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sav07_lecture_3_skeleton [2007/03/21 10:45] vkuncak |
sav07_lecture_3_skeleton [2007/03/21 10:59] vkuncak |
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This idea is important in static analysis. | This idea is important in static analysis. | ||
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==== Symbolic execution ==== | ==== Symbolic execution ==== | ||
- | Symbolic execution converts programs into formulas by going forward. It is therefore somewhat analogous to the way an [[interpreter]] for the language would work. It is based on the notion of strongest postcondition. | + | Symbolic execution converts programs into formulas by going forward. It is therefore somewhat analogous to the way an [[interpreter]] for the language would work. |
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+ | Avoid renaming all the time. | ||
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+ | SE(F,k, c1; c2) = SE(F & R(c1), k+1, c2) (update formula) | ||
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+ | SE(F,k,(c1 [] c2); c2) = SE(F, k, c1) | SE(F,k,c2) (explore both branches) | ||
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+ | Note: how many branches do we get? | ||
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+ | Strongest postcondition: | ||
\begin{equation*} | \begin{equation*} | ||
sp(P,r) = \{ s_2 \mid \exists s_1.\ s_1 \in P \land (s_1,s_2) \in r \} | sp(P,r) = \{ s_2 \mid \exists s_1.\ s_1 \in P \land (s_1,s_2) \in r \} | ||
\end{equation*} | \end{equation*} | ||
+ | Like composition of a set with a relation. It's called ''relational image'' of set $P$ under relation $r$. | ||
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+ | Note: when proving our verification condition, instead of proving that semantics of relation implies error=false, it's same as proving that the formula for set sp(U,r) implies error=false, where U is the universal relation, or, in terms of formulas, computing the strongest postcondition of formula 'true'. | ||
==== Weakest preconditions ==== | ==== Weakest preconditions ==== | ||
While symbolic execution computes formula by going forward along the program syntax tree, weakest precondition computes formula by going backward. | While symbolic execution computes formula by going forward along the program syntax tree, weakest precondition computes formula by going backward. | ||
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+ | wp(Q, x=t) = | ||
+ | wp(Q, assume F) = | ||
+ | wp(Q, assert F) = | ||
+ | wp(Q, c1 [] c2) = | ||
+ | wp(Q, c1 ; c2) = | ||
==== Inferring Loop Invariants ==== | ==== Inferring Loop Invariants ==== |