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sav07_lecture_3 [2007/03/29 21:05] kremena.diatchka |
sav07_lecture_3 [2007/04/18 09:39] kremena.diatchka |
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assume true = skip (does nothing) | assume true = skip (does nothing) | ||
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==== Composing formulas using relation composition ==== | ==== Composing formulas using relation composition ==== | ||
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In sequential composition we follow the rule for composition of relations. We want to get again formula with free variables x_0,y_0,x,y. So we need to do renaming. Let x_1,y_1,error_1 be fresh variables. | In sequential composition we follow the rule for composition of relations. We want to get again formula with free variables x_0,y_0,x,y. So we need to do renaming. Let x_1,y_1,error_1 be fresh variables. | ||
- | CR(c1 ; c2) = exists x_1,y_1,error_1. CR(c1)[x:=x_1,y:=y_1,error:=error_1] & CR(c2)[x:=x_1,y:=y_1,error:=error_1] | + | CR(c1 ; c2) = exists x_1,y_1,error_1. CR(c1)[x:=x_1,y:=y_1,error:=error_1] & CR(c2)[x_0:=x_1,y_0:=y_1,error_0:=error_1] |
The base case is | The base case is | ||
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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, [[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*} | ||
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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) | ||
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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]] for conjunctions of integer inequalities | ||
==== Full Presburger arithmetic ==== | ==== Full Presburger arithmetic ==== |