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| \begin{document} | |
| \title{Relational Reasoning for Markov Chains in a Probabilistic Guarded Lambda Calculus} | |
| \author{Alejandro Aguirre\inst{1}$^\text{(\Letter)}$ | |
| \and Gilles Barthe\inst{1} | |
| \and Lars Birkedal\inst{2} | |
| \and Ale\u s Bizjak\inst{2} | |
| \and \\ Marco Gaboardi\inst{3} | |
| \and Deepak Garg\inst{4}} | |
| \institute{IMDEA Software Institute | |
| \\ \texttt{alejandro.aguirre@imdea.org} | |
| \and Aarhus University | |
| \and University at Buffalo, SUNY | |
| \and MPI-SWS} | |
| \maketitle | |
| \begin{abstract} | |
| We extend the simply-typed guarded $\lambda$-calculus with discrete | |
| probabilities and endow it with a program logic for reasoning about | |
| relational properties of guarded probabilistic computations. This | |
| provides a framework for programming and reasoning about infinite | |
| stochastic processes like Markov chains. We demonstrate the logic | |
| sound by interpreting its judgements in the topos of trees and by | |
| using probabilistic couplings for the semantics of relational | |
| assertions over distributions on discrete types. | |
| The program logic is designed to support syntax-directed proofs in the | |
| style of relational refinement types, but retains the expressiveness | |
| of higher-order logic extended with discrete distributions, and the | |
| ability to reason relationally about expressions that have different | |
| types or syntactic structure. In addition, our proof system leverages | |
| a well-known theorem from the coupling literature to justify better | |
| proof rules for relational reasoning about probabilistic | |
| expressions. We illustrate these benefits with a broad range of | |
| examples that were beyond the scope of previous systems, including | |
| shift couplings and lump couplings between random walks. | |
| \end{abstract} | |
| \section{Introduction} | |
| Stochastic processes are often used in mathematics, physics, biology | |
| or finance to model evolution of systems with uncertainty. In | |
| particular, Markov chains are ``memoryless'' stochastic processes, in | |
| the sense that the evolution of the system depends only on the current | |
| state and not on its history. Perhaps the most emblematic example of a | |
| (discrete time) Markov chain is the simple random walk over the | |
| integers, that starts at 0, and that on each step moves one position | |
| either left or right with uniform probability. Let $p_i$ be the | |
| position at time $i$. Then, this Markov chain can be described as: | |
| \[ | |
| p_0 = 0 \quad\quad | |
| p_{i+1} = \begin{cases} | |
| p_i + 1\ \text{with probability}\ 1/2 \\ | |
| p_i - 1\ \text{with probability}\ 1/2 | |
| \end{cases} | |
| \] | |
| The goal of this paper is to develop a programming and reasoning | |
| framework for probabilistic computations over infinite objects, such | |
| as Markov chains. Although programming and reasoning frameworks for | |
| infinite objects and probabilistic computations are well-understood in | |
| isolation, their combination is challenging. In particular, one must | |
| develop a proof system that is powerful enough for proving interesting | |
| properties of probabilistic computations over infinite objects, and | |
| practical enough to support effective verification of these | |
| properties. | |
| \paragraph*{Modelling probabilistic infinite objects} | |
| A first challenge is to model probabilistic infinite objects. We focus | |
| on the case of Markov chains, due to its importance. A (discrete-time) | |
| Markov chain is a sequence of random variables $\{X_i\}$ over some | |
| fixed type $T$ satisfying some independence property. Thus, the | |
| straightforward way of modelling a Markov chain is as a \emph{stream | |
| of distributions} over $T$. Going back to the simple example | |
| outlined above, it is natural to think about this kind of | |
| \emph{discrete-time} Markov chain as characterized by the sequence of | |
| positions $\{p_i\}_{i \in \nat}$, which in turn can be described as an | |
| infinite set indexed by the natural numbers. This suggests that a | |
| natural way to model such a Markov chain is to use \emph{streams} in | |
| which each element is produced \emph{probabilistically} from the | |
| previous one. However, there are some downsides to this | |
| representation. First of all, it requires explicit reasoning about | |
| probabilistic dependency, since $X_{i+1}$ depends on $X_i$. Also, we | |
| might be interested in global properties of the executions of the | |
| Markov chain, such as ``The probability of passing through the initial | |
| state infinitely many times is 1''. These properties are naturally | |
| expressed as properties of the whole stream. For these reasons, we | |
| want to represent Markov chains as \emph{distributions over | |
| streams}. Seemingly, one downside of this representation is that the | |
| set of streams is not countable, which suggests the need for | |
| introducing heavy measure-theoretic machinery in the semantics of the | |
| programming language, even when the underlying type is discrete or | |
| finite. | |
| Fortunately, measure-theoretic machinery can be avoided | |
| (for discrete distributions) by developing a probabilistic extension | |
| of the simply-typed guarded $\lambda$-calculus and giving a semantic | |
| interpretation in the topos of trees~\cite{CBGB16}. Informally, the | |
| simply-typed guarded $\lambda$-calculus~\cite{CBGB16} extends the | |
| simply-typed lambda calculus with a \emph{later} modality, denoted by | |
| $\later$. The type $\later{A}$ ascribes expressions that are available | |
| one unit of logical time in the future. The $\later$ modality allows | |
| one to model infinite types by using \lq\lq | |
| finite\rq\rq\ approximations. For example, a stream of natural numbers | |
| is represented by the sequence of its (increasing) prefixes in the | |
| topos of trees. The prefix containing the first $i$ elements has the | |
| type $S_i \defeq \nat \times \later{\nat} \times \ldots \times | |
| \later^{(i-1)}{\nat}$, representing that the first element is | |
| available now, the second element a unit time in the future, and so | |
| on. This is the key to representing probability distributions over | |
| infinite objects without measure-theoretic semantics: We model | |
| probability distributions over non-discrete sets as discrete | |
| distributions over their (the sets') approximations. For example, a | |
| distribution over streams of natural numbers (which a priori would | |
| be non-discrete since the set of streams is uncountable) would be | |
| modelled by a \emph{sequence of distributions} over the finite | |
| approximations $S_1, S_2, \ldots$ of streams. Importantly, since each | |
| $S_i$ is countable, each of these distributions can be | |
| discrete. | |
| \dg{As Alejandro also noted after his example that I commented out, | |
| it's unclear what's so specific or special about the $\later$ | |
| modality. It seems that we could use \emph{any} approximating | |
| sequence of discrete sets to avoid the issue with | |
| measure-theory. What's special about $\later$ and guards in our | |
| context?} | |
| \lb{the use of $\later$ and guards just allow us to use types to | |
| ensure productivity (and contractiveness to ensure existence of | |
| fixed points)} | |
| \mg{My understanding is that there is nothing special but we need in the logic to have a way to talk about these approximations, and | |
| $\later$ and guards gives us a natural way to do this. no?} | |
| \paragraph*{Reasoning about probabilistic computations} | |
| Probabilistic computations exhibit a rich set of properties. One | |
| natural class of properties is related to probabilities of events, | |
| saying, for instance, that the probability of some event $E$ (or of an | |
| indexed family of events) increases at every iteration. However, | |
| several interesting properties of probabilistic computation, such as | |
| stochastic dominance or convergence (defined below) are relational, in | |
| the sense that they refer to two runs of two processes. In principle, | |
| both classes of properties can be proved using a higher-order logic | |
| for probabilistic expressions, e.g.\, the internal logic of the topos | |
| of trees, suitably extended with an axiomatization of finite | |
| distributions. However, we contend that an alternative approach | |
| inspired from refinement types is desirable and provides better | |
| support for effective verification. More specifically, reasoning in a | |
| higher-order logic, e.g.\, in the internal logic of the topos of trees, | |
| does not exploit the \emph{structure of programs} for non-relational | |
| reasoning, nor the \emph{structural similarities} between programs for | |
| relational reasoning. As a consequence, reasoning is more involved. | |
| To address this issue, we define a relational proof system that | |
| exploits the structure of the expressions and supports syntax-directed | |
| proofs, with necessary provisions for escaping the syntax-directed | |
| discipline when the expressions do not have the same structure. The | |
| proof system manipulates judgements of the form: | |
| \begin{equation*} | |
| \jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{A_1}{t_2}{A_2}{\phi} | |
| \label{eq:judgement-2} | |
| \end{equation*} | |
| where $\Delta$ and $\Gamma$ are two typing contexts, $\Sigma$ and | |
| $\Psi$ respectively denote sets of assertions over variables in these | |
| two contexts, $t_1$ and $t_2$ are well-typed expressions of type $A_1$ | |
| and $A_2$, and $\phi$ is an assertion that may contain the special | |
| variables $\res\ltag$ and $\res\rtag$ that respectively correspond to | |
| the values of $t_1$ and $t_2$. The context $\Delta$ and $\Gamma$, the | |
| terms $t_1$ and $t_2$ and the types $A_1$ and $A_2$ provide a | |
| specification, while $\Sigma$, $\Psi$, and $\phi$ are useful for | |
| reasoning about relational properties over $t_1,t_2$, their inputs and | |
| their outputs. This form of judgement is similar to that of Relational | |
| Higher-Order Logic \cite{ABGGS17}, from which our system draws inspiration. | |
| In more detail, our relational logic comes with typing rules that | |
| allow one to reason about relational properties by exploiting as much | |
| as possible the syntactic similarities between $t_1$ and $t_2$, and to | |
| fall back on pure logical reasoning when these are not available. In | |
| order to apply relational reasoning to guarded computations the logic | |
| provides relational rules for the later modality $\later{}$ and for a | |
| related modality $\square{}$, called ``constant''. These rules allow | |
| the relational verification of general relational properties that go | |
| beyond the traditional notion of program equivalence and, moreover, | |
| they allow the verification of properties of guarded computations over | |
| different types. The ability to reason about computations of different | |
| types provides significant benefits over alternative formalisms for | |
| relational reasoning. For example, it enables reasoning about | |
| relations between programs working on different data structures, | |
| e.g. a relation between a program working on a stream of natural | |
| numbers, and a program working on a stream of pairs of natural | |
| numbers, or having different structures, e.g. a relation between | |
| an application and a case expression. | |
| Importantly, our approach for reasoning formally about probabilistic | |
| computations is based on \emph{probabilistic couplings}, a standard | |
| tool from the analysis of Markov chains~\cite{Lindvall02,Thorisson00}. | |
| From a verification perspective, probabilistic couplings go beyond | |
| equivalence properties of probabilistic programs, which have been | |
| studied extensively in the verification literature, and yet support | |
| compositional reasoning~\cite{BartheEGHSS15,BartheGHS17}. The main | |
| attractive feature of coupling-based reasoning is that it limits the | |
| need of explicitly reasoning about the probabilities---this avoids | |
| complex verification conditions. We provide sound proof rules for | |
| reasoning about probabilistic couplings. Our rules make several | |
| improvements over prior relational verification logics based on | |
| couplings. First, we support reasoning over probabilistic processes of | |
| different types. Second, we use Strassen's | |
| theorem~\cite{strassen1965existence} a remarkable result about | |
| probabilistic couplings, to achieve greater expressivity. Previous | |
| systems required to prove a bijection between the sampling spaces to | |
| show the existence of a coupling~\cite{BartheEGHSS15,BartheGHS17}, | |
| Strassen's theorem gives a way to show their existence which is | |
| applicable in settings where the bijection-based approach cannot be | |
| applied. And third, we support reasoning with what are called shift | |
| couplings, coupling which permits to relate the states of two Markov | |
| chains at possibly different times (more explanations below). | |
| \paragraph*{Case studies} | |
| We show the flexibility of our formalism by verifying several examples | |
| of relational properties of probabilistic computations, and Markov | |
| chains in particular. These examples cannot be verified with existing | |
| approaches. | |
| First, we verify a classic example of probabilistic non-interference | |
| which requires the reasoning about computations at different | |
| types. Second, in the context of Markov chains, we verify an example | |
| about stochastic dominance which exercises our more general rule for | |
| proving the existence of couplings modelled by expressions of | |
| different types. Finally, we verify an example involving shift | |
| relations in an infinite computation. This style of reasoning is | |
| motivated by \lq\lq shift\rq\rq\ couplings in Markov chains. In | |
| contrast to a standard coupling, which relates the states of two | |
| Markov chains at the same time $t$, a shift coupling relates the | |
| states of two Markov chains at possibly different times. Our specific | |
| example relates a standard random walk (described earlier) to a | |
| variant called a lazy random walk; the verification requires relating | |
| the state of standard random walk at time $t$ to the state of the lazy | |
| random walk at time $2t$. We note that this kind of reasoning is | |
| impossible with conventional relational proof rules even in a | |
| non-probabilistic setting. Therefore, we provide a novel family of | |
| proof rules for reasoning about shift relations. At a high level, the | |
| rules combine a careful treatment of the later and constant modalities | |
| with a refined treatment of fixpoint operators, allowing us to relate | |
| different iterates of function bodies. | |
| \subsection*{Summary of contributions} | |
| With the aim of providing a general framework for programming and | |
| reasoning about Markov chains, the three main contributions of this work are: | |
| \begin{enumerate} | |
| \item A probabilistic extension of the guarded $\lambda$-calculus, | |
| that enables the definition of Markov chains as discrete | |
| probability distributions over streams. | |
| \item A relational logic based on coupling to reason in a | |
| syntax-directed manner about (relational) properties of Markov | |
| chains. This logic supports reasoning about programs that have | |
| different types and structures. Additionally, this logic uses results from the | |
| coupling literature to achieve greater expressivity than previous | |
| systems. | |
| \item An extension of the relational logic that allows to relate the | |
| states of two streams at possibly different times. This extension | |
| supports reasoning principles, such as shift couplings, that escape | |
| conventional relational logics. | |
| \end{enumerate} | |
| \section{Mathematical preliminaries} | |
| This section reviews the definition of discrete probability | |
| sub-distributions and introduces mathematical couplings. | |
| \begin{definition}[Discrete probability distribution] | |
| Let $C$ be a discrete (i.e., finite or countable) set. A (total) | |
| distribution over $C$ is a function $\mu : C \to [0,1]$ such that $ | |
| \sum_{x\in C} \mu(x) = 1 .$ The support of a distribution $\mu$ is the | |
| set of points with non-zero probability, $ \supp\ \mu \defeq \{x \in C | |
| \mid \mu(x) > 0 \} .$ We denote the set of distributions over $C$ as | |
| $\Distr(C)$. Given a subset $E \subseteq C$, the probability of | |
| sampling from $\mu$ a point in $E$ is denoted $\Pr_{x\leftarrow \mu}[x | |
| \in E]$, and is equal to $\sum_{x \in E} \mu(x)$. | |
| \end{definition} | |
| \begin{definition}[Marginals] | |
| Let $\mu$ be a distribution over a product space $C_1\times C_2$. The | |
| first (second) marginal of $\mu$ is another distribution | |
| $\Distr(\pi_1)(\mu)$ ($\Distr(\pi_2)(\mu)$) over $C_1$ ($C_2$) defined | |
| as: | |
| \[\Distr(\pi_1)(\mu)(x) = \sum_{y \in C_2} \mu(x,y) \qquad | |
| \left(\Distr(\pi_2)(\mu)(y) = \sum_{x \in C_1} \mu(x,y) \right) | |
| \] | |
| \end{definition} | |
| \subsubsection{Probabilistic couplings} | |
| Probabilistic couplings are a fundamental tool in the analysis of | |
| Markov chains. When analyzing | |
| a relation between two probability distributions it is sometimes useful to | |
| consider instead a distribution over the product space that somehow ``couples'' | |
| the randomness in a convenient manner. | |
| Consider for instance the case of the following Markov chain, which | |
| counts the total amount of tails observed when | |
| tossing repeatedly a biased coin with probability of tails $p$: | |
| \[ | |
| n_0 = 0 \quad\quad | |
| n_{i+1} = \left\{\begin{array}{l} | |
| n_i + 1\ \text{with probability}\ p \\ | |
| n_i\ \text{with probability}\ (1-p) | |
| \end{array}\right. | |
| \] | |
| If we have two biased coins with probabilities of tails $p$ and $q$ with $p\leq q$ and we | |
| respectively observe $\{n_i\}$ and $\{m_i\}$ we would expect that, in | |
| some sense, $n_i \leq m_i$ should hold for all $i$ (this property is | |
| known as stochastic dominance). A formal proof of this fact using | |
| elementary tools from probability theory would require to compute the | |
| cumulative distribution functions for $n_i$ and $m_i$ and then to | |
| compare them. The coupling method reduces this proof to showing a way | |
| to pair the coin flips so that if the first coin shows tails, so does | |
| the second coin. | |
| We now review the definition of couplings and state relevant | |
| properties. | |
| \begin{definition}[Couplings] | |
| Let $\mu_1\in\Distr(C_1)$ and $\mu_2\in\Distr(C_2)$, and | |
| $R\subseteq C_1\times C_2$. | |
| \begin{itemize} | |
| \item A distribution $\mu\in\Distr(C_1\times C_2)$ is a coupling for | |
| $\mu_1$ and $\mu_2$ iff its first and second marginals coincide with | |
| $\mu_1$ and $\mu_2$ respectively, i.e.\, $\Distr(\pi_1)(\mu)=\mu_1$ | |
| and $\Distr(\pi_2)(\mu)=\mu_2$. | |
| \item A distribution $\mu\in\Distr(C_1\times C_2)$ is a $R$-coupling | |
| for $\mu_1$ and $\mu_2$ if it is a coupling for $\mu_1$ and $\mu_2$ | |
| and, moreover, $\Pr_{(x_1,x_2)\leftarrow \mu} [R~x_1~x_2]=1$, i.e., if | |
| the support of the distribution $\mu$ is included in $R$.\end{itemize} | |
| Moreover, we write $\coupl{\mu_1}{\mu_2}{R}$ iff there exists a | |
| $R$-coupling for $\mu_1$ and $\mu_2$. | |
| \end{definition} | |
| Couplings always exist. For instance, the product distribution of two | |
| distributions is always a coupling. | |
| Going back to the example about the two coins, it can be proven by computation | |
| that the following is a coupling that lifts the less-or-equal relation | |
| ($0$ indicating heads and $1$ indicating tails): | |
| \[\left\{\begin{array}{lll} | |
| &(0,0) \ \text{w/ prob}\ (1-q)\quad\quad | |
| &(0,1) \ \text{w/ prob}\ (q-p) \\ | |
| &(1,0) \ \text{w/ prob}\ 0\quad\quad | |
| &(1,1) \ \text{w/ prob}\ p | |
| \end{array}\right.\] | |
| The following theorem in~\cite{strassen1965existence} gives a | |
| necessary and sufficient condition for the existence of $R$-couplings | |
| between two distributions. The theorem is remarkable in the sense that | |
| it proves an equivalence between an existential property (namely the | |
| existence of a particular coupling) and a universal property | |
| (checking, for each event, an inequality between probabilities). | |
| \begin{theorem}[Strassen's theorem] | |
| Consider $\mu_1\in\Distr(C_1)$ and $\mu_2\in\Distr(C_2)$, and $R\subseteq | |
| C_1 \times C_2$. Then $\coupl{\mu_1}{\mu_2}{R}$ iff for every | |
| $X \subseteq C_1$, $\Pr_{x_1\leftarrow \mu_1}[x_1\in X] \leq | |
| \Pr_{x_2\leftarrow \mu_2}[x_2\in R(X)]$, where $R(X)$ is the image | |
| of $X$ under $R$, i.e.\, $R(X) =\{ y \in C_2 \mid \exists x \in | |
| X.~R~x~y\}$. | |
| \end{theorem} | |
| An important property of couplings is closure under sequential | |
| composition. | |
| \begin{lemma}[Sequential composition couplings]\label{lem:sequential-composition-of-couplings} | |
| Let $\mu_1\in\Distr(C_1)$, $\mu_2\in\Distr(C_2)$, $M_1:C_1\rightarrow | |
| \Distr(D_1)$ and $M_2:C_2\rightarrow \Distr(D_2)$. Moreover, let | |
| $R\subseteq C_1\times C_2$ and $S\subseteq D_1\times D_2$. Assume: | |
| $(1)$ $\coupl{\mu_1}{\mu_2}{R}$; and | |
| $(2)$ for every $x_1\in C_1$ and $x_2\in C_2$ such that $R~x_1~x_2$, | |
| we have $\coupl{M_1(x_1)}{M_2(x_2)}{S}$. | |
| Then $\coupl{(\mathsf{bind}~\mu_1~M_1)}{(\mathsf{bind}~\mu_2~M_2)}{S}$, | |
| where $\mathsf{bind}~\mu~M$ is defined as | |
| $$(\mathsf{bind}~\mu~M)(y) =\sum_x \mu(x) \cdot M(x)(y)$$ | |
| \end{lemma} | |
| We conclude this section with the following lemma, which follows from | |
| Strassen's theorem: | |
| \begin{lemma}[Fundamental lemma of couplings]\label{lem:fun-coup} | |
| Let $R\subseteq C_1 \times C_2$, $E_1\subseteq C_1$ and $E_2\subseteq | |
| C_2$ such that for every $x_1\in E_1$ and $x_2\in C_2$, $R~x_1~x_2$ | |
| implies $x_2\in E_2$, i.e.\, $R(E_1)\subseteq E_2$. Moreover, let | |
| $\mu_1\in\Distr(C_1)$ and $\mu_2\in\Distr(C_2)$ such that | |
| $\coupl{\mu_1}{\mu_2}{R}$. Then | |
| $$\Pr_{x_1\leftarrow \mu_1} [x_1\in E_1] \leq \Pr_{x_2\leftarrow\mu_2} [x_2\in E_2]$$ | |
| \end{lemma} | |
| This lemma can be used to prove probabilistic inequalities | |
| from the existence of suitable couplings: | |
| \begin{corollary}\label{cor:fundamental} | |
| Let $\mu_1,\mu_2\in\Distr(C)$: | |
| \begin{enumerate} | |
| \item If $\coupl{\mu_1}{\mu_2}{(=)}$, then for all $x\in C$, $\mu_1(x) = \mu_2(x)$. | |
| \item If $C = \nat$ and $\coupl{\mu_1}{\mu_2}{(\geq)}$, then for all $n\in \nat$, | |
| $\Pr_{x\leftarrow \mu_1}[x\geq n] \geq \Pr_{x\leftarrow \mu_2}[x\geq n]$ | |
| \end{enumerate} | |
| \end{corollary} | |
| In the example at the beginning of the section, the property we want | |
| to prove is precisely that, for every $k$ and $i$, the following | |
| holds: | |
| \[ | |
| \Pr_{x_1\leftarrow n_i} [x_1 \geq k] \leq \Pr_{x_2\leftarrow m_i} [x_2 \geq k] | |
| \] | |
| Since we have a $\leq$-coupling, this proof is immediate. | |
| This example is formalized in \autoref{sec:proba-ex}. | |
| \section{Overview of the system} | |
| \label{sec:overview} | |
| In this section we give a high-level overview of our system, with the | |
| details on sections \ref{sec:syntax}, \ref{sec:ghol} and \ref{sec:grhol}. | |
| We start by presenting the base logic, and then we show how to extend it with | |
| probabilities and how to build a relational reasoning system on top of it. | |
| \subsection{Base logic: Guarded Higher-Order Logic} | |
| Our starting point is the Guarded Higher-Order Logic~\cite{CBGB16} | |
| (Guarded HOL) | |
| inspired by the topos of | |
| trees. In addition to the usual constructs of HOL to reason about lambda | |
| terms, this logic features the $\later$ and $\square$ modalities to reason | |
| about infinite terms, in particular streams. The $\later$ modality is used | |
| to reason about objects that will be available in the future, such as tails | |
| of streams. For instance, suppose we want to define an $\All(s,\phi)$ predicate, expressing that | |
| all elements of a stream $s \equiv \cons{n}{xs}$ satisfy a property $\phi$. This can be axiomatized as | |
| follows: | |
| \[\forall (xs: \later \Str{\nat}) (n : \nat). \phi\ n \Rightarrow \later[s\ot | |
| xs]{\All(s, x. \phi)} \Rightarrow \All(\cons{n}{xs}, x. \phi)\] | |
| We use $x. \phi$ to denote that the formula $\phi$ depends on a free variable | |
| $x$, which will get replaced by the first argument of $\All$. | |
| We have two antecedents. The first one states that the head $n$ satisfies | |
| $\phi$. The second one, $\later[s\ot xs]{\All(s, x. \phi)}$, states that all | |
| elements of $xs$ satisfy $\phi$. Formally, $xs$ is the tail of the stream and will | |
| be available in the future, so it has type $\later\Str{\nat}$. | |
| The \emph{delayed substitution} $\triangleright[s\ot xs]$ replaces $s$ of type | |
| $\Str{\nat}$ with $xs$ of type $\later\Str{\nat}$ inside $\All$ and shifts the | |
| whole formula one step into the future. | |
| In other words, $\later[s\ot xs]{\All(s, x. \phi)}$ | |
| states that $\All(-, x.\phi)$ will be satisfied by $xs$ in the future, once | |
| it is available. | |
| \subsection{A system for relational reasoning} | |
| \label{sec:overview:grhol} | |
| When proving relational properties it | |
| is often convenient to build proofs guided by the syntactic structure | |
| of the two expressions to be related. This style of reasoning is | |
| particularly appealing when the two expressions have the same | |
| structure and control-flow, and is appealingly close to the | |
| traditional style of reasoning supported by refinement types. At the | |
| same time, a strict | |
| adherence to the syntax-directed discipline is detrimental to the | |
| expressiveness of the system; for instance, it makes it difficult or | |
| even impossible to reason about structurally dissimilar terms. To | |
| achieve the best of both worlds, we present a relational proof system built | |
| on top of Guarded HOL, which we call Guarded RHOL. Judgements have the shape: | |
| \[\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{A_1}{t_2}{A_2}{\phi}\] | |
| where $\phi$ is a logical formula that may contain two distinguished | |
| variables $\res\ltag$ and $\res\rtag$ that respectively represent the | |
| expressions $t_1$ and $t_2$. This judgement subsumes two typing | |
| judgements on $t_1$ and $t_2$ and a relation $\phi$ on these two | |
| expressions. However, this form of judgement does not tie the logical | |
| property to the type of the expressions, and is key to achieving | |
| flexibility while supporting syntax-directed proofs whenever needed. | |
| The proof system combines rules of two different flavours: two-sided | |
| rules, which relate expressions with the same top-level constructs, | |
| and one-sided rules, which operate on a single expression. | |
| We then extend Guarded HOL with a modality $\diamond$ | |
| that lifts assertions over discrete types $C_1$ and $C_2$ to assertions over | |
| $\Distr(C_1)$ and $\Distr(C_2)$. | |
| Concretely, we define for every assertion $\phi$, variables | |
| $x_1$ and $x_2$ of type $C_1$ and $C_2$ respectively, and expressions | |
| $t_1$ and $t_2$ of type $\Distr(C_1)$ and $\Distr (C_2)$ respectively, | |
| the modal assertion $\diamond_{ [x_1\leftarrow t_1,x_2\leftarrow t_2]} | |
| \phi$ which holds iff the interpretations of $t_1$ and $t_2$ are | |
| related by the probabilistic lifting of the interpretation of | |
| $\phi$. We call this new logic Probabilistic Guarded HOL. | |
| We accordingly extend the relational proof system to support reasoning | |
| about probabilistic expressions by adding judgements of the form: | |
| \[\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{\Distr(C_1)} | |
| {t_2}{\Distr(C_2)}{ | |
| \diamond_{[x_1\leftarrow \res\ltag, x_2\leftarrow \res\rtag]} \phi}\] | |
| expressing that $t_1$ and $t_2$ are distributions related by a | |
| $\phi$-coupling. We call this proof system Probabilistic Guarded RHOL. | |
| These judgements can be built by using the following rule, | |
| that lifts relational judgements over discrete types | |
| $C_1$ and $C_2$ to judgements over distribution types $\Distr(C_1)$ | |
| and $\Distr(C_2)$ when the premises of Strassen's theorem are | |
| satisfied. | |
| \[ | |
| \infer[\sf COUPLING]{ | |
| \jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{\Distr(C_1)}{t_2}{\Distr(C_2)}{ | |
| \diamond_{[y_1\leftarrow \res_1, y_2\leftarrow | |
| \res_2]}\phi}}{ | |
| \jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\forall X_1 \subseteq C_1. | |
| \Pr_{y_1\ot t_1} [y_1\in X_1] \leq \Pr_{y_2\ot t_2} [\exists y_1 \in X_1. \phi]}} | |
| \] | |
| Recall that (discrete time) Markov chains are ``memoryless'' | |
| probabilistic processes, whose specification is given by a (discrete) | |
| set $C$ of states, an initial state $s_0$ and a probabilistic | |
| transition function $\mathsf{step}:C \rightarrow \Distr(C)$, where | |
| $\Distr(S)$ represents the set of discrete distributions over $C$. As | |
| explained in the introduction, a convenient modelling of Markov chains | |
| is by means of probabilistic streams, i.e.\, to model a Markov chain | |
| as an element of $\Distr(\Str{S})$, where $S$ is its underlying state | |
| space. To model Markov chains, we introduce a $\markov$ operator with | |
| type $C \to (C \to \Distr(C)) \to \Distr(\Str{C})$ that, given an | |
| initial state and a transition function, returns a Markov chain. We | |
| can reason about Markov chains by the \rname{Markov} rule (the context, omitted, | |
| does not change): | |
| \begin{gather*} | |
| \inferrule*[right=\sf Markov] | |
| {\jgrholnoc{t_1}{C_1}{t_2}{C_2}{\phi} \\\\ | |
| \jgrholnoc{h_1}{C_1 \to \Distr(C_1)}{h_2}{C_2 \to \Distr(C_2)}{\psi_3} \\\\ | |
| \vdash \psi_4} | |
| {\jgrholnoc{\operatorname{markov}(t_1,h_1)}{\Distr(\Str{D_1})}{\operatorname{markov}(t_2,h_2)}{\Distr(\Str{D_2})} | |
| {\diamond_{\left[\substack{y_1 \ot \res\ltag \\ y_2 \ot \res\rtag}\right]}\phi'}} | |
| \\[0.5em] | |
| \text{ where } | |
| \begin{cases} | |
| \psi_3 \equiv \forall x_1 x_2. \phi\defsubst{x_1}{x_2} \Rightarrow | |
| \diamond_{[ y_1 \ot \res\ltag\ x_1, y_2 \ot \res\rtag\ x_2]}\phi\defsubst{y_1}{y_2} \\ | |
| \psi_4 \equiv \forall x_1\ x_2\ xs_1\ xs_2. \phi\defsubst{x_1}{x_2} | |
| \Rightarrow \later[y_1 \ot xs_1, y_2 \ot xs_2]{\phi'} \Rightarrow \\ | |
| \quad\quad\quad\phi'\subst{y_1}{\cons{x_1}{xs_1}}\subst{y_2}{\cons{x_2}{xs_2}} | |
| \end{cases} | |
| \end{gather*} | |
| Informally, the rule stipulates the existence of an invariant $\phi$ | |
| over states. The first premise insists that the invariant hold on the | |
| initial states, the condition $\psi_3$ states that the transition | |
| functions preserve the invariant, and $\psi_4$ states that the | |
| invariant $\phi$ over pairs of states can be lifted to a stream | |
| property $\phi'$. | |
| Other rules of the logic are given in | |
| Figure~\ref{fig:phol}. The language construct $\mathsf{munit}$ creates a point distribution | |
| whose entire mass is at its argument. Accordingly, the [\textsf{UNIT}] rule creates a | |
| straightforward coupling. | |
| The [\textsf{MLET}] rule internalizes sequential composition of | |
| couplings (Lemma~\ref{lem:sequential-composition-of-couplings}) into | |
| the proof system. The construct $\mlet{x}{t}{t'}$ composes a | |
| distribution $t$ with a probabilistic computation $t'$ with one free | |
| variable $x$ by sampling $x$ from $t$ and running $t'$. | |
| The [\textsf{MLET-L}] rule supports one-sided reasoning about | |
| $\mlet{x}{t}{t'}$ and relies on the fact that couplings are closed | |
| under convex combinations. Note that one premise of the rule uses a | |
| unary judgement, with a non-relational modality | |
| $\diamond_{[x\leftarrow \res]} \phi$ whose informal meaning is that | |
| $\phi$ holds with probability $1$ in the distribution $\res$. | |
| The following table summarizes the different base logics we consider,the | |
| relational systems we build on top of them, including the ones presented in~\cite{ABGGS17}, and the equivalences between both sides: | |
| \begin{center} | |
| \small | |
| \begin{tabular}{lcl} | |
| Relational logic & & Base logic \\ \hline | |
| &&\\[-2mm] | |
| $\begin{array}{l}\text{RHOL~\cite{ABGGS17}} \\ | |
| \Gamma \mid \Psi \vdash t_1 \sim t_2 \mid \phi \end{array} $ | |
| & $\stackrel{\text{\cite{ABGGS17}}}{\Longleftrightarrow}$ & | |
| $\begin{array}{l} \text{HOL~\cite{ABGGS17}} \\ | |
| \Gamma \mid \Psi \vdash \phi\defsubst{t_1}{t_2} \end{array} $ \\[1em] | |
| $\begin{array}{l} \text{Guarded RHOL~\S\ref{sec:grhol}} \\ | |
| \Delta \mid \Sigma \mid \Gamma \mid \Psi \vdash t_1 \sim t_2 \mid \phi \end{array} $ | |
| & $\stackrel{\text{Thm~\ref{thm:equiv-rhol-hol}}}{\Longleftrightarrow}$ & | |
| $\begin{array}{l} \text{Guarded HOL~\cite{CBGB16}} \\ | |
| \Delta \mid \Sigma \mid \Gamma \mid \Psi \vdash \phi\defsubst{t_1}{t_2} \end{array} $ \\[1em] | |
| $\begin{array}{l} \text{Probabilistic Guarded RHOL~\S\ref{sec:grhol}} \\ | |
| \Delta \mid \Sigma \mid \Gamma \mid \Psi \vdash t_1 \sim t_2 \mid | |
| \diamond_{[y_1 \ot \res\ltag, y_2\ot\res\rtag]}.\phi \end{array} $ | |
| & \hspace{.3cm} | |
| $\stackrel{\text{Thm~\ref{thm:equiv-rhol-hol}}}{\Longleftrightarrow}$ | |
| \hspace{.3cm} & | |
| $\begin{array}{l} \text{Probabilistic Guarded HOL~\S\ref{sec:ghol}} \\ | |
| \Delta \mid \Sigma \mid \Gamma \mid \Psi \vdash | |
| \diamond_{[y_1 \ot t_1, y_2 \ot t_2]}.\phi \end{array} $ | |
| \end{tabular} | |
| \end{center} | |
| \begin{figure*}[!tb] | |
| \small | |
| \begin{gather*} | |
| \infer[\sf UNIT]{\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{\munit{t_1}}{\Distr(C_1)} | |
| {\munit{t_2}}{\Distr(C_2)}{ | |
| \diamond_{[x_1\leftarrow \res\ltag, x_2\leftarrow \res\rtag]} \phi}} | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{C_1}{t_2}{C_2}{\phi | |
| [\res\ltag/x_1, \res\rtag/x_2]}} | |
| \\[0.2em] | |
| \infer[\sf MLET]{ | |
| \jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{\mlet{x_1}{t_1}{t'_1}}{ | |
| \Distr(D_1)}{\mlet{x_2}{t_2}{t'_2}}{\Distr(D_2)}{\diamond_{\left[\substack{y_1\leftarrow \res_1 \\ y_2\leftarrow \res_2}\right]} \psi}} | |
| {\begin{array}{c}\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{\Distr(C_1)}{t_2}{\Distr(C_2)}{ | |
| \diamond_{[x_1\leftarrow \res_1, x_2\leftarrow \res_2]} \phi} \\ | |
| \jgrhol{\Delta}{\Sigma}{\Gamma,x_1:C_1,x_2:C_2}{\Psi,\phi | |
| }{t'_1}{\Distr(D_1)} {t'_2}{\Distr(D_2)}{\diamond_{[y_1\leftarrow \res_1, y_2\leftarrow \res_2]} \psi} | |
| \end{array}} | |
| \\[0.2em] | |
| \infer[\sf MLET-L]{ | |
| \jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{\mlet{x_1}{t_1}{t'_1}}{\Distr(D_1)}{t'_2}{\Distr(D_2)}{ | |
| \diamond_{[y_1\leftarrow \res_1, y_2\leftarrow \res_2]} \psi}} | |
| {\begin{array}{c} | |
| \jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{\Distr(C_1)}{\diamond_{[x\leftarrow \res]} \phi} | |
| \\ | |
| \jgrhol{\Delta}{\Sigma}{\Gamma,x_1:C_1}{\Psi,\phi | |
| }{t'_1}{\Distr(D_1)}{t'_2}{\Distr(D_2)}{\diamond_{[y_1\leftarrow \res_1, y_2\leftarrow \res_2]} \psi} | |
| \end{array}} | |
| \end{gather*} | |
| \shrinkcaption | |
| \caption{Proof rules for probabilistic constructs}\label{fig:phol} | |
| \end{figure*} | |
| \subsection{Examples}\label{sec:proba-ex} | |
| We formalize elementary examples from the literature on security and | |
| Markov chains. None of these examples can be verified in prior | |
| systems. Uniformity of \emph{one-time pad} and lumping of \emph{random | |
| walks} cannot even be stated in prior systems because the two | |
| related expressions in these examples have different types. The | |
| \emph{random walk vs lazy random walk} (shift coupling) cannot be | |
| proved in prior systems because it requires either asynchronous | |
| reasoning or code rewriting. Finally, the \emph{biased coin example} | |
| (stochastic dominance) cannot be proved in prior work because it | |
| requires Strassen's formulation of the existence of coupling (rather | |
| than a bijection-based formulation) or code rewriting. We give | |
| additional details below. | |
| \subsubsection{One-time pad/probabilistic non-interference} | |
| Non-interference~\cite{GoguenM82} is a baseline information flow | |
| policy that is often used to model confidentiality of computations. In | |
| its simplest form, non-interference distinguishes between public (or | |
| low) and private (or high) variables and expressions, and requires | |
| that the result of a public expression not depend on the value of its | |
| private parameters. This definition naturally extends to probabilistic | |
| expressions, except that in this case the evaluation of an expression | |
| yields a distribution rather than a value. There are deep connections | |
| between probabilistic non-interference and several notions of | |
| (information-theoretic) security from cryptography. In this paragraph, | |
| we illustrate different flavours of security properties for one-time | |
| pad encryption. Similar reasoning can be carried out for proving | |
| (passive) security of secure multiparty computation algorithms in the | |
| 3-party or multi-party setting~\cite{BogdanovNTW12,Cramer:2015:SMC}. | |
| One-time pad is a perfectly secure symmetric encryption scheme. Its | |
| space of plaintexts, ciphertexts and keys is the set | |
| $\{0,1\}^\ell$---fixed-length bitstrings of size $\ell$. The | |
| encryption algorithm is parametrized by a key $k$---sampled uniformly | |
| over the set of bitstrings $\{ 0,1 \}^\ell$---and maps every plaintext | |
| $m$ to the ciphertext $c = k \oplus m$, where the operator $\oplus$ | |
| denotes bitwise exclusive-or on bitstrings. We let $\mathsf{otp}$ | |
| denote the expression $\lambda | |
| m. \mlet{k}{\mathcal{U}_{\{0,1\}^\ell}}{\munit{k\oplus m}}$, where | |
| $\mathcal{U}_{X}$ is the uniform distribution over a finite set $X$. | |
| One-time pad achieves perfect security, i.e.\, the distributions of | |
| ciphertexts is independent of the plaintext. Perfect security can be | |
| captured as a probabilistic non-interference property: | |
| $$\jgrholnoc{\mathsf{otp}}{\{ 0,1 \}^\ell \rightarrow \Distr(\{ 0, 1 | |
| \}^\ell)}{\mathsf{otp}}{\{ 0,1 \}^\ell \rightarrow \Distr(\{ 0, 1 \}^\ell)}{ | |
| \forall m_1m_2. | |
| \res\ltag~m_1 \stackrel{\diamond}{=} \res\rtag~m_2 | |
| }$$ | |
| where $e_1 \stackrel{\diamond}{=} e_2$ is used as a shorthand for | |
| $\diamond_{[y_1\leftarrow e_1, y_2\leftarrow e_2]} y_1 = y_2$. The | |
| crux of the proof is to establish | |
| $$m_1,m_2: \{ 0,1 \}^\ell | |
| \jgrholnoc{\mathcal{U}_{\{0,1\}^\ell}}{\Distr(\{ 0, 1 \}^\ell)}{ | |
| \mathcal{U}_{\{0,1\}^\ell}}{\Distr(\{ 0, 1 \}^\ell)}{ | |
| \res\ltag \oplus m_2 \stackrel{\diamond}{=} \res\rtag \oplus m_1 | |
| }$$ | |
| using the [\textsf{COUPLING}] rule. It suffices to observe that the | |
| assertion induces a bijection, so the image of an arbitrary set $X$ | |
| under the relation has the same cardinality as $X$, and hence their | |
| probabilities w.r.t.\, the uniform distributions are equal. One can | |
| then conclude the proof by applying the rules for monadic | |
| sequenciation (\rname{MLET}) and abstraction (rule \rname{ABS} in appendix), using | |
| algebraic properties of $\oplus$. | |
| Interestingly, one can prove a stronger property: rather than proving | |
| that the ciphertext is independent of the plaintext, one can prove | |
| that the distribution of ciphertexts is uniform. This is captured by | |
| the following judgement: | |
| $$c_1, c_2: \{ 0,1 \}^\ell | |
| \jgrholnoc{\mathsf{otp}}{\{ 0,1 \}^\ell \rightarrow \Distr(\{ 0, 1 \}^\ell)} | |
| {\mathsf{otp}}{\{ 0,1 \}^\ell \rightarrow \Distr(\{ 0, 1 \}^\ell)}{\psi}$$ | |
| where | |
| $\psi\defeq \forall m_1\,m_2. m_1=m_2\Rightarrow | |
| \diamond_{[y_1\leftarrow \res\ltag~m_1, y_2\leftarrow \res\rtag ~m_2]} | |
| y_1=c_1 \Leftrightarrow y_2=c_2$. | |
| This style of modelling uniformity as a relational property is | |
| inspired from~\cite{BartheEGHS17}. The proof is similar to the | |
| previous one and omitted. However, it is arguably more natural to | |
| model uniformity of the distribution of ciphertexts by the judgement: | |
| $$\jgrholnoc{\mathsf{otp}}{\{ 0,1 \}^\ell \rightarrow \Distr(\{ 0, 1 \}^\ell)} | |
| {\mathcal{U}_{\{0,1\}^\ell}}{\Distr(\{ 0, 1 \}^\ell)}{ | |
| \forall m.~ | |
| \res\ltag~m \stackrel{\diamond}{=} \res\rtag | |
| }$$ | |
| This judgement is closer to the simulation-based notion of security | |
| that is used pervasively in cryptography, and notably in Universal | |
| Composability~\cite{canetti2001universally}. Specifically, the | |
| statement captures the fact that the one-time pad algorithm can be | |
| simulated without access to the message. It is interesting to note | |
| that the judgement above (and more generally simulation-based security) | |
| could not be expressed in prior works, since the two expressions of | |
| the judgement have different types---note that in this specific case, | |
| the right expression is a distribution but in the general case the | |
| right expression will also be a function, and its domain will be a | |
| projection of the domain of the left expression. | |
| The proof proceeds as follows. First, we prove | |
| $$ | |
| \jgrholnocnot{\mathcal{U}_{\{0,1\}^\ell}}{ | |
| \mathcal{U}_{\{0,1\}^\ell}}{ | |
| \forall m.~\diamond_{[y_1\leftarrow \res\ltag, y_2\leftarrow \res\rtag]} | |
| y_1 \oplus m = y_2}$$ | |
| using the [\textsf{COUPLING}] rule. Then, we apply the [\textsf{MLET}] | |
| rule to obtain | |
| $$\jgrholnocnot | |
| {\begin{array}{l}\mlet{k}{\mathcal{U}_{\{0,1\}^\ell}}{\\ \munit{k\oplus m}} \end{array}} | |
| {\begin{array}{l}\mlet{k}{\mathcal{U}_{\{0,1\}^\ell}}{\\ \munit{k}}\end{array}} | |
| {\diamond_{\left[y_1\leftarrow \res\ltag, | |
| y_2\leftarrow \res\rtag \right]} y_1 = y_2}$$ | |
| We have $\mlet{k}{\mathcal{U}_{\{0,1\}^\ell}}{\munit{k}} \equiv | |
| \mathcal{U}_{\{0,1\}^\ell}$; hence by equivalence (rule | |
| \rname{Equiv} in appendix), this entails | |
| $$ | |
| \jgrholnocnot{\mlet{k}{\mathcal{U}_{\{0,1\}^\ell}}{\munit{k\oplus m}}} | |
| {\mathcal{U}_{\{0,1\}^\ell}}{\diamond_{[y_1\leftarrow | |
| \res\ltag, y_2\leftarrow \res\rtag]} y_1 = y_2}$$ | |
| We conclude by applying the one-sided rule for abstraction. | |
| \subsubsection{Stochastic dominance} | |
| Stochastic dominance defines a partial order between random variables | |
| whose underlying set is itself a partial order; it has many different | |
| applications in statistical biology (e.g.\ in the analysis of the | |
| birth-and-death processes), statistical physics (e.g.\ in percolation | |
| theory), and economics. First-order stochastic dominance, which we | |
| define below, is also an important application of probabilistic | |
| couplings. We demonstrate how to use our proof system for proving | |
| (first-order) stochastic dominance for a simple Markov process which | |
| samples biased coins. While the example is elementary, the proof | |
| method extends to more complex examples of stochastic dominance, and | |
| illustrates the benefits of Strassen's formulation of the coupling | |
| rule over alternative formulations stipulating the existence of | |
| bijections (explained later). | |
| We start by recalling the definition of (first-order) stochastic | |
| dominance for the $\mathbb{N}$-valued case. The definition extends to | |
| arbitrary partial orders. | |
| \begin{definition}[Stochastic dominance] | |
| Let $\mu_1,\mu_2\in \Distr(\mathbb{N})$. We say that $\mu_2$ | |
| stochastically dominates $\mu_1$, written $\mu_1\leq_{\mathrm{SD}} | |
| \mu_2$, iff for every $n\in\mathbb{N}$, | |
| $$\Pr_{x\leftarrow \mu_1}[x\geq n] \leq \Pr_{x\leftarrow \mu_2}[x\geq n]$$ | |
| \end{definition} | |
| The following result, equivalent to \autoref{cor:fundamental}, | |
| characterizes stochastic dominance using probabilistic couplings. | |
| \begin{proposition} | |
| Let $\mu_1,\mu_2\in \Distr(\mathbb{N})$. Then $\mu_1\leq_{\mathrm{SD}} | |
| \mu_2$ iff $\coupl{\mu_1}{\mu_2}{(\leq)}$. | |
| \end{proposition} | |
| We now turn to the definition of the Markov chain. For $p\in [0,1]$, | |
| we consider the parametric $\mathbb{N}$-valued Markov chain | |
| $\mathsf{coins} \defeq \markov(0,h)$, with initial state $0$ and | |
| (parametric) step function: | |
| $$ h \defeq \lambda x. \mlet{b}{\bern{p}}{\munit{x+b}} | |
| $$ | |
| where, for $p \in [0,1]$, $\bern{p}$ is the Bernoulli distribution on | |
| $\{0,1\}$ with probability $p$ for $1$ and $1-p$ for $0$. Our goal is | |
| to establish that $\mathsf{coins}$ is monotonic, i.e.\, for every | |
| $p_1,p_2\in [0,1]$, $p_1\leq p_2$ implies $\mathsf{coins}~p_1 | |
| \leq_{\mathrm{SD}} \mathsf{coins}~p_2$. We formalize this statement as | |
| $$ | |
| \jgrholnoc{\mathsf{coins}}{[0,1] \rightarrow \Distr(\Str{\mathbb{N}})} | |
| {\mathsf{coins}}{[0,1] \rightarrow \Distr(\Str{\mathbb{N}})} | |
| {\psi} | |
| $$ | |
| where $\psi\defeq \forall p_1,p_2. p_1\leq p_2 \Rightarrow | |
| \diamond_{[y_1 \ot \res\ltag, y_2 \ot \res\rtag]} \All(y_1, y_2, z_1.z_2.z_1\leq z_2)$. | |
| The crux of the proof is to establish stochastic dominance for the | |
| Bernoulli distribution: | |
| $$ | |
| p_1:[0,1],p_2:[0,1]\mid p_1\leq p_2 \jgrholnoc{\bern{p_1}} | |
| {\Distr(\mathbb{N})} | |
| {\bern{p_2}} | |
| {\Distr(\mathbb{N})} | |
| {\res\ltag\stackrel{\diamond}{\leq} \res\rtag} | |
| $$ | |
| where we use $e_1 \stackrel{\diamond}{\leq} e_2$ as shorthand for | |
| $\diamond_{[y_1 \ot e_1, y_2 \ot e_2]} y_1\leq y_2$. This is proved | |
| directly by the \rname{COUPLING} rule and checking by simple | |
| calculations that the premise of the rule is valid. | |
| We briefly explain how to conclude the proof. Let $h_1$ and $h_2$ be | |
| the step functions for $p_1$ and $p_2$ respectively. It is clear from | |
| the above that (context omitted): | |
| $$ | |
| x_1\leq x_2 \jgrholnoc{h_1\ x_1}{\Distr(\bool)} | |
| {h_2\ x_2}{\Distr(\bool)} | |
| {\diamond_{[y_1 \ot \res\ltag, y_2 \ot \res\rtag]}. | |
| {y_1\leq y_2}} | |
| $$ | |
| and by the definition of $\All$: | |
| $$x_1 \leq x_2 \Rightarrow {\All(xs_1,xs_2,z_1.z_2.z_1\leq z_2)} | |
| \Rightarrow \All(\cons{x_1}{\later xs_1}, \cons{x_2}{\later xs_2}, z_1.z_2.z_1\leq z_2)$$ | |
| So, we can conclude by applying the \rname{Markov} rule. | |
| It is instructive to compare our proof with prior formalizations, and | |
| in particular with the proof in \cite{BartheEGHSS15}. Their proof is | |
| carried out in the \textsf{pRHL} logic, whose [\textsf{COUPLING}] rule | |
| is based on the existence of a bijection that satisfies some property, | |
| rather than on our formalization based on Strassen's Theorem. Their | |
| rule is motivated by applications in cryptography, and works well for | |
| many examples, but is inconvenient for our example at hand, which | |
| involves non-uniform probabilities. Indeed, their proof is based on | |
| code rewriting, and is done in two steps. First, they prove equivalence between sampling and | |
| returning $x_1$ from $\bern{p_1}$; and sampling $z_1$ from | |
| $\bern{p_2}$, $z_2$ from $\bern{\rfrac{p_1}{p_2}}$ and returning | |
| $z= z_1 \land z_2$. Then, they find a coupling between $z$ and | |
| $\bern{p_2}$. | |
| \subsubsection{Shift coupling: random walk vs lazy random walk} | |
| The previous example is an instance of a lockstep coupling, in that it | |
| relates the $k$-th element of the first chain with the $k$-th element | |
| of the second chain. Many examples from the literature follow this | |
| lockstep pattern; however, it is not always possible to establish | |
| lockstep couplings. Shift couplings are a relaxation of lockstep | |
| couplings where we relate elements of the first and second chains | |
| without the requirement that their positions coincide. | |
| We consider a simple example that motivates the use of shift | |
| couplings. Consider the random walk and lazy random walk (which, at | |
| each time step, either chooses to move or stay put), both defined as | |
| Markov chains over $\mathbb{Z}$. For simplicity, assume that both | |
| walks start at position 0. It is not immediate to find a coupling | |
| between the two walks, since the two walks necessarily get | |
| desynchronized whenever the lazy walk stays put. Instead, the trick is | |
| to consider a lazy random walk that moves two steps instead of | |
| one. The random walk and the lazy random walk of step 2 are defined by | |
| the step functions: | |
| $$\begin{array}{rcl} | |
| \operatorname{step} & \defeq & \lambda x.\mlet{z}{\mathcal{U}_{\{-1,1\}}}{\munit{z+x}} \\ | |
| \operatorname{lstep2} & \defeq & \lambda x.\mlet{z}{\mathcal{U}_{\{-1,1\}}}{\mlet{b}{\mathcal{U}_{\{0,1\}}}{\munit{x+2*z*b}}} | |
| \end{array}$$ | |
| After 2 iterations of $\operatorname{step}$, the position has either | |
| changed two steps to the left or to the right, or has returned to the | |
| initial position, which is the same behaviour $\operatorname{lstep2}$ | |
| has on every iteration. Therefore, the coupling we want to find should | |
| equate the elements at position $2i$ in $\operatorname{step}$ with the | |
| elements at position $i$ in $\operatorname{lstep2}$. The details on how | |
| to prove the existence of this coupling are in \autoref{sec:grhol}. | |
| \subsubsection{Lumped coupling: random walks on 3 and 4 dimensions} | |
| A Markov chain is \emph{recurrent} if it has probability 1 of | |
| returning to its initial state, and \emph{transient} otherwise. It is | |
| relatively easy to show that the random walk over $\mathbb{Z}$ is | |
| recurrent. One can also show that the random walk over $\mathbb{Z}^2$ | |
| is recurrent. However, the random walk over $\mathbb{Z}^3$ is | |
| transient. | |
| For higher dimensions, we can use a coupling | |
| argument to prove transience. Specifically, we can define a coupling | |
| between a lazy random walk in $n$ dimensions and a random walk in $n | |
| +m$ dimensions, and derive transience of the latter from transience of | |
| the former. We define the (lazy) random walks below, and sketch the | |
| coupling arguments. | |
| Specifically, we show here the particular case of the transience of | |
| the 4-dimensional random walk from the transience of the 3-dimensional | |
| lazy random walk. We start by defining the stepping functions: | |
| \[\begin{array}{rl} | |
| \operatorname{step}_4 &: \mathbb{Z}^4 \to \Distr(\mathbb{Z}^4) | |
| \defeq \lambda z_1. \mlet{x_1}{\unif{U_4}}{\munit{z_1 +_4 x_1}} | |
| \\ | |
| \operatorname{lstep}_3 &: \mathbb{Z}^3 \to \Distr(\mathbb{Z}^3) | |
| \defeq \lambda z_2. \mlet{x_2}{\unif{U_3}}{\mlet{b_2}{\bern{\rfrac{3}{4}}}{\munit{z_2 +_3 b_2*x_2 }}} | |
| \end{array} | |
| \] | |
| where $U_i=\{(\pm 1,0,\dots 0), \dots, (0,\dots,0,\pm 1)\}$ are the | |
| vectors of the basis of $\mathbb{Z}^i$ and their opposites. Then, the | |
| random walk of dimension 4 is modelled by $\operatorname{rwalk4} \defeq | |
| \markov(0, \operatorname{step_4})$, and the lazy walk of dimension 3 is | |
| modelled by $\operatorname{lwalk3} \defeq | |
| \markov(0, \operatorname{step_3})$. We want to prove: | |
| $$ | |
| \jgrholnoc{\operatorname{rwalk4}}{\Distr(\Str{\mathbb{Z}^{4}})} | |
| {\operatorname{lwalk3}}{\Distr(\Str{\mathbb{Z}^{3}})} | |
| {\diamond_{\left[\substack{y_1\leftarrow \res\ltag \\ y_2\leftarrow \res\rtag}\right]} | |
| \All(y_1, y_2, z_1.z_2.\operatorname{pr}^{4}_{3}(z_1) = z_2)} | |
| $$ | |
| where $\operatorname{pr}^{n_2}_{n_1}$ denotes the standard projection from | |
| $\mathbb{Z}^{n_2}$ to $\mathbb{Z}^{n_1}$. | |
| We apply the \rname{Markov} rule. The only interesting premise requires proving that the transition function preserves the coupling: | |
| \[p_2=\operatorname{pr}^{4}_{3}(p_1) \vdash \operatorname{step_4} \sim \operatorname{lstep}_3 | |
| \mid \forall x_1 x_2. x_2=\operatorname{pr}^4_3(x_1) \Rightarrow \diamond_{\left[\substack{y_1\ot \res\ltag\ x_1 \\ y_2\ot \res\rtag\ x_2}\right]} \operatorname{pr}^4_3(y_1)=y_2 \] | |
| To prove this, we need to find the appropriate coupling, i.e., one | |
| that preserves the equality. The idea is that the step in | |
| $\mathbb{Z}^3$ must be the projection of the step in | |
| $\mathbb{Z}^4$. This corresponds to the following judgement: | |
| \[\left.\begin{array}{rl} | |
| \lambda z_1. &\mlet{x_1}{\unif{U_4}} | |
| {\\ &\munit{z_1 +_4 x_1}} | |
| \end{array} | |
| \sim | |
| \begin{array}{rl} | |
| \lambda z_2. &\mlet{x_2}{\unif{U_3}} | |
| {\\ &\mlet{b_2}{\bern{\rfrac{3}{4}}} | |
| {\\ &\munit{z_2 +_3 b_2*x_2}}} | |
| \end{array} | |
| \; \right| \; | |
| \begin{array}{c} | |
| \forall z_1 z_2. \operatorname{pr}^4_3(z_1) = z_2 \Rightarrow \\ \operatorname{pr}^4_3(\res\ltag\ z_1) \stackrel{\diamond}{=} \res\rtag\ z_2 | |
| \end{array}\] | |
| which by simple equational reasoning is the same as | |
| \[\left.\begin{array}{rl} | |
| \lambda z_1. &\mlet{x_1}{\unif{U_4}} | |
| {\\ &\munit{z_1 +_4 x_1}} | |
| \end{array} | |
| \sim | |
| \begin{array}{rl} | |
| \lambda z_2. &\mlet{p_2}{\unif{U_3} \times \bern{\rfrac{3}{4}}} | |
| {\\ &\munit{z_2 +_3 \pi_1(p_2)*\pi_2(p_2)}} | |
| \end{array} | |
| \; \right| \; | |
| \begin{array}{c} | |
| \forall z_1 z_2. \operatorname{pr}^4_3(z_1) = z_2 \Rightarrow \\ \operatorname{pr}^4_3(\res\ltag\ z_1) \stackrel{\diamond}{=} \res\rtag\ z_2 | |
| \end{array} | |
| \] | |
| We want to build a coupling such that if we sample $(0,0,0,1)$ or | |
| $(0,0,0,-1)$ from $\unif{U_3}$, then we sample $0$ from | |
| $\bern{\rfrac{3}{4}}$, and otherwise if we sample | |
| $(x_1,x_2,x_3,0)$ from $\unif{U_4}$, we sample $(x_1,x_2,x_3)$ from | |
| $U_3$. Formally, we prove this with the \rname{Coupling} rule. | |
| Given $X:U_4 \to \bool$, by simple computation we show that: | |
| \[ \Pr_{z_1 \sim \unif{U_4}}[z_1 \in X] \leq \Pr_{z_2 \sim \unif{U_3} \times \bern{\rfrac{3}{4}}}[z_2 \in \{ y \mid \exists x \in X. {\sf pr}^4_3(x) = \pi_1(y)*\pi_2(y) \}]\] | |
| This concludes the proof. From the previous example, it follows that | |
| the lazy walk in 3 dimensions is transient, since the random walk in | |
| 3 dimensions is transient. By simple reasoning, we now conclude that | |
| the random walk in 4 dimensions is also transient. | |
| \section{Probabilistic Guarded Lambda Calculus} | |
| \label{sec:syntax} | |
| To ensure that a function on infinite datatypes is well-defined, one | |
| must check that it is \emph{productive}. This means that any finite | |
| prefix of the output can be computed in finite time. For instance, | |
| consider the following function on streams: | |
| \[ | |
| \mathtt{letrec\ bad\ (x:xs) = x : tail (bad\ xs)} | |
| \] | |
| This function is not productive since only the first element can be computed. We | |
| can argue this as follows: | |
| Suppose that the tail of a stream is available one unit of time after its head, | |
| and that that \texttt{x:xs} is available at time 0. How much time does it take | |
| for \texttt{bad} to start outputting its tail? Assume it takes $k$ units of | |
| time. This means that \texttt{tail(bad\ xs)} will be available at time $k+1$ , | |
| since \texttt{xs} is only available at time 1. But \texttt{tail(bad\ xs)} is | |
| exactly the tail of \texttt{bad(x:xs)}, and this is a contradiction, since | |
| \texttt{x:xs} is available at time 0 and therefore the tail of | |
| \texttt{bad(x:xs)} should be available at time $k$. Therefore, the tail of | |
| \texttt{bad} will never be available. | |
| The guarded lambda calculus solves the productivity problem by | |
| distinguishing at type level between data that is available now and | |
| data that will be available in the future, and restricting when | |
| fixpoints can be defined. Specifically, the guarded lambda calculus | |
| extends the usual simply typed lambda calculus with two modalities: | |
| $\later$ (pronounced \textit{later}) and $\square$ | |
| (\textit{constant}). The later modality represents data that will be | |
| available one step in the future, and is introduced and removed by the | |
| term formers $\later$ and $\prev$\! respectively. This modality is | |
| used to guard recursive occurrences, so for the calculus to remain | |
| productive, we must restrict when it can be eliminated. This is | |
| achieved via the constant modality, which expresses that all the data | |
| is available at all times. In the remainder of this section we present a probabilistic extension | |
| of this calculus. | |
| \paragraph*{Syntax} | |
| Types of the calculus are defined by the grammar | |
| \begin{align*} | |
| A,B ::= b \mid \nat \mid A \times B \mid A + B \mid A \to B \mid \Str{A} \mid \square~A \mid \later{A} \mid \Distr(C) | |
| \end{align*} | |
| where $b$ ranges over a collection of base types. | |
| $\Str{A}$ is the type of guarded streams of elements of type | |
| $A$. Formally, the type $\Str{A}$ is isomorphic to $A \times | |
| \later{\Str{A}}$. This isomorphism gives a way to introduce streams | |
| with the function $(\cons{}{}) : A \to \later\Str{A} \to | |
| \Str{A}$ and to eliminate them with the functions $\hd: \Str{A} \to A$ | |
| and $\tl: \Str{A} \to \later\Str{A}$. | |
| $\Distr (C)$ is the type of distributions over \emph{discrete types} | |
| $C$. Discrete types are defined by the following grammar, where $b_0$ | |
| are discrete base types, e.g., $\zint$. | |
| \begin{align*} | |
| C,D ::= b_0 \mid \nat \mid C \times D \mid C + D \mid \Str{C} | \later{C}. | |
| \end{align*} | |
| Note that, in particular, arrow types are not discrete but streams | |
| are. This is due to the semantics of streams as sets of finite | |
| approximations, which we describe in the next subsection. Also note | |
| that $\square\Str{A}$ is not discrete since it makes the full infinite | |
| streams available. | |
| We also need to distinguish between arbitrary types $A, B$ and | |
| constant types $S, T$, which are defined by the following grammar | |
| \begin{align*} | |
| S, T ::= b_C \mid \nat \mid S \times T \mid S + T \mid S \to T \mid \square~A | |
| \end{align*} | |
| where $b_C$ is a collection of constant base types. Note in | |
| particular that for any type $A$ the type $\square~A$ is constant. | |
| The terms of the language $t$ are defined by the following grammar | |
| \begin{align*} | |
| t &::= ~x | |
| \mid c | |
| \mid 0 \mid S t \mid \casenat{t}{t}{t} \mid \mu \mid \munit{t} \mid \mlet{x}{t}{t} \\ | |
| &\mid \langle t, t \rangle \mid \pi_1 t \mid \pi_2 t | |
| \mid \inj{1} t \mid \inj{2} t \mid \CASE t \OF \inj{1} x . t; \inj{2} y . t | |
| \mid \lambda x . t \mid t\,t \mid \fix{x}{t}\\ | |
| &\mid \cons{t}{ts} \mid \hd t \mid \tl t | |
| \mid \boxx{t} \mid \letbox{x}{t}{t} \mid \letconst{x}{t}{t} | |
| \mid \latern[\xi]t \mid \prev t | |
| \end{align*} | |
| where $\xi$ is a delayed substitution, a sequence of bindings | |
| $\hrt{x_1 \gets t_1, \ldots, x_n \gets t_n}$. | |
| The terms $c$ are constants corresponding to the base types used and | |
| $\munit{t}$ and $\mlet{x}{t}{t}$ are the introduction and sequencing | |
| construct for probability distributions. The meta-variable $\mu$ | |
| stands for base distributions like $\unif{C}$ and $\bern{p}$. | |
| Delayed substitutions were introduced in \cite{BGCMB16} in a dependent type theory to be able to work with types dependent on terms of type $\later{A}$. | |
| In the setting of a simple type theory, such as the one considered in this paper, delayed substitutions are equivalent to having the applicative structure~\cite{McBride:Applicative} $\app$ for the $\later$ modality. | |
| However, delayed substitutions extend uniformly to the level of propositions, and thus we choose to use them in this paper in place of the applicative structure. | |
| \paragraph*{Denotational semantics} | |
| The meaning of terms is given by a denotational model in the category $\trees$ of presheaves over $\omega$, the first infinite ordinal. | |
| This category $\trees$ is also known as the \emph{topos of trees}~\cite{Birkedal-et-al:topos-of-trees}. | |
| In previous work~\cite{CBGB16}, it was shown how to model most of the constructions of the guarded lambda calculus and its internal logic, with the notable exception of the probabilistic features. | |
| Below we give an elementary | |
| presentation of the semantics. | |
| Informally, the idea behind the topos of trees is to represent | |
| (infinite) objects from their finite approximations, which we observe | |
| incrementally as time passes. Given an object $x$, we can consider a | |
| sequence $\{x_i\}$ of its finite approximations observable at time | |
| $i$. These are trivial for finite objects, such as a natural number, | |
| since for any number $n$, $n_i = n$ at every $i$. But for infinite | |
| objects such as streams, the $i$th approximation is the prefix of | |
| length $i+1$. | |
| Concretely, the category $\trees$ consists of: | |
| \begin{itemize} | |
| \item Objects $X$: families of sets $\{X_i\}_{i\in\nat}$ together with \emph{restriction functions} $r_n^X : X_{n+1} \to X_n$. | |
| We will write simply $r_n$ if $X$ is clear from the context. | |
| \item Morphisms $X \to Y$ : families of functions $\alpha_n : X_n \to Y_n$ commuting with restriction functions in the sense of $r_n^Y \circ \alpha_{n+1} = \alpha_n \circ r_n^X$. | |
| \end{itemize} | |
| The full interpretation of types of the calculus can be found in | |
| \autoref{fig:sem-types} in the appendix. The main points we want to | |
| highlight are: | |
| \begin{itemize} | |
| \item Streams over a type $A$ are interpreted as sequences of finite | |
| prefixes of elements of $A$ with the restriction functions of $A$: | |
| $$\sem{\Str{A}} \defeq \sem{A}_0 \times \{*\} \xleftarrow{r_0 | |
| \times !} \sem{A}_1 \times \sem{\Str{A}}_0 | |
| \xleftarrow{r_1 \times r_0 \times !} | |
| \sem{A}_2\times \sem{\Str{A}}_1 \leftarrow \cdots$$ | |
| \item Distributions over a discrete object $C$ are | |
| defined as a sequence of distributions over each | |
| $\sem{C}_i$: $$\sem{\Distr(C)} \defeq \Distr(\sem{C}_0) | |
| \stackrel{\Distr(r_0)}{\longleftarrow} \Distr(\sem{C}_1) | |
| \stackrel{\Distr(r_1)}{\longleftarrow} \Distr(\sem{C}_2) | |
| \stackrel{\Distr(r_2)}{\longleftarrow} \ldots,$$ where $\Distr(\sem{C}_i)$ is | |
| the set of (probability density) functions $\mu : \sem{C}_i \to [0,1]$ such that $\sum_{x_\in X} \mu | |
| x = 1$, and $\Distr(r_i)$ adds the probability density of all the points | |
| in $\sem{C}_{i+1}$ that are sent by $r_i$ to the same point in the $\sem{C}_{i}$. In other words, | |
| $\Distr(r_i)(\mu)(x) = \Pr_{y \ot \mu}[r_i(y) = x]$ | |
| \end{itemize} | |
| An important property of the interpretation is that discrete types are interpreted as objects $X$ such that $X_i$ is finite or countably infinite for every $i$. | |
| This allows us to define distributions on these objects without the need for measure theory. | |
| In particular, the type of guarded streams $\Str{A}$ is discrete provided $A$ is, which is clear from the interpretation of the type $\Str{A}$. | |
| Conceptually this holds because $\sem{\Str{A}}_i$ is an approximation of real streams, consisting of only the first $i+1$ elements. | |
| An object $X$ of $\trees$ is \emph{constant} if all its restriction | |
| functions are bijections. Constant types are interpreted as constant | |
| objects of $\trees$ and for a constant type $A$ | |
| the objects $\sem{\square A}$ and $\sem{A}$ are isomorphic in $\trees$. | |
| \paragraph*{Typing rules} | |
| Terms are typed under a dual context $\Delta \mid \Gamma$, where $\Gamma$ is a usual context that binds variables to a type, and $\Delta$ is a constant context containing variables bound to types that are \textit{constant}. | |
| The term $\letconst{x}{u}{t}$ allows us to shift variables between constant and non-constant contexts. The typing rules can be found in \autoref{fig:glambda}. | |
| The semantics of such a dual context $\Delta \mid \Gamma$ is given as the product of types in $\Delta$ and $\Gamma$, except that we implicitly add $\square$ in front of every type in $\Delta$. | |
| In the particular case when both contexts are empty, the semantics of the dual context correspond to the terminal object $1$, which is the singleton set $\{\ast\}$ at each time. | |
| The interpretation of the well-typed term $\Delta \mid \Gamma \vdash t : A$ is | |
| defined by induction on the typing derivation, and can be found in | |
| \autoref{fig:sem-glc} in the appendix. | |
| \begin{figure*}[!tb] | |
| \small | |
| \begin{gather*} | |
| \inferrule | |
| {x : A \in \Gamma} | |
| {\Delta \mid \Gamma \vdash x : A} | |
| \qquad | |
| \inferrule | |
| {x : A \in \Delta} | |
| {\Delta \mid \Gamma \vdash x : A} | |
| \qquad | |
| \inferrule | |
| {\Delta \mid \Gamma, x : A \vdash t : B} | |
| {\Delta \mid \Gamma \vdash \lambda x . t : A \to B} | |
| \\ | |
| \inferrule | |
| {\Delta \mid \Gamma \vdash t : A \to B \and \Delta \mid \Gamma \vdash u : A} | |
| {\Delta \mid \Gamma \vdash t\,u : B} | |
| \qquad | |
| \inferrule | |
| {\Delta \mid \Gamma, f : \later{A} \vdash t : A} | |
| {\Delta \mid \Gamma \vdash \fix{f}{t} : A} | |
| \qquad | |
| \inferrule | |
| {\Delta \mid \cdot \vdash t : \later A} | |
| {\Delta \mid \Gamma \vdash \prev{t} : A} | |
| \\ | |
| \inferrule | |
| {\Delta \mid \cdot \vdash t : A} | |
| {\Delta \mid \Gamma \vdash \boxx{t} : \square A} | |
| \qquad | |
| \inferrule | |
| {\Delta \mid \Gamma \vdash u : \square B \and | |
| \Delta, x : B \mid \Gamma \vdash t : A} | |
| {\Delta \mid \Gamma \vdash \letbox{x}{u}{t} : A} | |
| \\ | |
| \inferrule | |
| {\Delta \mid \Gamma \vdash u : B \and | |
| \Delta, x : B \mid \Gamma \vdash t : A \and | |
| B \text{ constant}} | |
| {\Delta \mid \Gamma \vdash \letconst{x}{u}{t} : A} | |
| \\ | |
| \inferrule | |
| {\Delta \mid \Gamma, x_1 : A_1, \cdots x_n : A_n \vdash t : A | |
| \and | |
| \Delta \mid \Gamma \vdash t_i : \later{A_i}} | |
| {\Delta \mid \Gamma \vdash \later[x_1 \gets t_1, \ldots, x_n \gets t_n]{t} : \later{A}} | |
| \qquad | |
| \inferrule | |
| {\Delta \mid \Gamma \vdash t : A \and A \text{ discrete }} | |
| {\Delta \mid \Gamma \vdash \munit{t} : \Distr(A)} | |
| \\ | |
| \inferrule | |
| {\Delta \mid \Gamma \vdash t : \Distr(A) \and \Delta \mid \Gamma, x : A \vdash u : \Distr(B)} | |
| {\Delta \mid \Gamma \vdash \mlet{x}{t}{u} : \Distr(B)} | |
| \qquad | |
| \inferrule | |
| {\mu \text{ primitive distribution on type } A} | |
| {\Delta \mid \Gamma \vdash \mu : \Distr(A)} | |
| \end{gather*} | |
| \shrinkcaption | |
| \caption{A selection of the typing rules of the guarded lambda calculus. | |
| The rules for products, sums, and natural numbers are standard.} | |
| \label{fig:glambda} | |
| \end{figure*} | |
| \paragraph*{Applicative structure of the later modality} | |
| As in previous work we can define the operator $\app$ satisfying the typing rule | |
| \[ | |
| \inferrule | |
| {\Delta \mid \Gamma \vdash t : \later{(A \to B)} \and \Delta \mid \Gamma \vdash u : \later{A}} | |
| {\Delta \mid \Gamma \vdash t \app u : \later{B}} | |
| \] | |
| and the equation | |
| $(\later{t}) \app (\later{u}) \equiv \later(t\ u)$ | |
| as the term | |
| $t \app u \defeq \latern[\hrt{f \gets t, x \gets u}]{f\,x}$. | |
| \paragraph*{Example: Modelling Markov chains} | |
| As an application of $\app$ and an example of how to use guardedness | |
| and probabilities together, we now give the precise | |
| definition of the $\markov$ construct that we used to model Markov chains earlier: | |
| \[ | |
| \begin{array}{rl} | |
| \markov &: C \to (C \to \Distr(C)) \to \Distr(\Str{C}) \\ | |
| \markov &\defeq \fix{f}{\lambda x. \lambda h. \\ | |
| &\quad \mlet {z}{h\ x}{ \mlet{t}{\operatorname{swap_{\later\Distr}^{\Str{C}}}(f \app \later z \app \later h)} | |
| { \munit{\cons{x}{t}}}}} | |
| \end{array} | |
| \] | |
| The guardedness condition gives $f$ the type $\later( C \to (C \to | |
| \Distr(C)) \to \Distr(\Str{C}))$ in the body of the | |
| fixpoint. Therefore, it needs to be applied functorially (via $\app$) | |
| to $\later z$ and $\later h$, which gives us a term of type $\later | |
| \Distr(\Str{C})$. To complete the definition we need to build a term | |
| of type $\Distr(\later\Str{C})$ and then sequence it with $\cons{}{}$ | |
| to build a term of type $\Distr(\Str{C})$. To achieve this, we use the | |
| primitive operator $\operatorname{swap_{\later\Distr}^C} : | |
| \later\Distr(C) \to \Distr(\later C)$, which witnesses the isomorphism | |
| between $\later\Distr(C)$ and $\Distr(\later C)$. For this isomorphism | |
| to exist, it is crucial that distributions be total (i.e., we cannot | |
| use subdistributions). Indeed, the denotation for $\later\Distr(C)$ is | |
| the sequence $\{\ast\} \ot \Distr(C_1) \ot \Distr(C_2) \ot \dots$, | |
| while the denotation for $\Distr(\later C)$ is the sequence | |
| $\Distr(\{\ast\}) \ot \Distr(C_1) \ot \Distr(C_2) \ot \dots$, and | |
| $\{\ast\}$ is isomorphic to $\Distr(\{\ast\})$ in ${\sf Set}$ only if | |
| $\Distr$ considers only total distributions. | |
| \section{Guarded higher-order logic} | |
| \label{sec:ghol} | |
| We now introduce Guarded HOL (GHOL), which is a higher-order logic to reason about terms of the guarded lambda calculus. | |
| The logic is essentially that of~\cite{CBGB16}, but presented with the dual context formulation analogous to the dual-context typing judgement of the guarded lambda calculus. | |
| Compared to standard intuitionistic higher-order logic, the logic GHOL has two additional constructs, corresponding to additional constructs in the guarded lambda calculus. | |
| These are the later modality ($\later$) \emph{on propositions}, with delayed substitutions, | |
| which expresses that a proposition holds one time unit into the future, and the ``always'' | |
| modality $\square$, which expresses that a proposition holds at all times. | |
| Formulas are defined by the grammar: | |
| \[ \phi, \psi ::= \top \mid \phi \wedge \psi \mid \phi \vee \psi \mid \neg \psi | |
| \mid \forall x. \phi \mid \exists x. \phi \mid \later[x_1 \ot t_1 \dots x_n | |
| \ot t_n]\phi \mid \square \phi\] | |
| The basic judgement of the logic is $\Delta \mid \Sigma \mid \Gamma \mid \Psi \vdash \phi$ where $\Sigma$ is a logical context for $\Delta$ (that is, a list of formulas well-formed in $\Delta$) and $\Psi$ is another logical context for the dual context $\Delta\mid\Gamma$. | |
| The formulas in context $\Sigma$ must be \emph{constant} propositions. | |
| We say that a proposition $\phi$ is \emph{constant} if it is well-typed in context $\Delta \mid \cdot$ and moreover if every occurrence of the later modality in $\phi$ is under the $\square$ modality. | |
| Selected rules are displayed in Figure~\ref{fig:hol} on page~\pageref{fig:hol}. We highlight \rname{Loeb} induction, which is the key to reasoning about fixpoints: to prove that $\phi$ holds now, one | |
| can assume that it holds in the future. | |
| The interpretation of the formula $\Delta \mid \Gamma \vdash \phi$ is a subobject of the interpretation $\sem{\Delta \mid \Gamma}$. | |
| Concretely the interpretation $A$ of $\Delta \mid \Gamma \vdash \phi$ is a family $\left\{A_i\right\}_{i=0}^\infty$ of sets such that $A_i \subseteq \sem{\Delta \mid \Gamma}_i$. | |
| This family must satisfy the property that if $x \in A_{i+1}$ then $r_i(x) \in A_i$ where $r_i$ are the restriction functions of $\sem{\Delta \mid \Gamma}$. | |
| The interpretation of formulas is defined by induction on the typing derivation. | |
| In the interpretation of the context $\Delta \mid \Sigma \mid \Gamma \mid \Psi$ the formulas in $\Sigma$ are interpreted with the added $\square$ modality. | |
| Moreover all formulas $\phi$ in $\Sigma$ are typeable in the context $\Delta \mid \cdot \vdash \phi$ and thus their interpretations are subsets of $\sem{\square \Delta}$. | |
| We treat these subsets of $\sem{\Delta \mid \Gamma}$ in the obvious way. | |
| The cases for the semantics of the judgement $\Delta \mid \Gamma \vdash \phi$ can be found in the appendix. | |
| It can be shown that this logic is sound with respect to its model in the topos of trees. | |
| \begin{theorem}[Soundness of the semantics]\label{thm:sound-ghol} | |
| The semantics of guarded higher-order logic is sound: if $\Delta \mid \Sigma \mid \Gamma \mid \Psi \vdash \phi$ is derivable then for all $n \in \nat$, $\sem{\square\Sigma}_n \cap \sem{\Psi}_n \subseteq \sem{\phi}$. | |
| \end{theorem} | |
| In addition, Guarded HOL is expressive enough to axiomatize standard probabilities over discrete sets. | |
| This axiomatization can be used to define the $\diamond$ modality directly in Guarded HOL (as opposed to our relational proof system, | |
| were we use it as a primitive). | |
| Furthermore, we can derive from this axiomatization additional rules to reason about couplings, which can be seen | |
| in Figure~\ref{fig:prob-hol}. These rules will be the key to proving the soundness of the probabilistic fragment of the relational proof system, and can | |
| be shown to be sound themselves. | |
| \begin{proposition}[Soundness of derived rules]\label{thm:sound-prob-ghol} | |
| The additional rules are sound. | |
| \end{proposition} | |
| \begin{figure*}[!tb] | |
| \small | |
| \begin{gather*} | |
| \infer[\sf AX_U] | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\phi}} | |
| {\phi \in \Psi} | |
| \quad | |
| \infer[\sf AX_G] | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\phi}} | |
| {\phi \in \Sigma} | |
| \quad | |
| \infer[\sf CONV] | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{t=t'}} | |
| {\Gamma \vdash t:\tau & \Gamma \vdash t':\tau & t \equiv t'} | |
| \\[0.3em] | |
| \infer[\sf SUBST] | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\phi\subst{x}{u}}} | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\phi\subst{x}{t}} & \jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{t = u}} | |
| \quad\quad | |
| \infer[\sf Loeb] | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\phi}} | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi, \later\phi}{\phi}} | |
| \\[0.3em] | |
| \infer[\sf \later_I] | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\later[x_1 \ot t_1,\dots,x_n \ot t_n]\phi}} | |
| {\jghol{\Delta}{\Sigma}{\Gamma, x_1:A_1, \dots, x_n:A_n}{\Psi}{\phi} & | |
| \Delta\mid\Gamma\vdash t_1 : \later A_1 & \dots & | |
| \Delta\mid\Gamma\vdash t_n : \later A_n} | |
| \\[0.3em] | |
| \infer[\sf \later_E] | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\phi\subst{x_1}{\prev t_1}\dots\subst{x_n}{\prev t_n}}} | |
| {\jghol{\Delta}{\Sigma}{\cdot}{\cdot}{\later[x_1 \ot t_1 \dots x_n \ot t_n]\phi} & | |
| \Delta \mid \bullet \vdash t_1 : \later A_1 & \dots & \Delta \mid | |
| \bullet \vdash t_n : \later A_n} | |
| \\[0.3em] | |
| \inferrule | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\later[x_1\!\ot\!t_1,\dots,x_n\!\ot\!t_n]\psi} \\ | |
| \Delta \mid \Gamma \vdash t_1 : \later A_1 \; \dots\; \Delta \mid | |
| \Gamma \vdash t_n : \later A_n \\\\ | |
| \jghol{\Delta}{\Sigma}{\Gamma, x_1:A_1, \dots, x_n:A_n}{\Psi, \psi}{\phi}} | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\later[x_1 \ot t_1,\dots,x_n \ot t_n]\phi}} | |
| {\sf \later_{App}} | |
| \\[0.3em] | |
| \infer[\sf \square_I] | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\square\phi}} | |
| {\jghol{\Delta}{\Sigma}{\cdot}{\cdot}{\phi}} | |
| \quad\quad | |
| \infer[\sf \square_E] | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\phi}} | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\square\psi} & | |
| \jghol{\Delta}{\Sigma,\psi}{\Gamma}{\Psi}{\phi}} | |
| \end{gather*} | |
| \shrinkcaption | |
| \caption{Selected Guarded Higher-Order Logic rules}\label{fig:hol} | |
| \end{figure*} | |
| \begin{figure*}[!tb] | |
| \small | |
| \begin{gather*} | |
| \inferrule*[right = \sf MONO2] | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\diamond_{[x_1\leftarrow t_1, x_2 \leftarrow t_2]}\phi} \\ | |
| \jghol{\Delta}{\Sigma}{\Gamma, x_1 : C_1, x_2 : C_2}{\Psi, \phi}{\psi}} | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\diamond_{[x_1\leftarrow t_1, x_2\leftarrow t_2]}\psi}} | |
| \\[0.3em] | |
| \inferrule*[right = \sf UNIT2] | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\phi\subst{x_1}{t_1}\subst{x_2}{t_2}}} | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi} | |
| {\diamond_{[x_1\leftarrow \munit{t_1}, x_2\leftarrow \munit{t_2}]}\phi}} | |
| \\[0.3em] | |
| \inferrule*[right = \sf MLET2] | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\diamond_{[x_1\leftarrow t_1, x_2\leftarrow t_2]} \phi} | |
| \\ \jghol{\Delta}{\Sigma}{\Gamma,x_1:C_1,x_2:C_2}{\Psi,\phi} | |
| {\diamond_{[y_1\leftarrow t'_1, y_2\leftarrow t'_2]}\psi}} | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi} | |
| {\diamond_{[y_1\leftarrow \mlet{x_1}{t_1}{t'_1}, y_2\leftarrow \mlet{x_2}{t_2}{t'_2}]} \psi}} | |
| \\[0.3em] | |
| \inferrule*[right = \sf MLET{-}L] | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\diamond_{[x_1\leftarrow t_1]}\phi} | |
| \\ \jghol{\Delta}{\Sigma}{\Gamma,x_1:C_1}{\Psi,\phi} | |
| {\diamond_{[y_1\leftarrow t'_1, y_2\leftarrow t'_2]}\psi}} | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi} | |
| {\diamond_{[y_1\leftarrow \mlet{x_1}{t_1}{t'_1}, y_2\leftarrow t'_2]} \psi}} | |
| \end{gather*} | |
| \shrinkcaption | |
| \caption{Derived rules for probabilistic constructs}\label{fig:prob-hol} | |
| \end{figure*} | |
| \section{Relational proof system} | |
| \label{sec:grhol} | |
| We complete the formal description of the system by describing the | |
| proof rules for the non-probabilistic fragment of the relational proof | |
| system (the rules of the probabilistic fragment were described in | |
| Section~\ref{sec:overview:grhol}). | |
| \subsection{Proof rules} | |
| The rules for core $\lambda$-calculus constructs are | |
| identical to those of~\cite{ABGGS17}; for convenience, we present a selection | |
| of the main rules in Figure \ref{fig:sel-rhol} in the appendix. | |
| We briefly comment on the two-sided rules for the new constructs | |
| (Figure~\ref{fig:s-rhol}). | |
| The notation $\Omega$ abbreviates a context $\Delta\mid\Sigma\mid\Gamma\mid\Psi$. | |
| The rule \rname{Next} relates two terms that have a $\later$ term | |
| constructor at the top level. We require that both have one term in | |
| the delayed substitutions and that they are related pairwise. Then | |
| this relation is used to prove another relation between the main | |
| terms. This rule can be generalized to terms with more than one term | |
| in the delayed substitution. | |
| The rule \rname{Prev} proves a relation between terms from the same delayed relation by | |
| applying $\mathrm{prev}$ to both terms. | |
| The rule \rname{Box} proves a relation between two boxed terms if the | |
| same relation can be proven in a constant context. | |
| Dually, \rname{LetBox} uses a relation between two boxed terms to prove | |
| a relation between their unboxings. | |
| \rname{LetConst} is similar to \rname{LetBox}, but it requires instead a relation | |
| between two constant terms, rather than explicitly $\square$-ed terms. | |
| The rule \rname{Fix} relates two fixpoints following the \rname{Loeb} rule from Guarded HOL. | |
| Notice that in the premise, the fixpoints need to appear in the delayed substitution so that | |
| the inductive hypothesis is well-formed. | |
| The rule \rname{Cons} proves relations on streams | |
| from relations between their heads and tails, while | |
| \rname{Head} and \rname{Tail} behave as converses of \rname{Cons}. | |
| Figure~\ref{fig:a-rhol} contains the one-sided versions of the rules. | |
| We only present the left-sided versions as the right-sided versions | |
| are completely symmetric. The rule \rname{Next-L} relates at $\phi$ a | |
| term that has a $\later$ with a term that does not have a | |
| $\later$. First, a unary property $\phi'$ is proven on the term $u$ in | |
| the delayed substitution, and it is then used as a premise to prove | |
| $\phi$ on the terms with delays removed. Rules for proving unary | |
| judgements can be found in the appendix. | |
| Similarly, \rname{LetBox-L} proves a unary property on the term that gets unboxed | |
| and then uses it as a precondition. | |
| The rule \rname{Fix-L} builds a fixpoint just on the left, and relates it with | |
| an arbitrary term $t_2$ at a property $\phi$. Since $\phi$ may contain the | |
| variable $\res\rtag$ which is not in the context, it has to be replaced when | |
| adding $\later\phi$ to the logical context in the premise of the rule. | |
| The remaining rules are similar to their two-sided counterparts. | |
| \begin{figure*}[!htb] | |
| \small | |
| \begin{gather*} | |
| \mbox{}\hspace{-5mm}\inferrule*[right=\sf Next] | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma, x_1:A_1, x_2:A_2}{\Psi, \phi'\defsubst{x_1}{x_2}}{t_1}{A_1}{t_2}{A_2}{\phi} \\ | |
| \jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{u_1}{\later{A_1}}{u_2}{\later{A_2}} | |
| {\dsubst{\res\ltag, \res\rtag}{\res\ltag, \res\rtag}{\phi'}}} | |
| {\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi} | |
| {\nextt{x_1\!\leftarrow\!u_1}{t_1}}{\later{A_1}} | |
| {\nextt{x_2\!\leftarrow\!u_2}{t_2}}{\later{A_2}} | |
| {\triangleright[x_1\!\leftarrow\!u_1,x_2\!\leftarrow\!u_2, | |
| \res\ltag\!\leftarrow\!\res\ltag,\res\rtag\!\leftarrow\!\res\rtag].\phi}} | |
| \\[0.3em] | |
| \inferrule*[right=\sf Prev] | |
| {\jgrhol{\Delta}{\Sigma}{\cdot}{\cdot}{t_1}{\later A_1}{t_2}{\later A_2}{\dsubst{\res\ltag,\res\rtag}{\res\ltag,\res\rtag}{\phi}}} | |
| {\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{\prev{t_1}}{A_1}{\prev{t_2}}{A_2}{\phi}} | |
| \\[0.3em] | |
| \mbox{}\hspace{-2mm}\inferrule*[Right=\sf Box] | |
| {\jgrhol{\Delta}{\Sigma}{\cdot}{\cdot}{t_1}{A_1}{t_2}{A_2}{\phi}} | |
| {\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{\boxx{t_1}}{\square A_1}{\boxx{t_2}}{\square A_2} | |
| {\square \phi\defsubst{\letbox{x_1}{\res\ltag}{x_1}}{\letbox{x_2}{\res\rtag}{x_2}}}} | |
| \\[0.3em] | |
| \inferrule*[right=\sf LetBox] | |
| {\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{u_1}{\square B_1}{u_2}{\square B_2}{\square \phi\defsubst{\letbox{x_1}{\res\ltag}{x_1}}{\letbox{x_2}{\res\rtag}{x_2}}} \\ | |
| \jgrhol{\Delta, x_1 : B_1, x_2 : B_2}{\Sigma, \phi\defsubst{x_1}{x_2}}{\Gamma}{\Psi}{t_1}{A_1}{t_2}{A_2}{\phi'}} | |
| {\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{\letbox{x_1}{u_1}{t_1}}{A_1}{\letbox{x_2}{u_2}{t_2}}{A_2}{\phi'}} | |
| \\[0.3em] | |
| \inferrule*[right=\sf LetConst] | |
| {B_1,B_2,\phi\ \text{constant} \\ FV(\phi)\cap FV(\Gamma) = \emptyset \\ | |
| \jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{u_1}{B_1}{u_2}{B_2}{\phi} \\ | |
| \jgrhol{\Delta, x_1 : B_1, x_2 : B_2}{\Sigma, \phi\defsubst{x_1}{x_2}}{\Gamma}{\Psi}{t_1}{A_1}{t_2}{A_2}{\phi'}} | |
| {\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{\letconst{x_1}{u_1}{t_1}}{A_1}{\letconst{x_2}{u_2}{t_2}}{A_2}{\phi'}} | |
| \\[0.3em] | |
| \inferrule*[right=\sf Fix] | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma, f_1:\later{A_1}, f_2:\later{A_2}}{\Psi, \dsubst{\res\ltag,\res\rtag}{f_1,f_2}{\phi}} | |
| {t_1}{A_1}{t_2}{A_2}{\phi}} | |
| {\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{\fix{f_1}{t_1}}{A_1}{\fix{f_2}{t_2}}{A_2}{\phi}} | |
| \\[0.3em] | |
| \mbox{}\hspace{-5.2mm}\inferrule*[Right=\sf Cons] | |
| {\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{x_1}{A_1}{x_2}{A_2}{\phi_h} \\ | |
| \jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{xs_1}{\later{\Str{A_1}}}{xs_2}{\later{\Str{A_2}}}{\phi_t} \\ | |
| \jgholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{\forall x_1,x_2, s_1, s_2. \phi_h\defsubst{x_1}{x_2} \Rightarrow | |
| \phi_t\defsubst{s_1}{s_2} \Rightarrow \phi\defsubst{\cons{x_1}{s_1}}{\cons{x_2}{s_2}}}} | |
| {\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{\cons{x_1}{s_1}}{\Str{A_1}}{\cons{x_2}{s_2}}{\Str{A_2}}{\phi}} | |
| \\[0.3em] | |
| \inferrule*[right=\sf Head] | |
| {\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{\Str{A_1}}{t_1}{\Str{A_1}}{\phi\defsubst{hd\ \res\ltag}{hd\ \res\rtag}}} | |
| {\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{hd\ t_1}{A_1}{hd\ t_2}{A_2}{\phi}} | |
| \\[0.3em] | |
| \inferrule*[right=\sf Tail] | |
| {\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{\Str{A_1}}{t_2}{\Str{A_2}}{\phi\defsubst{tl\ \res\ltag}{tl\ \res\rtag}}} | |
| {\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{tl\ t_1}{\later{\Str{A_1}}}{tl\ t_2}{\later{\Str{A_2}}}{\phi}} | |
| \end{gather*} | |
| \shrinkcaption | |
| \caption{Two-sided rules for Guarded RHOL}\label{fig:s-rhol} | |
| \end{figure*} | |
| \begin{figure*}[!htb] | |
| \small | |
| \begin{gather*} | |
| \inferrule*[right=\sf Next{-}L] | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma, x_1:B_1}{\Psi, \phi'\subst{\res}{x_1}}{t_1}{A_1}{t_2}{A_2}{\phi} \\ | |
| \jguholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{u_1}{\later{B_1}}{\dsubst{\res}{\res}{\phi'}}} | |
| {\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi} | |
| {\later[x_1 \ot u_1]{t_1}}{\later{A_1}} | |
| {t_2}{A_2} | |
| {\later[x_1 \ot u_1,\res\ltag\leftarrow\res\ltag]{\phi}}} | |
| \\[0.3em] | |
| \inferrule*[right=\sf Prev{-}L ] | |
| {\jgrhol{\Delta}{\Sigma}{\cdot}{\cdot}{t_1}{\later A_1}{t_2}{A_2}{\dsubst{\res\ltag}{\res\ltag}{\phi}}} | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma_1; \Gamma_2}{\Psi_1 ; \Psi_2}{\prev{t_1}}{A_1}{t_2}{A_2}{\phi}} | |
| \\[0.3em] | |
| \inferrule*[right=\sf Box{-}L] | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma_2}{\Psi_2}{t_1}{A_1}{t_2}{A_2}{\phi} \\ FV(t_1)\not\subseteq FV(\Gamma_2)\\ | |
| FV(\Psi_2)\subseteq FV(\Gamma_2)} | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma_1; \Gamma_2}{\Psi_1; \Psi_2}{\boxx{t_1}}{\square A_1}{t_2}{A_2} | |
| {\square \phi\subst{\res\ltag}{\letbox{x_1}{\res\ltag}{x_1}}}} | |
| \\[0.3em] | |
| \inferrule*[right=\sf LetBox{-}L] | |
| {\jguholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{u_1}{\square B_1}{\square \phi\subst{\res}{\letbox{x_1}{\res\ltag}{x_1}}} \\ | |
| \jgrhol{\Delta, x_1 : B_1}{\Sigma, \phi\subst{\res}{x_1}}{\Gamma}{\Psi}{t_1}{A_1}{t_2}{A_2}{\phi'}} | |
| {\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{\letbox{x_1}{u_1}{t_1}}{A_1}{t_2}{A_2}{\phi'}} | |
| \\[0.3em] | |
| \inferrule*[right=\sf LetConst{-}L] | |
| {B_1,\phi\ \text{constant} \\ FV(\phi)\cap FV(\Gamma) = \emptyset \\ | |
| \jguholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{u_1}{B_1}{\phi} \\\\ | |
| \jgrhol{\Delta, x_1 : B_1}{\Sigma, \phi\subst{\res}{x_1}}{\Gamma}{\Psi}{t_1}{A_1}{t_2}{A_2}{\phi'}} | |
| {\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{\letconst{x_1}{u_1}{t_1}}{A_1}{t_2}{A_2}{\phi'}} | |
| \\[0.3em] | |
| \inferrule*[right=\sf Fix{-}L] | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma, f_1:\later{A_1}}{\Psi, \dsubst{\res\ltag}{f_1}{(\phi\subst{\res\rtag}{t_2})}} | |
| {t_1}{A_1}{t_2}{A_2}{\phi}} | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{\fix{f_1}{t_1}}{A_1}{t_2}{A_2}{\phi}} | |
| \\[0.3em] | |
| \inferrule*[right=\sf Cons{-}L] | |
| {\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{x_1}{A_1}{t_2}{A_2}{\phi_h} \\ | |
| \jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{xs_1}{\later{\Str{A_1}}}{t_2}{A_2}{\phi_t} \\ | |
| \jgholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{\forall x_1,x_2, xs_1. \phi_h\defsubst{x_1}{x_2} \Rightarrow | |
| \phi_t\defsubst{xs_1}{x_2} \Rightarrow \phi\defsubst{\cons{x_1}{xs_1}}{x_2}}} | |
| {\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{\cons{x_1}{xs_1}}{\Str{A_1}}{t_2}{A_2}{\phi}} | |
| \\[0.3em] | |
| \inferrule*[right=\sf Head{-}L] | |
| {\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{\Str{A_1}}{t_1}{A_2}{\phi\subst{\res\ltag}{hd\ \res\ltag}}} | |
| {\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{hd\ t_1}{A_1}{t_2}{A_2}{\phi}} | |
| \\[0.3em] | |
| \inferrule*[right=\sf Tail{-}L] | |
| {\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{\Str{A_1}}{t_2}{A_2}{\phi\subst{\res\ltag}{tl\ \res\ltag}}} | |
| {\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{tl\ t_1}{\later{\Str{A_1}}}{t_2}{A_2}{\phi}} | |
| \end{gather*} | |
| \shrinkcaption | |
| \caption{One-sided rules for Guarded RHOL}\label{fig:a-rhol} | |
| \end{figure*} | |
| \subsection{Metatheory} | |
| We review some of the most interesting metatheoretical properties of | |
| our relational proof system, highlighting the equivalence with Guarded HOL. | |
| \begin{theorem}[Equivalence with Guarded HOL] \label{thm:equiv-rhol-hol} | |
| For all contexts $\Delta,\Gamma$; types $\sigma_1,\sigma_2$; terms $t_1,t_2$; sets of assertions $\Sigma,\Psi,$; | |
| and assertions $\phi$: | |
| \[\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{\sigma_1}{t_2}{\sigma_2}{\phi} | |
| \quad\Longleftrightarrow\quad | |
| \jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\phi\subst{\res\ltag}{t_1}\subst{\res\rtag}{t_2}}\] | |
| \end{theorem} | |
| The forward implication follows by induction on the given derivation. | |
| The reverse implication is immediate from the rule which allows to | |
| fall back on Guarded HOL in relational proofs. (Rule \rname{SUB} in the appendix). | |
| The full proof is in the appendix. The consequence of this theorem is | |
| that the syntax-directed, relational proof system we have built | |
| on top of Guarded HOL does not lose expressiveness. | |
| The intended semantics of a judgement | |
| $\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{A_1}{t_2}{A_2}{\phi}$ is | |
| that, for every valuation $\delta \models \Delta$, $\gamma\models \Gamma$, | |
| if $\sem{\Sigma}(\delta)$ and $\sem{\Psi}(\delta,\gamma)$, then | |
| $$\sem{\phi}(\delta,\gamma[\res\ltag \ot \sem{t_1}(\delta,\gamma), | |
| \res\rtag\ot\sem{t_2}(\delta,\gamma)])$$ | |
| Since Guarded HOL is sound with respect to its semantics in | |
| the topos of trees, and our relational proof system is equivalent to | |
| Guarded HOL, we obtain that our relational proof system is also sound in | |
| the topos of trees. | |
| \begin{corollary}[Soundness and consistency]\label{cor:rhol:sound} | |
| If $\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{\sigma_2}{t_2}{\sigma_2}{\phi}$, then for every valuation | |
| $\delta \models \Delta$, $\gamma\models\Gamma$: | |
| \[\begin{array}{c} | |
| \sem{\Delta \vdash \square \Sigma}(\delta) \wedge \sem{\Delta \mid \Gamma \vdash \Psi}(\delta,\gamma) \Rightarrow | |
| \\ \sem{\Delta \mid \Gamma,\res\ltag:\sigma_1, \res\ltag:\sigma_2 \vdash \phi} | |
| (\delta, \gamma[\res\ltag \ot \sem{\Delta \mid \Gamma \vdash t_1}(\delta,\gamma)][\res\rtag \ot \sem{\Delta \mid \Gamma \vdash t_2}(\delta,\gamma)]) | |
| \end{array}\] | |
| In particular, there is no proof of | |
| $\jgrhol{\Delta}{\emptyset}{\Gamma}{\emptyset}{t_1}{\sigma_1}{t_2}{\sigma_2}{\bot}$. | |
| \end{corollary} | |
| \subsection{Shift couplings revisited} | |
| \label{sec:examples} | |
| We give further details on how to prove the example with shift couplings from Section~\ref{sec:proba-ex}. | |
| (Additional examples of relational reasoning on non-probabilistic streams can be | |
| found in the appendix.) | |
| Recall the step functions:$$\begin{array}{rcl} | |
| \operatorname{step} & \defeq & \lambda x.\mlet{z}{\mathcal{U}_{\{-1,1\}}}{\munit{z+x}} \\ | |
| \operatorname{lstep2} & \defeq & \lambda x.\mlet{z}{\mathcal{U}_{\{-1,1\}}}{\mlet{b}{\mathcal{U}_{\{0,1\}}}{\munit{x+2*z*b}}} | |
| \end{array}$$ | |
| We axiomatize the predicate $\All_{2,1}$, which relates the element at position $2i$ in one stream to the element at position $i$ in another stream, as follows. | |
| \[\begin{array}{l} | |
| \forall x_1 x_2 xs_1 xs_2 y_1. \phi\subst{x_1}{z_1}\subst{x_2}{z_2} \Rightarrow \\ | |
| \quad \later[ys_1\ot xs_1]{\later[zs_1 \ot ys_1, ys_2 \ot xs_2]{\All_{2,1}(zs_1, ys_2, z_1.z_2.\phi)}} \Rightarrow \\ | |
| \quad\quad \All_{2,1}(\cons{x_1}{\cons{y_1}{xs_1}},\cons{x_2}{xs_2}, z_1.z_2. \phi) | |
| \end{array} | |
| \] | |
| In fact, we can assume that, in general, we have a family of | |
| $\All_{m_1, m_2}$ predicates relating two streams at positions | |
| $m_1\cdot i$ and $m_2\cdot i$ for every $i$. | |
| We can now express the existence of a shift coupling by the statement: | |
| {\small | |
| \[p_1 = p_2 \vdash \markov(p_1, \operatorname{step}) \sim \markov(p_2, | |
| \operatorname{lstep2}) \mid | |
| \diamond_{\left[\substack{ y_1 \ot \res\ltag\\ y_2 \ot \res\rtag}\right]}\All_{2,1}(y_1,y_2,z_1.z_2.z_1=z_2) \]} | |
| For the proof, we need to introduce an asynchronous rule for Markov chains: | |
| {\small\[ | |
| \inferrule*[right=\sf Markov-2-1] | |
| {\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{C_1}{t_2}{C_2}{\phi} \\\\ | |
| \begin{array}{c} | |
| \jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{(\lambda x_1. \mlet{x'_1}{h_1\ x_1}{h_1\ x'_1})}{C_1\to\Distr(C_1)}{h_2}{C_2\to \Distr(C_2)} | |
| {\\ \forall x_1 x_2. \phi\subst{z_1}{x_1}\subst{z_2}{x_2} \Rightarrow \diamond_{[ z_1 \ot \res\ltag\ x_1, z_2 \ot \res\rtag\ x_2]}\phi} | |
| \end{array}} | |
| {\begin{array}{c}\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{\markov(t_1,h_1)}{\Distr(\Str{C_1})}{\markov(t_2,h_2)}{\Distr(\Str{C_2})} | |
| {\\ \diamond_{[y_1 \ot \res\ltag, y_2 \ot \res\rtag]}\All_{2,1}(y_1,y_2,z_1.z_2.\phi)} | |
| \end{array}} | |
| \]} | |
| This asynchronous rule for Markov chains shares the motivations of the | |
| rule for loops proposed in~\cite{BartheGHS17}. Note that one can define | |
| a rule \rname{Markov-m-n} for arbitrary $m$ and $n$ to prove a judgement | |
| of the form $\All_{m,n}$ on two Markov chains. | |
| We show the proof of the shift coupling. By equational reasoning, we get: | |
| \[ | |
| \begin{array}{rl} | |
| & \lambda x_1. \mlet{x'_1}{h_1\ x_1}{h_1\ x'_1}\; | |
| \equiv \; | |
| \lambda x_1. \mlet{z_1}{\unif{\{-1,1\}}} | |
| {h_1\ (z_1 + x_1)} \; \equiv \\ | |
| &\equiv \; \lambda x_1. \mlet{z_1}{\unif{\{-1,1\}}} | |
| {\mlet{z'_1}{\unif{\{-1,1\}}} | |
| {\munit{z'_1 + z_1 + x'_1}}} | |
| \end{array} | |
| \] | |
| and the only interesting premise of \rname{Markov-2-1} is: | |
| \[\left.\begin{array}{rl} | |
| \lambda x_1. &\mlet{z_1}{\unif{\{-1,1\}}} | |
| {\\ &\mlet{z'_1}{\unif{\{-1,1\}}} | |
| {\\ &\munit{z'_1 + z_1 + x'_1}}} | |
| \end{array} | |
| \sim | |
| \begin{array}{rl} | |
| \lambda x_2. &\mlet{z_2}{\unif{\{-1,1\}}} | |
| {\\ &\mlet{b_2}{\unif{\{1,0\}}} | |
| {\\ &\munit{x_2+2*b_2*z_2}}} | |
| \end{array} | |
| \;\right|\; | |
| \begin{array}{c} | |
| \forall x_1 x_2. x_1 = x_2 \Rightarrow \\ | |
| \res\ltag\ x_1 \stackrel{\diamond}{=} \res\rtag\ x_2 | |
| \end{array}\] | |
| Couplings between $z_1$ and $z_2$ and between $z_1'$ and $b_2$ can be found by simple computations. | |
| This completes the proof. | |
| \section{Related work} | |
| \label{sec:rw} | |
| Our probabilistic guarded $\lambda$-calculus and the associated logic | |
| Guarded HOL build on top of the guarded $\lambda$-calculus and its internal | |
| logic~\cite{CBGB16}. The guarded $\lambda$-calculus has been extended | |
| to guarded dependent type theory~\cite{BGCMB16}, which can be | |
| understood as a theory of guarded refinement types and as a foundation | |
| for proof assistants based on guarded type theory. These systems do | |
| not reason about probabilities, and do not support syntax-directed | |
| (relational) reasoning, both of which we support. | |
| Relational models for higher-order programming languages are often | |
| defined using logical relations. \cite{PlotkinA93} showed how to use | |
| second-order logic to define and reason about logical relations for | |
| the second-order lambda calculus. Recent work has extended this approach | |
| to logical relations for higher-order programming languages with | |
| computational effects such as nontermination, general references, and | |
| concurrency~\cite{DreyerAB11,caresl,ipm,Krogh-Jespersen17}. The | |
| logics used in \emph{loc. cit.} are related to our work in two ways: | |
| (1) the logics in \emph{loc. cit.} make use of the later modality for | |
| reasoning about recursion, and (2) the models of the logics in | |
| \emph{loc. cit.} can in fact be defined using guarded type theory. | |
| Our work is more closely related to Relational Higher Order | |
| Logic~\cite{ABGGS17}, which applies the idea of logic-enriched type | |
| theories~\cite{AczelG00,AczelG06} to a relational setting. There | |
| exist alternative approaches for reasoning about relational properties | |
| of higher-order programs; for instance,~\cite{GrimmMFHMPRSB17} have | |
| recently proposed to use monadic reification for reducing relational | |
| verification of $F^*$ to proof obligations in higher-order logic. | |
| A series of work develops reasoning methods for probabilistic | |
| higher-order programs for different variations of the lambda calculus. | |
| One line of work has focused on operationally-based techniques for | |
| reasoning about contextual equivalence of programs. The methods are | |
| based on probabilistic | |
| bisimulations~\cite{DBLP:conf/esop/CrubilleL14,DBLP:conf/popl/SangiorgiV16} | |
| or on logical relations~\cite{DBLP:conf/fossacs/BizjakB15}. Most of | |
| these approaches have been developed for languages with discrete | |
| distributions, but recently there has also been work on languages with | |
| continuous | |
| distributions~\cite{DBLP:conf/icfp/BorgstromLGS16,DBLP:conf/esop/CulpepperC17}. | |
| Another line of work has focused on denotational models, starting with | |
| the seminal work in~\cite{Jones:powerdomain-evaluations}. Recent work | |
| includes support for relational reasoning about equivalence of | |
| programs with continuous distributions for a total programming | |
| language~\cite{DBLP:conf/lics/StatonYWHK16}. Our approach is most | |
| closely related to prior work based on relational refinement types for | |
| higher-order probabilistic programs. These were initially considered | |
| by~\cite{BFGSSZ14} for a stateful fragment of $F^*$, and later | |
| by~\cite{BGGHRS15,BFGGGHS16} for a pure language. Both systems are | |
| specialized to building probabilistic couplings; however, the latter | |
| support approximate probabilistic couplings, which yield a natural | |
| interpretation of differential privacy~\cite{DR14}, both in its | |
| vanilla and approximate forms (i.e.\, $\epsilon$- and | |
| $(\epsilon,\delta)$-privacy). Technically, approximate couplings are | |
| modelled as a graded monad, where the index of the monad tracks the privacy budget ($\epsilon$ or $(\epsilon,\delta)$). | |
| Both systems are strictly syntax-directed, and cannot reason about | |
| computations that have different types or syntactic structures, while | |
| our system can. | |
| \section{Conclusion} | |
| \label{sec:conclusion} | |
| We have developed a probabilistic extension of the (simply typed) | |
| guarded $\lambda$-calculus, and proposed a syntax-directed proof | |
| system for relational verification. Moreover, we have verified a | |
| series of examples that are beyond the reach of prior work. Finally, | |
| we have proved the soundness of the proof system with respect to the | |
| topos of trees. | |
| There are several natural directions for future work. One first | |
| direction is to enhance the expressiveness of the underlying simply | |
| typed language. For instance, it would be interesting to introduce | |
| clock variables and some type dependency as in~\cite{BGCMB16}, and extend the proof system accordingly. | |
| This would allow us, for example, to type the function taking the $n$-th element of a \emph{guarded} stream, | |
| which cannot be done in the current system. | |
| Another exciting direction is to | |
| consider approximate couplings, as in~\cite{BGGHRS15,BFGGGHS16}, | |
| and to develop differential privacy for infinite streams---preliminary | |
| work in this direction, such as~\cite{KellarisPXP14}, considers very | |
| large lists, but not arbitrary streams. A final direction would be to | |
| extend our approach to continuous distributions to support | |
| other application domains. | |
| \subsubsection*{Acknowledgments.} | |
| We would like to thank the anonymous reviewers for their time and their helpful input. | |
| This research was supported in part by the ModuRes Sapere Aude Advanced Grant from The Danish Council for Independent Research for the Natural Sciences (FNU), | |
| by a research grant (12386, Guarded Homotopy Type Theory) from | |
| the VILLUM foundation, and by NSF under grant 1718220. | |
| \bibliography{refs} | |
| \appendix | |
| \newpage | |
| \section{Additional proof rules} | |
| \begin{figure*}[!htb] | |
| \small | |
| \begin{mathpar} | |
| \infer[\sf ABS] | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{\lambda x_1:\tau_1. t_1}{\tau_1 \to \sigma_1}{\lambda x_2:\tau_2. t_2}{\tau_2\to \sigma_2}{\forall x_1,x_2. \phi' \Rightarrow \phi\subst{\res\ltag}{\res\ltag\ x_1}\subst{\res\rtag}{\res\rtag\ x_2}}} | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma,x_1:\tau_1,x_2:\tau_2}{\Psi,\phi'}{t_1}{\sigma_1}{t_2}{\sigma_2}{\phi}} | |
| \and | |
| \infer[\sf APP]{\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1 u_1}{\sigma_1}{t_2 u_2}{\sigma_2}{\phi\subst{x_1}{u_1}\subst{x_2}{u_2}}} | |
| {\begin{array}{c} | |
| \jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{\tau_1\to \sigma_1}{t_2}{\tau_2\to \sigma_2}{ | |
| \forall x_1,x_2. \phi'\subst{\res\ltag}{x_1}\subst{\res\rtag}{x_2}\Rightarrow \phi\subst{\res\ltag}{\res\ltag\ x_1}\subst{\res\rtag}{\res\rtag\ x_2}}\\ | |
| \jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{u_1}{\tau_1}{u_2}{\tau_2}{ | |
| \phi'} | |
| \end{array}} | |
| \and | |
| \infer[\sf VAR]{\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{x_1}{\sigma_1}{x_2}{\sigma_2}{\phi}} | |
| {\jlc{\Delta\mid\Gamma}{x_1}{\sigma_1} & \jlc{\Delta\mid\Gamma}{x_2}{\sigma_2} & | |
| \jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\phi\subst{\res\ltag}{x_1}\subst{\res\rtag}{x_2}}} | |
| \and | |
| \infer[\sf SUB]{\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{\sigma_1}{t_2}{\sigma_2}{\phi}} | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{\sigma_1}{t_2}{\sigma_2}{\phi'} & | |
| \Delta \mid \Sigma \mid \Gamma \mid \Psi \vdash_{\sf GHOL} \phi'\defsubst{t_1}{t_2} \Rightarrow \phi\defsubst{t_1}{t_2}} | |
| \and | |
| \infer[\sf UHOL-L] | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{\sigma_1}{t_2}{\sigma_2}{\phi}} | |
| {\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{\sigma_1}{\phi\defsubst{\res}{t_2}} | |
| \\ \jlc{\Delta\mid\Gamma}{t_2}{\sigma_2}} | |
| \and | |
| \infer[\sf ABS{-}L] | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{\lambda x_1:\tau_1. t_1}{\tau_1 \to \sigma_1}{t_2}{\sigma_2}{\forall x_1. \phi' \Rightarrow \phi \subst{\res\ltag}{\res\ltag\ x_1}}} | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma,x_1:\tau_1}{\Psi, \phi'}{t_1}{\sigma_1}{t_2}{\sigma_2}{\phi}} | |
| \and | |
| \infer[\sf APP{-}L] | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1 u_1}{\sigma_1}{u_2}{\sigma_2}{\phi\subst{x_1}{u_1}}} | |
| {\begin{array}{c} | |
| \jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{\tau_1\to \sigma_1}{u_2}{\sigma_2}{\forall x_1. \phi'\subst{\res\ltag}{x_1} \Rightarrow \phi\subst{\res\ltag}{\res\ltag\ x_1}}\\ | |
| \jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{u_1}{\sigma_1}{\phi'} | |
| \end{array}} | |
| \and | |
| \infer[\sf VAR{-}L] | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{x_1}{\sigma_1}{t_2}{\sigma_2}{\phi}} | |
| {\phi\subst{\res\ltag}{x_1} \in \Psi & \res\rtag\not\in\ FV(\phi) & | |
| \jlc{\Delta\mid\Gamma}{t_2}{\sigma_2}} | |
| \and | |
| \inferrule*[Right=\sf Equiv] | |
| {\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{t'_1}{A_1}{t'_2}{A_2}{\Phi} \\ t_1 \equiv t_1' \\ t_2 \equiv t_2' \\ | |
| \Delta \mid \Gamma \vdash t_1 : A_1 \\ \Delta \mid \Gamma \vdash t_2 : A_2} | |
| {\jgrholsc{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{A_1}{t_2}{A_2}{\Phi}} | |
| \end{mathpar} | |
| \caption{Selected RHOL rules}\label{fig:sel-rhol} | |
| \end{figure*} | |
| \section{Denotational semantics} | |
| \subsection{Types and terms in context} | |
| The meaning of terms is given by the denotational model in the category $\trees$ of presheaves over $\omega$, the first infinite ordinal. | |
| This category $\trees$ is also known as the \emph{topos of trees}~\cite{Birkedal-et-al:topos-of-trees}. | |
| In previous work~\cite{CBGB16} it was shown how to model most of the constructions of the guarded lambda calculus and the associated logic, with the notable exception of the probabilistic features. | |
| Below we give an elementary and self-contained presentation of the semantics. | |
| Concretely, objects $X$ of $\trees$ are families of sets $X_i$ indexed over $\nat$ together with functions $r_n^X : X_{n+1} \to X_n$. | |
| These are called \emph{restriction functions}. | |
| We will write simply $r_n$ if $X$ is clear from the context. | |
| Moreover if $x \in X_i$ and $j \leq i$ we will write $x\restriction_j$ for the element $r_j(\cdots (r_{i-1}(x))\cdots) \in X_j$. | |
| Morphisms $X \to Y$ are families of functions $\alpha_n : X_n \to Y_n$ commuting with restriction functions in the sense of $r_n^Y \circ \alpha_{n+1} = \alpha_n \circ r_n^X$. | |
| One can see the restriction function $r_n : X_{n+1} \to X_n$ as mapping elements of $X_{n+1}$ to their approximations at time $n$. | |
| Semantics of types can be found on \autoref{fig:sem-types}, where | |
| $G\left(\sem{A}\right)$ consists of sequences $\{x_n\}_{n \in \nat}$ | |
| such that $x_i \in \sem{A}_i$ and $r_i(x_{i+1}) = x_i$ for all $i$, | |
| i.e., $\square\sem{A}$ is the set of so-called global sections of | |
| $\sem{A}$. | |
| The semantics of a dual context $\Delta \mid \Gamma$ is given as the product of types in $\Delta$ and $\Gamma$, except that we implicitly add $\square$ in front of every type in $\Delta$. | |
| In the particular case when both contexts are empty, the semantics of the dual context correspond to the terminal object $1$, which is the singleton set $\{\ast\}$ at each stage. A term in context $\Delta \mid \Gamma \vdash t : \tau$ is interpreted as a family of functions $\sem{t}_n : \sem{\Delta \mid \Gamma}_n \to \sem{\tau}_n$ commuting with restriction functions of $\sem{\Delta \mid \Gamma}$ and $\sem{\tau}$. | |
| Semantics of products, | |
| coproducts, and natural numbers is pointwise as in sets, so we | |
| omit writing it. The cases for the other constructs are in | |
| \autoref{fig:sem-glc} where $\mathsf{munit}$ and $\mathsf{mlet}$ | |
| are the standard unit and bind operations on discrete probabilities, | |
| i.e.\, | |
| \begin{align*} | |
| \begin{array}{rcl} | |
| \munit{c} & = & \lambda y.~\carac{c=y} \\ | |
| \mathsf{mlet}~x = \mu~\mathsf{in}~M & = & \lambda y.~\sum_{c\in C} \mu(c) \cdot M(c)(y) | |
| \end{array} | |
| \end{align*} | |
| The functions $\pi_0$ and $\pi_1$ are the first and second projections, respectively. | |
| \begin{figure*}[!htb] | |
| \begin{align*} | |
| \sem{b} &\defeq \text{ chosen object of } \trees\\ | |
| \sem{\nat} &\defeq \nat \xleftarrow{id} \nat \xleftarrow{id} \nat \xleftarrow{id} \cdots\\ | |
| \sem{A\times B} &\defeq \sem{A}_0\times\sem{B}_0 \xleftarrow{r_0 \times r_0} \sem{A}_1\times\sem{B}_1 \xleftarrow{r_1 \times r_1} \sem{A}_2\times\sem{B}_2 \xleftarrow{r_2 \times r_2} \cdots \\ | |
| \sem{A\to B} &\defeq \left(\sem{B}^{\sem{A}}\right)_0 \xleftarrow{\pi} \left(\sem{B}^{\sem{A}}\right)_1 \xleftarrow{\pi} \left(\sem{B}^{\sem{A}}\right)_2 \xleftarrow{\pi} \cdots \\ | |
| \sem{\Str{A}} &\defeq \sem{A}_0 \times \{*\} \xleftarrow{r_0 \times !} \sem{A}_1 \times \left(\sem{A}_0 \times \{*\}\right) | |
| \xleftarrow{r_1 \times r_0 \times !} \sem{A}_2\times\left(\sem{A}_1\times\left(\sem{A}_0 \times \{*\}\right)\right) \leftarrow \cdots \\ | |
| \sem{\later A} &\defeq \{*\} \xleftarrow{!} \sem{A}_0 \xleftarrow{r_0} \sem{A}_1 \xleftarrow{r_1} \cdots \\ | |
| \sem{\square A} &\defeq G(\sem{A}) \xleftarrow{id} G(\sem{A}) \xleftarrow{id} \cdots\\ | |
| \sem{\Distr(C)} &\defeq \Distr(\sem{C}_0) \stackrel{\Distr(r_0)}{\longleftarrow} \Distr(\sem{C}_1) \stackrel{\Distr(r_1)}{\longleftarrow} \Distr(\sem{C}_2) \stackrel{\Distr(r_2)}{\longleftarrow} \ldots | |
| \end{align*} | |
| \caption{Semantics of types in the topos of trees} | |
| \label{fig:sem-types} | |
| \end{figure*} | |
| \begin{figure*}[!htb] | |
| \begin{align*} | |
| &\sem{\Delta \mid \Gamma \vdash \lambda x:A. t : A \to B}_i(\delta,\gamma) | |
| \defeq (f_0,\ldots,f_i)\\ | |
| &\quad\quad\text{ where } f_i(x) = \sem{\Delta\mid \Gamma, x : A \vdash t : B}\left(\delta,(\gamma\restriction_i, x)\right)\\ | |
| &\sem{\Delta \mid \Gamma \vdash t_1\ t_2 : B}_i(\delta,\gamma) | |
| \defeq f_i\left(\sem{\Delta \mid\Gamma \vdash t_2 : A}(\delta,\gamma)\right)\\ | |
| &\quad\quad\text{ where } \sem{\Delta \mid \Gamma \vdash t_1 : A \to B}_i(\delta,\gamma) = (f_0,\ldots,f_i)\\ | |
| &\sem{\Delta \mid \Gamma \vdash \later[x_1\ot t_1, \dots, x_n\ot t_n]t : \later A}_0(\delta,\gamma) \defeq *\\ | |
| &\sem{\Delta \mid \Gamma \vdash \later[x_1\ot t_1, \dots, x_n\ot t_n]t : \later A}_{i+1}(\delta,\gamma) \defeq \\ | |
| &\quad\quad\sem{\Delta \mid \Gamma, \{x_k : A_k\}_{k=1}^n \vdash t : A}_i\left(\delta, (\gamma\restriction_i, \left\{\sem{\Delta \mid \Gamma \vdash t_k : \later A_k}_{i+1}\right\}_{k=1}^n(\delta,\gamma))\right)\\ | |
| &\sem{\Delta \mid \Gamma \vdash \prev t : A}_i(\delta,\gamma) \defeq \sem{\Delta \mid \cdot \vdash t : \later A}_{i+1}(\delta)\\ | |
| &\sem{\Delta \mid \Gamma \vdash \boxx t : \square A}_i(\delta,\gamma) \defeq \{\sem{\Delta \mid \cdot \vdash t: A}_j(\delta)\}_{j=0}^{\infty}\\ | |
| &\sem{\Delta \mid \Gamma \vdash \letbox{x}{u}{t} : A}_i(\delta,\gamma) \defeq \\ | |
| &\quad\quad\sem{\Delta, x : B \mid \Gamma \vdash t : A}_i\left(\left(\delta,\sem{\Delta \mid \Gamma \vdash u : \square B}_i(\delta,\gamma)\right),\gamma\right)\\ | |
| &\sem{\Delta \mid \Gamma \vdash \letconst{x}{u}{t} : A}_i(\delta,\gamma)\defeq\\ | |
| &\quad\quad\sem{\Delta, x : B \mid \Gamma \vdash t : A}_i\left(\left(\delta,\varepsilon^{-1}_i\left(\sem{\Delta \mid \Gamma \vdash u : \square B}_i(\delta,\gamma)\right)\right),\gamma\right)\\ | |
| &\sem{\Delta \mid \Gamma \vdash \hd t : A}_i(\delta,\gamma) \defeq \pi_0\left(\sem{\Delta \mid \Gamma \vdash t : \Str{A}}_i(\delta,\gamma)\right)\\ | |
| &\sem{\Delta \mid \Gamma \vdash \tl t : \later{\Str{A}}}_i(\delta,\gamma) \defeq \pi_1\left(\sem{\Delta \mid \Gamma \vdash t : \Str{A}}_i(\delta,\gamma)\right)\\ | |
| &\sem{\Delta \mid \Gamma \vdash \cons{t}{u} : \Str{A}}_i(\delta,\gamma) \defeq | |
| \left(\sem{\Delta \mid \Gamma \vdash t : A}_i(\delta,\gamma), \sem{\Delta \mid \Gamma \vdash u : \Str{A}}_i(\delta,\gamma)\right)\\ | |
| &\sem{\Delta \mid \Gamma \vdash \munit{t} : \Distr(C)}_i (\delta,\gamma) | |
| \defeq \munit{\sem{\Delta \mid \Gamma \vdash t:C}_i (\delta,\gamma)} \\ | |
| &\sem{\Delta \mid \Gamma \vdash \mlet{x}{t}{u} : \Distr(C)}_i (\delta,\gamma) | |
| \defeq | |
| \mathsf{mlet}~v= \sem{\Delta \mid \Gamma \vdash t : \Distr(D)}_i | |
| (\delta,\gamma)~\mathsf{in} \\ | |
| \mbox{}\hspace{0.5cm} | |
| &\quad\quad\sem{\Delta \mid \Gamma,x:D \vdash u : \Distr(C)}_i (\delta,\gamma[x:=v]) | |
| \end{align*} | |
| \caption{Semantics for the Guarded $\lambda$-calculus} | |
| \label{fig:sem-glc} | |
| \end{figure*} | |
| \subsection{Equational theory of the calculus} | |
| The denotational semantics validates the following equational theory in addition to the standard equational theory of the simply typed lambda calculus with sums and natural numbers.\\ | |
| \textbf{Rules for fixed points, always modality and streams} | |
| \begin{align*} | |
| \begin{split} | |
| \fix{f}{t} \quad &\equiv \quad t\subst{f}{\later(\fix{f}{t})} \\ | |
| \prev{(\later t)} \quad &\equiv \quad t \\ | |
| \letbox{x}{(\boxx{u})}{t} \quad &\equiv \quad t\subst{x}{u} \\ | |
| \letconst{x}{u}{t} \quad&\equiv\quad t\subst{x}{u} \\ | |
| \end{split} | |
| \begin{split} | |
| \hd\ (\cons{x}{xs}) \quad&\equiv\quad x \\ | |
| \tl\ (\cons{x}{xs}) \quad&\equiv\quad xs \\ | |
| \cons{\hd t}{\tl t} \quad&\equiv\quad t | |
| \end{split} | |
| \end{align*} | |
| \textbf{Rules for delayed substitutions} | |
| \begin{align*} | |
| \latern[\xi\hrt{x\gets t}]{u} &\equiv \latern[\xi]{u} & & \text{ if } x \text{ not in } u\\ | |
| \latern[\xi\hrt{x\gets t, y \gets s}\xi']{u} &\equiv\latern[\xi\hrt{y\gets s, x \gets t}\xi']{u}\\ | |
| \latern[\xi\hrt{x\gets \latern[\xi]{t}}]{u} &\equiv \latern[\xi]{\left(u\subst{x}{t}\right)}\\ | |
| \latern[\hrt{x\gets t}]x &\equiv t | |
| \end{align*} | |
| \textbf{Monad laws for distributions} | |
| \begin{align*} | |
| \mlet{x}{\munit{t}}{u} &\equiv u[t/x]\\ | |
| \mlet{x}{t}{\munit{x}} &\equiv t\\ | |
| \mlet{x_2}{(\mlet{x_1}{t_1}{t_2})}{u} &\equiv \mlet{x_1}{t_1}{(\mlet{x_2}{t_2}{u})} | |
| \end{align*} | |
| In particular, notice that fix does not reduce as usual, but instead the whole term is delayed before the substitution is performed. | |
| \subsection{Logical judgements} | |
| The cases for the semantics of the judgement $\Delta \mid \Gamma \vdash \phi$ of the non-probabilistic fragment are as follows (we omit writing the contexts if they are clear): | |
| \begin{align*} | |
| \sem{\top}_i &\defeq \sem{\Delta \mid \Gamma}_i\\ | |
| \sem{\phi \wedge \psi}_i &\defeq \sem{\phi}_i \cap \sem{\psi}_i\\ | |
| \sem{\phi \vee \psi}_i &\defeq \sem{\phi}_i \cup \sem{\psi}_i\\ | |
| \sem{\phi \To \psi}_i &\defeq | |
| \left\{ x \isetsep \forall j \leq i, x\restriction_j \in \sem{\phi}_j \To x\restriction_j \in \sem{\psi}_j\right\}\\ | |
| \sem{\forall x:A. \phi}_i &\defeq \left\{(\delta,\gamma) \isetsep \forall j \leq i, \forall x \in \sem{A}_j, \left(\delta, \left(\gamma\restriction_j\right), x\right) \in \sem{\phi}\right\}\\ | |
| \sem{\later[x_1 \ot t_1, \dots, x_n \ot t_n]\phi}_i &\defeq | |
| \left\{ (\delta, \gamma) \isetsep i > 0 \To \left(\delta,\gamma\restriction_{i-1}, \left\{\sem{t_k}_{i}(\delta,\gamma)\right\}_{k=1}^n\right) \in \sem{\phi}_{i-1}\right\}\\ | |
| \sem{\square \phi}_i &\defeq \left\{ x \isetsep \forall j, x \in \sem{\phi}_j\right\} | |
| \end{align*} | |
| \section{Additional background} | |
| One consequence of Strassen's theorem is | |
| that couplings are closed under convex combinations. | |
| \begin{lemma}[Convex combinations of couplings]\label{lem:convex-combination-of-couplings} | |
| Let $(\mu_i)_{i\in I}$ and $(\nu_i)_{i\in I}$ bet two families of | |
| distributions on $C_1$ and $C_2$ respectively, and let $(p_i)_{i\in I} | |
| \in [0,1]$ such that $\sum_{i\in I} p_i=1$. If | |
| $\coupl{\mu_i}{\nu_i}{R}$ for all $i\in I$ then $\coupl{(\sum_{i\in I} p_i | |
| \mu_i)}{(\sum_{i\in I} p_i \nu_i)}{R}$, where the convex | |
| combination $\sum_{i\in I} p_i \mu_i$ is defined by the clause | |
| $(\sum_{i\in I} p_i \mu_i)(x)=\sum_{i\in I} p_i \mu_i(x)$. | |
| \end{lemma} | |
| One obtains an asymmetric version of the lemma by observing that if | |
| $\mu_i=\mu$ for every $i\in I$, then $(\sum_{i\in I} p_i \mu_i)=\mu$. | |
| One can also show that couplings are closed under relation | |
| composition. | |
| \begin{lemma}[Couplings for relation composition] | |
| Let $\mu_1\in\Distr(C_1)$, $\mu_2\in\Distr(C_2)$, $\mu_3 | |
| \in\Distr(C_3)$. Moreover, let $R\subseteq C_1\times C_2$ and | |
| $S\subseteq C_2\times C_3$. If $\coupl{\mu_1}{\mu_2}{R}$ and | |
| $\coupl{\mu_2}{\mu_3}{R}$ then $\coupl{\mu_1}{\mu_3}{R}$. | |
| \end{lemma} | |
| \section{Proofs of the theorems} | |
| \subsection{Proof of Theorem~\ref{thm:sound-ghol}} | |
| \label{sec:soundness-of-ghol} | |
| The semantics of the guarded higher-order logic without the probabilistic fragment has been explained in previous work~\cite{Birkedal-et-al:topos-of-trees,CBGB16}. | |
| Thus we focus on showing soundness of the additional rules for the diamond modality, which will be useful for proving soundness of the relational proof system. | |
| Moreover we only describe soundness for the binary diamond modality, the soundness of the rules for the unary modality being entirely analogous. | |
| \noindent\textbf{Soundness of the rule \textsf{MONO2}} | |
| \begin{mathpar} | |
| \infer[\sf MONO2] | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\diamond_{[x_1\gets t_1, x_2\gets t_2]}\psi}} | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\diamond_{[x_1\gets t_1, x_2 \gets t_2]}\phi} \and | |
| \jghol{\Delta}{\Sigma}{\Gamma, x_1 : C_1, x_2 : C_2}{\Psi, \phi}{\psi} | |
| } | |
| \end{mathpar} | |
| Let $n \in \nat$ and $(\delta, \gamma) \in \sem{\Delta \mid \Sigma \mid \Gamma \mid \Psi}_n$. | |
| Then from the first premise we have $(\delta, \gamma) \in \sem{\diamond_{[x_1\gets t_1, x_2 \gets t_2]}\phi}_n$ and thus there exists an $\left\{(v, u) \isetsep (\delta, \gamma, v, u) \in \sem{\phi}_n\right\}$ coupling for the distributions $\sem{t_1}_n(\delta,\gamma)$ and $\sem{t_2}_n(\delta,\gamma)$. | |
| But since $(\delta, \gamma) \in \sem{\Delta \mid \Sigma \mid \Gamma \mid \Psi}_n$ we have from the second premise of the rule that | |
| \begin{align*} | |
| \left\{(v, u) \isetsep (\delta, \gamma, v, u) \in \sem{\phi}_n\right\} \subseteq | |
| \left\{(v, u) \isetsep (\delta, \gamma, v, u) \in \sem{\psi}_n\right\} | |
| \end{align*} | |
| and thus any $\left\{(v, u) \isetsep (\delta, \gamma, v, u) \in \sem{\phi}_n\right\}$ coupling is also an $\left\{(v, u) \isetsep (\delta, \gamma, v, u) \in \sem{\psi}_n\right\}$ coupling, which means there exists an $\left\{(v, u) \isetsep (\delta, \gamma, v, u) \in \sem{\psi}_n\right\}$ coupling for $\sem{t_1}_n(\delta,\gamma)$ and $\sem{t_2}_n(\delta,\gamma)$ as required. | |
| \noindent\textbf{Soundness of the rule \textsf{UNIT2}} | |
| \begin{mathpar} | |
| \infer[\sf UNIT2] | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{ \diamond_{[x_1\leftarrow \munit{t_1}, x_2\leftarrow \munit{t_2}]} \phi}} | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{ \phi[t_1/x_1][t_2/x_2]}} | |
| \end{mathpar} | |
| Let $n \in \nat$ and $(\delta, \gamma) \in \sem{\Delta \mid \Sigma \mid \Gamma \mid \Psi}_n$. | |
| We need to show the existence of a | |
| \begin{align*} | |
| \left\{(v, u) \isetsep (\delta, \gamma, v, u) \in \sem{\phi}_n\right\} | |
| \end{align*} | |
| coupling for the point-mass distributions concentrated at $\sem{t_1}_n(\delta,\gamma)$ and $\sem{t_2}_n(\delta,\gamma)$. | |
| The premise of the rule establishes the membership | |
| \begin{align*} | |
| \left(\sem{t_1}_n(\delta,\gamma), \sem{t_2}_n(\delta,\gamma)\right) | |
| \in | |
| \left\{(v, u) \isetsep (\delta, \gamma, v, u) \in \sem{\phi}_n\right\} | |
| \end{align*} | |
| and thus the point-mass distribution concentrated at $\left(\sem{t_1}_n(\delta,\gamma), \sem{t_2}_n(\delta,\gamma)\right)$ is a required coupling. | |
| \noindent\textbf{Soundess of the rule \textsf{MLET2}} | |
| \begin{mathpar} | |
| \infer[\sf MLET2] | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{ | |
| \diamond_{[y_1\leftarrow \mlet{x_1}{t_1}{t'_1}, y_2\leftarrow | |
| \mlet{x_2}{t_2}{t'_2}]} \psi}}{ | |
| \jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\diamond_{[x_1\leftarrow t_1, x_2\leftarrow t_2]} \phi} | |
| \and | |
| \jghol{\Delta}{\Sigma}{\Gamma,x_1:C_1,x_2:C_2}{\Psi,\phi}{ | |
| \diamond_{[y_1\leftarrow t'_1, y_2\leftarrow t'_2]}\psi}} | |
| \end{mathpar} | |
| Let $n \in \nat$ and $(\delta, \gamma) \in \sem{\Delta \mid \Sigma \mid \Gamma \mid \Psi}_n$. | |
| Then from the first premise we have that there exists an | |
| \begin{align*} | |
| \left\{(v, u) \isetsep (\delta, \gamma, v, u) \in \sem{\phi}_n\right\} | |
| \end{align*} | |
| coupling for the distributions $\sem{t_1}_n(\delta,\gamma)$ and $\sem{t_2}_n(\delta,\gamma)$. | |
| From the second premise we get that for every $v, u$ such that $(\delta, \gamma, v, u) \in \sem{\phi}_n$ | |
| there exists an | |
| \begin{align*} | |
| \left\{(v', u') \isetsep (\delta, \gamma, v, u, v', u') \in \sem{\psi}_n\right\} | |
| \end{align*} | |
| coupling for $\sem{t_1'}_n(\delta,\gamma, v)$ and $\sem{t_2'}_n(\delta,\gamma, u)$. | |
| Since $x_1$ and $x_2$ are fresh for $\psi$ the relation | |
| \begin{align*} | |
| \left\{(v', u') \isetsep (\delta, \gamma, v, u, v', u') \in \sem{\psi}_n\right\} | |
| \end{align*} | |
| is independent of $v, u$. | |
| Thus Lemma~\ref{lem:sequential-composition-of-couplings} instantiated with | |
| \begin{align*} | |
| \mu_1 &= \sem{t_1}_n(\delta,\gamma)\\ | |
| \mu_2 &= \sem{t_2}_n(\delta,\gamma)\\ | |
| M_1 &= \sem{t_1'}_n(\delta,\gamma, -)\\ | |
| M_2 &= \sem{t_2'}_n(\delta,\gamma, -)\\ | |
| R &= \left\{(v, u) \isetsep (\delta, \gamma, v, u) \in \sem{\phi}_n\right\}\\ | |
| S &= \left\{(v', u') \isetsep (\delta, \gamma, v, u, v', u') \in \sem{\psi}_n\right\} | |
| \end{align*} | |
| concludes the proof. | |
| \noindent\textbf{Soundness of the rule \textsf{MLET-L}} | |
| \begin{mathpar} | |
| \infer[\sf MLET-L] | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{ | |
| \diamond_{[y_1\leftarrow \mlet{x_1}{t_1}{t'_1}, y_2\leftarrow t'_2]} \psi}}{ | |
| \jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\diamond_{[x_1\leftarrow t_1]}\phi} | |
| \and | |
| \jghol{\Delta}{\Sigma}{\Gamma,x_1:C_1}{\Psi,\phi}{ | |
| \diamond_{[y_1\leftarrow t'_1, y_2\leftarrow t'_2]}\psi}} | |
| \end{mathpar} | |
| Let $n \in \nat$ and $(\delta, \gamma) \in \sem{\Delta \mid \Sigma \mid \Gamma \mid \Psi}_n$. | |
| Then from the first premise we have that the support of the distribution $\sem{t_1}_n(\delta,\gamma)$ is included in | |
| \begin{align*} | |
| \left\{v \isetsep (\delta, \gamma, v) \in \sem{\phi}_n\right\}. | |
| \end{align*} | |
| From the second premise we get that for every $v$ such that $(\delta, \gamma, v) \in \sem{\phi}_n$ there exists an | |
| \begin{align*} | |
| \left\{(v', u') \isetsep (\delta, \gamma, v, v', u') \in \sem{\psi}_n\right\} | |
| \end{align*} | |
| coupling for $\sem{t_1'}_n(\delta,\gamma, v)$ and $\sem{t_2'}_n(\delta,\gamma)$. | |
| Since $x_1$ is fresh for $\psi$ the relation | |
| \begin{align*} | |
| R \defeq \left\{(v', u') \isetsep (\delta, \gamma, v, v', u') \in \sem{\psi}_n\right\} | |
| \end{align*} | |
| is independent of $v$. | |
| Let $\mathcal{I} = \left\{v \isetsep (\delta, \gamma, v) \in \sem{\phi}_n\right\}$ and for any $v \in \mathcal{I}$ let $p_v = \sem{t_1}_n(\delta,\gamma)(v)$, $\mu_v = \sem{t_1'}_n(\delta,\gamma,v)$, and $\nu_v = \sem{t_2'}_n(\delta,\gamma)$. | |
| Then we have $\sum_{v \in \mathcal{I}} p_v = 1$ from the first premise of the rule and $\coupling{R}{\mu_v}{\nu_v}$ for all $v \in \mathcal{I}$ from the second premise. | |
| Lemma~\ref{lem:convex-combination-of-couplings} concludes the proof. | |
| \subsection{Proof of Theorem \ref{thm:equiv-rhol-hol}} | |
| The inverse implication follows immediately from the \rname{SUB} rule and | |
| the fact that we can always prove a judgement of the shape | |
| \[ \jgrhol{\Gamma}{\Sigma}{\Gamma}{\Psi}{t_1}{A_1}{t_2}{A_2}{\top} \] | |
| for well-typed $t_1$ and $t_2$. | |
| We will prove the direct implication by induction on the | |
| derivation. We will just prove the two-sided rules. The proofs for the one | |
| sided rule are similar. | |
| \\ \\ | |
| \noindent {\bf Case.} $\small\inferrule*[Right=\sf Next] | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma, x_1:A_1, x_2:A_2}{\Psi, \Phi'\defsubst{x_1}{x_2}}{t_1}{A_1}{t_2}{A_2}{\Phi} \\ | |
| \jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{u_1}{\later{A_1}}{u_2}{\later{A_2}}{\dsubst{\res\ltag, \res\rtag}{\res\ltag, \res\rtag}{\Phi'}}} | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi} | |
| {\nextt{x_1 \leftarrow u_1}{t_1}}{\later{A_1}} | |
| {\nextt{x_2 \leftarrow u_2}{t_2}}{\later{A_2}} | |
| {\triangleright[x_1\leftarrow u_1,x_2\leftarrow x_2,\res\ltag\leftarrow\res\ltag,\res\rtag\leftarrow\res\rtag]\Phi}}$ | |
| \\ \\ | |
| \noindent | |
| By I.H. $\jghol{\Delta}{\Sigma}{\Gamma, x_1:A_1, x_2:A_n}{\Psi, \Phi'\defsubst{x_1}{x_2}}{\Phi\defsubst{t_1}{t_2}}$, \hfill{}(H1) | |
| \\ | |
| and $\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\later[\res\ltag \ot u_1, \res\rtag \ot u_2]{\Phi'}}$ \hfill{}(H2) | |
| \\ | |
| To show: $\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\later[x_1 \ot u_1, x_2 \ot u_2, \res\ltag \ot \nextt{x_1 \leftarrow u_1}{t_1} , \res\rtag\ot \nextt{x_2 \leftarrow u_2}{t_2}]{\Phi}}$. \hfill{}(G) | |
| \\ | |
| By \rname{CONV} we can change the goal (G) into | |
| \\ | |
| $\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\later[x_1 \ot u_1, x_2 \ot u_2]{\Phi\defsubst{t_1}{t_2}}}$ \hfill{}(G') | |
| \\ | |
| and (H2) into: | |
| \\ | |
| $\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\later[x_1 \ot u_1, x_2 \ot u_2]{\Phi'\defsubst{x_1}{x_2}}}$ \hfill{}(H2') | |
| \\ | |
| Finally, by applying \rname{$\later_{App}$} to (H1) and (H2) we get (G') | |
| \\ | |
| \noindent {\bf Case.} $\small \inferrule*[Right=\sf Prev] | |
| {\jgrhol{\Delta}{\Sigma}{\cdot}{\cdot}{t_1}{\later A_1}{t_2}{\later A_2}{\dsubst{\res\ltag,\res\rtag}{\res\ltag,\res\rtag}{\Phi}}} | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{\prev{t_1}}{A_1}{\prev{t_2}}{A_2}{\Phi}}$ | |
| \\ \\ | |
| \noindent | |
| We just apply \rname{$\later_E$}. | |
| \\ | |
| \noindent {\bf Case.} $\small\inferrule*[Right=\sf Box] | |
| {\jgrhol{\Delta}{\Sigma}{\cdot}{\cdot}{t_1}{A_1}{t_2}{A_2}{\Phi}} | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{\boxx{t_1}}{\square A_1}{\boxx{t_2}}{\square A_2} | |
| {\square \Phi\defsubst{\letbox{x_1}{\res\ltag}{x_1}}{\letbox{x_2}{\res\rtag}{x_2}}}}$ | |
| \\ \\ | |
| \noindent | |
| By I.H. $\jghol{\Delta}{\Sigma}{\cdot}{\cdot}{\Phi\defsubst{t_1}{t_2}}$ | |
| \\ | |
| To show: $\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\square \Phi\defsubst{\letbox{x_1}{\boxx{t_1}}{x_1}}{\letbox{x_2}{\boxx{t_2}}{x_2}}}$ | |
| \\ | |
| By \rname{CONV} we can change the goal into: | |
| \\ | |
| $\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\square \Phi\defsubst{t_1}{t_2}}$ | |
| \\ | |
| And then we can prove it by \rname{$\square_I$}. | |
| \\ | |
| \noindent {\bf Case.}$\small\inferrule*[Right=\sf LetBox] | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{u_1}{\square B_1}{u_2}{\square B_2}{\square \Phi\defsubst{\letbox{x_1}{\res\ltag}{x_1}}{\letbox{x_2}{\res\rtag}{x_2}}} \\ | |
| \jgrhol{\Delta, x_1 : B_1, x_2 : B_2}{\Sigma, \Phi\defsubst{x_1}{x_2}}{\Gamma}{\Psi}{t_1}{A_1}{t_2}{A_2}{\Phi'}} | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{\letbox{x_1}{u_1}{t_1}}{A_1}{\letbox{x_2}{u_2}{t_2}}{A_2}{\Phi'}}$ | |
| \\ \\ | |
| \noindent | |
| By I.H. $\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\square \Phi\defsubst{\letbox{x_1}{u_1}{x_1}}{\letbox{x_2}{u_2}{x_2}}}$ \hfill{}(H1) | |
| \\ | |
| and $\jghol{\Delta, x_1 : B_1, x_2 : B_2}{\Sigma, \Phi\defsubst{x_1}{x_2}}{\Gamma}{\Psi}{\Phi'\defsubst{t_1}{t_2}}$. \hfill{}(H2) | |
| \\ | |
| To show: $\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\Phi'\defsubst{\letbox{x_1}{u_1}{t_1}}{\letbox{x_2}{u_2}{t_2}}}$ \hfill{}(G) | |
| \\ | |
| We instantiate (H2) into: | |
| \[\begin{array}{c} | |
| \jghol{\Delta}{\Sigma, \Phi\defsubst{\letbox{x_1}{u_1}{x_1}}{\letbox{x_2}{u_2}{x_2}}}{\Gamma}{\Psi} | |
| {\\ \Phi'\defsubst{t_1\subst{x_1}{\letbox{x_1}{u_1}{x_1}}}{t_2\subst{x_2}{\letbox{x_2}{u_2}{x_2}}}} | |
| \end{array}\] | |
| And by the equality $t\subst{x}{\letbox{x}{u}{x}} \equiv \letbox{x}{u}{t}$, we get: | |
| \[\begin{array}{c} | |
| \jghol{\Delta}{\Sigma, \Phi\defsubst{\letbox{x_1}{u_1}{x_1}}{\letbox{x_2}{u_2}{x_2}}}{\Gamma}{\Psi} | |
| {\\ \Phi'\defsubst{\letbox{x_1}{u_1}{t_1}}{\letbox{x_2}{u_2}{t_2}}} \end{array} \] | |
| and then, by applying \rname{$\square_E$} to (H1) and the previous judgement we get (G). | |
| \\ | |
| \noindent {\bf Case.} $\small \inferrule*[Right=\sf LetConst] | |
| {B_1,B_2,\Phi\ \text{constant} \\ FV(\Phi)\cap FV(\Gamma) = \emptyset \\ | |
| \jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{u_1}{B_1}{u_2}{B_2}{\Phi} \\ | |
| \jgrhol{\Delta, x_1 : B_1, x_2 : B_2}{\Sigma, \Phi\defsubst{x_1}{x_2}}{\Gamma}{\Psi}{t_1}{A_1}{t_2}{A_2}{\Phi'}} | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{\letconst{x_1}{u_1}{t_1}}{A_1}{\letconst{x_2}{u_2}{t_2}}{A_2}{\Phi'}}$ | |
| \\ \\ | |
| \noindent | |
| By I.H. $\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\Phi\subst{u_1}{u_2}}$, \hfill{}(H1) | |
| \\ | |
| and $\jghol{\Delta, x_1 : B_1, x_2 : B_2}{\Sigma, \Phi\defsubst{x_1}{x_2}}{\Gamma}{\Psi}{\Phi'\defsubst{t_1}{t_2}}$. \hfill{}(H2) | |
| \\ | |
| To show: $\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\Phi'\defsubst{\letconst{x_1}{u_1}{t_1}}{\letconst{x_2}{u_2}{t_2}}}$ \hfill{}(G) | |
| \\ | |
| From (H1) and the fact that $\Phi, B_1$ and $B_2$ are constant, we get: | |
| \\ | |
| $\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\square\Phi\defsubst{u_1}{u_2}}$ | |
| \\ | |
| The rest of the prove is analogous to the previous case. | |
| \\ | |
| \noindent {\bf Case.} $\small \inferrule*[Right=\sf Fix] | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma, f_1:\later{A_1}, f_2:\later{A_2}}{\Psi, \dsubst{\res\ltag,\res\rtag}{f_1,f_2}{\Phi}} | |
| {t_1}{A_1}{t_2}{A_2}{\Phi}} | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{fix f_1. t_1}{A_1}{fix f_2. t_2}{A_2}{\Phi}}$ | |
| \\ \\ | |
| \noindent | |
| By I.H. $\jghol{\Delta}{\Sigma}{\Gamma, f_1:\later{A_1}, f_2:\later{A_2}}{\Psi,\later[\res\ltag\ot f_1, \res\rtag\ot f_2]{\Phi}}{\Phi\defsubst{t_1}{t_2}}$. | |
| \\ | |
| To show: $\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\Phi\subst{\fix{f_1}{t_1}}{fix{f_2}{t_2}}}$ \\ | |
| Instantiating the I.H. with $f_1 = \later \fix{f_1}{t_1}$ and $f_2 = \later \fix{f_2}{t_2}$ we get: | |
| \\ | |
| $\jghol{\Delta}{\Sigma}{\Gamma}{\Psi,\later[\res\ltag \ot \later \fix{f_1}{t_1}, \res\rtag \ot \later \fix{f_2}{t_2}]{\Phi}} | |
| {\Phi\defsubst{t_1\subst{f_1}{\later \fix{f_1}{t_1}}}{t_2\subst{f_2}{\later \fix{f_2}{t_2}}}}$. | |
| \\ | |
| Since $t\subst{f}{\later \fix{f}{t}} \equiv \fix{f}{t}$, by \rname{CONV}: | |
| \\ | |
| $\jghol{\Delta}{\Sigma}{\Gamma}{\Psi,\later[\res\ltag \ot \later \fix{f_1}{t_1}, \res\rtag \ot \later \fix{f_2}{t_2}]{\Phi}}{\Phi\defsubst{\fix{f_1}{t_1}}{\fix{f_2}{t_2}}}$. | |
| \\ | |
| and since $\later[\res\ltag \ot \later \fix{f_1}{t_1}, \res\rtag \ot \later \fix{f_2}{t_2}]{\Phi} \Leftrightarrow \later \Phi\defsubst{\fix{f_2}{t_2}}{\fix{f_2}{t_2}}$, | |
| \\ | |
| $\jghol{\Delta}{\Sigma}{\Gamma}{\Psi,\later\Phi\defsubst{\fix{f_1}{t_1}}{fix{f_2}{t_2}}}{\Phi\defsubst{fix{f_1}{t_1}}{fix{f_2}{t_2}}}$, | |
| \\ | |
| and finally, by \rname{L\"ob} we get our goal. | |
| \\ | |
| \noindent {\bf Case.} $\small\inferrule*[Right=\sf Cons] | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{x_1}{A_1}{x_2}{A_2}{\Phi_h} \\ | |
| \jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{xs_1}{\later{\Str{A_1}}}{xs_2}{\later{\Str{A_2}}}{\Phi_t} \\ | |
| \jhol{\Gamma}{\Psi}{\forall x_1,x_2, xs_1, xs_2. \Phi_h\defsubst{x_1}{x_2} \Rightarrow | |
| \Phi_t\defsubst{xs_1}{xs_2} \Rightarrow \Phi\defsubst{\cons{x_1}{xs_1}}{\cons{x_2}{xs_2}}}} | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{\cons{x_1}{xs_1}}{\Str{A_1}}{\cons{x_2}{xs_2}}{\Str{A_2}}{\Phi}}$ | |
| \\ \\ | |
| Apply the I.H., \rname{$\forall_E$} and \rname{$\To_E$}. | |
| \\ | |
| \noindent {\bf Case.}$\small\inferrule*[Right=\sf Head] | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{\Str{A_1}}{t_1}{\Str{A_1}}{\Phi\defsubst{hd\ \res\ltag}{hd\ \res\rtag}}} | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{hd\ t_1}{A_1}{hd\ t_2}{A_2}{\Phi}}$ | |
| \\ \\ | |
| Trivial by I.H. | |
| \\ | |
| \noindent {\bf Case.} $\small\inferrule*[Right=\sf Tail] | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{\Str{A_1}}{t_2}{\Str{A_2}}{\Phi\defsubst{tl\ \res\ltag}{tl\ \res\rtag}}} | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{tl\ t_1}{\later{Str_{A_1}}}{tl\ t_2}{\later{Str_{A_2}}}{\Phi}}$ | |
| \\ \\ | |
| Trivial by I.H. | |
| \\ | |
| \noindent {\bf Case.} $\small\inferrule*[Right=\sf Equiv] | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t'_1}{A_1}{t'_2}{A_2}{\Phi} \\ t_1 \equiv t_1' \\ t_2 \equiv t_2' \\ | |
| \Delta \mid \Gamma \vdash t_1 : A_1 \\ \Delta \mid \Gamma \vdash t_2 : A_2} | |
| {\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{A_1}{t_2}{A_2}{\Phi}}$ | |
| \\ \\ | |
| Trivial by I.H. and \rname{Conv}. | |
| \\ | |
| Most of the proofs for the probabilistic fragment are a consequence of the proof of \autoref{thm:sound-ghol}. | |
| The only interesting case is \rname{Markov}. We do the proof directly in RHOL by showing we can derive it from \rname{Fix}. | |
| We have the premises: | |
| \begin{enumerate} | |
| \item $\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{C_1}{t_2}{C_2}{\phi}$ | |
| \item $\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{h_1}{C_1 \to \Distr(C_1)}{h_2}{C_2 \to \Distr(C_2)}{\psi_3}$ | |
| \item $\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\psi_4}$ | |
| \end{enumerate} | |
| where: | |
| \begin{align*} | |
| \psi_3 &\equiv \forall x_1 x_2. \phi\defsubst{x_1}{x_2} \Rightarrow \diamond_{[ y_1 \ot \res\ltag\ x_1, y_2 \ot \res\rtag\ x_2]}\phi\defsubst{y_1}{y_2} \\ | |
| \psi_4 &\equiv \forall x_1\ x_2\ xs_1\ xs_2. \phi\defsubst{x_1}{x_2} \Rightarrow \later[y_1 \ot xs_1, y_2 \ot xs_2]{\Phi} | |
| \Rightarrow \Phi\subst{y_1}{\cons{x_1}{xs_1}}\subst{y_2}{\cons{x_2}{xs_2}} | |
| \end{align*} | |
| If we inline the definition of unfold, we have to prove: | |
| \[\begin{aligned}[t] | |
| \fix{f}{&\lambda x_1. \lambda h_1. \mlet{z_1}{h_1\ x_1} | |
| { \mlet{t_1}{\operatorname{swap_{\later\Distr}^C}(f_1\app \later z_1 \app \later h_1)} | |
| { \munit{\cons{x_1}{t_1}}}} | |
| }\sim \\ \fix{f}{&\lambda x_2. \lambda h_2. \mlet{z_2}{h_2\ x_2} | |
| { \mlet{t_2}{\operatorname{swap_{\later\Distr}^C}(f_2\app \later z_2 \app \later h_2)} | |
| { \munit{\cons{x_2}{t_2}}}} | |
| }\\\mid\; &\forall x_1 x_2 h_1 h_2. \phi\defsubst{x_1}{x_2} \Rightarrow \psi_3\defsubst{h_1}{h_2} \Rightarrow \diamond_{[y_2 \ot\res\ltag, y_2 \ot\res\rtag]}\Phi | |
| \end{aligned} | |
| \] | |
| We apply \rname{FIX}, \rname{MLET} twice, and then \rname{MUNIT}. The main judgements we have to prove are: | |
| \begin{enumerate}[label*=(\alph*)] | |
| \item $\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{h_1\ x_1}{\Distr(C_1)}{h_2\ x_2}{\Distr(C_2)} | |
| {\diamond_{[\res\ltag\ot\res\ltag, \res\rtag\ot\res\rtag]}\phi}$ | |
| \item $\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{\operatorname{swap_{\later\Distr}^C}(f_1\app \later z_1 \app \later h_1)}{\Distr(\later C_1)} | |
| {\operatorname{swap_{\later\Distr}^C}(f_2\app \later z_2 \app \later h_2)}{\Distr(\later C_2)} | |
| {\diamond_{[z_1 \ot \res\ltag, z_2 \ot \res\rtag]} \later[y_1 \ot z_1, y_2 \ot z_2]{\Phi}}$ | |
| \item $\jgrhol{\Delta}{\Sigma}{\Gamma, x_1,x_2,t_1,t_2}{\Psi, \phi\defsubst{x_1}{x_2}, \later[y_1 \ot t_1, y_2 \ot t_2]{\Phi}}{\cons{y_1}{t_1}}{\Distr(\Str{C_1})}{\cons{y_2}{t_2}}{\Distr(\Str{C_2})} | |
| {\Phi\subst{y_1}{\res\ltag}\subst{y_2}{\res\rtag}}$ | |
| \end{enumerate} | |
| The judgement (a) is a direct consequence of premises (1) and (2), (b) is proven from the inductive hypothesis, | |
| and (d) is a direct consequence of (3). This completes the proof. | |
| \section{Examples} | |
| \subsection{Proof of ZipWith} | |
| This example, taken from \cite{BGCMB16}, proves a | |
| property about the $\operatorname{ZipWith}$ function, which takes two | |
| streams of type A, a function on pairs of elements, and ``zips'' the | |
| two streams by applying that function to the elements that are at the | |
| same position on the two streams. We want to show that if the | |
| function on the elements is commutative, zipping two streams with that | |
| function is commutative as well. | |
| We can define the zipWith function as: | |
| \begin{align*} | |
| \operatorname{zipWith} &: (\nat \to \nat \to \nat) \to \Str{\nat} \to \Str{\nat} \to \Str{\nat} \\ | |
| \operatorname{zipWith} &\defeq \fix{\operatorname{zipWith}}{\lambda f. \lambda xs. \lambda ys. \cons{(f\ (\hd\ xs)\ (\hd\ ys))}{(\operatorname{zipWith} \app (\tl\ xs) \app (\tl\ ys))}} | |
| \end{align*} | |
| We prove (omitting types of expressions): | |
| $$\jgrholnocnot{\operatorname{zipWith}}{\operatorname{zipWith}}{\Phi}$$ | |
| where | |
| \begin{align*} | |
| \Phi &\defeq \forall f_1 f_2. (f_1 = f_2 \wedge \forall x y. f_1 x y = f_1 y x) \Rightarrow | |
| \forall xs_1 xs_2. \forall ys_1 ys_2. (xs_1 = ys_2 \wedge xs_2 = ys_1)\\ | |
| & \Rightarrow | |
| \res\ltag\ f_1\ xs_1\ ys_1 = \res\rtag\ f_2\ xs_2\ ys_2 \\ | |
| A &\defeq (\nat \to \nat \to \nat) \to \Str{\nat} \to \Str{\nat} \to \Str{\nat} | |
| \end{align*} | |
| The proof proceeds by applying two-sided rules all the way. We invite | |
| interested readers to compare this proof with the one given | |
| in~\cite{BGCMB16} to see how the two approaches differ. | |
| We show how to derive the statement backwards. The derivation begins | |
| with the \rname{Fix} rule. Its premise is (omitting constant contexts): | |
| \[ | |
| \jrhol{\operatorname{zipWith}_1,\operatorname{zipWith}_2:\later A}{\later[\res\ltag \leftarrow \operatorname{zipWith}_1,\res\rtag \leftarrow \operatorname{zipWith}_2] \Phi}{\lambda f_1.(\cdots)}{A} | |
| {\lambda f_2.(\cdots)}{A}{\Phi} | |
| \] | |
| Then we apply the \rname{ABS} rule three times to introduce into the | |
| context the logical relations on $f_1,f_2,xs_1,xs_2,ys_1$, and $ys_2$. The premise | |
| we need to prove is then: | |
| \[ | |
| \begin{array}{c} | |
| \jrhol{\Gamma}{\Psi}{\cons{(f_1\ (hd\ xs_1)\ (hd\ ys_1))}{(\operatorname{zipWith}_1 \app (tl\ xs_1) \app (tl\ ys_1))}}{\Str{\nat}} | |
| {\\ \cons{(f_2\ (hd\ xs_2)\ (hd\ ys_2))}{(\operatorname{zipWith}_2 \app (tl\ xs_2) \app (tl\ ys_2))}}{\Str{\nat}}{\res\ltag = \res\rtag} | |
| \end{array} | |
| \] | |
| where | |
| \begin{align*} | |
| \Gamma &\defeq \operatorname{zipWith}_1,\operatorname{zipWith}_2 :\later A; f_1,f_2:(\nat\to\nat\to\nat); xs_1,xs_2,ys_1,ys_2: \Str{\nat} \\ | |
| \Psi &\defeq \later[\res\ltag \leftarrow \operatorname{zipWith}_1,\res\rtag \leftarrow \operatorname{zipWith}_2] \Phi, (f_1 = f_2 \wedge \forall x y. f x y = f y x), xs_1 = ys_2, xs_2 = ys_1 | |
| \end{align*} | |
| Now we can apply the \rname{Cons} rule, which has three premises: | |
| \begin{enumerate} | |
| \item $\jrhol{\Gamma}{\Psi}{f_1\ (hd\ xs_1)\ (hd\ ys_1)}{\nat} | |
| {f_2\ (hd\ xs_2)\ (hd\ ys_2)}{\nat}{\res\ltag = \res\rtag}$ | |
| \item $\jrhol{\Gamma}{\Psi}{\operatorname{zipWith}_1 \app (tl\ xs_1) \app (tl\ ys_1)}{\later \Str{\nat}} | |
| {\operatorname{zipWith}_2 \app (tl\ xs_2) \app (tl\ ys_2)}{\later \Str{\nat}}{\res\ltag = \res\rtag}$ | |
| \item $\jhol{\Gamma}{\Psi}{\forall x y xs ys. x=y \Rightarrow xs = ys \Rightarrow \cons{x}{xs} = \cons{y}{ys}}$ | |
| \end{enumerate} | |
| Premise (3) is easily provable in HOL. To prove premise (1) we first apply the \rname{App} rule twice, and we have to prove the judgments: | |
| \begin{itemize} | |
| \item $\jrhol{\Gamma}{\Psi}{f_1}{\nat\to\nat\to\nat} | |
| {f_2}{\nat\to\nat\to\nat}{ \forall vs_1 vs_2 ws_1 ws_2. | |
| \\ vs_1 = hd\ ys_2 \wedge vs_2 = hd\ ys_1 \Rightarrow | |
| ws_1 = hd\ xs_2 \wedge ws_2 = hd\ xs_1 \Rightarrow \res\ltag\ vs_1\ ws_1 = \res\rtag\ vs_2\ ws_2}$ | |
| \item $\jrhol{\Gamma}{\Psi}{hd\ xs_1}{\nat} | |
| {hd\ xs_2}{\nat}{\res\ltag = hd\ ys_2 \wedge \res\rtag = hd\ ys_1}$ | |
| \item $\jrhol{\Gamma}{\Psi}{hd\ ys_1}{\nat} | |
| {hd\ ys_2}{\nat}{\res\ltag = hd\ xs_2 \wedge \res\rtag = hd\ xs_1}$ | |
| \end{itemize} | |
| The three can be proven in HOL from the conditions imposed on $f_1,f_2$ and the equalities $xs_1 = ys_2$, $xs_2 = ys_1$. | |
| All that remains to prove is premise (2) of the \rname{Cons} application, which, by expanding the definition of $\app$ and | |
| using the equational theory of delayed substitutions, can be desugared to: | |
| \[\jrhol{\Gamma}{\Psi}{\latern[\xi_1]{(g_1\ t_1\ u_1)}}{\later \Str{\nat}} | |
| {\latern[\xi_2]{(g_2\ t_2\ u_2)}}{\later \Str{\nat}} | |
| {\latern[\xi_1, \xi_2, [\res\ltag\leftarrow\res\ltag, \res\rtag \leftarrow \res\rtag]]{(\res\ltag = \res\rtag)}}\] | |
| where, for $i=1,2$: | |
| \[ \xi_i = [g_i \leftarrow \operatorname{zipWith}_i, t_i\leftarrow(tl\ xs_i), u_i \leftarrow (tl\ ys_i)]\] | |
| We apply the \rname{Next} rule, and we have the four following premises: | |
| \begin{itemize} | |
| \item $\jrhol{\Gamma}{\Psi}{\operatorname{zipWith}_1}{\later A} | |
| {\operatorname{zipWith}_2}{\later A} | |
| {\later[\res\ltag\leftarrow\res\ltag, \res\rtag\leftarrow\res\rtag] (\res\ltag = \res\rtag \wedge \forall x y. \res\ltag x y = \res\ltag y x)}$ | |
| \item $\jrhol{\Gamma}{\Psi}{tl\ xs_1}{\later \Str{\nat}} | |
| {tl\ xs_2}{\later \Str{\nat}} | |
| {\later[\res\ltag\leftarrow\res\ltag, \res\rtag\leftarrow\res\rtag] (\res\ltag = tl\ ys_2 \wedge \res\rtag = tl\ ys_1)}$ | |
| \item $\jrhol{\Gamma}{\Psi}{tl\ ys_1}{\later \Str{\nat}} | |
| {tl\ ys_2}{\later \Str{\nat}} | |
| {\later[\res\ltag\leftarrow\res\ltag, \res\rtag\leftarrow\res\rtag] (\res\ltag = tl\ xs_2 \wedge \res\rtag = tl\ xs_1)}$ | |
| \item $\jrhol{\Gamma; g_1,g_2: A; t_1,t_2,u_1,u_2: \Str{\nat}} | |
| {\Psi, g_1 = g_2 \wedge \forall x y. g_1 x y = g_1 y x, t_1 = tl\ ys_2 \wedge t_2 = tl\ ys_1, | |
| \\ u_1 = tl\ xs_2 \wedge u_2 = tl\ xs_1} | |
| {g_1\ t_1\ u_1}{\Str{\nat}} | |
| {g_2\ t_2\ u_2}{\Str{\nat}} | |
| {\res\ltag = \res\rtag}$ | |
| \end{itemize} | |
| To prove the first premise we instantiate the inductive hypothesis we got | |
| from \rname{Fix}. To prove the second and the third premises we use the | |
| equalities $xs_1 = ys_2$, $xs_2 = xs_1$. Finally, the fourth premise is a | |
| simple derivation in HOL that follows from the same equalities plus | |
| the refinements of $g_1,g_2,t_1,t_2,u_1,u_2$. This concludes the | |
| proof. | |
| \subsection{Proof of approximation series} | |
| We now continue with another example that, while still being fully synchronous | |
| (i.e., uses only two-sided rules), goes beyond reasoning about equality of | |
| streams, and showcases the flexibility of streams to represent different kinds | |
| of information and structures. | |
| For instance, streams can be used to represent series of numbers. In | |
| this example, we illustrate an instance of a property about series | |
| that can be proven in our system. Consider the series $x_0, x_1, | |
| \ldots$ for any $p \geq \frac{1}{2}$ and any $a \geq 0$, where $x_0$ | |
| is given and: | |
| \[ x_{i+1} = px_i + (1-p) \frac{a}{x_i} \] | |
| It can be easily shown that if $x_0 \geq \sqrt{a}$, then this series | |
| converges \emph{monotonically} from the top to $\sqrt{a}$. In | |
| particular, $\displaystyle\lim_{i \to \infty} x_i = \sqrt{a}$. (For $p | |
| = \frac{1}{2}$, this is the standard Newton-Raphson series for | |
| square-root computation \cite{Scott11}) | |
| The interesting relational property is that for smaller $p$, this | |
| series converges faster. Concretely, define $f(p, a, x_0, i)$ as the | |
| $i$th element of the above series (for the given $p$, $a$ and | |
| $x_0$). Then, the relational property to prove is that: | |
| \[ \forall p_1\, p_2\, a\, x_0\, i.\, (\dfrac{1}{2}\leq p_1 \leq p_2 \conj x_0 \geq \sqrt{a}) \Rightarrow |f(p_1,a,x_0,i) - \sqrt{a}| \leq |f(p_2,a,x_0,i)- \sqrt{a}|\] | |
| We outline the proof of this property. First, note that because | |
| convergence is from the top, $|f(p_2,a,x_0,i) - \sqrt{a}| = | |
| f(p_2,a,x_0,i) - \sqrt{a}$. Therefore, the property above is the same | |
| as: | |
| \[ \forall p_1\, p_2\, a\, x_0\, i.\, (\dfrac{1}{2}\leq p_1 \leq p_2\conj x_0 \geq \sqrt{a}) \Rightarrow f(p_2,a,x_0,i) - f(p_1,a,x_0,i) \geq 0\] | |
| This is easy to establish by induction on $i$. | |
| (Note the importance of the assumption $p \geq \frac{1}{2}$: Without | |
| this assumption, convergence is not monotonic, and this relational | |
| property may not hold. If we start with $x_0 \leq \sqrt{a}$ instead of | |
| $x_0 \geq \sqrt{a}$, we need $p \leq \frac{1}{2}$ for convergence to | |
| be monotonic, this time from below.) | |
| Now we see how we can encode and prove this as a relational property | |
| of a pair of streams. We can define a stream whose elements are the | |
| elements of one of this series: | |
| \begin{align*} | |
| \operatorname{approx\_sqrt} &: \real \to \real \to \real \to \Str{\real} \\ | |
| \operatorname{approx\_sqrt} &\defeq \fix{f}{\lambda p. \lambda a. \lambda x. | |
| \cons{x}{(f \app \later p \app \later a \app \later (p*x + (1-p)*a/x))}} | |
| \end{align*} | |
| We prove: | |
| \[\jgrholnoc{\operatorname{approx\_sqrt_1}}{\real\to\real\to\real\to \Str{\real}}{\operatorname{approx\_sqrt_2}}{\real\to\real\to\real\to \Str{\real}} | |
| {\Phi} | |
| \] | |
| where | |
| \begin{align*} | |
| \Phi &\defeq \forall p_1 p_2. (\dfrac{1}{2}\leq p_1 \leq p_2) \Rightarrow \forall a_1 a_2. 0 \leq a_1 = a_2 \Rightarrow \forall x_1 x_2. (0 \leq x_1 \leq x_2 \wedge a_1\leq x_1*x_1) | |
| \\ &\Rightarrow \All(\res\ltag\ p_1\ a_1\ x_1, \res\rtag\ p_2\ a_2\ x_2, \lambda n_1 n_2. 0 \leq n_1 \leq n_2 \wedge a_1 \leq n_1 * n_1) | |
| \end{align*} | |
| and $\All$ is defined axiomatically as follows: | |
| \[\forall s_1,s_2,n_1,n_2. \phi n_1 n_2 \Rightarrow \later[s'_1\ot s_1, s'_2 \ot s_2]{\All(s'_1,s'_2,\lambda x_1 x_2. \phi)} \Rightarrow \All(\cons{n_1}{s_1}, \cons{n_2}{s_2}, \lambda x_1 x_2. \phi)\] | |
| The meaning of the judgement is that, if we have two approximation series for | |
| the square root of $a$ (formally, we write $a = a_1 = a_2$), with | |
| initial guesses $x_1 \leq x_2$, and parameters $1/2 \leq p_1 \leq | |
| p_2$, then, at every position, the first series is going to be closer | |
| to the root than the second one. Note that we have removed the square | |
| roots in the specification by squaring. | |
| Let $A =\defeq \real\to\real\to\real\to \Str{\real}$. We will show | |
| how to derive the judgment backwards. The proof starts by applying | |
| \rname{Fix} which has the premise (omitting constant contexts): | |
| \[\jrhol{f_1,f_2 : \later A}{\later[\res\ltag, \res\rtag \leftarrow f_1, f_2]\Phi}{\lambda p_1. \lambda a_1. \lambda x_1. \dots}{A}{\lambda p_2. \lambda a_2. \lambda x_2. \dots}{A} | |
| {\Phi} | |
| \] | |
| and after applying \rname{Abs} three times: | |
| \[\begin{array}{c} | |
| \jrhol{f_1,f_2 : \later A; p_1,p_2,a_1,a_2,x_1,x_2 : \real} | |
| {\\ \later[\res\ltag, \res\rtag \leftarrow f_1, f_2]\Phi, (\dfrac{1}{2}\leq p_1 \leq p_2) , 0 \leq a_1 = a_2 , 0 \leq x_1 \leq x_2, a_1 \leq x_1*x_1} | |
| {\\ (\lambda y_1.\cons{x_1}{(f_1\app \later p_1\app \later a_1\app \later y_1)})(p_1*x_1 + (1-p_1)*a_1/x_1)}{\Str{\real}} | |
| {\\ (\lambda y_2.\cons{x_2}{(f_2\app \later p_2\app \later a_2\app \later y_2)})(p_2*x_2 + (1-p_2)*a_2/x_2)}{\Str{\real}} | |
| {\\ All(\res\ltag, \res\rtag, \lambda n_1 n_2. n_1 \leq n_2)} | |
| \end{array} | |
| \] | |
| Let $\Gamma$ and $\Psi$ denote the typing and logical contexts in the previous judgement. Now we apply \rname{App}, which has two premises: | |
| \begin{itemize} | |
| \item $\jrhol{\Gamma}{\Psi} | |
| {\lambda y_1.\cons{x_1}{(f_1\app \later p_1\app \later a_1\app \later y_1)}}{\real\to \Str{\real}} | |
| {\lambda y_2.\cons{x_2}{(f_2\app \later p_2\app\later a_2\app\later y_2)}}{\real\to \Str{\real}} | |
| {\\ \forall y_1, y_2. (0 \leq y_1 \leq y_2 \wedge a_1 \leq y_1 * y_1) \Rightarrow All(\res\ltag\ y_1, \res\rtag\ y_2, \lambda n_1 n_2. 0\leq n_1\leq n_2 \wedge a_1 \leq n_1 * n_1)}$ | |
| \item $\jrhol{\Gamma}{\Psi} | |
| {p_1*x_1 + (1-p_1)*a_1/x_1}{\Str{\real}} | |
| {p_2*x_2 + (1-p_2)*a_2/x_2}{\Str{\real}} | |
| {\\ 0 \leq \res\ltag \leq \res\rtag \wedge a_1 \leq \res\ltag * \res\ltag}$ | |
| \end{itemize} | |
| The second premise can be established in Guarded HOL as an arithmetic | |
| property in our theory of reals. To prove the first one, we start by applying the \rname{Abs} | |
| rule, followed by the \rname{Cons} rule, which has three premises: | |
| \begin{enumerate} | |
| \item $\jrhol{\Gamma, y_1,y_2 : \real}{\Psi, (y_1 \leq y_2 \wedge y_1*y_1 \geq a_1)} | |
| {x_1}{\real} | |
| {x_2}{\real} | |
| {0 \leq \res\ltag \leq \res\rtag \wedge a \leq \res\ltag * \res\ltag}$ | |
| \item $\jrhol{\Gamma, y_1,y_2 : \real}{\Psi, (y_1 \leq y_2 \wedge y_1*y_1 \geq a_1)} | |
| {(f_1\app\later p_1\app\later a_1\app\later y_1)}{\later \Str{\real}} | |
| {\\ (f_2\app\later p_2\app\later a_2\app\later y_2)}{\later \Str{\real}} | |
| {\later[\res\ltag \ot \res\ltag, \res\rtag\ot\res\rtag]All(\res\ltag, \res\rtag, \lambda n_1 n_2. 0 \leq n_1 \leq n_2 \wedge a \leq n_1 * n_1)}$ | |
| \item $\jhol{\Gamma, y_1,y_2 : \real}{\Psi, (y_1 \leq y_2 \wedge y_1*y_1 \geq a_1)} | |
| {\forall h_1 h_2 t_1 t_2. 0 \leq h_1 \leq h_2 \Rightarrow a_1 \leq h_1*h_1 \Rightarrow\\ | |
| \later[\res\ltag\ot t_1, \res\rtag \ot t_2] All(\res\ltag, \res\rtag, \lambda n_1 n_2. 0\leq n_1 \leq n_2 \wedge a \leq n_1*n_1) \Rightarrow \\ | |
| All(\cons{h_1}{t_1}, \cons{h_2}{t_2}, \lambda n_1 n_2. 0\leq n_1 \leq n_2 \wedge a \leq n_1*n_1))}$ | |
| \end{enumerate} | |
| Premise (1) is just the refinement on $x_1,x_2$, while premise (3) is | |
| the axiomatization of $All$. To prove | |
| premise (2) one instantiates the induction hypothesis given by the | |
| \rname{Fix} rule. In order to do so, we first rewrite the two terms we are comparing to their desugared form: | |
| \[\later[f_1' \ot f_1, p_1' \ot \later p_1, a_1' \ot \later a_1, y_1' \ot \later y_1] f_1'\ p_1'\ a_1'\ y_1'\] | |
| and | |
| \[\later[f_2' \ot f_2, p_2' \ot \later p_2, a_2' \ot\later a_2, y_2' \ot\later y_2] f_2'\ p_2'\ a_2'\ y_2'\] | |
| We can also add by \rname{SUB} the same substitutions to the $\later$ | |
| in the conclusion, since the substituted variables do not appear in | |
| the formula. Then we can apply the \rname{Next} rule, which has the premises: | |
| \begin{itemize} | |
| \item $\jrhol{\Gamma, y_1,y_2 : \real}{\Psi, (y_1 \leq y_2 \wedge y_1*y_1 \geq a_1)} | |
| {f_1}{\later A} | |
| {f_2}{\later A} | |
| {\later[\res\ltag\ot\res\ltag, \res\rtag\ot\res\ltag]\Phi}$ | |
| \item $\jrhol{\Gamma, y_1,y_2 : \real}{\Psi, (y_1 \leq y_2 \wedge y_1*y_1 \geq a_1)} | |
| {\later p_1}{\real} | |
| {\later p_2}{\real} | |
| {\\ \later[\res\ltag\ot\res\ltag, \res\rtag\ot\res\ltag] \dfrac{1}{2}\leq \res\ltag \leq \res\rtag}$ | |
| \item $\jrhol{\Gamma, y_1,y_2 : \real}{\Psi, (y_1 \leq y_2 \wedge y_1*y_1 \geq a_1)} | |
| {\later a_1}{\real} | |
| {\later a_2}{\real} | |
| {\later[\res\ltag\ot\res\ltag, \res\rtag\ot\res\ltag] 0 \leq \res\ltag = \res\rtag}$ | |
| \item $\jrhol{\Gamma, y_1,y_2 : \real}{\Psi, (y_1 \leq y_2 \wedge y_1*y_1 \geq a_1)} | |
| {\later y_1}{\real} | |
| {\later y_2}{\real} | |
| {\\ \later[\res\ltag\ot\res\ltag, \res\rtag\ot\res\ltag] 0 \leq \res\ltag \leq \res\rtag \wedge a_1' \leq \res\ltag * \res\ltag}$ | |
| \item $\jrhol{\Gamma, y_1,y_2,p_1',p_2',a_1',a_2',y_1',y_2' : \real; f_1', f_2' : A} | |
| {\Psi, (y_1 \leq y_2 \wedge y_1*y_1 \geq a_1), \Phi\defsubst{f_1'}{f_2'},\\ \dfrac{1}{2}\leq p_1' \leq p_2', 0 \leq a_1' = a_2', 0 \leq y_1' \leq y_2' \wedge a_1' y_1' * y_1'} | |
| {\\ f_1'\ p_1'\ a_1'\ y_1'}{\Str{\real}} | |
| {f_2'\ p_2'\ a_2'\ y_2'}{\Str{\real}}{All(\res\ltag, \res\rtag, \lambda n_1 n_2. 0 \leq n_1 \leq n_2 \wedge a \leq n_1 * n_1)}$ | |
| \end{itemize} | |
| The first four can be proven simply by instantiating and then delaying | |
| one of the axioms. The last one is proven by applying \rname{App} three | |
| times. This concludes the proof. | |
| \subsection{Proof of Cassini's identity} | |
| We continue building on the idea from the previous example of using | |
| streams to represent series of numbers. This time, we prove a | |
| classical identity of the Fibonacci sequence. Since the example | |
| requires to observe the stream at different times, we will also have | |
| to deal with some asynchronicity on the delayed substitutions. | |
| Let $F_n$ be the $n$th Fibonacci number. Cassini's identity states | |
| that $F_{n-1} \cdot F_{n+1} - F_{n}^2 = (-1)^{n}$. Cassini's identity | |
| can be stated as a stream problem as follows. First, let $F$ be the | |
| Fibonnaci stream ($1,1,2,3,5,\ldots$) and $A$ be the stream | |
| $1,-1,1,-1,\ldots$ Let $\oplus$ and $\otimes$ be infix functions that | |
| add and multiply two streams pointwise. Cassini's identity can then be | |
| informally written as: | |
| \[ F \otimes \tl(\tl\; F) = \tl(F \otimes F) \oplus A \] | |
| In order to formalize Cassini's identity in our system, we first define: | |
| \begin{align*} | |
| \oplus &: \Str{\nat} \to \Str{\nat} \to \Str{\nat} &\otimes &: \Str{\nat} \to \Str{\nat} \to \Str{\nat} \\ | |
| \oplus &\defeq \begin{array}{l}\fix{f}{\lambda s.\lambda t\\ \cons{(\hd\ x + \hd\ y)}{(f \app (\tl\ x) \app (\tl\ y))}}\end{array} \quad \quad | |
| &\otimes &\defeq \begin{array}{l}\fix{f}{\lambda s.\lambda t\\ \cons{(\hd\ x * \hd\ y)}{(f \app (\tl\ x) \app (\tl\ y))}}\end{array} | |
| \end{align*} | |
| Then we define $F$ and $A$ as the fixpoints of the equations: | |
| \begin{align*} | |
| F &\defeq \fix{F}{\cons{1}{\later[ F' \ot F] (\cons{1}{\later [ T \ot \tl\ F' ] ( F' \oplus T)})}} \\ | |
| A &\defeq \fix{A}{\cons{1}{\later(\cons{-1}{A})}} | |
| \end{align*} | |
| We prove (using prefix notation for $\oplus$ and $\otimes$): | |
| \[\jgrholnoc | |
| {\later [ T_1 \ot \tl\ F] \otimes \app (\later F) \app \tl\ T_1 }{\later\later \Str{\nat}} | |
| {\oplus \app \tl(F \otimes F) \app (\later A)}{\later \Str{\nat}} | |
| { \res\ltag = \later \res\rtag} | |
| \] | |
| The proof combines applications of two-sided rules and one-sided | |
| rules; in particular, we use the rule \rname{NEXT-L} to proceed with | |
| the proof for a judgement where the left expression is delayed twice | |
| and the right expression is delayed once. | |
| By conversion, in the logic we can prove the following equalities: | |
| \[ | |
| \Psi \defeq \left\{\begin{aligned} | |
| F &= \cons{1}{\later(\cons{1}{\later [ T \ot tl\ F ] ( F \oplus T)})}, \\ | |
| A &= \cons{1}{\later(\cons{-1}{\later A})} | |
| \end{aligned}\right\} | |
| \] | |
| Using these equalities, and desugaring the applications, the judgment we want to prove is (omitting constant contexts): | |
| \[\begin{array}{c}\jrhol{F,A: \Str{\nat}}{\Psi} | |
| {\later [ T_1 \ot tl\ F] \later [T_1' \ot tl\ T_1] (F \otimes T_1') }{\later\later \Str{\nat}} | |
| {\\ \later [T_2 \ot tl(F \otimes F)] (T_2 \oplus A)}{\later \Str{\nat}} | |
| {\\ \later[\res\ltag' \ot \res\ltag, T_1 \ot tl\ F] \later [\res\ltag''\ot \res\ltag, \res\rtag'\ot\res\rtag, T_1' \ot tl\ T_1, T_2' \ot tl(F \otimes F)] \res\ltag'' = \res\rtag'} | |
| \end{array} | |
| \] | |
| Notice that on the left, since we want to apply tail twice to $F$, we | |
| need to delay the term twice so that $F$ and $tl\ tl\ F$ have the same | |
| type. On the right, we just need to delay the term once. As for the | |
| logical conclusion, $\res\ltag$ needs to be delayed twice, while | |
| $\res\rtag$ only once. The way to do this is by having $\res\ltag$ | |
| appear on the two substitutions but $\res\rtag$ only on the inner one. | |
| We start by applying \rname{NEXT-L}, which has the two following premises: | |
| \begin{itemize} | |
| \item $\juhol{F,A: \Str{\nat}}{\Psi}{tl\ F }{\later \Str{\nat}}{\later[\res' \ot \res] tl\ F = \later \res'}$ | |
| \item $\jrhol{F,A, T_1: \Str{\nat}}{\Psi, tl\ F = \later T_1}{\later [T_1' \ot tl\ T_1] (F \otimes T_1') }{\later \Str{\nat}} | |
| {\\ \later [T_2 \ot tl(F \otimes F)] (T_2 \oplus A)}{\later \Str{\nat}} | |
| {\later [\res\ltag''\ot \res\ltag, \res\rtag'\ot\res\rtag, T_1' \ot tl\ T_1, T_2' \ot tl(F \otimes F)] \res\ltag'' = \res\rtag'}$ | |
| \end{itemize} | |
| The first premise is trivial. We continue by applying \rname{NEXT} to the second, which has the following premises: | |
| \begin{itemize} | |
| \item $\jrhol{F,A, T_1: \Str{\nat}}{\Psi, tl\ F = \later T_1}{tl\ T_1}{\later \Str{\nat}}{tl(F \otimes F)}{\later \Str{\nat}} | |
| {\\ \later[\res\ltag' \ot \res\ltag, \res\rtag' \ot \res\rtag] T_1 = \later \res\ltag' \wedge \res\ltag'\otimes\res\ltag' = \res\rtag'}$ | |
| \item $\jrhol{F,A, T_1', T_2: \Str{\nat}}{\Psi,tl\ F = \later T_1 , tl\ T_1 = \later T_1', T_1'\otimes T_1' = T_2}{F \otimes T_1'}{\Str{\nat}}{T_2 \oplus A}{\Str{\nat}}{\res\ltag = \res\rtag}$ | |
| \end{itemize} | |
| Again, the first premise is trivial. We apply \rname{APP} twice to the second, and we have to prove: | |
| \begin{itemize} | |
| \item $\jrhol{F,A, T_1', T_2: \Str{\nat}}{\Psi,tl\ F = \later T_1 , tl\ T_1 = \later T_1', T_1'\otimes T_1' = T_2}{F}{\Str{\nat}}{A}{\Str{\nat}}{\\ \res\ltag = F \wedge \res\rtag = A}$ | |
| \item $\jrhol{F,A, T_1', T_2: \Str{\nat}}{\Psi,tl\ F = \later T_1 , tl\ T_1 = \later T_1', T_1'\otimes T_1' = T_2}{T_1'}{\Str{\nat}}{T_2}{\Str{\nat}} | |
| {\\ F = \cons{1}{\later(\cons{1}{\later T_1})} \wedge \res\ltag \otimes \res\rtag = \res\rtag}$ | |
| \item $\jrhol{F,A, T_1', T_2: \Str{\nat}}{\Psi,tl\ F = \later T_1 , tl\ T_1 = \later T_1', T_1'\otimes T_1' = T_2} | |
| {\\ \otimes}{\Str{\nat} \to \Str{\nat} \to \Str{\nat}}{\oplus}{\Str{\nat} \to \Str{\nat} \to \Str{\nat}} | |
| {\forall X_1 X_2 Y_1 Y_2. X_1 = F \wedge X_2 = A \Rightarrow F = \cons{1}{\later(\cons{1}{\later Y_1})} \wedge Y_1 \otimes Y_1 = Y_2 \Rightarrow \res\ltag\ X_1\ Y_1 = \res\rtag\ X_2\ Y_2}$ | |
| \end{itemize} | |
| The two first premises are easy to prove. We will show how to prove the last one. | |
| For this, we need a stronger induction hypothesis for $\hat{\oplus}$ and | |
| $\hat{\otimes}$. We propose the following: | |
| \[\begin{array}{c} | |
| \forall g_1,g_2,b_1,G,B. G = \conshat{g_1}{\conshat{g_2}{(G\hat{\oplus} (\hat{tl} G))}} \wedge b_1 = g_1^2 + g_1 g_2 - g_2^2 \wedge B = \conshat{b_1}{\conshat{-b_1}{B}} | |
| \\ \Rightarrow G \hat{\otimes} \hat{tl}(\hat{tl} G) = \hat{tl} (G \hat{\otimes} G) \hat{\oplus} B | |
| \end{array} | |
| \] | |
| We then use the \rname{SUB} rule to strengthen the inductive hypothesis, and now the new judgement to prove is: | |
| \[\begin{array}{c} | |
| \jrhol{F,A, T_1', T_2: \Str{\nat}}{\Psi,tl\ F = \later T_1 , tl\ T_1 = \later T_1', T_1'\otimes T_1' = T_2} | |
| {\\ \otimes}{\Str{\nat} \to \Str{\nat} \to \Str{\nat}}{\oplus}{\Str{\nat} \to \Str{\nat} \to \Str{\nat}} | |
| {\\ \forall X_1 X_2 Y_1 Y_2. (\exists g_1, g_2, b_1, G, B. G = \cons{g_1}{\later(\cons{g_2}{\later [ G' \ot tl\ G ] ( G \oplus G')})} | |
| \wedge b_1 = g_1^2 + g_1 g_2 - g_2^2 \wedge \\ B = \cons{b_1}{\later(\cons{-b_1}{\later B})} \wedge | |
| X_1 = G \wedge X_2 = B \wedge X_1 = \cons{1}{\later(\cons{1}{\later Y_1})} \wedge Y_1 \otimes Y_1 = Y_2) \Rightarrow \res\ltag\ X_1\ Y_1 = \res\rtag\ X_2\ Y_2} | |
| \end{array} | |
| \] | |
| Let $\Gamma'$, $\Psi'$ and $\Phi_{IH}$ denote respectively the typing context, logical context and logical conclusion of the previous judgement. The premise | |
| of the FIX rule is: | |
| \[\begin{array}{c} | |
| \jrhol{\Gamma; f_1,f_2 : \later(\Str{\nat} \to \Str{\nat} \to \Str{\nat})}{\Psi',\later[\res\ltag \ot f_1, \res\rtag \ot f_2] \Phi_{IH}} | |
| {\\ \fix{f_1}{\lambda X_1. \lambda Y_1. \dots}}{\Str{\nat} \to \Str{\nat} \to \Str{\nat}}{\fix{f_2}{\lambda X_2. \lambda Y_2. \dots}}{\Str{\nat} \to \Str{\nat} \to \Str{\nat}}{\Phi_{IH}} | |
| \end{array} | |
| \] | |
| Let $\Phi_E$ denote the existential clause in $\Phi_{IH}$. After applying \rname{ABS} twice, we have: | |
| \[\begin{array}{c} | |
| \jrhol{\Gamma; f_1,f_2 : \later(\Str{\nat} \to \Str{\nat} \to \Str{\nat}); X_1,X_2,Y_1,Y_2 : \Str{\nat}}{\Psi',\later[\res\ltag \ot f_1, \res\rtag \ot f_2] \Phi_{IH}, \Phi_E} | |
| {\\ \cons{(hd\ X_1)*(hd\ Y_1)}{f_1 \app (tl\ X_1) \app (tl\ Y_1)}}{\Str{\nat}}{\cons{(hd\ X_2)+(hd\ Y_2)}{f_2 \app (tl\ X_2) \app (tl\ Y_2)}}{\Str{\nat}}{\res\ltag = \res\rtag} | |
| \end{array}\] | |
| And then we apply \rname{Cons} to prove equality on the heads and the tails: | |
| \begin{itemize} | |
| \item $\jrhol{\Gamma; f_1,f_2 : \later(\Str{\nat} \to \Str{\nat} \to \Str{\nat}); X_1,X_2,Y_1,Y_2 : \Str{\nat}}{\Psi',\later[\res\ltag \ot f_1, \res\rtag \ot f_2] \Phi_{IH}, \Phi_E} | |
| {\\ (hd\ X_1)*(hd\ Y_1)}{\nat}{(hd\ X_2)+(hd\ Y_2)}{\nat}{\res\ltag = \res\rtag}$ | |
| \item $\jrhol{\Gamma; f_1,f_2 : \later(\Str{\nat} \to \Str{\nat} \to \Str{\nat}); X_1,X_2,Y_1,Y_2 : \Str{\nat}}{\Psi',\later[\res\ltag \ot f_1, \res\rtag \ot f_2] \Phi_{IH}, \Phi_E} | |
| {\\ f_1 \app (tl\ X_1) \app (tl\ Y_1)}{\later \Str{\nat}}{f_2 \app (tl\ X_2) \app (tl\ Y_2)}{\later \Str{\nat}}{\res\ltag = \res\rtag}$ | |
| \end{itemize} | |
| To prove the first one we notice that $hd X_1 * hd Y_1 = g_1 * (g_1 + g_2) = g_1^2 + g_2*g_1 = g_2^2 + g_1^2 + g_1 * g_2 - g_2^2 = hd X_2 * hd Y_2$. To prove the second one | |
| we need to check that $tl X_1,tl Y_1, tl X_2,tl Y_2$ satisfy the precondition of the inductive hypothesis. In particular, we need to check that | |
| \[-b_1 = -g_1^2 - g_1 g_2 + g_2^2 = g_2^2 + g_2 (g_1 + g_2) - (g_1 + g_2)^2\] | |
| which is can be proven by arithmetic computation. | |
| \section{Unary fragment} | |
| In this section we introduce a unary system to prove properties | |
| about a single term of the guarded lambda calculus. We will start | |
| by adding some definitions Guarded HOL for the unary diamond monad, | |
| following by the derivation rules for both the non-probabilistic | |
| and the probabilistic system, plus the metatheory and an example. | |
| \subsection{Unary fragment of GHOL} | |
| The unary semantics of the diamond monad are: | |
| \[\sem{\diamond_{[x\leftarrow t]}\phi}_i\defeq \left\{(\delta,\gamma) \isetsep | |
| \operatorname{Pr}_{v\ot \left(\sem{t}_i(\delta,\gamma)\right)} [(\delta, (\gamma, v))\in \sem{\phi}_i] = 1 \right\} \] | |
| The rules are on Figure~\ref{fig:u-prob-ghol} | |
| \begin{figure*}[!htb] | |
| \small | |
| \begin{mathpar} | |
| \infer[\sf MONO1] | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\diamond_{[x\gets t]}\psi}} | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\diamond_{[x\gets t]}\phi} \and | |
| \jghol{\Delta}{\Sigma}{\Gamma, x : C}{\Psi, \phi}{\psi} | |
| } | |
| \and | |
| \infer[\sf UNIT1]{\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{ | |
| \diamond_{[x\leftarrow \munit{t}]} \phi}} | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{ | |
| \phi[t/x]}} | |
| \and | |
| \infer[\sf MLET1] | |
| {\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{ | |
| \diamond_{[y\leftarrow \mlet{x}{t}{t'}]} \psi}}{ | |
| \jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\diamond_{[x\leftarrow t]} \phi} | |
| \and | |
| \jghol{\Delta}{\Sigma}{\Gamma,x:C}{\Psi,\phi}{ | |
| \diamond_{[y\leftarrow t']}\psi}} | |
| \end{mathpar} | |
| \caption{Rules for the unary diamond modality}\label{fig:u-prob-ghol} | |
| \end{figure*} | |
| \subsection{Guarded UHOL} | |
| We start by defining the Guarded UHOL system, which allows us to prove | |
| logical properties of a term of the Guarded Lambda Calculus. More | |
| concretely, judgements have the form: | |
| \[ \jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t}{\sigma}{\phi} \] | |
| where $t$ is a term well-typed in the dual context $\Delta \mid \Gamma$ and $\phi$ is a logical formula well-typed in the context | |
| $\Delta \mid \Gamma, \res : \sigma$ and that can refer to $t$ via the special variable $\res$. The logical contexts $\Sigma$ and | |
| $\Psi$ consist respectively of refinements over the contexts $\Delta$ and $\Gamma$. | |
| \subsection{Derivation rules} | |
| The rule \rname{Next} corresponds to the introduction of the later | |
| modality. A refinement $\Phi_i$ is proven on every term in the | |
| substitution, and using those as a premise, a refinement $\Phi$ is | |
| proven on $t$. In the notation $\later[\res \ot \res]\Phi$ the first | |
| $\res$ is the variable bound by the delayed substitution inside $\Phi$ | |
| while the second $\res$ is the distinguished variable in the refinement | |
| that refers to the term that is being typed. In other words, $t$ satisfies | |
| $\later[\res \ot \res]\Phi$ if $\later[\res \ot t]\Phi$. | |
| The rule \rname{Prev} corresponds to the elimination of the later modality. | |
| If we can prove $\later \phi$ in a constant context, then we can also prove | |
| $\phi$. | |
| The rule \rname{Box} applies the constant modality on a formula that can | |
| be proven on a constant context. | |
| The rule \rname{LetBox} removes the constant modality from a formula $\Phi$ | |
| by using it as a constant premise to prove another formula $\Phi'$. | |
| The rule \rname{LetConst} shifts constant terms between contexts. | |
| The rule \rname{Fix} introduces a fixpoint and proves a refinement on it by | |
| Loeb induction. | |
| The rule \rname{Cons} proves a property on a stream from a refinement on its | |
| head and its tail. | |
| The rule \rname{ConsHat} is the analogue of \rname{Cons} to build constant streams. | |
| In particular, the $\hat{::}$ operator can be defined as $\lambda x. \lambda s. \letbox{(y,t)}{(x,s)}{\boxx(\cons{y}{\later{t}})}$. | |
| Conversely the rules \rname{Head} and \rname{Tail} respectively prove a property | |
| on the head and the tail of a stream from a property on the full stream. | |
| \begin{figure*}[!htb] | |
| \small | |
| \begin{mathpar} | |
| \inferrule*[Right=\sf Next] | |
| {\jguhol{\Delta}{\Sigma}{\Gamma, x_1:A_1, \dots, x_n:A_n}{\Psi, \Phi_1\subst{\res}{x_1}, \dots, \Phi_n\subst{\res}{x_n}}{t}{A}{\Phi} \\ | |
| \jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{\later{A_1}}{\dsubst{\res}{\res}{\Phi_1}} | |
| \\ \dots \\ | |
| \jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_n}{\later{A_n}}{\dsubst{\res}{\res}{\Phi_n}}} | |
| {\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{\nextt{x_1 \leftarrow t_1, \dots, x_n \leftarrow t_n}{t}}{\later{A}}{\dsubst{x_1,\dots,x_n,\res}{t_1,\dots,t_n,\res}{\Phi}}} | |
| \inferrule*[Right=\sf Prev] | |
| {\jguhol{\Delta}{\Sigma}{\cdot}{\cdot}{t}{\later A}{\dsubst{\res}{\res}{\Phi}}} | |
| {\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{\prev{t}}{A}{\Phi}} | |
| \inferrule*[Right=\sf Box] | |
| {\jguhol{\Delta}{\Sigma}{\cdot}{\cdot}{t}{A}{\Phi}} | |
| {\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{\boxx{t}}{\square A}{\square \Phi\subst{\res}{\letbox{x}{\res}{x}}}} | |
| \inferrule*[Right=\sf LetBox] | |
| {\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{u}{\square B}{\square \Phi\subst{\res}{\letbox{x}{\res}{x}}} \\ | |
| \jguhol{\Delta, x : B}{\Sigma, \Phi\subst{\res}{x}}{\Gamma}{\Psi}{t}{A}{\Phi'}} | |
| {\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{\letbox{x}{u}{t}}{A}{\Phi'}} | |
| \inferrule*[Right=\sf LetConst] | |
| {\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{u}{B}{\Phi} \\ | |
| \jguhol{\Delta, x : B}{\Sigma, \Phi\subst{\res}{x}}{\Gamma}{\Psi}{t}{A}{\Phi'} \\ | |
| B,\Phi\ \text{constant} \\ FV(\Phi)\cap FV(\Gamma) = \emptyset} | |
| {\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{\letconst{x}{u}{t}}{A}{\Phi'}} | |
| \inferrule*[Right=\sf Fix] | |
| {\jguhol{\Delta}{\Sigma}{\Gamma, f:\later{A}}{\dsubst{\res}{f}{\Phi}}{t}{A}{\Phi}} | |
| {\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{fix f. t}{A}{\Phi}} | |
| \inferrule*[Right=\sf Cons] | |
| {\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{x}{A}{\Phi_h} \\ | |
| \jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{xs}{\later{\Str{A}}}{\Phi_t} \\ | |
| \jhol{\Gamma}{\Psi}{\forall x,xs. \Phi_h\subst{\res}{x} \Rightarrow \Phi_t\subst{\res}{xs} \Rightarrow \Phi\subst{\res}{\cons{x}{xs}}}} | |
| {\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{\cons{x}{xs}}{\Str{A}}{\Phi}} | |
| \inferrule*[Right=\sf ConsHat] | |
| {\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{x}{A}{\Phi_h} \\ | |
| \jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{xs}{\square \Str{A}}{\square \Phi_t} \\ | |
| \jhol{\Gamma}{\Psi}{\forall x,xs. \Phi_h\subst{\res}{x} \Rightarrow \Phi_t\subst{\res}{xs} \Rightarrow \Phi\subst{\res}{\conshat{x}{xs}}} \\ | |
| A,\Phi_h \text{ constant} } | |
| {\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{\conshat{x}{xs}}{\square \Str{A}}{\square \Phi}} | |
| \inferrule*[Right=\sf Head] | |
| {\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t}{\Str{A}}{\Phi\subst{\res}{hd\ \res}}} | |
| {\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{hd\ t}{A}{\Phi}} | |
| \inferrule*[Right=\sf Tail ] | |
| {\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t}{\Str{A}}{\Phi\subst{\res}{tl\ \res}}} | |
| {\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{tl\ t}{\later{Str_A}}{\Phi}} | |
| \end{mathpar} | |
| \caption{Guarded Unary Higher-Order Logic rules}\label{fig:uhol} | |
| \end{figure*} | |
| The intended meaning for a judgment $\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t}{\tau}{\phi}$ is: | |
| ``For every valuations $\delta$, $\gamma$ of $\Delta$ and $\Gamma$, | |
| \[\sem{\Delta\mid\Gamma\vdash \square\Sigma}(\delta,\gamma) \wedge \sem{\Delta\mid\Gamma\vdash \Psi}(\delta,\gamma) \To | |
| \sem{\Delta\mid\Gamma, \res:\tau\vdash \Sigma}(\delta,\langle \gamma, \sem{\Delta\mid\Gamma\vdash t}(\delta,\gamma)\rangle) \text{''} \] | |
| \subsection{Metatheory} | |
| We now the most interesting metatheoretical properties of Guarded UHOL. In particular, Guarded UHOL | |
| is equivalent to Guarded HOL: | |
| \begin{theorem}[Equivalence with Guarded HOL] \label{thm:equiv-uhol-hol} | |
| For every contexts $\Delta,\Gamma$, type $\sigma$, term $t$, sets of assertions $\Sigma,\Psi$ | |
| and assertion $\phi$, the following are equivalent: | |
| \begin{itemize} | |
| \item | |
| $\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t}{\sigma}{\phi}$ | |
| \item | |
| $\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{\phi\subst{\res}{t}}$ | |
| \end{itemize} | |
| \end{theorem} | |
| The proof is analogous to the relational case | |
| The previous result allows us to lift the soundness result from Guarded HOL to | |
| Guarded UHOL. | |
| \begin{corollary}[Soundness and consistency]\label{cor:uhol:sound} | |
| If $\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t}{\sigma}{\phi}$, then for every valuations | |
| $\delta \models \Delta$, $\gamma\models\Gamma$: | |
| \[\sem{\Delta \vdash \Sigma}(\delta) \wedge \sem{\Delta \mid \Gamma \vdash \Psi}(\delta,\gamma) \Rightarrow | |
| \sem{\Delta \mid \Gamma,\res:\sigma \vdash \phi}(\delta, \gamma[\res \ot \sem{\Delta \mid \Gamma \vdash t}(\delta,\gamma)])\] | |
| In particular, there is no proof of | |
| $\jguhol{\Delta}{\emptyset}{\Gamma}{\emptyset}{t}{\sigma}{\bot}$ in Guarded UHOL. | |
| \end{corollary} | |
| \subsection{Probabilistic extension} | |
| We comment on the rules, starting from the rules of the unary | |
| logic. There are three new rules for the probabilistic case, and they | |
| all establish that an expression $u$ of type $\Distr(D)$ satisfies the | |
| assertion $\diamond_{[y\leftarrow \res]} \phi$, i.e.\, for every | |
| element $v$ in the support of (the interpretation of) $u$, the | |
| interpretation of $\phi$ with the valuation $[y \mapsto v]$ is | |
| true. This intuition is captured by the rule $[\textsf{SUPP}]$, which | |
| can be used in particular in case $u$ is a primitive distribution. The | |
| rule [\textsf{UNIT}] considers the case where $u$ is of the form | |
| $\munit{t}$; in this case, it is clearly sufficient to know that | |
| $\phi[t/y]$ is valid. The rule [\textsf{MLET}] simply captures the | |
| fact that the support of $\mlet{x}{t}{t'}$ is the disjoint union of | |
| the support of $t'$ under all the assignments of $x$ to values in the | |
| support of $t$. | |
| \begin{figure*}[!htb] | |
| \small | |
| \textbf{Guarded UHOL} | |
| \begin{mathpar} | |
| \infer[\sf UNIT] | |
| {\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{\munit{t}}{\Distr(C)}{ | |
| \diamond_{[y\leftarrow \res]} \Phi}} | |
| {\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t}{C}{\Phi | |
| [\res/y]}} | |
| \and | |
| \infer[\sf MLET] | |
| {\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{\mlet{x}{t}{t'}}{ | |
| \Distr(D)}{\diamond_{[y\leftarrow \res]} \psi}} | |
| {\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t}{\Distr(C)}{\diamond_{[x\leftarrow r]} \Phi} | |
| \and | |
| \jguhol{\Delta}{\Sigma}{\Gamma,x:C}{\Psi,\phi | |
| }{t'}{\Distr(D)}{\diamond_{[y\leftarrow \res]} \psi}} | |
| \infer[\sf SUPP]{\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{u}{\Distr(D)}{ | |
| \diamond_{[y\leftarrow \res]}\phi}}{\jghol{\Delta}{\Sigma}{\Gamma}{\Psi}{ | |
| \Pr_{z\sim u}[\phi[z/y]]=1}} | |
| \end{mathpar} | |
| \caption{Proof rules for probabilistic constructs -- unary case}\label{fig:puhol} | |
| \end{figure*} | |
| Finally, we prove an embedding lemma for Guarded UHOL. The proof can be | |
| carried by induction on the structure of derivations, or using the | |
| equivalence between Guarded UHOL and Guarded HOL (Theorem~\ref{thm:equiv-uhol-hol}). | |
| \begin{lem}[Embedding lemma]\label{lem:emb-uhol-rhol} Assume that: | |
| \begin{itemize} | |
| \item $\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{\sigma_1}{\phi}$ | |
| \item $\jguhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_2}{\sigma_2}{\phi'}$ | |
| \end{itemize} | |
| Then | |
| $\jgrhol{\Delta}{\Sigma}{\Gamma}{\Psi}{t_1}{\sigma_1}{t_2}{\sigma_2}{ | |
| \phi\subst{\res}{\res\ltag}\land \phi'\subst{\res}{\res\rtag}}$. | |
| \end{lem} | |
| \subsection{Unary example: Every two} | |
| We define the $every2$ function, which receives a stream and returns another stream consisting | |
| of the elements at even positions in the input stream. Note that this function, while productive, | |
| cannot be built with the type $Str \to Str$, since we need to take twice the tail of the argument, | |
| which would have type $\later\later Str$, and then a $Str$ cannot be built. Instead, we need to use the constant modality | |
| as follows: | |
| \begin{align*} | |
| every2 &: \square Str \to Str \\ | |
| every2 &\defeq \fix{every2}{\lambda s. \cons{\hat{hd}(\hat{tl}\ s)}{(every2 \app {\rm next} (\hat{tl}(\hat{tl}\ s)))}} | |
| \end{align*} | |
| Where the $\hat{hd}$ and $\hat{tl}$ functions are not the native ones, but rather they are defined as: | |
| \begin{align*} | |
| \hat{hd} &: \square Str \to \nat &\hat{tl} &: \square Str \to \square Str \\ | |
| \hat{hd} &\defeq \lambda s. \letbox{x}{s}{hd\ x} \quad\quad &\hat{tl} &\defeq \lambda s. \letbox{x}{s}{\boxx{(\prev{(tl\ x)})}} | |
| \end{align*} | |
| The property we want to prove is: | |
| \[ | |
| \jguhol{\cdot}{\cdot}{ones : \square Str}{\Psi}{every2}{\square Str \to Str}{\forall s. s = ones \Rightarrow \res\ s = (\letbox{x}{s}{x})} | |
| \] | |
| where $ones$ is the constant stream containing only the number 1 defined as: | |
| \[ | |
| ones \defeq \boxx{(\fix{f}{\cons{1}{f}})} | |
| \] | |
| For which we can prove the following properties: | |
| \[ | |
| \Psi \defeq \hat{hd}\ ones = 1, \hat{tl}\ ones = ones | |
| \] | |
| In the rest of the proof we omit the empty contexts $\Delta$ and $\Sigma$. | |
| We start by applying the \rname{Fix} rule, which has the premise: | |
| \[ | |
| \begin{array}{c} | |
| \juhol{ones : \square Str, every2:\later{(\square Str \to Str)}}{\Psi, \later[\res \ot every2]{\forall s. s = ones \Rightarrow \res\ s = \letbox{x}{s}{x}}} | |
| {\\ \lambda s. (\cdots)}{\square Str \to Str}{\forall s. s = ones \Rightarrow (\res\ s) = \letbox{x}{s}{x}} | |
| \end{array} | |
| \] | |
| We apply the \rname{Abs} rule inmediately after: | |
| \[ | |
| \begin{array}{c} | |
| \juhol{ones : \square Str, every2:\later{(\square Str \to Str)}, s: \square Str} | |
| {\\ \Psi, \later[\res \ot every2]{\forall s. s = ones \Rightarrow \res\ s = \letbox{x}{s}{x}}, s = ones} | |
| {\\ \cons{\hat{hd}(\hat{tl}\ s)}{(every2 \app \later (\hat{tl}(\hat{tl}\ s)))}}{Str}{\res = \letbox{x}{s}{x}} | |
| \end{array} | |
| \] | |
| By \rname{SUB} and the equivalence $\letbox{x}{s}{x} \equiv \letbox{x}{ones}{x}$, we can change the | |
| conclusion of the judgement. | |
| Now we use the \rname{Cons} rule, which has three premises: | |
| \begin{enumerate} | |
| \item $\juhol{ones : \square Str, every2:\later{(\square Str \to Str)}, s: \square Str} | |
| {\\ \Psi, \later[\res \ot every2]{\forall s. s = ones \Rightarrow \res\ s = \letbox{x}{s}{x}}, s = ones} | |
| { \hat{hd}(\hat{tl}\ s)}{\nat}{\res = 1}$ | |
| \\ | |
| \item $\juhol{ones : \square Str, every2:\later{(\square Str \to Str)}, s: \square Str} | |
| {\\ \Psi, \later[\res \ot every2]{\forall s. s = ones \Rightarrow \res\ s = \letbox{x}{s}{x}}, s = ones} | |
| {\\(every2 \app \later(\hat{tl}(\hat{tl}\ s)))}{Str}{\dsubst{\res}{\res}{\res = \letbox{x}{ones}{x}}}$ | |
| \\ | |
| \item $\jhol{ones : \square Str, every2:\later{(\square Str \to Str)}, s: \square Str} | |
| {\\ \Psi, \dsubst{\res}{every2}{\forall s. s = ones \Rightarrow \res\ s = \letbox{x}{s}{x}}, s = ones} | |
| {\\ \forall y, ys. y = 1 \Rightarrow \later[zs \ot ys]{(zs = \letbox{x}{ones}{x})} \Rightarrow \cons{y}{ys} = (\letbox{x}{ones}{x})}$ | |
| \end{enumerate} | |
| Premises (1) is a consequence of the properties of $ones$. To prove premise (3) | |
| we reduce the letbox with the box inside $ones$, and do some reasoning using the definition of the fixpoint. To prove the | |
| premise (2) we first desugar the term we are typing: | |
| \[every2 \app \later(\hat{tl}(\hat{tl}\ s))) \defeq \later[g \leftarrow every2, t \leftarrow \later(\hat{tl}(\hat{tl}\ s))]{g t} \] | |
| and then we apply \rname{Next} which has the following premises: | |
| \begin{itemize} | |
| \item $\juhol{ones : \square Str, every2:\later{(\square Str \to Str)}, s: \square Str} | |
| {\\ \Psi, \dsubst{\res}{every2}{\forall s. s = ones \Rightarrow \res\ s = \letbox{x}{s}{x}}, s = ones} | |
| {\\ every2}{\later{(\square Str \to Str)}}{\dsubst{\res}{\res}{(\forall s = ones \Rightarrow \res\ s = \letbox{x}{s}{x})}}$ | |
| \\ | |
| \item $\juhol{ones : \square Str, every2:\later{(\square Str \to Str)}, s: \square Str} | |
| {\\ \Psi, \dsubst{\res}{every2}{(\forall s. s = ones \Rightarrow \res\ s = \letbox{x}{s}{x})}, s = ones} | |
| {\\ \later(\hat{tl}(\hat{tl}\ s))}{\later\square Str}{\dsubst{\res}{\res}{(\res = ones)}}$ | |
| \\ | |
| \item $\juhol{ones : \square Str, every2:\later{(\square Str \to Str)}, s: \square Str, g: \square Str \to Str, t: \square Str} | |
| {\\ \Psi, \dsubst{\res}{every2}{(\forall s. s = ones \Rightarrow \res\ s = \letbox{x}{s}{x})}, s = ones, | |
| \\ \forall s. s = ones \Rightarrow g\ s = \letbox{x}{s}{x}, t = ones} | |
| {g\ t}{Str}{\res = (\letbox{x}{ones}{x})}$ | |
| \end{itemize} | |
| The first premise is just an application of the \rname{Var} rule. The | |
| second premise can be proven as a consequence of the properties of | |
| $ones$. Finally, the third premise can be proven with some simple | |
| logical reasoning in HOL. This concludes the proof. | |
| \end{document} | |