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OperationalSemantics chapter: contextual
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Adam Chlipala
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2 changed files with 145 additions and 5 deletions
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@ -1018,9 +1018,7 @@ Module Concurrent.
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interp e v = 0
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-> step0 (v, While e body) (v, Skip)
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| Step0Par1 : forall v c,
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step0 (v, Parallel Skip c) (v, c)
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| Step0Par2 : forall v c,
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step0 (v, Parallel c Skip) (v, c).
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step0 (v, Parallel Skip c) (v, c).
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Inductive cstep : valuation * cmd -> valuation * cmd -> Prop :=
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| CStep : forall C v c v' c' c1 c2,
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@ -1137,8 +1135,6 @@ Module Concurrent.
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-> step (v, While e body) (v, Skip)
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| StepParSkip1 : forall v c,
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step (v, Parallel Skip c) (v, c)
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| StepParSkip2 : forall v c,
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step (v, Parallel c Skip) (v, c)
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| StepPar1 : forall v c1 c2 v' c1',
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step (v, c1) (v', c1')
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-> step (v, Parallel c1 c2) (v', Parallel c1' c2)
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144
frap_book.tex
144
frap_book.tex
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@ -1434,6 +1434,7 @@ The intuition behind the rules may come best from working out an example.
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Here is a step-by-step (literally!) derivation that finds $v$.
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$$\begin{array}{cl}
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& (\mupd{\mempty}{\mathtt{input}}{2}, \assign{\mathtt{output}}{1}; \mathtt{factorial\_loop}) \\
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\to & (\mupd{\mupd{\mempty}{\mathtt{input}}{2}}{\mathtt{output}}{1}, \skipe; \mathtt{factorial\_loop}) \\
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\to & (\mupd{\mupd{\mempty}{\mathtt{input}}{2}}{\mathtt{output}}{1}, \mathtt{factorial\_loop}) \\
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\to & (\mupd{\mupd{\mempty}{\mathtt{input}}{2}}{\mathtt{output}}{1}, (\assign{\mathtt{output}}{\mathtt{output} \times \mathtt{input}}; \assign{\mathtt{input}}{\mathtt{input} - 1}); \mathtt{factorial\_loop}) \\
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\to & (\mupd{\mupd{\mempty}{\mathtt{input}}{2}}{\mathtt{output}}{2}, (\skipe; \assign{\mathtt{input}}{\mathtt{input} - 1}); \mathtt{factorial\_loop}) \\
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@ -1519,6 +1520,149 @@ For instance, consider program $P = \while{\mathtt{n}}{\assign{\mathtt{a}}{\math
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\end{proof}
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\section{Contextual Small-Step Semantics}
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The reader may have noticed some tedium in certain rules of the small-step semantics, like this one.
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$$\infer{\bigstep{(v, c_1; c_2)}{v_2}}{
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\bigstep{(v, c_1)}{v_1}
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& \bigstep{(v_1, c_2)}{v_2}
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}$$
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This rule is an example of a \emph{congruence rule}\index{congruence rule}, which shows how to take a step and \emph{lift} it into a step within a larger command, whose other subcommands are unaffected.
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Complex languages can require many congruence rules, and yet we feel like we should be able to avoid repeating all this boilerplate logic somehow.
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A common way to do so is switching to \emph{contextual small-step semantics}\index{contextual small-step semantics}.
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We illustrate with our running example language.
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The first step is to define a set of \emph{evaluation contexts}\index{evaluation contexts}, which formalize the spots within a larger command where interesting steps are enabled.
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\encoding
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$$\begin{array}{rrcl}
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\textrm{Evaluation contexts} & C &::=& \Box \mid C; c
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\end{array}$$
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\newcommand{\plug}[2]{#1[#2]}
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We define the operator of \emph{plugging}\index{plugging evaluation contexts} an evaluation context in the natural way.
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\begin{eqnarray*}
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\plug{\Box}{c} &=& c \\
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\plug{(C; c_2)}{c} &=& \plug{C}{c}; c_2
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\end{eqnarray*}
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For this language, the only interesting case of evaluation contexts is the one that allows us to \emph{descend into the left subcommand}, because the old congruence rule invoked the step relation recursively for that position.
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\newcommand{\smallstepo}[2]{#1 \to_0 #2}
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The next ingredient is a reduced set of basic step rules, where we have dropped the congruence rule.
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$$\infer{\smallstepo{(v, \assign{x}{e})}{(\mupd{v}{x}{\denote{e}v}, \skipe)}}{}
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\quad \infer{\smallstepo{(v, \skipe; c_2)}{(v, c_2)}}{}$$
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$$\infer{\smallstepo{(v, \ifte{e}{c_1}{c_2})}{(v, c_1)}}{
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\denote{e}v \neq 0
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}
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\quad \infer{\smallstepo{(v, \ifte{e}{c_1}{c_2})}{(v, c_2)}}{
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\denote{e}v = 0
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}$$
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$$\infer{\smallstepo{(v, \while{e}{c_1})}{(v, c_1; \while{e}{c_1})}}{
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\denote{e}v \neq 0
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}
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\quad \infer{\smallstepo{(v, \while{e}{c_1})}{(v, \skipe)}}{
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\denote{e}v = 0
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}$$
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\newcommand{\smallstepc}[2]{#1 \to_\mathsf{c} #2}
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We regain the full coverage of the original rules with a new relation $\to_\mathsf{c}$, saying that we may apply $\to_0$ at the active subcommand within a larger command.
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$$\infer{\smallstepc{(v, C[c])}{(v', C[c'])}}{
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\smallstepo{(v, c)}{(v', c')}
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}$$
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Let's revisit last section's example, to see contextual semantics in action, especially to demonstrate how to express an arbitrary command as an evaluation context plugged with another command.
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\newcommand{\smallstepcs}[2]{#1 \to^*_\mathsf{c} #2}
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\begin{theorem}
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There exists valuation $v$ such that $\smallstepcs{(\mupd{\mempty}{\mathtt{input}}{2}, \mathtt{factorial})}{(v, \skipe)}$ and $\msel{v}{\mathtt{output}} = 2$.
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\end{theorem}
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\begin{proof}
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$$\begin{array}{cl}
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& (\mupd{\mempty}{\mathtt{input}}{2}, \assign{\mathtt{output}}{1}; \mathtt{factorial\_loop}) \\
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= & (\mupd{\mempty}{\mathtt{input}}{2}, \plug{(\Box; \mathtt{factorial\_loop})}{\assign{\mathtt{output}}{1}}) \\
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\to_\mathsf{c} & (\mupd{\mupd{\mempty}{\mathtt{input}}{2}}{\mathtt{output}}{1}, \skipe; \mathtt{factorial\_loop}) \\
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= & (\mupd{\mupd{\mempty}{\mathtt{input}}{2}}{\mathtt{output}}{1}, \plug{\Box}{\skipe; \mathtt{factorial\_loop}}) \\
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\to_\mathsf{c} & (\mupd{\mupd{\mempty}{\mathtt{input}}{2}}{\mathtt{output}}{1}, \mathtt{factorial\_loop}) \\
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= & (\mupd{\mupd{\mempty}{\mathtt{input}}{2}}{\mathtt{output}}{1}, \plug{\Box}{\mathtt{factorial\_loop}}) \\
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\to_\mathsf{c} & (\mupd{\mupd{\mempty}{\mathtt{input}}{2}}{\mathtt{output}}{1}, (\assign{\mathtt{output}}{\mathtt{output} \times \mathtt{input}}; \assign{\mathtt{input}}{\mathtt{input} - 1}); \mathtt{factorial\_loop}) \\
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= & (\mupd{\mupd{\mempty}{\mathtt{input}}{2}}{\mathtt{output}}{1}, \plug{((\Box; \assign{\mathtt{input}}{\mathtt{input} - 1}); \mathtt{factorial\_loop})}{\assign{\mathtt{output}}{\mathtt{output} \times \mathtt{input}}}) \\
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\to_\mathsf{c} & (\mupd{\mupd{\mempty}{\mathtt{input}}{2}}{\mathtt{output}}{2}, (\skipe; \assign{\mathtt{input}}{\mathtt{input} - 1}); \mathtt{factorial\_loop}) \\
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= & (\mupd{\mupd{\mempty}{\mathtt{input}}{2}}{\mathtt{output}}{2}, \plug{(\Box; \mathtt{factorial\_loop})}{\skipe; \assign{\mathtt{input}}{\mathtt{input} - 1})} \\
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\to_\mathsf{c} & (\mupd{\mupd{\mempty}{\mathtt{input}}{2}}{\mathtt{output}}{2}, \assign{\mathtt{input}}{\mathtt{input} - 1}; \mathtt{factorial\_loop}) \\
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= & (\mupd{\mupd{\mempty}{\mathtt{input}}{2}}{\mathtt{output}}{2}, \plug{\Box; \mathtt{factorial\_loop}}{\assign{\mathtt{input}}{\mathtt{input} - 1}}) \\
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\to_\mathsf{c} & (\mupd{\mupd{\mempty}{\mathtt{input}}{1}}{\mathtt{output}}{2}, \skipe; \mathtt{factorial\_loop}) \\
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= & (\mupd{\mupd{\mempty}{\mathtt{input}}{1}}{\mathtt{output}}{2}, \plug{\Box}{\skipe; \mathtt{factorial\_loop}}) \\
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\to^*_\mathsf{c} & \ldots \\
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\to_\mathsf{c} & (\mupd{\mupd{\mempty}{\mathtt{input}}{0}}{\mathtt{output}}{2}, \skipe)
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\end{array}$$
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Clearly the final valuation assigns $\mathtt{output}$ to 2.
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\end{proof}
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\subsection{Equivalence of Small-Step, With and Without Evaluation Contexts}
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This new semantics formulation is equivalent to the other two, as we establish now.
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\begin{theorem}
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If $\smallstep{(v, c)}{(v', c')}$, then $\smallstepc{(v, c)}{(v', c')}$.
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\end{theorem}
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\begin{proof}
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By induction on the derivation of $\smallstep{(v, c)}{(v', c')}$.
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\end{proof}
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\begin{lemma}
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If $\smallstepo{(v, c)}{(v', c')}$, then $\smallstep{(v, c)}{(v', c')}$.
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\end{lemma}
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\begin{proof}
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By cases on the derivation of $\smallstepo{(v, c)}{(v', c')}$.
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\end{proof}
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\begin{lemma}
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If $\smallstepo{(v, c)}{(v', c')}$, then $\smallstep{(v, C[c])}{(v', C[c'])}$.
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\end{lemma}
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\begin{proof}
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By induction on the structure of evaluation context $C$, appealing to the last lemma.
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\end{proof}
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\begin{theorem}
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If $\smallstepc{(v, c)}{(v', c')}$, then $\smallstep{(v, c)}{(v', c')}$.
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\end{theorem}
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\begin{proof}
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By inversion on the derivation of $\smallstepc{(v, c)}{(v', c')}$, followed by an appeal to the last lemma.
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\end{proof}
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\subsection{Evaluation Contexts Pay Off: Adding Concurrency}
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To showcase the convenience of contextual semantics, let's extend our example language with a simple construct for running two commands in parallel\index{parallel composition of threads}, implicitly extending the definition of plugging accordingly.
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$$\begin{array}{rrcl}
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\textrm{Commands} & c &::=& \ldots \mid c || c
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\end{array}$$
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To capture that idea that \emph{either} command in a parallel construct is allowed to step next, we extend evaluation contexts like so:
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\encoding
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$$\begin{array}{rrcl}
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\textrm{Evaluation contexts} & C &::=& \ldots \mid C || c \mid c || C
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\end{array}$$
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We need one more basic step rule, to ``garbage-collect'' threads that have finished.
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$$\infer{\smallstepo{(v, \skipe || c)}{(v, c)}}{}$$
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And that's it!
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The new system faithfully captures our usual idea of threads executing in parallel.
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All of the theorems proved previously about contextual steps continue to hold.
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In fact, in the accompanying Coq code, literally the same proof scripts establish the new versions of the theorems, with no new human proof effort.
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It's not often that concurrency comes for free in a rigorous proof!
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\appendix
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