We report results from large-scale 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 satisfiability-testing 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 NP-hard 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 NP-hard tasks degrades rapidly with small increases in input size, something will need to be given up to obtain acceptable behavior....

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 tree-like 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 width-size 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 quasi-polynomial in the smallest tree-like 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.

... quickly across a wide range of hard SAT problems than any other SAT tester in the literature on comparable platforms. On a Sun SPARCStation 10 running SunOS 4.1.3 U1, POSIT can solve hard random 400-variable 3-SAT problems in about 2 hours on the average. In general, it can solve hard n-variable random 3-SAT problems with search trees of size O(2 n=18:7 ). In addition to justifying these claims, this dissertation describes the most significant achievements of other researchers in this area, and discusses all of the widely known general techniques for speeding up SAT search algorithms. It should be useful to anyone interested in NP-complete problems or combinatorial optimization in general, and it should be particularly useful to researchers in either Artificial Intelligence or Operations Research.

We prove an exponential lower bound on the length of cutting plane proofs. The proof uses an extension of a lower bound for monotone circuits to circuits which compute with real numbers and use nondecreasing functions as gates. The latter result is of independent interest, since, in particular, it implies an exponential lower bound for some arithmetic circuits.

. 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, computer-aided 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 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.

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 non-trivial lower bounds are known. For example, we show that with probability approaching 1, any Resolution refutation of a randomly chosen 3-CNF 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.

We report results from large-scale 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 satisfiability-testing procedures. In Artificial Intelligence, 81 (19996) 17--29. 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 NP-hard 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 NP-hard tasks degrades rapidly with small increases in input size, something ...

Linear Pseudo-Boolean (LPB) constraints denote inequalities between arithmetic sums of weighted Boolean functions and provide a significant extension of the modeling power of purely propositional constraints. They can be used to compactly describe many discrete EDA problems with constraints on linearly combined, parameterized weights, yet also offer efficient search strategies for proving or disproving whether a satisfying solution exists. Furthermore, corresponding decision procedures can easily be extended for minimizing or maximizing an LPB objective function, thus providing a core optimization method for many problems in logic and physical synthesis. In this paper we review how recent advances in satisfiability (SAT) search can be extended for pseudo-Boolean constraints and describe a new LPB solver that is based on generalized constraint propagation and conflict-based learning. We present a comparison with other, state-of-the-art LPB solvers which demonstrates the overall efficiency of our method.

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