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sav07_lecture_3 [2007/03/22 16:21] yuanjianwz |
sav07_lecture_3 [2007/04/11 13:51] 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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| + | ==== Weakest preconditions ==== | ||
| + | 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. | ||
| + | \begin{equation*} | ||
| + | wp(r,P) = \{ s_1 \mid \forall s_2. (s_1,s_2) \in r \rightarrow s_2 \in P \} | ||
| + | \end{equation*} | ||
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| - | ==== Weakest preconditions ==== | ||
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| - | While symbolic execution computes formula by going forward along the program syntax tree, weakest precondition computes formula by going backward. | ||
| 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) | ||
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| Proof: small model theorem. | Proof: small model theorem. | ||
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| * [[http://www.cs.nyu.edu/acsys/cvc3/download.html|CVC3]] (successor of CVC Lite) | * [[http://www.cs.nyu.edu/acsys/cvc3/download.html|CVC3]] (successor of CVC Lite) | ||
| * [[http://combination.cs.uiowa.edu/smtlib/|SMT-LIB]] Standard for formulas, competition | * [[http://combination.cs.uiowa.edu/smtlib/|SMT-LIB]] Standard for formulas, competition | ||
| + | * [[http://doi.acm.org/10.1145/135226.135233|Omega test]] conjunctions of integer inequalities | ||
| ==== Full Presburger arithmetic ==== | ==== Full Presburger arithmetic ==== | ||
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| * Presburger Arithmetic (PA) bounds: {{papadimitriou81complexityintegerprogramming.pdf}} | * Presburger Arithmetic (PA) bounds: {{papadimitriou81complexityintegerprogramming.pdf}} | ||
| * Specializing PA bounds: http://www.lmcs-online.org/ojs/viewarticle.php?id=43&layout=abstract | * Specializing PA bounds: http://www.lmcs-online.org/ojs/viewarticle.php?id=43&layout=abstract | ||
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