Guarded Open Answer Set Programming with Generalized Literals

  • Stijn Heymans
  • Davy Van Nieuwenborgh
  • Dirk Vermeir
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3861)


We extend the open answer set semantics for programs with generalized literals. Such extended programs (EPs) have interesting properties, e.g. the ability to express infinity axioms – EPs that have but infinite answer sets. However, reasoning under the open answer set semantics, in particular satisfiability checking of a predicate w.r.t. a program, is already undecidable for programs without generalized literals. In order to regain decidability, we restrict the syntax of EPs such that both rules and generalized literals are guarded. Via a translation to guarded fixed point logic (μGF), in which satisfiability checking is 2-EXPTIME-complete, we deduce 2-EXPTIME-completeness of satisfiability checking in such guarded EPs (GEPs). Bound GEPs are restricted GEPs with EXPTIME-complete satisfiability checking, but still sufficiently expressive to optimally simulate computation tree logic (CTL). We translate Datalog LITE programs to GEPs, establishing equivalence of GEPs under an open answer set semantics, alternation-free μGF, and Datalog LITE. Finally, we discuss ω-restricted logic programs under an open answer set semantics.


Extended Program Boolean Formula Cardinality Constraint Open Answer Computation Tree Logic 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abiteboul, S., Hull, R., Vianu, V.: Foundations of Databases. Addison-Wesley, Reading (1995)zbMATHGoogle Scholar
  2. 2.
    Van Benthem, J.: Dynamic Bits and Pieces. In: ILLC research report. University of Amsterdam (1997)Google Scholar
  3. 3.
    Chandra, A.K., Harel, D.: Horn Clauses and the Fixpoint Query Hierarchy. In: Proc. of PODS 1982, pp. 158–163. ACM Press, New York (1982)Google Scholar
  4. 4.
    Emerson, E.A.: Temporal and Modal Logic. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, pp. 995–1072. Elsevier Science Publishers B.V., Amsterdam (1990)Google Scholar
  5. 5.
    Emerson, E.A., Clarke, E.M.: Using Branching Time Temporal Logic to Synthesize Synchronization Skeletons. Sciene of Computer Programming 2(3), 241–266 (1982)zbMATHCrossRefGoogle Scholar
  6. 6.
    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
  7. 7.
    Gelfond, M., Przymusinska, H.: Reasoning in Open Domains. In: Logic Programming and Non-Monotonic Reasoning, pp. 397–413. MIT Press, Cambridge (1993)Google Scholar
  8. 8.
    Gottlob, G., Grädel, E., Veith, H.: Datalog LITE: A deductive query language with linear time model checking. ACM Transactions on Computational Logic 3(1), 1–35 (2002)CrossRefGoogle Scholar
  9. 9.
    Grädel, E.: Guarded Fixed Point Logic and the Monadic Theory of Trees. Theoretical Computer Science 288, 129–152 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Grädel, E.: Model Checking Games. In: Proceedings of WOLLIC 2002. Electronic Notes in Theoretical Computer Science, vol. 67. Elsevier, Amsterdam (2002)Google Scholar
  11. 11.
    Grädel, E., Walukiewicz, I.: Guarded Fixed Point Logic. In: Proc. of LICS 1999, pp. 45–54. IEEE Computer Society, Los Alamitos (1999)Google Scholar
  12. 12.
    Halevy, A., Mumick, I., Sagiv, Y., Shmueli, O.: Static Analysis in Datalog Extensions. Journal of the ACM 48(5), 971–1012 (2001)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Heymans, S., Van Nieuwenborgh, D., Vermeir, D.: Nonmonotonic Ontological and Rule-Based Reasoning with Extended Conceptual Logic Programs. In: Gómez-Pérez, A., Euzenat, J. (eds.) ESWC 2005. LNCS, vol. 3532, pp. 392–407. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  14. 14.
    Heymans, S., Van Nieuwenborgh, D., Vermeir, D.: Guarded Open Answer Set Programming. In: Baral, C., Greco, G., Leone, N., Terracina, G. (eds.) LPNMR 2005. LNCS (LNAI), vol. 3662, pp. 92–104. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  15. 15.
    Huth, M.R.A., Ryan, M.: Logic in Computer Science: Modelling and Reasoning about Systems. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  16. 16.
    Lifschitz, V., Pearce, D., Valverde, A.: Strongly Equivalent Logic Programs. ACM Transactions on Computational Logic 2(4), 526–541 (2001)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Lloyd, J., Topor, R.: Making Prolog More Expressive. J. Log. Program. 1(3), 225–240 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Simons, P.: smodels homepage,
  19. 19.
    Syrjänen, T.: Omega-restricted Logic Programs. In: Eiter, T., Faber, W., Truszczyński, M. (eds.) LPNMR 2001. LNCS (LNAI), vol. 2173, pp. 267–279. Springer, Heidelberg (2001)Google Scholar
  20. 20.
    Syrjänen, T.: Cardinality Constraint Programs. In: Alferes, J.J., Leite, J. (eds.) JELIA 2004. LNCS (LNAI), vol. 3229, pp. 187–199. Springer, Heidelberg (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Stijn Heymans
    • 1
  • Davy Van Nieuwenborgh
    • 1
  • Dirk Vermeir
    • 1
  1. 1.Dept. of Computer ScienceVrije Universiteit Brussel, VUBBrusselsBelgium

Personalised recommendations