450 research outputs found
Message passing for quantified Boolean formulas
We introduce two types of message passing algorithms for quantified Boolean
formulas (QBF). The first type is a message passing based heuristics that can
prove unsatisfiability of the QBF by assigning the universal variables in such
a way that the remaining formula is unsatisfiable. In the second type, we use
message passing to guide branching heuristics of a Davis-Putnam
Logemann-Loveland (DPLL) complete solver. Numerical experiments show that on
random QBFs our branching heuristics gives robust exponential efficiency gain
with respect to the state-of-art solvers. We also manage to solve some
previously unsolved benchmarks from the QBFLIB library. Apart from this our
study sheds light on using message passing in small systems and as subroutines
in complete solvers.Comment: 14 pages, 7 figure
Hiding Satisfying Assignments: Two are Better than One
The evaluation of incomplete satisfiability solvers depends critically on the
availability of hard satisfiable instances. A plausible source of such
instances consists of random k-SAT formulas whose clauses are chosen uniformly
from among all clauses satisfying some randomly chosen truth assignment A.
Unfortunately, instances generated in this manner tend to be relatively easy
and can be solved efficiently by practical heuristics. Roughly speaking, as the
formula's density increases, for a number of different algorithms, A acts as a
stronger and stronger attractor. Motivated by recent results on the geometry of
the space of satisfying truth assignments of random k-SAT and NAE-k-SAT
formulas, we introduce a simple twist on this basic model, which appears to
dramatically increase its hardness. Namely, in addition to forbidding the
clauses violated by the hidden assignment A, we also forbid the clauses
violated by its complement, so that both A and complement of A are satisfying.
It appears that under this "symmetrization'' the effects of the two attractors
largely cancel out, making it much harder for algorithms to find any truth
assignment. We give theoretical and experimental evidence supporting this
assertion.Comment: Preliminary version appeared in AAAI 200
Delta-Complete Decision Procedures for Satisfiability over the Reals
We introduce the notion of "\delta-complete decision procedures" for solving
SMT problems over the real numbers, with the aim of handling a wide range of
nonlinear functions including transcendental functions and solutions of
Lipschitz-continuous ODEs. Given an SMT problem \varphi and a positive rational
number \delta, a \delta-complete decision procedure determines either that
\varphi is unsatisfiable, or that the "\delta-weakening" of \varphi is
satisfiable. Here, the \delta-weakening of \varphi is a variant of \varphi that
allows \delta-bounded numerical perturbations on \varphi. We prove the
existence of \delta-complete decision procedures for bounded SMT over reals
with functions mentioned above. For functions in Type 2 complexity class C,
under mild assumptions, the bounded \delta-SMT problem is in NP^C.
\delta-Complete decision procedures can exploit scalable numerical methods for
handling nonlinearity, and we propose to use this notion as an ideal
requirement for numerically-driven decision procedures. As a concrete example,
we formally analyze the DPLL framework, which integrates Interval
Constraint Propagation (ICP) in DPLL(T), and establish necessary and sufficient
conditions for its \delta-completeness. We discuss practical applications of
\delta-complete decision procedures for correctness-critical applications
including formal verification and theorem proving.Comment: A shorter version appears in IJCAR 201
Percolation of satisfiability in finite dimensions
The satisfiability and optimization of finite-dimensional Boolean formulas
are studied using percolation theory, rare region arguments, and boundary
effects. In contrast with mean-field results, there is no satisfiability
transition, though there is a logical connectivity transition. In part of the
disconnected phase, rare regions lead to a divergent running time for
optimization algorithms. The thermodynamic ground state for the NP-hard
two-dimensional maximum-satisfiability problem is typically unique. These
results have implications for the computational study of disordered materials.Comment: 4 pages, 4 fig
ILP Modulo Data
The vast quantity of data generated and captured every day has led to a
pressing need for tools and processes to organize, analyze and interrelate this
data. Automated reasoning and optimization tools with inherent support for data
could enable advancements in a variety of contexts, from data-backed decision
making to data-intensive scientific research. To this end, we introduce a
decidable logic aimed at database analysis. Our logic extends quantifier-free
Linear Integer Arithmetic with operators from Relational Algebra, like
selection and cross product. We provide a scalable decision procedure that is
based on the BC(T) architecture for ILP Modulo Theories. Our decision procedure
makes use of database techniques. We also experimentally evaluate our approach,
and discuss potential applications.Comment: FMCAD 2014 final version plus proof
A Logical Approach to Efficient Max-SAT solving
Weighted Max-SAT is the optimization version of SAT and many important
problems can be naturally encoded as such. Solving weighted Max-SAT is an
important problem from both a theoretical and a practical point of view. In
recent years, there has been considerable interest in finding efficient solving
techniques. Most of this work focus on the computation of good quality lower
bounds to be used within a branch and bound DPLL-like algorithm. Most often,
these lower bounds are described in a procedural way. Because of that, it is
difficult to realize the {\em logic} that is behind.
In this paper we introduce an original framework for Max-SAT that stresses
the parallelism with classical SAT. Then, we extend the two basic SAT solving
techniques: {\em search} and {\em inference}. We show that many algorithmic
{\em tricks} used in state-of-the-art Max-SAT solvers are easily expressable in
{\em logic} terms with our framework in a unified manner.
Besides, we introduce an original search algorithm that performs a restricted
amount of {\em weighted resolution} at each visited node. We empirically
compare our algorithm with a variety of solving alternatives on several
benchmarks. Our experiments, which constitute to the best of our knowledge the
most comprehensive Max-sat evaluation ever reported, show that our algorithm is
generally orders of magnitude faster than any competitor
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