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sav07_lecture_3_skeleton [2007/03/21 09:48] vkuncak |
sav07_lecture_3_skeleton [2007/03/21 10:15] vkuncak |
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| Proof: small model theorem. | Proof: small model theorem. | ||
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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 (ILP). | + | 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 {{papadimitriou81complexityintegerprogramming.pdf|Papadimitriou}}. | + | We can prove small model theorem for ILP - gives bound on search. |
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| + | 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 $M = n(ma)^{2m+1}$. | ||
| 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]]. | 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]]. | ||
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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 | ||
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| ==== Full Presburger arithmetic ==== | ==== Full Presburger arithmetic ==== | ||
| Full Presburger arithmetic is also decidable. | Full Presburger arithmetic is also decidable. | ||
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| + | Approaches: | ||
| + | * Quantifier-Elimination (Omega tool from Maryland) - see homework | ||
| + | * Automata Theoretic approaches: LASH, MONA (as a special case) | ||
| ===== Papers ===== | ===== Papers ===== | ||