Commit 7dad46a3 authored by Ralf Jung's avatar Ralf Jung

docs: fix adequacy for state invariant-style WP

parent 7b221fa8
......@@ -340,33 +340,31 @@ The purpose of the adequacy statement is to show that our notion of weakest prec
There are two properties we are looking for: First of all, the postcondition should reflect actual properties of the values the program can terminate with.
Second, a proof of a weakest precondition with any postcondition should imply that the program is \emph{safe}, \ie that it does not get stuck.
To express the adequacy statement for functional correctness, we assume we are given some set $V \subseteq \Val$ of legal return values.
Furthermore, we assume that the signature $\Sig$ adds a predicate $\pred$ to the logic which reflects $V$ into the logic:
\begin{defn}[Adequacy]
A program $\expr$ in some initial state $\state$ is \emph{adequate} for a set $V \subseteq \Val$ of legal return values ($\expr, \state \vDash V$) if for all $\tpool', \state'$ such that $([\expr], \state) \tpstep^\ast (\tpool', \state')$ we have
\begin{enumerate}
\item Safety: For any $\expr' \in \tpool'$ we have that either $\expr'$ is a
value, or \(\red(\expr'_i,\state')\):
\[ \All\expr'\in\tpool'. \toval(\expr') \neq \bot \lor \red(\expr', \state') \]
Notice that this is stronger than saying that the thread pool can reduce; we actually assert that \emph{every} non-finished thread can take a step.
\item Legal return value: If $\tpool'_1$ (the main thread) is a value $\val'$, then $\val' \in V$:
\[ \All \val',\tpool''. \tpool' = [\val'] \dplus \tpool' \Ra \val' \in V \]
\end{enumerate}
\end{defn}
To express the adequacy statement for functional correctness, we assume that the signature $\Sig$ adds a predicate $\pred$ to the logic which reflects the set $V$ of legal return values into the logic:
\[\begin{array}{rMcMl}
\Sem\pred &:& \Sem{\Val\,} \nfn \Sem\Prop \\
\Sem\pred &\eqdef& \Lam \val. \Lam \any. \setComp{n}{v \in V}
\end{array}\]
The signature can of course state arbitrary additional properties of $\pred$, as long as they are proven sound.
The adequacy statement now reads as follows:
\begin{align*}
&\All \mask, \expr, \val, \pred, \state, \state', \tpool'.
\\&( \ownPhys\state \proves \wpre{\expr}[\mask]{x.\; \pred(x)}) \Ra
\\&\cfg{\state}{[\expr]} \step^\ast
\cfg{\state'}{[\val] \dplus \tpool'} \Ra
\\&\val \in V
\end{align*}
The adequacy statement for safety says that our weakest preconditions imply that every expression in the thread pool either is a value, or can reduce further.
\begin{align*}
&\All \mask, \expr, \state, \state', \tpool'.
\\&(\All n. \melt \in \mval_n) \Ra
\\&( \ownPhys\state \proves \wpre{\expr}[\mask]{x.\; \pred(x)}) \Ra
\\&\cfg{\state}{[\expr]} \step^\ast
\cfg{\state'}{\tpool'} \Ra
\\&\All\expr'\in\tpool'. \toval(\expr') \neq \bot \lor \red(\expr', \state')
&\All \mask, \expr, \val, \pred, \state.
\\&( \TRUE \proves {\upd}_\mask \Exists S. S(\state) * \wpre{\expr}[\mask]{x.\; \pred(x)}) \Ra
\\&\expr, \state \vDash V
\end{align*}
Notice that this is stronger than saying that the thread pool can reduce; we actually assert that \emph{every} non-finished thread can take a step.
Notice that the state invariant $S$ used by the weakest precondition is chosen \emph{after} doing a fancy update, which allows it to depend on the names of ghost variables that are picked in that initial fancy update.
\paragraph{Hoare triples.}
It turns out that weakest precondition is actually quite convenient to work with, in particular when perfoming these proofs in Coq.
......
Markdown is supported
0% or
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment