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sav07_lecture_3_skeleton [2007/03/21 10:01] vkuncak |
sav07_lecture_3_skeleton [2007/03/21 10:42] vkuncak |
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| Alternative: | Alternative: | ||
| * decide that you will only loop for formulas of restricted form, as in abstract interpretation and data flow analysis (next week) | * decide that you will only loop for formulas of restricted form, as in abstract interpretation and data flow analysis (next week) | ||
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| Suppose that we obtain (one or more) verification conditions of the form | Suppose that we obtain (one or more) verification conditions of the form | ||
| + | \begin{equation*} | ||
| + | F\ \rightarrow\ (\mbox{error}=\mbox{false}) | ||
| + | \end{equation*} | ||
| + | |||
| + | whose validity we need to prove. We here assume that F contains only | ||
| + | |||
| + | Note: we can check satisfiability of $F\ \land\ (\mbox{error}=\mbox{true})$. | ||
| ==== Quantifier Presburger arithmetic ==== | ==== Quantifier Presburger arithmetic ==== | ||
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| Proof: small model theorem. | Proof: small model theorem. | ||
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| Line 160: | Line 173: | ||
| Next: reduce to integer linear programming: | Next: reduce to integer linear programming: | ||
| \begin{equation*} | \begin{equation*} | ||
| - | Ax = b, x \geq 0 | + | A\vec x = \vec b, \qquad \vec x \geq \vec 0 |
| \end{equation*} | \end{equation*} | ||
| where $A \in {\cal Z}^{m,n}$ and $x \in {\cal Z}^n$. | where $A \in {\cal Z}^{m,n}$ and $x \in {\cal Z}^n$. | ||
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| We can prove small model theorem for ILP - gives bound on search. | We can prove small model theorem for ILP - gives bound on search. | ||
| - | Short proof by {{papadimitriou81complexityintegerprogramming.pdf|Papadimitriou}}. | + | Short proof by {{papadimitriou81complexityintegerprogramming.pdf|Papadimitriou}}: |
| + | * solution of Ax=b (A regular) has as components rationals of form p/q with bounded p,q | ||
| + | * duality of linear programming | ||
| + | * obtains bound $M = n(ma)^{2m+1}$, which needs $(2m+1)(\log n + \log m + \log a)$ bits | ||
| + | * we could encode the problem into SAT: use circuits for addition, comparison etc. | ||
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| + | Note: if small model theorem applies to conjunctions, it also applies to arbitrary QFPA formulas. | ||
| - | Note: if small model theorem applies to conjunctions, it also applies to arbitrary QFPA formulas. Moreover, one can improve these bounds. One tool based on these ideas is [[http://www.cs.cmu.edu/~uclid/|UCLID]]. | + | Moreover, one can improve these bounds. One tool based on these ideas is [[http://www.cs.cmu.edu/~uclid/|UCLID]]. |
| - | Alternative: enumerate disjuncts of DNF on demand, each disjunct is a conjunction, then use ILP techniques (often first solve the underlying linear programming problem over reals). Most SMT tools are based on this idea (along with Nelson-Oppen combination: next class). | + | Alternative: enumerate disjuncts of DNF on demand, each disjunct is a conjunction, then use ILP techniques (often first solve the underlying linear programming problem over reals). Many SMT tools are based on this idea (along with Nelson-Oppen combination: next class). |
| * [[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 | ||