Loops and the corresponding loop formulas play an important role in answer set programming. On the one hand, they are used for guaranteeing correctness and completeness in SAT-based answer set solvers. On the other hand, they can be used by conventional answer set solvers for finding unfounded sets of atoms. Unfortunately, the number of loops is exponential in the worst case. We demonstrate that not all loops are actually needed for answer set computation. Rather, we characterize the subclass of elementary loops and show that they are sufficient and necessary for selecting answer sets among the models of a program’s completion. Given that elementary loops cannot be distinguished from general ones in atom dependency graphs, we show how the richer graph structure provided by body-head dependency graphs can be exploited for this purpose.


Logic Program Dependency Graph Deductive Database Elementary Closure General Loop 
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.
    Simons, P., Niemelä, I., Soininen, T.: Extending and implementing the stable model semantics. Artificial Intelligence 138, 181–234 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Leone, N., Faber, W., Pfeifer, G., Eiter, T., Gottlob, G., Koch, C., Mateis, C., Perri, S., Scarcello, F.: The DLV system for knowledge representation and reasoning. ACM Transactions on Computational Logic (2005) (to appear)Google Scholar
  3. 3.
    Babovich, Y., Lifschitz, V.: Computing answer sets using program completion. Unpublished draft (2003)Google Scholar
  4. 4.
    Lin, F., Zhao, Y.: Assat: computing answer sets of a logic program by sat solvers. Artificial Intelligence 157, 115–137 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Lierler, Y., Maratea, M.: Cmodels-2: Sat-based answer sets solver enhanced to non-tight programs. In: Lifschitz, V., Niemelä, I. (eds.) LPNMR 2004. LNCS (LNAI), vol. 2923, pp. 346–350. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  6. 6.
    Clark, K.: Negation as failure. In: Gallaire, H., Minker, J. (eds.) Logic and Data Bases, pp. 293–322. Plenum Press, New York (1978)Google Scholar
  7. 7.
    Fages, F.: Consistency of clark’s completion and the existence of stable models. Journal of Methods of Logic in Computer Science 1, 51–60 (1994)Google Scholar
  8. 8.
    Lifschitz, V., Razborov, A.: Why are there so many loop formulas? ACM Transactions on Computational Logic (to appear)Google Scholar
  9. 9.
    Linke, T.: Suitable graphs for answer set programming. In: Vos, M.D., Provetti, A. (eds.) Proceedings of the Second International Workshop on Answer Set Programming. CEUR Workshop Proceedings, pp. 15–28 (2003)Google Scholar
  10. 10.
    Apt, K., Blair, H., Walker, A.: Towards a theory of declarative knowledge. In: Minker, J. (ed.) Foundations of Deductive Databases and Logic Programming, pp. 89–148. Morgan Kaufmann Publishers, San Francisco (1987)Google Scholar
  11. 11.
    Przymusinski, T.: On the declarative semantics of deductive databases and logic programs. In: Minker, J. (ed.) Foundations of Deductive Databases and Logic Programming, pp. 193–216. Morgan Kaufmann Publishers, San Francisco (1988)Google Scholar
  12. 12.
  13. 13.
    Papadimitriou, C., Sideri, M.: Default theories that always have extensions. Artificial Intelligence 69, 347–357 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Janhunen, T.: Comparing the expressive powers of some syntactically restricted classes of logic programs. In: Lloyd, J., Dahl, V., Furbach, U., Kerber, M., Lau, K., Palamidessi, C., Pereira, L., Sagiv, Y., Stuckey, P. (eds.) Proceedings of the First International Conference on Computational Logic, pp. 852–866. Springer, Heidelberg (2000)Google Scholar
  15. 15.
    Dix, J., Furbach, U., Niemelä, I.: Nonmonotonic reasoning: Towards efficient calculi and implementations. In: Robinson, J., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 1241–1354. Elsevier/MIT Press (2001)Google Scholar
  16. 16.
    van Gelder, A., Ross, K., Schlipf, J.: The well-founded semantics for general logic programs. Journal of the ACM 38, 620–650 (1991)zbMATHGoogle Scholar
  17. 17.
    Lee, J., Lifschitz, V.: Loop formulas for disjunctive logic programs. In: Palamidessi, C. (ed.) ICLP 2003. LNCS, vol. 2916, pp. 451–465. Springer, Heidelberg (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Martin Gebser
    • 1
  • Torsten Schaub
    • 1
  1. 1.Institut für InformatikUniversität PotsdamPotsdam

Personalised recommendations