4,732 research outputs found
Some integer formula-encodings and related algorithms
We investigate the special class of formulas made up of arbitrary but finite
com- binations of addition, multiplication, and exponentiation gates. The
inputs to these formulas are restricted to the integral unit 1. In connection
with such formulas, we describe two essen- tially distinct families of
canonical formula-encodings for integers, respectively deduced from the decimal
encoding and the fundamental theorem of arithmetic. Our main contribution is
the de- tailed description of two algorithms which efficiently determine the
canonical formula-encodings associated with relatively large sets of
consecutive integers
Generalized Totalizer Encoding for Pseudo-Boolean Constraints
Pseudo-Boolean constraints, also known as 0-1 Integer Linear Constraints, are
used to model many real-world problems. A common approach to solve these
constraints is to encode them into a SAT formula. The runtime of the SAT solver
on such formula is sensitive to the manner in which the given pseudo-Boolean
constraints are encoded. In this paper, we propose generalized Totalizer
encoding (GTE), which is an arc-consistency preserving extension of the
Totalizer encoding to pseudo-Boolean constraints. Unlike some other encodings,
the number of auxiliary variables required for GTE does not depend on the
magnitudes of the coefficients. Instead, it depends on the number of distinct
combinations of these coefficients. We show the superiority of GTE with respect
to other encodings when large pseudo-Boolean constraints have low number of
distinct coefficients. Our experimental results also show that GTE remains
competitive even when the pseudo-Boolean constraints do not have this
characteristic.Comment: 10 pages, 2 figures, 2 tables. To be published in 21st International
Conference on Principles and Practice of Constraint Programming 201
On Tackling the Limits of Resolution in SAT Solving
The practical success of Boolean Satisfiability (SAT) solvers stems from the
CDCL (Conflict-Driven Clause Learning) approach to SAT solving. However, from a
propositional proof complexity perspective, CDCL is no more powerful than the
resolution proof system, for which many hard examples exist. This paper
proposes a new problem transformation, which enables reducing the decision
problem for formulas in conjunctive normal form (CNF) to the problem of solving
maximum satisfiability over Horn formulas. Given the new transformation, the
paper proves a polynomial bound on the number of MaxSAT resolution steps for
pigeonhole formulas. This result is in clear contrast with earlier results on
the length of proofs of MaxSAT resolution for pigeonhole formulas. The paper
also establishes the same polynomial bound in the case of modern core-guided
MaxSAT solvers. Experimental results, obtained on CNF formulas known to be hard
for CDCL SAT solvers, show that these can be efficiently solved with modern
MaxSAT solvers
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