[FMFP] Big and small step semantics, equivalence theorem

2026-04-28 16:19:23 +02:00 · 2026-04-24 18:01:57 +02:00
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+\newpage
+\subsubsection{Small-Step semantics}
+\paragraph{Structural Operational Semantics (SOS)}
+Expressing each individual steps of execution allows one to express the \bi{order of execution} of these steps,
+thus allowing us to describe properties of non-terminating programs and parallelization.
+
+$\gamma$ is used as the meta-variable for terminal and non-terminal configurations.
+For the transitions, we denote the relation $\rightarrow_1$, which can have two forms:
+\begin{itemize}
+    \item $\langle s, \sigma \rangle \rightarrow_1 \langle s', \sigma' \rangle$ is for any non-complete computation,
+          where the next computation is expressed by the intermediate configuration $\langle s', \sigma' \rangle$
+    \item $\langle s, \sigma \rangle \rightarrow_1 \sigma'$ is the final execution that reaches a terminal state.
+\end{itemize}
+
+Finally, a transition $\langle s, \sigma \rangle \rightarrow_1 \gamma$ describes the \bi{first step} of the execution of $s$ in state $\sigma$.
+
+A non-terminal configuration $\langle s, \sigma \rangle$ is \bi{stuck}, if there does not exist a configuration $\gamma$ such that $\langle s, \sigma \rangle \rightarrow_1 \gamma$.
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+\subparagraph{The rules}
+\[
+    \begin{prooftree}
+        \infer0[\textsc{Skip}$_{SOS}$]{\langle \texttt{skip}, \sigma \rangle \rightarrow_1 \sigma}
+    \end{prooftree}
+    \qquad
+    \begin{prooftree}
+        \infer0[\textsc{Ass}$_{SOS}$]{\langle x := e, \sigma \rangle \rightarrow_1 \sigma[x \mapsto \cA\llbracket e \rrbracket \sigma ]}
+    \end{prooftree}
+\]
+
+\shade{gray}{Sequential Composition} $s;s'$ ($s$ is executed in state $\sigma$, then $s'$ in resulting $\sigma'$, resulting in $\sigma''$).
+Then, either $s$ executes completely in one step (\textsc{Seq1}), or does not (\textsc{Seq2}).
+\[
+    \begin{prooftree}
+        \hypo{\langle s, \sigma \rangle \rightarrow_1 \sigma'}
+        \infer1[\textsc{Seq1}$_{SOS}$]{\langle s;s', \sigma \rangle \rightarrow_1 \langle s', \sigma' \rangle}
+    \end{prooftree}
+    \qquad
+    \begin{prooftree}
+        \hypo{\langle s', \sigma' \rangle \rightarrow_1 \langle s'', \sigma' \rangle}
+        \infer1[\textsc{Seq2}$_{SOS}$]{\langle s;s', \sigma \rangle \rightarrow_1 \langle s'';s', \sigma' \rangle}
+    \end{prooftree}
+\]
+
+\shade{gray}{Conditional Statements} $\texttt{if}\; b \; \texttt{then} \; s \; \texttt{else} \; s' \; \texttt{end}$ (If $b$ holds, execute $s$, otherwise execute $s'$)
+\[
+    \begin{prooftree}
+        \infer0[\textsc{IfT}$_{SOS}$]{\langle \texttt{if}\; b \; \texttt{then} \; s \; \texttt{else} \; s' \; \texttt{end}, \sigma \rangle \rightarrow_1 \langle s', \sigma' \rangle}
+    \end{prooftree}
+    \qquad
+    \begin{prooftree}
+        \infer0[\textsc{IfF}$_{SOS}$]{\langle \texttt{if}\; b \; \texttt{then} \; s \; \texttt{else} \; s' \; \texttt{end}, \sigma \rangle \rightarrow_1 \langle s', \sigma' \rangle}
+    \end{prooftree}
+\]
+Where the first rule applies if $\cB\llbracket b \rrbracket \sigma = \texttt{tt}$. Below a further two rules for the true case:
+\[
+    \begin{prooftree}
+        \hypo{\langle s, \sigma \rangle \rightarrow_1 \sigma'}
+        \infer1[\textsc{IfT1}$_{SOS}$]{\langle \texttt{if}\; b \; \texttt{then} \; s \; \texttt{else} \; s' \; \texttt{end}, \sigma \rangle \rightarrow_1 \sigma'}
+    \end{prooftree}
+    \qquad
+    \begin{prooftree}
+        \hypo{\langle s, \sigma \rangle \rightarrow_1 \langle s'', \sigma' \rangle}
+        \infer1[\textsc{IfT2}$_{SOS}$]{\langle \texttt{if}\;b \; \texttt{then} \; s \; \texttt{else} \; s' \; \texttt{end}, \sigma \rangle \rightarrow_1 \langle s''', \sigma' \rangle}
+    \end{prooftree}
+\]
+
+
+\shade{gray}{Loop statements} $\texttt{while} \; b \; \texttt{do} \; s \; \texttt{end}$ (If $b$ holds, execute $s$ once, whole statement executed in resulting state $\sigma$)
+\[
+    \begin{prooftree}
+        \infer0[\textsc{While}$_{SOS}$]{\langle \texttt{while} \; b \; \texttt{do} \; s \; \texttt{end}, \sigma \rangle \rightarrow_1 
+        \langle \texttt{if} \; b \; \texttt{then} \; s; \; \texttt{while} \; b \; \texttt{do} \; s \; \texttt{end else skip end}, \sigma \rangle}
+    \end{prooftree}
+\]
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+\paragraph{Multi-Step Execution}
+We use the definitions we have to define a \bi{$k$-step execution}, denoted $\gamma \rightarrow_1^k \gamma'$.
+Of course, this means intuitively that there is an execution of exactly $k$ steps from $\gamma$ to $\gamma'$.
+
+We define the relation inductively over $k$:
+\begin{itemize}
+    \item $\gamma \rightarrow_1^0 \gamma'$ if and only if $\gamma = \gamma'$
+    \item $\gamma \rightarrow_1^k \gamma'$ if and only if there exists $\gamma''$ such that both $\vdash \gamma \rightarrow_1$ and $\gamma'' \rightarrow_1^{k - 1} \gamma'$
+\end{itemize}
+Resulting from this, $\gamma \rightarrow_1^{k_1 + k_2} \gamma'$ if and only if $\exists \gamma'' . \gamma \rightarrow_1^{k_1} \gamma'' \land \gamma'' \rightarrow_1^{k_2} \gamma'$
+
+We write $\gamma \rightarrow_1^* \gamma'$ to signify that there is an execution from $\gamma$ to $\gamma'$ in some finite number of steps, or more formally:
+\[
+    \exists k. \gamma \rightarrow_1^k \gamma'
+\]
+
+
+\paragraph{Derivation Sequences}
+\inlinedefinition A \bi{derivation sequence} is a non-empty sequence of configurations $\gamma_0, \ldots$, for which $\gamma_i \rightarrow_1^1 \gamma_{i + 1}$ for each $0 \leq i$,
+such that $i + 1$ is in the range of the sequence. If the sequence is finite, then the last configuration in the sequence is either a terminal or stuck configuration.
+
+The \bi{length} of the derivation sequence is the number of transitions (thus number of states minus one!)
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+\paragraph{Termination}
+\inlinetheorem The execution of a statement $s$ in a state $\sigma$
+\begin{itemize}
+    \item \bi{terminates} if and only if there is a finite derivation sequence starting with $\langle s, \sigma \rangle$
+    \item \bi{runs forever} if and only if there is an infinite derivation sequence starting with $\langle s, \sigma \rangle$
+\end{itemize}
+
+\inlinetheorem The execution of statement $s$ in state $\sigma$ \bi{terminates} successfully, if and only if $\exists \sigma'. \langle s, \sigma \rangle \rightarrow^*_1 \sigma'$
+
+Of note is that these are properties of \bi{configurations} and not just statements.
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+\paragraph{Proving properties of Derivation Sequences}
+For reasoning about finite derivation sequences, we commonly reason about a multi-step execution $\gamma \rightarrow_1^k \gamma'$
+by \bi{strong induction on the number of steps $k$}, where we define $P(k) \equiv$ ``for all executions of length k, our property holds'' and we prove $P(k)$ for arbitrary $k$
+with the \bi{induction hypothesis} $\forall k' < k. P(k')$ holds
+
+After the setup, it \textit{often} proceeds by case distinction on the $0$ step and the other steps
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+\paragraph{Semantic Equivalence and Determinism}
+\inlinedefinition Under the small-step semantics, two statements $s_1$ and $s_2$ are \bi{semantically equivalent} if for all states $\sigma$ both:
+\begin{itemize}
+    \item for all stuck or terminal configurations $\gamma$ we have $\langle s_1, \sigma \rangle \rightarrow_1^* \gamma$ if and only if
+          $\langle s_2, \sigma \rangle \rightarrow_1^* \gamma$, and
+    \item there is and infinite derivation sequence starting in $\langle s_1, \sigma \rangle$ if and only if there is one starting in $\langle s_2, \sigma \rangle$
+\end{itemize}
+
+\inlinelemma The small-step semantics of IMP is \bi{deterministic}. That is:
+\[
+    \forall s, \sigma, \gamma, \gamma'. \vdash \langle s, \sigma \rangle \rightarrow_1 \gamma
+    \land \vdash \langle s, \gamma \rangle \rightarrow_1 \gamma'
+    \implies \gamma = \gamma'
+\]
+
+\inlinecorollary There is exactly one derivation sequence starting in configuration $\langle s, \sigma \rangle$
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+\paragraph{Extensions of IMP}
+
+\subparagraph{Local Variable Declarations}
+As already partially established in Section \ref{sec:big-step-local-var}, the steps to define a local variable are (for \texttt{var x:=e in s end}):
+\begin{multicols}{3}
+    \begin{enumerate}
+        \item Assign $e$ to $x$
+        \item Execute $s$ (possibly many steps)
+        \item Restore the initial value of $x$
+    \end{enumerate}
+\end{multicols}
+
+Since we need to somehow inject he restore instruction into the statements $s$, we extend the \texttt{Stm} category with a \texttt{restore} statement,
+defined as \texttt{restore (Var, Val)}.
+
+With that, we can define the rules:
+\[
+    \begin{prooftree}
+        \infer0[\textsc{Loc}$_{SOS}$]{\langle \texttt{var x:= e in s end}, \sigma \rangle \rightarrow_1 \langle \texttt{s;restore }(x, \sigma(x)),
+            \sigma[x \mapsto \cA \llbracket e \rrbracket \sigma] \rangle}
+    \end{prooftree}
+\]
+\[
+    \begin{prooftree}
+        \infer0[\textsc{Ret}$_{SOS}$]{\langle \texttt{restore }(x, \sigma(x)), \sigma \rangle \rightarrow_1 \sigma[x \mapsto v]}
+    \end{prooftree}
+\]
+
+Of course, we could also just model execution stacks, as most languages do.
+
+
+\subparagraph{Non-determinism}
+The rules here are analogous to the ones from the big-step rules
+\[
+    \begin{prooftree}
+        \infer0[\textsc{ND1}$_{SOS}$]{\langle s \bigbox s', \sigma \rangle \rightarrow_1 \langle s, \sigma \rangle}
+    \end{prooftree}
+    \qquad
+    \begin{prooftree}
+        \infer0[\textsc{ND2}$_{SOS}$]{\langle s \bigbox s', \sigma \rangle \rightarrow_1 \langle s, \sigma' \rangle}
+    \end{prooftree}
+\]
+
+
+\subparagraph{Parallelism}
+\[
+    \begin{prooftree}
+        \hypo{\langle s, \sigma \rangle \rightarrow_1 \langle s'', \sigma' \rangle}
+        \infer1[\textsc{Par1}$_{SOS}$]{\langle s \texttt{ par } s', \sigma \rangle \rightarrow_1 \langle s'' \texttt{ par } s', \sigma' \rangle}
+    \end{prooftree}
+    \qquad
+    \begin{prooftree}
+        \hypo{\langle s, \sigma \rangle \rightarrow_1 \sigma'}
+        \infer1[\textsc{Par2}$_{SOS}$]{\langle s \texttt{ par } s', \sigma \rangle \rightarrow_1 \langle s', \sigma' \rangle}
+    \end{prooftree}
+\]
+\[
+    \begin{prooftree}
+        \hypo{\langle s', \sigma \rangle \rightarrow_1 \langle s'', \sigma' \rangle}
+        \infer1[\textsc{Par3}$_{SOS}$]{\langle s \texttt{ par } s', \sigma \rangle \rightarrow_1 \langle s \texttt{ par } s'', \sigma' \rangle}
+    \end{prooftree}
+    \qquad
+    \begin{prooftree}
+        \hypo{\langle s', \sigma \rangle \rightarrow_1 \sigma'}
+        \infer1[\textsc{Par4}$_{SOS}$]{\langle s \texttt{ par } s', \sigma \rangle \rightarrow_1 \langle s, \sigma' \rangle}
+    \end{prooftree}
+\]