Skip to main content
Springer Nature Link
Log in

On the Continuity of Gelfond-Lifschitz Operator and Other Applications of Proof-Theory in ASP

  • Conference paper
Logic Programming (ICLP 2008)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 5366))

Included in the following conference series:

Abstract

Using a characterization of stable models of logic programs P as satisfying valuations of a suitably chosen propositional theory, called the set of reduced defining equations rΦ P , we show that the finitary character of that theory rΦ P is equivalent to a certain continuity property of the Gelfond-Lifschitz operator \({\mathit{GL}}_P\) associated with the program P. The introduction of the formula rΦ P leads to a double-backtracking algorithm for computation of stable models by reduction to satisfiability of suitably chosen propositional theories. This algorithm does not use the reduction via loop-formulas as proposed in [1] or its extension proposed in [2]. Finally, we discuss possible extensions of techniques proposed in this paper to the context of cardinality constraints.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Lin, F., Zhao, Y.: Assat: Computing answer sets of a logic program by sat solvers. In: Proceedings of AAAI 2002, pp. 112–117 (2002)

    Google Scholar 

  2. Ferraris, P., Lee, J., Lifschitz, V.: A generalization of Lin-Zhao theorem. Annals of Mathematics and Artificial Intelligence 47, 79–101 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Marek, V., Truszczyński, M.: Nonmonotonic Logic. Springer, Heidelberg (1993)

    Book  MATH  Google Scholar 

  4. Bonatti, P.: Reasoning with infinite stable models. Artificial Intelligence Journal 156, 75–111 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Milnikel, R.: Sequent calculi for skeptical reasoning in predicate default logic and other nonmonotonic systems. Annals of Mathematics and Artificial Intelligence 44, 1–34 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Lifschitz, V.: Foundations of logic programming. In: Principles of Knowledge Representation, pp. 69–127. CSLI Publications (1996)

    Google Scholar 

  7. Bondarenko, A., Toni, F., Kowalski, R.: An assumption-based framework for non-monotonic reasoning. In: Proceedings of LPNMR 1993, pp. 171–189. MIT Press, Cambridge (1993)

    Google Scholar 

  8. Marek, W., Nerode, A., Remmel, J.: Nonmonotonic rule systems I. Annals of Mathematics and Artificial Intelligence 1, 241–273 (1990)

    Article  MATH  Google Scholar 

  9. Marek, W., Nerode, A., Remmel, J.: Nonmonotonic rule systems II. Annals of Mathematics and Artificial Intelligence 5, 229–264 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Marek, W., Nerode, A., Remmel, J.: A context for belief revision: Normal logic programs. In: Proceedings, Workshop on Defeasible Reasoning and Constraint Solving, International Logic Programming Symposium (1991)

    Google Scholar 

  11. Marek, W., Nerode, A., Remmel, J.: How complicated is the set of stable models of a logic program? Annals of Pure and Applied Logic 56, 119–136 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  12. Marek, W., Nerode, A., Remmel, J.: The stable models of predicate logic programs. Journal of Logic Programming 21, 129–154 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Marek, W., Nerode, A., Remmel, J.: Context for belief revision: Forward chaining-normal nonmonotonic rule systems. Annals of Pure and Applied Logic 67, 269–324 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gebser, M., Schaub, T.: Generic tableaux for answer set programming. In: Proceedings of International Conference on Logic Programming, 2007, pp. 119–133 (2007)

    Google Scholar 

  15. Järvisalo, M., Oikarinen, E.: Extended asp tableaux and rule redundancy in normal logic programs. In: Dahl, V., Niemelä, I. (eds.) ICLP 2007. LNCS, vol. 4670, pp. 134–148. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  16. Gelfond, M., Lifschitz, V.: The stable model semantics for logic programming. In: Proceedings of the International Joint Conference and Symposium on Logic Programming, pp. 1070–1080 (1988)

    Google Scholar 

  17. Lloyd, J.: Foundations of Logic Programming. Springer, Heidelberg (1989)

    MATH  Google Scholar 

  18. Apt, K.: Logic programming. In: van Leeuven, J. (ed.) Handbook of Theoretical Computer Science, pp. 493–574. MIT Press, Cambridge (1990)

    Google Scholar 

  19. Dung, P., Kanchanasut, K.: On the generalized predicate completion of non-Horn programs. In: Logic programming. Proceedings of the North American Conference (1989)

    Google Scholar 

  20. Clark, K.: Negation as failure. In: Minker, J., Gallaire, H. (eds.) Logic and data bases, pp. 293–322. Plenum Press (1978)

    Google Scholar 

  21. Doets, K.: From Logic to Logic Programming. MIT Press, Cambridge (1994)

    MATH  Google Scholar 

  22. Jonsson, B., Tarski, A.: Boolean algebras with operators. American Journal of Mathematics 73, 891–939 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  23. Giunchiglia, E., Lierler, Y., Maratea, M.: Answer set programming based on propositional satisfiability. Journal of Automated Reasoning 36, 345–377 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lifschitz, V., Razborov, A.: Why are there so many loop formulas. Annals of Mathematics and Artificial Intelligence 7, 261–268 (2006)

    MathSciNet  Google Scholar 

  25. Simons, P., Niemelä, I., Soininen, T.: Extending and implementing the stable model semantics. Artificial Intelligence Journal 138, 181–234 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  26. Marek, V., Remmel, J.: Set constraints in logic programming. In: Lifschitz, V., Niemelä, I. (eds.) LPNMR 2004. LNCS (LNAI), vol. 2923, pp. 154–167. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  27. Marek, V., Niemelä, I., Truszczyński, M.: Logic programs with monotone abstract constraint atoms. Theory and Practice of Logic Programming 8, 167–199 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Liu, L., Truszczyński, M.: Properties of programs with monotone and convex constraints. In: Proceedings of the 20th National Conference on Artificial Intelligence, pp. 701–706 (2005)

    Google Scholar 

  29. Ferraris, P., Lifschitz, V.: Weight constraints as nested expressions. Theory and Practice of Logic Programming 5, 45–74 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lifschitz, V., Tang, L., Turner, H.: Nested expressions in logic programs. Annals of Mathematics and Artificial Intelligence 25, 369–389 (1999)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Kentucky, Lexington, KY 40506-0046, USA

    V. W. Marek

  2. Department of Mathematics, University of California, La Jolla, CA 92093, USA

    J. B. Remmel

Authors
  1. V. W. Marek
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. B. Remmel
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Clayton School of Information Technology, Monash University, Australia

    Maria Garcia de la Banda

  2. Department of Computer Science, New Mexico State University, NM 88003, Las Cruces, USA

    Enrico Pontelli

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Marek, V.W., Remmel, J.B. (2008). On the Continuity of Gelfond-Lifschitz Operator and Other Applications of Proof-Theory in ASP. In: Garcia de la Banda, M., Pontelli, E. (eds) Logic Programming. ICLP 2008. Lecture Notes in Computer Science, vol 5366. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89982-2_25

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-89982-2_25

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-89981-5

  • Online ISBN: 978-3-540-89982-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics