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Algorithms for the Satisfiability (SAT) Problem: A Survey
 DIMACS Series in Discrete Mathematics and Theoretical Computer Science
, 1996
"... . The satisfiability (SAT) problem is a core problem in mathematical logic and computing theory. In practice, SAT is fundamental in solving many problems in automated reasoning, computeraided design, computeraided manufacturing, machine vision, database, robotics, integrated circuit design, compute ..."
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Cited by 127 (3 self)
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. The satisfiability (SAT) problem is a core problem in mathematical logic and computing theory. In practice, SAT is fundamental in solving many problems in automated reasoning, computeraided design, computeraided manufacturing, machine vision, database, robotics, integrated circuit design, computer architecture design, and computer network design. Traditional methods treat SAT as a discrete, constrained decision problem. In recent years, many optimization methods, parallel algorithms, and practical techniques have been developed for solving SAT. In this survey, we present a general framework (an algorithm space) that integrates existing SAT algorithms into a unified perspective. We describe sequential and parallel SAT algorithms including variable splitting, resolution, local search, global optimization, mathematical programming, and practical SAT algorithms. We give performance evaluation of some existing SAT algorithms. Finally, we provide a set of practical applications of the sat...
Finding Hard Instances of the Satisfiability Problem: A Survey
, 1997
"... . Finding sets of hard instances of propositional satisfiability is of interest for understanding the complexity of SAT, and for experimentally evaluating SAT algorithms. In discussing this we consider the performance of the most popular SAT algorithms on random problems, the theory of average case ..."
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Cited by 114 (1 self)
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. Finding sets of hard instances of propositional satisfiability is of interest for understanding the complexity of SAT, and for experimentally evaluating SAT algorithms. In discussing this we consider the performance of the most popular SAT algorithms on random problems, the theory of average case complexity, the threshold phenomenon, known lower bounds for certain classes of algorithms, and the problem of generating hard instances with solutions.
Probabilistic Analysis Of A Generalization Of The Unit Clause Literal Selection Heuristic For The KSatisfiability Problem
 INFORMATION SCIENCE
, 1990
"... Two algorithms for the kSatisfiability problem are presented and a probabilistic analysis is performed. The analysis is based on an instance distribution which is parameterized to simulate a variety of sample characteristics. The algorithms assign values to literals appearing in a given instance of ..."
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Cited by 93 (9 self)
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Two algorithms for the kSatisfiability problem are presented and a probabilistic analysis is performed. The analysis is based on an instance distribution which is parameterized to simulate a variety of sample characteristics. The algorithms assign values to literals appearing in a given instance of kSatisfiability, one at a time, until a solution is found or it is discovered that further assignments cannot lead to finding a solution. One algorithm chooses the next literal from a unit clause if one exists and randomly from the set of remaining literals otherwise. The other algorithm uses a generalization of the UnitClause rule as a heuristic for selecting the next literal: at each step a literal is chosen randomly from a clause containing the least number of literals. The algorithms run in polynomial time and it is shown that they find a solution to a random instance of kSatisfiability with probability bounded from below by a constant greater than zero for two different ranges of...
Needed: An Empirical Science Of Algorithms
 Operations Research
, 1994
"... this article goes to press. Journal editors can be encouraged to seek out referees who have done rigorous empirical studies. Refereeing standards will evolve, particularly as the empirical science develops. ..."
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Cited by 73 (3 self)
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this article goes to press. Journal editors can be encouraged to seek out referees who have done rigorous empirical studies. Refereeing standards will evolve, particularly as the empirical science develops.
Results Related to Threshold Phenomena Research in Satisfiability: Lower Bounds
, 2000
"... We present a history of results related to the threshold phenomena which arise in the study of random Conjunctive Normal Form (CNF) formulas. In a companion paper [1] in this volume the major ideas used to achieve many of the lower bounds results on the location of the threshold are described in an ..."
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Cited by 20 (1 self)
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We present a history of results related to the threshold phenomena which arise in the study of random Conjunctive Normal Form (CNF) formulas. In a companion paper [1] in this volume the major ideas used to achieve many of the lower bounds results on the location of the threshold are described in an informal, intuitive manner.
Some Pitfalls for Experimenters with Random SAT
 Artificial Intelligence
, 1996
"... We consider the use of random CNF formulas in evaluating the performance of SAT testing algorithms, and in particular the role that the phase transition phenomenon plays in this use. Examples from the literature illustrate the importance of understanding the properties of formula distributions prior ..."
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Cited by 19 (3 self)
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We consider the use of random CNF formulas in evaluating the performance of SAT testing algorithms, and in particular the role that the phase transition phenomenon plays in this use. Examples from the literature illustrate the importance of understanding the properties of formula distributions prior to designing an experiment. We expect this to be of increasing importance in the field. 1 Introduction Satisfiability testing lies at the core of many computational problems and because of its close relationship to various reasoning tasks, this is especially so in Artificial Intelligence. Randomly generated CNF formulas are a popular class of test problems for evaluating the performance of SAT testing programs. Not surprisingly, the choice of formula distribution is crucial to the validity of any investigation using random formulas. In [23], we argued that some families of distributions were more useful sources of test material than others, and suggested choosing formulas from the "hard reg...
Elimination Of Infrequent Variables Improves Average Case Performance Of Satisfiability Algorithms
 SIAM J. Comput
, 1991
"... . We consider preprocessing a random instance I of CNF Satisfiability in order to remove infrequent variables (those which appear once or twice in an instance) from I. The model used to generate random instances is the popular randomclausesize model with parametersn, the number of clauses, r, the ..."
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Cited by 16 (5 self)
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. We consider preprocessing a random instance I of CNF Satisfiability in order to remove infrequent variables (those which appear once or twice in an instance) from I. The model used to generate random instances is the popular randomclausesize model with parametersn, the number of clauses, r, the number of Boolean variables from which clauses are composed, and p, the probability that a variable appears in a clause as a positive (or negative) literal. It is shown that exhaustive search over such preprocessed instances runs in polynomial average time over a significantly larger parameter space than has been shown for any other algorithm under the randomclausesize model when n = r ffl , ffl ! 1, and pr ! p fflr ln(r). Specifically, the results are that random instances of Satisfiability are "easy" in the average case if n = r ffl , 2=3 ? ffl ? 0, and pr ! (ln(n)=4) 1=3 r 2=3\Gammaffl ; or n = r ffl , 1 ? ffl 2=3, pr ! (1 \Gamma ffl \Gamma ffi) ln(n)=ffl for any ffi ? 0...
Using Neural Networks and Genetic Algorithms as Heuristics for NPComplete Problems
, 1983
"... Paradigms for using neural networks (NNs) and genetic algorithms (GAs) to heuristically solve boolean satisfiability (SAT) problems are presented. Since SAT is NPComplete, any other NPComplete problem can be transformed into an equivalent SAT problem in polynomial time, and solved via either parad ..."
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Cited by 15 (8 self)
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Paradigms for using neural networks (NNs) and genetic algorithms (GAs) to heuristically solve boolean satisfiability (SAT) problems are presented. Since SAT is NPComplete, any other NPComplete problem can be transformed into an equivalent SAT problem in polynomial time, and solved via either paradigm. This technique is illustrated for hamiltonian circuit (HC) problems. INTRODUCTION NPComplete problems are problems that are not currently solvable in polynomial time. However, they are polynomially equivalent in the sense that any NPComplete problem can be transformed into any other in polynomial time. Thus, if any NPComplete problem can be solved in polynomial time, they all can [Garey]. The canonical example of an NPComplete problem is the boolean satisfiability (SAT) problem: Given an arbitrary boolean expression of n variables, does there exist an assignment to those variables such that the expression is true? Other familiar examples include job shop scheduling, bin packing, a...
Probabilistic Performance of a Heuristic for the Satisfiability Problem
 Discrete Applied Mathematics
, 1986
"... An algorithm for the Satisfiability problem is presented and its probabilistic behavior is analysed when combined with two other algorithms studied earlier. The analysis is based on an instance distribution which is parameterized to simulate a variety of sample characteristics. The algorithm dynamic ..."
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Cited by 12 (6 self)
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An algorithm for the Satisfiability problem is presented and its probabilistic behavior is analysed when combined with two other algorithms studied earlier. The analysis is based on an instance distribution which is parameterized to simulate a variety of sample characteristics. The algorithm dynamically assigns values to literals appearing in a given instance until a satisfying assignment is found or the algorithm "gives up" without determining whether or not a solution exists. It is shown that if n clauses are constructed independently from r boolean variables where the probability that a variable appears in a clause as a positive literal is p and as a negative literal is p then almost all randomly generated instances of Satisfiability are solved in polynomial time if p ! :4 ln(n)=r or p ? ln(n)=r or p = c ln(n)=r, :4 ! c ! 1 and lim n;r!1 n 1\Gammac =r 1\Gammaffl ! 1 for any ffl ? 0. It is also shown that if p = c ln(n)=r, :4 ! c ! 1 and lim n;r!1 n 1\Gammac =r = 1 then almost ...
Theoretical analysis of DavisPutnam procedure and propositional satisfiability
 In Proceedings of the 14th International Joint Conference on Artificial Intelligence
, 1995
"... This paper presents a statistical analysis of the DavisPutnam procedure and propositional satisfiability problems (SAT). SAT has been researched in AI because of its strong relationship to automated reasoning and recently it is used as a benchmark problem of constraint satisfaction algorithms. The ..."
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Cited by 7 (0 self)
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This paper presents a statistical analysis of the DavisPutnam procedure and propositional satisfiability problems (SAT). SAT has been researched in AI because of its strong relationship to automated reasoning and recently it is used as a benchmark problem of constraint satisfaction algorithms. The DavisPutnam procedure is a wellknown satisfiability checking algorithm based on tree search technique. In this paper, I analyze two average case complexities for the DavisPutnam procedure, the complexity for satisfiability checking and the complexity for finding all solutions. I also discuss the probability of satisfiability. The complexities and the probability strongly depend on the distribution of formulas to be tested and I use the fixed clause length model as the distribution model. The result of the analysis coincides with the experimental result well. 1