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sav08:small_solutions_for_quantifier-free_presburger_arithmetic [2010/05/03 11:06] vkuncak |
sav08:small_solutions_for_quantifier-free_presburger_arithmetic [2015/04/21 17:30] (current) |
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QFPA is just PA without quantifiers: | QFPA is just PA without quantifiers: | ||

- | \[\begin{array}{l} | + | \begin{equation*}\begin{array}{l} |

F ::= A \mid F_1 \land F_2 \mid \lnot F_1 \mid F_1 \lor F_2 \\ | F ::= A \mid F_1 \land F_2 \mid \lnot F_1 \mid F_1 \lor F_2 \\ | ||

A ::= T_1 = T_2 \mid T_1 \le T_2 \mid (K|T) \\ | A ::= T_1 = T_2 \mid T_1 \le T_2 \mid (K|T) \\ | ||

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K ::= \ldots -2 \mid -1 \mid 0 \mid 1 \mid 2 \ldots | K ::= \ldots -2 \mid -1 \mid 0 \mid 1 \mid 2 \ldots | ||

\end{array} | \end{array} | ||

- | \] | + | \end{equation*} |

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 ++ | ||

- | |||

- | |||

===== Integer Linear Ineqautions ===== | ===== Integer Linear Ineqautions ===== | ||

In matrix form, integer linear inequation is | In matrix form, integer linear inequation is | ||

- | \[ | + | \begin{equation*} |

A \vec x \leq \vec b | A \vec x \leq \vec b | ||

- | \] | + | \end{equation*} |

where $A \vec x$ denotes matrix $A$ multiplied by vector $\vec x$. When $A$ is $m \times n$ matrix, this denotes the system of inequations: | where $A \vec x$ denotes matrix $A$ multiplied by vector $\vec x$. When $A$ is $m \times n$ matrix, this denotes the system of inequations: | ||

- | \[ | + | \begin{equation*} |

\bigwedge_{j=1}^m \sum_{i=1}^n a_{ij} x_i \le b_j | \bigwedge_{j=1}^m \sum_{i=1}^n a_{ij} x_i \le b_j | ||

- | \] | + | \end{equation*} |

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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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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===== Theorem on Solution Bounds ===== | ===== Theorem on Solution Bounds ===== | ||

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Consequence: it suffices to search for those solutions whose variables denote integers with the number of bits bounded by: | Consequence: it suffices to search for those solutions whose variables denote integers with the number of bits bounded by: | ||

- | \[ | + | \begin{equation*} |

\log (n(ma)^{2m+1}) = (2m+1) (\log(ma)) + \log n | \log (n(ma)^{2m+1}) = (2m+1) (\log(ma)) + \log n | ||

- | \] | + | \end{equation*} |

here | here | ||

* $a$ is the maximum of values of integers occuring in the problem | * $a$ is the maximum of values of integers occuring in the problem | ||

* $m$ is number of constraints | * $m$ is number of constraints | ||

* $n$ is number of variables | * $n$ is number of variables | ||

- | |||

===== Extending Theorem to QFPA ===== | ===== Extending Theorem to QFPA ===== | ||

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===== Reduction to SAT ===== | ===== Reduction to SAT ===== | ||

- | We observed before that if we have bounded integers we can encode formula into SAT. | + | Observe that if we have bounded integers we can encode the formula into SAT. |

- | | + | |

===== Implementation ===== | ===== Implementation ===== |