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sav08:small_solutions_for_quantifier-free_presburger_arithmetic [2009/04/23 10:36] vkuncak |
sav08:small_solutions_for_quantifier-free_presburger_arithmetic [2010/05/03 11:06] 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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Note that equations can be expressed as well by stating two inequations. | Note that equations can be expressed as well by stating two inequations. | ||
- | Conversely, if variables range over non-negative integers, then we can rewrite $A x \le b$ as $Ax + u = b$. | + | 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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+ | |||
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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}} | ||
+ | |||
+ | 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 | ||
+ | |||
===== 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". | ||
+ | |||
+ | We obtain a **polynomial-time reduction** from **quantifier-free Presburger arithmetic** to **SAT**. | ||
===== Reduction to SAT ===== | ===== Reduction to SAT ===== | ||
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But other solvers use solutions over rational numbers to help find solutions over integers, using techniques from integer linear programming. | But other solvers use solutions over rational numbers to help find solutions over integers, using techniques from integer linear programming. | ||
* {{projects:simplexdplltreport.pdf|Integrating Simplex with DPLL(T)}} | * {{projects:simplexdplltreport.pdf|Integrating Simplex with DPLL(T)}} | ||
+ | |||
+ | ===== Extensions ===== | ||
+ | |||
+ | Models get bigger if we allow bitvector operations: | ||
+ | * {{sav08:pa_bitvectors.pdf|PABitvectors}} |