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sav07_lecture_3 [2007/03/27 18:58] iulian.dragos |
sav07_lecture_3 [2007/03/30 21:52] vkuncak |
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| We can apply these rules to reduce the size of formulas. | We can apply these rules to reduce the size of formulas. | ||
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| - | |||
| ==== Approximation ==== | ==== Approximation ==== | ||
| - | If (F -> G) is valid, we say that F is stronger than F and we say G is weaker than F. | + | If (F -> G) is valid, we say that F is stronger than G and we say G is weaker than F. |
| When a formula would be too complicated, we can instead create a simpler approximate formula. To be sound, if our goal is to prove a property, we need to generate a *larger* relation, which corresponds to a weaker formula describing a relation, and a stronger verification condition. (If we were trying to identify counterexamples, we would do the opposite). | When a formula would be too complicated, we can instead create a simpler approximate formula. To be sound, if our goal is to prove a property, we need to generate a *larger* relation, which corresponds to a weaker formula describing a relation, and a stronger verification condition. (If we were trying to identify counterexamples, we would do the opposite). | ||
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| While symbolic execution computes formula by going forward along the program syntax tree, [[sav07_lecture_2#weakest_preconditions|weakest precondition]] computes formula by going backward. | While symbolic execution computes formula by going forward along the program syntax tree, [[sav07_lecture_2#weakest_preconditions|weakest precondition]] computes formula by going backward. | ||
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| + | \begin{equation*} | ||
| + | wp(r,P) = \{ s_1 \mid \forall s_2. (s_1,s_2) \in r \rightarrow s_2 \in P \} | ||
| + | \end{equation*} | ||
| + | |||
| We know that the weakest precondition holds following conditions for each relation r and sets Q1, Q2: | We know that the weakest precondition holds following conditions for each relation r and sets Q1, Q2: | ||
| wp(r, Q1 ∧ Q2) = wp(r,Q1) ∧ wp(r,Q2) | wp(r, Q1 ∧ Q2) = wp(r,Q1) ∧ wp(r,Q2) | ||