Abstract
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.
This work is supported by the french inter-PRC project “Classes Polynomiales”
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V. Arvind and S. Biswas. An O(n2) algorithm for the satisfiability problem of a subset of propositional sentences in CNF that includes horn sentences. Information Processing Letters, 24:67–69, 1987.
B. Aspvall. Recognizing Disguised NR(1) Instances of the Satisfiability Problem. Journal of Algorithms, 1:97–103, 1980.
B. Aspvall, M. Plass, and R. Tarjan. A Linear Time Algorithm for Testing the Truth of Certain Quantified Boolean Formulae. Information Processing Letter, 8(3):121–123, 1979.
G. Ausiello and G. Italiano. On-Line Algorithms for Polynoinially Solvable Satisfiability Problems. Journal of Logic Programming, 10:69–90, 1990.
A. Blass and Y. Gurevitch. On the unique satisfiability problem. Information and Control, 55:80–88, 1982.
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.
E. Boros, Y. Crama, and P.L. Hammer. Polynomial-time inference of all implications for Horn and related formulae. Annals of Mathematics and Artificial Intelligence, 1:21–32, 1990.
E. Boros, Y. Crama, P.L. Hammer, and M. Saks. A complexity index for satisfiability problems. SIAM Journ. Comp., 23:45–49, 1994.
E. Boros, P.L. Hammer, and X. Sun. Recognition of q-Horn formulae in linear time. Discrete Applied Mathematics, 55:1–13, 1994.
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.
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.
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.
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.
M.C. Cooper, D.A. Cohen, and P.G. Jeavons. Characterizing Tractable Constraints. Artificial Intelligence, 65:347–361, 1994.
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.
M. Davis, G. Logemann, and D. Loveland. A Machine Program for Theorem Proving. JACM, 5:394–397, 1962.
M. Davis and H. Putnam. A Computing Procedure for Quantification Theory. JACM, 7:201–215, 1960.
W.F. Dowling and J.H. Gallier. Linear-time Algorithms for Testing the Satisfiablity of Propositional Horn Formulae. J. Logic Programming, 3:267–284, 1984.
O. Dubois. On the r,s-SAT satisfiability problem and a conjecture of Tovey. Discrete Applied Mathematics, 26:51–60, 1990.
O. Dubois, P. André, Y. Boufkhad, and J. Carlier. SAT versus UNSAT, 1994. Position paper, DIMACS chalenge on Satisfiability Testing, to appear.
O. Dubois and J. Carlier. Sur le problème de satisfiabilité. Communication at the Barbizon Workshop on SAT, October 1991.
S. Even, A. Itai, and A. Shamir. On the Complexity of Timetable and Multicommodity Flow Problems. SIAM J. Comput., 5:691–703, 1976.
G. Gallo and M.G. Scutella. Polynomially Solvable Satisfiability Problems. Information Processing Letters, 29:221–227, 1988.
G. Gallo and G. Urbani. Algorithms for Testing the Satisfiablity of Propositional Formulae. Journal of Logic Programming, 7:45–61, 1989.
M.R. Garey and D.S. Johnson. Computer and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Fransisco, 1979.
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.
M. Ghallab and E. Escalada-Imaz. A linear control algorithm for a class of rule-based systems. Journal of Logic Programming, 11:117–132, 1991.
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.
P. Hansen and B. Jaumard. Uniquely solvable quadratic boolean equations. Discrete Applied Mathematics, 12:147–154, 1985.
J.-J. Hebrard. A linear algorithm for renaming a set of clauses as a Horn set. Theoretical Computer Science, 124:343–350, 1994.
L. Henschen and L. Wos. Unit refutations and Horn sets. JACM, 21(4):590–605, October 1974.
J.N. Hooker. Solving the Incremental Satisfiability Problem. Journal of Logic Programming, 15:177–186, 1993.
S. Jeannicot, L. Oxusoff, and A. Rauzy. Évaluation Sémantique en Calcul Propositionnel. Revue d'Intelligence Artificielle, 2:41–60, 1988.
R.J. Jeroslow and J. Wang. Solving Propositional Satisfiability Problems. Annals of Mathematics and Artificial Intelligence, 1:167–188, 1990.
D.S. Johnson. A Catalog of Complexity Classes. In J. Van Leeuwen, editor, Handbook of Theoretical Computer Science, volume A. Elsevier, 1990.
D.E. Knuth. Nested Satisfiability. Acta Informatica, 28, 1990.
T. Larrabee. Test Pattern Generation Using Boolean Satisfiability. IEEE Transactions on Computer-Aided Design, 11(1):4–15, January 1992.
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.
H.R. Lewis. Renaming a Set of Clauses as a Horn Set. JACM, 25(1):134–135, 1978.
G. Lindhorst and F. Shahroki. On renaming a set of clauses as a Horn set. Information Processing Letters, 30:289–293, 1989.
D. Loveland. Automated Theorem Proving: A Logical Basis. North Holland, 1978.
E.L. Lozinskii. A simple test improves checking satisfiability. Journal of Logic Programming, 15:99–111, 1993.
H. Mannila and K. Mehlorn. A fast algorithm for renaming a set of clauses as a Horn set. Information Processing Letters, 21:269–272, 1985.
M. Minoux. LTUR: A Simplified Linear-Time Unit Resolution Algorithm for Horn Formulae and its Computer Implementation. Information Processing Letter, 29:1–12, 1988.
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.
B. Monien and E. Speckenmeyer. Solving Satisfiability in Less than 2n Steps. Discrete Applied Math., 10:287–295, 1985.
R. Petreschi and B. Simeone. Experimental Comparison on 2-Satisfiability Algorithms. RAIRO Recherche Opérationelle, 25:241–264, 8 1991.
D. Pretolani. A linear time algorithm for unique Horn satisfiability. Information Processing Letters, 48:61–66, 1993.
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.
M.G. Scutella. A Note on Dowling and Gallier's Top-Down Algorithm for Propositional Horn Satisfiability. Journal of Logic Programming, 8:265–273, 1990.
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.
R.E. Tarjan. Depth First Search and Linear Graph Algorithms. SIAM J. Comput., 1:146–160, 1972.
C. A. Tovey. A Simplified NP-complete Satisfiability Problem. Discrete Applied Mathematics, 8:85–89, 1984.
A. van Gelder. Linear Time Unit Resolution for Propositional Formulas — in Prolog, Yet. submitted to the Journal of Logic Programming, 1994.
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.
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.
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Rauzy, A. (1995). Polynomial restrictions of SAT: What can be done with an efficient implementation of the Davis and Putnam's procedure?. In: Montanari, U., Rossi, F. (eds) Principles and Practice of Constraint Programming — CP '95. CP 1995. Lecture Notes in Computer Science, vol 976. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60299-2_31
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DOI: https://doi.org/10.1007/3-540-60299-2_31
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