237 research outputs found
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
A lower bound on CNF encodings of the at-most-one constraint
Constraint "at most one" is a basic cardinality constraint which requires
that at most one of its boolean inputs is set to . This constraint is
widely used when translating a problem into a conjunctive normal form (CNF) and
we investigate its CNF encodings suitable for this purpose. An encoding differs
from a CNF representation of a function in that it can use auxiliary variables.
We are especially interested in propagation complete encodings which have the
property that unit propagation is strong enough to enforce consistency on input
variables. We show a lower bound on the number of clauses in any propagation
complete encoding of the "at most one" constraint. The lower bound almost
matches the size of the best known encodings. We also study an important case
of 2-CNF encodings where we show a slightly better lower bound. The lower bound
holds also for a related "exactly one" constraint.Comment: 38 pages, version 3 is significantly reorganized in order to improve
readabilit
FourierSAT: A Fourier Expansion-Based Algebraic Framework for Solving Hybrid Boolean Constraints
The Boolean SATisfiability problem (SAT) is of central importance in computer
science. Although SAT is known to be NP-complete, progress on the engineering
side, especially that of Conflict-Driven Clause Learning (CDCL) and Local
Search SAT solvers, has been remarkable. Yet, while SAT solvers aimed at
solving industrial-scale benchmarks in Conjunctive Normal Form (CNF) have
become quite mature, SAT solvers that are effective on other types of
constraints, e.g., cardinality constraints and XORs, are less well studied; a
general approach to handling non-CNF constraints is still lacking. In addition,
previous work indicated that for specific classes of benchmarks, the running
time of extant SAT solvers depends heavily on properties of the formula and
details of encoding, instead of the scale of the benchmarks, which adds
uncertainty to expectations of running time.
To address the issues above, we design FourierSAT, an incomplete SAT solver
based on Fourier analysis of Boolean functions, a technique to represent
Boolean functions by multilinear polynomials. By such a reduction to continuous
optimization, we propose an algebraic framework for solving systems consisting
of different types of constraints. The idea is to leverage gradient information
to guide the search process in the direction of local improvements. Empirical
results demonstrate that FourierSAT is more robust than other solvers on
certain classes of benchmarks.Comment: The paper was accepted by Thirty-Fourth AAAI Conference on Artificial
Intelligence (AAAI 2020). V2 (Feb 24): Typos correcte
Native Cardinality Constraints: More Expressive, More Efficient Constraints
Boolean cardinality constraints are commonly translated (encoded) into Boolean CNF, a standard form for Boolean satisfiability problems, which can be solved using a standard SAT solving program. However, cardinality constraints are a simple generalization of clauses, and the complexity entailed by encoding them into CNF can be avoided by reasoning about cardinality constraints natively within a SAT solver. In this work, we compare the performance of two forms of native cardinality constraints against some of the best performing encodings from the literature. We designed a number of experiments, modeling the general use of cardinality constraints including crafted, random and application problems, to be run in parallel on a cluster of computers. Results show that native implementations substantially outperform CNF encodings on instances composed entirely of cardinality constraints, and instances that are mostly clauses with few cardinality constraints exhibit mixed results warranting further study
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