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On the Problem of Computing the Well-Founded Semantics

  • Zbigniew Lone
  • Miroslaw Truszczyński
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1861)

Abstract

The well-founded semantics is one of the most widely studied and used semantics of logic programs with negation. In the case of finite propositional programs, it can be computed in polynomial time, more specifically, in O(|At(P)| × size(P)) steps, where size(P) denotes the total number of occurrences of atoms in a logic program P. This bound is achieved by an algorithm introduced by Van Gelder and known as the alternating-fixpoint algorithm. Improving on the alternating-fixpoint algorithm turned out to be difficult. In this paper we study extensions and modifications of the alternating-fixpoint approach. We then restrict our attention to the class of programs whose rules have no more than one positive occurrence of an atom in their bodies. For programs in that class we propose a new implementation of the alternating-fixpoint method in which false atoms are computed in a top-down fashion. We show that our algorithm is faster than other known algorithms and that for a wide class of programs it is linear and so, asymptotically optimal.

Keywords

Logic Program Logic Programming Unicyclic Graph Nonmonotonic Reasoning Normal Logic Program 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    J.J. Alferes, C.V. Damásio, and L.M. Pereira. A logic programming system for nonmonotonic reasoning. Journal of Automated Reasoning, 14:93–147, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    K. Berman, J. Schlipf, and J. Franco. Computing the well-founded semantics faster. In Logic Programming and Nonmonotonic Reasoning (Lexington, KY, 1995), volume 928 of Lecture Notes in Computer Science, pages 113–125, Berlin, 1995. Springer.Google Scholar
  3. 3.
    S. Brass and J. Dix. Characterizations of the disjunctive well-founded semantics: confluent calculi and iterated GCWA. Journal of Automated Reasoning, 20(1):143–165, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    S. Brass and J. Dix and B. Freitag and U. Zukowski. Transformation-based bottom-up computation of the well-founded model. Manuscript.Google Scholar
  5. 5.
    W. Chen, T. Swift, and D.S. Warren. Efficient top-down computation of queries under the well-founded semantics. Journal of Logic Programming, 24(3):161–199, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    W. Chen and D.S. Warren. Tabled evaluation with delaying for general logic programs. Journal of the ACM, 43(1):20–74, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    W.F. Dowling and J.H. Gallier. Linear-time algorithms for testing the satisfiability of propositional Horn formulae. Journal of Logic Programming, 1(3):267–284, 1984.CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    M. C. Fitting. Fixpoint semantics for logic programming-a survey. Theoretical Computer Science, 1999. To appear.Google Scholar
  9. 9.
    M.C. Fitting. Well-founded semantics, generalized. In Logic programming (San Diego, CA, 1991, MIT Press Series in Logic Programming, pages 71–84, Cambridge, MA, 1991. MIT Press.Google Scholar
  10. 10.
    M. Gelfond and V. Lifschitz. The stable semantics for logic programs. In R. Kowalski and K. Bowen, editors, Proceedings of the 5th International Symposium on Logic Programming, pages 1070–1080, Cambridge, MA, 1988. MIT Press.Google Scholar
  11. 11.
    W. Marek and M. Truszczyński. Autoepistemic logic. Journal of the ACM, 38(3):588–619, 1991.zbMATHCrossRefGoogle Scholar
  12. 12.
    I. Niemelä and P. Simons. Efficient implementation of the well-founded and stable model semantics. In Proceedings of JICSLP-96. MIT Press, 1996.Google Scholar
  13. 13.
    P. Rao, I.V. Ramskrishnan, K. Sagonas, T. Swift, D. S. Warren, and J. Freire. XSB: A system for efficiently computing well-founded semantics. In Proceedings of LPNMR’97, pages 430–440. Berlin: Springer-Verlag, 1997. Lecture Notes in Computer Science, 1265.Google Scholar
  14. 14.
    V.S. Subrahmanian, D. Nau, and C. Vago. WFS + branch bound = stable models. IEEE Transactions on Knowledge and Data Engineering, 7:362–377, 1995.CrossRefGoogle Scholar
  15. 15.
    A. Van Gelder. The alternating fixpoints of logic programs with negation. In ACM symposium on principles of database systems, pages 1–10, 1989.Google Scholar
  16. 16.
    A. Van Gelder, K.A. Ross, and J.S. Schlipf. The well-founded semantics for general logic programs. Journal of the ACM, 38(3):620–650, 1991.zbMATHCrossRefGoogle Scholar
  17. 17.
    U. Zukowski, S. Brass, and B. Freitag. Improving the alternating fixpoint: the transformation approach. In Proceedings of LPNMR’97, pages 40–59. Berlin: Springer-Verlag, 1997. Lecture Notes in Computer Science, 1265.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Zbigniew Lone
    • 1
  • Miroslaw Truszczyński
    • 1
  1. 1.Department of Computer ScienceUniversity of KentuckyLexingtonUSA

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