Complexity Issues in the Davis and Putnam Scheme

  • G. Escalada-Imaz
  • R. Torres Velázquez
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1904)


The Davis and Putnam (D&P) scheme has been intensively studied during this last decade. Nowadays, its good empirical perfor- mances are well-known. Here, we deal with its theoretical side which has been relatively less studied until now. Thus, we propose a strictely lin- ear D&P algorithm for the most well known tractable classes: Horn-SAT and 2-SAT. Specifically, the strictely linearity of our proposed D&P algo- rithm improves significantly the previous existing complexities that were quadratic for Horn-SAT and even exponential for 2-SAT. As a conse- quence, the D&P algorithm designed to deal with the general SAT problem runs as fast (in terms of complexity) as the specialised algorithms designed to work exclusively with a specific tractable SAT subclass.


Automated Reasoning Computational Complexity Search Theorem Proving 


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  1. 1.
    G. Ausiello and G. F. Italiano. Online algorithms for poly normally solvable satisfiability problems. Journal of Logic Programming, 10(1), 1991.Google Scholar
  2. 2.
    C.M. Li and Anbulagan. Heuristics based on unit propagation for satisfiability problems. In Proceedings of the 15th IJCAI, pages 366–371, 1997.Google Scholar
  3. 3.
    J.M. Crawford and L. D. Auton. Experimental results on the crossover point in satisfiability problems. In Proc. of the Eleventh National Conference on Artificial Intelligence, AAAI-93, pages 21–27, 1993.Google Scholar
  4. 4.
    M. Davis, G. Logemann, and D. Loveland. A machine program for theorem proving. Comunnications of the ACM, 5:394–397, 1962.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    M. Davis and H. Putnam. A computing procedure for quantification theory. Journal of the ACM, 7:394–397, 1960.CrossRefMathSciNetGoogle Scholar
  6. 6.
    G. Davydov, I. Davydova, and H. K. Buning. An efficient algorithm for the minimal unsatisfiability problem for a classe of cnf. Annals of Mathematics and Artificial Intelligence, 23:229–245, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    R. Dechter and I. Rish. Rirectional resolution: the davis and putnam procedure revisited. In Proceedings of Knowledge Representattion International Conference, KR-94, pages 134–145, 1994.Google Scholar
  8. 8.
    W.F. Dowling and J. H. Gallier. Linear-time algorithms for testing the satisfiability of horn prepositional formulae. Journal of Logic Programming, 3:267–284, 1984.CrossRefMathSciNetGoogle Scholar
  9. 9.
    D. Dubois, P. Andre, Y. Boufkhad, and J. Carlier. Sat versus unsat. In Proceedings of the Second DIM ACS Challenge, 1993.Google Scholar
  10. 10.
    Z. Galil. On the complexity of Regular Resolution and the Davis-Putnam procedure. Theoretical Computer Sicence, 4:23–46, 1977.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    J.N. Hooker and V. Vinay. Branching rules for satisfiability. Journal of Automated Reasoning, 15:359–383, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    J.M. Crawford and L. D. Auton. Experimental Results on the Crossover Point in random 3-SAT. Artificial Intelligence, 81:31–57, 1996.CrossRefMathSciNetGoogle Scholar
  13. 13.
    R.E. Jeroslow and J. Wang. Solving prepositional satisfiability problems. Annals of Mathematics and Artificial Intelligence, 1:167–187, 1990.zbMATHCrossRefGoogle Scholar
  14. 14.
    H. Kautz and B. Selman. Planing as satisfiability. In Proceeding of the 10th EC AI, pages 359–363. European Conference on Artificial Intelligence, 1992.Google Scholar
  15. 15.
    A. Rauzy. Polynomial restrictions of sat: What ca be done with an efficient implementation of the davis and putnam’s procedure. In U. Mntanari and F. Rossi, editors, Principles and Practice of Constraint Programming, CP’95, volume 976 of Lecture Notes in Computer Science, pages 515–532. Fisrt International Conference on Principles and Practice of Constraint Programming, Cassis, France, Springer-verlag, 1995.Google Scholar
  16. 16.
    T.E. Tarjan. Amortized computational complexity. SIAM J. Algebraic Discrete Methods, 6:306–318, 1985.zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    H. Zhang. Sato: An efficient prepositional prover. In proceedings of the 13th Conference on Automated Deduction, pages 272–275, 1997.Google Scholar
  18. 18.
    H. Zhang and M. E. Stickel. An efficient algorithm for unit propagation. In International Symposium on Artificial Intelligence and Mathematics, 1996.Google Scholar
  19. 19.
    H Zhang and M. E. Stickel. Implementing the davis-putnam algorithm by tries. Technical report, The University of Iowa, 1994.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • G. Escalada-Imaz
    • 1
  • R. Torres Velázquez
    • 1
  1. 1.Artificial Intelligence Research Institute (IIIA)Scientific Research Spanish Council (CSIC)BellaterraSpain

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