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sav07_lecture_3_skeleton [2007/03/21 09:37] vkuncak |
sav07_lecture_3_skeleton [2007/03/21 09:48] vkuncak |
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| Proof: small model theorem. | Proof: small model theorem. | ||
| - | ==== Small model theorem for quantifier-free Presburger arithmetic ==== | + | |
| + | |||
| + | ==== Small model theorem for Quantifier-Free Presburger Arithmetic (QFPA) ==== | ||
| First step: transform to disjunctive normal form. | First step: transform to disjunctive normal form. | ||
| Line 161: | Line 163: | ||
| where $A \in {\cal Z}^{m,n}$ and $x \in {\cal Z}^n$. | where $A \in {\cal Z}^{m,n}$ and $x \in {\cal Z}^n$. | ||
| - | Then use small model theorem for integer linear programming. | + | Then use small model theorem for integer linear programming (ILP). |
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| + | Short proof by {{papadimitriou81complexityintegerprogramming.pdf|Papadimitriou}}. | ||
| - | Short proof by | + | 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]]. |
| - | Tools: | + | 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). |
| - | * [[http://www.cs.cmu.edu/~uclid/|UCLID]] | + | * [[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 | ||
| ==== Full Presburger arithmetic ==== | ==== Full Presburger arithmetic ==== | ||