Logic Programs With Monotone Cardinality Atoms

  • Victor W. Marek
  • Ilkka Niemelä
  • Mirosław Truszczyński
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2923)


We investigate mca-programs, that is, logic programs with clauses built of monotone cardinality atoms of the form kX, where k is a non-negative integer and X is a finite set of propositional atoms. We develop a theory of mca-programs. We demonstrate that the operational concept of the one-step provability operator generalizes to mca-programs, but the generalization involves nondeterminism. Our main results show that the formalism of mca-programs is a common generalization of (1) normal logic programming with its semantics of models, supported models and stable models, (2) logic programming with cardinality atoms and with the semantics of stable models, as defined by Niemelä, Simons and Soininen, and (3) of disjunctive logic programming with the possible-model semantics of Sakama and Inoue.


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  1. 1.
    Babovich, Y., Lifschitz, V.: Cmodels (2002), http://www.cs.utexas.edu/users/tag/cmodels.html
  2. 2.
    Denecker, M., Marek, V., Truszczyński, M.: Approximations, stable operators, well-founded fixpoints and applications in nonmonotonic reasoning. In: Minker, J. (ed.) Logic-Based Artificial Intelligence, pp. 127–144. Kluwer Academic Publishers, Dordrecht (2000)Google Scholar
  3. 3.
    Denecker, M., Marek, V., Truszczyński, M.: Ultimate approximations in nonmonotonic knowledge representation systems. In: Principles of Knowledge Representation and Reasoning, Proceedings of the Eighth International Conference (KR 2002), pp. 177–188. Morgan Kaufmann Publishers, San Francisco (2002)Google Scholar
  4. 4.
    East, D., Truszczyński, M.: Propositional satisfiability in answer-set programming. In: Baader, F., Brewka, G., Eiter, T. (eds.) KI 2001. LNCS (LNAI), vol. 2174, pp. 138–153. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  5. 5.
    Erdem, E., Lifschitz, V.: Tight logic programs. Theory and Practice of Logic Programming 3(4-5), 499–518 (2003)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Fages, F.: Consistency of Clark’s completion and existence of stable models. Journal of Methods of Logic in Computer Science 1, 51–60 (1994)Google Scholar
  7. 7.
    Fitting, M.C.: Fixpoint semantics for logic programming – a survey. Theoretical Computer Science 278, 25–51 (2002)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gelfond, M., Lifschitz, V.: The stable semantics for logic programs. In: Kowalski, R., Bowen, K. (eds.) Proceedings of the 5th International Conference on Logic Programming, pp. 1070–1080. MIT Press, Cambridge (1988)Google Scholar
  9. 9.
    Liu, L., Truszczyński, M.: Local-search techniques in propositional logic extended with cardinality atoms. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 495–509. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  10. 10.
    Niemelä, I., Simons, P.: Efficient implementation of the well-founded and stable model semantics. In: Proceedings of JICSLP 1996. MIT Press, Cambridge (1996)Google Scholar
  11. 11.
    Niemelä, I., Simons, P.: Extending the smodels system with cardinality and weight constraints. In: Minker, J. (ed.) Logic-Based Artificial Intelligence, pp. 491–521. Kluwer Academic Publishers, Dordrecht (2000)Google Scholar
  12. 12.
    Niemelä, I., Simons, P., Soininen, T.: Stable model semantics of weight constraint rules. In: Gelfond, M., Leone, N., Pfeifer, G. (eds.) LPNMR 1999. LNCS (LNAI), vol. 1730, pp. 317–331. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  13. 13.
    Przymusinski, T.C.: The well-founded semantics coincides with the three-valued stable semantics. Fundamenta Informaticae 13(4), 445–464 (1990)MATHMathSciNetGoogle Scholar
  14. 14.
    Sakama, C., Inoue, K.: An alternative approach to the semantics of disjunctive logic programs and deductive databases. Journal of Automated Reasoning 13, 145–172 (1984)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Simons, P., Niemelä, I., Soininen, T.: Extending and implementing the stable model semantics. Artificial Intelligence 138, 181–234 (2002)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Victor W. Marek
    • 1
  • Ilkka Niemelä
    • 2
  • Mirosław Truszczyński
    • 1
  1. 1.Department of Computer ScienceUniversity of KentuckyLexingtonUSA
  2. 2.Department of Computer Science and EngineeringHelsinki University of TechnologyFinland

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