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sav07_lecture_3 [2007/03/21 18:51] yuanjianwz |
sav07_lecture_3 [2007/03/22 16:21] yuanjianwz |
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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 F 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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| 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) | ||
| - | wp(r, Q1 ∨ Q2) = wp(r,Q1) ∨ wp(r,Q2) | ||
| But for statements, we have: | But for statements, we have: | ||
| wp(Q, x=t) = Q[x:=t] | wp(Q, x=t) = Q[x:=t] | ||
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| wp(Q, c1 [] c2) = wp(Q,c1) ∧ wp(Q,c2) | wp(Q, c1 [] c2) = wp(Q,c1) ∧ wp(Q,c2) | ||
| wp(Q, c1 ; c2) = wp(wp(Q,c2),c1) | wp(Q, c1 ; c2) = wp(wp(Q,c2),c1) | ||
| - | How to prove: wp(Q,c1 [] c2) = wp(Q,c1) ∧ wp(Q,c2)? | + | wp(Q, havoc(x)) = ∀x.Q (or introduce a fresh variable) |
| + | 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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| Proof: small model theorem. | Proof: small model theorem. | ||
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| ==== Small model theorem for Quantifier-Free Presburger Arithmetic (QFPA) ==== | ==== Small model theorem for Quantifier-Free Presburger Arithmetic (QFPA) ==== | ||
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| + | The idea is to reduce the case, for example: | ||
| + | //∃x,y,z.F// | ||
| + | reduce to | ||
| + | //∃x≤ M,y≤ M,z ≤ M.F// | ||
| + | Then we try to figure out the boundary M. | ||
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| + | Another example: | ||
| + | //¬t1 < t2// | ||
| + | reduce to | ||
| + | //t2+1≤t1// | ||
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| + | How about the not equal ? | ||
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| + | //t1≠t2// | ||
| + | can be reduced to | ||
| + | //(t1 < t2 ) ∨ (t2 < t1) => (t1 ≤ t2-1) ∨ (t2 ≤ t1-1)// | ||
| First step: transform to disjunctive normal form. | First step: transform to disjunctive normal form. | ||