Lower bound for the Complexity of the Boolean Satisfiability Problem (original) (raw)

A novel algorithm for solving the Decision Boolean Satisfiability Problem without algebra

Cornell University - arXiv, 2016

This paper depicts an algorithm for solving the Decision Boolean Satisfiability Problem using the binary numerical properties of a Special Decision Satisfiability Problem, parallel execution, object oriented, and short termination. The two operations: expansion and simplification are used to explains why using algebra grows the resolution steps. It is proved that its complexity has an upper bound of 2 n−1 where n is the number of logical variables of the given problem.

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...

Solving satisfiability in less than 2n steps

Discrete Applied Mathematics, 1985

In this paper we describe and analyse an algorithm for solving the satisfiability problem. If E is a boolean formula in conjunctive normal form with n variables and r clauses, then we will show that this algorithm solves the satisfiability problem for formulas with at most k literals per clause in time O(]F[. a~), where a k is the greatest number satisfying ak=2-1/a~ I (in the case of 3-satisfiability a 3 = 1,6181).

Combinatorial optimization of boolean satisfiability problems

2013

The purpose of this thesis is to design, implement and analyse a hybrid genetic algorithm for solving the MAX-3SAT problem. The problem is a well known NP-complete decision problem. Solving this type of problems is highly motivated by its practical use in industry. The main field of 3SAT solver application ranges from integrated circuit delay optimization to FPGA routing. See section 1.9 for details on practical applications of the MAX- 3SAT solver. The idea behind this hybrid genetic algorithm, is to see just when and how much of local search can still prove beneficial to the overall solution quality. In the results section 4.4, we have shown that this algorithm performs best when the local search is applied periodically and with our best run results (presented in section 4.6) we came very close to a similar implementation described in [14]. Our best average run time on the biggest test instances was trailing behind aforementioned article by about 30 percent. The reason, why our av...

A Hardware/Software Approach to Accelerate Boolean Satisfiability

Workshop on Design and Diagnostics of Electronic Circuits and Systems, 2000

This paper proposes a new algorithm for solving the Boolean satisfiability (SAT) problem. On the basis of this algorithm a software/reconfigurable hardware SAT solver was designed, implemented and compared to a similar realization of the Davis-Putnam-like method. The satisfier suggested uses an application-specific approach, thus an instance- specific hardware compilation is completely avoided.

The Complexity of the Boolean Formula Value Problem

We examine the complexity of the formula value problem for Boolean formulas, which is the following decision problem: Given a Boolean formula without variables, does it evaluate to true? We show that the complexity of this problem is determined by certain closure properties of the connectives allowed to build the formula, and achieve a complete classification: The formula value problem is either in LOGTIME, complete for one of the classes NLOGTIME, coNLOGTIME or NC 1 , or equivalent to counting modulo 2 under very strict reductions.

Generalizing Boolean Satisfiability III: Implementation

Journal of Artificial Intelligence Research

This is the third of three 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 has been to define a representation in which this structure is apparent and can be exploited to improve computational performance. The first paper surveyed existing work that (knowingly or not) exploited problem structure to improve the performance of satisfiability engines, and the second paper showed that this structure could be understood in terms of groups of permutations acting on individual clauses in any particular Boolean theory. We conclude the series by discussing the techniques needed to implement our ideas, and by reporting on their performance on a variety of problem instances.

A Variable Dependency-Based Decision Strategy for Boolean Satisfiability Problems

Many decision procedures for Boolean Satisfiability problems solve a instance by searching assignments to its variables. They repeatedly select a free (unassigned) variable based on some heuristics, and assign a truth value to it. Such algorithms are highly dependent on the order of selecting variables. In this work, we propose a method for a variable ordering reflecting dependencies among variables, which gives priority to independent variables, rather than dependent variables of which values can be indirectly determined by assigning truth values to other variables. Our method calculates the order of variables statically, and can be given as a hint to existing state-of-the-art SAT solvers easily. Our method captures structured properties of instances which are derived from state transition models, and experimental results show that the fast SAT solver "zChaff" which incorporates our decision strategy runs about 200 times faster than the original version on the instance derived from 8-bit integer multiplier.