Stability, Supportedness, Minimality and Kleene Answer Set Programs

  • Patrick Doherty
  • Andrzej Szałas
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9060)


Answer Set Programming is a widely known knowledge representation framework based on the logic programming paradigm that has been extensively studied in the past decades. The semantic framework for Answer Set Programs is based on the use of stable model semantics. There are two characteristics intrinsically associated with the construction of stable models for answer set programs. Any member of an answer set is supported through facts and chains of rules and those members are in the answer set only if generated minimally in such a manner. These two characteristics, supportedness and minimality, provide the essence of stable models. Additionally, answer sets are implicitly partial and that partiality provides epistemic overtones to the interpretation of disjunctive rules and default negation. This paper is intended to shed light on these characteristics by defining a semantic framework for answer set programming based on an extended first-order Kleene logic with weak and strong negation. Additionally, a definition of strongly supported models is introduced, separate from the minimality assumption explicit in stable models. This is used to both clarify and generate alternative semantic interpretations for answer set programs with disjunctive rules in addition to answer set programs with constraint rules. An algorithm is provided for computing supported models and comparative complexity results between strongly supported and stable model generation are provided.


Logic Program Logic Programming Semantic Theory Stable Model Closed World Assumption 
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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  1. 1.
    Alviano, M., Faber, W., Leone, N., Perri, S., Pfeifer, G., Terracina, G.: The disjunctive datalog system DLV. In: de Moor, O., Gottlob, G., Furche, T., Sellers, A. (eds.) Datalog 2010. LNCS, vol. 6702, pp. 282–301. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  2. 2.
    Baral, C.: Knowledge Representation, Reasoning, and Declarative Problem Solving. Cambridge University Press (2003)Google Scholar
  3. 3.
    Bonatti, P., Calimeri, F., Leone, N., Ricca, F.: Answer set programming. In: Dovier, A., Pontelli, E. (eds.) GULP. LNCS, vol. 6125, pp. 159–182. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  4. 4.
    Brewka, G.: Preferences, contexts and answer sets. In: Dahl, V., Niemelä, I. (eds.) ICLP 2007. LNCS, vol. 4670, pp. 22–22. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Brewka, G., Eiter, T., Truszczynski, M.: Answer set programming at a glance. Commun. ACM 54(12), 92–103 (2011)CrossRefGoogle Scholar
  6. 6.
    Brewka, G., Niemelä, I., Truszczynski, M.: Answer set optimization. In: Gottlob, G., Walsh, T. (eds.) Proc. 18th IJCAI, pp. 867–872. Morgan Kaufmann (2003)Google Scholar
  7. 7.
    Cabalar, P., Pearce, D., Valverde, A.: A revised concept of safety for general answer set programs. In: Erdem, E., Lin, F., Schaub, T. (eds.) LPNMR 2009. LNCS, vol. 5753, pp. 58–70. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  8. 8.
    Denecker, M., Marek, V., Truszczynski, M.: Stable operators, well-founded fixpoints and applications in nnonmonotonic reasoning. In: Minker, J. (ed.) Logic-based Artificial Intelligence, pp. 127–144. Kluwer Academic Pub. (2000)Google Scholar
  9. 9.
    Eiter, T., Gottlob, G.: Complexity results for disjunctive logic programming and application to nonmonotonic logics. In: Miller, D. (ed.) Proceedings of the 1993 International Symposium on Logic Programming, pp. 266–278 (1993)Google Scholar
  10. 10.
    Fages, F.: A new fixpoint sematics for general logic programs compared with the well-founded and stable model semantics. New Generation Computing 9, 425–443 (1991)CrossRefMATHGoogle Scholar
  11. 11.
    Fages, F.: Consistency of Clark’s completion and existence of stable models. Methods of Logic in Computer Science 1, 51–60 (1994)Google Scholar
  12. 12.
    Fenstad, J.E.: Situations, Language and Logic. D. Reidel Publishing Company (1987)Google Scholar
  13. 13.
    Ferraris, P., Lifschitz, V.: On the minimality of stable models. In: Balduccini, M., Son, T.C. (eds.) Logic Programming, Knowledge Representation, and Nonmonotonic Reasoning. LNCS, vol. 6565, pp. 64–73. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  14. 14.
    Fitting, M.: A Kripke-Kleene semantics for logic programs. J. Logic Programming 2(4), 295–312 (1985)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Fitting, M.: The family of stable models. J. Logic Programming 17(2/3&4), 197–225 (1993)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Gelfond, M., Kahl, Y.: Knowledge Representation, Reasoning, and the Design of Intelligent Agents - The Answer-Set Programming Approach. Cambridge University Press (2014)Google Scholar
  17. 17.
    Gelfond, M., Lifschitz, V.: The stable model semantics for logic programming. In: Kowalski, R., Bowen, K. (eds.) Proc. of Int’l Logic Programming, pp. 1070–1080. MIT Press (1988)Google Scholar
  18. 18.
    Kleene, S.C.: On a notation for ordinal numbers. Symbolic Logic 3, 150–155 (1938)CrossRefMATHGoogle Scholar
  19. 19.
    Lifschitz, V.: Thirteen definitions of a stable model. In: Blass, A., Dershowitz, N., Reisig, W. (eds.) Fields of Logic and Computation. LNCS, vol. 6300, pp. 488–503. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  20. 20.
    Lonc, Z., Truszczynski, M.: Computing minimal models, stable models and answer sets. TPLP 6(4), 395–449 (2006)MathSciNetMATHGoogle Scholar
  21. 21.
    Pearce, D.: Equilibrium logic. Annals of Mathematics and AI 47(1-2), 3–41 (2006)MathSciNetMATHGoogle Scholar
  22. 22.
    Przymusinski, T.: Stable semantics for disjunctive programs. New Generation Comput. 9(3/4), 401–424 (1991)CrossRefMATHGoogle Scholar
  23. 23.
    Shepherdson, J.C.: A sound and complete semantics for a version of negation as failure. Theoretical Computer Science 65(3), 343–371 (1989)MathSciNetCrossRefMATHGoogle Scholar

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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Patrick Doherty
    • 1
  • Andrzej Szałas
    • 1
    • 2
  1. 1.Dept. of Computer and Information ScienceLinköping UniversityLinköpingSweden
  2. 2.Institute of InformaticsUniversity of WarsawWarsawPoland

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