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sav07_lecture_3 [2007/03/22 13:44] yuanjianwz |
sav07_lecture_3 [2007/03/29 21:05] kremena.diatchka |
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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 value, 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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| 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'. | 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'. | ||
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| ==== 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, [[sav07_lecture_2#weakest_preconditions|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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| wp(Q, c1 ; c2) = wp(wp(Q,c2),c1) | wp(Q, c1 ; c2) = wp(wp(Q,c2),c1) | ||
| wp(Q, havoc(x)) = ∀x.Q (or introduce a fresh variable) | wp(Q, havoc(x)) = ∀x.Q (or introduce a fresh variable) | ||
| - | How to prove: wp(Q,c1 [] c2) = wp(Q,c1) ∧ wp(Q,c2)? | + | The idea to get : wp(Q,c1 [] c2) = wp(Q,c1) ∧ wp(Q,c2) |
| CR(c1 [] c2) = CR(c1) ∨ CR(c2) | CR(c1 [] c2) = CR(c1) ∨ CR(c2) | ||
| CR(c1 [] c2) -> error = false (it's valid) | CR(c1 [] c2) -> error = false (it's valid) | ||
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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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