Advertisement

Approximation Fixpoint Theory and the Semantics of Logic and Answers Set Programs

  • Marc Denecker
  • Maurice Bruynooghe
  • Joost Vennekens
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7265)

Abstract

Approximation Fixpoint Theory was developed as a fixpoint theory of lattice operators that provides a uniform formalization of four main semantics of three major nonmonotonic reasoning formalisms. This paper clarifies how this fixpoint theory can define the stable and well-founded semantics of logic programs. It investigates the notion of strong equivalence underlying this semantics. It also shows the remarkable power of this theory for defining natural and elegant versions of these semantics for extensions of logic and answer set programs. In particular, we here consider extensions with general rule bodies, general interpretations (also non-Herbrand interpretations) and aggregates. We also investigate the relationship with the equilibrium semantics of nested answer set programs, on the formal and the informal level.

Keywords

Logic Program Logic Programming Stable Model Predicate Symbol Stable Model Semantic 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Belnap, N.D.: A useful four-valued logic. In: Dunn, J.M., Epstein, G. (eds.) Modern Uses of Multiple-Valued Logic, pp. 8–37. Reidel, Dordrecht (1977); Invited papers from the Fifth International Symposium on Multiple-Valued Logic, held at Indiana University, Bloomington, Indiana, May 13-16 (1975)Google Scholar
  2. 2.
    Clark, K.L.: Negation as failure. In: Logic and Data Bases, pp. 293–322. Plenum Press (1978)Google Scholar
  3. 3.
    Denecker, M., Marek, V.W., Truszczyński, M.: Approximating operators, stable operators, well-founded fixpoints and applications in non-monotonic reasoning. In: Logic-based Artificial Intelligence. The Kluwer International Series in Engineering and Computer Science, pp. 127–144. Kluwer Academic Publishers, Boston (2000)CrossRefGoogle Scholar
  4. 4.
    Denecker, M., Marek, V.W., Truszczynski, M.: Uniform semantic treatment of default and autoepistemic logics. Artif. Intell. 143(1), 79–122 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Eiter, T., Fink, M., Tompits, H., Traxler, P., Woltran, S.: Replacements in non-ground answer-set programming. In: Doherty, P., Mylopoulos, J., Welty, C.A. (eds.) KR, pp. 340–351. AAAI Press (2006)Google Scholar
  6. 6.
    Feferman, S.: Toward useful type-free theories. Journal of Symbolic Logic 49(1), 75–111 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fitting, M.: A Kripke-Kleene semantics for logic programs. Journal of Logic Programming 2(4), 295–312 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fitting, M.: Fixpoint semantics for logic programming a survey. Theoretical Computer Science 278(1-2), 25–51 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gelfond, M.: Representing Knowledge in A-Prolog. In: Kakas, A.C., Sadri, F. (eds.) Computational Logic: Logic Programming and Beyond. LNCS (LNAI), vol. 2408, pp. 413–451. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  10. 10.
    Gelfond, M., Lifschitz, V.: The stable model semantics for logic programming. In: Kowalski, R.A., Bowen, K.A. (eds.) ICLP/SLP, pp. 1070–1080. MIT Press (1988)Google Scholar
  11. 11.
    Janhunen, T., Oikarinen, E., Tompits, H., Woltran, S.: Modularity aspects of disjunctive stable models. J. Artif. Intell. Res (JAIR) 35, 813–857 (2009)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Kleene, S.C.: Introduction to Metamathematics. Van Nostrand (1952)Google Scholar
  13. 13.
    Lifschitz, V., Pearce, D., Valverde, A.: Strongly equivalent logic programs. ACM Trans. Comput. Log. 2(4), 526–541 (2001)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lin, F., Chen, Y.: Discovering classes of strongly equivalent logic programs. J. Artif. Intell. Res (JAIR) 28, 431–451 (2007)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Marek, V.W., Truszczyński, M.: Logic programs with abstract constraint atoms. In: Proceedings of the 19th National Conference on Artificial Intelligence (AAAI 2004), pp. 86–91. AAAI Press (2004)Google Scholar
  16. 16.
    Pearce, D.: A New Logical Characterisation of Stable Models and Answer Sets. In: Dix, J., Przymusinski, T.C., Moniz Pereira, L. (eds.) NMELP 1996. LNCS, vol. 1216, pp. 57–70. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  17. 17.
    Pearce, D.: Equilibrium logic. Ann. Math. Artif. Intell. 47(1-2), 3–41 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Pelov, N., Denecker, M., Bruynooghe, M.: Well-founded and stable semantics of logic programs with aggregates. Theory and Practice of Logic Programming (TPLP) 7(3), 301–353 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Przymusinski, T.C.: The well-founded semantics coincides with the three-valued stable semantics. Fundamenta Informaticae 13(4), 445–463 (1990)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Van Gelder, A.: The alternating fixpoint of logic programs with negation. Journal of Computer and System Sciences 47(1), 185–221 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Van Gelder, A., Ross, K.A., Schlipf, J.S.: The well-founded semantics for general logic programs. Journal of the ACM 38(3), 620–650 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Vennekens, J., Mariën, M., Wittocx, J., Denecker, M.: Predicate introduction for logics with a fixpoint semantics. Part I: Logic programming. Fundamenta Informaticae 79(1-2), 187–208 (2007)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Marc Denecker
    • 1
  • Maurice Bruynooghe
    • 1
  • Joost Vennekens
    • 1
  1. 1.Department of Computer ScienceKULeuvenBelgium

Personalised recommendations