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sav07_lecture_3_skeleton [2007/03/21 09:37] vkuncak |
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| Proof: small model theorem. | Proof: small model theorem. | ||
| - | ==== Small model theorem for quantifier-free Presburger arithmetic ==== | + | |
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| + | ==== Small model theorem for Quantifier-Free Presburger Arithmetic (QFPA) ==== | ||
| First step: transform to disjunctive normal form. | First step: transform to disjunctive normal form. | ||
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| 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 solve integer linear programming (ILP) problem |
| + | * [[wk>Integer Linear Programming]] | ||
| + | * online book chapter on ILP | ||
| + | * [[http://www.gnu.org/software/glpk/|GLPK]] tool | ||
| - | Short proof by | + | We can prove small model theorem for ILP - gives bound on search. |
| + | |||
| + | Short proof by {{papadimitriou81complexityintegerprogramming.pdf|Papadimitriou}}. | ||
| + | |||
| + | 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]]. | ||
| + | |||
| + | 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.nyu.edu/acsys/cvc3/download.html|CVC3]] (successor of CVC Lite) | ||
| + | * [[http://combination.cs.uiowa.edu/smtlib/|SMT-LIB]] Standard for formulas, competition | ||
| - | Tools: | ||
| - | * [[http://www.cs.cmu.edu/~uclid/|UCLID]] | ||
| ==== Full Presburger arithmetic ==== | ==== Full Presburger arithmetic ==== | ||
| Full Presburger arithmetic is also decidable. | Full Presburger arithmetic is also decidable. | ||
| + | |||
| + | Approaches: | ||
| + | * Quantifier-Elimination (Omega tool from Maryland) - see homework | ||
| + | * Automata Theoretic approaches: LASH, MONA (as a special case) | ||
| ===== Papers ===== | ===== Papers ===== | ||