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sav08:small_solutions_for_quantifier-free_presburger_arithmetic [2009/04/23 10:35] vkuncak |
sav08:small_solutions_for_quantifier-free_presburger_arithmetic [2009/04/23 10:40] vkuncak |
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| We also do not need divisibility constraints: $K|t$ is satisfiable iff ++| $t= K q$ is satisfiable, for $q$ fresh ++ | We also do not need divisibility constraints: $K|t$ is satisfiable iff ++| $t= K q$ is satisfiable, for $q$ fresh ++ | ||
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| ===== Integer Linear Ineqautions ===== | ===== Integer Linear Ineqautions ===== | ||
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| Note that equations can be expressed as well by stating two inequations. | Note that equations can be expressed as well by stating two inequations. | ||
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| + | Conversely, if variables range over non-negative integers, then we can rewrite $A x \le b$ as $Ax + u = b$. | ||
| Relatively well studied problem | Relatively well studied problem | ||
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| Each QFPA formula can be reduced to a disjunction of linear integer inequations. | Each QFPA formula can be reduced to a disjunction of linear integer inequations. | ||
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| Proof uses duality in integer linear programming and is given here: | Proof uses duality in integer linear programming and is given here: | ||
| * [[http://doi.acm.org/10.1145/322276.322287|On the complexity of integer programming]], {{:papadimitriou81complexityintegerprogramming.pdf|pdf}} | * [[http://doi.acm.org/10.1145/322276.322287|On the complexity of integer programming]], {{:papadimitriou81complexityintegerprogramming.pdf|pdf}} | ||
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| + | Consequence: it suffices to search for those solutions whose variables denote integers with the number of bits bounded by: | ||
| + | \[ | ||
| + | \log (n(ma)^{2m+1}) = (2m+1) (\log(ma)) + \log n | ||
| + | \] | ||
| + | here | ||
| + | * $a$ is the maximum of values of integers occuring in the problem | ||
| + | * $m$ is number of constraints | ||
| + | * $n$ is number of variables | ||
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| ===== Extending Theorem to QFPA ===== | ===== Extending Theorem to QFPA ===== | ||
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| We therefore obtain a "small model theorem": if there are solutions, there are "small" solutions. In this case, small means "polynomially many bits". | We therefore obtain a "small model theorem": if there are solutions, there are "small" solutions. In this case, small means "polynomially many bits". | ||
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| + | We obtain a **polynomial-time reduction** from **quantifier-free Presburger arithmetic** to **SAT**. | ||
| ===== Reduction to SAT ===== | ===== Reduction to SAT ===== | ||