diff --git "a/papers/1511/1511.09230.tex" "b/papers/1511/1511.09230.tex" new file mode 100644--- /dev/null +++ "b/papers/1511/1511.09230.tex" @@ -0,0 +1,3810 @@ +\documentclass[a4paper,UKenglish]{lipics} + + +\usepackage{microtype} + +\usepackage{stmaryrd} +\usepackage{bussproofs} +\usepackage{xspace} +\usepackage{mdframed} +\usepackage{fancybox} +\usepackage{nicefrac} +\usepackage{xypic} +\xyoption{tips} +\SelectTips{cm}{} \usepackage{multicol} +\usepackage{booktabs} + + + +\bibliographystyle{plain} + +\title{A Type Theory for Probabilistic and Bayesian Reasoning\footnote{This work was supported by ERC Advanced Grant QCLS: Quantum Computation, Logic and Security.}} + + +\author[1]{Robin Adams} +\author[1]{Bart Jacobs} +\affil[1]{Institute for Computing and Information Sciences,\\ + Radboud University, the Netherlands\\ + \texttt{\{r.adams,bart\}@cs.ru.nl}} +\authorrunning{R. Adams and B. Jacobs} + +\Copyright{Robin Adams and Bart Jacobs} + +\subjclass{F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logic --- Lambda calculus and related systems; G.3 [Probability and Statistics]: Probabilistic algorithms; F.3.1 [Logics and Meanings of Programs]: Specifying and Verifying and Reasoning about Programs}\keywords{Probability theory, type theory, effect module, Bayesian reasoning} + +\serieslogo{}\volumeinfo {Billy Editor and Bill Editors}{2}{Conference title on which this volume is based on}{1}{1}{1}\EventShortName{} +\DOI{10.4230/LIPIcs.xxx.yyy.p} + +\newcommand{\COMETgrammar}{ +} +\newcommand{\Rvar}{ + \text{(var)\xspace} +} +\newcommand{\Tvar}{ + \TBvar + \DisplayProof +} +\newcommand{\TTvar}{ + \begin{prooftree} + \TBvar + \end{prooftree} +} +\newcommand{\TBvar}{ + \LeftLabel{\Rvar} + \AxiomC{$ x : A \in \Gamma$} + \UnaryInfC{$\Gamma \vdash x : A$} +} +\newcommand{\Rexch}{ + \text{(exch)\xspace} +} +\newcommand{\Texch}{ + \TBexch + \DisplayProof +} +\newcommand{\TTexch}{ + \begin{prooftree} + \TBexch + \end{prooftree} +} +\newcommand{\TBexch}{ + \LeftLabel{\Rexch} + \AxiomC{$\Gamma, x : A, y : B, \Delta \vdash \mathcal{J}$} + \UnaryInfC{$\Gamma, y : B, x : A, \Delta \vdash \mathcal{J}$} +} +\newcommand{\Rref}{ + \text{(ref)\xspace} +} +\newcommand{\Tref}{ + \TBref + \DisplayProof +} +\newcommand{\TTref}{ + \begin{prooftree} + \TBref + \end{prooftree} +} +\newcommand{\TBref}{ + \LeftLabel{\Rref} + \AxiomC{$\Gamma \vdash t : A$} + \UnaryInfC{$\Gamma \vdash t = t : A$} +} +\newcommand{\Rsym}{ + \text{(sym)\xspace} +} +\newcommand{\Tsym}{ + \TBsym + \DisplayProof +} +\newcommand{\TTsym}{ + \begin{prooftree} + \TBsym + \end{prooftree} +} +\newcommand{\TBsym}{ + \LeftLabel{\Rsym} + \AxiomC{$\Gamma \vdash s = t : A$} + \UnaryInfC{$\Gamma \vdash t = s : A$} +} +\newcommand{\Rtrans}{ + \text{(trans)\xspace} +} +\newcommand{\Ttrans}{ + \TBtrans + \DisplayProof +} +\newcommand{\TTtrans}{ + \begin{prooftree} + \TBtrans + \end{prooftree} +} +\newcommand{\TBtrans}{ + \LeftLabel{\Rtrans} + \AxiomC{$\Gamma \vdash r = s : A$} + \AxiomC{$\Gamma \vdash s = t : A$} + \BinaryInfC{$\Gamma \vdash r = t : A$} +} +\newcommand{\Rmagic}{ + \text{(magic)\xspace} +} +\newcommand{\Tmagic}{ + \TBmagic + \DisplayProof +} +\newcommand{\TTmagic}{ + \begin{prooftree} + \TBmagic + \end{prooftree} +} +\newcommand{\TBmagic}{ + \LeftLabel{\Rmagic} + \AxiomC{$\Gamma \vdash t : 0$} + \UnaryInfC{$\Gamma \vdash \magic{t} : A$} +} +\newcommand{\Retazero}{ + \text{($\eta 0$)\xspace} +} +\newcommand{\Tetazero}{ + \TBetazero + \DisplayProof +} +\newcommand{\TTetazero}{ + \begin{prooftree} + \TBetazero + \end{prooftree} +} +\newcommand{\TBetazero}{ + \LeftLabel{\Retazero} + \AxiomC{$\Gamma \vdash s : 0$} + \AxiomC{$\Gamma \vdash t : A$} + \BinaryInfC{$\Gamma \vdash \magic{s} = t : A$} +} +\newcommand{\Runit}{ + \text{(unit)\xspace} +} +\newcommand{\Tunit}{ + \TBunit + \DisplayProof +} +\newcommand{\TTunit}{ + \begin{prooftree} + \TBunit + \end{prooftree} +} +\newcommand{\TBunit}{ + \LeftLabel{\Runit} + \AxiomC{$$} + \UnaryInfC{$\Gamma \vdash * : 1$} +} +\newcommand{\Retaone}{ + \text{($\eta 1$)\xspace} +} +\newcommand{\Tetaone}{ + \TBetaone + \DisplayProof +} +\newcommand{\TTetaone}{ + \begin{prooftree} + \TBetaone + \end{prooftree} +} +\newcommand{\TBetaone}{ + \LeftLabel{\Retaone} + \AxiomC{$\Gamma \vdash t : 1$} + \UnaryInfC{$\Gamma \vdash t = * : 1$} +} +\newcommand{\Rinl}{ + \text{(inl)\xspace} +} +\newcommand{\Tinl}{ + \TBinl + \DisplayProof +} +\newcommand{\TTinl}{ + \begin{prooftree} + \TBinl + \end{prooftree} +} +\newcommand{\TBinl}{ + \LeftLabel{\Rinl} + \AxiomC{$\Gamma \vdash t : A$} + \UnaryInfC{$\Gamma \vdash \inl{t} : A + B$} +} +\newcommand{\Rinleq}{ + \text{(inl-eq)\xspace} +} +\newcommand{\Tinleq}{ + \TBinleq + \DisplayProof +} +\newcommand{\TTinleq}{ + \begin{prooftree} + \TBinleq + \end{prooftree} +} +\newcommand{\TBinleq}{ + \LeftLabel{\Rinleq} + \AxiomC{$\Gamma \vdash t = t' : A$} + \UnaryInfC{$\Gamma \vdash \inl{t} = \inl{t'} : A + B$} +} +\newcommand{\Rinr}{ + \text{(inr)\xspace} +} +\newcommand{\Tinr}{ + \TBinr + \DisplayProof +} +\newcommand{\TTinr}{ + \begin{prooftree} + \TBinr + \end{prooftree} +} +\newcommand{\TBinr}{ + \LeftLabel{\Rinr} + \AxiomC{$\Gamma \vdash t : B$} + \UnaryInfC{$\Gamma \vdash \inr{t} : A + B$} +} +\newcommand{\Rinreq}{ + \text{(inr-eq)\xspace} +} +\newcommand{\Tinreq}{ + \TBinreq + \DisplayProof +} +\newcommand{\TTinreq}{ + \begin{prooftree} + \TBinreq + \end{prooftree} +} +\newcommand{\TBinreq}{ + \LeftLabel{\Rinreq} + \AxiomC{$\Gamma \vdash t = t' : B$} + \UnaryInfC{$\Gamma \vdash \inr{t} = \inr{t'} : A + B$} +} +\newcommand{\Rcase}{ + \text{(case)\xspace} +} +\newcommand{\Tcase}{ + \TBcase + \DisplayProof +} +\newcommand{\TTcase}{ + \begin{prooftree} + \TBcase + \end{prooftree} +} +\newcommand{\TBcase}{ + \LeftLabel{\Rcase} + \AxiomC{$\Gamma \vdash r : A + B$} + \AxiomC{$\Delta, x : A \vdash s : C$} + \AxiomC{$\Delta, y : B \vdash t : C$} + \TrinaryInfC{$\Gamma, \Delta \vdash \pcase{r}{x}{s}{y}{t} : C$} +} +\newcommand{\Rcaseeq}{ + \text{(case-eq)\xspace} +} +\newcommand{\Tcaseeq}{ + \TBcaseeq + \DisplayProof +} +\newcommand{\TTcaseeq}{ + \begin{prooftree} + \TBcaseeq + \end{prooftree} +} +\newcommand{\TBcaseeq}{ + \LeftLabel{\Rcaseeq} + \AxiomC{$\Gamma \vdash r = r' : A + B$} + \AxiomC{$\Delta, x : A \vdash s = s' : C$} + \AxiomC{$\Delta, y : B \vdash t = t': C$} + \TrinaryInfC{$\Gamma, \Delta \vdash \pcase{r}{x}{s}{y}{t} = \pcase{r'}{x}{s'}{y}{t'} : C$} +} +\newcommand{\Rbetaplustwo}{ + \text{($\beta+_2$)\xspace} +} +\newcommand{\Tbetaplustwo}{ + \TBbetaplustwo + \DisplayProof +} +\newcommand{\TTbetaplustwo}{ + \begin{prooftree} + \TBbetaplustwo + \end{prooftree} +} +\newcommand{\TBbetaplustwo}{ + \LeftLabel{\Rbetaplustwo} + \AxiomC{$\Gamma \vdash r : B$} + \AxiomC{$\Delta, x : A \vdash s : C$} + \AxiomC{$\Delta, y : B \vdash t : C$} + \TrinaryInfC{$\Gamma, \Delta \vdash \pcase{\inr{r}}{x}{s}{y}{t} = t[y:=r] : C$} +} +\newcommand{\Rbetaplusone}{ + \text{($\beta+_1$)\xspace} +} +\newcommand{\Tbetaplusone}{ + \TBbetaplusone + \DisplayProof +} +\newcommand{\TTbetaplusone}{ + \begin{prooftree} + \TBbetaplusone + \end{prooftree} +} +\newcommand{\TBbetaplusone}{ + \LeftLabel{\Rbetaplusone} + \AxiomC{$\Gamma \vdash r : A$} + \AxiomC{$\Delta, x : A \vdash s : C$} + \AxiomC{$\Delta, y : B \vdash t : C$} + \TrinaryInfC{$\Gamma, \Delta \vdash \pcase{\inl{r}}{x}{s}{y}{t} = s[x:=r] : C$} +} +\newcommand{\Retaplus}{ + \text{($\eta+$)\xspace} +} +\newcommand{\Tetaplus}{ + \TBetaplus + \DisplayProof +} +\newcommand{\TTetaplus}{ + \begin{prooftree} + \TBetaplus + \end{prooftree} +} +\newcommand{\TBetaplus}{ + \LeftLabel{\Retaplus} + \AxiomC{$\Gamma \vdash t : A + B$} + \UnaryInfC{$\Gamma \vdash t = \pcase{t}{x}{\inl{x}}{y}{\inr{y}} : A + B$} +} +\newcommand{\Rcasecase}{ + \text{(case-case)\xspace} +} +\newcommand{\Tcasecase}{ + \TBcasecase + \DisplayProof +} +\newcommand{\TTcasecase}{ + \begin{prooftree} + \TBcasecase + \end{prooftree} +} +\newcommand{\TBcasecase}{ + \LeftLabel{\Rcasecase} + \AxiomC{ + $\begin{array}{ccc} + \Gamma \vdash r : A + B & \Delta, x : A \vdash s : C + D & \Delta, y : B \vdash s' : C + D\\ + \multicolumn{3}{c}{\Theta, z : C \vdash t : E \qquad \Theta, w : D \vdash t' : E} + \end{array}$ + } + \UnaryInfC{$\Gamma, \Delta, \Theta \vdash +\begin{array}[t]{l} +\case r \of \inl{x} \mapsto \pcase{s}{z}{t}{w}{t'} \mid \\ +\qquad \inr{y} \mapsto \pcase{s'}{z}{t}{w}{t'} \\ += \case (\pcase{r}{x}{s}{y}{s'}) \\ +\qquad \of \inl{z} \mapsto t \mid \inr{w} \mapsto t' : E +\end{array}$} +} +\newcommand{\Rcasepair}{ + \text{(case-$\sotimes$)\xspace} +} +\newcommand{\Tcasepair}{ + \TBcasepair + \DisplayProof +} +\newcommand{\TTcasepair}{ + \begin{prooftree} + \TBcasepair + \end{prooftree} +} +\newcommand{\TBcasepair}{ + \LeftLabel{\Rcasepair} + \AxiomC{$\Gamma \vdash r : A + B$} + \AxiomC{$\Delta, x : A \vdash s : C$} + \AxiomC{$\Delta, y : A \vdash s' : C$} + \AxiomC{$\Theta \vdash t : D$} + \QuaternaryInfC{$\Gamma, \Delta, \Theta \vdash +\begin{array}[t]{l} +(\pcase{r}{x}{s}{y}{s'}) \sotimes t = \\ +\pcase{r}{x}{s \sotimes t}{y}{s' \sotimes t} : D +\end{array}$} +} +\newcommand{\Rletcase}{ + \text{(let-case)\xspace} +} +\newcommand{\Tletcase}{ + \TBletcase + \DisplayProof +} +\newcommand{\TTletcase}{ + \begin{prooftree} + \TBletcase + \end{prooftree} +} +\newcommand{\TBletcase}{ + \LeftLabel{\Rletcase} + \AxiomC{ + $\begin{array}{cc} + \Gamma \vdash r : A + B & \Delta, z : A \vdash s : C \otimes D\\ + \Delta, w : B \vdash s' : C \otimes D & \Theta, x : C, y : D \vdash t : E + \end{array}$ + } + \UnaryInfC{$\Gamma, \Delta, \Theta \vdash +\begin{array}[t]{l} +\plet{x}{y}{\pcase{r}{z}{s}{w}{s'}}{t} = \\ +\pcase{r}{z}{\plet{x}{y}{s}{t}}{w}{\plet{x}{y}{s'}{t}} : E +\end{array}$} +} +\newcommand{\Rinlr}{ + \text{(inlr)\xspace} +} +\newcommand{\Tinlr}{ + \TBinlr + \DisplayProof +} +\newcommand{\TTinlr}{ + \begin{prooftree} + \TBinlr + \end{prooftree} +} +\newcommand{\TBinlr}{ + \LeftLabel{\Rinlr} + \AxiomC{$\Gamma \vdash s : A + 1$} + \AxiomC{$\Gamma \vdash t : B + 1$} + \AxiomC{$\Gamma \vdash s \downarrow = t \uparrow : \mathbf{2}$} + \TrinaryInfC{$\Gamma \vdash \inlr{s}{t} : A + B$} +} +\newcommand{\Rinlreq}{ + \text{(inlr-eq)\xspace} +} +\newcommand{\Tinlreq}{ + \TBinlreq + \DisplayProof +} +\newcommand{\TTinlreq}{ + \begin{prooftree} + \TBinlreq + \end{prooftree} +} +\newcommand{\TBinlreq}{ + \LeftLabel{\Rinlreq} + \AxiomC{$\Gamma \vdash s = s' : A + 1$} + \AxiomC{$\Gamma \vdash t = t' : B + 1$} + \AxiomC{$\Gamma \vdash s \downarrow = t \uparrow : \mathbf{2}$} + \TrinaryInfC{$\Gamma \vdash \inlr{s}{t} = \inlr{s'}{t'} : A + B$} +} +\newcommand{\Rbetainlrone}{ + \text{($\beta$inlr$_1$)\xspace} +} +\newcommand{\Tbetainlrone}{ + \TBbetainlrone + \DisplayProof +} +\newcommand{\TTbetainlrone}{ + \begin{prooftree} + \TBbetainlrone + \end{prooftree} +} +\newcommand{\TBbetainlrone}{ + \LeftLabel{\Rbetainlrone} + \AxiomC{$\Gamma \vdash s : A + 1$} + \AxiomC{$\Gamma \vdash t : B + 1$} + \AxiomC{$\Gamma \vdash s \downarrow = t \uparrow : \mathbf{2}$} + \TrinaryInfC{$\Gamma \vdash \rhd_1(\inlr{s}{t}) = s : A + 1$} +} +\newcommand{\Rbetainlrtwo}{ + \text{($\beta$inlr$_1$)\xspace} +} +\newcommand{\Tbetainlrtwo}{ + \TBbetainlrtwo + \DisplayProof +} +\newcommand{\TTbetainlrtwo}{ + \begin{prooftree} + \TBbetainlrtwo + \end{prooftree} +} +\newcommand{\TBbetainlrtwo}{ + \LeftLabel{\Rbetainlrtwo} + \AxiomC{$\Gamma \vdash s : A + 1$} + \AxiomC{$\Gamma \vdash t : B + 1$} + \AxiomC{$\Gamma \vdash s \downarrow = t \uparrow : \mathbf{2}$} + \TrinaryInfC{$\Gamma \vdash \rhd_2(\inlr{s}{t}) = t : B + 1$} +} +\newcommand{\Retainlr}{ + \text{($\eta$inlr)\xspace} +} +\newcommand{\Tetainlr}{ + \TBetainlr + \DisplayProof +} +\newcommand{\TTetainlr}{ + \begin{prooftree} + \TBetainlr + \end{prooftree} +} +\newcommand{\TBetainlr}{ + \LeftLabel{\Retainlr} + \AxiomC{$\Gamma \vdash t : A + B$} + \UnaryInfC{$\Gamma \vdash t = \inlr{\rhd_1(t)}{\rhd_2(t)} : A + B$} +} +\newcommand{\Rleft}{ + \text{(left)\xspace} +} +\newcommand{\Tleft}{ + \TBleft + \DisplayProof +} +\newcommand{\TTleft}{ + \begin{prooftree} + \TBleft + \end{prooftree} +} +\newcommand{\TBleft}{ + \LeftLabel{\Rleft} + \AxiomC{$\Gamma \vdash t : A + B$} + \AxiomC{$\Gamma \vdash \inlprop{t} = \top : \mathbf{2}$} + \BinaryInfC{$\Gamma \vdash \lft{t} : A$} +} +\newcommand{\Rlefteq}{ + \text{(left-eq)\xspace} +} +\newcommand{\Tlefteq}{ + \TBlefteq + \DisplayProof +} +\newcommand{\TTlefteq}{ + \begin{prooftree} + \TBlefteq + \end{prooftree} +} +\newcommand{\TBlefteq}{ + \LeftLabel{\Rlefteq} + \AxiomC{$\Gamma \vdash t = t' : A + B$} + \AxiomC{$\Gamma \vdash \inlprop{t} = \top : \mathbf{2}$} + \BinaryInfC{$\Gamma \vdash \lft{t} = \lft{t'} : A$} +} +\newcommand{\Rbetaleft}{ + \text{($\beta$left)\xspace} +} +\newcommand{\Tbetaleft}{ + \TBbetaleft + \DisplayProof +} +\newcommand{\TTbetaleft}{ + \begin{prooftree} + \TBbetaleft + \end{prooftree} +} +\newcommand{\TBbetaleft}{ + \LeftLabel{\Rbetaleft} + \AxiomC{$\Gamma \vdash t : A + B$} + \AxiomC{$\Gamma \vdash \inlprop{t} = \top : \mathbf{2}$} + \BinaryInfC{$\Gamma \vdash \inl{\lft{t}} = t : A + B$} +} +\newcommand{\Retaleft}{ + \text{($\eta$left)\xspace} +} +\newcommand{\Tetaleft}{ + \TBetaleft + \DisplayProof +} +\newcommand{\TTetaleft}{ + \begin{prooftree} + \TBetaleft + \end{prooftree} +} +\newcommand{\TBetaleft}{ + \LeftLabel{\Retaleft} + \AxiomC{$\Gamma \vdash t : A$} + \UnaryInfC{$\Gamma \vdash \lft{\inl{t}} = t : A$} +} +\newcommand{\RJMprime}{ + \text{(JM)\xspace} +} +\newcommand{\TJMprime}{ + \TBJMprime + \DisplayProof +} +\newcommand{\TTJMprime}{ + \begin{prooftree} + \TBJMprime + \end{prooftree} +} +\newcommand{\TBJMprime}{ + \LeftLabel{\RJMprime} + \AxiomC{ + $\begin{array}{cc} + \Gamma \vdash s : (A + A) + 1 & \Gamma \vdash t : (A + A) + 1\\ + \Gamma \vdash s \goesto \rhd_1 = t \goesto \rhd_1 : A + 1 & \Gamma \vdash s \goesto \rhd_2 = t \goesto \rhd_2 : A + 1 + \end{array}$ + } + \UnaryInfC{$\Gamma \vdash s = t : (A + A) + 1$} +} +\newcommand{\Rpair}{ + \text{($\sotimes$)\xspace} +} +\newcommand{\Tpair}{ + \TBpair + \DisplayProof +} +\newcommand{\TTpair}{ + \begin{prooftree} + \TBpair + \end{prooftree} +} +\newcommand{\TBpair}{ + \LeftLabel{\Rpair} + \AxiomC{$\Gamma \vdash s : A$} + \AxiomC{$\Delta \vdash t : B$} + \BinaryInfC{$\Gamma, \Delta \vdash s \sotimes t : A \otimes B$} +} +\newcommand{\Rpaireq}{ + \text{(paireq)\xspace} +} +\newcommand{\Tpaireq}{ + \TBpaireq + \DisplayProof +} +\newcommand{\TTpaireq}{ + \begin{prooftree} + \TBpaireq + \end{prooftree} +} +\newcommand{\TBpaireq}{ + \LeftLabel{\Rpaireq} + \AxiomC{$\Gamma \vdash s = s' : A$} + \AxiomC{$\Delta \vdash t = t': B$} + \BinaryInfC{$\Gamma, \Delta \vdash s \sotimes t = s' \sotimes t' : A \otimes B$} +} +\newcommand{\Rlett}{ + \text{(lett)\xspace} +} +\newcommand{\Tlett}{ + \TBlett + \DisplayProof +} +\newcommand{\TTlett}{ + \begin{prooftree} + \TBlett + \end{prooftree} +} +\newcommand{\TBlett}{ + \LeftLabel{\Rlett} + \AxiomC{$\Gamma \vdash s : A \otimes B$} + \AxiomC{$\Delta, x : A, y : B \vdash t : C$} + \BinaryInfC{$\Gamma, \Delta \vdash \plet{x}{y}{s}{t} : C$} +} +\newcommand{\Rleteq}{ + \text{(leteq)\xspace} +} +\newcommand{\Tleteq}{ + \TBleteq + \DisplayProof +} +\newcommand{\TTleteq}{ + \begin{prooftree} + \TBleteq + \end{prooftree} +} +\newcommand{\TBleteq}{ + \LeftLabel{\Rleteq} + \AxiomC{$\Gamma \vdash s = s' : A \otimes B$} + \AxiomC{$\Delta, x : A, y : B \vdash t = t' : C$} + \BinaryInfC{$\Gamma, \Delta \vdash (\plet{x}{y}{s}{t}) = (\plet{x}{y}{s'}{t'}) : C$} +} +\newcommand{\Rbeta}{ + \text{($\beta \otimes$)\xspace} +} +\newcommand{\Tbeta}{ + \TBbeta + \DisplayProof +} +\newcommand{\TTbeta}{ + \begin{prooftree} + \TBbeta + \end{prooftree} +} +\newcommand{\TBbeta}{ + \LeftLabel{\Rbeta} + \AxiomC{$\Gamma \vdash r : A$} + \AxiomC{$\Delta \vdash s : B$} + \AxiomC{$\Theta, x : A, y : B \vdash t : C$} + \TrinaryInfC{$\Gamma, \Delta, \Theta \vdash (\plet{x}{y}{r \sotimes s}{t}) = t[x:=r,y:=s] : C$} +} +\newcommand{\Reta}{ + \text{($\eta \otimes$)\xspace} +} +\newcommand{\Teta}{ + \TBeta + \DisplayProof +} +\newcommand{\TTeta}{ + \begin{prooftree} + \TBeta + \end{prooftree} +} +\newcommand{\TBeta}{ + \LeftLabel{\Reta} + \AxiomC{$\Gamma \vdash t : A \otimes B$} + \UnaryInfC{$\Gamma \vdash t = (\plet{x}{y}{t}{x \sotimes y}) : A \otimes B$} +} +\newcommand{\Rletlet}{ + \text{(let-let)\xspace} +} +\newcommand{\Tletlet}{ + \TBletlet + \DisplayProof +} +\newcommand{\TTletlet}{ + \begin{prooftree} + \TBletlet + \end{prooftree} +} +\newcommand{\TBletlet}{ + \LeftLabel{\Rletlet} + \AxiomC{$\Gamma \vdash r : A \otimes B$} + \AxiomC{$\Delta, x : A, y : B \vdash s : C \otimes D$} + \AxiomC{$\Theta, z : C, w : D \vdash t : E$} + \TrinaryInfC{$\Gamma, \Delta, \Theta \vdash +\begin{array}[t]{l} +\plet{x}{y}{r}{(\plet{z}{w}{s}{t})} \\ += \plet{z}{w}{(\plet{x}{y}{r}{s})}{t} +: E +\end{array}$} +} +\newcommand{\Rletpair}{ + \text{(let-$\sotimes$)\xspace} +} +\newcommand{\Tletpair}{ + \TBletpair + \DisplayProof +} +\newcommand{\TTletpair}{ + \begin{prooftree} + \TBletpair + \end{prooftree} +} +\newcommand{\TBletpair}{ + \LeftLabel{\Rletpair} + \AxiomC{$\Gamma \vdash r : A \otimes B$} + \AxiomC{$\Delta, x : A, y : B \vdash s : C$} + \AxiomC{$\Theta \vdash t : D$} + \TrinaryInfC{$\Gamma, \Delta, \Theta \vdash +\plet{x}{y}{r}{(s \sotimes t)} = (\plet{x}{y}{r}{s}) \sotimes t : D$} +} +\newcommand{\RleqI}{ + \text{(order)\xspace} +} +\newcommand{\TleqI}{ + \TBleqI + \DisplayProof +} +\newcommand{\TTleqI}{ + \begin{prooftree} + \TBleqI + \end{prooftree} +} +\newcommand{\TBleqI}{ + \LeftLabel{\RleqI} + \AxiomC{ + $\begin{array}{cc} + \Gamma \vdash s : A + 1 & \Gamma \vdash t : A + 1\\ + \Gamma \vdash b : (A + A) + 1 & \Gamma \vdash \doo{x}{b}{\rhd_1(x)} = s : A + 1\\ + \multicolumn{2}{c}{\Gamma \vdash \doo{x}{b}{\return \nabla(x)} = t : A + 1} + \end{array}$ + } + \UnaryInfC{$\Gamma \vdash s \leq t : A + 1$} +} +\newcommand{\Rinstr}{ + \text{(instr)\xspace} +} +\newcommand{\Tinstr}{ + \TBinstr + \DisplayProof +} +\newcommand{\TTinstr}{ + \begin{prooftree} + \TBinstr + \end{prooftree} +} +\newcommand{\TBinstr}{ + \LeftLabel{\Rinstr} + \AxiomC{$x : A \vdash t : \mathbf{n}$} + \AxiomC{$\Gamma \vdash s : A$} + \BinaryInfC{$\Gamma \vdash \instr_{\lambda x t}(s) : n \cdot A$} +} +\newcommand{\Rnablainstr}{ + \text{($\nabla$-instr)\xspace} +} +\newcommand{\Tnablainstr}{ + \TBnablainstr + \DisplayProof +} +\newcommand{\TTnablainstr}{ + \begin{prooftree} + \TBnablainstr + \end{prooftree} +} +\newcommand{\TBnablainstr}{ + \LeftLabel{\Rnablainstr} + \AxiomC{$x : A \vdash t : \mathbf{n}$} + \AxiomC{$\Gamma \vdash s : A$} + \BinaryInfC{$\Gamma \vdash \nabla(\instr_{\lambda x t}(s)) = s : A$} +} +\newcommand{\Rinstrtest}{ + \text{(instr-test)\xspace} +} +\newcommand{\Tinstrtest}{ + \TBinstrtest + \DisplayProof +} +\newcommand{\TTinstrtest}{ + \begin{prooftree} + \TBinstrtest + \end{prooftree} +} +\newcommand{\TBinstrtest}{ + \LeftLabel{\Rinstrtest} + \AxiomC{$x : A \vdash t : \mathbf{n}$} + \AxiomC{$\Gamma \vdash s : A$} + \BinaryInfC{$\Gamma \vdash \case_{i=1}^n \instr_{\lambda x t}(s) \of \nin{i}{n}{\_} \mapsto i = t[x:=s] : \mathbf{n}$} +} +\newcommand{\Retainstr}{ + \text{($\eta$instr)\xspace} +} +\newcommand{\Tetainstr}{ + \TBetainstr + \DisplayProof +} +\newcommand{\TTetainstr}{ + \begin{prooftree} + \TBetainstr + \end{prooftree} +} +\newcommand{\TBetainstr}{ + \LeftLabel{\Retainstr} + \AxiomC{$x : A \vdash r : n \cdot A$} + \AxiomC{$x : A \vdash \nabla(r) = x : A$} + \AxiomC{$\Gamma \vdash s : A$} + \TrinaryInfC{$\Gamma \vdash \instr_{\lambda x. \case_{i=1}^n r \of \nin{i}{n}{\_} \mapsto i}(s) = r[x:=s] : n \cdot A$} +} +\newcommand{\Rinstreq}{ + \text{(instr-eq)\xspace} +} +\newcommand{\Tinstreq}{ + \TBinstreq + \DisplayProof +} +\newcommand{\TTinstreq}{ + \begin{prooftree} + \TBinstreq + \end{prooftree} +} +\newcommand{\TBinstreq}{ + \LeftLabel{\Rinstreq} + \AxiomC{$x : A \vdash t = t' : \mathbf{n}$} + \AxiomC{$\Gamma \vdash s = s' : A$} + \BinaryInfC{$\Gamma \vdash \instr_{\lambda x t}(s) = \instr_{\lambda x t'}(s') : n \cdot A$} +} +\newcommand{\Rcomm}{ + \text{(comm)\xspace} +} +\newcommand{\Tcomm}{ + \TBcomm + \DisplayProof +} +\newcommand{\TTcomm}{ + \begin{prooftree} + \TBcomm + \end{prooftree} +} +\newcommand{\TBcomm}{ + \LeftLabel{\Rcomm} + \AxiomC{$x : A \vdash p : \mathbf{2}$} + \AxiomC{$x : A \vdash q : \mathbf{2}$} + \AxiomC{$\Gamma \vdash t : A$} + \TrinaryInfC{$\Gamma \vdash \begin{array}[t]{l} +\assert_{\lambda x p}(t) \goesto \assert_{\lambda x q} = \assert_{\lambda x q}(t) \goesto \assert_{\lambda x p} : A + 1 +\end{array}$} +} +\newcommand{\Roneovern}{ + \text{($1 / n$)\xspace} +} +\newcommand{\Toneovern}{ + \TBoneovern + \DisplayProof +} +\newcommand{\TToneovern}{ + \begin{prooftree} + \TBoneovern + \end{prooftree} +} +\newcommand{\TBoneovern}{ + \LeftLabel{\Roneovern} + \AxiomC{$$} + \UnaryInfC{$\Gamma \vdash 1 / n : \mathbf{2}$} +} +\newcommand{\Rntimesoneovern}{ + \text{($n \cdot 1 / n$)\xspace} +} +\newcommand{\Tntimesoneovern}{ + \TBntimesoneovern + \DisplayProof +} +\newcommand{\TTntimesoneovern}{ + \begin{prooftree} + \TBntimesoneovern + \end{prooftree} +} +\newcommand{\TBntimesoneovern}{ + \LeftLabel{\Rntimesoneovern} + \AxiomC{$$} + \UnaryInfC{$\Gamma \vdash n \cdot 1 / n = \top : \mathbf{2}$} +} +\newcommand{\Rdivide}{ + \text{(divide)\xspace} +} +\newcommand{\Tdivide}{ + \TBdivide + \DisplayProof +} +\newcommand{\TTdivide}{ + \begin{prooftree} + \TBdivide + \end{prooftree} +} +\newcommand{\TBdivide}{ + \LeftLabel{\Rdivide} + \AxiomC{$\Gamma \vdash n \cdot t = \top : \mathbf{2}$} + \UnaryInfC{$\Gamma \vdash t = 1 / n : \mathbf{2}$} +} +\newcommand{\Rnorm}{ + \text{(nrm)\xspace} +} +\newcommand{\Tnorm}{ + \TBnorm + \DisplayProof +} +\newcommand{\TTnorm}{ + \begin{prooftree} + \TBnorm + \end{prooftree} +} +\newcommand{\TBnorm}{ + \LeftLabel{\Rnorm} + \AxiomC{$\vdash t : A + 1$} + \AxiomC{$\vdash 1 / n \leq t : \mathbf{2}$} + \BinaryInfC{$\Gamma \vdash \norm{t} : A$} +} +\newcommand{\Rbetanorm}{ + \text{($\beta$nrm)\xspace} +} +\newcommand{\Tbetanorm}{ + \TBbetanorm + \DisplayProof +} +\newcommand{\TTbetanorm}{ + \begin{prooftree} + \TBbetanorm + \end{prooftree} +} +\newcommand{\TBbetanorm}{ + \LeftLabel{\Rbetanorm} + \AxiomC{$\vdash t : A + 1$} + \AxiomC{$\vdash 1 / n \leq t \downarrow : \mathbf{2}$} + \BinaryInfC{$\Gamma \vdash t = \doo{\_}{t}{\return{\norm{t}}} : A + 1$} +} +\newcommand{\Retanorm}{ + \text{($\eta$nrm)\xspace} +} +\newcommand{\Tetanorm}{ + \TBetanorm + \DisplayProof +} +\newcommand{\TTetanorm}{ + \begin{prooftree} + \TBetanorm + \end{prooftree} +} +\newcommand{\TBetanorm}{ + \LeftLabel{\Retanorm} + \AxiomC{$\vdash t : A + 1$} + \AxiomC{$\vdash 1 / n \leq t \downarrow : \mathbf{2}$} + \AxiomC{$\vdash \rho : A$} + \AxiomC{$\vdash t = \doo{\_}{t}{\return{\rho}} : A + 1$} + \QuaternaryInfC{$\Gamma \vdash \rho = \norm{t} : A$} +} +\newcommand{\Rrhdoneboundmn}{ + \text{($\rhd_1-b_{mn}$)\xspace} +} +\newcommand{\Trhdoneboundmn}{ + \TBrhdoneboundmn + \DisplayProof +} +\newcommand{\TTrhdoneboundmn}{ + \begin{prooftree} + \TBrhdoneboundmn + \end{prooftree} +} +\newcommand{\TBrhdoneboundmn}{ + \LeftLabel{\Rrhdoneboundmn} + \AxiomC{$$} + \RightLabel{$\left(1 \leq m < n\right)$} + \UnaryInfC{$\Gamma \vdash \doo{x}{b_{mn}}{\rhd_1(x)} = m \cdot 1 / n : \mathbf{2}$} +} +\newcommand{\Rrhdtwoboundmnprime}{ + \text{($\rhd_2-b_{mn}$)\xspace} +} +\newcommand{\Trhdtwoboundmnprime}{ + \TBrhdtwoboundmnprime + \DisplayProof +} +\newcommand{\TTrhdtwoboundmnprime}{ + \begin{prooftree} + \TBrhdtwoboundmnprime + \end{prooftree} +} +\newcommand{\TBrhdtwoboundmnprime}{ + \LeftLabel{\Rrhdtwoboundmnprime} + \AxiomC{$$} + \RightLabel{$\left(1 \leq m < n\right)$} + \UnaryInfC{$\Gamma \vdash \doo{x}{b_{mn}}{\return \nabla(x)} = 1 / n : \mathbf{2}$} +} +\newcommand{\Rboundmn}{ + \text{($b_{mn}$)\xspace} +} +\newcommand{\Tboundmn}{ + \TBboundmn + \DisplayProof +} +\newcommand{\TTboundmn}{ + \begin{prooftree} + \TBboundmn + \end{prooftree} +} +\newcommand{\TBboundmn}{ + \LeftLabel{\Rboundmn} + \AxiomC{$$} + \RightLabel{$\left(1 \leq m < n\right)$} + \UnaryInfC{$\Gamma \vdash b_{mn} : \mathbf{3}$} +} +\newcommand{\Roveeprime}{ + \text{($\ovee$)\xspace} +} +\newcommand{\Toveeprime}{ + \TBoveeprime + \DisplayProof +} +\newcommand{\TToveeprime}{ + \begin{prooftree} + \TBoveeprime + \end{prooftree} +} +\newcommand{\TBoveeprime}{ + \LeftLabel{\Roveeprime} + \AxiomC{ + $\begin{array}{cc} + \Gamma \vdash s : A + 1 & \Gamma \vdash t : A + 1\\ + \Gamma \vdash b : (A + A) + 1 & \Gamma \vdash \doo{x}{b}{\rhd_1(x)} = s : A + 1\\ + \multicolumn{2}{c}{\Gamma \vdash \doo{x}{b}{\rhd_2(x)} = t : A + 1} + \end{array}$ + } + \UnaryInfC{$\Gamma \vdash s \ovee t : A + 1$} +} +\newcommand{\Roveedef}{ + \text{($\ovee$-def)\xspace} +} +\newcommand{\Toveedef}{ + \TBoveedef + \DisplayProof +} +\newcommand{\TToveedef}{ + \begin{prooftree} + \TBoveedef + \end{prooftree} +} +\newcommand{\TBoveedef}{ + \LeftLabel{\Roveedef} + \AxiomC{ + $\begin{array}{cc} + \Gamma \vdash s : A + 1 & \Gamma \vdash t : A + 1\\ + \Gamma \vdash b : (A + A) + 1 & \Gamma \vdash \doo{x}{b}{\rhd_1(x)} = s : A + 1\\ + \multicolumn{2}{c}{\Gamma \vdash \doo{x}{b}{\rhd_2(x)} = t : A + 1} + \end{array}$ + } + \UnaryInfC{$\Gamma \vdash s \ovee t = \doo{x}{b}{\return{\nabla(x)}} : A + 1$} +} +\newcommand{\Rletsub}{ + \text{(letsub)\xspace} +} +\newcommand{\Tletsub}{ + \TBletsub + \DisplayProof +} +\newcommand{\TTletsub}{ + \begin{prooftree} + \TBletsub + \end{prooftree} +} +\newcommand{\TBletsub}{ + \LeftLabel{\Rletsub} + \AxiomC{$\Gamma \vdash r : A \otimes B$} + \AxiomC{$\Delta, x : A, y : B \vdash s : C$} + \AxiomC{$\Theta, z : C \vdash t : D$} + \TrinaryInfC{$\Gamma, \Delta, \Theta \vdash t[z:=\plet{x}{y}{r}{s}] = \plet{x}{y}{r}{t[z:=s]} : D$} +} +\newcommand{\Rcasesub}{ + \text{(case-sub)\xspace} +} +\newcommand{\Tcasesub}{ + \TBcasesub + \DisplayProof +} +\newcommand{\TTcasesub}{ + \begin{prooftree} + \TBcasesub + \end{prooftree} +} +\newcommand{\TBcasesub}{ + \LeftLabel{\Rcasesub} + \AxiomC{$\Gamma \vdash r : A + B$} + \AxiomC{$\Delta, x : A \vdash s : C$} + \AxiomC{$\Delta, y : B \vdash s' : C$} + \AxiomC{$\Theta, z : C \vdash t : D$} + \QuaternaryInfC{$\Gamma, \Delta, \Theta \vdash \begin{array}[t]{l} +t[z:=\pcase{r}{x}{s}{y}{s'}] \\ += \pcase{r}{x}{t[z:=s]}{y}{t[z:=s']} : D +\end{array}$} +} +\newcommand{\Rassert}{ + \text{(assert)\xspace} +} +\newcommand{\Tassert}{ + \TBassert + \DisplayProof +} +\newcommand{\TTassert}{ + \begin{prooftree} + \TBassert + \end{prooftree} +} +\newcommand{\TBassert}{ + \LeftLabel{\Rassert} + \AxiomC{$\Gamma \vdash t : A$} + \AxiomC{$x : A \vdash p : \mathbf{2}$} + \BinaryInfC{$\Gamma \vdash \assert_{\lambda x p}(t) : A + 1$} +} +\newcommand{\Rgoesto}{ + \text{($\goesto$)\xspace} +} +\newcommand{\Tgoesto}{ + \TBgoesto + \DisplayProof +} +\newcommand{\TTgoesto}{ + \begin{prooftree} + \TBgoesto + \end{prooftree} +} +\newcommand{\TBgoesto}{ + \LeftLabel{\Rgoesto} + \AxiomC{$\Gamma \vdash t : A + 1$} + \AxiomC{$\Delta, x : A \vdash f(x) : B + 1$} + \BinaryInfC{$\Gamma, \Delta \vdash t \goesto f : B + 1$} +} \newcommand{\Lweak}{ + \begin{lm} + \label{lm:weak} + The following rule of deduction is admissible. \TTweak + \end{lm} +} +\newcommand{\Lsub}{ + \begin{lm} + \label{lm:sub} + The following rule of deduction is admissible. \TTsub + \end{lm} +} +\newcommand{\Leqval}{ + \begin{lm} + \label{lm:eqval} + If $\Gamma \vdash s = t : A$ then $\Gamma \vdash s : A$ and $\Gamma \vdash t : A$. + \end{lm} +} +\newcommand{\Lineqval}{ + \begin{lm} + \label{lm:ineqval} + If $\Gamma \vdash s \leq t : A$ then $\Gamma \vdash s : A$ and $\Gamma \vdash t : A + \end{lm} +} +\newcommand{\Lfunc}{ + \begin{lm} + \label{lm:func} + If $\Gamma \vdash r = s : A$ and $\Delta, x : A \vdash t : B$ then $\Gamma, \Delta \vdash t[x:=r] = t[x:=s] : B$. + \end{lm} +} +\newcommand{\Sets}{\mathbf{Sets}} +\newcommand{\Kl}{\mathcal{K}{\kern-.2ex}\ell} +\newcommand{\Dst}{\mathcal{D}} +\newcommand{\Giry}{\mathcal{G}} + +\newcommand{\Pred}{\mathbf{Pred}} + +\newcommand{\assert}{\mathsf{assert}} +\newcommand{\case}{\mathsf{case}\ } +\newcommand{\elsen}{\ \mathsf{else}\ } +\newcommand{\idmap}[1][]{\ensuremath{\mathrm{id}_{#1}}} +\newcommand{\id}{\idmap} +\newcommand{\cond}[3]{\ifn {#1} \thenn {#2} \elsen {#3}} +\newcommand{\ifn}{\mathsf{if}\ } +\newcommand{\inln}{\mathsf{inl}} +\newcommand{\inlprop}[1]{\mathsf{inl?} \left( {#1} \right)} +\newcommand{\inlrn}{\mathsf{inlr}} +\newcommand{\inl}[1]{\inln \left( {#1} \right)} +\newcommand{\inn}{\ \mathsf{in}\ } +\newcommand{\inrn}{\mathsf{inr}} +\newcommand{\inrprop}[1]{\mathsf{inr?} \left( {#1} \right)} +\newcommand{\inr}[1]{\inrn \left( {#1} \right)} +\newcommand{\lett}{\mathsf{let}\ } +\newcommand{\lftn}{\mathsf{left}} +\newcommand{\lft}[1]{\lftn \left( {#1} \right)} +\newcommand{\instr}{\mathsf{instr}} +\newcommand{\measure}{\mathbf{TODO}} +\newcommand{\meas}{\mathsf{measure}} +\newcommand{\norm}[1]{\ensuremath{\mathsf{nrm} \left( {#1} \right)}} +\newcommand{\of}{\ \mathsf{of}\ } +\newcommand{\pcase}[5]{\case {#1} \of \inl{#2} \mapsto {#3}\mid \inr{#4} \mapsto {#5}} +\newcommand{\rgtn}{\mathsf{right}} +\newcommand{\rgt}[1]{\rgtn \left( {#1} \right)} +\newcommand{\swapper}[1]{\ensuremath{\mathsf{swap} \left( {#1} \right)}} +\newcommand{\thenn}{\ \mathsf{then}\ } +\newcommand{\type}{\ \mathrm{type}} +\newcommand{\nin}[3]{\mathsf{in}_{#1}^{#2} \left( {#3} \right)} +\newcommand{\return}[1]{\mathsf{return}\ {#1}} +\newcommand{\fail}{\mathsf{fail}} +\newcommand{\doo}[3]{\mathsf{do}\ {#1} \leftarrow {#2} ; {#3}} +\newcommand{\ind}[1]{\ensuremath{\mathsf{index} \left( {#1} \right)}} +\newcommand{\intest}[2]{\mathsf{in}_{#1} ? \left( {#2} \right)} +\newcommand{\condn}[2]{\mathsf{cond} \left( {#1} , {#2} \right)} +\newcommand{\bang}{\mathord{!}} + +\newcommand{\inlr}[2]{\ensuremath{\text{\guillemotleft} {#1} , {#2} \text{\guillemotright}}} +\newcommand{\eqdef}{\mathrel{\smash{\stackrel{\text{def}}{=}}}} +\newcommand{\fromInit}{\,\mathop{\text{\rm \textexclamdown}}} +\newcommand{\magic}[1]{\fromInit{#1}} +\newcommand{\sotimes}{\mathrel{\raisebox{.05pc}{$\scriptstyle\otimes$}}} +\newcommand{\plet}[4]{\lett {#1} \sotimes {#2} = {#3} \inn {#4}} +\newcommand{\slet}[3]{\lett {#1} = {#2} \inn {#3}} +\newcommand{\ifte}[3]{\ifn {#1} \thenn {#2} \elsen {#3}} +\newcommand{\bigovee}{\mathop{\vphantom{\sum}\mathchoice {\vcenter{\hbox{\huge $\ovee$}}}{\vcenter{\hbox{\Large $\ovee$}}}{\ovee}{\ovee}}\displaylimits} +\newcommand{\goesto}{\ensuremath{\gg\!\!=}} +\newcommand{\andthen}{\mathrel{\&}} +\renewcommand{\ker}[1]{{#1}\!\uparrow} +\newcommand{\dom}[1]{{#1}\!\downarrow} +\newcommand{\after}{\circ} +\newcommand{\supp}{\mathop{\mathrm{supp}}} + +\newcommand{\brackets}[1]{\left[ \! \left[ {#1} \right] \! \right]} +\newcommand{\Prob}[1]{\mathrm{Pr} \left( {#1} \right)} + +\newcommand{\COMET}{\mathbf{COMET}} + +\renewcommand{\arraystretch}{1.3} +\setlength{\arraycolsep}{2pt} + + +\theoremstyle{plain} +\newtheorem{proposition}[theorem]{Proposition} + +\begin{document} + +\maketitle + +\begin{abstract} +This paper introduces a novel type theory and logic for probabilistic +reasoning. Its logic is quantitative, with fuzzy predicates. It +includes normalisation and conditioning of states. This conditioning +uses a key aspect that distinguishes our probabilistic type theory +from quantum type theory, namely the bijective correspondence between +predicates and side-effect free actions (called instrument, or assert, +maps). The paper shows how suitable computation rules can be derived +from this predicate-action correspondence, and uses these rules for +calculating conditional probabilities in two well-known examples of +Bayesian reasoning in (graphical) models. Our type theory may thus +form the basis for a mechanisation of Bayesian inference. +\end{abstract} + + +\section{Introduction} + +A probabilistic program is understood (semantically) as a stochastic +process. A key feature of probabilistic programs as studied in the +1980s and 1990s is the presence of probabilistic choice, for instance +in the form of a weighted sum $x +_{r} y$, where the number $r \in +[0,1]$ determines the ratio of the contributions of $x$ and $y$ to the +result. This can be expressed explicitly as a convex sum $r\cdot x + +(1-r)\cdot y$. Some of the relevant sources +are~\cite{Kozen81,Kozen85}, and~\cite{JonesP89}, +and~\cite{MorganMS96}, and also~\cite{TixKP05} for the combination of +probability and non-determinism. In the language of category theory, a +probabilistic program is a map in the Kleisli category of the +distribution monad $\Dst$ (in the discrete case) or of the Giry monad +$\Giry$ (in the continuous case). + +In recent years, with the establishement of Bayesian machine learning +as an important area of computer science, the meaning of probabilistic +programming shifted towards conditional inference. The key feature is +no longer probabilistic choice, but normalisation of distributions +(states), see \textit{e.g.}~\cite{Borgstroem2011}. Interestingly, this +can be done in basically the same underlying models, where a program +still produces a distribution --- discrete or continuous --- over its +output. + +This paper contributes to this latest line of work by formulating a +novel type theory for probabilistic and Bayesian reasoning. We list +the key features of our type theory. +\begin{itemize} +\item It includes a logic, which is quantitative in nature. This means + that its predicates are best understood as `fuzzy' predicates, + taking values in the unit interval $[0,1]$ of probabilities, instead + of in the two-element set $\{0,1\}$ of Booleans. + +\item As a result, the predicates of this logic do not form Boolean + algebras, but effect modules (see \emph{e.g.}~\cite{Jacobs15d}). The + double negation rule does hold, but the sum $\ovee$ is a partial + operation. Moreover, there is a scalar multiplication $s\cdot p$, + for a scalar $s$ and a predicate $p$, which produces a scaled + version of the predicate $p$. + +\item This logic is a special case of a more general quantum type + theory~\cite{Adams2014}. What we describe here is the probabilistic + subcase of this quantum type theory, which is characterised by a + bijective correspondence between predicates and side-effect free + assert maps (see below for details). + +\item The type theory includes normalisation (and also probabilistic + choice). Abstractly, normalisation means that each non-zero + `substate' in the type theory can be turned into a proper state + (like in~\cite{JacobsWW15a}). This involves, for instance, turning a + \emph{sub}distribution $\sum_{i}r_{i}x_{i}$, where the probabilities + $r_{i}\in [0,1]$ satisfy $0 < r \leq 1$ for $r \eqdef + \sum_{i}r_{i}$, into a proper distribution + $\sum_{i}\frac{r_i}{r}x_{i}$ --- where, by construction, + $\sum_{i}\frac{r_i}{r} = 1$. + +\item The type theory also includes conditioning, via the combination + of assert maps and normalisation (from the previous two points). + Hence, we can calculate conditional probabilities inside the type + theory, via appropriate (derived) computation rules. In contrast, in + the language of~\cite{Borgstroem2011}, probabilistic (graphical) + models can be formulated, but actual computations are done in the + underlying mathematical models. Since these computation are done + inside our calculus, our type theory can form the basis for + mechanisation. +\end{itemize} + +The type theory that we present is based on a new categorical +foundation for quantum logic, called effectus theory, +see~\cite{Jacobs15d,JacobsWW15a,Cho15a,ChoJWW15}\footnote{A general + introduction to effectus theory~\cite{Cho} will soon be + available.}. This theory involves a basic duality between states and +effects (predicates), which is implicitly also present in our type +theory. A subclass of `commutative' effectuses can be defined, forming +models for probabilistic computation and logic. Our type theory +corresponds to these commutative effectuses, and will thus be called +$\COMET$, as abbreviation of COMmutative Effectus Theory. This +$\COMET$ can be seen as an internal language for commutative +effectuses. + +A key feature of quantum theory is that observations have a +side-effect: measuring a system disturbs it at the quantum level. In +order to perform such measurements, each quantum predicate comes with +an associated `measurement' instrument operation which acts on the +underlying space. Probabilistic theories also have such instruments +\ldots but they are side-effect free! + +The idea that predicates come with an associated action is familiar in +mathematics. For instance, in a Hilbert space $\mathscr{H}$, a closed +subspace $P \subseteq \mathscr{H}$ (a predicate) can equivalently be +described as a linear idempotent operator $p\colon \mathscr{H} +\rightarrow \mathscr{H}$ (an action) that has $P$ has image. We sketch +how these predicate-action correspondences also exist in the models +that underly our type theory. + +First, in the category $\Sets$ of sets and functions, a predicate $p$ +on a set $X$ can be identified with a subset of $X$, but also with a +`characteristic' map $p\colon X \rightarrow 1+1$, where $1+1 = 2$ is +the two-element set. We prefer the latter view. Such a predicate +corresponds bijectively to a `side-effect free' instrument +$\instr_{p} \colon X \rightarrow X+X$, namely to: +$$\begin{array}{rcl} +\instr_{p}(x) +& = & +\left\{\begin{array}{ll} +\inl{x} \mbox{\quad} & \mbox{if } p(x) = 1 \\ +\inr{x} & \mbox{if } p(x) = 0 \\ +\end{array}\right. +\end{array}$$ + +\noindent Here we write $X+X$ for the sum (coproduct), with left and +right coprojections (also called injections) $\inl{\_}, \inr{\_} +\colon X \rightarrow X+X$. Notice that this instrument merely makes a +left-right distinction, as described by the predicate, but does not +change the state $x$. It is called side-effect free because it +satisfies $\nabla \after \instr_{p} = \idmap$, where $\nabla = +[\idmap,\idmap] \colon X+X \rightarrow X$ is the codiagonal. It easy +to see that each map $f\colon X \rightarrow X+X$ with $\nabla \after f += \idmap$ corresponds to a predicate $p\colon X \rightarrow 1+1$, +namely to $p = (\bang+\bang) \after f$, where $\bang \colon X +\rightarrow 1$ is the unique map to the final (singleton, unit) set +$1$. + +Our next example describes the same predicate-action correspondence in +a probabilistic setting. It assumes familiarity with the discrete +distribution monad $\Dst$ --- see~\cite{Jacobs15d} for details, and +also Subsection~\ref{section:dpc} --- and with its Kleisli category +$\Kl(\Dst)$. A predicate map $p\colon X \rightarrow 1+1$ in +$\Kl(\Dst)$ is (essentially) a fuzzy predicate $p\colon X \rightarrow +[0,1]$, since $\Dst(1+1) = \Dst(2) \cong [0,1]$. There is also an +associated instrument map $\instr_{p} \colon X \rightarrow X+X$ in +$\Kl(\Dst)$, given by the function $\instr_{p} \colon X \rightarrow +\Dst(X+X)$ that sends an element $x\in X$ to the distribution +(formal convex combination): +$$\begin{array}{rcl} +\instr_{p}(x) +& = & +p(x)\cdot \inl{x} + (1-p(x))\cdot \inr{x}. +\end{array}$$ + +\noindent This instrument makes a left-right distinction, with the +weight of the distinction given by the fuzzy predicate $p$. Again we +have $\nabla \after \instr_{p} = \idmap$, in the Kleisli category, +since the instrument map does not change the state. It is easy to see +that we get a bijective correspondence. + +These instrument maps $\instr_{p} \colon X \rightarrow X+X$ can in +fact be simplified further into what we call assert maps. The +(partial) map $\assert_{p} \colon X \rightarrow X+1$ can be defined as +$\assert_{p} = (\idmap+\bang) \after \instr_{p}$. We say that such a +map is side-effect free if there is an inequality $\assert_{p} \leq +\inl{\_}$, for a suitable order on the homset of partial maps $X +\rightarrow X+1$. Given assert maps for $p$, and for its +orthosupplement (negation) $p^{\bot}$, we can define the associated +instrument via a partial pairing operation as $\instr_{p} = +\inlr{\assert_p}{\assert_{p^\bot}}$, see below for details. + +The key aspect of a probabilistic model, in contrast to a quantum model, +is that there is a bijective correspondence between: +\begin{itemize} +\item predicates $X \rightarrow 1+1$ +\item side-effect free instruments $X \rightarrow X+X$ --- or + equivalently, side-effect free assert maps $X \rightarrow X+1$. +\end{itemize} + +\noindent We shall define conditioning via normalisation after assert. +More specifically, for a state $\omega\colon X$ and a predicate $p$ on +$X$ we define the conditional state $\omega|_{p} = \condn{\omega}{p}$ +as: +$$\begin{array}{rcl} +\condn{\omega}{p} +& = & +\norm{\assert_{p}(\omega)}, +\end{array}$$ + +\noindent where $\norm{-}$ describes normalisation (of substates to +states). This description occurs, in semantical form +in~\cite{JacobsWW15a}. Here we formalise it at a type-theoretic level +and derive suitable computation rules from it that allow us to do +(exact) conditional inference. + +The paper is organised as follows. Section~\ref{section:overview} +provides an overview of the type theory, with some key results, +without giving all the details and +proofs. Section~\ref{section:examples} takes two familiar examples of +Bayesian reasoning and formalises them in our type theory $\COMET$. +Subsequently, Section~\ref{section:metatheorems} explores the type +theory in greater depth, and provides justification for the +computation rules in the examples. Next, +Section~\ref{section:semantics} sketches how our type theory can be +interpreted in set-theoretic and probabilistic +models. Appendix~\ref{section:rules} contains a formal presentation of +the type theory $\COMET$. + + + +\section{Syntax and Rules of Deduction} +\label{section:overview} + +We present here the terms and types of $\COMET$. We shall describe the system +at a high level here, giving the intuition behind each construction. The complete list of +the rules of deduction of $\COMET$ is given in Appendix \ref{section:rules}, and the +properties that we use are all proved in Section \ref{section:metatheorems}. + +\subsection{Syntax} + +Assume we are given a set of +\emph{type constants} $\mathbf{C}$, representing the base data types needed for each example. (These may typically include for instance $\mathbf{bool}$, $\mathbf{nat}$ and $\mathbf{real}$.) +Then the types of $\COMET$ are the following. +$$ \begin{array}{lrcll} +\text{Type} & A & ::= & \mathbf{C} \mid & \text{constant type} \\ +& & & 0 \mid & \text{empty type} \\ +& & & 1 \mid & \text{unit type} \\ +& & & A + B \mid & \text{disjoint union} \\ +& & & A \otimes B & \text{pairs} +\end{array} $$ + +The \emph{terms} of $\COMET$ are given by the following grammar. + +$$ \begin{array}{lrcll} +\text{Term} & t & ::= & x \mid & \text{variable} \\ +& & & * \mid & \text{element of unit type} \\ +& & & t \sotimes t \mid & \text{pair} \\ +& & & \plet{x}{y}{t}{t} \mid & \text{decomposing a pair} \\ +& & & \magic{t} \mid & \text{eliminate element of empty type} \\ +& & & \inl{t} \mid \inr{t} \mid & \text{elements of a disjoint union} \\ +& & & (\pcase{t}{x}{t}{x}{t}) \mid & \text{case distinction over union} \\ +& & & \inlr{s}{t} \mid & \text{partial pairing} \\ +& & & \lft{t} \mid & \text{extract element of union} \\ +& & & \instr_{\lambda x t}{t} \mid & \text{instrument map} \\ +& & & 1/n \mid & \text{constant scalar} (n \geq 2) \\ +& & & \norm{t} \mid & \text{normalised substate} \\ +& & & s \ovee t & \text{partial sum} +\end{array}$$ + +The variables $x$ and $y$ are bound within $s$ in $\plet{x}{y}{s}{t}$. The variable $x$ is bound within $s$ and $y$ within $t$ in $\pcase{r}{x}{s}{y}{t}$, and $x$ is bound within $t$ in $\instr_{\lambda x t}(s)$. +We identify terms up to $\alpha$-conversion (change of bound variable). We write $t[x:=s]$ for the result of substituting $s$ for $x$ within $t$, renaming bound variables to avoid variable capture. +We shall write $\_$ for a vacuous bound variable; for example, we write $\pcase{r}{\_}{s}{y}{t}$ for $\pcase{r}{x}{s}{y}{t}$ when $y$ does not occur free in $s$. + +We shall also sometimes abbreviate our terms, for example writing $\instr_{\mathsf{inl}}(t)$ when we should strictly write $\instr_{\lambda x \inl{x}}(t)$. Each time, the meaning should be clear from context. + +The typing rules for these terms are given in Figure \ref{fig:typing}. (Note that some of these rules make use of defined +expressions, which will be introduced in the sections below.) + +\begin{figure} +\begin{mdframed} +$$ \Tvar \; \Tunit \; \Tpair $$ +\TTlett +$$ \Tmagic \; \Tinl \; \Tinr $$ +\TTcase +\TTinlr +$$ \Tleft \; \Tinstr $$ +$$ \Toneovern \; \Tnorm $$ +\TToveeprime +\end{mdframed} +\caption{Typing rules for $\COMET$} +\label{fig:typing} +\end{figure} + +The typing rule for the term $\magic{t}$ +says that from an inhabitant $t:0$ we can produce an inhabitant +$\magic{t}$ in any type $A$. Intuitively, this says `If the empty type is inhabited, +then every type is inhabited', which is vacuously true. + +A term of type $A$ is intended to represent a \emph{total} computation, that always terminates and returns a value of type $A$. +We can think of a term of type $A + 1$ as a \emph{partial} computation that may return a value $a$ of type $A$ +(by outputting $\inl{a}$) or diverge (by outputting $\inr{*}$). The judgement $s \leq t$ should be understood as: +the probability that $s$ returns $\inl{a}$ is $\leq$ the probability that $t$ returns $\inl{a}$, for all $a$. The rule for this +ordering relation is given in Figure \ref{fig:ordering}. + +\begin{figure} +\begin{mdframed} +\TTleqI +\end{mdframed} +\caption{Rule for Ordering in $\COMET$} +\label{fig:ordering} +\end{figure} + +The term $\inlr{s}{t}$ is understood intuitively as follows. We are +given two partial computations $s$ and $t$, and we have derived the +judgement $\dom{s} = \ker{t}$, which tells us that exactly one of +$s$ and $t$ converges on any given input. We may then form the +computation $\inlr{s}{t}$ which, given an input $x$, returns either +$s(x)$ or $t(x)$, whichever of the two converges. + +For the term $\lft{t}$: if we have a term $t : A + B$ and we have derived the judgement $\inlprop{t} = \top$, then we know +that $t$ has the form $\inl{a}$ for some term $a : A$. We denote this unique term $a$ by $\lft{t}$. + +For the term $\instr_{\lambda x t}(s)$: think of the type $\mathbf{n}$ as the set $\{ 1, \ldots, n \}$. The elements of the type $A + \cdots + A$ consist of $n$ copies of each element $a$ of $A$, +denoted $\nin{1}{n}{a}$, \ldots, $\nin{n}{n}{a}$. Then $\instr_{\lambda x t}(s)$ is the object $\nin{t[x:=s]}{n}{s}$. It maps $s$ into one of the $n$ copies of $A$, which one being +determined by the test $t$. + +The term $1 / n$ represents the probability distribution on $\mathbf{2} = \{ \top, \bot \}$ which returns $\top$ with probability $1 / n$ and $\bot$ with probability $(n - 1) / n$. It can +be thought of as a coin toss, with a weighted coin that returns heads with probability $1 / n$. + +For the term $\norm{t}$: the term $t : A + 1$ represents a distribution on $A + 1$. Let $s$ denote the probability that $t$ terminates (i.e. returns a term of the form $\inl{a}$), and let +$\omega(a)$ denote the probability that $t$ returns $a$. Then $\norm{t}$ returns $a$ with probability $\omega(a) / s$. Thus, $\norm{t}$ is the distribution resulting from normalising +the subdistribution given by $t$. + +The term $s \ovee t$ is the `sum' of $s$ and $t$ in the following sense. It is defined on a given input if and only if, for any $a$, the probability that $s$ and $t$ both return $\inl{a}$ is $\leq 1$. +In this case, the probability that $s \ovee t$ returns $\inl{a}$ is the sum of these two probabilities. + +The computation rules that these terms obey are given in Figure \ref{fig:equations}. + +\begin{figure} +\begin{mdframed} +\begin{gather*} +\plet{x}{y}{r \sotimes s}{t} = t[x:=r,y:=s] \tag*{\Rbeta} \\ +\pcase{\inl{r}}{x}{s}{y}{t} = s[x:=r] \tag*{\Rbetaplusone} \\ +\pcase{\inr{r}}{x}{s}{y}{t} = t[y:=r] \tag*{\Rbetaplustwo} \\ +\rhd_1(\inlr{s}{t}) = s \tag*{\Rbetainlrone} \\ +\rhd_2(\inlr{s}{t}) = t \tag*{\Rbetainlrtwo} \\ +\inl{\lft{t}} = t \tag*{\Rbetaleft} \\ +\lft{\inl{t}} = t \tag*{\Retaleft} \\ +\ind{\instr_{\lambda x p}(t)} = p[x:=t] \tag*{\Rinstrtest} \\ +\nabla(\instr_{\lambda x p}(t)) = t \tag*{\Rnablainstr} \\ +\text{if } \nabla(t) = x \text{ then } \instr_{\lambda x \ind{t}}(s) = t[x:=s] \tag*{\Retainstr} \\ +\text{if } t : 1 \text{ then } * = t \tag*{\Retaone} \\ +\text{if } t : A \otimes B \text{ then } \plet{x}{y}{t}{x \sotimes y} = t \tag*{\Reta} \\ +\text{if } t : A + B \text{ then } t\case t \of \inl{x} \mapsto \inl{x} \mid \inr{y} \mapsto \inr{y} = t \tag*{\Retaplus} \\ +\text{if } t : A + B \text{ then } \inlr{\rhd_1(t)}{\rhd_2(t)} = t \tag*{\Retainlr} \\ +\text{if } t \text{ is well-typed then } \doo{\_}{t}{\return{\norm{t}}} = t \tag*{\Rbetanorm} \\ +\text{if } t = \doo{\_}{t}{\return{\rho}} \text{ and } 1 / n \leq t, \text{ then } \rho = \norm{t} +\tag*{\Retanorm} \\ +n \cdot 1 / n = \top + \tag*{\Rntimesoneovern} \\ +\text{if } n \cdot t = \top \text{ then } t = 1/n + \tag*{\Rdivide} +\end{gather*} +\end{mdframed} +\caption{Computation rules for $\COMET$} +\label{fig:equations} +\end{figure} + +Figures \ref{fig:typing} and \ref{fig:equations} should be understood +simultaneously. So the term $\inlr{s}{t}$ is well-typed if and only +if we can type $s : A + 1$ and $t : B + 1$ (using the rules in Figure +\ref{fig:typing}), \emph{and} derive the equation $\dom{s} = \ker{t}$ +using the rules in Figure~\ref{fig:equations}. + +The full set of rules of deduction for the system is given in Appendix \ref{section:rules}. + + + + + + + + + + + +\subsection{Linear Type Theory} + +Note the form of several of the typing rules in Figure \ref{fig:typing}, including\Rpair and\Rlett. These rules do not allow +a variable to be duplicated; in particular, we cannot derive the judgement $x : A \vdash x \sotimes x : A \otimes A$. The \emph{contraction} rule does not hold in our type theory --- it is not the case in general that, if $\Gamma, x : A, y : B \vdash \mathcal{J}$, then $\Gamma, z : A \vdash \mathcal{J}[x:=z,y:=z]$. Our theory is thus similar to a \emph{linear} type theory (see for example \cite{Benton93aterm}). + +The reason is that these judgements do not behave well with respect to substitution. For example, take the computation $x : \mathbf{2} \vdash x \sotimes x : 2 \otimes 2$. +If we apply this computation to the scalar $1 / 2$, we presumably wish the result to be $\top \sotimes \top$ with probability $1/2$, and $\bot \sotimes \bot$ with probability $1/2$. But +this is not the semantics for the term $\vdash 1/2 \sotimes 1/2 : 2 \otimes 2$. This term assigns probability $1/4$ to all four possibilities $\top \sotimes \top$, $\top \sotimes \bot$, $\bot \sotimes \top$, +$\top \sotimes \top$. + +\subsection{Defined Constructions} +We can define the following types and computations from the primitive constructions given above. + +\subsubsection{States, Predicates and Scalars} + +A closed term $\vdash t : A$ will be called a \emph{state} of type $A$, and intuitively it represents a probability distribution over the elements of $A$. + +A \emph{predicate} on type $A$ is a proposition of the form $x : A \vdash p : \mathbf{2}$. These shall be the formulas of the logic of $\COMET$ (see Section \ref{section:logic}). + +A \emph{scalar} is a term $s$ such that $\vdash s : \mathbf{2}$. +The closed terms $t$ such that $\vdash t : \mathbf{2}$ are called \emph{scalars}, and represent the \emph{probabilities} or \emph{truth values} of our system. In our intended semantics for discrete and continuous probabilities, these +denote elements of the real interval $[0,1]$. + +Given a state $\vdash t : A$ and a predicate $x : A \vdash p : \mathbf{2}$, we can find the probability that $p$ is true when measured on $t$; this probability is simply the scalar $p[x:=t]$. + +\subsubsection{Coproducts and Copowers} +\label{section:copowers} + +Since we have the coproduct $A + B$ of two types, we can construct the disjoint union of $n$ types $A_1 + \cdots + A_n$ in the obvious way. We write $\nin{1}{n}{}$, \ldots, $\nin{n}{n}{}$ +for its constructors; thus, if $a : A_i$ then $\nin{i}{n}{a} : A_1 + \cdots + A_n$. And given $t : A_1 + \cdots + A_n$, we can eliminate it as: +\[ \case t \of \nin{1}{n}{x_1} \mapsto t_1 \mid \cdots \mid \nin{n}{n}{x_n} \mapsto t_n \enspace . \] +We abbreviate this expression as $\case_{i=1}^n\ t \of \nin{i}{n}{x_i} \mapsto t_i$. + +For the special case where all the types are equal, we write $n \cdot A$ for the type $A + \cdots + A$, where there are $n$ copies of $A$. In category +theory, this is known as the $n$th \emph{copower} of $A$. (We +include the special cases $0 \cdot A \eqdef 0$ and $1 \cdot A \eqdef A$.) + +The \emph{codiagonal} $\nabla(t) : A$ for $t : n \cdot A$ is defined by +\[ \nabla(t) = \case_{i=1}^n\ t \of \nin{i}{n}{x} \mapsto x \enspace . \] +This computation extracts the value of type $A$ and discards the information about which of the $n$ copies it came from. + +We write $\mathbf{n}$ for $n \cdot 1$. Intuitively, this is a finite type with $n$ canonical elements. We denote these elements by $1$, $2$, \ldots, $n$: +\[ i \eqdef \nin{i}{n}{*} : \mathbf{n} \qquad (1 \leq i \leq n) \enspace . \] +For $t : n \cdot A$, we define +\[ \ind{t} = \case_{i=1}^n t \of \nin{i}{n}{\_} \mapsto i : \mathbf{n} \enspace . \] +Thus, if $t = \nin{i}{n}{a}$, then $\ind{t}$ extracts the index $i$ and throws away the value $a$. + +We have the $\lft{}$ construction, which extracts a term of type $A$ from a term of type $A + B$. +We have a similar $\rgt{}$ construction, but there is no need to give primitive rules for this one, as it can be defined in terms of $\lft{}$: +\[ \rgt{t} \eqdef \lft{\swapper{t}} \] +where $\swapper{t} = \pcase{t}{x}{\inr{x}}{y}{\inl{y}}$. + + +\subsubsection{Partial Functions} + +We may see a term $\Gamma \vdash t : A + 1$ as denoting a \emph{partial function} into $A$, which has some probability of terminating (returning a value of form $\inl{s}$) and some probability of diverging (returning $\inr{*}$). +We shall introduce the following notation for dealing with partial functions. + +We define: +\begin{itemize} +\item If $\Gamma \vdash t : A$ then $\Gamma \vdash \return{t} \eqdef \inl{t} : A + 1$. This program converges with probability 1. +\item $\Gamma \vdash \fail \eqdef \inr{*} : A + 1$. This program diverges with probability 1. +\item If $\Gamma \vdash s : A + 1$ and $\Delta, x : A \vdash t : B + 1$ then \\ +$\Gamma, \Delta \vdash \doo{x}{s}{t} \eqdef \pcase{s}{x}{t}{\_}{\fail}$. +\item We introduce the following abbreviation. If $f$ is an expression (such as $\inln$, $\inrn$) such that $f(x)$ is a term, then we write $t \goesto f$ for $\doo{x}{t}{f(x)}$. +\end{itemize} + +The term $\doo{x}{s}{t}$ should be read as the following computation: Run $s$. If $s$ returns a value, pass this as input $x$ to the computation $t$; otherwise, diverge. + +These constructions satisfy these computation rules (Lemma \ref{lm:do}): +\begin{align*} +\doo{x}{\return{s}}{t} & = t[x:=s] \\ +\doo{x}{\fail}{t} & = \fail \\ +\doo{x}{r}{\return{x}} & = r \\ +\doo{\_}{r}{\fail} & = \fail \\ +\doo{x}{r}{(\doo{y}{s}{t})} & = \doo{y}{(\doo{x}{r}{s})}{t} +\end{align*} + +This construction also allows us to define \emph{scalar multiplication}. Given a scalar $\vdash s : \mathbf{2}$ and a substate $\vdash t : A + 1$, the result of multiplying or scaling $t$ by $s$ is $\vdash \doo{\_}{s}{t} : A + 1$. + +\paragraph{Partial Projections} + +Recall that $n \cdot A$ has, as objects, $n$ copies of each object $a : A$, namely $\nin{1}{n}{a}$, \ldots, $\nin{n}{n}{a}$. +Given $t : n \cdot A$, the \emph{partial projection} $\rhd_{i_1 i_2 \cdots i_k}^{n}(t) : A + 1$ is the partial computation that: +\begin{itemize} +\item given an element $\nin{i_r}{n}{a}$, returns $a$; +\item given an element $\nin{j}{n}{a}$ for $j \neq i_1, \ldots, i_k$, diverges. +\end{itemize} + +Formally, we define +\[ \rhd_{i_1 i_2 \cdots i_k}^{n}(t) \eqdef \case_{i=1}^n t \of \nin{i}{n}{x} \mapsto \begin{cases} +\return{x} & \text{if}\ i = i_1, \ldots, i_k \\ +\fail & \text{otherwise} +\end{cases} \] + +\paragraph{Partial Sum} +\label{section:ordering} + +Let $\Gamma \vdash s,t : A + 1$. If these have disjoint domains (i.e. given any input $x$, the sum of the probability that $s$ and $t$ return $a$ is never greater than 1), then we may form the computation $\Gamma \vdash s \ovee t$, +the \emph{partial sum} of $s$ and $t$. The probability that this program converges with output $a$ is the sum of the probability that $s$ returns $a$, and the probability that $t$ returns $a$. The definition is given by the rule \Roveedef; see Section \ref{section:psum}. + +We write $n \cdot t$ for the sum $t \ovee \cdots \ovee t$ with $n$ summands. (We include the special cases $0 \cdot t = \fail$ and $1 \cdot t = t$.) + +With this operation, the partial functions +in $A + 1$ form a \emph{partial commutative monoid} (PCM) (see Lemma \ref{lm:ordering}). + +\subsection{Logic} +\label{section:logic} + +The type $\mathbf{2} = 1 + 1$ shall play a special role in this type theory. It is the type of \emph{propositions} or \emph{predicates}, and its objects shall be used as the formulas of our logic. + +We define $\top \eqdef \inl{*}$ and $\bot \eqdef \inr{*}$. +We also define the \emph{orthosupplement} of a predicate $p$, which roughly corresponds to negation: +\[ p^\bot \eqdef \pcase{p}{\_}{\bot}{\_}{\top} \] + +We immediately have that $p^{\bot \bot} = p$, $\top^\bot = \bot$ and $\bot^\bot = \top$. + +The ordering on $\mathbf{2}$ shall play the role of the \emph{derivability} relation in our logic: $p \leq q$ will indicate that $q$ is derivable from $p$, or that $p$ implies $q$. The rules for this logic +are not the familiar rules of classical or intuitionistic logic. Rather, the predicates over any context form an \emph{effect algebra} (Proposition \ref{prop:logic}). + +In the case of two predicates $p$ and $q$, the partial sum can be thought of as the proposition `$p$ or $q$'. However, +it differs from disjunction in classical or intuitionistic logic as it is a \emph{partial} operation: it is only defined if $p \leq q^\bot$ (Proposition \ref{prop:logic}.\ref{prop:ortho}). +This condition can be thought of as expressing that $s$ and $t$ are \emph{disjoint}; that is, they are never both true. + +\subsubsection{$n$-tests} + +An \emph{$n$-test} in a context $\Gamma$ is an $n$-tuple of predicates $(p_1, \ldots, p_n)$ on $A$ such that +\[ \Gamma \vdash p_1 \ovee \cdots \ovee p_n = \top : \mathbf{2} \enspace . \] + +Intutively, this can be thought of as a set of $n$ fuzzy predicates whose probabilities always sum to 1. We can think of this as a test that +can be performed on the types of $\Gamma$ with $n$ possible outcomes; and, indeed, there is a one-to-one correspondence between +the $n$-tests of $\Gamma$ and the terms of type $\mathbf{n}$ (Lemma \ref{lm:ntest}). + +\subsubsection{Instrument Maps} + +Let $x : A \vdash t : \mathbf{n}$ and $\Gamma \vdash s : A$. +The term $\instr_{\lambda x t}(s) : n \cdot A$ is interpreted as follows: we read the computation $x : A \vdash t : \mathbf{n}$ as a test on the type $A$, with $n$ possible outcomes. +The computation $\instr_{\lambda x t}(s)$ runs $t$ on (the output of) $s$, and returns either $\nin{i}{n}{s}$, where $i$ is the outcome of the test. + + + +Given an $n$-test $(p_1, \ldots, p_n)$ on $A$, we can write a program that tests which of $p_1$, \ldots, $p_n$ is true of its input, and performs one of $n$ different calculations +as a result. We write this program as +\[ \Gamma \vdash \mathsf{measure}\ p_1 \mapsto t_1 \mid \cdots \mid p_n \mapsto t_n \enspace . \] +It will be defined in Definition \ref{df:measure}. + +If $x : A \vdash p : \mathbf{2}$ and $\Gamma, x : A \vdash s,t : A$, we define +\[ \Gamma \vdash (\cond{p}{s}{t}) = \meas\ p \mapsto s \mid p^\bot \mapsto t \enspace . \] +In the case where $s$ and $t$ do not depend on $x$, +we have the following fact (Lemma \ref{lm:measuretwo}.\ref{lm:measurecond}): +\[ \cond{p}{s}{t} = \pcase{p}{\_}{s}{\_}{t} \] + +\subsubsection{Assert Maps} + +If $x : A \vdash p : \mathbf{2}$ is a predicate, we define +\[ \Gamma \vdash \assert_{\lambda x p}(t) \eqdef \pcase{\instr_{\lambda x p}(t)}{x}{\return{x}}{\_}{\fail} : A + 1 \] +The computation $\assert_p(t)$ is a partial computation with output type $A$. It tests whether $p$ is true of $t$; if so, it leaves $t$ unchanged; if not, it diverges. +That is, if $p[x:=t]$ returns $\top$, the computation converges and returns $t$; if not, it diverges. + +These constructions satisfy the following computation rules (see Section \ref{section:assert} below for the proofs). + +\newcommand{\Rassertdown}{(assert$\downarrow$)\xspace} +\newcommand{\Rassertscalar}{(assert-scalar)\xspace} +\newcommand{\Rinstrplus}{(instr$+$)\xspace} +\newcommand{\Rassertplus}{(assert$+$)\xspace} +\newcommand{\Rinstrm}{(instr $m$)\xspace} +\newcommand{\Rassertm}{(assert $m$)\xspace} +\begin{description} +\item[\Rassertdown] +$\dom{(\assert_{\lambda x p}(t))} = p[x:=t]$ +\item[\Rassertscalar] +For a scalar $\vdash s : \mathbf{2}$: $\assert_{\lambda \_ s}(*) = \instr_{\lambda \_ s}(*) = s : \mathbf{2}$. +\item[\Rinstrplus] +For $x : A + B \vdash t : \mathbf{n}$: +\begin{align*} +\instr_{\lambda x t}(s) = +\case s \of +& \inl{y} \mapsto \case_{i=1}^n \instr_{\lambda a. t[x:=\inl{a}]}(y) \of \nin{i}{n}{z} \mapsto \nin{i}{n}{\inl{z}} \\ +& \inr{y} \mapsto \case_{i=1}^n \instr_{\lambda b.t[x:=\inl{b}]}(y) \of \nin{i}{n}{z} \mapsto \nin{i}{n}{\inr{z}} \\ +\end{align*} +\item[\Rassertplus] +For $x : A + B \vdash p : \mathbf{2}$: +\begin{align*} +\assert_{\lambda x p}(t) = \case t \of +& \inl{x} \mapsto \doo{z}{\assert_{\lambda a. p[x:=\inl{a}]}(x)}{\return{\inl{z}}} \mid \\ +& \inr{y} \mapsto \doo{z}{\assert_{\lambda b.p[x:=\inr{b}]}(y)}{\return{\inr{z}}} +\end{align*} +\item[\Rinstrm] +For $x : \mathbf{m} \vdash t : \mathbf{n}$: +\[ \instr_{\lambda x t}(s) = \case_{i=1}^m s \of i \mapsto \case_{j=1}^n t[x:=i] \of j \mapsto \nin{j}{n}{i} \] +\item[\Rassertm] +For $x : \mathbf{m} \vdash p : \mathbf{2}$: +\[ \assert_{\lambda x p}(t) = \case_{i=1}^m t \of i \mapsto \cond{p[x:=i]}{\return{i}}{\fail} \] +\end{description} + +In particular, we have $\assert_{\inln?}(t) = \rhd_1(t)$ and $\assert_{\inrn?}(t) = \rhd_2(t)$. + +\subsubsection{Sequential Product} + +Given two predicates $x : A \vdash p,q : \mathbf{2}$, we can define their \emph{sequential product} +\[ x : A \vdash p \andthen q \eqdef \doo{x}{\assert_p(x)}{q} : \mathbf{2} \enspace . \] +The probability of this predicate being true at $x$ is the product of the probabilities of $p$ and $q$. +This operation has many of the familiar properties of conjunction --- including commutativity --- but not all: in particular, we do not have $p \andthen p^\bot = \bot$ in all cases. (For example, $1/2 \andthen (1 / 2)^\bot = 1/4$.) + +\subsubsection{Coproducts} + +We can define predicates which, given a term $t : A + B$, test which of $A$ and $B$ the term came from. We write these as $\inlprop{t}$ and $\inrprop{t}$. (Compare these with the operators +$FstAnd$ and $SndAnd$ defined in \cite{Jacobs14}.) They are defined by + +\begin{align*} +\inlprop{t} & \eqdef \pcase{t}{\_}{\top}{\_}{\bot} \\ +\inrprop{t} & \eqdef \pcase{t}{\_}{\bot}{\_}{\top} +\end{align*} + +\subsubsection{Kernels} + +The predicate $\inrprop{}$ is particularly important for partial maps. + +Let $\Gamma \vdash t : A + 1$. The \emph{kernel} of the map denoted by $t$ is +\[ \ker{t} \eqdef \inrprop{t} \eqdef \pcase{t}{\_}{\bot}{\_}{\top} \] +Intuitively, if we think of $t$ as a partial computation, then +$\ker{t}$ is the proposition `$t$ does not terminate', or the function +that gives the probability that $t$ will diverge on a given input. + +Its orthosupplement, $(\ker{t})^\bot = \inlprop{t}$, which we shall +also write as $\dom{t}$, is also called the \emph{domain + predicate} of $t$, and represents the proposition that $t$ +terminates. We note that it is equal to $\doo{\_}{t}{\top}$. + + + + + + + + + + + + + +\subsubsection{Normalisation}\label{subsec:normalisation} + +We have a representation of all the rational numbers in our system: let $m / n$ be the term +\[ \overbrace{1 / n \ovee \cdots \ovee 1 / n}^{m} \enspace . \] +The usual arithmetic of rational numbers (between 0 and 1) can be carried out in our system (see Section \ref{sec:scalars}). In particular, for rational numbers $q$ and $r$, we have that if $q \leq r$ then the judgement $q \leq r$ is derivable; +$q \ovee r$ is well-typed if and only if $q + r \leq 1$, in which case $q \ovee r$ is equal to $q + r$; and $q \andthen r = qr$. + +Now, let $\vdash t : A + 1$. Then $t$ represents a \emph{substate} of $A$. +As long as the probability $\dom{t}$ is non-zero, we can +\emph{normalise} this program over the probability of non-termination. The result is the state denoted by $\norm{t}$. Intuitively, the probability that $\norm{t}$ will output $a$ is the probability +that $t$ will output $\inl{a}$, conditioned on the event that $t$ terminates. + +In order to type $\norm{t}$, we must first prove that $t$ has a +non-zero probability of terminating by deriving an inequality of the +form $1 / n \leq \dom{t}$ for some positive integer $n \geq 2$. + +If $\vdash t : A$ and $x : A \vdash p(x) : \mathbf{2}$, we write $\condn{t}{p}$ for +\[ \condn{t}{p} \eqdef \norm{\assert_p(t)} \enspace . \] +The term $t$ denotes a computation whose output is given by a probability distribution over $A$. Then $\condn{t}{p}$ gives the result of normalising that conditional probability distribution +with respect to $p$. + +\subsubsection{Marginalisation} + +The tensor product of type $A \otimes B$ comes with two \emph{projections}. Given $\Gamma \vdash t : A \otimes B$, define +\begin{align*} + \Gamma \vdash \pi_1(t) \eqdef \plet{x}{\_}{t}{x} : A \\ +\Gamma \vdash \pi_2(t) \eqdef \plet{\_}{y}{t}{y} : B +\end{align*} +If $t$ is a state (i..e $\Gamma$ is the empty context), then $\pi_1(t)$ denotes the result of \emph{marginalising} $t$, as +a probability distribution over $A \otimes B$, to a probability distribution over $A$. + +\subsubsection{Local Definition} + +In our examples, we shall make free use of \emph{local definition}. This is not a part of the syntax of $\COMET$ itself, +but part of our metalanguage. We write $\lett x = s \inn t$ for $t[x:=s]$. We shall also locally define functions: we +write $\lett f(x) = s \inn t$ for the result of replacing every subterm of the form $f(r)$ with $s[x:=r]$ in $t$. + +\section{Examples} +\label{section:examples} + +This section describes two examples of (Bayesian) reasoning in our +type theory $\COMET$. The first example is a typical exercise in +Bayesian probability theory. Since such kind of reasoning is not very +intuitive, a formal calculus is very useful. The second example +involves a simple graphical model. + + +\begin{example} + (See also \cite{Yudkowsky2003,Borgstroem2011}) Consider the + following situation. + \begin{quote} + 1\% of a population have a disease. 80\% of subjects with the + disease test positive, and 9.6\% without the disease also test + positive. If a subject is positive, what are the odds they have + the disease? + \end{quote} + +\noindent This situation can be described as a very simple graphical +model, with associated (conditional) probabilities. +$$\vcenter{\xymatrix@R-1pc@C-2pc{ +\ovalbox{HasDisease}\ar[d] +\\ +\ovalbox{PositiveResult} +}} +\qquad\qquad +{\setlength\tabcolsep{0.3em}\begin{tabular}{|c|} +\hline +$\Prob{HD}$ \\ +\hline\hline +$0.01$ \\ +\hline +\end{tabular}} +\qquad +{\setlength\tabcolsep{0.3em}\begin{tabular}{|c|c|} +\hline +$HD$ & $\Prob{PR}$ \\ +\hline\hline +$t$ & $0.8$ \\ +\hline +$f$ & $0.096$ \\ +\hline +\end{tabular}}$$ + +\newcommand{\subj}{\textsf{subject}} +\newcommand{\pr}{\textsf{positive\_result}} + +\noindent In our type theory $\COMET$, we use the following description. +$$\begin{array}{l} +\slet{\subj}{0.01}{} \\ +\qquad\slet{\pr(x)}{(\ifte{x}{0.8}{0.096})}{} \\ +\condn{\subj}{\pr} +\end{array}$$ + +\noindent We thus obtain a state $\subj : \mathbf{2}$, +conditioned on the predicate $\pr$ on $\mathbf{2}$. We calculate the +outcome in semi-formal style. The conditional state +$\condn{\subj}{\pr}$ is defined via normalisation of assert, see +Subsection~\ref{subsec:normalisation}. +We first show what this assert +term is, using the rule \Rassertm and \Rassertscalar: +$$\begin{array}{rcl} +\assert_{\pr}(x) +& = & +\ifn\ x \begin{array}[t]{ll} +\thenn & \doo{\_}{\assert_{\pr(\top)}(x)}{\return{\top}} \\ +\elsen & \doo{\_}{\assert_{\pr(\bot)}(x)}{\return{\bot}} +\end{array} \\ +& = & +\ifn\ x \begin{array}[t]{ll} +\thenn & \doo{\_}{\assert_{0.8}(x)}{\return{\top}} \\ +\elsen & \doo{\_}{\assert_{0.096}(x)}{\return{\bot}} +\end{array} \\ +& = & \ifn x {\begin{array}[t]{rl} +\thenn & \cond{0.8}{\return{\top}}{\fail} \\ +\elsen & \cond{0.096}{\return{\bot}}{\fail} +\end{array}} +\end{array}$$ +\noindent Conditioning requires that the domain of the substate +$\assert_{\pr}(\subj)$ is non-zero. We compute this domain as: +$$\begin{array}{rcll} +\dom{\assert_{\pr}(\subj)} +& = & +\pr(\subj) & (\text{Rule \Rassertdown}) \\ +& = & +\cond{0.01}{0.8}{0.096} \\ +& = & +0.01 \andthen 0.8 \ovee 0.99 \andthen 0.096 \mbox{\qquad} & + (\text{Lemma \ref{lm:measuretwo}.\ref{lm:measurecond}}) \\ +& = & +0.10304 & (\text{Lemma \ref{lm:rational}}) +\end{array}$$ + +\noindent Hence we can choose (for example) $n = 10$, to get +$\frac{1}{n} \leq 0.10304 = \dom{\assert_{\pr}(\subj)}$. + +We now proceed to calculate the result, answering the question in +the beginning of this example. +$$\begin{array}{rcll} +\assert_{\pr}(\subj) +& = & +\ifn 0.01 {\begin{array}[t]{rl} +\thenn & \cond{0.8}{\return{\top}}{\fail} \\ +\elsen & \cond{0.096}{\return{\bot}}{\fail} +\end{array}} \\ +& = & +\meas\ {\begin{array}[t]{lcl} +0.01 \andthen 0.8 +& \mapsto & +\return{\top} \\ +0.01 \andthen 0.8^\bot +& \mapsto & +\fail \\ +0.01^\bot \andthen 0.096 +& \mapsto & +\return{\bot} \\ +0.01^\bot \andthen 0.096^\bot +& \mapsto +& \fail +\end{array}} & (\text{Lemma \ref{lm:measure}.\ref{lm:measureand}}) +\\ +& = & +\meas\ {\begin{array}[t]{lcl} +0.008 +& \mapsto & +\return{\top} \\ +0.09504 +& \mapsto & +\return{\bot} \\ +0.89696 +& \mapsto & +\fail +\end{array}} & (\text{Lemma \ref{lm:measure}.\ref{lm:measureor}}) +\\ +\condn{\subj}{\pr} +& \eqdef & +\norm{\assert_{\pr}(\subj)} \\ +& = & +\meas {\begin{array}[t]{lcl} +0.0776 +& \mapsto & +\top \\ +0.9224 +& \mapsto & +\bot +\end{array}} & (\text{Corollary \ref{cor:normmeasure}}) \\ +& = & +0.0776. & (\text{Lemma \ref{lm:measuretwo}.\ref{lm:measuretwo'}}) +\end{array}$$ + +\noindent Hence the probability of having the disease after a positive +test result is 7.8\%. + + + + + + + + + + +\end{example} + + + + +\begin{example}[Bayesian Network] +The following is a standard example of a problem in Bayesian networks, +created by~\cite[Chap.~14]{RusselN03}. + + + +\begin{quote} +I’m at work, neighbor John calls to say my alarm is ringing. Sometimes +it’s set off by minor earthquakes. Is there a burglar? +\end{quote} + +We are given that the situation is as described by the following +Bayesian network. +$$\vcenter{\xymatrix@R-1pc@C-2pc{ +\ovalbox{Burglary}\ar[dr] & & \ovalbox{Earthquake}\ar[dl] +\\ +& \ovalbox{Alarm}\ar[dl]\ar[dr] & +\\ +\ovalbox{JohnCalls} & & \ovalbox{MaryCalls} +}} +\qquad +\begin{tabular}{c} +{\setlength\tabcolsep{0.3em}\begin{tabular}{|c|} +\hline +$\Prob{B}$ \\ +\hline\hline +$\frac{1}{1000}$ \\ +\hline +\end{tabular}} +\\[2em] +{\setlength\tabcolsep{0.3em}\begin{tabular}{|c|c|} +\hline +$A$ & $\Prob{J}$ \\ +\hline\hline +$t$ & $\frac{9}{10}$ \\ +\hline +$f$ & $\frac{1}{20}$ \\ +\hline +\end{tabular}} +\end{tabular} +\quad +{\setlength\tabcolsep{0.3em}\begin{tabular}{|cc|c|} +\hline +$B$ & $E$ & $\Prob{A}$ \\ +\hline\hline +$t$ & $t$ & $\frac{95}{100}$ \\ +\hline +$t$ & $f$ & $\frac{94}{100}$ \\ +\hline +$f$ & $t$ & $\frac{29}{100}$ \\ +\hline +$f$ & $f$ & $\frac{1}{1000}$ \\ +\hline +\end{tabular}} +\quad +\begin{tabular}{c} +{\setlength\tabcolsep{0.3em}\begin{tabular}{|c|} +\hline +$\Prob{E}$ \\ +\hline\hline +$\frac{1}{500}$ \\ +\hline +\end{tabular}} +\\[2em] +{\setlength\tabcolsep{0.3em}\begin{tabular}{|c|c|} +\hline +$A$ & $\Prob{M}$ \\ +\hline\hline +$t$ & $\frac{7}{10}$ \\ +\hline +$f$ & $\frac{1}{100}$ \\ +\hline +\end{tabular}} +\end{tabular}$$ + +\noindent The probability of each event given its preconditions is as +given in the tables --- for example, the probability that the alarm +rings given that there is a burglar but no earthquake is 0.94. + +We model the above question in $\COMET$ as follows. +$$\begin{array}{l} +\slet{b}{0.01}{\slet{e}{0.002}{}} \\ +\qquad\lett a(x,y) = (\ifn x {\begin{array}[t]{l} + \thenn (\cond{y}{0.95}{0.94}) \\ + \elsen (\cond{y}{0.29}{0.001})) \inn + \end{array}} \\ +\qquad\qquad \slet{j(z)}{(\cond{z}{0.9}{0.05})}{} \\ +\qquad\qquad\qquad \slet{m(z)}{(\cond{z}{0.7}{0.01})}{} \\ +\pi_{1}\big(\condn{b\sotimes e}{j \after a}\big) +\end{array}$$ + +\noindent We first elaborate the predicate $j\after a$, given in +context as $x\colon \mathbf{2}, y\colon \mathbf{2} \vdash j(a(x,y)) +\colon \mathbf{2}$. It is: +$$\begin{array}{rcl} +j(a(x,y)) +& = & +\cond{a(x,y)}{0.90}{0.05} \\ +& = & +\ifn x {\begin{array}[t]{l} + \thenn (\cond{y}{(\cond{0.95}{0.90}{0.05})}{(\cond{0.94}{0.90}{0.05})} \\ + \elsen (\cond{y}{(\cond{0.29}{0.90}{0.05})}{(\cond{0.001}{0.90}{0.05})} +\end{array}} +\\ +& = & +\ifn x {\begin{array}[t]{l} + \thenn (\cond{y}{0.95 \andthen 0.90 \ovee 0.95^{\bot} \andthen 0.05} + {0.94 \andthen 0.90 \ovee 0.94^{\bot} \andthen 0.05}) \\ + \elsen (\cond{y}{0.29\andthen 0.90 \ovee 0.29^{\bot} \andthen 0.05} + {0.001 \andthen 0.90 \ovee 0.001^{\bot} \andthen 0.05} +\end{array}} +\\ +& = & +\ifn x \thenn (\cond{y}{0.8575}{0.849}) \elsen (\cond{y}{0.2965}{0.05085}) +\end{array}$$ + +\noindent The associated assert map is: +$$\begin{array}{rcl} +\assert_{j \after a}(b,e) +& = & +\meas\ {\begin{array}[t]{lcl} +0.001 \andthen 0.002 \andthen 0.8575 +& \mapsto & +\return{\top \sotimes \top} \\ +0.001 \andthen 0.998 \andthen 0.849 +& \mapsto & +\return{\top \sotimes \bot} \\ +0.999 \andthen 0.002 \andthen 0.2965 +& \mapsto & +\return{\bot \sotimes \top} \\ +0.999 \andthen 0.998 \andthen 0.05085 +& \mapsto & +\return{\bot \sotimes \bot} \\ +0.052138976^\bot +& \mapsto & +\fail +\end{array}} \\ +& = & +\meas\ {\begin{array}[t]{lcl} +0.000001715 & \mapsto & +\return{\top \sotimes \top} \\ +0.000847302 & \mapsto & +\return{\top \sotimes \bot} \\ +0.000592407 & \mapsto & +\return{\bot \sotimes \top} \\ +0.050697552 & \mapsto & +\return{\bot \sotimes \bot} \\ +0.052138976^\bot & \mapsto & +\fail +\end{array}} +\end{array}$$ + +\noindent Hence by Corollary~\ref{cor:normmeasure} we obtain the +marginalised conditional: +$$\begin{array}{rcl} +\pi_{1}\big(\condn{b\sotimes e}{j \after a}\big) +& = & +\pi_{1}\big(\norm{\assert_{j \after a}(b,e)}\big) \\ +& = & +\pi_{1}\big(\meas\ {\begin{array}[t]{lcl} +\nicefrac{0.000001715}{0.052138976} +& \mapsto & +\top \sotimes \top \\ +\nicefrac{0.000847302}{0.052138976} +& \mapsto & +\top \sotimes \bot \\ +\nicefrac{0.000592407}{0.052138976} +& \mapsto & +\bot \sotimes \top \\ +\nicefrac{0.050697552}{0.052138976} +& \mapsto & +\bot \sotimes \bot\,\big) \\ +\end{array}} \\ +& = & +\meas\ {\begin{array}[t]{lcl} +0.000032893 +& \mapsto & +\pi_{1}(\top \sotimes \top) \\ +0.016250837 +& \mapsto & +\pi_{1}(\top \sotimes \bot) \\ +0.011362078 +& \mapsto & +\pi_{1}(\bot \sotimes \top) \\ +0.972354194 +& \mapsto & +\pi_{1}(\bot \sotimes \bot) \\ +\end{array}} +\\ +& = & +\meas\ {\begin{array}[t]{lcl} +0.000032893 +& \mapsto & +\top \\ +0.016250837 +& \mapsto & +\top \\ +0.011362076 +& \mapsto & +\bot \\ +0.972354194 +& \mapsto & +\bot \\ +\end{array}} +\\ +& = & +\meas\ {\begin{array}[t]{lcl} +0.01628373 +& \mapsto & +\top \\ +0.98371627 +& \mapsto & +\bot \\ +\end{array}} +\\ +& = & +0.01628373 \end{array}$$ + +\noindent We conclude that there is a 1.6\% chance of a burglary when +John calls. + + + +\end{example} + +\section{Metatheorems} +\label{section:metatheorems} + +We presented an overview of the system in Section \ref{section:overview}, and gave the intuitive meaning of the terms of $\COMET$. +In this section, we proceed to a more formal development of the theory, and investigate what can be proved within the system. + +The type theory we have presented enjoys the following standard properties. + +\begin{lemma} +\label{lm:meta} +$ $ +\begin{enumerate} +\item \textbf{Weakening} +\label{lm:weak} + If $\Gamma \vdash \mathcal{J}$ and $\Gamma \subseteq \Delta$ then $\Delta \vdash \mathcal{J}$. +\item \textbf{Substitution} + If $\Gamma \vdash t : A$ and $\Delta, x : A \vdash \mathcal{J}$ then $\Gamma, \Delta \vdash \mathcal{J}[x:=t]$. +\item \textbf{Equation Validity} + If $\Gamma \vdash s = t : A$ then $\Gamma \vdash s : A$ and $\Gamma \vdash t : A$. +\item \textbf{Inequality Validity} +If $\Gamma \vdash s \leq t : A + 1$ then $\Gamma \vdash s : A + 1$ and $\Gamma \vdash t : A + 1$. +\item \textbf{Functionality} +If $\Gamma \vdash r = s : A$ and $\Delta, x : A \vdash t : B$ then $\Gamma, \Delta \vdash t[x:=r] = t[x:=s] : B$. +\end{enumerate} +\end{lemma} + +\begin{proof} +The proof in each case is by induction on derivations. Each case is straightforward. +\end{proof} + +The following lemma shows that substituting within our binding operations works as desired. + +\begin{lemma} +\label{lm:sub} + \begin{enumerate} + \item \label{lm:letsub}If $\Gamma \vdash r : A \otimes B$; $\Delta, x : A, y : B \vdash s : C$; and $\Theta, z : C \vdash t : D$ +then $\Gamma, \Delta, \Theta \vdash t[z:=\plet{x}{y}{r}{s}] = \plet{x}{y}{r}{t[z:=s]} : D$. +\item \label{lm:casesub} If $\Gamma \vdash r : A + B$; $\Delta, x :A \vdash s : C$; $\Delta, y : B \vdash s' : C$; and $\Theta, z : C \vdash t : D$ then +$$\Gamma, \Delta, \Theta \vdash \begin{array}[t]{l} +t[z:=\pcase{r}{x}{s}{y}{s'}] \\ += \pcase{r}{x}{t[z:=s]}{y}{t[z:=s']} : D +\end{array} \enspace . $$ + \end{enumerate} +\end{lemma} + +\begin{proof} + For part 1, we us the following `trick' to simulate local definition (see \cite{Adams2014}): +\begin{align*} +\lefteqn{t[z := \pcase{r}{x}{s}{y}{s'}]} \\ +& = \plet{z}{\_}{(\pcase{r}{x}{s}{y}{s'}) \sotimes *}{t} & \Rbeta \\ +& = \plet{z}{\_}{\pcase{r}{x}{s \sotimes *}{y}{s' \sotimes *}}{t} & \Rcasepair \\ +& = \pcase{r}{x}{\plet{z}{\_}{s \sotimes *}{t}}{y}{\plet{z}{\_}{s' \sotimes *}{t}} & \Rletcase \\ +& = \pcase{r}{x}{t[z:=s]}{y}{t[z:=s']} & \Rbeta +\end{align*} +Part 2 is proven similarly using\Rletpair and\Rletlet. +\end{proof} + +\begin{corollary} +\label{cor:vacsub} + \begin{enumerate} + \item If $\Gamma \vdash s : A \otimes B$ and $\Delta \vdash t : C$ then +$\Gamma, \Delta \vdash \plet{\_}{\_}{s}{t} = t : C$. + \item \label{cor:vaccase} If $\Gamma \vdash s : A + B$ and $\Delta \vdash t : C$ then $\Gamma, \Delta \vdash \pcase{s}{\_}{t}{\_}{t} = t : C$. + \end{enumerate} +\end{corollary} + +\begin{proof} + These are both the special case where $z$ does not occur free in $t$. +\end{proof} + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +\subsection{Coproducts} + +We generalise the $\inlprop{}$ and $\inrprop{}$ constructions as follows. +Define the predicate $\intest{i}{}$ on $n \cdot A$, which tests whether a term +comes from the $i$th component, as follows. +\[ \intest{i}{t} \eqdef \case_{j=1}^n t \of \nin{j}{n}{\_} \mapsto \begin{cases} +\top & \text{if } i = j \\ +\bot & \text{if } i \neq j +\end{cases} \] + + + + + + + + +\subsection{The Do Notation} + +Our construction $\doo{x}{s}{t}$ satisfies the following laws. + +\begin{lemma} +\label{lm:do} +Let $\Gamma \vdash r : A + 1$, $\Delta, x : A \vdash s : B + 1$, and $\Theta, y : B \vdash t : C$. Let +also $\Gamma \vdash r' : A$. Then +\begin{align*} +\Gamma, \Delta \vdash \doo{x}{\return{r'}}{s} & = t[x:=s] : B + 1 \\ +\Gamma, \Delta \vdash \doo{x}{\fail}{s} & = \fail : B + 1 \\ +\Gamma \vdash \doo{x}{r}{\return{x}} & = r : A + 1 \\ +\Gamma \vdash \doo{\_}{r}{\fail} & = \fail : B + 1 \\ +\Gamma, \Delta, \Theta \vdash \doo{x}{r}{(\doo{y}{s}{t})} & = \doo{y}{(\doo{x}{r}{s})}{t} : C +\end{align*} +\end{lemma} + +\begin{proof} + These all follow easily from the rules for coproducts\Rbetaplusone,\Rbetaplustwo,\Retaplus and\Rcasecase. +\end{proof} + +\subsection{Kernels} + +\begin{lemma}$ $ +\label{lm:kernel} +\begin{enumerate} +\item +If $\Gamma \vdash t : A + 1$ then $\Gamma \vdash \dom{t} = (\doo{\_}{t}{\top}) : \mathbf{2}$ +\item +\label{lm:kernel2} +Let $\Gamma \vdash t : A + 1$. Then +$\Gamma \vdash \dom{t} = \bot : \mathbf{2}$ if and only if $\Gamma \vdash t = \fail : A + 1$. +\item +Let $\Gamma \vdash s : A + 1$ and $\Delta, x : A \vdash t : B + 1$. Then +$\Gamma, \Delta \vdash \dom{(\doo{x}{s}{t})} = \doo{x}{s}{\dom{t}} : \mathbf{2}$. +\end{enumerate} +\end{lemma} + +\begin{proof}$ $ +\begin{enumerate} +\item +This holds just by expanding definitions. +\item +Obviously, $(\dom{\fail}) = \bot$. For the converse, if $\dom{t} = +\bot$ then $\ker{t} = \top$ and so $t = \inr{\rgt{t}} = \inr{*}$ by\Retaone. +\item +$ \begin{aligned}[t] +(\dom{\pcase{s}{x}{t}{\_}{\fail}}) & = \pcase{s}{x}{\dom{t}}{\_}{\dom{\fail}} \\ +& = \pcase{s}{x}{\dom{t}}{\_}{\bot} +\end{aligned} $ +\end{enumerate} +\end{proof} + +\subsection{Finite Types} + +\begin{lemma} +\label{lm:rhdfin} +Let $\Gamma \vdash t : \mathbf{n}$ and $i \leq n$. +If $\Gamma \vdash \rhd_i(t) = \top : \mathbf{2}$ then $\Gamma \vdash t = i : \mathbf{n}$. +\end{lemma} + +\begin{proof} +Define $x : \mathbf{n} \vdash f(x) : 1 + \mathbf{n-1}$ by +\[ f(x) \eqdef \case_{j=1}^n x \of \begin{cases} +\inr{j} & \text{if } j < i \\ +\inl{*} & \text{if } j = i \\ +\inr{j-i} & \text{if } j > i +\end{cases} \] +Then $\Gamma \vdash \inlprop{f(t)} = \top : \mathbf{2}$, hence +\[ f(t) = \inl{\lft{f(t)}} = \inl{*} \] +We can define an inverse to $f$: given $x : 1 + \mathbf{n-1}$, define +\[ f^{-1}(x) \eqdef \case x \of \inl{\_} \mapsto i \mid \inr{t} \mapsto \case_{j=1}^{n-1} t \of j \text{ if } j < i \mid j + 1 \text{ if } j \geq i \] +Then $x : \mathbf{n} \vdash f^{-1}(f(x)) = x : 1 + \mathbf{n-1}$ and so $\Gamma \vdash t = f^{-1}(f(t)) = f^{-1}(\inl{*}) = i : \mathbf{n}$. +\end{proof} + +\subsection{Ordering on Partial Maps and the Partial Sum} +\label{section:psum} +Note that, from the rules\Roveeprime and\Roveedef, we have $\Gamma \vdash s \ovee t : A + 1$ if and only if there exists $\Gamma \vdash b : (A + A) + 1$ such that +\[ \Gamma \vdash b \goesto \rhd_1 = s : A + 1, \qquad \Gamma \vdash b \goesto \rhd_2 = t : A + 1 \enspace , \] +in which case $\Gamma \vdash s \ovee t = \doo{x}{b}{\return{\nabla(x)}} : A + 1$. We say that such a term $b$ is a \emph{bound} for $s \ovee t$. By +the rule\RJMprime, this bound is unique if it exists. + +\begin{lemma} +For predicates $\Gamma \vdash p, q : \mathbf{2}$, we have that $\Gamma \vdash b : \mathbf{3}$ is a bound for $p \ovee q$ if and only if $\rhd_1(b) = p$ and $\rhd_2(b) = q$. +\end{lemma} + +\begin{proof} + This holds because $b \goesto \rhd_1 = \rhd_1(b)$ and $b \goesto \rhd_2 = \rhd_2(b)$, as can be seen just from expanding definitions. +\end{proof} + +The set of \emph{partial} maps $A \rightarrow B + 1$ between any two types $A$ and $B$ form a \emph{partial commutative monoid} (PCM) with least element $\fail$, as shown by the following results. + +\begin{lemma}$ $ + \label{lm:ordering} + \begin{enumerate} + \item \label{lm:zerolaw} If $\Gamma \vdash t : A + 1$ then $\Gamma \vdash t \ovee \fail = t : A + 1$. + \item (\textbf{Commutativity}) If $\Gamma \vdash s \ovee t : A + 1$ then $\Gamma \vdash t \ovee s : A + 1$ and $\Gamma \vdash s \ovee t = t \ovee s : A + 1$. + \item (\textbf{Associativity}) $\Gamma \vdash (r \ovee s) \ovee t : A + 1$ if and only if $\Gamma \vdash r \ovee (s \ovee t) : A + 1$, in which case $\Gamma \vdash r \ovee (s \ovee t) = (r \ovee s) \ovee t : A + 1$. \label{lm:assoc} + \end{enumerate} +\end{lemma} + +\begin{proof} + \begin{enumerate} + \item The bound is $\doo{x}{t}{\return{\inl{x}}}$. +\item +Let $b$ be a bound for $s \ovee t$. Then $\doo{x}{b}{\return{\swapper{x}}}$ is a bound for $t \ovee s$ and we have +\begin{align*} +t \ovee s & = \doo{y}{(\doo{x}{b}{\return{\swapper{x}}})}{\return{\nabla(y)}} \\ +& = \doo{x}{b}{\doo{y}{\return{\swapper{x}}}{\return{\nabla(y)}}} \\ +& = \doo{x}{b}{\return{\nabla(\swapper{x})}} = \doo{x}{b}{\return{\nabla(x)}} \\ +& = s \ovee t +\end{align*} +\item +This is proved in Appendix \ref{section:associativity} +\end{enumerate} +\end{proof} + +\begin{lemma} +\label{lm:oveeleq} +Let $\Gamma \vdash r : A + 1$ and $\Gamma \vdash s : A + 1$. Then $\Gamma \vdash r \leq s : A + 1$ if and only if there exists $t$ such that $\Gamma \vdash r \ovee t = s : A + 1$. +\end{lemma} + +\begin{proof} +Suppose $r \leq s$. If $b$ is such that $\doo{x}{b}{\rhd_1(x)} = r$ and $\doo{x}{b}{\return{\nabla(x)}} = s$ then take $t = \doo{x}{b}{\rhd_2(x)}$. + +Conversely, if $r \ovee t = s$, then inverting the derivation of $\Gamma \vdash r \ovee t : A + 1$ we have that there exists $b$ such that $r = +\doo{x}{b}{\rhd_1(x)}$, $t = \doo{x}{b}{\rhd_2(x)}$ and $s = r \ovee t = \doo{x}{b}{\return{\nabla(x)}}$. Therefore, $r \leq s$ by\RleqI. +\end{proof} + +\begin{corollary} + Let $\Gamma \vdash r : A + 1$ and $\Gamma \vdash s : A + 1$. Then $\Gamma \vdash r \leq s : A + 1$ if and only if there exists $b$ such that $\Gamma \vdash b : (A + A) + 1$, +$\Gamma \vdash b \goesto \rhd_1 = s : A + 1$, and $\Gamma \vdash \doo{x}{b}{\return{\nabla(x)}} = s : A + 1$. +\end{corollary} + +This term $b$ is called a \emph{bound} for $s \leq t$. + +Using this characterisation of the ordering relation, we can read off several properties directly from Lemma \ref{lm:ordering}. + +\begin{lemma} +\begin{enumerate} + \item If $\Gamma \vdash s \ovee t : A + 1$ then $\Gamma \vdash s \leq s \ovee t : A + 1$ and $\Gamma \vdash t \leq s \ovee t : A + 1$. \label{lm:leqovee} + \item If $\Gamma \vdash t : A + 1$ then $\Gamma \vdash t \leq t : A + 1$. + \item If $\Gamma \vdash t : A + 1$ then $\Gamma \vdash \fail \leq t : A + 1$. + \item If $\Gamma \vdash r \leq s : A + 1$ and $\Gamma \vdash s \leq t : A + 1$ then $\Gamma \vdash r \leq t : A + 1$. + \item If $\Gamma \vdash r \leq s : A + 1$ and $\Gamma \vdash s \ovee t : A + 1$ then $\Gamma \vdash r \ovee t \leq s \ovee t : A + 1$. +\end{enumerate} +\end{lemma} + +\begin{proof} +\begin{enumerate} +\item +From Lemma \ref{lm:oveeleq} and Commutativity. +\item From Lemma \ref{lm:oveeleq} and Lemma \ref{lm:ordering}.\ref{lm:zerolaw}. +\item From Lemma \ref{lm:oveeleq} and Lemma \ref{lm:ordering}.\ref{lm:zerolaw}. +\item From Lemma \ref{lm:oveeleq} and Associativity. +\item +Let $r \ovee x = s$. Then $r \ovee x \ovee t = s \ovee t$ and so $r \ovee t \leq s \ovee t$. +\end{enumerate} +\end{proof} + +On the predicates, we have the following structure, which shows that they form an \emph{effect algebra}. (In fact, they have more structure: +they form an \emph{effect module} over the scalars, as we will prove in Proposition \ref{prop:effmod}.) + +\begin{proposition} +\label{prop:logic} +Let $\Gamma \vdash p,q,r : \mathbf{2}$. + \begin{enumerate} + \item If $\Gamma \vdash p : \mathbf{2}$ then $\Gamma \vdash p \ovee p^\bot = \top : \mathbf{2}$. + \item If $\Gamma \vdash p \ovee q = \top : \mathbf{2}$ then $\Gamma \vdash q = p^\bot : \mathbf{2}$. + \item (\textbf{Zero-One Law}) If $\Gamma \vdash p \ovee \top : \mathbf{2}$ then $\Gamma \vdash p = \bot : \mathbf{2}$. + \item \label{prop:ortho} $\Gamma \vdash p \ovee q : \mathbf{2}$ if and only if $\Gamma \vdash p \leq q^\bot : \mathbf{2}$. + \end{enumerate} +\end{proposition} + +\begin{proof} +\begin{enumerate} +\item +The term $\inl{p} : \mathbf{2} + 1$ is a bound for $p \ovee p^\bot$, and $\doo{x}{\inl{p}}{\return{\nabla(x)}} = \top$. +\item +Let $b$ be a bound for $p \ovee q$. We have +\begin{align*} +\top & = \doo{x}{b}{\return{\nabla(x)}} = \doo{x}{b}{\top} & \text{using \Retaone} \\ +& = \dom{b} +\end{align*} +Therefore, $b = \inl{\lft{b}}$ by\Rbetaleft, and so +\begin{align*} + p & = \rhd_1(\lft{b}), \qquad q = \rhd_2(\lft{b}) = \rhd_1(\lft{b})^\bot = p^\bot +\end{align*} +\item Let $b$ be a bound for $p \ovee \top$. Then $\rhd_2(b) = \top$ and so $b = 2 : \mathbf{3}$ by Lemma \ref{lm:rhdfin}. Therefore, $p = \rhd_1(b) = \bot$. +\item +Suppose $p \ovee q : \mathbf{2}$. Then $p \ovee q \ovee (p \ovee q)^\bot = \top$, hence $p \ovee (p \ovee q)^\bot = q^\bot$, and +thus $p \leq q^\bot$. + +Conversely, if $p \leq q^\bot$, let $p \ovee x = q^\bot$. Then $\top = q \ovee q^\bot = p \ovee q \ovee x$, and so $p \ovee q : \mathbf{2}$. +\end{enumerate} +\end{proof} + +\begin{corollary} + \begin{enumerate} + \item (\textbf{Cancellation}) If $\Gamma \vdash p \ovee q = p \ovee r : \mathbf{2}$ then $\Gamma \vdash q = r : \mathbf{2}$. + \item (\textbf{Positivity}) If $\Gamma \vdash p \ovee q = \bot : \mathbf{2}$ then $\Gamma \vdash p = \bot : \mathbf{2}$ and $\Gamma \vdash q = \bot : \mathbf{2}$. + \item If $\Gamma \vdash p : \mathbf{2}$ then $\Gamma \vdash p \leq \top : \mathbf{2}$. + \item If $\Gamma \vdash p \leq q : \mathbf{2}$ then $\Gamma \vdash q^\bot \leq p^\bot : \mathbf{2}$. + \end{enumerate} +\end{corollary} + +\begin{proof} + \begin{enumerate} + \item We have + \begin{gather*} + p \ovee q \ovee (p \ovee q)^\bot = p \ovee r \ovee (p \ovee q)^\bot = \top \\ +\therefore q = r = (p \ovee (p \ovee q)^\bot)^\bot + \end{gather*} + \item +If $p \ovee q = \bot$ then $p \ovee q \ovee \top : \mathbf{2}$, hence $p \ovee \top : \mathbf{2}$ by Associativity, and so $p = \bot$ by the Zero-One Law. + \item +We have $p \ovee p^\bot = \top$ and so $p \leq \top$. +\item +Let $p \ovee x = q$. Then $\top = q \ovee q^\bot = p \ovee x \ovee q^\bot$, and so $p^\bot = x \ovee q^\bot$. Thus, $q^\bot \leq p^\bot$. + \end{enumerate} +\end{proof} + +Our next lemma shows how $\ovee$ and $\mathsf{case}$ interact. + +\begin{lemma} +\label{lm:caseovee} +Suppose $\Gamma \vdash r : A + B$ and $\Delta, x : A \vdash s \ovee t : C + 1$ and $\Delta, y : B \vdash s' \ovee t' : C + 1$. Then +\[ \Gamma, \Delta \vdash \begin{array}[t]{l} +\pcase{r}{x}{s \ovee t}{y}{s' \ovee t'} \\ += (\pcase{r}{x}{s}{y}{s'}) \ovee (\pcase{r}{x}{t}{y}{t'}) : C + 1 +\end{array} \] +\end{lemma} + +\begin{proof} + Let $b(x)$ be a bound for $s \ovee t$ in $\Delta, x : A$, and $c(y)$ a bound for $s' \ovee t'$ in $\Delta, y : B$. Then +\[ \pcase{r}{x}{b(x)}{y}{c(y)} : (B + B) + 1 \] +is a bound for $(\pcase{r}{x}{s}{y}{s'}) \ovee (\pcase{r}{x}{t}{y}{t'})$, and so +\begin{align*} +\lefteqn{(\pcase{r}{x}{s}{y}{s'}) \ovee (\pcase{r}{x}{t}{y}{t'})} \\ +& = \doo{z}{\pcase{r}{x}{b(x)}{y}{c(y)}}{\return{\nabla(z)}} \\ +& = \pcase{r}{x}{\doo{z}{b(x)}{\return{\nabla(z)}}}{y}{\doo{z}{c(y)}{\return{\nabla(z)}}} \\ +& = \pcase{r}{x}{s \ovee t}{y}{s' \ovee t'} +\end{align*} +\end{proof} + +\begin{corollary} +\label{cor:doovee} + If $\Gamma \vdash r : A + 1$ and $\Delta, x : A \vdash s \ovee t : B + 1$ then +\[ \Gamma, \Delta \vdash \doo{x}{r}{s \ovee t} = (\doo{x}{r}{s}) \ovee (\doo{x}{r}{t}) : B + 1 \enspace . \] +\end{corollary} + +\begin{proof} + \begin{align*} + \doo{x}{r}{s \ovee t} = & \pcase{r}{x}{s \ovee t}{\_}{\fail \ovee \fail} \\ += & (\pcase{r}{x}{s}{\_}{\fail}) \ovee \\ +& (\pcase{r}{x}{t}{\_}{\fail}) + \end{align*} +\end{proof} + +The following lemma relates the structures on partial maps and predicates via the domain operator. + +\begin{lemma} + If $\Gamma \vdash s \ovee t : A + 1$ then $\Gamma \vdash \dom{(s \ovee t)} = \dom{s} \ovee \dom{t} : \mathbf{2}$. +\end{lemma} + +\begin{proof} + Let $b$ be a bound for $s \ovee t$. Then +\[ \dom{(s \ovee t)} = \dom{(\doo{x}{b}{\return{\nabla(x)}})} = \doo{x}{b}{\top} = \dom{b} \] +We also have +\begin{align*} + \dom{s} & = \doo{x}{b}{\inlprop{x}}, \qquad \dom{t} = \doo{x}{b}{\inrprop{x}} \\ +\therefore \dom{s} \ovee \dom{t} & = \doo{x}{b}{\inlprop{x} \ovee \inrprop{x}} & (\text{previous part}) \\ +& = (\doo{x}{b}{\top}) = \dom{b} +\end{align*} +\end{proof} + +Using this, we can conclude several properties about partial maps immediately from the fact that they hold for predicates: + +\begin{lemma} +\begin{enumerate} + \item (\textbf{Restricted Cancellation Law}) If $\Gamma \vdash s \ovee t = t : A + 1$ then $\Gamma \vdash s = \fail : A + 1$. + \item (\textbf{Positivity}) If $\Gamma \vdash s \ovee t = \fail : A + 1$ then $\Gamma \vdash s = \fail : A + 1$ and $\Gamma \vdash t = \fail : A + 1$. + \item If $\Gamma \vdash s \leq t : A + 1$ and $\Gamma \vdash t \leq s : A + 1$ then $\Gamma \vdash s = t : A + 1$. +\end{enumerate} +\end{lemma} + +\begin{proof} +\begin{enumerate} +\item +Suppose $\Gamma \vdash s \ovee t = t : A + 1$. Then $\Gamma \vdash \dom{(s \ovee t)} = \dom{s} \ovee \dom{t} = \dom{t} : \mathbf{2}$, +and so $\Gamma \vdash \dom{s} = \bot : \mathbf{2}$ and $\Gamma \vdash s = \fail : A + 1$ by Lemma \ref{lm:kernel}.\ref{lm:kernel2}. +\item Suppose $\Gamma \vdash s \ovee t = \fail$. Then $\dom{(s \ovee + t)} = \dom{s} \ovee \dom{t} = \bot$, and so $\dom{s} = \bot$ and +$\dom{t} = \bot$. Therefore, $s = \fail$ and $t = \fail$ by Lemma +\ref{lm:kernel}.\ref{lm:kernel2}. +\item +Let $s \ovee b = t$ and $t \ovee c = s$. Then $s \ovee b \ovee c = s$ and so $b \ovee c = \fail$ by the Restricted Cancellation Law, hence $b = c = \fail$ by Positivity. +Thus, $s = s \ovee \fail = t$. + \end{enumerate} +\end{proof} + +Finally, we can show that the partial projections on copowers behave as expected with respect to $\ovee$. + +\begin{lemma} +For $t : n \cdot A$, +\[ \rhd_{i_1, \ldots, i_k}(t) = \rhd_{i_1}(t) \ovee \cdots \ovee \rhd_{i_k}(t) \] +\end{lemma} + +\begin{proof} +The proof is by induction on $k$. Take \[ b = \case_{i=1}^n t \of \nin{i}{n}{\_} \mapsto \begin{cases} 1 & \text{if } i = i_1, \ldots, i_k\\ +2 & \text{if } i = i_{k+1} \\ +3 & \text{otherwise} +\end{cases}\] +Then $\rhd_1(b) = \rhd_{i_1\cdots i_k}(t)$, $\rhd_2(b) = \rhd_{i_{k+1}}(t)$, and $\rhd_{12}(b) = \rhd_{i_1\cdots i_ki_{k+1}}(t)$. Therefore, +\[ \rhd_{i_1i_2\cdots i_{k+1}}(t) = \rhd_{i_1\cdots i_k}(t) \ovee \rhd_{i_{k+1}}(t) = \rhd_{i_1}(t) \ovee \cdots \ovee \rhd_{i_{k+1}}(t) \] +by the induction hypothesis. +\end{proof} + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +\subsubsection{Assert Maps} +\label{section:assert} + +Recall that, for $x : A \vdash p : \mathbf{2}$ and $\Gamma \vdash t : A$, we define $\Gamma \vdash \assert_{\lambda x p}(t) \eqdef \rhd_1(\instr_{\lambda x p}(t)) : A + 1$. + +This operation $\assert$ forms a bijection between: +\begin{itemize} +\item +the terms $p$ such that $x : A \vdash p : \mathbf{2}$ (the predicates on $A$); and +\item +the terms $t$ such that $x : A \vdash t \leq \return{x} : A + 1$ +\end{itemize} + +This is proven by the following result. + +\begin{lemma} +\label{lm:assert} +If $x : A \vdash p : 1 + 1$ and $\Gamma \vdash t : A$, then +\begin{enumerate} +\item +$\Gamma \vdash \assert_{\lambda x p}(t) : A + 1$ +\item +$\Gamma \vdash \assert_{\lambda x p}(t) \leq \inl{t} : A + 1$. +\item \textbf{\Rassertdown} +\label{lm:assertdown} +$\Gamma \vdash \dom{\assert_{\lambda x p}(t)} = [t/x] p : \mathbf{2}$ +\item +If $x : A \vdash t \leq \inl{x} : A + 1$ then $x : A \vdash t = \assert_{\lambda x (\dom{t})}(x) : A + 1$. +\end{enumerate} +\end{lemma} + +\begin{proof} +\begin{enumerate} +\item +An easy application of the rules\Rinstr,\Rcase,\Rinl,\Rinr and\Runit. +\item +The term $\inl{\instr_{\lambda x p}(t)}$ is a bound for this inequality. +\item +\begin{align*} +\dom{\assert_{\lambda x p}(t)} & \eqdef \dom{\rhd_1(\instr_{\lambda x p}(t))} = \inlprop{\instr_{\lambda x p}(t)} \\ +& = p[x:=t] & \text{by\Rinstrtest} +\end{align*} +\item +Let $b$ be a bound for the inequality $t \leq \inl{x}$, so $(b \goesto \rhd_1) = t$ and $\doo{x}{b}{\return{\nabla(x)}} = \inl{x}$. +Then +\[ \dom{b} = \dom{(\doo{x}{b}{\return{\nabla(x)}})} = \dom{\inl{x}} = \top . \] +Hence we can define $c = \lft{b}$. We therefore have $\rhd_1(c) = t$ and $\nabla(c) = x$. +Now, the rule\Retainstr gives us +\begin{gather*} +c = \instr_{\lambda x \inlprop{c}}(x) = \instr_{\dom{\lambda x t}}(x) \\ +\therefore t = \rhd_1(c) = \assert_{\dom{\lambda x t}}(x) +\end{gather*} +\end{enumerate} +\end{proof} + +We now give rules for calculating $\instr_{\lambda x p}$ and $\assert_{\lambda x p}$ directed by the type. + +\begin{lemma}[\Rassertscalar] +If $\vdash s : \mathbf{2}$ then +\[ \vdash \assert_{\lambda \_ s}(*) = \instr_{\lambda \_ s}(*) = s : \mathbf{2} \] +\end{lemma} + +\begin{proof} +We have $\nabla(s) = *$ by\Retaone and $\dom{s} = s$ by\Retaplus. The +result follows by\Retainstr. +\end{proof} + +\begin{lemma}[\Rinstrplus,\Rassertplus] + If $x : A + B \vdash p : \mathbf{2}$ and $\Gamma \vdash t : A + B$ then +\begin{align*} +\Gamma \vdash \instr_{\lambda x p}(t) = \case t \of +& \inl{y} \mapsto (\inln + \inln)(\instr_{\lambda a. p[x:=\inl{a}]}(y)) \mid \\ +& \inr{z} \mapsto (\inrn + \inrn)(\instr_{\lambda b. p[x:=\inr{b}]}(z)) \\ +\Gamma \vdash \assert_{\lambda x p}(t) = \case t \of +& \inl{y} \mapsto \doo{w}{\assert_{\lambda a. p[x:=\inl{a}]}(y)}{\return{\inl{w}}} \mid \\ +& \inr{z} \mapsto \doo{w}{\assert_{\lambda b.p[x:=\inr{b}]}(z)}{\return{\inr{w}}} +\end{align*} +where $(\inln + \inln)(t) \eqdef \pcase{t}{x}{\inl{x}}{y}{\inl{y}}$, and $(\inrn + \inrn)(t)$ is defined similarly. +\end{lemma} + +\begin{proof} +For $x : A + B$, let us write $f(x)$ for +\begin{align*} +f(x) \eqdef +\case x \of +& \inl{y} \mapsto (\inln + \inln)(\instr_{\lambda a.p[\inl{a}]}(y)) \mid \\ +& \inr{z} \mapsto (\inrn + \inrn)(\instr_{\lambda b.p[\inr{b}]}(z)) +\end{align*} +We shall prove $f(x) = \instr_{\lambda x p}(x)$. + +We have +\begin{align*} +\nabla(f(x)) & = \case x \of \begin{array}[t]{l} \inl{y} \mapsto +\inl{\nabla(\assert_{\lambda a.p[x:=\inl{a}]}(y))} \mid \\ +\inr{z} \mapsto \inr{\nabla(\assert_{\lambda b.p[\inr{b}]}(z))} +\end{array} \\ +& = \pcase{x}{y}{\inl{y}}{z}{\inr{z}} \\ +& = x & \text{by \Retaplus} \\ + \dom{f(x)} & = \pcase{x}{y}{\dom{\instr_{\lambda a.p[x:=\inl{a}]}(y)}}{z}{\instr_{\lambda b.p[\inr{b}]}(z)} \\ +& = \pcase{x}{y}{p[x:=\inl{y}]}{z}{p[x:=\inr{z}]} \\ +& = p & \text{by Corollary \ref{cor:vacsub}.\ref{cor:vaccase}} +\end{align*} +Hence $f(x) = \instr_p(x)$ by\Retainstr. +\end{proof} + +\begin{corollary}[\Rinstrm,\Rassertm] +\label{cor:assertn} +\begin{enumerate} +\item +Given $x : \mathbf{m} \vdash t : \mathbf{n}$ and $\Gamma \vdash s : \mathbf{m}$, +\[ \instr_{\lambda x t}(s) = \case_{i=1}^m\ s \of i \mapsto \case_{j=1}^n\ t[x:=i] \of j \mapsto \nin{j}{n}{i} \enspace . \] +\item +Given $x : \mathbf{n} \vdash p : \mathbf{2}$ and $\Gamma \vdash t : \mathbf{n}$, +\[ \assert_p(t) = \case_{i=1}^n t \of i \mapsto \cond{p[x:=i]}{\return{i}}{\fail} \enspace . \] +\end{enumerate} +\end{corollary} + +\subsection{Sequential Product} + +We do not have conjunction or disjunction in our language for predicates over the same type, as this would involve duplicating variables. However, we do have +the following \emph{sequential product}. +(This was called the `and-then' test operator in Section 9 in \cite{Jacobs14}.) + +Let $x : A \vdash p,q : \mathbf{2}$. We define the \emph{sequential product} $p \andthen q$ by +\[ x : A \vdash p \andthen q \eqdef \doo{x}{\assert_{\lambda x p}(x)}{q} : \mathbf{2} \enspace . \] + +\begin{proposition}$ $ +\label{prop:testops} +\label{prop:effmod} +Let $x : A \vdash p,q : \mathbf{2}$. +\begin{enumerate} +\item $\instr_{p \andthen q}(x) = \pcase{\instr_p(x)}{x}{\instr_q(x)}{y}{\inr{y}}$ +\item $\assert_{p \andthen q}(x) = \doo{x}{\assert_p(x)}{\assert_q(x)} \eqdef \assert_p(x) \goesto \assert_q$ \label{prop:assertand} + \item \label{prop:odotcomm} (\textbf{Commutativity}) +$p \andthen q = q \andthen p$. + \item $(p \ovee q) \andthen r = p \andthen r \ovee q \andthen r$ and $p \andthen (q \ovee r) = p \andthen q \ovee p \andthen r$. + \item $p \andthen \bot = \bot \andthen q = \bot$ + \item $p \andthen \top = p$ and $\top \andthen q = q$ + \item $p \andthen (q \andthen r) = (p \andthen q) \andthen r$ + \item Let $x : A \vdash p : \mathbf{2}$. If $x$ does not occur in $q$, then $p \andthen q = \pcase{p}{\_}{q}{\_}{\bot}$. +\end{enumerate} +\end{proposition} + +\begin{proof} +\begin{enumerate} +\item +We have +\begin{align*} +& \inlprop{\pcase{\instr_p(x)}{x}{\instr_q(x)}{y}{\inr{y}}} \\ +& = \pcase{\instr_p(x)}{x}{q}{y}{\bot} \\ +& = \doo{x}{\assert_p(x)}{q} = p \andthen q +\end{align*} +and +\begin{align*} +& \nabla(\pcase{\instr_p(x)}{x}{\instr_q(x)}{y}{\inr{y}}) \\ +& = \pcase{\instr_p(x)}{x}{x}{y}{y} \\ +& = \nabla(\instr_p(x)) = x +\end{align*} +so the result follows by\Retainstr. +\item This follows immediately from the previous part. +\item This follows from the previous part and the rule\Rcomm (Appendix \ref{section:instruments}). +\item +$p \andthen (q \ovee r) = (p \andthen q) \ovee (p \andthen r)$ by Corollary \ref{cor:doovee}. The other case follows by Commutativity. +\item +$\bot \andthen p = \bot$ by Lemma \ref{lm:do}. +\item +$\top \andthen q = q$ by Lemma \ref{lm:do}. +\item $ +\begin{aligned}[t] +(p \andthen q) \andthen r +& \eqdef \doo{x}{\assert_{p \andthen q}(x)}{r} \\ +& = \doo{x}{(\assert_p(x) \goesto \assert_q)}{r} & \text{by part \ref{prop:assertand}} \\ +& = \doo{x}{\assert_p(x)}{\doo{x}{\assert_q(x)}{r}} & \text{by Lemma \ref{lm:do}} \\ +& \eqdef p \andthen (q \andthen r) + \end{aligned} $ +\item +$p \andthen q = \doo{\_}{\assert_p(x)}{q} = \pcase{\assert_p(x)}{\_}{q}{\_}{\bot} = \pcase{\dom{(\assert_p(x))}}{\_}{q}{\_}{\bot} = \cond{p}{q}{\bot}$. +\item +Let $b : \mathbf{3}$ be given by +\[ b \eqdef \cond{p}{\cond{q}{1}{3}}{\cond{r}{2}{3}} \] +Then +\begin{align*} + b \goesto \rhd_1 & = \cond{p}{\cond{q}{\top}{\bot}}{\cond{r}{\bot}{\bot}} \\ +& = \cond{p}{q}{\bot} = p \andthen q \\ +b \goesto \rhd_2 & = \cond{p}{\bot}{r} & \text{similarly} \\ +& = \cond{p^\bot}{r}{\bot} = p^\bot \andthen r +\end{align*} +Thus, $b$ is a bound for $p \andthen q \ovee p^\bot \andthen r$. We also have +\begin{align*} +\doo{x}{b}{\return{\nabla(x)}} & \eqdef \cond{p}{\cond{q}{\top}{\bot}}{\cond{r}{\top}{\bot}} \\ +& = \cond{p}{q}{r} +\end{align*} +and the result is proved. +\end{enumerate} +\end{proof} + +These results show that the scalars form an \emph{effect monoid}, and the predicates on any type form an \emph{effect module} over that effect monoid (see \cite{Jacobs14} Lemma 13 and Proposition 14). + +\subsection{n-tests} +\label{section:ntest} + +Recall that an \emph{$n$-test} on a type $A$ is an $n$-tuple $(p_1, \ldots, p_n)$ such that +\[ x : A \vdash p_1 \ovee \cdots \ovee p_n = \top : \mathbf{2} \] + +The following lemma shows that there is a one-to-one correspondance between the $n$-tests on $A$, and +the maps $A \rightarrow \mathbf{n}$. + +\begin{lemma} +\label{lm:ntest} + For every $n$-test $(p_1, \ldots, p_n)$ on $A$, there exists a term $x : A \vdash t(x) : \mathbf{n}$, +unique up to equality, such that +\[ x : A \vdash p_i(x) = \rhd_i(t(x)) : \mathbf{2} \] +\end{lemma} + +\begin{proof} +The proof is by induction on $n$. The case $n = 1$ is trivial. + +Suppose the result is true for $n$. Take an $n+1$-test $(p_1, \ldots, p_{n+1})$. Then \\ +$(p_1, p_2, \ldots, p_n \ovee p_{n+1})$ is an $n$-test. By the induction hypothesis, there exists $t : \mathbf{n}$ such that +\[ \rhd_i(t) = p_i \; (i < n), \qquad \rhd_n(t) = p_n \ovee p_{n+1} \enspace . \] +Let $b : \mathbf{3}$ be the bound for $p_n \ovee p_{n+1}$, so +\[ \rhd_1(b) = p_n, \qquad \rhd_2(b) = p_{n+1}, \qquad \rhd_{12}(b) = p_n \ovee p_{n+1} \enspace . \] +Reading $t$ and $b$ as partial functions in $\mathbf{n-1} + 1$ and $\mathbf{2} + 1$, we have that +$\ker{t} = \dom{b} = p_n \ovee p_{n+1}$. +Hence $\inlr{b}{t} : \mathbf{2} + \mathbf{n - 1}$ exists. Reading it as a term of type $\mathbf{n+1}$, we have that +\[ \rhd_1(\inlr{b}{t}) = p_n, \quad \rhd_2(\inlr{b}{t}) = p_{n+1}, \quad \rhd_{i + 2}(\inlr{b}{t}) = p_i \; (i < n) \enspace . \] +From this it is easy to construct the term of type $\mathbf{n + 1}$ required. +\end{proof} + +We write $\instr_{(p_1, \ldots, p_n)}(s)$ for $\instr_t(s)$, where $t$ is the term such that $\rhd_i(t) = p_i$ for each $i$. +We therefore have + +\begin{lemma} +\label{lm:instrn} + $\instr_{(p_1, \ldots, p_n)}(x)$ is the unique term such that $\intest{i}{\instr_{(p_1, \ldots, p_n)}(x)} = p_i$ for all $i$ +and +$\nabla(\instr_{(p_1, \ldots, p_n)}(x)) = x$. +\end{lemma} + +\begin{proof} +Let $t : \mathbf{n}$ be the term such that $\rhd_i(t) = p_i$ for all $i$. +By the rules for instruments, $\instr_{(p_1, \ldots, p_n)}(x)$ is the unique term such that +\begin{align*} + (\case_{i=1}^n \instr_{(p_1, \ldots, p_n)}(x) \of \nin{i}{n}(\_) \mapsto i) & = t \\ +\nabla(\instr_{(p_1, \ldots, p_n)}(x)) & = x +\end{align*} +It is therefore sufficient to prove that, given terms $\Gamma \vdash s, t : \mathbf{n}$, +\[ \Gamma \vdash s = t : \mathbf{n} \Leftrightarrow \forall i. \Gamma \vdash \rhd_i(s) = \rhd_i(t) : \mathbf{2} \] +This fact is proven by induction on $n$, with the case $n = 2$ holding by the rules\Rbetainlrone,\Rbetainlrtwo and\Retainlr. +\end{proof} + +\begin{lemma} +\label{lm:assertpi} +\begin{align*} +\instr_{p_i}(x) & = \case_{j=1}^n \instr_{(p_1, \ldots, p_n)}(x) \of \nin{j}{n}{x} \mapsto +\begin{cases} +\inl{x} & \text{if } i = j \\ +\inr{x} & \text{if } i \neq j +\end{cases} \\ +\assert_{p_i}(x) & = \case_{j=1}^n \instr_{(p_1, \ldots, p_n)}(x) \of \nin{j}{n}{x} \mapsto \begin{cases} +\return x & \text{if } i = j \\ +\fail & \text{if } i \neq j +\end{cases} +\end{align*} +\end{lemma} + +\begin{proof} +The first formula holds because $\inlprop{}$ maps the right-hand side to $\intest{i}{\instr_{(p_1, \ldots, p_n)}(x)} = p_i$, +and $\nabla$ mapst the right-hand side to $x$. +The second formula follows immediately from the first. +\end{proof} + +\begin{lemma} + \item If $(p,q)$ is a 2-test, then $q = p^\bot$, and $\mathsf{instr}_{(p,q)}(t) = \mathsf{instr}_{p}(t)$. +\end{lemma} + +\begin{proof} + If $(p,q)$ is a 2-test then $p \ovee q = \top$ and so $q = p^\bot$ by Proposition \ref{prop:logic}.\ref{prop:ortho}. Then +$\mathsf{instr}_{(p,q)}(t) = \mathsf{instr}_p(t)$ by\Retainstr, since $\inlprop{\mathsf{instr}_{(p,q)}(x)} = \langle p ? \rangle \top \ovee \langle q ? \rangle \bot = p$ +and $\nabla(\mathsf{instr}_{(p,q)}(x)) = x$. +\end{proof} + + + + + +We can now define the program that divides into $n$ branches depending on the outcome of an $n$-test: + +\begin{definition} +\label{df:measure} +Given $x : A \vdash p_1(x) \ovee \cdots \ovee p_n(x) = \top : \mathbf{2}$, define +\begin{align*} +x : A & \vdash \meas\ p_1(x) \mapsto t_1(x) \mid \cdots \mid p_n(x) \mapsto t_n(x) \\ +& \eqdef \case \mathsf{instr}_{(p_1, \ldots, p_n)}(x) \of \inn_1(x) \mapsto t_1(x) \mid \cdots \mid \inn_n(x) \mapsto t_n(x) +\end{align*} +\end{definition} + + + +\begin{lemma} +\label{lm:measure} +The $\meas$ construction satisfies the following laws. + \begin{enumerate} + \item \label{lm:measuretop} $(\meas\ \top \mapsto t) = t$ + \item \label{lm:measurebot} $(\meas\ p_1 \mapsto t_1 \mid \cdots \mid p_n \mapsto t_n \mid \bot \mapsto t_{n+1}) = (\meas\ p_1 \mapsto t_1 \mid \cdots \mid p_n \mapsto t_n)$ + \item \label{lm:measureand} $(\meas_i\ p_i \mapsto \meas_j\ q_{ij} \mapsto t_{ij}) = (\meas_{i,j}\ p_i \andthen q_{ij} \mapsto t_{ij})$ + \item \label{lm:measureperm} For any permutation $\pi$ of $\{1, \ldots, n\}$, $\meas_i\ p_i \mapsto t_i = \meas_i\ p_{\pi(i)} \mapsto t_{\pi(i)}$. + \item \label{lm:measureor} If $t_n = t_{n+1}$ then \\ $\meas_{i=1}^n p_i \mapsto t_i = \meas\ p_1 \mapsto t_1 \mid \cdots \mid p_{n-1} \mapsto t_{n-1} \mid p_n \ovee p_{n+1} \mapsto t_n$. + \end{enumerate} +\end{lemma} + +\begin{proof} + \begin{enumerate} + \item +$ \begin{aligned}[t] + \meas \top \mapsto t(x) & \eqdef \case \instr_{(\top)}(x) \of \nin{1}{1}{x} \mapsto t(x) \\ +& = t(\instr_{(\top)}(x)) + \end{aligned}$. + +So it suffices to prove $\instr_{(\top)}(s) = s$. +This holds by the uniqueness of Lemma \ref{lm:instrn}, since we have $\intest{1}{x} = \top$ and $\nabla(x) = x$. +\item +It suffices to prove $\instr_{(p_1, \ldots, p_n, \bot)}(x) = \case_{i=1}^n \instr_{(p_1, \ldots, p_n)}(x) \of \nin{i}{n}{x} \mapsto \nin{i}{n+1}{x}$. +Let $R$ denote the right-hand side. Then +\begin{align*} +\intest{i}{R} & = \intest{i}{\instr_{(p_1, \ldots, p_n)}(x)} = p_i \\ + \nabla(R) & = \case_{i=1}^n \instr_{(p_1, \ldots, p_n)}(x) \of \nin{i}{n}{x} \mapsto x \\ +& = \nabla(\instr_{(p_1, \ldots, p_n)}(x)) = x +\end{align*} +\item +Let us write $\nin{i,j}{}{}$ ($1 \leq i \leq m$, $1 \leq j \leq n_i$) for the constructors of $(n_1 + \cdots + n_m) \cdot A$, +and $\intest{i,j}{}$ for the corresponding predicates. + +It suffices to prove that +\[ \instr_{(p_i \andthen q_{ij})_{i,j}}(x) = \case_{i=1}^m\ \instr_{\vec{p}}(x) \of \nin{i}{m}{x} \mapsto +\case_{j=1}^{n_1}\ \instr_{\vec{q_i}}(x) \of \nin{j}{n_i}{x} \mapsto \nin{i,j}{}{x} \enspace . \] + Let $R$ denote the right-hand side. We have +\begin{align*} +\intest{i,j}{R} & = \case_{i'=1}^m\ \instr_{\vec{p}}(x) \of \nin{i'}{m}{x} \mapsto \begin{cases} +\intest{j}{\instr_{\vec{q_i}}(x)} & \text{if } i = i' \\ +\bot & \text{if } i \neq \i' +\end{cases} \\ +& = \case_{i'=1}^m\ \instr_{\vec{p}}(x) \of \nin{i'}{m}{x} \mapsto \begin{cases} +q_{ij} & \text{if } i = i' \\ +\bot & \text{if } i \neq i' +\end{cases} \\ +& = \doo{x}{\left( \case_{i'=1}^m\ \instr_{\vec{p}}(x) \of \nin{i'}{m}{x} \mapsto \begin{cases} +\return x & \text{if } i = i' \\ +\fail & \text{if } i \neq i' +\end{cases} \right)}{q_{ij}} \\ +& = \doo{x}{\assert_{p_i}(x)}{q_{ij}} \\ +& \qquad \qquad \text{(by Lemma \ref{lm:assertpi})} \\ +& = p_i \andthen q_{ij} +\end{align*} +and +\begin{align*} +\nabla(R) & = \case_{i=1}^m\ \instr_{\vec{p}}(x) \of \nin{i}{m}{x} \mapsto \nabla(\instr_{\vec{q_i}}(x)) \\ +& = \case_{i=1}^m\ \instr_{\vec{p}}(x) \of \nin{i}{m}{x} \mapsto x = \nabla(\instr_{\vec{p}}(x)) = x +\end{align*} +\item +It is sufficient to prove that +\[ \instr_{(p_1, \ldots, p_n)}(x) = \case_{i=1}^n \instr_{(p_{\pi(1)}, \ldots, p_{\pi(n)})}(x) \of \nin{i}{n}{x} \mapsto \nin{\pi^{-1}(i)}{n}{x} +\enspace . \] +Let $R$ denote the right-hand side. We have +\begin{align*} +\intest{i}{R} & = \intest{\pi^{-1}(i)}{\instr_{(p_{\pi(1)}, \ldots, p_{\pi(n)})}(x)} = p_i \\ + \nabla(R) & = \nabla(\instr_{(p_{\pi(1)}, \ldots, p_{\pi(n)})}(x)) = x +\end{align*} +\item +It suffices to prove $\instr_{(p_1, \ldots, p_{n-1}, p_n \ovee p_{n+1})} = \case_{i=1}^{n+1} \instr_{\vec{p}}(x) \of \nin{i}{n}{x} \mapsto \begin{cases} +\nin{i}{n}{x} & \text{if } i < n \\ \nin{i}{n}{x} & \text{if } i \geq n \end{cases}$. Let $R$ denote the right-hand side. We have, for $i < n$: +\begin{align*} +\intest{i}{R} & = \intest{i}{\instr_{\vec{p}}(x)} = p_i + \intest{n}{R} & = \rhd_{n,n+1}(\ind{\instr_{\vec{p}}(x)}) \\ +& = \intest{n}{\instr_{\vec{p}}(x)} \ovee \intest{n+1}{\instr_{\vec{p}}(x)} = p_n \ovee p_{n+1} +\nabla(R) & = x \enspace . +\end{align*} + \end{enumerate} +\end{proof} + + + + + +Let $x : A \vdash p : \mathbf{2}$ and $\Gamma, x : A \vdash s,t : B$. We define +\[ \cond{p}{s}{t} \eqdef \meas\ p \mapsto s \mid p^\bot \mapsto t : B \enspace . \] + +\begin{lemma} +\label{lm:measuretwo} + \begin{enumerate} + \item If $x : A \vdash p_1 \ovee \cdots \ovee p_n = \top : \mathbf{2}$ and $x : A \vdash q_1, \ldots, q_n : \mathbf{2}$, then +\[ (\mathsf{measure}\ p_1 \mapsto q_1 \mid \cdots \mid p_n \mapsto q_n) = p_1 \andthen q_1 \ovee \cdots \ovee p_n \andthen q_n \enspace . \] + \item Let $x : A \vdash p : \mathbf{2}$ and $\Gamma \vdash q,r : B$ where $x \notin \Gamma$. Then $\cond{p}{q}{r} = \pcase{p}{\_}{q}{\_}{r} : B$. \label{lm:measurecond} + \item\label{lm:measuretwo'} Let $x : A \vdash p : \mathbf{2}$. Then $x : A \vdash \cond{p}{\top}{\bot} = p : \mathbf{2}$. + \end{enumerate} +\end{lemma} + +\begin{proof} + \begin{enumerate} + \item + Immediate from Lemma \ref{lm:instrn}. + \item We have + \begin{align*} + \meas\ p \mapsto q \mid p^\bot \mapsto r & \eqdef \case \instr_{\lambda x p}(x) \of \inl{\_} \mapsto q \mid \inr{\_} \mapsto r \\ +& = \case \inlprop{\instr_{\lambda x p}(x)} \of \inl{\_} \mapsto q \mid \inr{\_} \mapsto r \\ +& = \case p \of \inl{\_} \mapsto q \mid \inr{\_} \mapsto r + \end{align*} + \item $\cond{p}{\top}{\bot} = \pcase{p}{\_}{\top}{\_}{\bot} = p$ by\Retaplus. + \end{enumerate} +\end{proof} + +\subsection{Scalars} +\label{sec:scalars} + +From the rules given in Figure \ref{fig:equations}, the usual algebra of the rational interval from 0 to 1 follows. + +\begin{lemma} + If $p / q = m / n$ as rational numbers, then $\vdash p \cdot (1 / q) = m \cdot (1 / n) : \mathbf{2}$. +\end{lemma} + +\begin{proof} +We first prove that $\vdash a \cdot (1 / a b) = 1 / b : \mathbf{2}$ for all $a$, $b$. This holds because $ab \cdot (1 / ab) = \top$ by\Rntimesoneovern, +hence $a \cdot (1 / ab) = 1/b$ by\Rdivide. + +Hence we have $p \cdot (1 / q) = pn \cdot (1 / nq) = qm \cdot (1 / n q) = m \cdot (1 /n)$. +\end{proof} + +Recall that within $\COMET$, we are writing $m / n$ for the term $m \cdot (1 / n)$. + +\begin{lemma} +\label{lm:rational} +Let $q$ and $r$ be rational numbers in $[0,1]$. +\begin{enumerate} +\item If $q \leq r$ in the usual ordering, then $\vdash q \leq r : \mathbf{2}$. +\item $\vdash q \ovee r : \mathbf{2}$ iff $q + r \leq 1$, in which case $\Gamma \vdash q \ovee r = q + r : \mathbf{2}$. +\item $\vdash q \andthen r = qr : \mathbf{2}$. +\end{enumerate} +\end{lemma} + +\begin{proof} + By the previous lemma, we may assume $q$ and $r$ have the same denominator. Let $q = a / n$ and $r = b / n$. + \begin{enumerate} + \item We have $a \leq b$, hence $\vdash a \cdot (1 / n) \leq b \cdot (1 / n) : \mathbf{2}$ by Lemma \ref{lm:ordering}.\ref{lm:leqovee}. + \item If $q + r \leq 1$ then $\vdash a \cdot (1 / n) \ovee b \cdot (1 / n) = (a + b) \cdot (1 / n) : \mathbf{2}$ by Associativity. + +For the converse, suppose $\vdash q \ovee r : \mathbf{2}$, so $\vdash (a + b) \cdot (1 / n) : \mathbf{2}$, and suppose for a contradiction $q + r > 1$. Then we have +\[ \vdash \top \ovee (a + b - n) \cdot (1 / n) : \mathbf{2} \] +and so $\vdash (1 / n) = 0 : \mathbf{2}$ by the Zero-One Law, hence $\vdash \top = n \cdot (1 / n) = n \cdot 0 = \bot : \mathbf{2}$. This contradicts Corollary \ref{cor:consistency}. +\item We first prove $(1 / a) \andthen (1 / b) = 1 / ab : \mathbf{2}$. This holds because $ab \cdot (1 / a) \andthen (1 / b) = (a \cdot (1 / a)) \andthen (b \cdot (1 / b)) = \top \andthen \top = \top$. + +Now we have, $(m / n) \andthen (p / q) = mp \cdot ((1 / n) \andthen (1 /q)) = mp \cdot (1 / nq)$ as required. + \end{enumerate} +\end{proof} + +\subsection{Normalisation} + +The following lemma gives us a rule that allows us to calculate the normalised form of a substate in many cases, including the examples in Section \ref{section:examples}. + +\begin{lemma} +Let $\vdash t : A + 1$, $\vdash p_1 \ovee \cdots \ovee p_n = \top : \mathbf{2}$, and $\vdash q : \mathbf{2}$. Let $\vdash s_1, \ldots, s_n : A$. Suppose $\vdash 1 / m \leq q : \mathbf{2}$. If +\[ \vdash t = \meas\ p_1 \andthen q \mapsto \return{s_1} \mid \cdots \mid p_n \andthen q \mapsto \return{s_n} \mid q^\bot \mapsto \fail : A + 1 \] +then +\[ \vdash \norm{t} = \meas\ p_1 \mapsto s_1 \mid \cdots \mid p_n \mapsto s_n : A \] +\end{lemma} + +\begin{proof} +Let $\rho \eqdef \meas_{i=1}^n p_i \mapsto s_i$. + By the rule\Retanorm, it is sufficient to prove that $t = \doo{\_}{t}{\return{\rho}}$. +We have +\begin{align*} + \doo{\_}{t}{\return{\rho}} +& = \meas\ p_1 \andthen q \mapsto \return{\rho} \mid \cdots \mid p_n \andthen q \mapsto \return{\rho} \mid q^\bot \mapsto \fail \\ +& = \meas\ (p_1 \ovee \cdots \ovee p_n) \andthen q \mapsto \return{\rho} \mid q^\bot \mapsto \fail \\ +& = \meas\ q \mapsto \return{\rho} \mid q^\bot \mapsto \fail \\ +& = \meas\ q \mapsto \meas_{i=1}^n p_i \mapsto \return{s_i} \mid q^\bot \mapsto \fail \\ +& = \meas_{i=1}^n\ q \andthen p_i \mapsto \return{s_i} \mid q^\bot \mapsto \fail \\ +& = t +\end{align*} +(We used the commutativity of $\andthen$ in the last step.) +\end{proof} + +\begin{corollary} +\label{cor:normmeasure} +Let $\alpha_1$, \ldots, $\alpha_n$, $\beta$ be rational numbers that sum to 1, with $\beta \neq 1$. If +\[ \vdash t = \meas\ \alpha_1 \mapsto \return{s_1} \mid \cdots \mid \alpha_n \mapsto \return{s_n} \mid \beta \mapsto \fail : A + 1 \] +then +\[ \vdash \norm{t} = \meas\ \alpha_1 / (\alpha_1 + \cdots + \alpha_n) \mapsto s_1 \mid \cdots \mid \alpha_n / (\alpha_1 + \cdots + \alpha_n) \mapsto s_n : A \] +\end{corollary} + +\section{Semantics} +\label{section:semantics} + +The terms of $\COMET$ are intended to represent probabilistic programs. +We show how to give semantics to our system in three different ways: using discrete and continuous probability distributions, and +simple set-theoretic semantics for deterministic computation. + +\subsection{Discrete Probabilistic Computation} +\label{section:dpc} + +We give an interpretation that assigns, to each term, a discrete probability distribution over its output type. + +\begin{definition} +Let $A$ be a set. +\begin{itemize} +\item +The \emph{support} of a function $\phi : A \rightarrow [0,1]$ is $\supp \phi = \{ a \in A : \phi(a) \neq 0 \}$. +\item +A \emph{(discrete) probability distribution} over $A$ is a function $\phi : A \rightarrow \phi$ with finite support +such that $\sum_{a \in A} \phi(a) = 1$. +\item +Let $\mathcal{D} A$ be the set of all probability distributions on $A$. +\end{itemize} +\end{definition} + +We shall interpret every type $A$ as a set +$\brackets{A}$. Assume we are given a set $\brackets{\mathbf{C}}$ for each type constant $\mathbf{C}$. +Define a set $\brackets{A}$ for each type $A$ thus: +\[ \brackets{0} = \emptyset \qquad \brackets{1} = \{ * \} \qquad \brackets{A + B} = \brackets{A} \uplus \brackets{B} \qquad \brackets{A \sotimes B} = \brackets{A} \times \brackets{B} \] +where $A \uplus B = \{ a_1 : a \in A \} \cup \{ b_2 : b \in B \}$. We extend this to contexts by defining $\brackets{x_1 : A_1, \ldots, x_n : A_n} = \brackets{A_1} \times \cdots \times \brackets{A_n}$. + +Now, to every term $x_1 : A_1, \ldots, x_n : A_n \vdash t : B$, we assign a function +$\brackets{t} : \brackets{A_1} \times \cdots \times \brackets{A_n} \rightarrow \mathcal{D} \brackets{B}$. +The value $\brackets{t}(a_1, \ldots, a_n)(b) \in [0,1]$ will be written as $P(t(a_1, \ldots, a_n) = b)$, and should be thought of as the probability +that $b$ will be the output if $a_1$, \ldots, $a_n$ are the inputs. + +\begin{figure} +\begin{mdframed} +\begin{multicols}{2} +$$\begin{aligned} +P(x_i(\vec{a}) = b) & = \begin{cases} +1 \text{ if } b = a_i \\ + 0 \text{ if } b \neq a_i +\end{cases} \\ \midrule +P(*(\vec{a}) = *) & = 1 \\ \midrule +\multicolumn{2}{l}{$P((s \sotimes t)(\vec{g}, \vec{d}) = (a,b))$} \\ +& = P(s(\vec{g}) = a) P(t(\vec{d}) = b) \\ \midrule +P((\magic{t})(\vec{g}) = a) & = 0 \\ \midrule +P(\inl{t}(\vec{g}) = a_1) & = P(t(\vec{g}) = a) \\ +P(\inl{t}(\vec{g}) = b_2) & = 0 \\ \midrule +P(\inr{t}(\vec{g}) = a_1) & = 0 \\ +P(\inr{t}(\vec{g}) = b_2) & = P(t(\vec{g}) = b) \\ \midrule +P(\inlr{s}{t}(\vec{g}) = a_1) & = P(s(\vec{g}) = a_1) \\ +P(\inlr{s}{t}(\vec{g}) = b_2) & = P(t(\vec{g}) = b_1) +\end{aligned}$$ + +$\begin{aligned} +\multicolumn{2}{l}{$P(\lft{t}(\vec{g}) = a) = P(t(\vec{g}) = a_1)$} \\ \midrule +\multicolumn{2}{l}{$P(\instr_{\lambda x t}(s)(\vec{g}) = a_i)$} \\ & = P(s(\vec{g}) = a) P(t(a) = i) \\ \midrule +\multicolumn{2}{l}{$P(1 / n(\vec{g}) = \top) = 1 / n$} \\ +\multicolumn{2}{l}{$P(1 / n(\vec{g}) = \bot) = (n - 1) / n$} \\ \midrule +\multicolumn{2}{l}{$P(\norm{t}(\vec{g}) = a)$} \\ & \ = P(t(\vec{g}) = a_1) / (1 - P(t(\vec{g}) = *_2)) \\ \midrule +\multicolumn{2}{l}{$P((s \ovee t)(\vec{g}) = a_1)$} \\ & \ = P(s(\vec{g}) = a_1) + P(t(\vec{g}) = a_1) \\ +\multicolumn{2}{l}{$P((s \ovee t)(\vec{g}) = *_2)$} \\ & \ = P(s(\vec{g}) = *_2) + P(t(\vec{g}) = *_2) - 1 +\end{aligned}$ +\end{multicols} +$\begin{aligned} +& P((\plet{x}{y}{s}{t})(\vec{g},\vec{d}) = c) = \sum_a \sum_b P(s(\vec{g}) = (a,b)) P(t(\vec{d},a,b) = c) \\ \midrule +& P(\pcase{r}{x}{s}{y}{t}(\vec{g},\vec{d}) = c) \\ +& = \sum_a P(r(\vec{g}) = a_1) P(s(\vec{d}, a) = c) + +\sum_b P(r(\vec{g}) = b_2) P(t(\vec{d}, b) = c) +\end{aligned}$ +\end{mdframed} +\end{figure} + +The sums involved here are all well-defined because, for all $t$ and $\vec{g}$, the function $P(t(\vec{g}) = -)$ has finite support. + +\begin{lemma} +Let $\Gamma \vdash s : A$ and $\Delta, x : A \vdash t : B$, so that $\Gamma, \Delta \vdash t[x:=s] : B$. Then +\[ P(t[x:=s](\vec{g}, \vec{d}) = b) = \sum_{a \in \brackets{A}} P(s(\vec{g}) = a) P(t(\vec{d},a) = b) \] +\end{lemma} + +\begin{proof} +The proof is by induction on $t$. We do here the case where $t \equiv x$: +\[ P(x[x:=s](\vec{g}) = b) = P(s(\vec{g}) = b) \] +and +\[ \sum_a P(s(\vec{g}) = a) P(x(a) = b) = P(s(\vec{g}) = b) \] +since $P(x(a) = b)$ is 0 if $a \neq b$ and 1 if $a = b$. +\end{proof} + +\begin{theorem}[Soundness] + \begin{enumerate} + \item If $\Gamma \vdash t : A$ is derivable, then for all $\vec{g} \in \brackets{\Gamma}$, we have $P(t(\vec{g}) = -)$ is a +probability distribution on $\brackets{A}$. +\item If $\Gamma \vdash s = t : A$, then $P(s(\vec{g}) = a) = P(t(\vec{g}) = a)$. + \end{enumerate} +\end{theorem} + +\begin{proof} + The proof is by induction on derivations. We do here the case of the rule\Rinstrtest: + \begin{align*} +& P((\case_i\ \instr_{\lambda x t}(s) \of \nin{i}{n}{\_} \mapsto i)(\vec{g}) = i) \\ +& = \sum_{j = 1}^n \sum_{a \in \brackets{A}} P(\instr_{\lambda x t}(s)(\vec{g}) = a_j) P(\nin{i}{n}{*}() = *_j) \\ +& = \sum_{a \in \brackets{A}} P(\instr_{\lambda x t}(s)(\vec{g}) = a_i) \\ +& = \sum_{a \in \brackets{A}} P(s(\vec{g}) = a) P(t(a) = i) \\ +& = P(t[x:=s](\vec{g}) = i) + \end{align*} +by the lemma. +\end{proof} + +\begin{corollary} + If $\Gamma \vdash s \leq t : A + 1$ then $P(s(\vec{g}) = a) \leq P(t(\vec{g}) = a)$ for all $\vec{g}$, $a$. +\end{corollary} + +As a corollary, we know that $\COMET$ is non-degenerate: + +\begin{corollary} +\label{cor:consistency} + Not every judgement is derivable; in particular, the judgement $\vdash \top = \bot : \mathbf{2}$ is not derivable. +\end{corollary} + +With these definitions, we can calculate the semantics of each of our defined constructions. For example, +the semantics of $\mathsf{assert}$ are given by +\[ P(\assert_{\lambda x p}(t)(\vec{g}) = a_1) = P(t(\vec{g}) = a)P(p(a) = \top) \] +\[ P(\assert_{\lambda x p}(t)(\vec{g}) = *_2) = \sum_a P(t(\vec{g}) = a) P(p(a) = \bot) \] + + + + + + + + + + + + + + + + + + + + + + + + + + + +\subsection{Alternative Semantics} + +It is also possible to give semantics to $\COMET$ using continuous probabilities. We assign a measurable space $\brackets{A}$ to every type $A$. Each term then gives a measurable function $\brackets{A_1} \times \cdots \times \brackets{A_n} \rightarrow \mathcal{G} \brackets{B}$, where $\mathcal{G} X$ is the space of all probability distributions over the measurable space $X$. ($\mathcal{G}$ here is the \emph{Giry monad} \cite{Jacobs13a}.) + +If we remove the constants $1 / n$ from the system, we can give \emph{deterministic} semantics to the subsystem, in which we assign a set to every type, and a function $\brackets{A_1} \times \cdots \times \brackets{A_n} \rightarrow \brackets{B}$. + +More generally, we can give an interpretation of $\COMET$ in any \emph{commutative monoidal effectus with normalisation} +in which there exists a scalar $s$ such that $n \cdot s = 1$ for all positive integers $n$ \cite{Cho}. The discrete and continuous semantics we have described are two instances of this interpretation. + +\section{Conclusion} + +The system $\COMET$ allows for the specification of probabilistic programs and reasoning about their properties, both within the same syntax. + +There are several avenues for further work and research. +\begin{itemize} +\item The type theory that we describe can be interpreted both in + discrete and in continuous probabilistic models, that is, both in + the Kleisli category $\Kl(\Dst)$ of the distribution monad $\Dst$ + and in the Kleisli category $\Kl(\Giry)$ of the Giry monad $\Giry$. + On a finite type each distribution is discrete. The discrete semantics were exploited in + the current paper in the examples in Section~\ref{section:examples}. + In a follow-up version we intend to elaborate also continuous + examples. + +\item The normalisation and conditioning that we use in this paper can + in principle also be used in a quantum context, using the + appropriate (non-side-effect free) assert maps that one has + there. This will give a form of Bayesian quantum theory, as also + explored in~\cite{LeiferS13}. + +\item A further ambitious follow-up project is to develop tool support + for $\COMET$, so that the computations that we carry out here by + hand can be automated. This will provide a formal language for + Bayesian inference. +\end{itemize} + +\subparagraph*{Acknowledgements} + +Thanks to Kenta Cho for discussion and suggestions during the writing of this paper, and very detailed proofreading. Thanks to Bas Westerbaan for discussions about effectus theory. + +\bibliography{probable} + +\appendix + +\section{Formal Presentation of $\COMET$} +\label{section:rules} + +The full set of rules of deduction for $\COMET$ are given below. + +\subsection{Structural Rules} +\label{section:structural} +$$ \Texch \qquad \Tvar $$ + +The exchange rule says that the order of the variables in the context does not matter. This holds +for all types of judgements J on the right hand side of the turnstile. The weakening rule is admissible (see Lemma \ref{lm:meta}.\ref{lm:weak}), and says +that one may add (unused) assumptions to the context. + +However, we do \emph{not} have the contraction rule in our type theory. In particular, the judgement $x : A \vdash x \otimes x : A \otimes A$ is \emph{not} derivable. +Thus, in our probabilistic settings, information may be discarded, but cannot be duplicated. + +$$ \Tref \; \Tsym \; \Ttrans $$ + +These rules simply ensure that the judgement equality is an equivalence relation. + +\subsection{The Singleton Type} + +$$ \Tunit \quad \Tetaone $$ + +These ensure that the type $1$ is a type with only one object up to equality. + +\subsection{Tensor Product} + +$$ \Tpair \; \Tlett $$ +$$ \Tpaireq $$ +$$ \Tleteq $$ + +Notice that in rule\Rpair the contexts $\Gamma$ and +$\Delta$ of the two terms $s$, $t$ are put together in the +conclusion. Thus, the tensor $s \sotimes t$ on terms is a form of +parallel composition. This is a so-called \emph{introduction rule} for the +tensor type, since it tells us how to produce terms in a tensor type +$A\otimes B$ on the right hand side of the turnstile $\vdash$. The +rule\Rlett is an \emph{elimination rule} since it tells us how to use terms +of tensor type. + +$$ \Tbeta $$ +$$ \Teta $$ + +Rule\Rbeta tells how a let +term should decompose a term $r \sotimes s$, namely by simultaneously +substituting $r$ for $x$ and $s$ for $y$ in as described in the term +$t[x:=r,y:=s]$. Rule\Reta is its dual, and says that decomposing an object then +immediately recomposing it does nothing. + +$$ \Tletlet $$ +$$ \Tletpair $$ + +\noindent Our final set of rules are so-called commuting conversion +rules described above. They regulate the proper interaction between +the term constructs let, case and $\sotimes$. It looks like several interactions are missing here (a $\lett$ on the right +of a tensor, a $\lett$ inside a $\case$, etc.), but in fact, the rules for all the other cases can be derived from these four, as we show in +Lemma \ref{lm:sub}.\ref{lm:letsub}. + +\subsection{Empty Type} + +$$ \Tmagic \quad \Tetazero $$ + +The rule\Rmagic +says that from an inhabitant $M:0$ we can produce an inhabitant +$\magic{M}$ in any type $A$. Intuitively, this says `If the empty type is inhabited, then every type is inhabited', which is vacuously true. +And\Retazero says that vacuously, if the empty type $0$ is inhabited, then all terms of any type are equal. + +\subsection{Binary Coproducts} +\label{section:coproducts} +$$ \Tinl \quad \Tinr $$ +$$ \Tinleq \quad \Tinreq $$ +$$\Tcase $$ +$$\Tcaseeq $$ + +For the coproduct type $A+B$ we have two introduction rules\Rinl +and\Rinr which produce terms $\inl{s}, \inr{t} : A+B$, coming from +$s:A$ and $t:B$. These operations $\inl{-}$ and $\inr{-}$ are often +called \emph{coprojections} or \emph{injections}. + +The associated elimination +rule\Rcase produces a term that uses a term $r:A+B$ by distinguishing +whether or not $r$ is of the form $\inl{-}$ or $\inr{-}$. In the first +case the outcome of $r$ is used in term $s$, and in the second case in +term $t$. + +$$ \Tbetaplusone $$ +$$ \Tbetaplustwo $$ +$$ \Tetaplus $$ + +There are two $\beta$-conversions\Rbetaplusone and\Rbetaplustwo +for the coproduct type, describing how a $\mathsf{case}$ term should handle a +term of form $\inl{r}$ or $\inr{r}$. Again this this +done via the expected substitution, using the appropriate variable +($x$ or $y$). + +In rule\Retaplus, if the decomposition of $t$ into $\inl{-}$ and $\inr{-}$ +is then immediately reconstituted, then the input is unchanged. +$$ \Tcasecase $$ +$$ \Tcasepair $$ +$$ \Tletcase $$ +These rules for commuting conversions show how the eliminators for $\otimes$ and $+$ interact. Again, +the other cases can be derived from the primitive rules given here (Lemma \ref{lm:sub}). + +\subsection{Partial Pairing} +\label{section:effectus} + +We now come to the constructions that are new to our type theory. These possess a feature that is unique to this type theory: +we allow typing judgements (of the form $t : A$) to depend on equality judgements (of the form $s = t : A$). + +$$ \Tinlr $$ +$$ \Tinlreq $$ + +The term $\inlr{s}{t}$ can be understood in this way. Consider a term $\Gamma \vdash t : A + 1$ as a partial computation: +it may output a value of type $A$, or it may diverge (if it reduces to $\inr{*}$.) If the judgement $s \downarrow = t \uparrow$ holds, +then we know that exactly one of the computations $s$ and $t$ will terminate on any input. The term $\inlr{s}{t}$ intuitively denotes the following computation: +given an input, decide which of $s$ or $t$ will terminate. If $s$ will terminate, run $s$; otherwise, run $t$. + + + +We have the following $\beta$- and $\eta$-rules for the $\inlrn$ construction: + +$$ \Tbetainlrone $$ +$$ \Tbetainlrtwo $$ +$$ \Tetainlr $$ + +\subsection{The $\lft{}$ Construction} + +$$ \Tleft $$ +$$ \Tlefteq $$ + +The term $\lft{t}$ should be understood as follows: if we have a term $t : A + B$ and a `proof' that $t = \inl{s}$ for some term $s : A$, then +$\lft{t}$ is that term $s$. The computation rules for this construction are: + +$$ \Tbetaleft \Tetaleft $$ + +\subsection{Joint Monicity Condition} +\label{section:JM} + +We need the following rule for technical reasons. It corresponds to the condition that the two maps $\rhd_!$ and $\rhd_2$ from $A + A$ to $A$ are jointly monic +in the partial form of the effectus (see \cite{Jacobs14} Assumption 1 or \cite{Cho} Lemma 49.4). + +\TTJMprime + +It is used in the proof of the associativity of $\ovee$ (Lemma \ref{lm:ordering}.\ref{lm:assoc}). + +\subsection{Instruments} +\label{section:instruments} + +The \emph{instrument} map $\instr_{\lambda x t}(s)$ should be understood as follows: it denotes the value $\nin{i}{n}{s}$ if $t[x:=s]$ returns the value $i : \mathbf{n}$. + +If we were allowed to simply duplicate data, we could have defined $\measure{x}{p}{t}$ to be $\pcase{[t/x]p}{\_}{\inl{t}}{\_}{\inr{t}}$. This cannot be done in our system, as it would involve duplicating the variables in $t$. + +The computation rules for this construction are as follows. + +$$ \Tinstr \quad \Tnablainstr $$ +$$ \Tinstrtest $$ +$$ \Tetainstr $$ +$$ \Tinstreq $$ + +We also introduce the following rule, which ensures that the sequential product $\andthen$ is commutative. + +\TTcomm + +\subsection{Scalar Constants} + +For any natural number $n \geq 2$, we have the following rules. + +$$ \Toneovern \; \Tntimesoneovern $$ +$$ \Tdivide \; \Tboundmn $$ +$$ \Trhdoneboundmn $$ +$$ \Trhdtwoboundmnprime $$ + +These ensure that $1 / n$ is the unique scalar whose sum with itself $n$ times is $\top$. +The term $b_{mn}$ is required to ensure that the term $1 / n \ovee \cdots \ovee 1 / n$ is well-typed. + +\subsection{Normalisation} + +Finally, we have these rules for normalisation. + +$$ \Tnorm \; \Tbetanorm $$ +$$ \Tetanorm $$ + +These ensure that, if $t$ is a non-zero state in $A + 1$, then $\rho$ is the unique state in $A$ such that +$t = \doo{\_}{t}{\return{\rho}}$. + +\section{Proof of Associativity} +\label{section:associativity} + +\begin{theorem} +If $\Gamma \vdash (r \ovee s) \ovee t : A + 1$, then $\Gamma \vdash r \ovee (s \ovee t) : A + 1$ and $\Gamma \vdash r \ovee (s \ovee t) = (r \ovee s) \ovee t : A + 1$. +\end{theorem} + +(Note: this proof follows the proofs that $\ovee$ is associative in an effectus, found in \cite{Jacobs14} Proposition 12 or \cite{Cho} Proposition 13.) + +\begin{proof} +Let $b$ be a bound for $r \ovee s$ and $c$ a bound for $(r \ovee s) \ovee t$, so that +\begin{align} +b \goesto \rhd_1 & = r \label{eq:axiom1} \\ +b \goesto \rhd_2 & = s \label{eq:axiom2} \\ +\doo{x}{b}{\return{\nabla(x)}} & = r \ovee s \label{eq:axiom3} \\ +c \goesto \rhd_1 & = r \ovee s \label{eq:axiom4} \\ +c \goesto \rhd_2 & = t \label{eq:axiom5} \\ +\doo{x}{c}{\return{\nabla(x)}} & = (r \ovee s) \ovee t \label{eq:axiom6} +\end{align} + +Define $d : (A + 1) + 1$ by +\[ d = \case c \of \inl{\inl{\_}} \mapsto \fail \mid \inl{\inr{x}} \mapsto \return{\inl{x}} \mid \inr{\_} \mapsto \return{\inr{*}} \] +We wish to form the term $\inlr{b}{d}$. To do this, we must prove +$\dom{b} = \ker{d}$. We do this by proving both are equal to $\dom{(r + \ovee s)}$. + +We have +\begin{align*} +\dom{(r \ovee s)} & = \dom{(\doo{x}{b}{\return{\nabla(x)}})} = \doo{x}{b}{\dom{(\return{\nabla(x)})}} = \doo{x}{b}{\top} = \dom{b} +\end{align*} +and +\begin{align*} +\dom{(r \ovee s)} & = \dom{(\doo{x}{c}{\rhd_1(x)})} = \doo{x}{c}{\dom{(\rhd_1(x))}} = \doo{x}{c}{\inlprop{x}} \\ +\ker{d} & = \case c \of \inl{\inl{\_}} \mapsto \top \mid \inl{\inr{\_}} \mapsto \bot \mid \inr{\_} \mapsto \bot \\ +& = \doo{x}{c}{\inlprop{y}} \\ +\therefore \dom{b} & = \ker{d} +\end{align*} +So, let $e = \inlr{b}{d} : (A + A) + (A + 1)$. We claim +\begin{align} + \label{eq:transitivity} +c = \case e \of & \inl{\inl{a}} \mapsto \return{\inl{a}} \mid \inl{\inr{a}} \mapsto \return{\inl{a}} \mid \\ +& \inr{\inl{a}} \mapsto \return{\inr{a}} \mid \inr{\inr{\_}} \mapsto \fail \nonumber +\end{align} + +We prove the claim using\RJMprime. Writing $R$ for the right-hand side of (\ref{eq:transitivity}), we have +\begin{align*} +(RHD \goesto \rhd_1) +& = \doo{x}{\rhd_1(e)}{\return \nabla(x)} = \doo{x}{b}{\return \nabla(x)} = r \ovee s & \text{by (\ref{eq:axiom3})} \\ +(c \goesto \rhd_1) & = r \ovee s & \text{by (\ref{eq:axiom4})} \\ +(R \goesto \rhd_2) +& = (\doo{x}{\rhd_2(e)}{x}) = (\doo{x}{d}{x}) = (c \goesto \rhd_2) +\end{align*} +and so (\ref{eq:transitivity}) follows by\RJMprime. + +Now that the claim (\ref{eq:transitivity}) is proved, we return to the main proof. Define $e' : (A + A) + 1$ by +\begin{align*} +e' = \case e \of & \inl{\inl{\_}} \mapsto \fail \mid \inl{\inr{a}} \mapsto \return{\inl{a}} \mid \\ +& \inr{\inl{a}} \mapsto \return{\inr{a}} \mid \inr{\inr{\_}} \mapsto \fail +\end{align*} +We claim $e'$ is a bound for $s \ovee t$. We have +\begin{align*} +(e' \goesto \rhd_1) +& = \case e \of \inl{\inl{\_}} \mapsto \fail \mid \inl{\inr{a}} \mapsto \return{a} \mid \inr{\_} \mapsto \fail \\ +& = (\rhd_1(e) \goesto \rhd_2) = (b \goesto \rhd_2) = s & \text{by (\ref{eq:axiom2})} \\ +(e' \goesto \rhd_2) & = \case e \of \inl{\_} \mapsto \fail \mid \inr{\inl{a}} \mapsto \return{a} \mid \inr{\inr{\_}} \mapsto \fail \\ +& = (\rhd_2(e) \goesto \rhd_1) = (d \goesto \rhd_1) \\ +& = \case c \of \inl{\inl{\_}} \mapsto \fail \mid \inl{\inr{x}} \mapsto \return{x} \mid \inr{\_} \mapsto \fail \\ +& = (c \goesto \rhd_2) = t & \text{by (\ref{eq:axiom5})} +\end{align*} +and so +\begin{align} +s \ovee t = & \doo{x}{e'}{\return{\nabla(x)}} \label{eq:soveet} \\ += & \case e \of \inl{\inl{\_}} \mapsto \fail \mid \inl{\inr{a}} \mapsto \return{a} \mid \nonumber \\ +& \inr{\inl{a}} \mapsto \return{a} \mid \inr{\inr{\_}} \mapsto \fail \nonumber \\ +\end{align} +Now, define $e'' : (A + A) + 1$ by +\begin{align*} + e'' = \case e \of & \inl{\inl{a}} \mapsto \return{\inl{a}} \mid \inl{\inr{a}} \mapsto \return{\inr{a}} \\ +& \inr{\inl{a}} \mapsto \return{\inr{a}} \mid \inr{\inr{\_}} \mapsto \fail +\end{align*} +We will prove that $e''$ is a bound for $r \ovee (s \ovee t)$. We have +\begin{align*} +(e'' \goesto \rhd_1) +& = \case e \of \begin{array}[t]{l} +\inl{\inl{a}} \mapsto \return{a} \\ +\mid \inl{\inr{\_}} \mapsto \fail \\ +\mid \inr{\inl{\_}} \mapsto \fail \\ +\mid \inr{\inr{\_}} \mapsto \fail +\end{array} \\ +& = (\rhd_1(e) \goesto \rhd_1) = (b \goesto \rhd_1) = r & \text{by (\ref{eq:axiom1})} \\ +(e'' \goesto \rhd_2) & = \case e \of \begin{array}[t]{l} + \inl{\inl{\_}} \mapsto \fail \\ +\mid \inl{\inr{a}} \mapsto \return{a} \\ +\mid \inr{\inl{a}} \mapsto \return{a} \\ +\mid \inr{\inr{\_}} \mapsto \fail +\end{array} \\ +& = s \ovee t & \text{by (\ref{eq:soveet})} \\ +\doo{x}{e''}{\return{\nabla(x)}} & = \case e \of \begin{array}[t]{l} + \inn_1(a) \mapsto \return{a} \\ +\inn_2(a) \mapsto \return{a} \\ +\inn_3(a) \mapsto \return{a} \\ +\inn_4(\_) \mapsto \fail +\end{array} \\ +& = \mathsf{do}\ x \leftarrow \case e \of \begin{array}[t]{l} \inn_1(a) \mapsto \return{\inl{a}} \\ +\inn_2(a) \mapsto \return{\inl{a}} \\ +\inn_3(a) \mapsto \return{\inr{a}} \\ +\inn_4(a) \mapsto \fail; \return{\nabla(x)} +\end{array} \\ +& = \doo{x}{c}{\return{\nabla(x)}} & \text{by (\ref{eq:transitivity})} \\ +& = (r \ovee s) \ovee t +\end{align*} +Thus, $r \ovee (s \ovee t) = \doo{x}{e''}{\return{\nabla(x)}} = (r \ovee s) \ovee t$. +\end{proof} + +\end{document}