c. The state of the search is kept in a machine word matches the end of the text read up to now (i.e. the state numbered i in Figure 1 is active). Therefore, a match is reported whenever dm is zero. D is set to all ones originally, and for each new text character T j , D is updated using the formula The formula is correct because the i-th bit is zero if and only if the (i \Gamma 1)-th bit was zero for the previous text character and the new text character matches the pattern at position i. In other words, T j It is possible to relate this formula to the movement that occurs in the non-deterministic automaton for each new text character: each state gets the value of the previous state, but this happens only if the text character matches the corresponding arrow. For patterns longer than the computer word (i.e. m ? w), the algorithm uses dm=we computer words for the simulation (not all them are active all the time). The algorithm is O(mn=w) worst case time, and the preprocessing is O(m On average, the algorithm is O(n) even when m ? w, since only the first O(1) states of the automaton have active states on average (and hence the first O(1) computer words need to be updated on average). It is easy to extend Shift-Or to handle classes of characters. In this extension, each position in the pattern matches with a set of characters rather than with a single character. The classical string matching algorithms are not so easily extended. In Shift-Or, it is enough to set the i-th bit of B[c] for every c 2 P i (P i is a set now). For instance, to search for "survey" in case-insensitive form, we just set the first bit of B["s"] and of B["S"] to "match" (zero), and the same with the rest. Shift-Or can also search for multiple patterns (where the complexity is O(mn=w) if we consider that m is the total length of all the patterns) by arranging many masks B and D in the same machine word. Shift-Or was later enhanced [34] to support a larger set of extended patterns and even regular expressions. Recently, in [25], Shift-Or was combined with a sublinear string matching algorithm, obtaining the same flexibility and an efficiency competitive against the best classical algorithms. Many on-line text algorithms can be seen as implementations of clever automata (classically, in their deterministic form). Bit-parallelism has since its invention became a general way to simulate simple non-deterministic automata instead of converting them to deterministic. It has the advantage of being much simpler, in many cases faster (since it makes better usage of the registers of the computer word), and easier to extend to handle complex patterns than its classical counterparts. Its main disadvantage is the limitations it imposes with regard to the size of the computer word. In many cases its adaptations to cope with longer patterns are not so efficient. 2.3 Bit-parallelism for Approximate Pattern Matching We present now an application of bit-parallelism to approximate pattern matching, which is especially relevant for the present work. Consider the NFA for searching "patt" with at most errors shown in Figure 2. Every row denotes the number of errors seen. The first one 0, the second one 1, and so on. Every column represents matching the pattern up to a given position. At each iteration, a new text character is considered and the automaton changes its states. Horizontal arrows represent matching a character (they can only be followed if the corresponding match occurs). All the others represent errors, as they move to the next row. Vertical arrows represent inserting a character in the pattern (since they advance in the text and not in the pattern), solid diagonal arrows represent replacing a character (since they advance in the text and the pattern), and dashed diagonal arrows represent deleting a character of the pattern (since, as ffl-transitions, they advance in the pattern but not in the text). The loop at the initial state allows considering any character as a potential starting point of a match. The automaton accepts a character (as the end of a match) whenever a rightmost state is active. Initially, the active states at row i (i 2 0::k) are those at the columns from 0 to i, to represent the deletion of the first i characters of the pattern P 1::m . a a a no errors Figure 2: An NFA for approximate string matching. We show the active states after reading the text "pait". An interesting application of bit-parallelism is to simulate this automaton in its nondeterministic form. A first approach [34] obtained O(kdm=wen) time, by packing each automaton row in a machine word and extending the Shift-Or algorithm to account for the vertical and diagonal arrows. Note that even if all the states fit in a single machine word, the k have to be sequentially updated because of the ffl-transitions. The same happens in the classical dynamic programming algorithm [26], which can be regarded as a column-wise simulation of this NFA. In this paper we are interested in a more recent simulation technique [6], where we show that by packing diagonals of the automaton instead of rows or columns all the new values can be computed in one step if they fit in a computer word. We give a brief description of the idea. Because of the ffl-transitions, once a state in a diagonal is active, all the subsequent states in that diagonal become active too, so we can define the minimal active row of each diagonal, D i (diagonals are numbered by looking the column they start at, e.g. D 1 and D 2 are enclosed in dotted lines in Figure 2). The new values for D i (i after we read a new text character c can be computed by where it always holds and we report a match whenever Dm\Gammak k. The formula for accounts for replacements, insertions and matches, respectively. Deletions are accounted for by keeping the minimum active row. All the interesting matches are caught by considering only the diagonals D 1 :::D m\Gammak . We use bit-parallelism to represent the D i 's in unary. Each one is hold in k (plus an overflow bit) and stored sequentially inside a bit mask D. Interestingly, the effect is the same if we read the diagonals bottom-up and exchange 0 $ 1, with each bit representing a state of the NFA. The update formula can be seen either as an arithmetic implementation of the previous formula in unary or as a logical simulation of the flow of bits across the arrows of the NFA. As in Shift-Or, a table of (m bits long) masks b[ ] is built representing match or mismatch against the pattern. A table B[c] is built by mapping the bits of b[ ] to their appropriate positions inside D. Figure 3 shows how the states are represented inside the masks D and B. separator separator final state t a p t t a Figure 3: Bit-parallel representation of the NFA of Figure 2. This representation requires k+2 bits per diagonal, so the total number of bits is (m \Gamma k)(k+2). If this number of bits does not exceed the computer word size w, the update can be done in O(1) operations. The resulting algorithm is linear and very fast in practice. For our purposes, it is important to realize that the only connection between the pattern and the algorithm is given by the b[ ] table, and that the pattern can use classes of characters just as in the Shift-Or algorithm. We use this property next to search for multiple patterns. 3 Superimposed Automata In this section we describe an approach based on the bit-parallel simulation of the NFA just described Suppose we have to search r patterns . We are interested in the occurrences of any one of them, with at most k errors. We can extend the previous bit-parallelism approach by building the automaton for each one, and then "superimpose" all the automata. Assume that all patterns have the same length (otherwise, truncate them to the shortest one). Hence, all the automata have the same structure, differing only in the labels of the horizontal arrows. The superimposition is defined as follows: we build the b[ ] table for each pattern, and then take the bitwise-and of all the tables (recall that 0 means match and 1 means mismatch). The resulting table matches at position i with the i-th character of any of the patterns. We then build the automaton as before using this table. The resulting automaton accepts a text position if it ends an occurrence of a much more relaxed pattern with classes of characters, namely for example, if the search is for "patt" and "wait", as shown in Figure 4, the string "pait" is accepted with zero errors. or w no errors errors a t a t t or i a t t or i t or i or w or w Figure 4: An NFA to filter the search for "patt" and "wait". For a moderate number of patterns, the filter is strict enough at the same cost of a single search. Each occurrence reported by the automaton has to be verified for all the involved patterns (we use the single-pattern automaton for this step). That is, we have to retraverse the last m+ characters to determine if there is actually an occurrence of some of the patterns. If the number of patterns is too large, the filter will be too relaxed and will trigger too many verifications. In that case, we partition the set of patterns into groups of r 0 patterns each, build the automaton of each group and perform dr=r 0 e independent searches. The cost of this search is O(r=r 0 n), where r 0 is small enough to make the cost of verifications negligible. This r 0 always exists, since for r we have a single pattern per automaton and no verification is needed. When grouping, we use the heuristic of sorting the patterns and packing neighbors in the same group, trying to have the same first characters. 3.1 Hierarchical Verification The simplest verification alternative (which we call "plain") is that, once a superimposed automaton reports a match, we try the individual patterns one by one in the candidate area. However, a smarter verification technique (which we call hierarchical) is possible. Assume first that r is a power of two. Then, when the automaton reports a match, run two new automata over the candidate area: one which superimposes the first half of the patterns and another with the second half. Repeat the process recursively with each of the two automata that finds again a match. At the end, the automata will represent single patterns and if they find a match we know that their patterns have been really found (see Figure 5). Of course the automata for the required subsets of patterns are all preprocessed. Since they correspond to the internal nodes of a binary tree of r leaves, they are so the space and preprocessing cost does not change. If r is not a power of two then one of the halves may have one more pattern than the other.2424 Figure 5: The hierarchical verification method for 4 patterns. Each node of the tree represents a check (the root represents in fact the global filter). If a node passes the check, its two children are tested. If a leaf passes the check, its pattern has been found. The advantage of hierarchical verification is that it can remove a number of candidates from consideration in a single test. Moreover, it can even find that no pattern has really matched before actually checking any specific pattern (i.e. it may happen that none of the two halves match in a spurious match of the whole group). The worst-case overhead over plain verification is just a constant factor, that is, twice as many tests over the candidate area of r). On average, as we show later analytically and experimentally, hierarchical verification is by far superior to plain verification. 3.2 Automaton Partitioning Up to now we have considered short patterns, whose NFA fit into a computer word. If this is not the case (i.e. (m we partition the problem. In this subsection and the next we adapt the two partitioning techniques described in [6]. The simplest technique to cope with a large automaton is to use a number of machine words for the simulation. The idea is as follows: once the (large) automata have been superimposed, we partition the superimposed automaton into a matrix of subautomata, each one fitting in a computer word. Those subautomata behave slightly differently than the simple one, since they must propagate bits to their neighbors. Figure 6 illustrates. Once the automaton is partitioned, we run it over the text updating its subautomata. Each step takes time proportional to the number of cells to update, i.e. O(k(m \Gamma k)=w). Observe, however, that it is not necessary to update all the subautomata, since those on the right may not have any active state. Following [31], we keep track of up to where we need to update the matrix of subautomata, working only on the "active" cells. Information flow Affected area000000000111111111111111111000111111 I rows J columns c r Figure large NFA partitioned into a matrix of I \Theta J computer words, satisfying (' r +1)' c w. 3.3 Pattern Partitioning This technique is based on Lemma 1 of Section 2.1. We can reduce the size of the problem if we divide the pattern in j parts, provided we search all the sub-patterns with bk=jc errors. Each match of a sub-pattern must be verified to determine if it is in fact a complete match. To perform the partition, we pick the smallest j such that the problem fits in a single computer word (i.e. (dm=je w). The limit of this method is reached for since in that case we search with zero errors. The algorithm for this case is qualitatively different and is described in Section 4. We divide each pattern in j subpatterns as evenly as possible. Once we partition all the r patterns, we are left with j \Theta r subpatterns to be searched with bk=jc errors. We simply group them as if they were independent patterns to search with the general method. The only difference is that, after determining that a subpattern has appeared, we have to verify its complete pattern. Another kind of hierarchical verification, which we call "hierarchical piece verification", is applied in this case too. As shown in [23, 24], the single-pattern algorithm can verify hierarchically whether the complete pattern matches given that a piece matches (see Figure 7). That is, instead of checking the complete pattern we check the concatenation of two pieces containing the one that matched, and if it matches then we check the concatenation of four pieces, and so on. This works because Lemma 1 applies at each level of the tree of Figure 7. The method is orthogonal to our hierarchical verification idea because hierarchical piece verification works bottom-up instead of top-down and operates on pieces of the pattern rather than on sets of patterns. As we are using our hierarchical verification on the sets of pattern pieces to determine which piece matched given that a superimposition of them matched, we are coupling two different hierarchical verification techniques in this case: we first use our new mechanism to determine which piece matched from the superimposed group and then use hierarchical piece verification to determine the occurrence of the complete pattern the piece belongs to. Figure 8 illustrates the whole process. aaabbbcccddd aaabbb cccddd ccc ddd bbb aaa Figure 7: The hierarchical piece verification method for a pattern split in 4 parts. The boxes (leaves) are the elements which are actually searched, and the root represents the whole pattern. At least one pattern at each level must match in any occurrence of the complete pattern. If the bold box is found, all the bold lines may be verified. p22 p22 p22 each one is split in 4 3 pieces to search superimposed groups the pieces are arranged in hierarchical verif. p22 is found and searched hierarchical piece verif. P2 is finally found Figure 8: The whole process of pattern partitioning with hierarchical verifications. Partitioning into Exact Searching This technique (called "exact partitioning" for short) is based on a single-pattern filter which reduces the problem of approximate searching to a problem of multipattern exact searching. The algorithm first appeared in [34], and was later improved in [7, 6, 24]. We first present the single- pattern version and then our extension to multiple patterns. 4.1 A Filter Based on Exact Searching A particular case of Lemma 1 shows that if a pattern matches a text position with k errors, and we split the pattern in k+1 pieces, then at least one of the pieces must be present with no errors in each occurrence (this is a folklore property which has been used several times [34, 18, 12]). Searching with zero errors leads to a completely different technique. Since there are efficient algorithms to search for a set of patterns exactly, we partition the pattern in k similar length), and apply a multipattern exact search for the pieces. Each occurrence of a piece is verified to check if it is surrounded by a complete match. If there are not too many verifications, this algorithm is extremely fast. From the many algorithms for multipattern search, an extension of Sunday's algorithm [27] gave us the best results. We build a trie with the sub-patterns. From each text position we search the text characters into the trie, until a leaf is found (match) or there is no path to follow (mismatch). The jump to the next text position is precomputed as the minimum of the jumps allowed in each sub-pattern by the Sunday algorithm. As in [24], we use the same technique for hierarchical piece verification of a single pattern presented in Section 3.3. 4.2 Searching Multiple Patterns Observe that we can easily add more patterns to this scheme. Suppose we have to search for r patterns We cut each one into search in parallel for all the r(k pieces. When a piece is found in the text, we use a classical algorithm to verify its pattern in the candidate area. Note an important difference with superimposed automata. In this multipattern search we know which piece has matched. This is not the case in superimposed automata, where not only we do not know which piece matched, but it is even possible that no piece has really matched. The work to determine which is the matching piece (carried out by hierarchical verification in superimposed automata) is not necessary here. Moreover, we only detect real matches, so there are no more matches in the union of patterns than the sum of the individual matches. Therefore, there is no point in separating the search for the r(k in groups. The only reason to superimpose less patterns is that the shifts of the Sunday algorithm are reduced as the number of patterns grow, but as we show in the experiments, this never justifies in practice splitting one search into two. 5 A Counting Filter We present now a filter based on counting letters in common between the pattern and a text window. This filter was first presented in [14] (a simple variant of [13]), but we use a slightly different version here. Our variant uses a fixed-size instead of variable-size text window (a possibility already noted in [32]), which makes it better suited for parallelization. We first explain the single-pattern filter and then extend it to handle many patterns using bit-parallelism. 5.1 A Simple Counter This filter is based in Lemma 2 of Section 2.1. It passes over the text examining an m-letters long window. It keeps track of how many characters of P are present in the current text window (accounting for multiplicities too). If, at a given text position j, more characters of P are in the window T j \Gammam+1::j , the window area is verified with a classical algorithm. We implement the filtering algorithm as follows. We keep a counter count of pattern characters appearing in the text window. We also keep a table A[ ] where, initially, the number of times that each character c appears in P is kept in A[c]. Throughout the algorithm, each entry A[c] indicates how many occurrences of c can still be taken as belonging to P . For example, if 'h' appears once in P , we count only one of the 'h's of the text window as belonging to P . When A[c] is negative, it means that c must exit the text window \GammaA[c] times before we take it again as belonging to P . For example, if we run the pattern "aloha" over the text "aaaaaaaa", it will hold and the value of the counter will be 2. This is independent on k. To advance the window, we must include the new character T j+1 and exclude the last character, To include the new character, we subtract one from A[T j+1 ]. If it was greater than zero before being decremented, it is because the new character T j+1 is in P , so we increment count. To exclude the old character T j \Gammam+1 , we add one to A[T j \Gammam+1 ]. If its is greater than zero after being incremented, it is because T j \Gammam+1 was considered to be in P , so we decrement count. Whenever count reaches we verify the preceding area. As can be seen, the algorithm is not only linear (excluding verifications), but the number of operations per character is very small. 5.2 Keeping Many Counters in Parallel To search r patterns in the same text, we use bit-parallelism to keep all the counters in a single machine word. We must do that for the A[ ] table and for count. The values of the entries of A[ ] lie in the range [\Gammam::m], so we need exactly 1)e bits to store them. This is also enough for count, since it is in the range [0::m]. Hence, we can pack patterns of length m in a single search (recall that w is the number of bits in the computer word). If the patterns have different lengths, we can either truncate them to the shortest length or use a window size of the longest length. If we have more patterns, we must divide the set in subsets of maximal size and search each subset separately. We focus our attention on a single subset now. The algorithm simulates the simple one as follows. We have a table MA[ ] that packs all the tables. Each entry of MA[ ] is divided in bit areas of length ' + 1. In the area of the machine word corresponding to each pattern, we store its normal A[ ] value, set to 1 the most significant bit of the area, and subtract 1 (i.e. we store in the algorithm, we have to add or subtract 1 to all A[ ]'s, we can easily do it in parallel without causing overflow from an area to the next. Moreover, the corresponding A[ ] value is not positive if and only if the most significant bit of the area is zero. We have also a parallel counter Mcount, where the areas are aligned with MA[ ]. It is initialized by setting to 1 the most significant bit of each area and then subtracting at each one, i.e. we store Later, we can add or subtract 1 in parallel without causing overflow. Moreover, the window must be verified for a pattern whenever the most significant bit of its area reaches 1. The condition can be checked in parallel, but when some of the most significant bits reach 1, we need to sequentially check which one it was. Finally, observe that the counters that we want to selectively increment or decrement correspond exactly to the MA[ ] areas that have a 1 in their most significant bit (i.e. those whose A[ ] value is positive). This allows an obvious bit mask-shift-add mechanism to perform this operation in parallel on all the counters. Figure 9 illustrates. Mcount MA [a] MA [l] MA [o] MA [h] MA [e] count m\Gammak ? (false) Figure 9: The bit-parallel counters. The example corresponds to the pattern "aloha" searched with 1 error and the text window "hello". The A values are A[ 0 a 6 Analysis We are interested in the complexity of the presented algorithms, as well as in the restrictions that ff and r must satisfy for each mechanism to be efficient in filtering most of the unrelevant part of the text. To this effect, we define two concepts. First, we say that a multipattern search algorithm is optimal if it searches r patterns in the same time it takes to search one pattern. If we call C n;r the cost to search r patterns in a text of size n, then an algorithm is optimal if C we say that a multipattern search algorithm is useful if it searches r patterns in less than the time it takes to search them one by one with the corresponding sequential algorithm, i.e. C n;r ! r C n;1 . As we work with filters, we are interested in the average case analysis, since in the worst case none is useful. We compare in Table 1 the complexities and limits of applicability of all the algorithms. Muth & Manber are included for completeness. The analysis leading to these results is presented later in this section. Algorithm Complexity Optimality Usefulness Simple Superimp. r oe Automaton Part. ffm 2 r oe Pattern Part. mr oe w(1\Gammaff) Part. Exact Search ffoe 1=ff log oe (rm)+\Theta(log oe log oe (rm)) log oe m+\Theta(log oe log oe m) Counting r log m Muth & Manber mn Table 1: Complexity, optimality and limit of applicability for the different algorithms. We present in Figure 10 a schematical representation of the areas where each algorithm is the best in terms of complexity. We show later how the experiments match those figures. Exact partitioning is the fastest choice in most reasonable scenarios, for the error levels where it can be applied. First, it is faster than counting for m= log m ! ffoe 1=ff =w, which does not hold asymptotically but holds in practice for reasonable values of m. Second, it is faster than superimposing automata for min( which is true in most practical cases. ffl The only algorithm which can be faster than exact partitioning is that of Muth & Manber [17], namely for r ? ffoe 1=ff . However, it is limited to ffl For increasing m, counting is asymptotically the fastest algorithm since its cost grows as O(log m) instead of O(m) thanks to its optimal use of the bits of the computer word. However, its applicability is reduced as m grows, being useless at the point where it wins over exact partitioning. ffl When the error level is too high for exact partitioning, superimposing automata is the only remaining alternative. Automaton partitioning is better for m while pattern partitioning is asymptotically better. Both algorithms have the same limit of usefulness, and for higher error levels no filter can improve over a sequential search. Pattern Partitioning Partitioning Automaton Partitioning into Exact Search oe 1= log oe m ff oe 1= log oe m Partitioning into Exact Search Superimposed Automata r Muth-Manber ffoe 1=ff ff Figure 10: The areas where each algorithm is better, in terms of ff, m and r. In the left plot (varying m), we have assumed a moderate r (i.e. less than 50). 6.1 Superimposed Automata Suppose that we search r patterns. As explained before, we can partition the set in groups of r 0 patterns each, and search each group separately (with its r 0 automata superimposed). The size of the groups should be as large as possible, but small enough for the verifications to be not significant. We analyze which is the optimal value for r 0 and which is the complexity of the search. In [6] we prove that the probability of a given text position matching a random pattern with error level ff is O(fl m ), where It is also proved that oe, and experimentally shown that this holds very precisely in practice if we replace e by 1.09. In fact, a very abrupt phenomenon occurs, since the matching probability is very low for oe and very high otherwise. In this formula, 1=oe stands for the probability of a character crossing a horizontal edge of the automaton (i.e. the probability of two characters being equal). To extend this result, we note that we have r 0 characters on each edge now, so the above mentioned probability is which is smaller than r 0 =oe. We use this upper bound as a pessimistic approximation (which stands for the case of all the r 0 characters being different, and is tight for r 0 !! oe). As the single-pattern algorithm is O(n) time, the multipattern algorithm is optimal on average whenever the total cost of verifications is O(1) per character. Since each verification costs O(m) (because we use a linear-time algorithm on an area of length O(m)), we need that the total number of verifications performed is O(1=m) per character, on average. If we used the plain verification scheme, this would mean that the probability that a superimposed automaton matches a text position should be O(1=(mr)), as we have to perform r verifications. If hierarchical verification is not used we have that, as r increases, matching becomes more probable (because it is easier to cross a horizontal edge of the automaton) and it costs more (because we have to check the r patterns one by one). This results in two different limits on the maximum allowable r, one for each of the two facts just stated. The limit due to the increased cost of each verification is more stringent than that of increased matching probability. The resulting analysis without hierarchical verification is very complex and is omitted here because hierarchical verification yields considerably better results and a simpler analysis. As we show in Appendix A, the average cost to verify a match of the superimposed automaton is O(m) when hierarchical verification is used, instead of the O(rm) cost of plain verification. That is, the cost does not grow as the number of patterns increases. Hence, the only limit that prevents us from superimposing all the r patterns is that the matching probability becomes higher. That is, if ff r=oe, then the matching probability is too high and we will spend too much time verifying almost all text positions. On the other hand, we can superimpose as much as we like before that limit is reached. This tells that the best r (which we call r ) is the maximum one not reaching the limit, i.e. r (1) Since we partition in sets small enough to make the verifications not significant, the cost is simply O(r=r n) This means that the algorithm is optimal for (taking the error level as a constant), or alternatively ff r=oe. On the other hand, for oe, the cost is O(rn), not better than the trivial solution (i.e. r hence no superimposition occurs and the algorithm is not useful). Figure 11 illustrates. Automaton Partitioning: the analysis for this case is similar to the simple one, except because each step of the large automaton takes time proportional to the total number of subautomata, i.e. rts pr oe r ff r oe Figure 11: Behavior of superimposed automata. On the left, the cost increases linearly with r, with slope depending on ff. On the right, the cost of a parallel search (t p ) approaches r single searches In fact, this is a worst case since on average not all cells are active, but we use the worst case because we superimpose all the patterns we can until the worst case of the search is almost reached. Therefore, the cost formula is rn This is optimal for constant ff), or alternatively for ff r=oe. It is useful for ff oe. Pattern Partitioning: we have now jr patterns to search with bk=jc errors. The error level is the same for subproblems (recall that the subpatterns are of length m=j). To determine which piece matched from the superimposed group, we pay O(m) independently of the number of pieces superimposed (thanks to the hierarchical verification). Hence the limit for our grouping is given by Eq. (1). In both the superimposed and in the single-pattern algorithm, we also pay to verify if the match of the piece is part of a complete match. As we show in [23], this cost is negligible for ff oe, which is less strict than the limit given by Eq. (1). As we have jr pieces to search, we need an analytical expression for j. Since j is just large enough so that the subpatterns fit in a computer word, where d(w; ff) can be shown to be O(1= w) by maximizing it in terms of ff (see [23]). Therefore, the complexity is oe rn On the other hand, the search cost of the single-pattern algorithm is O(jrn). With respect to the simple algorithm for short patterns, both costs have been multiplied by j, and therefore the limits for optimality and usefulness are the same. If we compare the complexities of pattern versus automaton partitioning, we have that pattern partitioning is better for k ? w. This means that for constant ff and increasing m, pattern partitioning is asymptotically better. 6.2 Partitioning into Exact Searching In [6] we analyze this algorithm as follows. Except for verifications, the search time can be made O(n) in the worst case by using an Aho-Corasick machine [1], and O(ffn) in the best case if we use a multipattern Boyer-Moore algorithm. This is because we search pieces of length m=(k+1) 1=ff. We are interested in analyzing the cost of verifications. Since we cut the pattern in k +1 pieces, they are of length bm=(k 1)e. The probability of each piece matching is at most 1=oe bm=(k+1)c . Hence, the probability of any piece matching is at most (k We can easily extend that analysis to the case of multiple search, since we have now r(k pieces of the same length. Hence, the probability of verifying is r(k We check the matches using a classical algorithm such as dynamic programming. Note that in this case we know which pattern to verify, since we know which piece matched. As we show in [23], the total verification cost if the pieces are of length ' is O(' 2 ) (in our case, Hence, the search cost is O ffoe 1=ff where the "1" must be changed to "ff" if we consider the best case. We consider optimality and usefulness now. An optimal algorithm should pay O(n) total search time, which holds for The algorithm is always useful, since it searches at the same cost independently on the number of patterns, and the number of verifications triggered is exactly the same as if we searched each pattern separately. However, if ff ? 1=(log oe m+ \Theta(log oe log oe m)), then both algorithms (single and multipattern) work as much as dynamic programming and hence the multipattern search is not useful. The other case when the algorithm could not be useful is when the shifts of a Boyer-Moore search are shortened by having many patterns up to the point where it is better to perform separate searches. This never happens in practice. 6.3 Counting If the number of verifications is negligible, each pass of the algorithms is O(n). In the case of multiple patterns, only O(w= log m) patterns can be packed in a single search, so the cost to search r patterns is O(rn log(m)=w). The difficult part of the analysis is the maximum error level ff that the filtration scheme can tolerate while keeping the number of verifications low. We assume that we use dynamic programming to verify potential matches. We call the probability of verifying. If log(m)=(wm 2 ) the algorithm keeps linear (i.e. optimal) on average. The algorithm is always useful since the number of verifications triggered with the multipattern search is the same as for the single-pattern version. However, if 1=m both algorithms work O(rmn) as for dynamic programming and hence the filter is not useful. We derive in Appendix B a pessimistic bound for the limit of optimality and usefulness, namely grows, we can tolerate smaller error levels. This limit holds for any condition of the type independently of the constant c. In our case, we need usefulness. 7 Experimental Results We experimentally study our algorithms and compare them against previous work. We tested with megabytes of lower-case English text. The patterns were randomly selected from the same text. We use a Sun UltraSparc-1 running Solaris 2.5.1, with 64 megabytes of RAM, and Each data point was obtained by averaging the Unix's user time over 10 trials. We present all the times in tenths of seconds per megabyte. We do not present results on random text to avoid an excessively lengthly exposition. In general, all the filters improve as the alphabet size oe grows. Lower-case English text behaves approximately as random text with which is the inverse of the probability that two random letters are equal. Figure compares the plain and hierarchical verification methods against a sequential application of the r searches, for the case of superimposed automata when the automaton fits in a computer word. We show the cases of increasing r and of increasing k. It is clear that hierarchical verification outperforms plain verification in all cases. Moreover, the analysis for hierarchical verification is confirmed since the maximum r up to where the cost of the parallel algorithm does not grow linearly is very close to r . On the other hand, the algorithm with simple verification degrades sooner, since the verification cost grows with r. The mentioned maximum r value is the point where the parallelism ratio is maximized. That is, if we have to search for 2r patterns, it is better to split them in two groups of size r and search each group sequentially. To stress this point, Figure 12 (right) shows the quotient between the parallel and the sequential algorithms, where the optimum is clear for superimposed automata. On the other hand, the parallelism ratio of exact partitioning keeps improving as r grows, as predicted by the analysis (there is an optimum for larger m, related to the Sunday shifts, but it still does not justify to split a search in two). When we compare our algorithms against the others, we consider only hierarchical verification and use this r value to obtain the optimal grouping for the superimposed automata algorithms. The exact partitioning, on the other hand, performs all the searches in a single pass. In counting, it is clear that the speedup is optimal and we pack as many patterns as we can in a single search. Notice that the plots which depend on r show the point where r should be selected. Those which depend on k (for fixed r), on the other hand, just show how the parallelization works as the error level increases, which cannot be controlled by the algorithm. We compare now our algorithms among them and against others. We begin with short patterns whose NFA fit in a computer word. Figure 13 shows the results for increasing r and for increasing k. For low and moderate error levels, exact partitioning is the fastest algorithm. In particular, it is faster than previous work [17] when the number of patterns is below 50 (for English text). When r r rts r r rts rts Sequential NFA Superimposed, plain verif. Exact Partitioning Superimposed, hierarchical verif. Figure 12: Comparison of sequential and multipattern algorithms for 9. The rows correspond to respectively. The left plots show search time and the right plots show the ratio between the parallel (t p ) and the sequential time (r \Theta t s ). the error level increases, superimposed automata is the best choice. This agrees with the analysis. r r Exact Partitioning Superimposed Automata Counting Figure 13: Comparison among algorithms for 9. The top plots show increasing r for 3. The bottom plots show increasing k for We consider longer patterns now 14 shows the results for increasing r and for increasing k. As before, exact partitioning is the best where it can be applied, and improves over previous work [17] for r up to 90-100. For these longer patterns the superimposed automata technique also degrades, and only rarely is it able to improve over exact partitioning. In most cases it only begins to be the best when it (and all the others) are no longer useful. Figure summarizes some of our experimental results, becoming a practical version of the theoretical Figure 10. The main differences are that exact partitioning is better in practice than what its complexity suggests, and that there is no clear winner between pattern and automaton Exact Partitioning Pattern Partitioning Automaton Partitioning Counting Figure 14: Comparison among algorithms for 30. The top plots show, for increasing r, 4. The bottom plots show, for increasing k, partitioning is not run for would resort to exact partitioning. Partitioning into Exact Search Superimposed Automata r Muth-Manber aPartitioning into Exact Search 9 Superimposed Automata Figure 15: The areas where each algorithm is better, in practice, on English text. In the right plot we assumed 9. Compare with Figure 10. Conclusions We have presented a number of different filtering algorithms for multipattern approximate search- ing. These are the only algorithms that allow an arbitrary number of errors. On the other hand, the only previous work allows just one error and we have outperformed it when the number of patterns to search is below 50-100 on English text, depending on the pattern length. We have explained, analyzed and experimentally tested our algorithms. We have also presented a map of the best algorithms for each case. Many of the ideas we propose here can be used to adapt other single-pattern approximate searching algorithms to the case of multipattern searching. For instance, the idea of superimposing automata can be adapted to most bit-parallel algorithms, such as [19]. Another fruitful idea is that of exact partitioning, where a multipattern exact search is easily adapted to search the pieces of many patterns. There are many other filtering algorithms of the same type, e.g. [28]. On the other hand, other exact multipattern search algorithms may be better suited to other search parameters (e.g. working better on many patterns). A number of practical optimizations to our algorithms are possible, for instance ffl If the patterns have different lengths, we truncate them to the shortest one when superimposing automata. We can select cleverly the substrings to use, since having the same character at the same position in two patterns improves the filtering mechanism. ffl We used simple heuristics to group subpatterns in superimposed automata. These can be improved to maximize common letters too. A more general technique could group patterns which are similar in terms of number of errors needed to convert one into the other (i.e. a clustering technique). ffl We are free to partition each pattern in k pieces as we like in exact partitioning. This is used in [24] to minimize the expected number of verifications when the letters of the alphabet do not have the same probability of occurrence (e.g. in English text). An O(m 3 ) dynamic programming algorithm is presented there to select the best partition, and this could be applied to multipattern search. Acknowledgements We thank Robert Muth and Udi Manber for their implementation of [17]. We also thank the anonymous referees for their detailed comments that improved this work. --R Efficient string matching: an aid to bibliographic search. Efficient Text Searching. Text retrieval: Theory and practice. A new approach to text searching. Multiple approximate string matching. Faster approximate string matching. Fast and practical approximate pattern matching. A fast string searching algorithm. Theoretical and empirical comparisons of approximate string matching algorithms. Sublinear approximate string matching and biological applications. An improved algorithm for approximate string matching. Simple and efficient string matching with k mismatches. A comparison of approximate string matching algo- rithms Fast parallel and serial approximate string matching. Approximate multiple string search. A sublinear algorithm for approximate keyword searching. A fast bit-vector algorithm for approximate pattern matching based on dynamic progamming Multiple approximate string matching by counting. Approximate Text Searching. A guided tour to approximate string matching. Improving an algorithm for approximate pattern matching. Very fast and simple approximate string matching. The theory and computation of evolutionary distances: pattern recognition. A very fast substring search algorithm. On using q-gram locations in approximate string matching Approximate Boyer-Moore string matching Algorithms for approximate string matching. Finding approximate patterns in strings. Approximate string matching with q-grams and maximal matches Approximate string matching using within-word parallelism Fast text searching allowing errors. --TR Algorithms for approximate string matching Fast parallel and serial approximate string matching Efficient text searching A very fast substring search algorithm An improved algorithm for approximate string matching A new approach to text searching Fast text searching Approximate string-matching with <italic>q</italic>-grams and maximal matches Approximate Boyer-Moore string matching Approximate string matching using within-word parallelism A comparison of approximate string matching algorithms Very fast and simple approximate string matching A fast string searching algorithm Efficient string matching A guided tour to approximate string matching Text-Retrieval Multiple Approximate String Matching Fast and Practical Approximate String Matching Theoretical and Empirical Comparisons of Approximate String Matching Algorithms Approximate Multiple Strings Search A Bit-Parallel Approach to Suffix Automata A Fast Bit-Vector Algorithm for Approximate String Matching Based on Dynamic Programming On Using q-Gram Locations in Approximate String Matching --CTR Atsuhiro Takasu, An approximate multi-word matching algorithm for robust document retrieval, Proceedings of the 15th ACM international conference on Information and knowledge management, November 06-11, 2006, Arlington, Virginia, USA Kimmo Fredriksson, On-line Approximate String Matching in Natural Language, Fundamenta Informaticae, v.72 n.4, p.453-466, December 2006 Kimmo Fredriksson , Gonzalo Navarro, Average-optimal single and multiple approximate string matching, Journal of Experimental Algorithmics (JEA), v.9 n.es, 2004 Josu Kuri , Gonzalo Navarro , Ludovic M, Fast Multipattern Search Algorithms for Intrusion Detection, Fundamenta Informaticae, v.56 n.1-2, p.23-49, January Josu Kuri , Gonzalo Navarro , Ludovic M, Fast multipattern search algorithms for intrusion detection, Fundamenta Informaticae, v.56 n.1,2, p.23-49, July Federica Mandreoli , Riccardo Martoglia , Paolo Tiberio, A syntactic approach for searching similarities within sentences, Proceedings of the eleventh international conference on Information and knowledge management, November 04-09, 2002, McLean, Virginia, USA

multipattern search;search allowing errors;string matching

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Web-conscious storage management for web proxies.

Many proxy servers are limited by their file I/O needs. Even when a proxy is configured with sufficient I/O hardware, the file system software often fails to provide the available bandwidth to the proxy processes. Although specialized file systems may offer a significant improvement and overcome these limitations, we believe that user-level disk management on top of industry-standard file systems can offer similar performance advantages. In this paper, we study the overheads associated with file I/O in web proxies, we investigate their underlying causes, and we propose Web-Conscious Storage Management, a set of techniques that exploit the unique reference characteristics of web-page accesses in order to allow web proxies to overcome file I/O limitations. Using realistic trace-driven simulations, we show that these techniques can improve the proxy's secondary storage I/O throughput by a factor of 15 over traditional open-source proxies, enabling a single disk to serve over 400 (URL-get) operations per second. We demonstrate our approach by implementing Foxy, a web proxy which incorporates our techniques. Experimental evaluation suggests that Foxy outperforms traditional proxies, such as SQUID, by more than a factor of four in throughput, without sacrificing response latency.

Introduction World Wide Web proxies are being increasingly used to provide Internet access to users behind a firewall and to reduce wide-area network traffic by caching frequently used URLs. Given that web traffic still increases exponentially, web proxies are one of the major mechanisms to reduce the overall traffic at the core of the Internet, protect network servers from traffic surges, and improve the end user experience. Today's typical web proxies usually run on UNIX-like operating systems and their associated file systems. While UNIX-like file systems are widely available and highly reliable, they often result in poor performance for web proxies. For instance, Rousskov and Soloviev [35] observed that disk delays in web proxies contribute about 30% toward total hit response time. Mogul [28] observed that the disk I/O overhead of caching turns out to be much higher than the latency improvement from cache hits at the web proxy at Digital Palo Alto firewall. Thus, to save the disk I/O overhead the proxy is typically run in non-caching mode. These observations should not be surprising, because UNIX-like file-systems are optimized for general-purpose workloads [27, 30, 21], while web proxies exhibit a distinctly different workload. For example, while read operations outnumber write operations in traditional UNIX-like file systems [3], web accesses induce a write- dominated workload [25]. Moreover, while several common files are frequently updated, most URLs are rarely (if ever at all) updated. In short, traditional UNIX-like file access patterns are different from web access patterns, and therefore, file systems optimized for UNIX-like workloads do not necessarily perform well for web-induced workloads. In this article we study the overheads associated with disk I/O for web proxies, and propose Web-Conscious Storage Management (WebCoSM), a set of techniques designed specifically for high performance. We show that the single most important source of overhead is associated with storing each URL in a separate file, and we propose methods to aggregate several URLs per file. Once we reduce file management overhead, we show that the next largest source of overhead is associated with disk head movements due to file write requests in widely scattered disk sectors. To improve write throughput, we propose a file space allocation algorithm inspired from log-structured file systems [34]. Once write operations proceed at maximum speed, URL read operations emerge A3 Cache Clients URL Response Request to Web Server URL Request Lookup Cache Web Server Proxy Server Response from [3]00111100110000011111 Figure 1: Typical Web Proxy Action Sequence. as the next largest source of overhead. To reduce the disk read overhead we identify and exploit the locality that exists in URL access patterns, and propose algorithms that cluster several read operations together, and reorganize the layout of the URLs on the file so that URLs accessed together are stored in nearby file locations. We demonstrate the applicability of our approach by implementing Foxy, a web proxy that incorporates our WebCoSM techniques that successfully remove the disk bottlenecks from the proxy's critical path. To evaluate the performance of our approach we use a combination of trace-driven simulation and experimental evaluation. Using simulations driven by real traces we show that the file I/O throughput can be improved by a factor of 18, enabling a single disk proxy to serve around 500 (URL-get) operations per second. Finally, we show that our implementation outperforms traditional proxies such as SQUID by more than a factor of 7 under heavy load. We begin by investigating the workload characteristics of web proxies, and their implications to the file system performance. Then, in section 2 we motivate and describe our WebCoSM techniques, and in section 3 we present comprehensive performance results that show the superior performance potential of these techniques. We verify the validity of the WebCoSM approach in section 4 by outlining the implementation of a proof-of-concept proxy server and by evaluating its performance through experiments. We then compare the findings of this paper to related research results in section 5, discuss issues regarding WebCoSM in section 6 and summarize our findings in section 7. Web-Conscious Storage Management Techniques Traditional web proxies frequently require a significant number of operations to serve each URL request. Consider a web proxy that serves a stream of requests originating from its clients. For each request it receives, the proxy has to look-up its cache for the particular URL. If the proxy does not have a local copy of the URL (cache miss), it requests the URL from the actual server, saves a copy of the data in its local storage, and delivers the data to the requesting client. If, on the other hand, the proxy has a local copy of the URL (cache hit), it reads the contents from its local storage, and delivers them to the client. This action sequence is illustrated in Figure 1. To simplify storage management, traditional proxies [42], [8] store each URL on a separate file, which induces several file system operations in order to serve each request. Indeed, at least three file operations are needed to serve each URL miss: (i) an old file (usually) needs to be deleted in order to make space for the new URL, (ii) a new file needs to be created in order to store the contents of this URL, and (iii) the new file needs to be filled with the URLs contents. Similarly, URL hits are costly operations, because, even though they do not change the file's data or meta-data, they invoke one file read operation, which is traditionally expensive due to disk head movements. Thus, for each URL request operation, and regardless of hit or miss, the file system is faced with an intensive stream of file operations. File operations are usually time consuming and can easily become a bottleneck for a busy web proxy. For instance, it takes more than 20 milliseconds to create even an empty file, as can be seen in Figure 2 1 . To make matters worse, it takes another 10 milliseconds to delete this empty file. This file creation and deletion cost approaches milliseconds for large files (10 Kbytes). Given that for each URL-miss the proxy should create and delete a file, the proxy will only be able to serve one URL miss every 60 milliseconds, or equivalently, less than To quantify performance limitations of file system operations, we experimented with the HBENCH-OS file system benchmark [6]. HBENCH-OS evaluates file creation/deletion cost by creating 64 files of a given size in the same directory and then deleting them in reverse- of-creation order. We run HBENCH-OS on an UltraSparc-1 running Solaris 5.6, for file sizes of 0 to 10 Kbytes. Our results, plotted in Figure 2, suggest that the time required to create and delete a file is very high. File Size (in Kbytes)1030Time per operation (in File Create File Delete Figure 2: File Management Overhead. The figure plots the cost of file creation and file deletion operations as measured by the HBENCH-OS (latfs). The benchmark creates 64 files and then deletes them in the order of creation. The same process is repeated for files of varying sizes. We see that the time to create a file is more than 20 msec, while the time to delete a file is between 10 and 20 msec. per second which provide clients with no more than 100-200 Kbytes of data. Such a throughput is two orders of magnitude smaller than most modern magnetic disks provide. This throughput is even smaller than most Internet connections. Consequently, the file system of a web-proxy will not be able to keep up with the proxy's Internet requests. This disparity between the file system's performance and the proxy's needs is due to the mismatch between the storage requirements needed by the web proxy and the storage guarantees provided by the file system. We address this semantic mismatch in two ways: meta-data overhead reduction, and data-locality exploitation. 2.1 Meta-Data Overhead Reduction Most of the meta-data overhead that cripples web proxy performance can be traced to the storage of each URL in a separate file. To eliminate this performance bottleneck we propose a novel URL-grouping method (called BUDDY), in which we store all the URLs into a small number of files. To simplify space management within each of these files, we use the URL size as the grouping criterion. That is, URLs that need one block of disk space (512 bytes) are all stored in the same file, URLs that need two blocks of disk space (1024 bytes) are all stored in another file, and so forth. In this way, each file is composed of fixed-sized slots, each large enough to contain a URL. Each new URL is stored in the first available slot of the appropriate file. The detailed behavior of BUDDY is as follows: Initially, BUDDY creates one file to store all URLs that are smaller than one block, another file to store all URLs that are larger than a block, but smaller than two, and so on. URLs larger than a predefined threshold are stored in separate files - one URL per file. ffl On a URL-write request for a given size, BUDDY finds the first free slot on the appropriate file, and stores the contents of the new URL there. If the size of the contents of the URL is above the threshold (128 Kbytes in most of our experiments), BUDDY creates a new file to store this URL only. ffl When a URL needs to be replaced, BUDDY marks the corresponding slot in the appropriate file as free. This slot will be reused at a later time to store another URL. ffl On a URL-read request, BUDDY finds the slot in the appropriate file and reads the content of the requested URL. The main advantage of BUDDY is that it practically eliminates the overhead of file creation/deletion operations by storing potentially thousands of URLs per file. The URLs that occupy a whole file of their own, are large enough and represent a tiny percentage of the total number of URLs, so that their file creation/deletion overhead is not noticeable overall. 2.2 Data-Locality Exploitation Although BUDDY reduces the file management overhead, it makes no special effort to layout data intelligently on disk in a way that improves write or read performance. However, our experience suggests that a significant amount of locality exists in the URL reference streams; identifying and exploiting this locality can result in large performance improvements. 2.2.1 Optimizing Write throughput Both traditional proxies and our proposed BUDDY technique write new URL data in several different files scattered all over the disk, possibly requiring a head movement operation for each URL write. The number of these head movements can be significantly reduced if we write the data to the disk in a log-structured fashion. Thus, instead of writing new data in some free space on the disk, we continually append data to it until we reach the end of the disk, in which case we continue from the beginning. This approach has been widely used in log-structured file systems [5, 17, 29, 38]. However, unlike previous implementations of log-structured file systems, we use a user-level log-structured file management, which achieves the effectiveness of log-structured file systems on top of commercial operating systems. Towards this end, we develop a file-space management algorithm (called STREAM) that (much-like log-structured file systems) streams write operations to the disk: The web proxy stores all URLs in a single file organized in slots of 512 bytes long. Each URL occupies an integer number of (usually) contiguous slots. URL- read operations read the appropriate portions of the file that correspond to the given URL. URL-write operations continue appending data to the file until the end of the file, in which case, new URL-write operations continue from the beginning of the file writing on free slots. URL-delete operations mark the space currently occupied the URL as free, so they can later be reused by future URL-write operations. 2.2.2 Improving Read Requests While STREAM does improve the performance of write operations, URL-read operations still suffer from disk seek and rotational overhead, because the disk head must move from the point it was writing data to disk to the point from where it should read the data. To make matters worse, once the read operation is completed, the head must move back to the point it was before the read operation and continue writing its data onto the disk. For this reason, each read operation within a stream of writes, induces two head movements: the first to move the head to the reading position, and the second to restore the head in the previous writing position, resulting in a ping-pong effect. To reduce this overhead, we have developed a LAZY-READS technique which extends STREAM so that it batches read operations. When a URL-read operation is issued, it is not serviced immediately, but instead, it is sent into an intermediate buffer. When the buffer fills up with read requests (or when a timeout period expires), the pending read requests are forwarded to the file system, sorted according to the position (in the file) of the data they want to read. Using LAZY-READS, a batch of N URL-read requests can be served with at most N movements. Figure 3 illustrates the movements of the heads before and after LAZY-READS. Although LAZY-READS appear to increase the latency of URL-read operations, a sub-second timeout period guarantees unnoticeable latency increase for read operations. 2.2.3 Preserving the Locality of the URL stream The URL requests that arrive at a web proxy exhibit a significant amount of spatial locality. For example, consider the case of an HTML page that contains several embedded images. Each time a client requests the HTML page, it will probably request all the embedded images as well. That is, the page and its embedded images are usually accessed together as if they were a single object. An indication of this relationship is that all these requests are sent to the proxy server sequentially within a short time interval. Unfortunately, these requests do not arrive Head Position Time Read Operations (6 head movements) Stream of Write Operations (a) Without LAZY-READS Head Position Time Read Operations (4 head movements) Stream of Write Operations (b) With LAZY-READS Figure 3: Disk Head Distance. Using the STREAM technique, the disk receives a stream of write requests to contiguous blocks interrupted only by read requests which cause the ping-pong effect. In Part (a) three read requests are responsible for six long head movements. In Part (b) the LAZY-READS technique services all three read requests together, with only four disk head movements. sequentially to the proxy server; instead, they arrive interleaved with requests from several other clients. Therefore, web objects requested contiguously by a single client, may be serviced and stored in the proxy's disk sub-system interleaved with web objects requested from totally unrelated clients. This interleaved disk layout may result in significant performance degradation because future accesses to each one of the related web objects may require a separate disk head movement. To make matters worse, this interleaving may result in serious disk fragmentation: when the related objects are evicted from the disk, they will leave behind a set of small, non-contiguous portions To recover the lost locality, we augmented the STREAM and LAZY-READS techniques with an extra level of buffers (called locality buffers) between the proxy server and the file system. Each locality buffer is associated with a web server, and accumulates objects that originate from that web server. Instead of immediately writing each web object in the next available disk block, the proxy server places the object into the locality buffer associated with its origin server. If no such locality buffer can be found, the proxy empties one of the locality buffers by flushing it to the disk, and creates a new association between the newly freed locality buffer and the object's web server. When a buffer fills-up, or is evicted, its contents are sent to the disk and are probably written in contiguous disk locations. Figure 4 outlines the operation of a proxy server augmented with locality buffers. The figure shows three clients, each requesting a different stream of web objects (A1-A3, B1-B2, and C1-C2). The requests arrive interleaved at the proxy server, which will forward them over the Internet to the appropriate web servers. Without locality buffers, the responses will be serviced and stored to the disk in an interleaved fashion. The introduction of the locality buffers groups the requested web objects according to their origin web server and stores the groups to the disk as contiguously as possible, reducing fragmentation and interleaving. Future read operations will benefit from the reduced interleaving through the use of prefetching techniques that will read multiple related URLs with a single disk I/O. Even future write operations will benefit from the reduced fragmentation, since they will be able to write more data on the disk in a contiguous fashion. 3 Simulation-based Evaluation We evaluate the disk I/O performance of web proxies using a combination of simulation and experimental evaluation. In the simulation study, we model a web proxy with a storage system organized as a two level cache: a main-memory and a disk cache. Using traces obtained from busy web proxies, we drive the two-level cache simulator, which in turn generates the necessary secondary storage requests. These requests are translated into file system operations and are sent to a Solaris UFS file system and from it, to an actual magnetic disk. Therefore, in our methodology we use simulation to identify cache read, write, and delete operations, which are then sent to a real storage system in order to accurately measure their execution time. Thus, our methodology combines the ease A3 A2 A1 A3 A2 C1 Proxy Server A3 Client Requests A3 Locality Buffers Magnetic2 Figure 4: Streaming into Locality Buffers. The sequences of client requests arrive interleaved at the proxy server. The proxy server groups the requested objects in the stream into the available locality buffers, and writes the rearranged stream to the disk. For the sake of simplicity we have omitted the forwarding of the client requests over the Internet to the appropriate web server, and the web server's responses. URL-delete File Space Simulator Simulation Disk Cache Simulation Main Memory Traces Disks Magnetic URL-write Disk Cache URL-read Main Memory File Access Proxy Traces File Access Calls mmap/write lseek, read, (Simulated) Figure 5: Evaluation Methodology. Traces from the DEC's web proxy are fed into a 512-Mbyte main memory LRU cache simulator. URLs that miss the main memory cache are fed into a 2-Gbyte disk LRU cache simulator. URLs that miss this second-level cache are assumed to be fetched from the Internet. These misses generate URL-write requests because once they fetch the URL from the Internet they save it on the disk. Second-level URL hits generate URL-read requests, since they read the contents of the URL from the disk. To make space for the newly arrived URLs, the LRU replacement policy deletes non-recently accessed URLs resulting in URL-delete requests. All URL-write, URL-read, and URL-delete requests are fed into a file space simulator which maps URLs into files (or blocks within files) and sends the appropriate calls to the file system. Request Type Number of Requests Percentage URL-read 42,085 4% URL-write 678,040 64% Main Memory Hits 336,081 32% Total 1,058,206 100% Table 1: Trace Statistics. of trace-driven simulation with the accuracy of experimental evaluation. 3.1 Our traces come from a busy web proxy located at Digital Equipment Corporation (ftp://ftp.digital.com/pub/DEC/traces/proxy/webtraces.html). We feed these traces to a 512 Mbyte-large first-level main memory cache simulator that uses the LRU replacement policy 2 . URL requests that hit in the main memory cache do not need to access the second-level (disk) cache. The remaining URL requests are fed into a second-level cache simulator, whose purpose is to generate a trace of URL-level disk requests: URL-read, URL-write, and URL-delete. URL-read requests are generated as a result of a second-level cache hit. Misses in the second-level cache are assumed to contact the appropriate web server over the Internet, and save the server's response in the disk generating a URL-write request. Finally, URL-delete requests are generated when the secondary storage runs out of space, and an LRU replacement algorithm is invoked to delete unused URLs. The generated trace of URL-read, URL-delete, and URL-write requests is sent to a file-space management simulator which forwards them to a Solaris UFS file system that reads, deletes, and writes the contents of URLs as requested. The file-space management simulator implements several secondary storage management policies, ranging from a simple "one URL per file" (SQUID-like), to STREAM, LAZY-READS, and LOCALITY-BUFFERS. The Solaris UFS system runs on top of an ULTRA-1 workstation running Solaris 5.6, equipped with a Seagate ST15150WC 4-Gbyte disk with 9 millisecond average latency, 7200 rotations per minute, on which we measured a maximum write throughput of 4.7 Mbytes per second. Figure 5 summarizes our methodology. In all our experiments, we feed the simulator pipeline with a trace of one million URL read, write, and delete requests that are generated by 1,058,206 URL-get requests. Table 1 summarizes the trace statistics. The performance metric we use, is the total completion time needed to serve all one million requests. This time is inversely proportional to the system's throughput (operations per second) and thus is a direct measure of it. If, for example, the completion time reported is 2000 seconds, then the throughput of the system is URL-get requests per second. Another commonly used performance metric of web proxies, especially important for the end user, is the service latency of each URL request. However, latency (by itself) can be a misleading performance metric for our work because significant relative differences in latency can be unnoticeable to the end user, if they amount to a small fraction of a second. For example, a proxy that achieves request latency may appear twice as good as a proxy that achieves 60 millisecond average request latency, but the end user will not perceive any difference. We advocate that, as long as the latency remains within unperceivable time ranges, the proxy's throughput is a more accurate measure of the system's performance. 3.2 Evaluation We start our experiments by investigating the performance cost of previous approaches that store one URL per file and comparing them with our proposed BUDDY that stores several URLs per file, grouped according to their size. We consider three such approaches, SINGLE-DIRECTORY, SQUID, and MULTIPLE-DIRS: ffl SINGLE-DIRECTORY, as the name implies, uses a single directory to store all the URL files. ffl SQUID (used by the SQUID proxy server) uses a two-level directory structure. The first level contains directories (named 0.F), each of which contains 256 sub-directories (named 00.FF). Files are stored in the second level directories in a round robin manner. Although more sophisticated policies than LRU have been proposed they do not influence our results noticeably. Number of URL requests 1:00:00 2:00:00 3:00:00 5:00:00 Completion time SQUID MULTIPLE-DIRS Figure File Management Overhead for Web Proxies. The figure plots the overhead of performing 300,000 URL-read/URL-write/URL-delete operations that were generated by 398,034 URL-get requests for a 1-Gbyte large disk. It is clear that BUDDY, improves the performance considerably compared to all other approaches. ffl MULTIPLE-DIRS creates one directory per server: all URLs that correspond to the same server are stored in the same directory. Our experimental results confirm that BUDDY improves performance by an order of magnitude compared to previous approaches. Indeed, as Figure 6 shows, BUDDY takes forty minutes to serve 300,000 URL requests, while the other approaches require from six to ten times more time to serve the same stream of URL requests. BUDDY is able to achieve such an impressive performance improvement because it does not create and delete files for URLs smaller than a predefined threshold. Choosing an appropriate threshold value can be important to the performance of BUDDY. A small threshold will result in frequent file create and delete operations, while a large threshold will require a large number of BUDDY files that may increase the complexity of their management. Figure 7 plots the completion time as a function of the threshold under the BUDDY management policy. We see that as the threshold increases, the completion time of BUDDY improves quickly, because an increasing number of URLs are stored in the same file, eliminating a significant number of file create and delete operations. When the threshold reaches 256 blocks (i.e. 128 Kbytes), we get (almost) the best performance. Further increases do not improve performance noticeably. URLs larger than 128 Kbytes should be given a file of their own. Such URLs are rare and large, so that the file creation/deletion overhead is not noticeable. 3.3 Optimizing Write Throughput Although BUDDY improves performance by an order of magnitude compared to traditional SQUID-like approaches, it still suffers from significant overhead because it writes data into several different files, requiring (potentially long) disk seek operations. Indeed, a snapshot of the disk head movements (shown in Figure 8 taken with TazTool [9]) reveals that the disk head traverses large distances to serve the proxy's write requests. We can easily see that the head moves frequently within a region that spans both the beginning of the disk (upper portion of the figure) and the end of the disk (lower portion of the figure). Despite the clustering that seems to appear at the lower quarter of the disk and could possibly indicate some locality of accesses, the lower portion of the graph that plots the average and maximum disk head distance, indicates frequent and long head movements. To eliminate the long head movements incurred by write operations in distant locations, STREAM stores all URLs in a single file and writes data to the file as contiguously as possible, much like log-structured file systems THRESHOLD (Kbytes)50.0150.0250.0Completion time (minutes) Figure 7: Performance of BUDDY as a function of threshold. The figure plots the completion time as a function of BUDDY's threshold parameter. The results suggest that URLs smaller than 64-128 Kbytes should be ``buddied'' together. URLs larger than that limit can be given a file of their own (one URL per file) without any (noticeable) performance penalty. do. Indeed, a snapshot of the disk head movements (Figure shows that STREAM accesses data on the disk mostly sequentially. The few scattered accesses (dots) that appear on the snapshot, are not frequent enough to undermine the sequential nature of the accesses. Although STREAM obviously achieves minimal disk head movements, this usually comes at the cost of extra disk space. Actually, to facilitate long sequential write operations, both log-structured file systems and STREAM never operate with a (nearly) full disk. It is not surprising for log-structured file systems to operate at a disk utilization factor of 60% or even less [34], because low disk utilization increases the clustering of free space, and allows more efficient sequential write operations 3 Fortunately, our experiments (shown in Figure 10) suggest that STREAM can operate very efficiently even at 70% disk utilization, outperforming BUDDY by more than a factor of two. As expected, when the disk utilization is high (90%-95%), the performance of BUDDY and STREAM are comparable. However, when the disk utilization decreases, the performance of STREAM improves rapidly. When we first evaluated the performance of STREAM, we noticed that even when there was always free disk space available and even in the absence of read operations, STREAM did not write to disk at maximum throughput. We traced the problem and found that we were experiencing a small-write performance problem: writing a small amount of data to the file system, usually resulted in both a disk-read and a disk-write operation. The reason for this peculiar behavior is the following: if a process writes a small amount of data in a file, the operating system will read the corresponding page from the disk (if it is not already in the main memory), perform the write in the main memory page, and then, at a later time, write the entire updated page to the disk. To reduce these unnecessary read operations incurred by small writes, we developed a packetized version of STREAM, STREAM-PACKETIZER, that works just like STREAM with the following difference: URL-write operations are not forwarded directly to the file system - instead they are accumulated into a page-boundary-aligned one-page-long packetizer buffer, as long as they are stored contiguously to the previous URL-write request. Once the packetizer fills up, or if the current request is not contiguous to the previous one, the packetizer is sent to the file system to be written to the disk. 3 Fortunately, recent measurements suggest that most file systems are about half-full on the average [11], and thus, log-structured approaches for file management may be more attractive than ever, especially at the embarrassingly decreasing cost of disk space [10]. Time Distance Disk Head Range Figure 8: Disk Access Pattern of BUDDY. This snapshot was taken with TazTool, a disk head position plotting Time Distance Disk Head Range Figure 9: Disk Access Pattern of STREAM. This snapshot was taken with TazTool, a disk head position plotting Figure 10: Performance of BUDDY and STREAM as a function of disk (space) utilization. The figure plots the completion time for serving 1,000,000 URL operations as a function of disk utilization. As expected, the performance of BUDDY is unaffected by the disk utilization, and the performance of STREAM improves as disk utilization decreases. When the disk utilization is around 70% STREAM outperforms BUDDY by more than a factor of two. STREAM-PACKETIZER Figure 11: Performance of STREAM and STREAM-PACKETIZER as a function of disk (space) utilization. The figure plots the completion time for serving 1,000,000 URL operations as a function of disk utilization. STREAM consistently outperforms STREAM-PACKETIZER, by as much as 20% for low disk utilizations. In this way, STREAM-PACKETIZER instead of sending a large number of small sequential write operations to the file system (like STREAM does), it sends fewer and larger (page size long) write operations to the file system. Figure 11 plots the performance of STREAM and STREAM-PACKETIZER as a function of disk utilization. STREAM-PACKETIZER performs consistently better than STREAM, by as much as 20% when disk utilization is low, serving one million requests in less than three thousand seconds, achieving a service rate of close to 350 URL-get operations per second. 3.4 Improving Read Requests While STREAM improves the performance of URL-write operations, URL-read operations still suffer from seek and rotational latency overhead. A first step towards improving the performance of read operations, LAZY- READS, reduce this overhead by clustering several read operations and by sending them to the disk together. This grouping of read requests not only reduces the disk head ping-pong effect but also presents the system with better opportunities for disk head scheduling. Figure 12 shows that LAZY-READS consistently improve the performance over STREAM-PACKETIZER by about 10% 4 . Although a 10% performance improvement may not seem impressive at first glance, we believe that the importance of LAZY-READS will increase in the near future. In our experimental environment, read requests represent a small percentage (a little over 6%) of the total disk operations. Therefore, 4 The careful reader will notice however, that LAZY-READS may increase operation latency. However, we advocate that such an increase will be unnoticeable by end-users. Our trace measurements show that STREAM-PACKETIZER augmented with LAZY-READS is able to serve 10-20 read requests per second (in addition) to the write requests. Thus LAZY-READS will delay the average read operation only by a fraction of the second. Given that the average web server latency may be several seconds long [2], LAZY-READS impose an unnoticeable overhead. To make sure that no user ever waits an unbounded amount of time to read a URL from the disk even in an unloaded system, LAZY-READS can also be augmented with a time out period. If the time out elapses then all the outstanding read operations are sent to disk. STREAM-PACKETIZER LAZY-READS Figure 12: Performance of LAZY-READS. The figure plots the completion time for serving 1,000,000 URL operations as a function of 2-Gbyte disk utilization. LAZY-READS gathers reads requests ten-at-a-time and issues them all at the same time to the disk reducing the disk head movements between the write stream and the data read. The figure shows that LAZY-READS improves the performance of STREAM-PACKETIZER by 10%. even significant improvements in the performance of read requests will not necessarily yield significant overall performance gains. With the increasing size of web caches, the expected hit rates will probably increase, and the percentage of disk read operations will become comparable to (if not higher than) the percentage of disk write operations. In this case, optimizing disk read performance through LAZY-READS or other similar techniques will be increasingly important. 3.5 Preserving the Locality of the URL stream 3.5.1 The Effects of Locality Buffers on Disk Access Patterns To improve the performance of disk I/O even further, our locality buffers policies (LAZY-READS-LOC and STREAM- improve the disk layout by grouping requests according to their origin web server before storing them to the disk. Without locality buffers, the available disk space tends to be fragmented and spread all over the disk. With locality buffers, the available disk space tends to be clustered in one large empty portion of the disk. Indeed, the two-dimensional disk block map in Figure 13(b), shows the available free space as a long white stripe. On the contrary, in the absence of locality buffers, free space tends to be littered with used blocks shown as black dots Figure 13(a)). Even when we magnify a mostly allocated portion of the disk (Figure 13 (a) and (b) right), small white flakes begin to appear within the mostly black areas, corresponding to small amounts of free disk space within large portions of allocated space. We see that locality buffers are able to cluster the white (free) space more effectively into sizable and square white patches, while in the absence of locality buffers, the free space if clustered into small and narrow white bands. Figure 14 confirms that locality buffers result in better clustering of free space, plotting the average size of Algorithm Performance URL-get operations per second Table 2: Performance of traditional and Web-Conscious Storage Management techniques (in URL-get operations per second). chunks of contiguous free space as a function of time. After the warm-up period (about 300 thousand requests), locality buffers manage to sustain an average free chunk size of about 145 Kbytes. On the contrary, the absence of locality buffers (STREAM) exhibits a fluctuating behavior with an average free chunk size of only about Kbytes. Locality buffers not only cluster the free space more effectively, they also populate the allocated space with clusters of related documents by gathering URLs originating from each web server into the same locality buffer, and in (probably) contiguous disk blocks. Thus, future read requests to these related web documents will probably access nearby disk locations. To quantify the effectiveness of this related object clustering, we measure the distance (in file blocks) between successive disk read requests. Our measurements suggest that when using locality buffers, a larger fraction of read requests access nearby disk locations. Actually, as many as l,885 read requests refer to the immediately next disk block to their previous read request, compared to only 611 read requests in the absence of locality buffers (as can be seen from Figure 15). Furthermore, locality buffers improve the clustering of disk read requests significantly: as many as 8,400 (17% of total) read requests fall within ten blocks of their previous read request, compared to only 3,200 (6% of total) read requests that fall within the same range for STREAM-PACKETIZER. We expect that this improved clustering of read requests that we observed, will eventually lead to performance improvements, possibly through the use of prefetching. 3.5.2 Performance Evaluation of LOCALITY BUFFERS Given the improved disk layout of LOCALITY BUFFERS, we expect that the performance of LAZY-READS-LOC to be superior to that of LAZY-READS and of STREAM-PACKETIZER. In our experiments we vary the number of locality buffers from 8 up to 128; each locality buffer is 64-Kbytes large. Figure 16 shows that as few as eight locality buffers (LAZY-READS-LOC-8) are sufficient to improve performance over LAZY-READS between 5% and 20%, depending on the disk utilization. However, as the number of locality buffers increases, the performance advantage of LAZY-READS-LOC increases even further. Actually, at 76% disk utilization, LAZY-READS-LOC with 128 locality buffers performs 2.5 times better than both LAZY-READS and STREAM-PACKETIZER. We summarize our performance results in Table 2 presenting the best achieved performance (measured in URL-get operations per second) for each of the studied techniques. We see that the Web-Conscious Storage Management techniques improve performance by more than an order of magnitude, serving close to 500 URL-get operations per second, on a single-disk system. Actually, in our experimental environment, there is little room for any further improvement. LAZY-READS-LOC-128 transfers 7.6 Gbytes of data (both to and from secondary storage) in 2,020 seconds, which corresponds to a sustained throughput of 3.7 Mbytes per second. Given that the disk used in our experiments can sustain a maximum write throughput of 4.7 Mbytes per second, we see that our WebCoSM techniques achieve up to 78% of the maximum (and practically unreachable) upper limit in performance. Therefore, any additional, more sophisticated techniques are not expected to result in significant performance improvements (at least in our experimental environment). (b) Figure 13: Disk Fragmentation Map. The Figure plots a two-dimensional disk block allocation map at the end of our simulations for STREAM (a) and LOCALITY BUFFERS (b). We plot allocated blocks with black dots and free blocks with white dots. The beginning of the disk is plotted at the lower left corner, the rest of the disk is plotted following a column-major order, and finally, the end of the disk is plotted at the top right corner. Average size of a Hole (Kbytes) Time (Thousands of URL requests) Locality Buffers Figure 14: Average size of free disk blocks. The Figure plots the average size of chunks of contiguous free space as a function of time.200600100014001800-30 -20 Number of Read Operations Distance from Previous Read Request (in File Blocks) Locality Buffers Figure 15: Distribution of distances for Read Requests. The Figure plots the histogram of the block distances between successive read operations for STREAM-PACKETIZER and STREAM-LOC (using 128 Locality Buffers). Disk Space Utilization (%)1000300050007000 Completion time STREAM-PACKETIZER LAZY-READS Figure Performance of LAZY-READS-LOC. The figure plots the completion time for serving 1,000,000 URL operations as a function of disk utilization. LAZY-READS-LOC attempts to put URLs from the same server in nearby disk locations by clustering them in locality buffers before sending them to the disk. Implementation To validate our Web-Conscious Storage Management approach, we implemented a lightweight user-level web proxy server called Foxy. Our goal in developing Foxy was to show that WebCoSM management techniques (i) can be easily implemented, (ii) can provide significant performance improvements, and (iii) require neither extensive tuning nor user involvement. Foxy consists of no more that 6,000 lines of C code, and implements the basic functionality of an HTTP web proxy. For simplicity and rapid prototyping, Foxy implements the HTTP protocol but not the not-so-frequently used protocols like FTP, ICP, etc. Foxy, by being first and foremost a caching proxy, uses a two level caching approach. The proxy's main memory holds the most frequently accessed documents, while the rest reside in the local disk. To provide an effective and transparent main memory cache, Foxy capitalizes on the existing file buffer cache of the operating system. The secondary-storage system management of Foxy stores all URLs in a single file, according to the STREAM- PACKETIZER policy: URLs are contiguously appended to the disk. When the disk utilization reaches a high watermark, a cache replacement daemon is invoked to reduce the utilization below a low watermark. The replacement policy used is LRU-TH [1], which replaces the least recently used documents from the cache. In order to prevent large documents from filling-up the cache, LRU-TH does not cache documents larger than a prespecified threshold. Foxy was developed on top of Solaris 5.7, and has also being tested on top of Linux 2.2. 4.1 Design of the Foxy Web Proxy The design and architecture of the Foxy Web Cache follows the sequence of operations the proxy must perform in order to serve a user request (mentioned here also as "proxy request" or "proxy connection"). Each user request is decomposed into a sequence of synchronous states that form a finite state machine (FSM) shown in Figure 17. Based on this FSM diagram, Foxy functions as a pipelined scheduler in an infinite loop. At every step of the loop, the scheduler selects a ready proxy request, performs the necessary operations, and advances the request into the next state. The rectangles in Figure 17 represent the various states that comprise the processing of a request, while the clouds represent important asynchronous actions taking place between two states. Finally, lines represent transitions between states and are annotated to indicate the operations that must be completed in order to transition Incomplete Transfer While Incomplete Transfer While New Client Connection Object IS NOT in Cache Lookup Object in Index Connection Failed (Retry) Receive WWW Object Parse HTTP Response Object IS in Cache Send the Object to Client Read Object from Cache Established HTTP Response Receiving Connection Close TCP Connection Connection or DNS Store (?) Object to Cache HTTP or TCP Finished Transfer Object Read Cache Read FINAL Sent to Client Object Received & message to client EXCEPTION Request Open TCP Connection to remote Web Server Object to Client Request Figure 17: The Foxy Processing Finite State Machine Diagram. Number of Clients 100 Number of Servers 4 Server Response Time 1 second Document Hit Rate 40% Request Distribution zipf(0.6) Document Size Distribution exp(5 Kbytes) Cacheable Objects 100% Table 3: The WebPolygraph Parameters Used in our Experiments. to occur. Every request begins at "STATE 1", where the HTTP request is read and parsed. Then, Foxy searches its index for the requested object. The index contains the metadata of the objects stored in the Foxy's Cache (e.g. name and size of each object, its storage location, the object's retrieval time, etc. If the search for the requested object is successful, Foxy issues a read request to the cache. When the object is read from the cache, a transition to "STATE 5" is made, where the object is returned to the client. After a successful transfer, the client TCP/IP connection is closed in "STATE 6", and the processing of the HTTP request is completed. If the search for the requested object is unsuccessful, Foxy, in "STATE 2", performs a DNS lookup for the IP address of the remote Web server and then opens a TCP/IP connection to it. When the TCP/IP connection is established, Foxy, in "STATE 3", sends an HTTP request to the origin web server. The server response is parsed in "STATE 4", and Foxy receives the object's content over the (possibly slow) Internet connection. Foxy passes the object through to the requesting client as it receives it from the remote server, until all the object's contents are transferred. Then, the Foxy uses a cache admission policy (LRU-TH) to decide whether this object should be cached, and is so, the object's content is stored in the (disk) cache using the STREAM-PACKETIZER algorithm, and its corresponding metadata are stored in the index. Finally, the TCP connection is closed in "STATE 6", and the processing of the HTTP request is completed. 4.2 Experiments To measure the performance of Foxy and compare it to that of SQUID, we used the Web Polygraph proxy performance benchmark (version 2.2.9), a de facto industry standard for benchmarking proxy servers [33]. We configured Web Polygraph with 100 clients and four web servers each responding with a simulated response time of one second. Table 3 summarizes the Web Polygraph configuration used in our experiments. We run Web Polygraph on a SUN Ultra 4500 machine with four UltraSparc-2 processors at 400MHz. The machine runs Solaris 5.7, and is equipped with a Seagate ST318203FSUN1-8G, 18-Gbyte disk. The secondary storage size that both SQUID and Foxy use is 8 Gbytes. We used the 2.2.STABLE4 version of SQUID, and we configured it according to the settings used in the Second Polygraph Bake-off [36]. To reduce the effects of a possibly slow network, we run all processes on the same computer. The results from our experiments are presented in Figures 18, 19, and 20. Figure plots the throughput of SQUID and Foxy, as a function of the client demands (input load) that ranges from 40 to 350 requests per second. We see that for small input load (less than 80 requests per second), the throughput of both proxies increases linearly. However, the throughput of SQUID quickly levels-off and decreases at 90 requests per second, while Foxy sustains the linear increase up to 340 requests per second, giving a factor of four improvement over SQUID. The deficiencies of SQUID are even more pronounced in Figure 19, which plots the average response time achieved by the two proxies as a function of the input load. We can easily see that SQUID's latency increases exponentially for input load higher than 50 requests per second. Thus, although Figure suggests that SQUID can achieve a throughput of 90 requests per second, this throughput comes at a steep increase of the request latency as seen by the end user (almost 8 seconds). The maximum throughput that SQUID can sustain without introducing a noticeable increase in latency is around 50 requests per second. On the contrary, Foxy manages to serve more than 340 requests per second without any noticeable increase in the latency. In fact, Foxy can serve up to 340 requests per second, with a user latency of about 0.7 seconds. Therefore, for the acceptable (sub-second) latency ranges, Foxy achieves almost 7 times higher throughput than SQUID. To make matters worse, SQUID not only increases the perceived end-user latency, but it also increases the network traffic required. In fact, when the disk sub-system becomes overloaded, SQUID, in an effort to off-load the disk, may forward URL requests to the origin server, even if it has a local copy in its disk cache. This behavior Observed Throughput (requests/second) Input Load (requests/second) FOXY SQUID Figure 18: Throughput of SQUID and FOXY.13579 Avg Response Time per Request (sec) Input Load (requests/second) SQUID FOXY Figure 19: Latency of SQUID and FOXY. Network Bandwidth (KBytes/Sec) Input Load (requests/second) SQUID FOXY Figure 20: Network Bandwidth Requested by SQUID and FOXY. may increase network traffic significantly. As Figure 20 shows when the input load is less than 50 requests per second, both SQUID and Foxy require the same network bandwidth. When the input load increases beyond 50 requests per second, the network bandwidth required by Foxy increases linearly with the input load as expected. However, the network bandwidth required by SQUID increases at a higher rate, inducing 45% more network traffic than Foxy at 110 URL requests per second. 5 Previous Work Caching is being extensively used on the web. Most web browsers cache documents in main memory or in local disk. Although this is the most widely used form of web caching, it rarely results in high hit rates, since the browser caches are typically small and are accessed by a single user only [1]. To further improve the effectiveness of caching, proxies are being widely used at strategic places of the Internet [8, 42]. On behalf of their users, caching proxies request URLs from web servers, store them in a local disk, and serve future requests from their cache whenever possible. Since even large caches may eventually fill up, cache replacement policies have been the subject of intensive research [1, 7, 23, 26, 32, 37, 43, 44]. Most of the proposed caching mechanisms improve the user experience, reduce the overall traffic, and protect network servers from traffic surges [15]. To improve hit rates even further and enhance the overall user experience, some proxies may even employ intelligent prefetching methods [4, 12, 16, 41, 31, 40]. As the Internet traffic grew larger, it was realized that Internet servers were bottlenecked by their disk subsystem. Fritchie found that USENET news servers spend a significant amount of time storing news articles in files "one file per article" [14]. To reduce this overhead he proposes to store several articles per file and to manage each file as a cyclic buffer. His implementation shows that storing several news articles per file results in significant performance improvement. Much like news servers, web proxies also spend a significant percentage of their time performing disk I/O. Rousskov and Soloviev observed that disk delays contribute as much as 30% towards total hit response time [35]. Mogul suggests that disk I/O overhead of disk caching turns out to be much higher than the latency improvement from cache hits [28]. Thus, to save the disk I/O overhead the proxy is typically run in its non-caching mode [28]. To reduce disk I/O overhead, Soloviev and Yahin suggest that proxies should have several disks [39] in order to distribute the load among them, and that each disk should have several partitions in order to localize accesses to related data. Unfortunately, Almeida and Cao [2] suggest that adding several disks to exiting traditional web proxies usually offers little (if any) performance improvement. Maltzahn, Richardson and Grunwald [24] measured the performance of web proxies and found that the disk subsystem is required to perform a large number of requests for each URL accessed and thus it can easily become the bottleneck. In their subsequent work they propose two methods to reduce disk I/O for web proxies [25]: ffl they store URLs of the same origin web server in the same proxy directory (SQUIDL), and ffl they use a single file to store all URLs less than 8 Kbytes in size (SQUIDM). Although our work and [25] shares common goals and approaches toward reducing disk meta-data accesses in web proxies, our work presents a clear contribution towards improving both data and meta-data access overhead: ffl We propose and evaluate STREAM and STREAM-PACKETIZER, two file-space management algorithms that (much like log-structured file systems) optimize write performance by writing data contiguously on the disk. ffl We propose and evaluate LAZY-READS and LAZY-READS-LOC, two methods that reduce disk seek overhead associated with read operations. The performance results reported both in [25] and this paper, verify that meta-data reduction methods improve performance significantly. For example, Maltzahn et al. report that on a Digital AlphaStation 250 4/266 with 512 Mbytes of RAM and three magnetic disks, SQUID is able to serve around 50 URL-get requests per second, and their best performing SQUIDMLA approach is able to serve around 150 requests per second. Similarly, on our Sun Ultra-1 at 166 MHz with 384 Mbytes RAM and a single magnetic disk, SQUID is able to serve around 27 URL- get requests per second, while BUDDY, the simplest WebCoSM technique that reduces only meta-data overhead, achieves around 133 requests per second. Furthermore, the remaining WebCoSM techniques that improve not only meta-data, but also the data accesses, are able to achieve close to 500 URL-get requests per second. Most web proxies have been implemented as user-level processes on top of commodity (albeit state-of-the- art) file-systems. Some other web proxies were built on top of custom-made file systems or operating systems. NetCache was build on top of WAFL, a file system that improves the performance of write operations [18]. Inktomi's traffic server uses UNIX raw devices [20]. CacheFlow has developedCacheOS, a special-purpose operating system for proxies [19]. Similarly, Novell has developed a special-purpose system for storing URLs: the Cache Object Store [22]. Unfortunately very little information has been published about the details and performance of such custom-made web proxies, and thus a direct quantitative comparison between our approach and theirs is very difficult. Although custom-made operating systems and enhanced file-systems can offer significant performance improvements, we choose to explore the approach of running a web proxy as a user-application on top of a commodity UNIX-like operating system. We believe that our approach will result in lower software development and maintenance costs, easier deployment and use, and therefore a quicker and wider acceptance of web proxies. The main contributions of this article are: ffl We study the overheads associated with file I/O in web proxies, investigate their underlying causes, and propose WebCoSM, a set of new techniques that overcome file I/O limitations. ffl We identify locality patterns that exist in web accesses, show that web proxies destroy these patterns, and propose novel mechanisms that restore and exploit them. ffl The applicability of our approach is shown through Foxy, a user-level web proxy implementation on top of a commercial UNIX operating system. ffl Comprehensiveperformancemeasurements using both simulation and experimental evaluation show that our approach can easily achieve an order of magnitude performance improvement over traditional approaches. 6 Discussion 6.1 Reliability Issues Although SQUID-like policies that store one URL per file do not perform well, they appear to be more robust in case of a system crash than our WebCoSM approach, because they capitalize on the expensive reliability provided by the file system. For example, after a system crash, SQUID can scan all the directories and files in its disk cache, and recreate a complete index of the cached objects. On the contrary, WebCoSM methods store meta-data associated with each URL in main memory for potentially long periods of time, increasing the cache's vulnerability to system crashes. For example, STREAM stores all URLs in a single file, which contains no indication of the beginning block of each URL. This information is stored in main memory, and can be lost in a system crash. Fortunately, the seemingly lost reliability does not pose any real threat to the system's operation: ffl WebCoSM methods can periodically (i.e. every few minutes) write their metadata information on safe storage, so that in the case of a crash they will only lose the work of the last few minutes. Alternatively, they can store along with each URL, its name, size, and disk blocks. In case of a crash, after the system reboots, the disk can be scanned, and the information about which URLs exist on the disk can be recovered. ffl Even if a few cached documents are lost due to a crash, they can be easily retrieved from the web server where they permanently reside. Thus, a system crash does not lose information permanently; it just loses the local copy of some data (i.e. a few minutes worth) which can be easily retrieved from the web again. 6.2 Lessons Learned During the course of this research we were called to understand the intricate behavior of a web proxy system. Some of the most interesting lessons that we learned include: ffl User requests may invoke counter-intuitive operating system actions. For example, we observed that small write requests in STREAM surprisingly invoked disk read operations. In this case, there is no intuitive and direct correspondence between what the user requests and what the operating system actually does. This mismatch between the user requests and the operating system actions not only hurts performance, but also undermines the user's understanding of the system. ffl System bottlenecks may appear in places least expected. It is a popular belief that proxies are bottlenecked by their network subsystem. On the contrary, we found that the secondary storage management system is also a major (if not more significant) bottleneck because it has not been designed to operate efficiently with web workloads. For example, traditional file systems used to serve no more than 10 concurrent users, requesting no more than 50 Kbytes per second each [3]. On the contrary, a busy web proxy (especially a country-wide proxy), may be required to serve hundreds of concurrent users requesting data totaling several Mbytes per second [13]. ffl Optimizing the performance of read operations will be one of the major factors in the design of secondary storage management systems of web proxies. Write operations can usually be implemented efficiently and proceed at disk bandwidth. On the contrary, read operations (even asynchronous ones), involve expensive disk seek and rotational latencies, which are difficult if not impossible to avoid. As disk bandwidth improves much faster than disk latency, read operations will become an increasing performance bottleneck. ffl Locality can manifest itself even when not expected. We found that clients exhibit a significant amount spatial locality, requesting sequences of related URLs. Traditional proxies tend to destroy this locality, by interleaving requests arriving from different clients. Identifying and exploiting the existing locality in the URL stream are challenging tasks that should be continuously pursued. Summary -Conclusions In this paper we study the disk I/O overhead of world-wide web proxy servers. Using a combination of trace-driven simulation and experimental evaluation, we show that busy web proxies are bottlenecked by their secondary storage management subsystem. To overcome this limitation, we propose WebCoSM, a set of techniques tailored for the management of web proxy secondary storage. Based on our experience in designing, implementing, and evaluating WebCoSM, we conclude: ffl The single largest source of overhead in traditional web proxies is the file creation and file deletion costs associated with storing each URL on a separate file. Relaxing this one-to-one mapping between URLs and files, improves performance by an order of magnitude. ffl Web clients exhibit a locality of reference in their accesses because they usually access URLs in clusters. By interleaving requests from several clients, web proxies tend to conceal this locality. Restoring the locality in the reference stream results in better layout of URLs on the disk, reduces fragmentation, and improves performance by at least 30%. ffl Managing the mapping between URLs and files in user level improves performance over traditional web proxies by a factor of 20 overall, leaving little room for improvement by specialized kernel-level implementations We believe that our results are significant today and they will be even more significant in the future. As disk bandwidth improves at a much higher rate than disk latency for more that two decades now [10], methods like WebCoSM that reduce disk head movements and stream data to disk will result in increasingly larger performance improvements. Furthermore, web-conscious storage management methods will not only result in better performance, but they will also help to expose areas for further research in discovering and exploiting the locality in the Web. Acknowledgments This work was supported in part by the Institute of Computer Science of Foundation for Research and Technology -Hellas, in part by the University of Crete through project "File Systems for Web servers" (1200). We deeply appreciate this financial support. Panos Tsirigotis was a source of inspiration and of many useful comments. Manolis Marazakis and George Dramitinos gave us useful comments in earlier versions of this paper. Katia Obraczka provided useful comments in an earlier version of the paper. P. Cao provided one of the simulators used. We thank them all. --R Caching Proxies: Limitations and Potentials. Measuring Proxy Performance with the Wisconsin Proxy Benchmark. Measurements of a Distributed File System. Speculative Data Dissemination and Service to Reduce Server Load Heuristic Cleaning Algorithms for Log-Structured File Systems Operating System Benchmarking in the Wake of Lmbench: A Case Study of the Performance of NetBSD on the Intel x86 Architecture. A Hierarchical Internet Object Cache. tracing revisited. Serverless Network File Systems. A Large-Scale Study of File System Contents Prefetching Hyperlinks. The Measured Access Characteristics of World-Wide-Web Client Proxy Caches The Cyclic News Filesystem: Getting INN To Do More With Less. The Global Internet Project. The Zebra Striped Network File System. File System Design for an NFS File Server Appliance. Cache Flow Inc. Inktomi Inc. Workload Requirements for a Very High-Capacity Proxy Cache Design Replacement Policies for a Proxy Cache Performance Issues of Enterprise Level Web Proxies. Reducing the Disk I/O of Web Proxy Server Caches. Main Memory Caching of Web Documents. A Fast File System for UNIX. Speedier Squid: A Case Study of an Internet Server Performance Problem. Caching in the Sprite Network File System. A Trace-Driven Analysis of the UNIX 4.2 BSD File System Using Predictive Prefetching to Improve World Wide Web Latency. A Simple Web Polygraph. The Design and Implementation of a Log-Structured File System On Performance of Caching Proxies. The Second IRCache Web Cache Bake-off A Case for Delay-Conscious Caching of Web Documents An Implementation of a Log-Structured File System for UNIX File Placement in a Web Cache Server. Defining High Speed Protocols Fast World-Wide Web Browsing Over Low-Bandwidth Links Squid Internet Object Cache Removal Policies in Network Caches for World-Wide Web Documents Proxy Caching that Estimates Page Load Delays. --TR The design and implementation of a log-structured file system Measurements of a distributed file system The Zebra striped network file system Disk-directed I/O for MIMD multiprocessors Removal policies in network caches for World-Wide Web documents Performance issues of enterprise level web proxies Operating system benchmarking in the wake of <italic>lmbench</italic> On performance of caching proxies (extended abstract) Proxy caching that estimates page load delays Measuring proxy performance with the Wisconsin Proxy benchmark Let''s put NetApp and CacheFlow out of business! The Cyclic News Filesystem Storage Management for Web Proxies Efficient Algorithms for Persistent Storage Allocation Serverless network file systems --CTR Abdolreza Abhari , Sivarama P. Dandamudi , Shikharesh Majumdar, Web object-based storage management in proxy caches, Future Generation Computer Systems, v.22 n.1, p.16-31, January 2006

web performance;web caching;web proxies;secondary storage

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Multicast-based inference of network-internal delay distributions.

Packet delay greatly influences the overall performance of network applications. It is therefore important to identify causes and locations of delay performance degradation within a network. Existing techniques, largely based on end-to-end delay measurements of unicast traffic, are well suited to monitor and characterize the behavior of particular end-to-end paths. Within these approaches, however, it is not clear how to apportion the variable component of end-to-end delay as queueing delay at each link along a path. Moreover, there are issues of scalability for large networks.In this paper, we show how end-to-end measurements of multicast traffic can be used to infer the packet delay distribution and utilization on each link of a logical multicast tree. The idea, recently introduced in [3] and [4], is to exploit the inherent correlation between multicast observations to infer performance of paths between branch points in a tree spanning a multicast source and its receivers. The method does not depend on cooperation from intervening network elements; because of the bandwidth efficiency of multicast traffic, it is suitable for large-scale measurements of both end-to-end and internal network dynamics. We establish desirable statistical properties of the estimator, namely consistency and asymptotic normality. We evaluate the estimator through simulation and observe that it is robust with respect to moderate violations of the underlying model.

Introduction Background and Motivation. Monitoring the performance of large communications networks is essential for diagnosing the causes of performance degradation. There are two broad approaches to monitoring. In the internal approach, direct measurements are made at or between network elements, e.g. of packet loss or delay. In the external approach, measurements are made across a network on end-to-end or edge-to-edge paths. The internal approach has a number of potential limitations. Due to the commercial sensitivity of performance measurements, and the potential load incurred by the measurement process, it is expected that measurement access to network elements will be limited to service providers and, possibly, selected peers and users. The internal approach assumes sufficient coverage, i.e. that measurements can be performed at all relevant elements on paths of interest. In practice, not all elements may possess the required functionality, or it may be disabled at heavily utilized elements in order reduce CPU load. On the other hand, arranging for complete coverage of larger networks raises issues of scale, both in the in the gathering of measurement data, and joining data collected from a large number of elements in order to form a composite view of end-to-end performance. This motivates external approaches, network diagnosis through end-to-end measurements, without necessarily assuming the cooperation of network elements on the path. There has been much recent experimental work to understand the phenomenology of end-to-end performance (e.g., see [3, 9, 19, 26, 27, 29]). Several research efforts are working on the developments of measurement infrastructure projects (Felix [13], IPMA [15], NIMI [18] and Surveyor [35]) with the aim to collect and analyze end-to-end measurements across a mesh of paths between a number of hosts. Standard diagnostic tools for IP networks, ping and traceroute report roundtrip loss and de- lay, the latter incrementally along the IP path by manipulating the time-to-live (TTL) field of probe packets. A recent refinement of this approach, pathchar [17], estimates hop-by-hop link capac- ities, packet delay and loss rates. pathchar is still under evaluation; initial experience indicates many packets are required for inference leading to either high load of measurement traffic or long measurement intervals, although adaptive approaches can reduce this [10]. More broadly, measurement approaches based on TTL expiry require the cooperation of network elements in returning Internet Control Message Protocol (ICMP) messages. Finally, the success of active measurement approaches to performance diagnosis may itself cause increased congestion if intensive probing techniques are widely adopted. In response to some of these concerns, a multicast-based approach to active measurement has been proposed recently in [4, 5]. The idea behind the approach is that correlation in performance seen on intersecting end-to-end paths can be used to draw inferences about the performance characteristics of the common portion (the intersection) of the paths, without the cooperation of network elements on the path. Multicast traffic is particular well suited for this since a given packet only occurs once on a given link in the (logical) multicast tree. Thus characteristics such as loss and end-to-end delay of a given multicast packet as seen at different endpoints are highly correlated. Another advantage of using multicast traffic is scalability. Suppose packets are exchanged on a mesh of paths between a collection of N measurement hosts stationed in a network. If the packets are unicast, then the load on the network may grow proportionally to N 2 in some parts of the network, depending on the topology. For multicast traffic the load grows proportionally only to N . Contribution The work of [4, 5] showed how multicast end-to-to measurements can be used to per link loss rates in a logical multicast tree. In this paper we extend this approach to infer the probability distribution of the per link variable delay. Thus we are not concerned with propagation delay on a link, but rather the distribution of the additional variable delay that is attributable to either queuing in buffers or other processing in the router. A key part of the method is an analysis that relates the probabilities of certain events visible from end-to-end measurements (end-to-end delays) to the events of interest in the interior of the network (per-link delays). Once this relation is known, we can estimate the delay distribution on each link from the measured distributions of end-to-end delays of multicast packets. For a glimpse of how the relations between end-to-end delay and per link delays could be found, consider a multicast tree spanning a source of multicast probes (identified as the root of the tree) and a set of receivers (one at each leaf of the tree). We assume the packets are potentially subject to queuing delay and even loss at each link. Focus on a particular node k in the interior of the tree. If, for a given packet, the source-to-leaf delay does not exceed a given value on any leaf descended from k, then clearly the delay from the root to the node k was less than that value. The stated desired relation between the distributions of per-link and source-to-leaf delays is obtained by a careful enumeration of the different ways in which end-to-end delay can be split between the portion of the path above or below the node in question, together with the assumption that per-link delays are independent between different links and packets. We shall comment later upon the robustness of our method to violation of this independence assumption. We model link delay by non-parametric discrete distributions. The choice of non parametric distributions rather than a parameterized delay model is dictated by the lack of knowledge of the distribution of link delays in networks. While there is significant prior work on the analysis and characterization of end-to-end delay behavior (see [2, 24, 27]), to the best of our knowledge there is no general model for per link delays. The use of a non-parametric model provides the flexibility to capture broadly different delay distributions, albeit at the cost of increasing the number of quantities to estimate (i.e. the weights in the discrete distribution). Indeed, we believe that our inference technique can shed light on the behavior and dynamics of per link delays and so provide useful results for the analysis and modeling; this we will consider in future work. The discrete distribution can be a regarded as binned or discretized version of the (possibly continuous) true delay distribution. Use of a discrete rather than a continuous distribution allows us to perform the calculations for inference using only algebra. Formally, there is no difficulty in formulating a continuous version of the inference algorithm. However, it proceeds via inversion of Laplace transforms, a procedure that is in practice implemented numerically. In the discrete approach we can explicitly trade-off the detail of the distribution with the cost of calculation; the cost is inversely proportional to the bin widths of the discrete distribution. The principle results of the analysis are as follows. Based on the independent delay model, we derive an algorithm to estimate the per link discrete delay distributions and utilization from the measured end-to-end delay distributions. We investigate the statistical properties of the estimator, and show it to be strongly consistent, i.e., it converges to the true distribution as the number of probes grows to infinity. We show that the estimator is asymptotically normal; this allows us to compute the rate of convergence of the estimator to its true value, and to construct confidence intervals for the estimated distribution for a given number of probes. This is important because the presence of large scale routing fluctuation (e.g. as seen in the Internet; see [26]) sets a timescale within which measurement must be completed, and hence the accuracy that can be obtained when sending probes at a given rate. We evaluated our approach through extensive simulation in two different settings. The first set used a model simulation in which packet delays obey the independence assumption of the model. We applied the inference algorithm to the end-to-end delays generated in the simulation and compared the (true) model delay distribution. We verified the convergence to the model distribution, and also the rate of convergence, as the number of probes increased. In the second set of experiments we conducted an ns simulation of packets on a multicast tree. Packet delays and losses were entirely due to queueing and packet discard mechanisms, rather than model driven. The bulk of the traffic in the simulations was background traffic due to TCP and UDP traffic sources; we compared the actual and predicted delay distributions for the probe traffic. Here we found rapid convergence, although with some persistent differences with respect to the actual distributions. These differences appear to be caused by violation of the model due to the presence of spatial dependence (i.e., dependence between delays on different links). In our simulations we find that when this type of dependence occurs, it is usually between the delays on child and parent links. However, it can extend to entire paths. As far as we know there are no experimental results concerning the magnitude of such dependence in real networks. In any case, by explicitly introducing spatial correlations into the model simulations, we were able to show that small violations of the independence assumption lead to only small inaccuracies of the estimated distribution. This continuity property of the deformation in inference due to correlations is also to be expected on theoretical grounds. We also verified the presence of temporal dependence, i.e., dependence between the delays between successive probes on the same link. This is to be expected from the phenomenology of queueing: when a node is idle, many consecutive probes can experience constant delay; during congestion, probes can experience the same delay if their interarrival time is smaller than the congestion timescale. This poses no difficulty as all that is required for consistency of the estimator is ergodicity of the delay process, a far weaker assumption than independence. However, dependence can decrease the rate of convergence of the estimators. In our experiments, inferred values closely tracked the actual ones despite the presence of temporal dependence. Implementation Requirements Since the data for delay inference comprises one-way packet delays, the primary requirement is the deployment of measurement hosts with synchronized clocks. Global Positioning System (GPS) systems afford one way to achieve a synchronization to within tenths of microseconds; it is currently used or planned in several of the measurement infrastructures mentioned earlier. More widely deployed is the Network Time Protocol (NTP) [20]. However, this provides accuracy only on the order of milliseconds at best, a resolution at least as coarse as the queueing delays in practice. An alternative approach that could supplement delay measurement from unsynchronized or coarsely synchronized clocks has been developed in [28, 30, 21]. These authors propose algorithms to detect clock adjustments and rate mismatches and to calibrate the delay measurements. Another requirement is knowledge of the multicast topology. There is a multicast-based measurement tool, mtrace [23], already in use in the Internet. mtrace reports the route from a multicast source to a receiver, along with other information about that path such as per-hop loss and rate. Presently it does not support delay measurements. A potential drawback for larger topologies is that mtrace does not scale to large numbers of receivers as it needs to run once for each receiver to cover the entire multicast tree. In addition, mtrace relies on multicast routers responding to explicit measurement queries; a feature that can be administratively disabled. An alternative approach that is closely related to the work on multicast-based loss inference [4, 5] is to infer the logical multicast topology directly from measured probe statistics; see [31] and [7]. This method does not require cooperation from the network. Structure of the Paper. The remaining sections of the paper are organized as follows. In Section 2 we describe the delay model and in Section 3 we derive the delay estimator. In Section 4 we describe the algorithm used to compute the estimator from data. In Section 5 we present the model and network simulations used to evaluate our approach. Section 6 concludes the paper. Model & Framework 2.1 Description of the Logical Multicast Tree We identify the physical multicast tree as comprising actual network elements (the nodes) and the communication links than join them. The logical multicast tree comprises the branch points of the physical tree, and the logical links between them. The logical links comprise one or more physical links. Thus each node in the logical tree, except for the leaf nodes and possibly the root, must have 2 or more children. We can construct the logical tree from the physical tree by deleting all links with one child (except for the root) and adjusting the links accordingly by directly joining its parent and child. denote the logical multicast tree, consisting of the set of nodes V , including the source and receivers, and the set of links L, which are ordered pairs (j; of nodes, indicating a link from j to k. We will denote f0g. The set of children of node j is denoted by these are the nodes whose parent is j. Nodes are said to be siblings if they have the same parent. For each node j, other than the root 0, there is a unique node f(j), the parent of j, such that (f(j); Each link can therefore be also identified by its "child" endpoint. We shall define f n (k) recursively by f n We say that j is a descendant of the corresponding partial order in V as j OE k. For each node j we define its level '(j) to be the non-negative integer such that f '(j) root represents the source of the probes and the set of leaf nodes R ae V (i.e., those with no children) represents the receivers. 2.2 Modeling Delay and Loss of Probe Packets Probe packets are sent down the tree from the root node 0. Each probe that arrives at node k results in a copy being sent to every child of k. We associate with each node k a random variable D k taking values in the extended positive real line R+ [ f1g. By convention D is the random delay that would be encountered by a packet attempting to traverse the link (f(k); L. The value indicates that the packet is lost on the link. We assume that the D k are independent. The delay experienced on the path from the root 0 to a node k is Y . Note that Y i.e. if the packet was lost on some link between node 0 and k. Unless otherwise stated, we will discretize each link delay D k to a set f0; q; Here q is the bin width, is the number of bins, and the point 1 is interpreted as "packet lost" or "encountered delay greater than i max q". The distribution of D k is denoted by ff k , where the probability that D 1. For each link, we denote u k the link utilization; then, u (0), the probability that a packet experience delay or it is lost in traversing link k. For each k 2 V , the cumulative delay process Y k , k 2 V , takes values in f0; q; i.e., it supports addition in the ranges of the constituent D j . We set A k A k (1) the probability that Y 1. Because of delay independence, for finite i, A k by convention A 0 We consider only canonical delay trees. A delay tree consists of the pair (T ; ff), delay tree is said to be canonical if ff k (0) ? 0, 8k 2 U , i.e., if there is a non-zero probability that a probe experiences no delay in traversing each link. 3 Delay Distribution Estimator and its Properties Consider an experiment in which n probes are sent from the source node down the multicast tree. As result of the experiment we collect the set of source-to-leaf delays (Y k;l ) k2R;l=1;:::;n . Our goal is to infer the internal delay characteristics solely from the collected end-to-end measurements. In this section we state the main analytic results on which inference is based. In Section 3.1 we establish the key property underpinning our delay distribution estimator, namely the one-to- one correspondence between the link delay distributions and the probabilities of a well defined set of observable events. Applying this correspondence to measured leaf delays allows us to obtain an estimate of the link delay distribution. We show that the estimator is strongly consistent and asymptotically normal. In Section 3.2 we present the proof of the main result which also provides the construction of the algorithm to compute the estimator we present in Section 4. In Section 3.4 we analyze the rate of convergence of the estimator as the number of probes increase. 3.1 The Delay Distribution Estimator denote the subtree rooted at node k and of receivers which descend from k. denote the event fmin j2R(k) Y j iqg that the end-to-end delay is no greater than iq for at least least one receiver in R(k) . Let fl k P[\Omega k (i)] denote its probability. Finally let \Gamma denote the mapping associating the link distributions k2U;i2f0;:::;imaxg to the probabilities of the . The proof of the next result is given in the following section. Theorem 1 Let \Gamma(ff)g. \Gamma is a bijection from A to G which is continuously differentiable and has a continuously differentiable inverse. Estimate fl by the empirical probabilities bfl , where denotes the indicator function of the set S and ( b are the subsidiary quantities Our estimate of ff k (i) is b (i). We estimate link k utilization by b Let A denote the open interior of A. The following holds: Theorem 2 When almost surely to ff, i.e., the estimator is strongly consistent. is continuous on \Gamma(A (1) ) and A (1) is open in A, it follows that \Gamma(A (1) ) is an open set in \Gamma(A). By the Strong Law of large numbers, since bfl is the mean of n independent random variables, bfl converges to fl almost surely for n !1. Therefore, when exists n 0 such that bfl 2 \Gamma(A (1) Then, the continuity of \Gamma \Gamma1 insures that b ff converges almost surely to ff as n !1. 3.2 Proof of Theorem 1 To prove the Theorem, we first express fl as function of ff and then show that the mapping from A to G is injective. 3.2.1 Relating fl to ff Denote obeys the recursion Then, by observing that readily obtain The set of equations (5) completely identifies the mapping \Gamma from A to G. The mapping is clearly continuously differentiable. Observe that the above expressions can be regarded as a generalization of those derived for the loss estimator in [4] (by identifying the event no delay with the event no loss). 3.2.2 Relating ff to fl It remains to show that the mapping from A to G is injective. To this end, below we derive an algorithm for inverting (5). We postpone to Appendix A the proof that the inverse is unique and continuously differentiable. For sake of clarity we separate the algorithm into two parts: in the first we derive the cumulative delay distributions A from fl; then, we deconvolve A to obtain ff. Computing A Step 0: Solve (5) for amounts solving the equation Y and This equation is formally identical to the one of the loss estimator [4]. From [4], we have that the solution of (6) exists and is unique in (0; 1) provided that which holds for canonical delay trees. We then compute fi k Step i: Given A k (j) and fi k (j), k 2 U , 1, in this step we compute A k (i) and fi k (i), . For k 2 U n R, in expression (5) we replace fi d (i) with fl d (i)\Gamma A k (0) (from (4)) and obtain the following equation ae Q A k (0) oe (the unknown term A k (i) is highlighted in boldface). This is a polynomial in A k (i) of degree #d(k). As shown in Appendix A we consider the second largest solution of (8). For directly compute A k (i) from (5), A k (j). Then we compute A f(k) (0) Computing ff Once step i max is completed, we compute ff k (i), k 2 U as follows A k (0) A A k (i)\Gamma A 3.3 Example: the Two-leaf Tree In this section we illustrate the application of the results of Section 3.1 to the two-leaf tree of Figure 1. We assume that on each link, a probe either suffers no delay, a unit amount of delay, or is otherwise lost; for k 2 f1; 2; 3g, therefore, delay takes values in f0; 1; 1g. For this example, equations (6) and (8) can be solved explicitly; combined with (9) we obtain Figure 1: TWO-LEAF MULTICAST TREE.0752 Figure 2: FOUR-LEAF MULTICAST TREE. the estimates 3.4 Rates of Convergences of the Delay Distribution Estimator 3.4.1 Asymptotic Behavior of the Delay Distribution Estimator In this section, we study the rate of convergence of the estimator. Theorem 2 states that b ff converges to ff with probability 1 as n grows to infinity; but it provides no information on the rate of convergence. Because of the mild conditions satisfied by \Gamma \Gamma1 , we can use Central Limit Theorem to establish the following asymptotic result Theorem 3 When converges in distribution to a multivariate normal random variable with mean vector 0 and covariance matrix denotes the transpose. Proof: By the Central Limit Theorem, it follows that the random variables bfl are asymptotically Gaussian as n !1 with Here D denotes convergence in distribution. Following the same lines of the proof of Theorem continuously differentiable on G, the Delta method (see Chapter 7 of [34]) yields that b is also asymptotically Gaussian as n !1: Theorem 3 allows us to compute confidence intervals of the estimates, and therefore their accuracy and their convergence rate to the true values as n grows. This is relevant in assessing: (i) the number of probes required to obtain a desired level of accuracy of the estimate; (ii) the likely accuracy of the estimator from actual measurements by associating confidence intervals to the estimates. For large n, the estimator b ff k (i) will lie in the interval r (k;i)(k;i) where z ffi=2 is the quantile of the standard distribution and the interval estimate is a 100(1 \Gamma ffi)% confidence interval. To obtain the confidence interval for b ff derived from measured data from n probes, we estimate by b and D(bff) is the Jacobian of the inverse computed for ff. We then use confidence intervals of the form 3.4.2 Dependence of the Delay Distribution Estimator on Topology The estimator variance determines the number of probes required to obtain a given level of ac- curacy. Therefore, it is important to understand how the variance is affected by the underlying a (a)0.20.61 a (b) Figure 3: ASYMPTOTIC ESTIMATOR VARIANCE AND TREE DEPTH. Binary tree with depth 2, 3 and 4. Left: Minimum and Maximum Variance of the estimates b ff k (0) (a) and b ff k (1) (b) over all links. parameters, namely the delay distributions and the multicast tree topology. The following Theo- rem, the proof of which we postpone to Appendix C, characterizes the behavior of the variance for small delays. Set k ff Theorem 4 As k ff k ! 0, Theorem 4 states that the estimator variance is, to first order, independent of the topology. To explore higher order dependencies, we computed the asymptotic variance for a selection of trees with different depths and branching ratio. We use the notation T (r to denote a tree of apart from node 0 that has one descendent, nodes at level j have exactly children. For simplicity, we consider the case when link delay takes values in f0; 1g, i.e., we consider no loss, and study the behavior as function of ff k In Figure 3 we show the dependence on tree depth for binary trees of depth 2, 3 and 4. We plot the maximum value of the variance over the links max k Var(bff k (0)) (a) and max k Var(bff k (1)) (b). In these examples, the variance increases with the tree depth. In Figure 4 we show the dependence Variance a (a)0.20.61 Variance a (b) Figure 4: ASYMPTOTIC ESTIMATOR VARIANCE AND BRANCHING RATIO. Binary tree with depth 2 and 2, 4 or 6 receivers. Left: Variance of b ff k (0) (a) and b ff k (1) (b) for link 1 (common link) and 2 (generic receiver). on branching ratio for a tree of level 2. We plot the estimator variance for both link 1 (the common link) and link 2 (a generic receiver). In these examples, increasing the branching ratio decreases the variances, especially those of the common link estimates which increases less than linearly for ff up to 0.7 when the branching ratio is larger than 3. In all cases, the variance of b ff k (1) is larger than b In all cases, as predicted by Theorem 4, the estimator variance is asymptotically linear in ff independently of the topology as ff ! 0. As ff increases, the behavior is affected by different factors: increasing the branch ratio results in a reduction of the variance, while increasing the tree depth results in variance increase. The first can be explained in terms of the increased number of measurements available for the estimation as the number of receivers sharing a given link increases; the second appears to be the effect of cumulative errors that accrue as the number of links along a path increases (ff is computed iteratively on the tree). We also observe that the variance increases with the delay lag; this appears to be caused by the iterative computation on the number of bins that progressively cumulate errors. 4 Computation of the Delay Distribution Estimator In this section we describe an algorithm for computing the delay distribution estimate from measurements based on the results presented in the previous section. We also discuss its suitability for distributed implementation and how to adapt the computation to handle different bin sizes. We assume the experimental data of source-to-leaf delays (Y k;m ) k2R;m=1;:::n from n probes, as collected at the leaf nodes k 2 R. Two steps must be initially performed to render the data into a form suitable for the inference algorithms: (i) removal of fixed delays and (ii) choosing a bin size q and computing the estimate bfl . The first step is necessary since it is generally not possible to apportion the deterministic component of the source-to-leaf delays between interior links. (To see this, it is sufficient to consider the case of the two receiver tree; expressing the link fixed delays in terms of the source-to-leaf fixed delays results in two equations in three unknowns). Thus we normalize each measurement by subtracting the minimum delay seen at the leaf. Observe that to interpret the observed minimum delay as the transmission delay assumes that at least one probe has experienced no queuing delay along the path). The second step is to choose the bin size q and discretize the delays measurements accordingly. This introduces a quantization error which affects the accuracy of the estimates. As our results have shown, the accuracy increases as q decreases (we have obtained accurate results over a significant range of values of q up to the same order of magnitude of the links average delay). The choice of q represents a trade-off between accuracy and cost of the computation as a smaller bin size entails a higher computational cost due to the higher dimensionality of the binned distributions. These two steps are carried out as follows. From the measured data (Y k ) k2R , we recursively construct the auxiliary vector process b m2f1;:::;ng The binned estimates of bfl are with Y k;m Here dxe denotes the smallest integer greater than x and N k 1g. Observe that i max represents the largest value at which the estimate b The estimate can be computed iteratively over the delay lag and recursively over the tree. The pseudo code for carrying out the computation is found in Figure 5. The procedure find y calculates b Y k and bfl k , with b Y k;l initialized to Y k;l \Gamma minm2f1;:::;ng Y k;m for k 2 R and 1 (a value procedure main f find y foreach procedure find y foreach foreach ng ) foreach return b procedure infer delay ( k, i A k [i] == else f A k [0] ae Q oe A k [j] A A f(k) [0] A A f(k) [0] foreach Figure 5: PSEUDOCODE FOR INFERENCE OF DELAY DISTRIBUTION. larger than any observed delay suffices) otherwise. The procedure infer delay calculates b for a fixed i recursively on the tree, with b initialized to 0, except for A 0 [0] set to 1. The output of the algorithm are the estimates b Within the code, an empty product (which occurs when the first argument of infer is a leaf) is assumed to be zero. The routines solvefor1 and solvefor2 return the value of the first symbolic argument that solves the equation in the second argument. solvefor1 returns a solution in (0; 1); from Lemma 1 in [4] this is known to be unique. solvefor2 returns the unique solution if the second argument is linear in b A k (i) ( this happen only if k is a leaf-node), otherwise it returns the second largest solution. 4.1 Distributed Implementation As with the loss estimator [4] the algorithm is recursive on trees. In particular, observe that the computation of bfl and b A k only requires the knowledge of ( b are computed recursively on the the tree starting from the receivers. Therefore it is possible to distribute the computation among the nodes of the tree (or representative nodes of subtrees), with each node k being responsible for the aggregation of the measurements of its child nodes through (14) and for the computation of b A k . 4.2 Adopting Different Bin Sizes Following the results of the previous section, we presented the algorithm using a fixed value of q for all links. This can be quite restrictive in a heterogeneous environment, where links may differ significantly in terms of speed and buffer sizes; a single value of q could be at the same time too coarse grained for describing the delay of a high bandwidth link but too fine-grained to efficiently capture the essential characteristics of the delay experienced along a low bandwidth link. A simple way to overcome this limitation is to run the algorithm for different values of q, each best suited for the behavior of a different group of links, and retain each time only the solutions for those links. A drawback of this approach is that each distribution is computed for all the different bin sizes. The distributed nature of the algorithm suggests we can do better; indeed, since A k , can be computed independently from one another, it is possible to compute each link delay distribution only for the bin size best suited to its delay characteristics. More precisely, let q k denote the bin size adopted for link k. In order to compute b ff k with bin size q k we need to compute both b A k and b A f(k) with bin size q k . Thus, the overall computation requires calculating each cumulative distribution b A k only for the bin sizes q j , only for the bin sizes adopted for the links terminating at node k and at all its child nodes rather than for bin sizes adopted for all links. In an implementation, we envision that a fixed value for all links is used first. This can be chosen based on the measurements spread and the tree topology or delay past history. Then, with a better idea of each link delay spread, it would be possible to refine the value of the bin size on a link by link basis. 5 Experimental Evaluation We evaluated our delay estimator through extensive simulation. Our first set of experiments focus on the statistical properties of the estimator. We perform model simulation, where delay and loss are determined by random processes that follow the model on which we based our analysis. In our second set of experiments we we investigate the behavior of the estimators in a more realistic setting where the model assumption of independence may be violated. To this end, we perform TCP/UDP simulation, using the ns simulator. Here delay and loss are determined by queuing delay and queue overflows at network nodes as multicast probes compete with traffic generated by TCP/UDP traffic sources. 5.1 Comparing Inferred vs. Sample Distributions Before examining the results of our experiments, we describe our approach to assessing the accuracy of the inferred distributions. Given an experiment in which n probes are sent from the source to the receivers, for k 2 V , the inferred distribution b is computed from the end-to-end measurements using the algorithm described in Section 4. Its accuracy must be measured against the actual data, represented by a finite sequence of delays fD k;m g n experienced by the probes in traversing (reaching) that link. For simplicity of notation we assume, hereafter, that each set of data has been already normalized by subtracting the minimum delay from the sequence. We compare summary statistics of link delay, namely the mean and the variance. A finer evaluation of the accuracy lies in a direct comparison of the inferred and sample distributions. To this end, we also compute the largest absolute deviation between the inferred and sample c.d.f.s. This measure is used in statistics for the Kolmogoroff-Smirnoff test for goodness of fit of a theoretical with a sample distribution. A small value for this measure indicates that the theoretical distribution provides a good fit to the sample distribution; a large value leads to the rejection of the hypoth- esis. We cannot directly apply the test as we deal with an inferred rather than a sample c.d.f.; however, we will use the largest absolute deviation as a global measure of accuracy of the inferred distributions. We compute the sample distributions ~ ff and ~ A using the same bin size q of the estimator. More precisely, we compute (~ff k ) k2V and ( ~ (Observe that in computing (~ff k ) k2V , the sum is carried out only over N f(k) 1g, the set over which the delay along link k is defined either finite or infinite.) The largest absolute deviation between the inferred and sample c.d.f.s is, then, ff k (i)j. In other words, \Delta k is the smallest nonnegative number such that lies between . The same result holds for the tail probabilities, (a)0.050.150.250 2000 4000 6000 8000 10000 a (1) n. of probes link 1 link 2 link 3 a k (1)=0.2 (b) Figure Simulation topology. (b): Convergence of b ff k (1) to ff k (1). a (1) n. of probes a 1 (1) a 1 (1) 2 std a 2 (1) a 2 (1) 2 std (a)0.050.150.250 2000 4000 6000 8000 10000 a (1) n. of probes a 1 (1)=a 2 (1) a 1 (1) 2 std a 2 (1) 2 std (b) Figure 7: AGREEMENT BETWEEN SIMULATED AND THEORETICAL CONFIDENCE INTERVALS. (a): Results from 100 model simulations. (b): Prediction from (10). The graphs show two-sided confidence interval at 2 standard deviation for link 1 and 2. Parameters are ff k links. 5.2 Model Simulation We first consider the two-leaf topology of Figure 6(a), with source 0 and receivers 2 and 3. Link delays are independent, taking values in f0; 1; 1g; if a probe is not lost it experiences either no delay or unit delay. In Figure 6(b) we plot the estimate b versus the model values for a run comprising 10000 probes. The estimate converges within to 2% of the model value within 4000 probes. In Figure 7 we compare the empirical and theoretical 95% confidence intervals. The theoretical intervals are computed from (10). The empirical intervals are computed over 100 independent simulations. The agreement between simulation and theory is close: the two sets of curves are almost indistinguishable. Next we consider the topology of Figure 8. Delays are independently distributed according to a truncated geometric distribution taking values in f0; (in ms) . This topology is also used in subsequent TCP/UDP simulations, and the link average delay and loss probability are chosen to match the values obtained from these. The average delay range between 1 and 2ms for the slower edge links and between 0:2 and 0:5ms for the interior faster links; the link losses range from 1% to 11%. In Figure 9 we plot the estimated average link delay and standard deviation with the empirical 95% confidence interval computed over 100 simulations. The results are very accurate even for several hundred probes: the theoretical average delay always lies within the confidence interval and the standard deviation does so for 1500 or more probes. To compare the inferred and sample distributions, we computed the largest absolute deviation between the inferred and sample c.d.f.s. The results are summarized in Figure 10 where we plot the minimum, median and the maximum largest absolute deviation in 100 simulations computed over all links as a function of n (a) and link by link for (b). The accuracy increases with the number of probes as 1= n with a spread of two orders of magnitude between the minimum and maximum. For more than 3000 probes, the average largest deviation over all links is less than 1%. The accuracy varies from link to link: when the number of probes is at one extreme we have link 4 with 0:18% \Delta 4 0:8% and at the other extreme link 6 with simulations. We observe that the inferred distributions are less accurate as we go down the tree. This is in agreement with the results of Section 3.4 and is explained in terms of the larger inferred probabilities variances of downstream with respect to upstream nodes. 5.3 TCP/UDP Simulations We used the topology shown in Figure 8. To capture the heterogeneity between edges and core of a WAN, interior links have higher capacity (5Mb/sec) and propagation delay (50ms) then at the 1Mb/sec, 10ms 5Mb/sec, 50ms6 79 11 Figure 8: Simulation Topology: Link are of two types: edge links of 1MB/s capacity and 10ms latency, and interior links of 5Mb/s capacity and 50ms latency.0.20.611.41.82.2 Average Delay (ms) n. of probes link 1 - estimated link 6 - estimated link 8 - estimated link 11 - estimated link 1 - model link 6 - model link 8 - model link 11 - model (a)12345 Standard Deviation (ms) n. of probes link 1 - estimated link 6 - estimated link 8 - estimated link 11 - estimated link 1 - model link 6 - model link 8 - model link 11 - model (b) Figure 9: MODEL SIMULATION: TOPOLOGY OF FIGURE 8. ESTIMATED VERSUS THEORETICAL DELAY AVERAGE AND STANDARD DEVIATION WITH 95% CONFIDENCE INTERVAL COMPUTED OVER 100 MODEL SIMULATIONS. 1e-031e-010 2000 4000 6000 8000 10000 Largest Absolute Vertical Deviation n. of probes Maximum Median Minimum (a)1e-031e-01 Largest Absolute Vertical Deviation Link Maximum Median Minimum (b) Figure 10: MODEL SIMULATION: TOPOLOGY OF FIGURE 8. ACCURACY OF THE ESTIMATED DISTRIBUTION. LARGEST VERTICAL ABSOLUTE DEVIATION BETWEEN ESTIMATED AND SAMPLE C.D.F. Minimum, median and the maximum largest absolute deviation in 100 simulations computed over all links as function of n (a) and link by link for edge (1Mb/sec and 10ms). Each link is modeled as a FIFO queue with a 4-packet capacity. probes as a 20Kbit/s stream comprising 40 byte UDP packets according to a Poisson process with a mean interarrival time of 16ms; this represents 2% of the smallest link capacity. Observe that even for this simple topology with 8 end-points, a mesh of unicast measurements with the same traffic characteristics would require an aggregate bandwidth of 160Kbit/s at the root. The background traffic comprises a mix of infinite data source TCP connections and exponential on-off sources using UDP. Averaged over the different simulations, the link loss ranges between 1% and 11% and link utilization ranges between 20% and 60%. For a single experiment, Figure 11 compares the estimated versus the sample average delay for representative selected links. The analysis has been carried out using 1ms (a) and 0:1ms (b). In this example, we practically obtain the same accuracy despite a tenfold difference in resolution. (Observe that 1ms is of the same order of magnitude of the average delays.) The inferred averages rapidly converge to the sample averages even though we have persistent systematic errors in the inferred values due to consistent spatial correlation. We shall comment upon this later. In order to show how the inferred values not only quickly converge, but also exhibit good dynamics tracking, in Figure 12 we plot the inferred versus the sample average delay for 3 links (1, 3 and 10) computed over a moving window of two different sizes with jumps of half its width. To allow greater dynamics, here we arranged background sources with random start and stop times. Under both window sizes (approximately 300 and 1200 probes are used, respectively), the esti- 0Average Delay (ms) n. of probes link 1 - estimated link 6 - estimated link 9 - estimated link 11 - estimated link 1 - sample link 6 - sample link 9 - sample link 11 - sample (a)0.51.52.5 Average Delay (ms) n. of probes link 1 - estimated link 6 - estimated link 9 - estimated link 11 - estimated link 1 - sample link 6 - sample link 9 - sample link 11 - sample (b) Figure TCP/UDP SIMULATIONS. (a): bin-size 0:1ms. The graphs shows how the inferred values closely track the sample average delays.0.51.52.53.54.5 Average Delay (ms) Seconds link 1 - estimated link 3 - estimated link estimated link 1 - sample link 3 - sample link (a)0.51.52.53.54.5 Average Delay (ms) Seconds link 1 - estimated link 3 - estimated link estimated link 1 - sample link 3 - sample link (b) Figure 12: DYNAMIC ACCURACY OF INFERENCE. Sample and Inferred average delay on links of the multicast tree in Figure 8. (a): 5 seconds window. (b): 20 seconds windows. Background traffic has random start stop times. RMS normalized error (Average n. of probes link 1 link 3 link 5 link 6 link link 11 (a)0.050.150.250.350.450 2000 4000 6000 8000 10000 RMS normalized error (Average n. of probes link 1 link 3 link 5 link 6 link link 11 (b) Figure 13: ACCURACY OF INFERENCE: AVERAGE DELAY. Left: 1ms. Right: 0:1ms. The graphs show the normalized Root Mean Square error between the estimated and sample average delay over 100 simulations. mates of the average delays of links 1 and 10 show good agreement and a quick response to delay variability revealing a good convergence rate of the estimator. For link 3 with a smaller average delay, the behavior is rather poor, especially for the 5 seconds windows size. For a selection of links, in Figure 13 we plot the Root Mean Square (RMS) normalized error between the estimated and sample average delays calculated over 100 simulations using and 0:1ms. The two plots demonstrate that the error drops significantly up to 2000 probes after which it becomes almost constant. In this example, increasing the resolution by a factor of ten improves, although not significantly, the overall accuracy of the estimates especially for those links that enjoy smaller delays. After 10000 probes the relative error ranges from 1% to 23%. The higher values occur when link average delays are small due to the fact that for these links the same absolute error results in a more pronounced relative error. The persistence of systematic errors we observe in Figure 13 is due to the presence of spatial correlation. In our simulations, a multicast probe is more likely to experience similar level of congestion on consecutive links or on sibling links than is dictated by the independence assumption. We also verified the presence of temporal correlation among successive probes on the same link caused by consecutive probes experiencing the same congestion level at a node. To assess the extent to which our real traffic simulations violate the model assumptions, we computed the delay correlation between links pairs and among packets on the same link. The analysis revealed the presence of significant spatial correlations up to 0:3 0:4 between consecutive links. The smallest values are observed for link 5 which always exhibits a correlation with its parent node that lies below 0.1. From Figure 13 we verify that, not surprisingly node 5 enjoys the smallest relative error. We believe that these high correlations are a result of the small scale of the simulated network. We have observed smaller correlations in large simulations as would be expected in real networks because of the wide traffic and link diversity. The autocorrelation function rapidly decreases and can be considered negligible for a lag larger than (approximatively 2 seconds). The presence of short-term correlation does not alter the key property of convergence of the estimator as it suffices that the underlying processes be stationary and ergodic (this happens for example, when recurrence conditions are satisfied). The price of correlation, however, is that the convergence rate is slower than when delay are independent. Now we turn our attention to the inferred distributions. For an experiment of 300 seconds during which approximately 18000 probes were generated, we plot the complementary c.d.f. conditioned on the delay being finite in Figures 14. In Figure 15 we also plot the complement c.d.f of the node cumulative delay. (we show only the internal links as b 1ms. From these two sets of plots, it is striking to note the differences between the accuracy of the estimated cumulative delay distributions b A k and the estimated link delay distributions b while the former are all very close to the actual distributions, the latter results are inaccurate in many cases. This is explained by observing that in presence of significant correlations, the convolution among A k , ff k , and A f(k) , used in the model, does not well capture the relationship between the actual distributions. We verified this by convolving ff k and A f(k) and comparing the result with A k ; as expected, in the presence of strong local correlation, the results exhibit significant differences that account for the discrepancies of the inferred distributions. Nevertheless, results should be affected in a continuous way with small violations leading to small inaccuracies. Indeed, we have good agreement for the inferred distributions of links 4, 5, 10 and 12 that are the nodes with smallest spatial correlations. Unfortunately it is not easy to determine whether the correlations are strong and therefore assess the expected accuracy of the estimates, even though pathological shapes of the inferred distributions could provide evidence of strong local correlations 1 . A solution could be the extension of the model to explicitly account for the presence of spatial correlation in the analysis. This will be the focus of future research. The accuracy of the inferred cumulative delay distributions, on the other hand, derives from the fact that even in presence of significant local correlations, equation (8), which assumes inde- 1 To this end, we observed that under strong spatial correlation inaccuracies of the estimator b ff are often associated to the existence of significant increasing behavior portions in the complement c.d.f. that reveals the presence of negative inferred probabilities with possibly non negligible absolute values. Complement of the Cumulative Density Function Delay (ms) Estimated vs. sample node 1 delay c.d.f sample Complement of the Cumulative Density Function Delay (ms) Estimated vs. sample node 2 delay c.d.f sample Complement of the Cumulative Density Function Delay (ms) Estimated vs. sample node 4 delay c.d.f sample Complement of the Cumulative Density Function Delay (ms) Estimated vs. sample node 7 delay c.d.f sample Complement of the Cumulative Density Function Delay (ms) Estimated vs. sample node 10 delay c.d.f sample Complement of the Cumulative Density Function Delay (ms) Estimated vs. sample node 12 delay c.d.f sample estimated Figure 14: Sample vs. Estimated Delay c.d.f. for selected links. Complement of the Cumulative Density Function Delay (ms) Estimated vs. sample node 1 cumulative delay c.d.f sample Complement of the Cumulative Density Function Delay (ms) Estimated vs. sample node 2 cumulative delay c.d.f sample Complement of the Cumulative Density Function Delay (ms) Estimated vs. sample node 3 cumulative delay c.d.f sample Complement of the Cumulative Density Function Delay (ms) Estimated vs. sample node 6 cumulative delay c.d.f sample Complement of the Cumulative Density Function Delay (ms) Estimated vs. sample node 7 cumulative delay c.d.f sample estimated Figure 15: Sample vs. Estimated node k cumulative delay c.d.f. c.d.f. Largest Absolute Vertical Deviation n. of probes Maximum Median Minimum (a)1e-021e+00 c.d.f. Largest Absolute Vertical Deviation Link Maximum Median Minimum (b) Figure 8. ACCURACY OF THE ESTIMATED DISTRIBUTION. LARGEST VERTICAL ABSOLUTE DEVIATION BETWEEN ESTIMATED AND THEORETICAL C.D.F. Minimum, median and the maximum largest absolute deviation in 100 simulations computed over all links as function of n (a) and link by link for pendence, is still accurate. This can be explained by observing that (8) is equivalent to (4) which consists of a convolution between A f(k) and fi k ; we expect the correlation between the delay accrued by a probe in reaching node f(k) and the minimum delay accrued from node f(k) to reach any receiver be rather small, especially as the tree size grows, as these delays span the entire multicast tree. Finally in Figure 16 we plotted the minimum, median and maximum largest deviation between inferred and theoretical c.d.f. over 100 simulations computed over all links as function of n (left) and link by link as for (right). Due to spatial correlation, the largest deviation level off after the first 2000 probes with the median that stabilize at 5%. The accuracy again exhibits a negative trend as we descend the tree. 6 Conclusions and Future Work In this paper, we introduced the use of end-to-end multicast measurements to infer network internal delay in a logical multicast tree. Under the assumption of delay independence, we derived an algorithm to estimate the per link discrete delay distributions and utilization from the measured end-to-end delay distributions. We investigated the statistical properties of the estimator, and show it to be strongly consistent and asymptotically normal. We evaluated our estimator through simulation. Within model simulation we verified the accuracy and convergence of the inferred to the actual values as predicted by our analysis. In real traffic simulations, we found rapid convergence, although some persistent difference to the actual distributions because of spatial correlation. We are extending our delay distribution analysis in several directions. First we plan to do more extensive simulations, exploring larger topologies, different node behavior, background traffic and probe characteristics. Moreover, we are exploring how probe delay is representative of the delay suffered by other applications and protocols, for examples TCP. Second, we are analyzing the effect of spatial correlation among delays and we are planning to extend the model by directly taking into account the presence of correlation. Moreover, we are studying the effect of the choice of the bin size on the accuracy of the results. To deal with continuously distributed delay, we derived a continuous version of the inference algorithm we are currently investigating. Finally, we believe that our inference technique can shed light on the behavior and dynamics of per link delay and so allow us to develop accurate link delay models. This will be also object of future works. We feel that multicast based delay inference is an effective approach to perform delay mea- surements. The techniques developed are based on rigorous statistical analysis and, as our results show, yield representative delay estimates for all traffic which receive the same per node behavior of multicast probes. The approach does not depend on cooperation from networks elements and because of bandwidth efficiency of multicast traffic is well suited to cope with the growing size of today networks. --R "The Laplace Transform" "Characterizing End-to-End Packet Delay and Loss in the Internet." "The case for FEC-based error control for packet audio in the Internet" "Multicast-Based Inference of Network Internal Loss Characteristics" "Multicast-Based Inference of Network Internal Loss Characteristics: Accuracy of Packet Estimation" "Inferring Link-Level Performance from End- to-End Measurements" "Loss-Based Inference of Multicast Network Topology" "Measurements Considerations for Assessing Unidirectional Latencies" "Measuring Bottleneck Link Speed in Packet-Switched Networks," "Multicast Inference of Packet Delay Variance at Interior Networks Links" "Probabilistic Inference Methods for Multicast Network Topology" Felix: Independent Monitoring for Network Survivability. "Random Early Detection Gateways for Congestion Avoidance," IPMA: Internet Performance Measurement and Analysis. IP Performance Metrics Working Group. "Creating a Scalable Architecture for Internet Mea- surement," "Diagnosing Internet Congestion with a Transport Layer Performance Tool," "Network Time Protocol (Version 3): Specification, Implementation and Analysis" "Estimation and Removal of Clock Skew from Network Delay Measurements" "Correlation of Packet Delay and Loss in the Internet" "On the Dynamics and Significance of Low Frequency Components of Internet Load" "End-to-End Routing Behavior in the Internet," "End-to-End Internet Packet Dynamics," "Measurements and Analysis of End-to-End Internet Dynamics," "Automated Packet Trace Analysis of TCP Implementations," "On calibrating measurements of Packet Transit Times" "Inference of Multicast Routing Tree Topologies and Bottleneck Bandwidths using End-to-end Measurements" "Experimental assessment of end-to-end behavior on Internet" "Study of network dynamics" "Theory of Statistics" --TR Measuring bottleneck link speed in packet-switched networks End-to-end routing behavior in the Internet End-to-end Internet packet dynamics Automated packet trace analysis of TCP implementations On calibrating measurements of packet transit times Using pathchar to estimate Internet link characteristics Network Delay Tomography from End-to-End Unicast Measurements Multicast-Based Inference of Network-Internal Delay Distributions TITLE2: --CTR Fabio Ricciato , Francesco Vacirca , Martin Karner, Bottleneck detection in UMTS via TCP passive monitoring: a real case, Proceedings of the 2005 ACM conference on Emerging network experiment and technology, October 24-27, 2005, Toulouse, France Earl Lawrence , George Michailidis , Vijay N. Nair, Local area network analysis using end-to-end delay tomography, ACM SIGMETRICS Performance Evaluation Review, v.33 n.3, December 2005 N. G. Duffield , V. Arya , R. Bellino , T. Friedman , J. Horowitz , D. Towsley , T. Turletti, Network tomography from aggregate loss reports, Performance Evaluation, v.62 n.1-4, p.147-163, October 2005 Zhen Liu , Laura Wynter , Cathy H. Xia , Fan Zhang, Parameter inference of queueing models for IT systems using end-to-end measurements, Performance Evaluation, v.63 n.1, p.36-60, January 2006 Omer Gurewitz , Israel Cidon , Moshe Sidi, One-way delay estimation using network-wide measurements, IEEE/ACM Transactions on Networking (TON), v.14 n.SI, p.2710-2724, June 2006 N. G. Duffield , Francesco Lo Presti, Network tomography from measured end-to-end delay covariance, IEEE/ACM Transactions on Networking (TON), v.12 n.6, p.978-992, December 2004 Nick Duffield , Francesco Lo Presti , Vern Paxson , Don Towsley, Network loss tomography using striped unicast probes, IEEE/ACM Transactions on Networking (TON), v.14 n.4, p.697-710, August 2006 Azer Bestavros , John W. Byers , Khaled A. Harfoush, Inference and Labeling of Metric-Induced Network Topologies, IEEE Transactions on Parallel and Distributed Systems, v.16 n.11, p.1053-1065, November 2005

network tomography;queueing delay;estimation theory;multicast tree;end-to-end measurements

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A data and task parallel image processing environment.

The paper presents a data and task parallel low-level image processing environment for distributed memory systems. Image processing operators are parallelized by data decomposition using algorithmic skeletons. Image processing applications are parallelized by task decomposition, based on the image application task graph. In this way, an image processing application can be parallelized both by data and task decomposition, and thus better speed-ups can be obtained. We validate our method on the multi-baseline stereo vision application.

Introduction Image processing is widely used in many application areas including the lm industry, medical imaging, industrial manufacturing, weather forecasting etc. In some of these areas the size of the images is very large yet the processing time has to be very small and sometimes real-time processing is required. Therefore, during the last decade there has been an increasing interest in the developing and the use of parallel algorithms in image processing. Many algorithms have been developed for parallelizing dierent image operators on dierent parallel architectures. Most of these parallel image processing algorithms are either architecture dependent, or specically developed for dierent applications and hard to implement for a typical image processing user without enough knowledge of parallel computing. In this paper we present an approach of adding data and task parallelism to an image processing library using algorithmic skeletons [3, 4, 5] and the Image Application Task Graph (IATG). Skeletons are algorithmic abstractions common to a series of applications, which can be implemented in parallel. Skeletons are embedded in a sequential host language, thus being the only source of parallelism in a program. Using skeletons we create a data parallel image processing framework which is very easy to use for a typical image processing user. It is already known that exploiting both task and data parallelism in a program to solve very large computational problems yields better speedups compared to either pure task parallelism or pure data parallelism [7, 8]. The main reason is that both data and task parallelism are relatively limited, and therefore using only one of them limits the achievable performance. Thus, exploiting mixed task and data parallelism has emerged as a natural solution. For many applications from the eld of signal and image processing, data set sizes are limited by physical constraints and cannot be easily increased. In such cases the amount of available data parallelism is limited. For example, in the multi-baseline stereo application described in Section 5, the size of an image is determined by the circuitry of the video cameras and the throughput of the camera interface. Increasing the image size means buying new cameras and building a faster interface, which may not be feasible. Since the data parallelism is limited, additional parallelism may come from tasking. By coding the image processing application using skeletons and having the IATG we obtain a both data and task parallel environment. The paper is organized as follows. Section 2 brie y presents a description of algorithmic skeletons and a survey of related work. Section 3 presents a classication of low-level image operators and skeletons for parallel low-level image processing on a distributed memory system. Section 4 presents some related work and describes the Image Application Task Graph used in the task parallel framework. The multi-baseline stereo vision application together with its data parallel code using skeletons versus sequential code and the speedup results for the data parallel approach versus the data and task parallel approach is presented in Section 5. Finally, concluding remarks are made in Section 6. Skeletons and related work Skeletons are algorithmic abstractions which encapsulate dierent forms of parallelism, common to a series of applications. The aim is to obtain environments or languages that allow easy parallel programming, in which the user does not have to handle with problems as com- munication, synchronization, deadlocks or non-deterministic program runs. Usually, they are embedded in a sequential host language and they are used to code and hide the parallelism from the application user. The concept of algorithmic skeletons is not new and a lot of research is done to demonstrate their usefulness in parallel programming. Most skeletons are polymorphic higher-order functions, and can be dened in functional languages in a straightforward way. This is the reason why most skeletons are build upon a functional language [3, 4]. Work has also been done in using skeletons in image processing. In [5] Serot et al. presents a parallel image processing environment using skeletons on top of CAML functional language. In this paper we develop algorithmic skeletons to create a parallel image processing environment ready to use for easy implementation/development of parallel image processing applications. The dierence from the previous approach [5] is that we allow the application to be implemented in a C programming environment and that we allow the possibility to use/implement dierent scheduling algorithms for obtaining the minimum execution time. 3 Skeletons for low-level image processing 3.1 A classication of low-level image operators Low-level image processing operators use the values of the image pixels to modify the image in some way. They can be divided into point operators, neighborhood operators and global operators [1, 2]. Below, we disscus in detail about all these three types of operators. 1. Point operators Image point operators are the most powerful functions in image processing. A large group of operators falls in this category. Their main characteristics is that a pixel from the output image depends only on the corresponding pixel from the input image. Point operators are used to copy an image from one memory location to another, in arithmetic and logical operations, table lookup, image compositing. We will discuss in detail arithmetic and logic operators, classifying them from the point of view of the number of images involved, this being an important issue in developing skeletons for them. Arithmetic and logic operations Image ALU operations are fundamental operations needed in almost any imaging product for a variety of purposes. We refer to operations between an image and a constant as monadic operations, operations between two images as dyadic operations and operations involving three images as triadic operations. { Monadic image operations Monadic image operators are ALU operators between an image and a constant. These operations are shown in Table 1 - s(x; y) and d(x; y) are the source and destination pixel values at location (x; y), and K is the constant. Table 1 Monadic image operations Function Operation Add constant d(x; Subtract constant d(x; Multiply constant d(x; Divide by constant d(x; Or constant d(x; And constant d(x; Xor constant d(x; Absolute value d(x; Monadic operations are useful in many situations. For instance, they can be used to add or subtract a bias value to make a picture brighter or darker. { Dyadic image operators Dyadic image operators are arithmetic and logical functions between the pixels of two source images producing a destination image. These functions are shown below in Table are the two source images that are used to create the destination image d(x; y). Table operations Function Operation Add Subtract Multiply Divide Min Or And Dyadic operators have many uses in image processing. For example, the subtraction of one image from another is useful for studying the ow of blood in digital subtraction angiography or motion compensation in video coding. Addition of images is a useful step in many complex imaging algorithms like development of image restoration algorithms for moddeling additive noise, and special eects, such as image morphing, in motion pictures. { Triadic image operators Triadic operators use three input images for the computation of an output image. An example of such an operation is alpha blending. Image compositing is a useful function for both graphics and computer imaging. In graphics, compositing is used to combine several images into one. Typically, these images are rendered separately, possibly using dierent rendering algorithms. For example, the images may be rendered separately, possibly using dierent types of rendering hardware for dierent algorithms. In image processing, compositing is needed for any product that needs to merge multiple pictures into one nal image. All image editing programs, as well as programs that combine synthetically generated images with scanned images, need this function. In computer imaging, the term alpha blend can be dened using two source images and S2, an alpha image and a destination image D, see formula (1). Another example of a triadic operator is the squared dierence between a reference image and two shifted images, an operator used in the multi-baseline stereo vision application, described in Section 5. Table 3 Triadic image operations Function Operation Alpha blend d(x; Squared di d(x; 2. Local neighborhood operators Neighborhood operators (lters) create a destination pixel based on the criteria that depend on the source pixel and the value of pixels in the \neighborhood" surrounding it. Neighborhood lters are largely used in computer imaging. They are used for enhancing and changing the appearance of images by sharpening, blurring, crispening the edges, and noise removal. They are also useful in image processing applications as object recognition, image restoration, and image data compression. We dene a lter as an operation that changes pixels of the source image based on their values and those of their surrounding pixels. We may have linear and nonlinear lters. Linear ltering versus nonlinear ltering Generally speaking, a lter in imaging refers to any process that produces a destination image from a source image. A linear lter has the property that a weighted sum of the source images produces a similarly weighted sum of the destination images. In contrast to linear lters, nonlinear lters are somewhat more dicult to characterize. This is because the output of the lter for a given input cannot be predicted by the impulse response. Nonlinear lters behave dierently for dierent inputs. Linear ltering using two-dimensional discrete convolution In imaging, two-dimensional convolution is the most common way to implement a linear lter. The operation is performed between a source image and a two-dimensional convolution kernel to produce a destination image. The convolution kernel is typically much smaller than the source image. Starting at the top of the image (the top left corner which is also the origin of the image), the kernel is moved horizontally over the image, one pixel at a time. Then it is moved down one row and moved horizontally again. This process is continued until the kernel has traversed the entire image. For the destination pixel at row m and column n, the kernel is centered at the same location in the source image. Mathematically, two-dimensional discrete convolution is dened as a double summation. Given an MN image f(m; n) and KL convolution kernel h(k; l), we dene the origin of each to be at the top left corner. We assume that f(m; n) is much larger than h(k; l). Then, the result of convolving f(m; n) by h(k; l) is the image g(m; n) given by formula In the above formula we assume that K;L are odd numbers and we extend the image by (K 1)=2 lines in each vertical direction and by (L 1)=2 columns in each horizontal direction. The sequential time complexity of this operation is O(MNKL). As it can be observed, this is a time consuming operation, very well tted to the data parallel approach. 3. Global operators Global operators create a destination pixel based on the entire image information. A representative example of an operator within this class is the Discrete Fourier Transform (DFT). The Discrete Fourier Transform converts an input data set from the tempo- ral/spatial domain to the frequency domain, and vice versa. It has a lot of applications in image processing, being used for image enhancement, restoration, and compression. In image processing the input is a set of pixels forming a two-dimensional function that is already discrete. The formula for the output pixel X lm is the following: x jk e 2i( jl where j and k are column coordinates, 0 j N 1 and 0 k M 1. We also include in the class of global operators, operators like the histogram transform, which do not have an image as output, but another data structure. 3.2 Data parallelism of low-level image operators From the operator description given in the previous section we conclude that point, neighborhood and global image processing operators can be parallelized using the data parallel paradigm with a host/node approach. A host processor is selected for splitting and distributing the data to the other nodes. The host also processes a part of the image. Each node processes its received part of the image and then the host gathers and assembles the image back together. In Figures 1, 2 and 3 we present the data parallel paradigm with the host/node approach for point, neighborhood and global operators. For global operators we send the entire image to the corresponding nodes but each node will process only a certain part of the image. In order to avoid extra inter-processor communication due to the border information exchange for neighborhood operators we extend and partition the image as showed in Figure 2. In this way, each node processor receives all the data needed for applying the neighborhood operator. Original Image Processed Image Master 0 Master 0 node 1 node 2 node n-1 Master 0 Fig. 1. DCG skeleton for point operators Original Image Extended Image Master Master node 1 node 2 node n-1 Master0Processed Image Fig. 2. DCG skeleton for neighborhood operator Original Image Processed area at 0 Processed area at 1 Processed area at 2 Processed area at n-1 Processed Image Master Master (0) (0) Fig. 3. DCG skeleton for global operators Based on the above observations we identify a number of skeletons for parallel processing of low-level image processing operators. They are named according to the type of the low-level operator and the number of images involved in the operation. Headers of some skeletons are shown below. All of them are based on a "Distribute Compute and Gather" (DCG) main skeleton, previously known as the map skeleton [4], suitable for regular applications as the low-level operators from image processing. The implementation of all the skeletons is based on the ideea described in the above paragraph, see Figures 1, 2 and 3. Each skeleton can run on a set of processors. From this set of processors a host processor is selected to split and distribute the image(s) to the other nodes, each other node from the set receives a part of the image(s) and the image operator which should be applied on it, then the computation takes place and the result is sent back to the host processor. The skeletons are implemented in C using MPI-Panda library [19, 20]. The implementation is transparent to the user. void ImagePointDist_1IO(unsigned int n,char *name,void(*im_op)()); // DCG skeleton for monadic point operators - one Input/Output void ImagePointDist_1IO_C(unsigned int n,char *name, void(*im_op)(),float ct); // DCG skeleton for monadic point operators which need a constant value as pararameter // one Input/Output void ImagePointDist_1I_1O(unsigned int n,char *name1,char *name2,void(*im_op)()); // DCG skeleton for monadic/dyadic point operators - one Input and one Output void ImagePointDist_1IO_1I(unsigned int n,char *name1,char *name2,void(*im_op)()); // DCG skeleton for monadic/dyadic point operators - one Input/Output and one Input. void ImagePointDist_2I_1O(unsigned int n,char *name1,char *name2,char *name3,void(*im_op)()); // DCG skeleton for dyadic/triadic point operators - 2 Inputs and one Output void ImagePointDist_2I_2O(unsigned int n,char *name1,char *name2,char *name3,char *name4, Inputs and 2 Outputs void ImagePointDist_3I_1O(unsigned int n,char *name1,char *name2,char *name3,char *name4, // DCG skeleton for triadic point operators - 3 Inputs and one Output void ImageWindowDist_1IO(unsigned int n,char *name,Window *win,void(*im_op)()); // DCG skeleton for neighborhood operators - one Input/Output void ImageWindowDist_1I_1O(unsigned int n,char *name1,char *name2,Window *win,void(*im_op)()); // DCG skeleton for neighborhood operators - one Input and one Output void ImageGlobalDist_1IO(unsigned int n,char *name,void(*im_op)()); // DCG skeleton for global operators - one Input/Output We develop several types of skeletons, which depend on the type of the low-level operator (point, neighborhood, global) and the number of input/output images. With each skeleton we associate a parameter which represents the task number corresponding to that skeleton. This is used by the task parallel framework. Depending on the skeleton type, one or more identiers of the images are given as parameters. The last argument is the point operator for processing the image(s). So, each skeleton is used for a number of low-level image processing operators which perform in a similar way (for instance all dyadic point operators take two input images, combine and process them depending on the operator type and then produce an output image). Depending on the operator type and the skeleton type, there might exist additional parameters necessary for the image operator. For point operators we assigned the ImagePointDist skeletons, for neighborhood operators we assigned the ImageWindowDist skeletons, and for global operators we assigned the ImageGlobalDist skeletons. Some of the skeletons modify the input image (ImagePointDist 1IO, ImageWindowDist 1IO, ImageGlob- alDist 1IO, so 1IO stands for 1 Input/Output image), other skeletons take a number of input images and create a new output image, for example the ImagePointDist 2I 1O skeleton for point operators takes 2 input images and creates a new output image. This skeleton is necessary for dyadic point operators (like addition, subtraction, etc., see Table 2) which create a new image by processing two input images. Similarly, the skeleton ImagePointDist 3I 1O for point operators takes 3 input images and creates a new output image. An example of a low-level image processing operator suitable for this type of skeleton is the squared dierence between one reference image and two disparity images, operator used in the multi-baseline stereo vision application, see Table 3 and Section 5. Similar skeletons exist also for local neighborhood and global operators. ImagePointDist 1IO C is a skeleton for monadic point operators which need a constant value as parameter, for processing the input image, see Table 1. Below we present an example of using the skeletons to code a very simple image processing application in a data-parallel way. It is an image processing application of edge detection using Laplace and Sobel operators. First we read the input image and we create the two output images and a 3 3 window, and then we apply the Laplace and Sobel operators on the num nodes number of processors. num nodes is the number of nodes on which the application is run and is detected on the rst line of the partial code showed below. image in is the name of the input image given as input parameter to both skeletons and image l, image s are the output parameters (images) for each skeleton. We have used a ImageWindowDist 1I 1O skeleton to perform both operators. The last two parameters are the window used (which contains information about the size and the data of the window) and the image operator that is applied via the skeleton. 4 The task parallel framework Recently, it has been shown that exploiting both task and data parallelism in a program to solve very large computational problems yields better speedups compared to either pure data parallelism or either pure task parallelism [7, 8]. The main reason is that both task and data parallelism are relatively limited, and therefore using only one of them bounds the achievable performance. Thus, exploiting mixed task and data parallelism has emerged as a natural solution. We show that applying both data and task parallelism can improve the speedup at the application level. There have been considerable eort in adding task-parallel support to data-parallel lan- guages, as in Fx [10], Fortran M [11] or Paradigm HPF [7], or adding data-parallel support to task-parallel languages such as in Orca [12]. In order to fully exploit the potential advantage of the mixed task and data parallelism, ecient support for task and data parallelism is a critical issue. This can be done not only at the compiler level, but also at the application level and applications from the image processing eld are very suitable for this technique. Mixed task and data parallel techniques use a directed acyclic graph, in the literature also called a Macro Data ow Graph (MDG) [7], in which data parallel tasks (in our case the image processing operators) are the nodes and the precedence relationships are the edges. For the purpose of our work we change the name of this graph to the Image Application Task Graph (IATG). 4.1 The Image Application Task Graph model A task parallel program can be modeled by a Macro Data ow communication Graph [7], which is a directed acyclic graph c), where: { V is the nite set of nodes which represents tasks (image processing operators) { E is the set of directed edges which represent precedence constraints between tasks: { w is the weight function which gives the weight (processing time) of each node (task). Task weights are positive integers. { c is the communication function c which gives the weight (communication time) of each edge. Communication weights are positive integers. An Image processing Application Task Graph (IATG) is, in fact, an MDG in which each node stands for an image processing operator and each edge stands for a precedence constraint between two adjacent operators. In this case, a node represents a larger entity that in the MDG where a node can be any simple instruction from the program. Some important properties of the IATG are: { It is a weighted directed acyclic graph. { Nodes represent image processing operators and edges represent precedence constraints between them. { The are two distinguished nodes: START precedes all other nodes and STOP succeeds all other nodes. We dene a well balanced IATG as an application task graph which has the same type of tasks (image operators) on each level. An example is the IATG of the multi-baseline stereo vision application, described in Section 5 Figure 7, which on the rst level has the squared dierence operator applied to 3 images for each task and on the second level the error operator is executed by all the tasks. Moreover, the graph edges form a regular pattern. The weights of nodes and edges in the IATG are based on the concepts of processing and communication costs. Processing costs account for the computation and communication costs of data parallel tasks - image processing operators corresponding to nodes, and depend on the number of processors allocated to the node. Communication costs account for the costs of data communication between nodes. 4.2 Processing cost model A node in the IATG represents a processing task (an image processing operator applied via a DCG skeleton, as described in Section 3.2) that runs non-preemptively on any number of processors. Each task i is assumed to have a computation cost, denoted T exec (i; p i ), which is a function of the number of processors. The computation cost function of the task can be obtained either by estimation or by proling. For cost estimation we use Amdahl's law. According to it, the execution time of the task is: where i is the task number, p i is the number of processors on which task i is executed, is the task's execution time on a single processor and is the fraction of the task that executes serially. If we use proling, the task's execution costs are either tted to a function similar to the one described above (in the case that data is not available for all processors), or the proled values can be used directly through a table. The values are simple to determine, we measure the execution times of the basic image processing operators implemented in the image processing library and we tabulate their values. 4.3 Communication cost model Data communication (redistribution) is essential for implementing an execution scheme which uses both data and task parallelism. Individual tasks are executed in a data parallel fashion on subsets of processors and the data dependences between tasks may necessitate not only changing the set of processors but also the distribution scheme. Figure 4 illustrates a classical approach of redistribution between a pair of tasks. Task TaskA is executed using seven processors and reads from data D. Task TaskB is executed using four processors and reads from the same data D. This necessitates the redistribution of the data D from the seven processors executing task TaskA to the four processors executing task TaskB. In addition to changing the set of processors we could also change the distribution scheme of the data D. For instance, if D is a two dimensional data then TaskA might use a block distribution for D, whereas TaskB might use a row-stripe distribution. Processors executing TaskB Processors executing TaskA Redistribution Fig. 4. Data redistribution between two tasks Processors executing TaskB Processors executing TaskA master A master B Fig. 5. Image communication between two host processors We reduce the complexity of the problem rst by allowing only one type of distribution scheme (row-stripe) and second by sending images only between two processors (the selected host processors from the two sets of processors), as shown in Figure 5. An edge in the IATG corresponds to a precedence relationship and has associated a communication cost, denoted through Tcomm (i; which depends on the network characteristics (latency, bandwidth) and the amount of data to be transferred. It should be emphasized that there are two types of communication times. First, we have internal communication time which represents the time for internal transfer of data between the processors allocated to a task. This quantity is part of the term of the execution time associated to a node of the graph. Secondly, we have external communication time which is the time of transferring data, i.e. images, between two processors. These two processors represent the host processors for the two associated image processing tasks (corresponding to the two adjacent graph nodes). This quantity is actually the communication cost of an edge of the graph. In this case we can also use either cost estimation or proling to determine the communication time. In state-of-the-art of distributed memory systems the time to send a message containing L units of data from a processor to another processor can be modeled as: are the startup and per byte cost for point-to-point communication and L is the length of the message, in bytes. We run our experiments on a distributed memory system which consists of a cluster of Pentium Pro/200Mhz PCs with 64Mb RAM running Linux, and connected through Myrinet in a 3D-mesh topology, with dimension order routing [16]. Figure 6 shows the performance of point-to-point communication operations and the predicted communication time. The reported time is the minimum time obtained over 20 executions of the same code. It is reasonable to select the minimum value because of the possible interference caused by other users' trac in the network. From these measurements we perform a linear tting and we extract the communication parameters t s and t b . In Figure 6 we see that the predicted communication time, based on the above formula, approximates very good the measured communication time. measured predicted time(microseconds) message size (2 Fig. 6. Performance of point-to-point communication on DAS 4.4 IATG cost properties A task with no input edges is called an entry task and a task with no output edges is called an exit task. The length of a path from the graph is the sum of the computation and communication costs of all nodes and edges belonging to the path. We dene the Critical Path [7] (CP) as the longest path in the graph. If we have a graph with n nodes, where n is the last node of the graph and t i represents the nish time of node i, T exec (i; p i ) is the execution time of task i on a set of p i nodes then the critical path is given by the formulas and (7), where PRED i is the set of immediate predecessor nodes of node i. We dene the Average Area [7] (A) of an IATG with n nodes (tasks) for a P processor system as in formula (8), where p i is the number of processors allocated to task T i . A =P The critical path represents the longest path in the IATG and the average area provides a measure of the processor-time area required by the IATG. Based on these two formulas, processors are allocated to tasks according to the results obtained by solving the following minimization problem: subject to After solving the allocation problem, a scheduler is needed to schedule the tasks to obtain a minimum execution time. The classical approach is the well-known list scheduling paradigm [13] introduced by Graham, which schedules one processor tasks (tasks running only on one processor). Scheduling is known to be NP-complete for one processor tasks. Since then several other list scheduling algorithms were proposed, and the scheduling problem was also extended to multiple processor tasks (tasks that run non-preemptively on any number of processors) [7]. Therefore, multiple processor task scheduling is also NP-complete and heuristics are used. The intuition behind minimizing in equation (9) is that represents a theoretical lower bound on the time required to execute the image processing application corresponding to the IATG. The execution time of the application can neither be smaller than the critical path of the graph nor be less than the average area of the graph. As the TSAS's convex programming algorithm [7] for determining the number of processors for each task was not available, we have used in the experimental part of Section 5 the nonlinear solver based on SNOPT [17] available on the internet [18] for solving the previous problem. For solving the scheduling problem, the proposed scheduling algorithm presented in [7] is used. Another possibility is to use scheduling algorithms developed for data and task parallel graphs [8, 9]. 5 Experiments To evaluate the benets of the propose data parallel framework based on skeletons and also of the task parallel framework based on the IATG, we rst compare the code of the multi-baseline stereo vision algorithm with and without using skeletons (with and without data parallelism). Then we compare the speed-ups obtained by applying only data parallelism to the application, with the speed-ups obtained with both data and task parallelism. The multi-baseline stereo vision application uses an algorithm developed by Okutomi and Kanade [6] and described by Webb and al. [14, 15], that gives greater accuracy in depth through the use of more than two cameras. Input consists of three n n images acquired from three horizontally aligned, equally spaced cameras. One image is the reference image, the other two are named match images. For each of 16 disparities, 15, the rst match image is shifted by d pixels, the second image is shifted by 2d pixels. A dierence image is formed by computing the sum of squared dierences between the corresponding pixels of the reference image and the shifted match images. Next, an error image is formed by replacing each pixel in the dierence image with the sum of the pixels in a surrounding 13 13 window. A disparity image is then formed by nding, for each pixel, the disparity that minimizes error. Finally, the depth of each pixel is displayed as a simple function of its disparity. Figure 7 presents the IATG of this application. It can be observed that the computation of the dierence images requires point op- erators, while the computation of the error images requires neighborhood operators. The computation of the disparity image requires also a point operator. Input: ref, m1, m2 (the reference and the two match images) for d=0,15 Task T1,d: m1 shifted by d pixels Task T2,d: m2 shifted by 2*d pixels Task T5: Disparity image = d which minimizes the err image Pseudocode of the multi-baseline stereo vision application broadcast diff0 diff1 diff2 diff15 err0 err1 err2 reduce ref disparity image13171933 Fig. 7. Multi-baseline stereo vision IATG Below we present the sequential code of the application versus the data parallel code of the application. Coding the application by just combining a number of skeletons doesn't require much eort from the image processing user, yet it parallelizes the application. The data and task parallel code is slightly more dicult and we do not present it here. { Sequential code { { { ImagePointDist_3I_1O(d,"im","ref","m1", DT-PIPE code based on skeletons Besides creating the images on the host processor, the code is nearly the same, only the function headers dier. The skeleton have as parameters the name of the images, the window and the image operator, while in the sequential version operator headers have as parameters the images and the window. The skeletons are implemented in C using MPI [19]. The results of the data parallel approach are compared with the results obtained using data and task parallelism on a distributed memory system which consists of a cluster of Pentium Pro/200Mhz PCs with 64Mb RAM running Linux [16], and connected through Myrinet in a 3D-mesh topology, with dimension order routing. In the task parallel framework we use a special mechanism to register the images on the processors where they are rst created. Moreover, each skeleton has associated the task number to which it corresponds. We use 1, 2, 4, 8, 16, 32 and 64 processing nodes in the pool. Three articial reference images of sizes 256 256, 512 512 and 1024 1024 are used. The code is written using C and MPI message passing library. The multi-baseline stereo vision algorithm is an example of a regular well balanced application in which task parallelism can be applied without the need of an allocator of scheduler. Just for comparison reasons, we have used the algorithm described in [7] and we have obtained identical results (we divide the number of nodes to the number of tasks and we obtain the number of the nodes on which each task should run). In Figure 8 we show the speed-ups obtained for the data parallel approach for dierent image sizes. Figure 9 shows the speed-up of the same application using the data and task parallel approach, also for dierent image sizes. We can observe that the speed-ups become quickly saturated for the data-parallel approach while the speed-ups for the data and task parallel approach perform very good. In fact, we have pure task parallelism up to 16 processors and data and task parallelism from 16 on. So, the pure task parallel speed-ups will become attened from 16 processors on because at this type of application is better to rst apply task parallelism and then to add data parallelism. Using both data and task parallelism is more ecient than using only data parallelism. Processors Fig. 8. Speed-up for the data-parallel approac Processors81624324048 Fig. 9. Speed-up for the data and task parallel approach 6 Conclusions We have presented an environment for data and task parallel image processing. The data parallel framework, based on algorithmic skeletons, is easy to use for any image processing user. The task parallel environment is based on the Image Application Task Graph and computing the IATG communication and processing costs. If the IATG is a regular well balanced graph task parallelism can be applied without the need of these computations. We showed an example of using skeletons and the task parallel framework for the multi-baseline stereo vision application. The multi-baseline stereo vision is an example of an image processing application which contain parallel tasks, each of the tasks being a very simple image point or neighborhood operator. Using both data and task parallelism is more ecient than using only data parallelism. Our code for the data and task parallel environment, using C and the MPI-Panda library [19, 20] can be easily ported to other parallel machines. --R Parallel Algorithms for Digital Image Processing Parallel Programming "Algorithmic skeletons: structured management of parallel computations" Skeletons for structured parallel composition A multiple-baseline stereo A framework for exploiting task and data parallelism on distributed memory multicomputers Optimal use of mixed task and data parallelism for pipelined compu- tations CPR: Mixed Task and Data Parallel Scheduling for Distributed Systems A new model for integrated nested task and data parallel program- ming Fortran M: A language for modular parallel programming A task and data parallel programming language based on shared objects Bounds on multiprocessing timing anomalies Implementation and Performance of Fast Parallel Multi-Baseline Stereo Vision The Distributed ASCI supercomputer (DAS) site User's guide for snopt 5.3: A fortran package for large-scale nonlinear programming Lucent Technologies AMPL site "MPI - The Complete Reference, vol.1, The MPI Core" Experience with a portability layer for implementing parallel programming systems --TR Algorithmic skeletons Parallel algorithms Fortran M A new model for integrated nested task and data parallel programming A Framework for Exploiting Task and Data Parallelism on Distributed Memory Multicomputers A task- and data-parallel programming language based on shared objects Optimal use of mixed task and data parallelism for pipelined computations MPI-The Complete Reference A Multiple-Baseline Stereo CPR --CTR Development platform for parallel image processing, Proceedings of the 6th WSEAS International Conference on Signal, Speech and Image Processing, p.31-36, September 22-24, 2006, Lisbon, Portugal Antonio Plaza , David Valencia , Javier Plaza , Pablo Martinez, Commodity cluster-based parallel processing of hyperspectral imagery, Journal of Parallel and Distributed Computing, v.66 n.3, p.345-358, March 2006 Frank J. Seinstra , Dennis Koelma , Andrew D. Bagdanov, Finite State Machine-Based Optimization of Data Parallel Regular Domain Problems Applied in Low-Level Image Processing, IEEE Transactions on Parallel and Distributed Systems, v.15 n.10, p.865-877, October 2004

data parallelism;skeletons;image processing;task parallelism

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Approaches to zerotree image and video coding on MIMD architectures.

The wavelet transform is more and more widely used in image and video compression. One of the best known algorithms in image compression is the set partitioning in hierarchical trees algorithm which involves the wavelet transform. As today the parallelisation of the wavelet transform is sufficiently investigated, this work deals with the parallelisation of the compression algorithm itself as a next step. Two competitive approaches are presented: one is a direct parallelisation and the other uses an altered algorithm which suits better to the parallel architecture.

Introduction Image and video coding methods that use wavelet transforms have been successful in providing high rates of compression while maintaining good image quality and have generated much interest in the scientic community as competitors to DCT based compression schemes in the context of the MPEG-4 and JPEG2000 standardisation processes. Most video compression algorithms rely on 2-D based schemes employing motion compensation techniques. On the other hand, rate-distortion e-cient 3-D algorithms exist which are able to capture temporal redundancies in a more natural way [10, 5, 4, 16, 1]. Unfortunately, these 3-D algorithms often show prohibitive computational and memory demands (especially for real-time applications). Therefore, MIMD architectures seem to be an interesting choice for such an algorithm. A signicant amount of work has already been done on parallel wavelet transform algorithms for all sorts of high performance computers. We nd various kinds of suggestions for 1-D, 2-D and 3-D algorithms on MIMD computers for decomposition only [8, 19, 15, 13, 3, 6] as well as in connection with image compression schemes [9, 2]. Zero-tree based coding algorithms e-ciently encode approximations of wavelet coe-cients by encoding collections of neglectable (insignicant) coe-cients through single symbols. These collections are called zero-trees because of the tree-like arrangement of wavelet coe-cients which exploits the wavelet transform's self-similarity property. A parallelisation of the EZW algorithm { an important zero-tree coding scheme { is presented in [2] where two approaches are proposed: One is a straight-forward parallelisation which performs the EZW algorithm locally on each processing element (PE) for distinct blocks. This makes the resulting bit-stream (BS) incompatible with the sequential algorithm. The other approach reserves one PE for the collection of the symbols that have to be encoded. This PE performs a reordering of the symbols before it encodes them. This approach is similar to the approaches presented in this work and (as far as the author understands) is compatible with the sequential EZW. An improvement of the EZW is the SPIHT algorithm (Set Partitioning In Hierarchical Trees [14]). It is a well known, fast and e-cient algorithm which can also be used as 3D-variant in video compression [5]. However, this algorithm makes use of lists of coe-cients, which makes it hard to parallelise. This means that although consecutive list entries initially point to neighbouring wavelet transform coe-cients, the algorithm jumbles the entries after a while. Thus, there is no easy data driven parallelisation. An approach to packetise the SPIHT algorithm is presented in [17]. This is similar to the rst approach in [2]. The output of several executions of SPIHT on spacial blocks is multiplexed based on a rate allocation technique that approximates distortion reduction by fast SPIHT-specic statistics. However, the resulting bit-stream (BS) is not compatible with sequential SPIHT. This work concentrates on the parallelisation of the SPIHT algorithm and presents two competitive approaches. The rst is a direct parallelisation, i.e. the sequential algorithm is mapped to the parallel architecture without alteration of the sequential algorithm. Several \tricks" have to be found to overcome involved parallelisation di-culties. The second approach introduces a variant of SPIHT that involves a more spacially oriented coe-cient scan order and, thus, avoids the problems of the rst approach. Similar algorithms are proposed in [18, 11, 12]. Although a breadth-rst scan order through the trees of coe-cients [18] is PSNR-optimal, a depth-rst scan order [11, 12] is preferred here because of better spacial separability. 1.1 Parallel Wavelet Transform The fast wavelet transform can be e-ciently implemented by a pair of appropriately designed Quadrature Mirror Filters (QMF) consisting of a low-pass and a high-pass lter which decompose the original data set into two frequency-bands. These sub-bands are down-sampled by 2 and the same procedure is recursively applied to the coarse scale (low-pass ltered) sub-band. In the 2-D case consecutive ltering of rows and columns produces four sub-bands (eight in the 3-D case). Only the one which is ltered with the low-pass lter in each dimension is decomposed further. This is called the pyramidal wavelet transform. To perform the wavelet transform in parallel, the data has to be distributed among the PEs in some way. In this work data is splitted into slices (in the time domain). The ltering is performed in parallel on local data. Border data has to be exchanged before each decomposition step between neighbouring PEs due to the lter length. After that, transformed data is found distributed as shown in Figure 1. In contrast to the parallelisation of the wavelet transform as presented in a previous paper[7], the parallel wavelet transform used here dispenses with video data distribution as well as collection of transformed data. Initial data distribution is not necessary because input is performed in parallel (i.e. each PE reads its own part of the video data). Note that the speedups reported in this work do not include I/O operations as I/O is not viewed as a part of the algorithm. Another advantage of this is that we can drop the host-node paradigm because there is no extra single PE responsible for data distribution. Likewise, the collection of transformed data is not necessary because data is passed on to the coding part of the algorithm which is also performed in parallel. There is no redistribution of data required as we will see in section 2.2 and 3.2. (a) 2-D case (b) 3-D case Figure 1: Distribution of coe-cients or list entries after parallel wavelet transform. Dierent colours indicate dierent PEs 1.2 Zero-Trees Zero-tree based algorithms arrange the coe-cients of a wavelet transform in a tree-like manner (as in Figure 2), i.e. each coe-cient has a certain number of child coe-cients in another sub-band (mostly 4 in the 2-D, 8 in the 3-D case). We will use the following notations: o(p) The direct ospring of a coe-cient p, i.e. all coe-cients whose parent coe-cient is p. desc(p) All descendants of a coe-cient p. This includes o(p), o(o(p)) and so on. parent(p) The parent coe-cient of p. p 2 o(parent(p)). Furthermore, a zero-tree is a sub-tree which entirely consists of insignicant coe-cients. The signicance of a coe-cient is relative to a threshold which plays an important role in the SPIHT algorithm: The statistical properties of transformed image or video data (self-similarity) ensures the existence of many zero-trees. Sets of insignicant coe-cients can be encoded e-ciently with the help of zero- trees. We will see that sometimes the root coe-cient of the subtree (or even its direct ospring) does not have to be insignicant. Zero-trees can be viewed as a collection of coe-cients with approximately equal spacial position. While this fact implies that the coe-cient's signicances are statistically related, which is exploited by the SPIHT algorithm, this also means that zero-trees are local objects, corresponding to the data distribution produced by the parallel wavelet transform (see Figure 2). This can be exploited by the parallelisation of the zero-tree algorithms (see Section 2.2 and 3.2). Parallelisation without Algorithm Alteration 2.1 The SPIHT Algorithm Although the SPIHT algorithm is su-ciently explained in the original paper[14], it is helpful in this context to reformulate the algorithm. Figure 2: After parallel decomposition, data is distributed in a way so that each zero-tree resides on a single PE threshold ll LIS, LIP with approximation subband set LSP empty for each renement step threshold threshold =2 process LIS process LIP process LSP (a) Pseudo code LIS LIP LSP Coefficients process process process BS sig., sign ref.-bits sig., sign (b) Data ow graph Figure 3: The SPIHT algorithm Signicance information is represented by three lists: LIS List of insignicant set of pixels An entry in this list can be of two types: Type Type LIP List of insignicant pixels LSP List of signicant pixels The LIS basically contains all zero-tree roots. An entry of type A corresponds to an insignicant sub-tree without its root. An entry of type B corresponds to an insignicant sub-tree without its root and the root's direct ospring. The LIP contains all insignicant coe-cients that are not part of any zero-tree in the LIS. The LSP contains all signicant coe-cients. LIS LIP LSP BS Initialisation Before reading separator After processing separator End LIS LIP LSP BS LIS LIP LSP BS LIS LIP LSP BS Figure 4: Functionality of separators. Four states of the three lists and the bit-stream while processing the LIS. The algorithm is shown at a coarse level in Figure 3. Initially, the threshold is greater than all coe-cients. Thus, the LIS and the LIP are lled with the approximation sub-band's coe-cients, and the LSP is empty. After that, each entry of each list has to be tested for a change of signicance, and the result of the test has to be encoded as a bit in the bit-stream (BS). If, for instance, a type A entry of the LIS turns out not to to be insignicant any more (to be precise: its descendants), a zero bit has to be written into the bit-stream, and the entry has to be deleted from the LIS, inserted as a type B entry at the end of the LIS and its direct ospring has to be inserted at the end of the LIP. All entries inserted at the end of a list are also processed in the same renement step until no more entries are left. Figure 3(b) shows a data ow graph for a renement step. The decoding process performs the same algorithm. It does not, however, evaluate the signi- cance of the list entries, but simply reads this information from the bit-stream and approximates the value of the corresponding coe-cient as good as it can. 2.2 SPIHT Parallelisation When parallelising the SPIHT algorithm, we have to face the problem that it uses lists of coe-cient positions and is, therefore, inherently sequential. The reason is that it is hard to perform general list operations on distributed lists. Nevertheless, the set of distinct list operations involved in the SPIHT algorithm is limited. This enables us to develop an e-cient way to manage distributed lists of coe-cients. 2.2.1 Separators The basic operations of the algorithm are: Moving an iterator all through a list, deleting elements at iterator position and appending elements at the end of a list. The aim is to distribute the list so that each PE-local entry corresponds to a local coe-cient where coe-cients are distributed among the PEs as shown in Figure 1. This is a simple task for initial distribution. However, as coe-cients are appended to the end of lists, one has to provide a mechanism to indicate which parts of a list belong to which PE { or from a PE's view: where a sequence of local coe-cients ends and parts of another PE's list should be inserted. This work is done by separators (see Figure 4). The idea is to insert a separator at the end of each part of the list which entirely belongs to a single PE. Initially, the approximation sub-band is split into equal slices. Each slice is assigned to a single PE. On each PE, the (local) lists LIS and LIP are lled with all coe-cients from the PE's (local) slice, and a separator is appended to the end of each list. From here on, the sequential algorithm is performed locally with one exception: Each time the iterator meets a separator, the separator is copied to the end of each destination list. A destination list is a list into which entries are potentially inserted during the current list processing (see Figure 3(b)). Applying this principle, the lists L i on PE i are split by separators into parts L ij such that the assembled list is identical to the list the sequential algorithm would produce. The same is true for the bit-stream. This enables the parallel algorithm to assemble the bit-stream correctly after each PE has encoded its part of the wavelet coe-cients. An important question is when the processing of a list (i.e. a renement step) is completed. Essentially, the procedure can stop if it has processed the last non-separator entry in the list. Unfortunately, this does not guarantee that each PE produces the same number of separators. However, this is a necessary condition for the correctness of the parallel algorithm because, other- wise, the correct order of the list-parts would be lost. Therefore, the global maximum number of separators has to be calculated (which unfortunately synchronises the PEs), and the lists have to be lled up with separators before the algorithm continues with the next renement step (in fact even before each processing of a list). As a matter of fact, the number of separators grows exponentially with the number of renement steps, and very often separators appear in a row together in the list. To avoid unnecessary memory demands, consecutive separators should be kept together in a single entry associated with a counter. This means, if a separator entry (containing a counter) is inserted at the end of a list where a separator entry (also containing a counter) is already sitting, their counters can simply be added. 2.2.2 Algorithm Termination Another problem is the termination of the whole algorithm. In the sequential case, the process terminates when the required number of bits have been written to the bit-stream. In the parallel case, this test (as it is a global test) can only be executed at the end of a renement step. Thus, the parallel algorithm potentially generates too much bits which of course decreases the speedup in inconvenient cases. If necessary, super uous bits can simply be cut after assembling the bit-stream (due to the nature of the SPIHT algorithm). The procedure of assembling the bit-stream (after collecting the PE-local bit-streams) is the only sequential part of the algorithm. Unfortunately, it gets more complicated and, therefore, consumes more calculation time when the number of PEs is increased. The result will be a signicant decrease in speedup. 2.2.3 Parallel SPIHT Decompression Note that for the reverse algorithm { the reconstruction of video data from the bit-stream { the methods described above are not applicable. Although it is not part of this work, we can shortly outline the ideas how to implement a parallel SPIHT decoder: First of all, the whole bit-stream has to be copied to all PEs. Also, each PE has to process all bits of the bit-stream, independent of whether they belong to local coe-cients. Therefore, the global lists have to be kept at each PE. The only speedup potentials are: Not adjusting non-local coe-cients which involves oating point operations. threshold _ _ desc(p) off(p) Figure 5: Predicates used in the algorithm Keeping consecutive non-local list entries (entries belonging to non-local coe-cients) together in a single entry associated with a counter (similar to separators). This is possible because position information is not needed for non-local entries. Although this seems to be a very simple approach, it does not imply the necessity of process synchronisation and it does not contain a sequential part. 3 Parallelisation with Algorithm Alteration The approach described above reveals some drawbacks as e.g. complicated bit-stream handling, additional communication needs and non-neglectable sequential code parts. This is a direct consequence of the fact that the SPIHT algorithm is inherently sequential. Therefore, we will modify the sequential algorithm itself. Although the resulting bit-stream will not be compatible with SPIHT, the parallelisation of the altered algorithm will, of course, be compatible with its sequential version. The basic idea is to substitute the lists of coe-cient positions involved in the algorithm by bitmaps indicating the membership of each coe-cient to a certain list. As a result, list iteration, which is used frequently to process the list entries, is turned into a normal scan of coe-cients that follows a certain spacial direction. Thus, the data driven parallelisation can be performed more easily by a loop parallelisation of the coe-cient scan. 3.1 Zero-Tree Compression with Signicance Maps (SM) In the following, we will use three logical predicates A(p), B(p) and C(p) which are dened as in Figure 5. A(p) simply denotes the signicance of the coe-cient p. B(p) is true if and only if at least one of p's descendants is signicant, while C(p) denotes the same but does not include the direct ospring of p. This is visualised in Figure 5. The state of signicance of a given set of coe-cients can be described by these predicates in terms of zero-trees. Corresponding to these predicates, we will use the mappings a, b and c which essentially represent the same as A, B and C. The dierence is that A, B and C immediately change their values if the threshold is changed and are, therefore, implemented as a function/procedure in the used programming language. a, b and c have to be updated explicitly and are, therefore, implemented as array of Boolean values. We will call a, b and c \signicance maps" (SM). They substitute the lists of coe-cients (see section 2.1). The algorithm is responsible for the equality of a, b, c and A, B, C respectively while the ProcessAll := threshold set a, b and c to all false for each renement step threshold threshold =2 for p in approximation-subband c) := if a p then Rene(p) else a p A(p) if for q in o(p) Figure based Zero-tree coding algorithm threshold is successively decreased by a factor of 1 2 . This should be done by avoiding the evaluation of A, B and C as far as possible because { following the idea of the SPIHT algorithm { the result of each evaluation will be encoded into the bit-stream as one bit to allow the decoder to reproduce the decisions the encoder has made. The algorithm that obeys these rules is shown in Figure 6. The outer loop is the renement loop which divides the threshold by 2 in each iteration. This is exactly the same as in the original algorithm. Within this loop, the algorithm navigates through the set of coe-cients along trees of coe-cients in a depth-rst manner (this is the major dierence between the SM based algorithm and SPIHT) starting at the set of coe-cients contained in the approximation sub-band. This is accomplished by the recursive procedure \ProcessCoe". For each coe-cient p, the state of a, b and c is checked one after another: a p A(p) has to be evaluated only if a p is false because the transition true 7! false is not possible for a p . If A(p) is true then the sign of p has to be encoded as well. If a p is already true the procedure \Rene" is called which en/decodes another bit of the coe-cients value to rene the decoded approximation. p B(p) has to be evaluated only if, again, b p is false and ^ C(p) has to be evaluated only if c p is false and b p is true because last, the recursion to the child coe-cients only has to be performed if b p is true (for an obvious reason). The decoding algorithm looks exactly the same. The only dierence is that instead of encoding the results of the evaluation of A, B and C, this information is read from the bit-stream. Together with the sign- and renement-bits, this is enough information to enable the decoder to perform the same steps as the encoder and approximate the coe-cients with an error below the threshold. Note that this algorithm encodes the same information (in fact the same bits) as the SPIHT algorithm. The order of the bits is the only dierence. This means that at the end of each renement step, the compression performance is equal to that of SPIHT. In between, the order of the bits that are written to the bit-stream is crucial because the bits can have dierent eect on the decoded image. Thus, it is important to encode bits with greater eect rst. Figure 7 shows the comparison of the PSNR-performance for the well-known \Lena"-image (2-D case). The algorithm shows major drawbacks with respect to the original SPIHT which, nevertheless, can almost be overcome by scanning the set of coe-cients in several passes. The rst pass should only process those coe-cients that are not part of zero-trees but not signicant. Subsequent passes check the state of zero-tree roots and process those coe-cients that emerge from decomposed zero-trees. This method is denoted \sophisticated" in Figure 7. Nevertheless, this improvement of the algorithm is not used in this work in parallelisation investigations. PSNR bpp SM simple sophisticated Figure 7: PSNR of the SM based algorithm compared to the original SPIHT 3.2 Parallelisation In contrast to the original SPIHT algorithm, the parallelisation of the SM based algorithm is easy. Again, it is based on the fact that after the parallel wavelet transform, data is distributed in a way so that zero-trees are local objects (see Figure 2). All we have to do is to parallelise the inner loop in the procedure ProcessAll (which reads \for p in approximation-subband") according to the data distribution of the approximation sub-band (see Figure 1). All other computations within a renement step are localised, i.e. PE-local computations do not depend on data of neighbouring PEs. So, no communication is required within a renement step. Each PE produces one continuous part of the bit-stream for each renement step. At the end, these parts have to be collected by a single PE and assembled properly (i.e. in an alternating way). As in the direct SPIHT parallelisation, this is a major bottleneck. However, the number of bit-stream parts is reduced signicantly, which should speed up the bit-stream assembly. Because of the termination problem (see section 2.2) the PEs again have to synchronise at the end of each renement step to determine if the global number of bits produced so far is su-cient. However, this synchronisation can be dropped if the termination condition is not the bit-stream size but a xed number of renement steps. 4 Experimental Results Experimental results were conducted on a Cray T3E-900/LC at the Edinburgh Parallel Computing Centre using MPI. Video data size is always 864 frames with 88 by 72 pixels. The video sequence used here is the U-part of \grandma". The wavelet transform is performed up to a level of 3. Note that the data size is limited by memory constraints in the case when the number of PEs is 1 and a single PE has to hold all video data. The number of frames has to be high to enable uniform data distribution for parallelisation as well as down-scaling for the wavelet transform. output bpp overall coding Figure 8: Sequential speedup of the SM based algorithm with respect to SPIHT Thus, the frame size is small. In a real-world scenario, however, the frame size can be bigger. For the same reason, frame size scalability is di-cult to measure. Nevertheless, linear scalability is assumed due to the authors experiences and the fact that the execution time of the coding part does not depend on the video data size but on the number of output bits only. First of all, we have to look at the sequential performance of the SM based algorithm because if it was slower than the original SPIHT, its parallelisation would not make sense. However, Figure 8 shows that it outperforms the original SPIHT especially for higher bit-rates. On the other hand, this means that it is even harder to get reasonable speedups. Figure 9 shows speedups for a xed compression rate: 0:14 bpp (bits per pixel, pixels in dierent frames are counted as dierent pixels). The fact that the speedup curves are not smooth, i.e. have discontinuities at and 54, is caused by the divisibility of 864 { the length of the video sequence which determines the size of the local data sub-sets. Note that due to the depth of the wavelet transform (3), this size is divided by 8 and the resulting number is then divided by the number of PEs which is not always possible without remainder. The sequential bit-stream assembly takes more and more execution time for higher numbers of PEs. Its share in execution time gets higher than 50% of the coding part. Note at this point that in a particular hardware implementation, the bit-stream assembly can be integrated in the output module and separated from the actual coding. The speedups of the two dierent algorithms are about the same. This shows that the complicated bit-stream assembly - which is the main problem of a direct SPIHT parallelisation - could be solved e-ciently. Some dominant problems, such as PE synchronisation and sequential code parts, are present in both approaches. However, the SM based algorithm is expected to gain a lower parallelisation e-ciency for two reasons: It owns the same communication overhead while the sequential algorithm is faster. There is less potential for positive caching eects in the parallelisation because optimal cache utilisation (due to the spacially oriented coe-cient scan) is supposed to be the main reason that the SM based algorithm is faster. overall spiht (a) overall coding (b) SM based algorithm Figure 9: Speedups for varying #PE and xed compression rate (0.14 bpp). The fact that the parallelisation e-ciency is about the same strongly suggests that the bit-stream assembly in the parallel SM based algorithm is more e-cient. Figure shows speedup curves for xed #PE and varying compression rate. Of course, the execution time of the wavelet decomposition does not depend on the compression rate. The reason for the speedup breakdowns at certain compression rates is the termination problem of the parallel algorithm (see section 2.2.2). The parallel algorithm is optimal only at compression rates achieved at the end of a renement step. Note that although the speedup of the coding part increases with the bit-rate, the overall speedup remains constant or drops slightly because the share in execution time of the coding part increases with the bit-rate. The problem of unevenly distributed complexity is illustrated in Figure 11. Here, approximately the rst half of the video sequence is substituted by the \car-phone" sequence which contains much more motion than the \grandma" sequence. This causes bigger coe-cient values at higher frequency sub-bands for the more complex video parts. Thus, more coe-cients have to be processed within a renement step, which makes the algorithm consume more computation time. A load balancing problem is the consequence. One can clearly see that the necessity of process synchronisation at several points in the algorithm leads to an increase of idle times of PEs waiting for other PEs. Figure 12 shows the speedups that are { compared to Figure 9 { slightly reduced because of this problem. Conclusions We have seen how an inherently sequential zero-tree coding algorithm can be parallelised. Although the speedups are not overwhelming, the presented way of parallelisation prevents the necessity to perform the coding sequentially. Thus, reasonable speedups are possible for higher numbers of processing elements and the whole range of compression rates. There are two methods for parallelisation: Either by the use of so called separators or by output bpp decomposition overall spiht (a) SPIHT, 8 PEs26100.004 output bpp decomposition overall coding (b) SM based, 8 PEs5150.004 output bpp decomposition overall spiht (c) SPIHT, output bpp decomposition overall coding (d) SM based, output bpp decomposition overall spiht output bpp decomposition overall coding Figure 10: Speedup for decomposition, coding and overall speedup for varying compression rate. (a) evenly distributed complexity13579 (b) unevenly distributed complexity Figure Execution scheme of decomposition and SPIHT coding for 10 PEs. Time (on horizontal axis) is measured in milliseconds. (Nearly) vertical black lines indicate data transfer. Horizontal grey bars indicate calculation phases.515253510 20 overall spiht Figure 12: Speedups for video with unevenly distributed complexity (at xed compression rate (0.14 bpp)). rewriting the algorithm to t better into the parallel architecture (signicance map based algo- rithm). The rst method is more complicated but guarantees compatibility with original SPIHT bit-streams. The second method shows very similar speedup results but better execution times. Although unevenly distributed image/motion complexity can decrease the speedup potential, this eect seems to keep within limits. Acknowledgments The authors would like to acknowledge the support of the European Commission through TMR grant number ERB FMGE CT950051 (the TRACS Programme at EPCC). The author was also supported by the Austrian Science Fund FWF, project no. P13903. --R Image coding using parallel implementations of the embedded zerotree wavelet algorithm. On the scalability of 2D discrete wavelet transform algo- rithms An embedded wavelet video coder using three-dimensional set partitioning in hierarchical trees (SPIHT) Parallel algorithm for the two-dimensional discrete wavelet transform Hardware and software aspects for 3-D wavelet decomposition on shared memory MIMD computers Optimization of 3-d wavelet decomposition on multiprocessors Parallelization of the 2D fast wavelet transform with a space- lling curve image scan Video compression using 3D wavelet transforms. Listless zerotree coding for color images. 3d listless zerotree coding for low bit rate video. Scalability of 2-D wavelet transform algorithms: analytical and experimental results on coarse-grain parallel computers Vector and parallel implementations of the wavelet transform. Multirate 3-D subband coding of video image compression without lists. Parallel discrete wavelet transform on the Paragon MIMD machine. --TR On the Scalability of 2-D Discrete Wavelet Transform Algorithms Hardware and Software Aspects for 3-D Wavelet Decomposition on Shared Memory MIMD Computers An Embedded Wavelet Video Coder Using Three-Dimensional Set Partitioning in Hierarchical Trees (SPIHT) --CTR Roland Norcen , Andreas Uhl, High performance JPEG 2000 and MPEG-4 VTC on SMPs using OpenMP, Parallel Computing, v.31 n.10-12, p.1082-1098, October - December 2005

video coding;wavelets;zerotree;MIMD

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Parallel computation of pseudospectra by fast descent.

The pseudospectrum descent method (PsDM) is proposed, a new parallel method for the computation of pseudospectra. The idea behind the method is to use points from an already existing pseudospectrum level curve to generate in parallel the points of a new level curve such that > . This process can be continued for several steps to approximate several pseudospectrum level curves lying inside the original curve. It is showed via theoretical analysis and experimental evidence that PsDM is embarrassingly parallel, like GRID, and that it adjusts to the geometric characteristics of the pseudospectrum; in particular it captures disconnected components. Results obtained on a parallel system using MPI validate the theoretical analysis and demonstrate interesting load-balancing issues.

a region of the complex plane and ii) compute min (zI A) for every node zh . The boundary curves @ (A) are obtained as the contour plots of the smallest singular values on the mesh h . The obvious advantage of GRID is its straightforward simplicity and robustness. If we let for the moment C min be This work has been partially supported by the Greek General Secretariat for Research and Development, Project ENE 99-07 a measure of the average cost for the computation of min (zI A), it is easily seen that the total cost of GRID can be approximated by where denotes the number of nodes of h . The total cost quickly becomes prohibitive with the increase of either the number of nodes or the size of A. It has been observed that the cost formula (2) readily indicates two major methods for accelerating the computation: a) reducing the number of nodes z and hence the number of evaluations of min , and b) reducing the cost of each evaluation of min (zI A). Both approaches are the subject of active research; see [15] for a comprehensive survey of recent eorts. The use of path following, a powerful tool in many areas of applied mathe- matics, in order to compute a single boundary curve @ (A) was suggested by Kostin in [8]. It was M. Bruhl in [5] who presented an algorithm to that end and showed that signicant savings can be achieved compared to GRID when seeking a small number of boundary curves. The key is that tracing a single boundary curve, drastically reduces the number of min evaluations. Bekas and Gallopoulos carried this work further in Cobra [2], a method that ameliorated two weaknesses of the original path following method, in particular i) its lack of large-grain parallelism and ii) its frequent failure near sharp turns or neighboring curves. In particular, parallelism was introduced by incorporating multiple, corrections. More recently, D. Mezher and B. Philippe suggested PAT, a new path following method that reliably traces contours [13]. Despite its advantages over the original method of [5], Cobra maintains two of the path following approach vis-a-vis GRID. These are a) that in each run, a single boundary curve @ (A) is computed, b) that is closed. Therefore, only one curve is computed at a time and its disconnected components are not cap- tured, at least in a single run. While we can consider the concurrent application of multiple path following procedures to remedy these problems, the solution is less straightforward than it sounds, especially for (b). In the remainder of this paper, when our results do not depend on the exact version of path following that we choose to use, we denote these methods collectively by PF. In this paper we propose the Pseudospectrum Descent Method (PsDM) that takes an approach akin to PF but results in a set of points dening pseu- dospectrum boundaries for several values of . PsDM starts from an initial contour @ (A), approximated by N points z k computed by a PF method. These points are corrected towards directions of steepest descent, to N points . The process can be repeated recursively, without the need to reuse PF, and computes pseudospectra contours. PsDM can be viewed as a \dynamic" version of GRID where the stride from mesh point to mesh point has been replaced by a PF Prediction-Correction step. To illustrate this fact and thus provide the reader with an immediate feeling of the type of information obtained using the method, we show in Figure 1 the results from the application of PsDM to matrix kahan of order 100 which has been obtained from the Test Matrix Toolbox ([7]) and is a typical example of matrices with interesting Y Figure 1: Pseudospectrum contours and trajectories of points computed by PsDM for @ (A); kahan of order 100. Arrows show the directions used in preparing the outermost curve with path following and the directions used in marching from the outer to the inner curves with PsDM. See section 4 for further results with this matrix. pseudospectra. The plot shows i) the trajectories of the points undergoing the steepest descent and ii) the corresponding level curves. The intersections are the actual points computed by PsDM. We note that the idea for plotting the pseudospectrum using descent owes to an original idea of I. Koutis for the parallel computation of eigenvalues using descent described in [3]. The rest of this paper is organized as follows. Section 2 brie y reviews path following. Section 3 describes PsDM. Section 4 illustrates the characteristics of PsDM as well as its parallel performance using an MPI implementation. We also introduce adaptivity and show that the method can reveal disconnected components. Section 5 presents our conclusions. 2 Review of Path Following Consider the function G According to denition of the pseudospectrum, the zeros of G(z) are points of the boundary @ (A). Allgower and Georg in [1] describe a generic procedure Figure 2: A generic Prediction-Correction PF scheme. to numerically trace solution curves of equations such as (3). In Table 1 we outline the algorithm and illustrate one step in Figure 2. See [2, 5] for more details. z 0 on @ (A). for k=1,. (* Prediction phase *) 1.1 Determine a prediction direction p k . 1.2 Choose a steplength h and predict the point ~ (* Correction phase *) 2.1 Determine a correction direction c k . 2.2 Correct ~ z k to z k using Newton iteration. Table 1: PF generic procedure for the computation of a single contour. In the sequel, as in [5], we would be identifying the complex plane C with R 2 and frequently use the notation G(z) for G(x; y). Critical to the eective use of PF is the availability of the gradient rG(x; y); that this becomes available at little cost follows from the following important result (cf. [5, 6]): Theorem 2.1 Let z 2 C n(A). Then G(x,y) is real analytic in a neighborhood of is a simple singular value. Then the gradient of G at z is min u min ); =(v min u min )): (4) where denote the left and right singular vectors corresponding to min (zI A). Therefore, assuming that the minimum singular triplet A (in the sequel we would be referring to it simply as \the triplet at z") has been computed, the gradient is available with one inner product. From the above it follows that the dominant eort in one step of the PF scheme proposed in [5] amounts to i) the estimation of the prediction direction and ii) the Newton iteration of the correction procedure. Regarding (ii) we note that it is su-cient (cf. [5, 2]) to use a single Newton step, requiring one triplet evaluation at ~ z k . Then, as proposed in [5], ~ z k is corrected towards the direction of steepest descent c k , i.e. z k min where the triplet ( min ; u min ; v min ) is associated with ~ z k . Regarding (i), note that selecting p k to be tangential to the curve at z k 1 requires the triplet at z k 1 . Since this would double the cost, it has been shown acceptable to take p k orthogonal to the previous correction direction c k 1 ; cf. [5, 2]. Therefore the overall cost of a single step of the original method of [5] is approximately equal to the cost for computing the triplet. 3 The Pseudospectrum Descent Method Let us now assume that an initial contour @ (A) is available in the form of a some approximation (e.g. piecewise linear) based on N points z k previously computed using some version of PF. In order to obtain a new set of points that dene an inner level curve we proceed in two steps: 1: Start from z k and compute an intermediate point ~ w k by a single modied Newton step towards a steepest descent direction d k obtained earlier. Step 2: Correct ~ w k to w k using a Newton step along the direction l k of steepest descent at ~ Figure 3 illustrates the basic idea for a single initial point. Applying one Newton step at z k would require r min ((x k therefore, a triplet evaluation at z k . To avoid this extra cost, we apply the same idea as PF and use instead min q min ), that is already available from the original path following procedure. In essence we approximated the gradient based at z k with the gradient based at ~ z k . Vectors g min ; q min are the right and left singular vectors associated with min (~z k I A). Applying correction as in (5) it follows that ~ min g min Figure 3: Computing @ - (A); - < . is the new pseudospectrum boundary we are seeking. Once we have computed ~ we perform a second Newton step that yields w min ( ~ where the triplet used is associated with ~ w k . These steps can be applied to all N points in what we call one sweep of PsDM; we denote it by PsDM and outline it in the following Table. Starting from an initial contour @ (A) we have shown of the initial contour @ (A). on the target contour @ - (A); - < . for k=1,. ,N 1. Compute the intermediate point ~ according to (6). 2. Compute the target point w k using (7). Table 2: PsDM: One sweep of PsDM. how to compute points that approximate a nearby contour @ - (A); - < . Assume now that the new points computed with one sweep of PsDM dene satisfactory approximations of @ - (A) (cf. end of section). We ask whether it would it be practical to use these points to march one further step to approximate another As noted in the previous discussion, the application of the sweep PsDM uses r min ((~x k i.e. the triplet at ~ z k . This is readily available when the curve @ (A) is obtained via PF. Observe now that as the sweep proceeds to compute @ ~ - (A) from @ - (A), it also computes the triplet at ~ w k . Therefore enough derivative information is available for the sweep PsDM to proceed one more step with starting points computed Method PsDM approximating a contour @ (A). approximating M contours for i=1,. ,M Compute points of @ - i by PsDM on the N points of @ - i 1 Table 3: The PsDM method. via the previous application of PsDM. Therefore, it is not necessary to run PF again. Continuing with this repeated application of sweeps PsDM, we obtain the promised pseudospectrum descent method, outlined in Table 3 and illustrated in Figure 4. It is worth noting that each sweep can be viewed as a map that takes as input N in points approximating - i (A) and produces N out points approximating In the description so far N in = N out , but as we show in Section 4.4, this is not necessarily an optimal strategy. We nally note that the two step process described above was found to be necessary. By contrast, the more straightforward approach in which compute the exact steepest descent directions from each point at the starting curve and use those values directly to approximate the next set of points in one step required a much smaller stepsize to be successful. We did not experiment further with this procedure. Cost considerations It is evident that the cost for the computations of the intermediate points ~ relatively small since the derivatives at ~ z have already been computed by PF or a previous sweep of PsDM. Furthermore, we have assumed that min (z k I the points z k approximate @ (A). On the other hand, computing the nal points evaluations. To avoid proliferation of symbols, we use C min to now denote the average cost for computing the triplet. Therefore, we approximate the cost of a single sweep of PsDM by C NC min . The computation of each target point from the computation of all other target points w k. On a system with P processors, we can assign the computation of at most dN=P e target points to each processor; one sweep will then proceed with no need for synchronization and communication and its total cost is approximated by C We discuss this issue in more detail in Section 4.3. We emphasize that typically, the number of points N on each curve is expected to be large, therefore the algorithm is scalable. This is better than the PF methods described in the introduction: Cobra allows only a moderate number of independent calculations of min per step while in the original version of PF, coarse-grain parallelism Figure 4: Descent process for a single point becomes available only if we attempt to compute dierent curves in parallel. analysis for a single sweep of PsDM In order to gauge the quality of the approximation of the curve obtained in a single sweep of PsDM, it is natural to use as a measure the value which we call \stepsize of the sweep". Let be the point approximated starting from a point z 2 @ - i (A). Dene the function G 0 be the point obtained using a single Newton step at ~ as depicted in Figure real analytic in a domain D if the minimum singular value min (wI A) is simple for all w 2 D. Let all ve points ~ z; z; ~ Fig. 5) lie in the interior of such a domain. From the relations z and ~ z it follows that w 0 can also be considered as the outcome of two exact Newton steps originating from ~ z. From the identity (w 0 I and standard singular value inequalities, it follows that min (w 0 I and therefore From results of Levin and Ben-Israel (see [12]) regarding Newton's method for underdetermined systems, under standard assumptions there will be a region in which the Newton iteration used to produce w 0 from ~ w will converge quadratically; we next assume that we are within this region. Then jw 0 wj 1 jw 0 ~ w y )k); note that 1=krG 0 ( ~ w y )k is at least 1, since G 0 ( ~ This gure can be considered as an enlargement of Figure 3 that reveals that the point computed with a single Newton step in the sweep produces only an approximation of w k . Figure 5: The transformation of z to w 0 via PsDM. is the inner product of singular vectors (cf. Theorem 2.1). Therefore, using relations (7) and (9) it follows that wI A). It follows that if this minimum is not much smaller than 1 and the assumptions made above hold, then the error induced by one sweep of PsDM will be bounded by a moderate multiple of the square of the stepsize of the sweep. A full scale analysis of the global error, for multiple sweeps of PsDM, is the subject of current work. 4 Renements and numerical experiments We conducted experiments with matrices that have been used in the literature to benchmark pseudospectra algorithms. Our results, presented below, indicate that PsDM returns the same levels of accuracy as GRID at a fraction of the cost. We then describe a parallel implementation of PsDM. The speedups obtained underscore the parallel nature of the algorithm. We also show that the application of PsDM to large matrices for which the triplets have to be computed via some iterative method is likely to suer from load imbalance and speedups that are than what one would expect from an embarassingly parallel algorithm; we describe a simple heuristic to address this problem. Finally we discuss some other properties that underline the exibility of PsDM. We show in particular that: i) adaptation of the number of points computed in each sweep of PsDM can lead to signicant cost savings, and ii) PsDM can capture disconnected components of the pseudospectrum lying inside the initial boundary computed via PF. 4.1 System conguration We performed our experiments on a SGI Origin 2000 system with 8 MIPS R10000 processors. The system had a total of 768 MB RAM and 1MB cache per processor, running IRIX 6.5. The codes were written in Fortran-90 using F90/77 MIPSpro version 7.2.1 compilers. For the parallelization we used the MPI programming paradigm, implemented by SGI's MPT 1.2.1.0. We used ARPACK [11] to approximate the triplets and SPARSKIT [14], suitably modied to handle double precision complex arithmetic, for sparse matrix{vector multiplies. All our experiments were conducted in single-user mode. 4.2 Numerical experiments with PsDM We remind the reader that for matrices with real elements, the pseudospectrum curves are symmetric with respect to the real axis, and it has become standard in the pseudospectrum literature to measure the success of methods for pseudospectra by direct comparison of the one half of the gure computed with a new method and the other half with GRID. Therefore, in subsequent experiments with any method we only compute points and curves lying on the upper or lower half of the complex plane. We start with the same matrix used to obtain Figure 1, that is kahan of order 100. We used the parallel version of PF, namely Cobra, to obtain points to approximate (one half of) the pseudospectrum curve corresponding to asked PsDM to compute 60 contours corresponding to values of ranging from - using a stepsize that remained equal 6. The upper half of gure 6 illustrates the contours corresponding to while the the lower half illustrates the same contours computed using GRID on a 100 100 mesh of equidistant points. We chose this resolution in order to allow GRID to oer a level of detail that is between the minimum and maximum resolution oered by PsDM. In particular, the distance between neighboring points of GRID was selected near the median of the smallest and largest distance of neighboring points computed in the course of PsDM. The pictures are virtually indistinguishable and indicate that PsDM achieves an accuracy comparable to GRID. On the other hand, the cost is much smaller; in particular PsDM approximates the contours using 2700 points while GRID used 10000 mesh points. It is also worth noting that despite the fact that GRID does not require the computation of singular vectors, in the context of iterative methods such as ARPACK the extra cost is not signicant and PsDM is expected to be far less expensive. In order to further analyze the accuracy of PsDM, we computed the relative error jmin at each computed point z of the (approximate) curves Y PsDM GRID Figure Selected pseudospectrum contours @ (A); kahan of order 100 computed by PsDM (top) and GRID (bottom). produced by the algorithm; from these values, we obtained the maximum and mean relative errors for each contour point and show the results in Figure 7. Note that the best way to read this gure is from left to right, as this shows how the maximum and mean errors per curve develop as we consider the curves from the outermost to the innermost. The maximum relative error appears to be satisfactory except in a few cases, where the maximum error even then, however, the mean error remains two orders of magnitude smaller. This highlights one observation we made in our experi- ments, namely that only a very limited number of points suer from increased relative error. We also note that there is an apparent increase in the error as we approach very small values of . This does not re ect a weakness of PsDM, but the di-culty of the underlying SVD method, ARPACK in this case, to compute approximations of the minimum singular value with very small relative error. 4.3 Parallel performance One important advantage of PsDM is that, by construction, it is embarrassingly parallel: Each sweep can be split into a number of tasks equal to the number of points it handles and each task can proceed independently with its work, most of it being triplet computations. Furthermore, assuming that no adaptation is used, no communication is needed between sweeps. Let us assume that this is the case and that the number of points computed in each sweep and therefore the number of tasks is constant, say N . It is thus natural to use static partitioning e Relative error Maximum relative error Mean relative error Figure 7: Maximum and mean relative errors for each curve (60 total) computed via PsDM for (*Input*) N points z k on the starting number of processors 1. For each processor, select and assign the sweep computations for dN=P e points. 2. Each processor proceeds with PsDM on its own set of points. 3. A marked processor gathers the results of all processors. Table 4: Parallel PsDM with static partitioning. of tasks, allocating to each of the P processor those tasks handling the sweep for approximately dN=P e points. Table 4 outlines the method. The next question is how to allocate tasks to processors. One natural idea is to use \static block partitioning" and allocate to each processor the computations corresponding to dN=P e consecutive points, say z We applied PsDM on matrix kahan(100) from Section 4.2, starting from points on the initial curve @ 0:1 (A) and computed curves. Therefore, the total number of points that are computed by the end of the run was In these and subsequent performance results we did not take into account the time taken by Cobra to approximate the initial curve. Table 5 depicts the corresponding execution times and speedups. The improvements are sunstantial and the speedups re ect the parallel nature of the # of Processors Time (secs) 264 135 70 36 Table 5: Performance of parallel implementation of PsDM for kahan(100)). Table Number of points computed by each processor by the time the rst processor nishes its share of the workload under static block partitioning. processors are used to compute curves for kahan(100). Numbers in boldface denote the number of points computed by the processor that nished rst. algorithm. In order to better understand the eect of the static task allocation policy, we marked which processor would nish rst and then examined how much work had been accomplished by then in each of the remaining processors. The results are shown in Table 6, where in boldface are shown the number of points that have been completed by the processor that nished rst. Line 1, for instance, shows that processor 0 nished rst (the number 600 is in boldface) while, by that time, processor 1 had accomplished the computation of 591 points. Since each processor had to deal with this meant that processor 1 still had to accomplish work for 9 points. As each row shows, the work is reasonably well balanced, thus justifying the good speedups reported in Table 5. We next applied PsDM on matrix gre 1107 (1107 1107 sparse, real and unsymmetric) from the Harwell-Boeing collection. The initial curve @ 0:1 (A) was approximated by 64 points computed by Cobra. We computed 20 pseu- dospectrum curves corresponding to log therefore each processor was allocated d64 20=P e consecutive points. Rows 3 and 4 of Table 7 depict the times and speedups. Even though computing time is reduced, the speedups are far inferior than those reported for kahan in Table 5. To explain this phenomenon, we prepared for gre 1107 a table similar to Table 6. Results are tabulated in rows 3-5 of Table 8 and reveal severe load imbalance. For example, notice that when using 2 processors in PsDM, (2nd row of Table 8), by the time processor 1 has nished with all its allocated points, processor 0 has nished with only 75% them. Similar patterns hold when using 4 and 8 processors. This load imbalance is due to the varying level of di-culty that an iterative method, such as ARPACK, has when computing the triplet corresponding to min (zI A) as z moves from point to point. Two ways to resolve this problem are a) a system-level approach to dispatch Static block partitioning Time (secs) 6000 3750 2500 1390 4.3 Static cyclic partitioning Time (secs) 6000 3060 1560 840 Table 7: Performance of parallel implementation of PsDM using static block and cyclic partitionings for gre 1107. proc. / proc. id. Static block partitioning Static cyclic partitioning Table 8: As Table 6 for matrix gre 1107 using static block and cyclic partitionings to compute 20 pseudospectrum cuves. tasks to the processors from a queue as processors are freed, and/or b) a problem- level approach in which we estimate the work involved in each task and partition the tasks so as to achieve acceptable load balance. The former approach has the potential for better load balance, at additional system-level overhead; the latter approach has little overhead and is simpler to implement, but appears to require a priori estimates for the workload of each task. Even though this is di-cult to know beforehand when using iterative methods, it helps to note that when interested in load balancing, what is important is not information regarding the amount of time taken by each task, but information regarding work dierential between dierent tasks. Based on this idea, we tried the heuristic that the number of iterations required for a triplet based at z is likely to be similar for neighboring values of z. It is therefore reasonable to allocate points in an interleaved fashion. This leads to \static cyclic partitioning", in which processor are initialized with the points z Therefore, by interleaving the initial points we aim to shue the \di-cult" points, along the descent of SpDM and distribute them to all the processors. Rows 6-7 of Table 7 and rows 7-9 of Table 8 depict the speedups and load distribution obtained from static cyclic partitioning. The improvement in load balance is evident and leads to much better speedups. We also experimented with \static block cyclic partitioning" in which points were partitioned in dN=be blocks of b consecutive points each and then the blocks were allocated in a cyclic fashion. We experimented with blocks of size 2 and 4 and found that performance was becoming worse with increasing blocksize, and were all inferior than the static cyclic case. We nally show, in Figure 8, the contours obtained for gre 1107. Y Figure 8: gre 1107. @ (A) contours for log 4.4 Adapting to decreasing contour lengths It is known that for a given matrix and varying values of , the -pseudospectrum forms a family of nested sets on the complex plane. Consequently, for any given and smaller -, the area of - (A) is likely to be smaller than the area of (A); similarly, the length of the boundary is likely to change; it would typically become smaller, unless there is separation and creation of disconnected components whose total perimeter exceeds that of the original curve, in which case it might increase. Let us assume now that any contour is approximated by the polygonal path dened by the points computed by PsDM. Then we can readily compute the lengths of the approximating paths. See for example the lengths corresponding to dierent values of for matrix kahan in Figure 6 and the lengths of the corresponding polygonal paths in Table 9 (row 2). It is clear, log approx. lengths 4.48 2.63 1.85 1.48 1.29 1.18 1.1 # points 46 29 24 22 22 21 20 Table 9: Approximate contour lengths of the pseudospectra of kahan(100) and number of points used to approximate each curve following the point reduction policy described in this section. in this case, that the polygonal path lengths become smaller as we move inwards. Nevertheless, in the PsDM algorithm we described so far, the same number of points (46) was used to approximate both curves. It would seem appropriate, then, to monitor any signicant length reduction or increase, and adapt accordingly the number of points computed by PsDM. In the remainder of this section we examine the case of reduced path lengths, noting that similar policies could also be adapted if we needed to increase, rather than reduce, the number of points. The following general scheme describes the transition from @ -m (A) to @ -m+1 (A). We denote by N -m the points dening the contour @ -m (A). 1. Exclude K points of @ -m (A) according to a curve length criterion. 2. Proceed to compute the next contour @ -m+1 (A); - m+1 < - m , starting from the N -m K remaining points of @ -m (A). The scheme allows one to use a variety of curve length criteria, possibly chosen dynamically as PsDM proceeds. In Table 10, we show a single step of PsDM together with the implementation of such a strategy, based on the variation of the minimum distance between consecutive points on subsequent contours. Steps 1 to 6 implement the point reduction and also compute the next minimum distance, l m+1 , between consecutive points of @ -m (A), while step 7 is the standard sweep described in Table 2. First, the algorithm computes the distances consecutive points on @ -m (A), and then drops any point z k from the curve if it nds that d k + d k+1 < 2l m , unless the previous point z k 1 had already been dropped. In the algorithm presented in Table 10, this test is implemented by means of the boolean variables k ; k and a boolean recurrence specied in line 4.1. Our implementation keeps xed the rst and last points on @ -m (A), though this is easily modied. We also take advantage of the fact that A is real so that we need to compute only one half of the curve. Figure 9 illustrates the underlying idea. In the top curve, all points, z k , inside a box or circle are candidates for dropping, because they satisfy jz k 1 z (inside circle) are dropped because they are the only ones for which the preceding point, z k 1 , was not dropped. The remaining points are shown in the bottom curve, renumbered to indicate their relative position within the curve. distance between consecutive points on @ -m 1 (A); Points distance between consecutive points on @ -m (A); Points 1. for 2. for 3. 4. for 4.1 4.2 if ( 4.3 5. N 6. l z 7. Apply single sweep of PsDM on f~z i 2 @ -m (A); to compute fw g. Table 10: Single sweep of PsDM with adaptive point reduction. We implemented PsDM with the point reduction policy described in Table on our parallel platform. We specically required the parallel version to produce the same points that would have been produced, had the above policy ran serially. To achieve this, processors synchronize so that the sweep outlined in Table starts only when all input data is available to all. Due to the low complexity of this procedure relative to the remaining work, we decided to use a simple approach in which steps 1 to 6 are executed by a single, master, processor instead of a more complicated parallel reduction policy. In particular, processors send back points to the master processor who applies point reduc- tion. The new points are then ready for allocation to the processors and the application of PsDM (step 7 of Table 10). The question arises then, how to allocate these points. As before, we could use a queue or a static approach. Given the success of the latter in the previous experiments, and its lack of overhead, we experimented with static approaches described earlier, assigning approximately e points to each processor. We rst applied this adaptive version of PsDM on kahan(100). This almost halved the number of triplet evaluations, reducing them from 2760 to 1466. Table 9 (3d row) depicts how the number of points varied for selected values of . The contours computed using this strategy are depicted in Figure 10 vs. Figure 9: Adaptive point reduction: Inscribed points are candidates for dropping but only the encircled ones drop successfully. kahan gre 1107 static block static cyclic static block static cyclic Table Execution times in seconds and speedups (in parentheses) for parallel PsDM with point reduction for matrices kahan(100) (left) and gre 1107 (right). the same contours computed using PsDM but holding the number of points per contour constant. It is clear that adaptation is very eective without visibly aecting the quality of the output. Columns 2 and 3 of Table 11 depict the execution times and speedups (in parentheses). Once again, static cyclic is superior to block assignment. Furthermore, the corresponding speedups are similar, though, naturally, not as high as PsDM without point reduction (cf. Table 5). We also performed the same experiment on matrix gre 1107 and show the times and corresponding speedups in columns 4 and 5 of Table 11. We started with 74 points on @ 0:1 (A) and computed 20 curves as deep as @ 0:001 (A), where we concluded with points. The total number of triplet evaluations was reduced to 1300 compared to that would have been required if no point reduction had been performed. In this case too, cyclic allocation performs better than block. Furthermore, the achieved speedups are satisfactory, as they are very close to those reported for the no drop policy in rows 7-9 of Table 8. The above discussion shows that PsDM can be enhanced to adapt to the geo- -0.4 Y PsDM REDUCED PsDM Figure 10: Pseudospectra contours, log Up: normal PsDM. Down: PsDM with point reduction. metric features of the contours, in which case there is a corresponding reduction in cost as we move inwards. It is clear that this idea can serve as springboard for the design of alternative adaptation strategies. 4.5 Capturing disconnected components One documented weakness of path following methods is that they cannot readily capture disconnected components of the pseudospectrum. One question is how does PsDM handle this di-culty? In this section we show that PsDM manages to trace disconnected curves, as long as they lie inside the initial curve @ (A). In that respect, the performance is the same as that of GRID, where the level curves traced are only those that lie within the area discretized by h . Consider matrix grcar(50) from the Test Matrix Toolbox. We begin from 192 points proceed with a xed step 0.1 to compute curves. Thus the inner curve is @ 10 6(A). The pseudospectra for disconnected components. Figure 11 demonstrates that PsDM manages to retrieve Y Figure Capturing disconnected components for grcar(50). Curves from outer to inner: this intricate structure. Of course this procedure does not return components that were already outside the initial @ (A) curve. If we want to assure that all components are captured, we can start with an initial curve that is large enough using techniques such as those presented in [4]. Conclusions We presented PsDM, a new method for the computation of the pseudospectrum of a matrix A, that combines the appealing characteristics of the traditional method GRID and the versatility of path following methods. We saw that PsDM automatically generates new curves starting, for instance, from one application of PF. The method is such that the curves adapt to the geometric properties of the pseudospectrum and is able to capture disconnected components. Experimental results showed that the method achieves a signicant reduction of the number of necessary triplet evaluations vs. GRID even though, like GRID, it also computes the pseudospectrum for several values of . Given the parallel nature of PsDM, we implemented it using MPI. These experiments also revealed the advantages of a simple heuristic designed to achieve load balance. An OpenMP implementation of PsDM is currently underway and is expected to serve as a platform for the investigation of a variety of alternative policies. It is worth noting that PsDM computes successively, using path following but in the direction of steepest descent, points dening the nested curves @ (A). This idea holds great promise for pseudospectra and eigenvalues as described in [3, 9, 10]. Overall, we believe that the approach used in PsDM will be useful in the construction of an adaptive algorithm for the computation of pseudospectra that will be based on path following. We note that the codes used in this paper are available from URL http://www.hpclab.ceid.upatras.gr/scgroup/pseudospectra.html. Acknowledgments We thank Bernard Philippe, for his astute comments regarding the paper. We also thank the referees for their valuable suggestions that helped improve the paper as well as our colleague, E Kokiopoulou, for her comments and support. --R Numerical Continuation Methods: An Introduction Cobra: Parallel path following for computing the matrix pseudospectrum. Parallel algorithms for the computation of pseudospectra. Using the An algorithm for computing the distance to uncontrollability. The Test Matrix Toolbox for MATLAB (version 3.0). Iterations on domains for computing the matrix (pseudo) spectrum. Hermitian methods for computing eigenvalues. Arpack User's Guide: Solution of Large-Scale Eigenvalue Problems With Implicitly Restarted Arnoldi Methods Directional Newton methods in n variables. SPARSKIT: A basic tool-kit for sparse matrix computations (version 2) Computation of pseudospectra. --TR Numerical continuation methods: an introduction An algorithm for computing the distance to uncontrollability Directional Newton Methods in n Variables --CTR C. Bekas , E. Kokiopoulou , E. Gallopoulos, The design of a distributed MATLAB-based environment for computing pseudospectra, Future Generation Computer Systems, v.21 n.6, p.930-941, June 2005

pseudospectra;parallel computation;newton's method;ARPACK