Analysing and Extending Well-Founded and Partial Stable Semantics Using Partial Equilibrium Logic

  • Pedro Cabalar
  • Sergei Odintsov
  • David Pearce
  • Agustín Valverde
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4079)


In [4] a nonmonotonic formalism called partial equilibrium logic (PEL) was proposed as a logical foundation for the well-founded semantics (WFS) of logic programs. PEL consists in defining a class of minimal models, called partial equilibrium (p-equilibrium), inside a non-classical logic called HT 2. In [4] it was shown that, on normal logic programs, p-equilibrium models coincide with Przymusinki’s partial stable (p-stable) models. This paper begins showing that this coincidence still holds for the more general class of disjunctive programs, so that PEL can be seen as a way to extend WFS and p-stable semantics to arbitrary propositional theories. We also study here the problem of strong equivalence for various subclasses of p-equilibrium models, investigate transformation rules and nonmonotonic inference, and consider a reduction of PEL to equilibrium logic. In addition we examine the behaviour of PEL on nested logic programs and its complexity in the general case.


Logic Program Stable Model Double Negation Disjunctive Program Partial Equilibrium Model 
These keywords were added by machine and not by the authors.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alcantara, J., Damasio, C., Pereira, L.M.: A Frame-based Characterisation of the Paraconsistent Well-founded Semantics with Explicit Negation (Unpublished draft), available at:
  2. 2.
    Bochman, A.: A logical foundation for logic programming I: Biconsequence relations and nonmonotonic completion, II: Semantics for logic programs. Journal of Logic Programming 35, 151–194, 171–194 (1998)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Brass, S., Dix, J.: A disjunctive semantics based on unfolding and bottom-up evaluation. In: IFIP 1994 Congress, Workshop FG2: Disjunctive Logic Programming and Disjunctive Databases, pp. 83–91 (1994)Google Scholar
  4. 4.
    Cabalar, P., Odintsov, S., Pearce, D.: Logical Foundations of Well-Founded Semantics. In: Proceedings KR 2006 (to appear, 2006)Google Scholar
  5. 5.
    Eiter, T., Fink, M., Tompits, H., Woltran, S.: Simplifying Logic Programs Under Uniform and Strong Equivalence. In: Lifschitz, V., Niemelä, I. (eds.) LPNMR 2004. LNCS (LNAI), vol. 2923, pp. 87–99. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  6. 6.
    Eiter, T., Leone, N., Saccà, D.: On the Partial Semantics for Disjunctive Deductive Databases. Ann. Math, & Artificial Intelligence 17, 59–96 (1997)CrossRefGoogle Scholar
  7. 7.
    Eiter, T., Leone, N., Saccà, D.: Expressive Power and Complexity of Partial Models for Disjunctive Deductive Databases. Theoretical Computer Science 206, 181–218 (1998)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gelfond, M., Lifschitz, V.: The stable model semantics for logic programming. In: Proc. of ICLP 1988, pp. 1070–1080. MIT Press, Cambridge (1988)Google Scholar
  9. 9.
    Janhunen, T., Niemelä, I., Seipel, D., Simons, P., You, J.-H.: Unfolding partiality and disjunctions in stable model semantics. ACM Transactions on Computational Logic (to appear)Google Scholar
  10. 10.
    Lifschitz, V., Pearce, D., Valverde, A.: Strongly equivalent logic programs. ACM Transactions on Computational Logic 2(4), 526–541 (2001)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Lifschitz, V., Tang, L.R., Turner, H.: Nested expressions in logic programs. Annals of Mathematics and Artificial Intelligence 25(3-4), 369–389 (1999)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Osorio, M., Navarro, J.A., Arrazola, J.: Equivalence in Answer Set Programming. In: Pettorossi, A. (ed.) LOPSTR 2001. LNCS, vol. 2372, pp. 57–75. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  13. 13.
    Pearce, D.: A new logical characterization 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
  14. 14.
    Pereira, L.M., Alferes, J.J.: Well Founded Semantics for Logic Programs with Explicit Negation. In: Neumann, B. (ed.) European Conference on Artificial Intelligence, pp. 102–106. John Wiley & Sons, Chichester (1992)Google Scholar
  15. 15.
    Przymusinski, T.: Stable semantics for disjunctive programs. New Generation Computing 9, 401–424 (1991)CrossRefGoogle Scholar
  16. 16.
    Routley, R., Routley, V.: The Semantics of First Degree Entailment. Noûs 6, 335–359 (1972)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Ruiz, C., Minker, J.: Compuing Stable and Partial Stable Models of Extended Disjunctive Logic Programs. In: Przymusinski, T.C., Dix, J., Moniz Pereira, L. (eds.) ICLP-WS 1994. LNCS, vol. 927, pp. 205–229. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  18. 18.
    Seipel, D., Minker, J., Ruiz, C.: A Characterization of the Partial Stable Models for Disjunctive Deductive Databases. In: Int. Logic Programming Symp., pp. 245–259. MIT Press, Cambridge (1997)Google Scholar
  19. 19.
    Valverde, A.: tabeql: A Tableau Based Suite for Equilibrium Logic. In: Alferes, J.J., Leite, J. (eds.) JELIA 2004. LNCS (LNAI), vol. 3229, pp. 734–737. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  20. 20.
    van Gelder, A., Ross, K.A., Schlipf, J.S.: Unfounded sets and well-founded semantics for general logic programs. JACM 38(3), 620–650 (1991)MATHGoogle Scholar
  21. 21.
    Wang, K., Zhou, L.: Comparisons and computation of well-founded semantics for disjunctive logic programs. ACM Transactions on Computational Logic 6(2), 295–327 (2005)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Woltran, S.: Characterizations for Relativized Notions of Equivalence in Answer Set Programming. In: Alferes, J.J., Leite, J.A. (eds.) JELIA 2004. LNCS (LNAI), vol. 3229, pp. 161–173. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  23. 23.
    You, J., Yuan, L.Y.: Three-valued formalization of logic programming: is it needed? In: Proc. PODS 1990, pp. 172–182. ACM Press, New York (1990)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Pedro Cabalar
    • 1
  • Sergei Odintsov
    • 2
  • David Pearce
    • 3
  • Agustín Valverde
    • 4
  1. 1.Corunna UniversityCorunnaSpain
  2. 2.Sobolev Institute of MathematicsNovosibirskRussia
  3. 3.Universidad Rey Juan CarlosMadridSpain
  4. 4.University of MálagaMálagaSpain

Personalised recommendations