434 research outputs found
A Dichotomy Theorem for the Inverse Satisfiability Problem
The inverse satisfiability problem over a set of Boolean relations Gamma (Inv-SAT(Gamma)) is the computational decision problem of, given a set of models R, deciding whether there exists a SAT(Gamma) instance with R as its set of models. This problem is co-NP-complete in general and a dichotomy theorem for finite ? containing the constant Boolean relations was obtained by Kavvadias and Sideri. In this paper we remove the latter condition and prove that Inv-SAT(Gamma) is always either tractable or co-NP-complete for all finite sets of relations Gamma, thus solving a problem open since 1998. Very little of the techniques used by Kavvadias and Sideri are applicable and we have to turn to more recently developed algebraic approaches based on partial polymorphisms. We also consider the case when ? is infinite, where the situation differs markedly from the case of SAT. More precisely, we show that there exists infinite Gamma such that Inv-SAT(Gamma) is tractable even though there exists finite Delta is subset of Gamma such that Inv-SAT(Delta) is co-NP-complete
The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies
Boolean satisfiability problems are an important benchmark for questions
about complexity, algorithms, heuristics and threshold phenomena. Recent work
on heuristics, and the satisfiability threshold has centered around the
structure and connectivity of the solution space. Motivated by this work, we
study structural and connectivity-related properties of the space of solutions
of Boolean satisfiability problems and establish various dichotomies in
Schaefer's framework.
On the structural side, we obtain dichotomies for the kinds of subgraphs of
the hypercube that can be induced by the solutions of Boolean formulas, as well
as for the diameter of the connected components of the solution space. On the
computational side, we establish dichotomy theorems for the complexity of the
connectivity and st-connectivity questions for the graph of solutions of
Boolean formulas. Our results assert that the intractable side of the
computational dichotomies is PSPACE-complete, while the tractable side - which
includes but is not limited to all problems with polynomial time algorithms for
satisfiability - is in P for the st-connectivity question, and in coNP for the
connectivity question. The diameter of components can be exponential for the
PSPACE-complete cases, whereas in all other cases it is linear; thus, small
diameter and tractability of the connectivity problems are remarkably aligned.
The crux of our results is an expressibility theorem showing that in the
tractable cases, the subgraphs induced by the solution space possess certain
good structural properties, whereas in the intractable cases, the subgraphs can
be arbitrary
The Connectivity of Boolean Satisfiability: Dichotomies for Formulas and Circuits
For Boolean satisfiability problems, the structure of the solution space is
characterized by the solution graph, where the vertices are the solutions, and
two solutions are connected iff they differ in exactly one variable. In 2006,
Gopalan et al. studied connectivity properties of the solution graph and
related complexity issues for CSPs, motivated mainly by research on
satisfiability algorithms and the satisfiability threshold. They proved
dichotomies for the diameter of connected components and for the complexity of
the st-connectivity question, and conjectured a trichotomy for the connectivity
question. Recently, we were able to establish the trichotomy [arXiv:1312.4524].
Here, we consider connectivity issues of satisfiability problems defined by
Boolean circuits and propositional formulas that use gates, resp. connectives,
from a fixed set of Boolean functions. We obtain dichotomies for the diameter
and the two connectivity problems: on one side, the diameter is linear in the
number of variables, and both problems are in P, while on the other side, the
diameter can be exponential, and the problems are PSPACE-complete. For
partially quantified formulas, we show an analogous dichotomy.Comment: 20 pages, several improvement
Generalized Satisfiability Problems via Operator Assignments
Schaefer introduced a framework for generalized satisfiability problems on
the Boolean domain and characterized the computational complexity of such
problems. We investigate an algebraization of Schaefer's framework in which the
Fourier transform is used to represent constraints by multilinear polynomials
in a unique way. The polynomial representation of constraints gives rise to a
relaxation of the notion of satisfiability in which the values to variables are
linear operators on some Hilbert space. For the case of constraints given by a
system of linear equations over the two-element field, this relaxation has
received considerable attention in the foundations of quantum mechanics, where
such constructions as the Mermin-Peres magic square show that there are systems
that have no solutions in the Boolean domain, but have solutions via operator
assignments on some finite-dimensional Hilbert space. We obtain a complete
characterization of the classes of Boolean relations for which there is a gap
between satisfiability in the Boolean domain and the relaxation of
satisfiability via operator assignments. To establish our main result, we adapt
the notion of primitive-positive definability (pp-definability) to our setting,
a notion that has been used extensively in the study of constraint satisfaction
problems. Here, we show that pp-definability gives rise to gadget reductions
that preserve satisfiability gaps. We also present several additional
applications of this method. In particular and perhaps surprisingly, we show
that the relaxed notion of pp-definability in which the quantified variables
are allowed to range over operator assignments gives no additional expressive
power in defining Boolean relations
The number of clones determined by disjunctions of unary relations
We consider finitary relations (also known as crosses) that are definable via
finite disjunctions of unary relations, i.e. subsets, taken from a fixed finite
parameter set . We prove that whenever contains at least one
non-empty relation distinct from the full carrier set, there is a countably
infinite number of polymorphism clones determined by relations that are
disjunctively definable from . Finally, we extend our result to
finitely related polymorphism clones and countably infinite sets .Comment: manuscript to be published in Theory of Computing System
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