Polynomial restrictions of SAT: What can be done with an efficient implementation of the Davis and Putnam's procedure?
Constraint Solving Problems are NP-Complete and thus computationaly intractable. Two approaches have been used to tackle this intractability: the improvement of general purpose solvers and the research of polynomial time restrictions. An interesting question follows: what is the behavior of the former solvers on the latter restrictions?
In this paper, we examplify this problem by studying both theoretical and practical complexities of the Davis and Putnam's procedure on the two main polynomial restrictions of SAT, namely Horn-SAT and 2-SAT. We propose an efficient implementation and an improvement that make it quadratic in the worst case on these sub-classes. We show that this complexity is never reached in practice where linear times are observed, making the Davis and Putnam's as efficient as specialized algorithms.
Unable to display preview. Download preview PDF.
- 6.J.M. Boï and A. Rauzy. Two algorithms for constraints system solving in propositional calculus and their implementation in prologIII. In P. Jorrand and V. Sugrev, editors, Proceedings Artificial Intelligence IV Methodology, Systems, Applications (AIMSA'90), pages 139–148. North-Holand, September 1990. Alba-Varna bulgarie.Google Scholar
- 10.V. Chandru, C.R. Coulard, P.L. Hammer, M. Montanez, and X. Sun. On renamable Horn and generalized Horn functions. In Annals of Mathematics and Artificial Intelligence, volume 1. J.C. Baltzer AG, Scientific Publishing Company, Basel Switzerland, 1990.Google Scholar
- 11.P. Cheeseman, B. Kanefsky, and W.M. Taylor. Where the Really Hard Problems Are. In Proceedings of the International Joint Conference of Artificial Intelligence, IJCAI'91, 1991.Google Scholar
- 12.V. Chvátal and B. Reed. Miks gets some (the odds are on his side). In Proceedings of the 33rd IEEE Symp. on Foundations of Computer Science, pages 620–627, 1992.Google Scholar
- 13.S.A. Cook. The Complexity of Theorem Proving Procedures. In Proceedings of the 3rd Ann. Symp. on Theory of Computing, ACM, pages 151–158, 1971.Google Scholar
- 15.J.M. Crawford and L.D. Anton. Experimental results on the crossover point in satisfiability problems. In Proceedings of the Eleventh National Conference on Artificial Intelligence (Washington, D.C., AAAI'1993), pages 21–27, 1993.Google Scholar
- 16.M. Davis, G. Logemann, and D. Loveland. A Machine Program for Theorem Proving. JACM, 5:394–397, 1962.Google Scholar
- 20.O. Dubois, P. André, Y. Boufkhad, and J. Carlier. SAT versus UNSAT, 1994. Position paper, DIMACS chalenge on Satisfiability Testing, to appear.Google Scholar
- 21.O. Dubois and J. Carlier. Sur le problème de satisfiabilité. Communication at the Barbizon Workshop on SAT, October 1991.Google Scholar
- 25.M.R. Garey and D.S. Johnson. Computer and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Fransisco, 1979.Google Scholar
- 26.I.P. Gent and T. Walsh. The SAT Phase Transition. In A.G. Cohn, editor, Proceedings of 11th European Conference on Artificial Intelligence, ECAI'94, pages 105–109. Wiley, 1994.Google Scholar
- 28.A. Goerdt. A treshold for unsatifiability. In I.M. Havel and V. Koubek, editors, Proceedings of Mathematical Foundations of Computer Science, MFCS'92, pages 264–272, August 1994.Google Scholar
- 33.S. Jeannicot, L. Oxusoff, and A. Rauzy. Évaluation Sémantique en Calcul Propositionnel. Revue d'Intelligence Artificielle, 2:41–60, 1988.Google Scholar
- 35.D.S. Johnson. A Catalog of Complexity Classes. In J. Van Leeuwen, editor, Handbook of Theoretical Computer Science, volume A. Elsevier, 1990.Google Scholar
- 36.D.E. Knuth. Nested Satisfiability. Acta Informatica, 28, 1990.Google Scholar
- 38.T. Larrabee and Y. Tsuji. Evidence for a satisfiability threshold for random 3cnf formulas. In H. Hirsh and al., editors, Proceedings of Spring Symposium on Artificial Intelligence and NP-Hard Problems (Stanford CA 1993), pages 112–118, 1993.Google Scholar
- 41.D. Loveland. Automated Theorem Proving: A Logical Basis. North Holland, 1978.Google Scholar
- 45.D. Mitchell, B. Selman, and H. Levesque. Hard and Easy Distributions of SAT Problems. In Proceedings Tenth National Conference on Artificial Intelligence (AAAI'92), 1992.Google Scholar
- 47.R. Petreschi and B. Simeone. Experimental Comparison on 2-Satisfiability Algorithms. RAIRO Recherche Opérationelle, 25:241–264, 8 1991.Google Scholar
- 49.A. Rauzy. On the Complexity of the Davis and Putnam's Procedure on Some Polynomial Sub-Classes of SAT. Technical Report 806-94, LaBRI, URA CNRS 1304, Université Bordeauxl, 9 1994.Google Scholar
- 51.B. Selman, H. Levesque, and D. Mitchell. A New Method for Solving Hard Satisfiability Problems. In Proceedings of the 10th National Conference on Artificial Intelligence (AAAI'92), 1992.Google Scholar
- 54.A. van Gelder. Linear Time Unit Resolution for Propositional Formulas — in Prolog, Yet. submitted to the Journal of Logic Programming, 1994.Google Scholar
- 55.S. Yamasaki and S. Doshita. The satisfiability problem for a class consisting of horn sentences and some non-horn sentences in propositional logic. Information and Computation, 59:1–12, 1983.Google Scholar
- 56.R. Zabih and D. Mac Allester. A rearrangement search strategy for determining propositional satisfiability. In Proceedings of the National Conference on Artificial Intelligence, AAAI'88, pages 155–160, 1988.Google Scholar