The Connectivity of Boolean Satisfiability: Dichotomies for Formulas and Circuits (original) (raw)
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A Computational Trichotomy for Connectivity of Boolean Satisfiability
Journal on Satisfiability, Boolean Modeling and Computation
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. Motivated by research on heuristics and the satisfiability threshold, in 2006, Gopalan et al. studied connectivity properties of the solution graph and related complexity issues for constraint satisfaction problems [11]. They found dichotomies for the diameter of connected components and for the complexity of the stconnectivity question, and conjectured a trichotomy for the connectivity question. Their results could be improved based on findings by Makino et al. [15]. Building on this work, we here prove the trichotomy for the connectivity question. Also, we correct a minor mistake in [11], which leads to a slight shift of the boundaries towards the hard side.
The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies
SIAM Journal on Computing, 2009
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 stconnectivity 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.
A Dichotomy Theorem within Schaefer for the Boolean Connectivity Problem
Eccc, 2007
Gopalan et al. studied in [14] connectivity properties of the solution-space of Boolean formulas, and investigated complexity issues on connectivity problems in Schaefer's framework [26]. A set S of logical relations is Schaefer if all relations in S are either bijunctive, Horn, dual Horn, or affine. They conjectured that the connectivity problem for Schaefer is in P. We disprove their conjecture by showing that it is coN P-complete for Horn and dual Horn relations. This, together with the results in [14], implies a dichotomy theory within Schaefer and a trichotomy theory for the connectivity problem. We also show that the connectivity problem for bijunctive relations can be solved in O(min{n|ϕ|, T (n)}) time, where n denotes the number of variables, ϕ denotes the corresponding 2-CNF formula, and T (n) denotes the time needed to compute the transitive closure of a directed graph of n vertices. Furthermore, we investigate a tractable aspect of Horn and dual Horn relations.
An upper bound for the circuit complexity of existentially quantified Boolean formulas
Theoretical Computer Science, 2010
The expressive power of existentially quantified Boolean formulas ∃CNF with free variables is investigated. We introduce a hierarchy of subclasses ∃MU * (k) of ∃CNF formulas based on the maximum deficiency k of minimal unsatisfiable subformulas of the bound part of the formulas. We will establish an upper bound of the size of minimally equivalent circuits. It will be shown, that there are constants a and b, such that for every formula in ∃MU * (k) of length m of the bound part and length l of the free part of the formula there is an equivalent circuit of size less than l + a • m b(log 2 (m)+k) 2 .
Beating Exhaustive Search for Quantified Boolean Formulas and Connections to Circuit Complexity
Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, 2014
We study algorithms for the satisfiability problem for quantified Boolean formulas (QBFs), and consequences of faster algorithms for circuit complexity. • We show that satisfiability of quantified 3-CNFs with m clauses, n variables, and two quantifier blocks (one existential block and one universal) can be solved deterministically in time 2 n−Ω(√ n) • poly(m). For the case of multiple quantifier blocks (alternations), we show that satisfiability of quantified CNFs of size poly(n) on n variables with q quantifier blocks can be solved in 2 n−n 1/(q+1) • poly(n) time by a zero-error randomized algorithm. These are the first provable improvements over brute force search in the general case, even for quantified polynomial-sized CNFs with two quantifier blocks. A second zero-error randomized algorithm solves QBF on circuits of size s in 2 n−Ω(q) • poly(s) time when the number of quantifier blocks is q. • We complement these algorithms by showing that improvements on them would imply new circuit complexity lower bounds. For example, if satisfiability of quantified CNF formulas with n variables, poly(n) size and at most q quantifier blocks can be solved in time 2 n−n ωq (1/q) , then the complexity class NEXP does not have O(log n) depth circuits of polynomial size. Furthermore, solving satisfiability of quantified CNF formulas with n variables, poly(n) size and O(log n) quantifier blocks in time 2 n−ω(log(n)) time would imply the same circuit complexity lower bound. The proofs of these results proceed by establishing strong relationships between the time complexity of QBF satisfiability over CNF formulas and the time complexity of QBF satisfiability over arbitrary Boolean formulas.
An exact algorithm for the Boolean connectivity problem for k-CNF
Theoretical Computer Science, 2011
We present an exact algorithm for a PSPACE-complete problem, denoted by CONNkSAT, which asks if the solution space for a given k-CNF formula is connected on the n-dimensional hypercube. The problem is known to be PSPACE-complete for k ≥ 3, and polynomial solvable for k ≤ 2 [6]. We show that CONNkSAT for k ≥ 3 is solvable in time O((2 − k) n) for some constant k > 0, where k depends only on k, but not on n. This result is considered to be interesting due to the following fact shown by [5]: QBF-3-SAT, which is a typical PSPACE-complete problem, is not solvable in time O((2 −) n) for any constant > 0, provided that the SAT problem (with no restriction to the clause length) is not solvable in time O((2 −) n) for any constant > 0.
Complexity results for quantified boolean formulae based on complete propositional languages
2006
Several propositional fragments have been considered so far as target languages for knowledge compilation and used for improving computational tasks from major AI areas (like inference, diagnosis and planning); among them are the ordered binary decision diagrams, prime implicates, prime implicants, "formulae" in decomposable negation normal form. On the other hand, the validity problem val(QPROP P S ) for Quantified Boolean Formulae (QBF) has been acknowledged for the past few years as an important issue for AI, and many solvers have been designed. In this paper, the complexity of restrictions of the validity problem for QBF obtained by imposing the matrix of the input QBF to belong to propositional fragments used as target languages for compilation, is identified. It turns out that this problem remains hard (PSPACE-complete) even under severe restrictions on the matrix of the input. Nevertheless some tractable restrictions are pointed out.
2005
Several propositional fragments have been considered so far as target languages for knowledge compilation and used for improving computational tasks from major AI areas (like inference, diagnosis and planning); among them are the ordered binary decision diagrams, prime implicates, prime implicants, "formulae" in decomposable negation normal form. On the other hand, the validity problem val(QPROP P S ) for Quantified Boolean Formulae (QBF) has been acknowledged for the past few years as an important issue for AI, and many solvers have been designed. In this paper, the complexity of restrictions of the validity problem for QBF obtained by imposing the matrix of the input QBF to belong to propositional fragments used as target languages for compilation, is identified. It turns out that this problem remains hard (PSPACE-complete) even under severe restrictions on the matrix of the input. Nevertheless some tractable restrictions are pointed out.
Satisfiability in multi-valued circuits
Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, 2018
Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates. We go outside Boolean realm and consider circuits built of any fixed set of gates on an arbitrary large finite domain. From the complexity point of view this is strictly connected with the problems of solving equations (or systems of equations) over finite algebras. The research reported in this work was motivated by a desire to know for which finite algebras A there is a polynomial time algorithm that decides if an equation over A has a solution. We are also looking for polynomial time algorithms that decide if two circuits over a finite algebra compute the same function. Although we have not managed to solve these problems in the most general setting we have obtained such a characterization for a very broad class of algebras from congruence modular varieties. This class includes most known and well-studied algebras such as groups, rings, modules (and their generalizations like quasigroups, loops, near-rings, nonassociative rings, Lie algebras), lattices (and their extensions like Boolean algebras, Heyting algebras or other algebras connected with multi-valued logics including MV-algebras). This paper seems to be the first systematic study of the computational complexity of satisfiability of non-Boolean circuits and solving equations over finite algebras. The characterization results provided by the paper is given in terms of nice structural properties of algebras for which the problems are solvable in polynomial time.
Generalizing Boolean Satisfiability I: Background and Survey of Existing Work
Journal of Artificial Intelligence Research
This is the first of three planned papers describing ZAP, a satisfiability engine that substantially generalizes existing tools while retaining the performance characteristics of modern high-performance solvers. The fundamental idea underlying ZAP is that many problems passed to such engines contain rich internal structure that is obscured by the Boolean representation used; our goal is to define a representation in which this structure is apparent and can easily be exploited to improve computational performance. This paper is a survey of the work underlying ZAP, and discusses previous attempts to improve the performance of the Davis-Putnam-Logemann-Loveland algorithm by exploiting the structure of the problem being solved. We examine existing ideas including extensions of the Boolean language to allow cardinality constraints, pseudo-Boolean representations, symmetry, and a limited form of quantification. While this paper is intended as a survey, our research results are contained i...