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Formula Caching in DPLL

Published:01 March 2010Publication History
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Abstract

We consider extensions of the DPLL approach to satisfiability testing that add a version of memoization, in which formulas that the algorithm has previously shown to be unsatisfiable are remembered for later use. Such formula caching algorithms have been suggested for satisfiability and stochastic satisfiability by several authors. We formalize these methods by developing extensions of the fruitful connection that has previously been developed between DPLL algorithms for satisfiability and tree-like resolution proofs of unsatisfiability. We analyze a number of variants of these formula caching methods and characterize their strength in terms of proof systems. These proof systems are new and simple, and have a rich structure. We compare them to several studied proof systems: tree-like resolution, regular resolution, general resolution, Res(k), and Frege systems and present both simulation and separations. One of our most interesting results is the introduction of a natural and implementable form of DPLL with caching, FCWreason. This system is surprisingly powerful: we prove that it can polynomially simulate regular resolution, and furthermore, it can produce short proofs of some formulas that require exponential-size resolution proofs.

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            Manoj Kumar Raut

            The DPLL algorithm is a satisfiability-based algorithm. The authors consider in this paper the extension of the DPLL approach to satisfiability checking, by adding memoization, which means saving previously solved subproblems for later use (also called caching). The authors analyze a number of variants of the memoized DPLL algorithm, such as basic formula caching (FC), formula caching with weakening (FC-w), formula caching with weakening and subsumption (FC-ws), formula caching with returned reasons for unsatisfiability, and formula caching with nondeterministic rules (FC-ws-nondet). These are different ways to introduce caching of unsatisfiable formulas into the DPLL algorithm. Then, the authors "characterize the strength of these nondeterministic algorithms in terms of [existing] proof systems," such as tree-like resolution, regular resolution, general resolution, Res( k ) for each k ?2, depth-2 Frege (F2), and extended Frege. They define a new proof system called contradiction-caching proof system (CC+T), where T stands for weakening and subsumption, and relate this proof system to the basic FC algorithm. They also introduce an implementable form of DPLL with caching. This system is very powerful and polynomially simulates regular resolution; polynomial simulation means efficient proofs in one system can be translated to efficient proofs in another system. This system produces short proofs of some formulas that require exponential-size resolution proofs. Online Computing Reviews Service

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            • Published in

              cover image ACM Transactions on Computation Theory
              ACM Transactions on Computation Theory  Volume 1, Issue 3
              March 2010
              64 pages
              ISSN:1942-3454
              EISSN:1942-3462
              DOI:10.1145/1714450
              Issue’s Table of Contents

              Copyright © 2010 ACM

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              Association for Computing Machinery

              New York, NY, United States

              Publication History

              • Published: 1 March 2010
              • Revised: 1 December 2009
              • Accepted: 1 December 2009
              • Received: 1 July 2008
              Published in toct Volume 1, Issue 3

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