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sav07_lecture_3_skeleton [2007/03/21 10:01] vkuncak |
sav07_lecture_3_skeleton [2007/03/21 10:17] vkuncak |
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Proof: small model theorem. | Proof: small model theorem. | ||
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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). Most SMT tools are based on this idea (along with Nelson-Oppen combination: next class). |