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sav08:small_solutions_for_quantifier-free_presburger_arithmetic [2012/05/21 09:57] 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: | ||
- | \[\begin{array}{l} | + | \begin{equation*}\begin{array}{l} |
F ::= A \mid F_1 \land F_2 \mid \lnot F_1 \mid F_1 \lor F_2 \\ | F ::= A \mid F_1 \land F_2 \mid \lnot F_1 \mid F_1 \lor F_2 \\ | ||
A ::= T_1 = T_2 \mid T_1 \le T_2 \mid (K|T) \\ | A ::= T_1 = T_2 \mid T_1 \le T_2 \mid (K|T) \\ | ||
Line 16: | Line 16: | ||
K ::= \ldots -2 \mid -1 \mid 0 \mid 1 \mid 2 \ldots | K ::= \ldots -2 \mid -1 \mid 0 \mid 1 \mid 2 \ldots | ||
\end{array} | \end{array} | ||
- | \] | + | \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 ++ | ||
Line 22: | Line 22: | ||
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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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 |