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160
GSAT and Dynamic Backtracking
 Journal of Artificial Intelligence Research
, 1994
"... There has been substantial recent interest in two new families of search techniques. One family consists of nonsystematic methods such as gsat; the other contains systematic approaches that use a polynomial amount of justification information to prune the search space. This paper introduces a new te ..."
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Cited by 360 (14 self)
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There has been substantial recent interest in two new families of search techniques. One family consists of nonsystematic methods such as gsat; the other contains systematic approaches that use a polynomial amount of justification information to prune the search space. This paper introduces a new technique that combines these two approaches. The algorithm allows substantial freedom of movement in the search space but enough information is retained to ensure the systematicity of the resulting analysis. Bounds are given for the size of the justification database and conditions are presented that guarantee that this database will be polynomial in the size of the problem in question. 1 INTRODUCTION The past few years have seen rapid progress in the development of algorithms for solving constraintsatisfaction problems, or csps. Csps arise naturally in subfields of AI from planning to vision, and examples include propositional theorem proving, map coloring and scheduling problems. The probl...
Hard and Easy Distributions of SAT Problems
, 1992
"... We report results from largescale experiments in satisfiability testing. As has been observed by others, testing the satisfiability of random formulas often appears surprisingly easy. Here we show that by using the right distribution of instances, and appropriate parameter values, it is possible to ..."
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Cited by 219 (17 self)
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We report results from largescale experiments in satisfiability testing. As has been observed by others, testing the satisfiability of random formulas often appears surprisingly easy. Here we show that by using the right distribution of instances, and appropriate parameter values, it is possible to generate random formulas that are hard, that is, for which satisfiability testing is quite difficult. Our results provide a benchmark for the evaluation of satisfiabilitytesting procedures. Introduction Many computational tasks of interest to AI, to the extent that they can be precisely characterized at all, can be shown to be NPhard in their most general form. However, there is fundamental disagreement, at least within the AI community, about the implications of this. It is claimed on the one hand that since the performance of algorithms designed to solve NPhard tasks degrades rapidly with small increases in input size, something will need to be given up to obtain acceptable behavior....
Experimental Results on the Crossover Point in Satisfiability Problems
 In Proceedings of the Eleventh National Conference on Artificial Intelligence
, 1993
"... Determining whether a propositional theory is satisfiable is a prototypical example of an NPcomplete problem. Further, a large number of problems that occur in knowledge representation, learning, planning, and other areas of AI are essentially satisfiability problems. This paper reports on a series ..."
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Cited by 202 (3 self)
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Determining whether a propositional theory is satisfiable is a prototypical example of an NPcomplete problem. Further, a large number of problems that occur in knowledge representation, learning, planning, and other areas of AI are essentially satisfiability problems. This paper reports on a series of experiments to determine the location of the crossover point  the point at which half the randomly generated propositional theories with a given number of variables and given number of clauses are satisfiable  and to assess the relationship of the crossover point to the difficulty of determining satisfiability. We have found empirically that, for 3sat, the number of clauses at the crossover point is a linear function of the number of variables. This result is of theoretical interest since it is not clear why such a linear relationship should exist, but it is also of practical interest since recent experiments [ Mitchell et al. 92; Cheeseman et al. 91 ] indicate that the most comput...
Short Proofs are Narrow  Resolution made Simple
 Journal of the ACM
, 2000
"... The width of a Resolution proof is de ned to be the maximal number of literals in any clause of the proof. In this paper we relate proof width to proof length (=size), in both general Resolution, and its treelike variant. The following consequences of these relations reveal width as a crucial "reso ..."
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Cited by 181 (15 self)
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The width of a Resolution proof is de ned to be the maximal number of literals in any clause of the proof. In this paper we relate proof width to proof length (=size), in both general Resolution, and its treelike variant. The following consequences of these relations reveal width as a crucial "resource" of Resolution proofs. In one direction, the relations allow us to give simple, unified proofs for almost all known exponential lower bounds on size of resolution proofs, as well as several interesting new ones. They all follow from width lower bounds, and we show how these follow from natural expansion property of clauses of the input tautology. In the other direction, the widthsize relations naturally suggest a simple dynamic programming procedure for automated theorem proving  one which simply searches for small width proofs. This relation guarantees that the running time (and thus the size of the produced proof) is at most quasipolynomial in the smallest treelike proof. This algorithm is never much worse than any of the recursive automated provers (such as DLL) used in practice. In contrast, we present a family of tautologies on which it is exponentially faster.
Analysis Of Two Simple Heuristics On A Random Instance Of kSAT
 Journal of Algorithms
, 1996
"... We consider the performance of two algorithms, GUC and SC studied by Chao and Franco [2], [3], and Chv'atal and Reed [4], when applied to a random instance ! of a boolean formula in conjunctive normal form with n variables and bcnc clauses of size k each. For the case where k = 3, we obtain the exa ..."
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Cited by 139 (4 self)
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We consider the performance of two algorithms, GUC and SC studied by Chao and Franco [2], [3], and Chv'atal and Reed [4], when applied to a random instance ! of a boolean formula in conjunctive normal form with n variables and bcnc clauses of size k each. For the case where k = 3, we obtain the exact limiting probability that GUC succeeds. We also consider the situation when GUC is allowed to have limited backtracking, and we improve an existing threshold for c below which almost all ! is satisfiable. For k 4, we obtain a similar result regarding SC with limited backtracking. 1 Introduction Given a boolean formula ! in conjunctive normal form, the satisfiability problem (sat) is to determine whether there is a truth assignment that satisfies !. Since sat is NPcomplete, one is interested in efficient heuristics that perform well "on average," or with high probability. The choice of the probabilistic space is crucial for the significance of such a study. In particular, it is easy to ...
Experimental Results on the Crossover Point in Random 3sat
 Artificial Intelligence
, 1996
"... Determining whether a propositional theory is satisfiable is a prototypical example of an NPcomplete problem. Further, a large number of problems that occur in knowledgerepresentation, learning, planning, and other ares of AI are essentially satisfiability problems. This paper reports on the most ..."
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Cited by 137 (5 self)
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Determining whether a propositional theory is satisfiable is a prototypical example of an NPcomplete problem. Further, a large number of problems that occur in knowledgerepresentation, learning, planning, and other ares of AI are essentially satisfiability problems. This paper reports on the most extensive set of experiments to date on the location and nature of the crossover point in satisfiability problems. These experiments generally confirm previous results with two notable exceptions. First, we have found that neither of the functions previously proposed accurately models the location of the crossover point. Second, we have found no evidence of any hard problems in the underconstrained region. In fact the hardest problems found in the underconstrained region were many times easier than the easiest unsatisfiable problems found in the neighborhood of the crossover point. We offer explanations for these apparent contradictions of previous results. This work has been supported ...
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...
Heuristics based on unit propagation for satisfiability problems
, 1997
"... The paper studies new unit propagation based heuristics for DavisPutnamLoveland (DPL) procedure. These are the novel combinations of unit propagation and the usual "Maximum Occurrences in clauses of Minimum Size " heuristics. Based on the experimental evaluations of di erent alternatives ..."
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Cited by 121 (10 self)
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The paper studies new unit propagation based heuristics for DavisPutnamLoveland (DPL) procedure. These are the novel combinations of unit propagation and the usual "Maximum Occurrences in clauses of Minimum Size " heuristics. Based on the experimental evaluations of di erent alternatives a new simple unit propagation based heuristic is put forward. This compares favorably with the heuristics employed in the current stateoftheart DPL implementations (CSAT, Tableau, POSIT). 1
Simplified and Improved Resolution Lower Bounds
 IN PROCEEDINGS OF THE 37TH IEEE FOCS
, 1996
"... We give simple new lower bounds on the lengths of Resolution proofs for the pigeonhole principle and for randomly generated formulas. For random formulas, our bounds significantly extend the range of formula sizes for which nontrivial lower bounds are known. For example, we show that with probabili ..."
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Cited by 102 (7 self)
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We give simple new lower bounds on the lengths of Resolution proofs for the pigeonhole principle and for randomly generated formulas. For random formulas, our bounds significantly extend the range of formula sizes for which nontrivial lower bounds are known. For example, we show that with probability approaching 1, any Resolution refutation of a randomly chosen 3CNF formula with at most n 6=5\Gammaffl clauses requires exponential size. Previous bounds applied only when the number of clauses was at most linear in the number of variables. For the pigeonhole principle our bound is a small improvement over previous bounds. Our proofs are more elementary than previous arguments, and establish a connection between Resolution proof size and maximum clause size.
Generating Hard Satisfiability Problems
 Artificial Intelligence
, 1996
"... We report results from largescale experiments in satisfiability testing. As has been observed by others, testing the satisfiability of random formulas often appears surprisingly easy. Here we show that by using the right distribution of instances, and appropriate parameter values, it is possible ..."
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Cited by 98 (2 self)
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We report results from largescale experiments in satisfiability testing. As has been observed by others, testing the satisfiability of random formulas often appears surprisingly easy. Here we show that by using the right distribution of instances, and appropriate parameter values, it is possible to generate random formulas that are hard, that is, for which satisfiability testing is quite difficult. Our results provide a benchmark for the evaluation of satisfiabilitytesting procedures. In Artificial Intelligence, 81 (19996) 1729. 1 Introduction Many computational tasks of interest to AI, to the extent that they can be precisely characterized at all, can be shown to be NPhard in their most general form. However, there is fundamental disagreement, at least within the AI community, about the implications of this. It is claimed on the one hand that since the performance of algorithms designed to solve NPhard tasks degrades rapidly with small increases in input size, something ...