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On Acyclic and Head-Cycle Free Nested Logic Programs

  • Thomas Linke
  • Hans Tompits
  • Stefan Woltran
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3132)

Abstract

We define the class of head-cycle free nested logic programs, and its proper subclass of acyclic nested programs, generalising similar classes originally defined for disjunctive logic programs. We then extend several results known for acyclic and head-cycle free disjunctive programs under the stable-model semantics to the nested case. Most notably, we provide a propositional semantics for the program classes under consideration. This generalises different extensions of Fages’ theorem, including a recent result by Erdem and Lifschitz for tight logic programs. We further show that, based on a shifting method, head-cycle free nested programs can be rewritten into normal programs in polynomial time and space, extending a similar technique for head-cycle free disjunctive programs. All this shows that head-cycle free nested programs constitute a subclass of nested programs possessing a lower computational complexity than arbitrary nested programs, providing the polynomial hierarchy does not collapse.

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References

  1. 1.
    Ben-Eliyahu, R., Dechter, R.: Propositional Semantics for Disjunctive Logic Programs. Annals of Mathematics and Artificial Intelligence 12, 53–87 (1994)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bidoit, N., Froidevaux, C.: Negation by Default and Unstratifiable Logic Programs. Theoretical Computer Science 78, 85–112 (1991)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Brass, S., Dix, J.: Semantics of (Disjunctive) Logic Programs Based on Partial Evaluation. Journal of Logic Programming 38(3), 167–213 (1999)Google Scholar
  4. 4.
    Brignoli, G., Costantini, S., D’Antona, O., Provetti, A.: Characterizing and Computing Stable Models of Logic Programs: The Non-stratified Case. In: Proc. of the 2nd International Conference on Information Technology (CIT 1999), pp. 197–201 (1999)Google Scholar
  5. 5.
    Clark, K.L.: Negation as Failure. In Logic and Databases, pp. 293–322. Plenum, New York (1978)Google Scholar
  6. 6.
    Dix, J., Gottlob, G., Marek, V.: Reducing Disjunctive to Non-Disjunctive Semantics by Shift-Operations. Fundamenta Informaticae XXVIII(1/2), 87–100 (1996)MathSciNetGoogle Scholar
  7. 7.
    Eiter, T., Faber, W., Leone, N., Pfeifer, G.: Declarative Problem-Solving Using the DLV System. In: Logic-Based Artificial Intelligence, pp. 79–103. Kluwer, Dordrecht (2000)Google Scholar
  8. 8.
    Eiter, T., Fink, M., Tompits, H., Woltran, S.: On Eliminating Disjunctions in Stable Logic Programming. In: Proc. KR 2004 (2004) (to appear)Google Scholar
  9. 9.
    Eiter, T., Gottlob, G.: On the Computational Cost of Disjunctive Logic Programming: Propositional Case. Annals of Mathematics and Artificial Intelligence 15(3-4), 289–323 (1995)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Erdem, E., Lifschitz, V.: Fages’ Theorem for Programs with Nested Expressions. In: Codognet, P. (ed.) ICLP 2001. LNCS, vol. 2237, pp. 242–254. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  11. 11.
    Erdem, E., Lifschitz, V.: Tight Logic Programs. Theory and Practice of Logic Programming 3(4-5), 499–518 (2003)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Fages, F.: Consistency of Clark’s Completion and Existence of Stable Models. Methods of Logic in Computer Science 1, 51–60 (1994)Google Scholar
  13. 13.
    Gelfond, M., Lifschitz, V., Przymusinska, H., Truszczyński, M.: Disjunctive Defaults. In: Proc. KR 1991, pp. 230–237. Morgan Kaufmann, San Francisco (1991)Google Scholar
  14. 14.
    Inoue, K., Sakama, C.: Negation as Failure in the Head. Journal of Logic Programming 35(1), 39–78 (1998)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Janhunen, T.: On the Effect of Default Negation on the Expressiveness of Disjunctive Rules. In: Eiter, T., Faber, W., Truszczyński, M. (eds.) LPNMR 2001. LNCS (LNAI), vol. 2173, pp. 93–106. Springer, Heidelberg (2001)Google Scholar
  16. 16.
    Konczak, K., Linke, T., Schaub, T.: Graphs and Colorings for Answer Set Programming: Abridged Report. In: Lifschitz, V., Niemelä, I. (eds.) LPNMR 2004. LNCS (LNAI), vol. 2923, pp. 127–140. Springer, Heidelberg (2003)CrossRefGoogle 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
  18. 18.
    Lifschitz, V., Tang, L., Turner, H.: Nested Expressions in Logic Programs. Annals ofMathematics and Artificial Intelligence 25(3-4), 369–389 (1999)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Lin, F., Zhao, Y.: ASSAT: Computing Answer Sets of a Logic Program by SAT Solvers. In: Proc. AAAI 2002, pp. 112–117 (2002)Google Scholar
  20. 20.
    Linke, T.: Graph Theoretical Characterization and Computation of Answer Sets. In: Proc. IJCAI 2001, pp. 641–645. Morgan Kaufmann, San Francisco (2001)Google Scholar
  21. 21.
    Linke, T.: Suitable Graphs for Answer Set Programming. In: Proc. ASP 2003. CEUR Workshop Proceedings, vol. 78, pp. 15–28 (2003)Google Scholar
  22. 22.
    Linke, T., Anger, C., Konczak, K.: More on noMoRe. In: Flesca, S., Greco, S., Leone, N., Ianni, G. (eds.) JELIA 2002. LNCS (LNAI), vol. 2424, pp. 468–480. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  23. 23.
    Lloyd, J., Topor, R.: Making Prolog More Expressive. Journal of Logic Programming 3, 225–240 (1984)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Marek, W., Truszczyński, M.: Autoepistemic Logic. Journal of the ACM 38, 588–619 (1991)MATHCrossRefGoogle Scholar
  25. 25.
    Pearce, D., Sarsakov, V., Schaub, T., Tompits, H., Woltran, S.: A Polynomial Translation of Logic Programs with Nested Expressions into Disjunctive Logic Programs: Preliminary Report. In: Stuckey, P.J. (ed.) ICLP 2002. LNCS, vol. 2401, pp. 405–420. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  26. 26.
    Pearce, D., Tompits, H., Woltran, S.: Encodings for EquilibriumLogic and Logic Programs with Nested Expressions. In: Proc. EPIA 2001. LNCS, vol. 2285, pp. 306–320. Springer, Heidelberg (2001)Google Scholar
  27. 27.
    Simons, P., Niemelä, I., Soininen, T.: Extending and Implementing the Stable Model Semantics. Artificial Intelligence 138, 181–234 (2002)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Tarjan, R.: Depth-first Search and Linear Graph Algorithms. SIAM Journal on Computing 1, 146–160 (1972)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    You, J., Yuan, L., Zhang, M.: On the Equivalence Between Answer Sets and Models of Completion for Nested Logic Programs. In: Proc. IJCAI 2003, pp. 859–865 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Thomas Linke
    • 1
  • Hans Tompits
    • 2
  • Stefan Woltran
    • 2
  1. 1.Institut für InformatikUniversität PotsdamPotsdamGermany
  2. 2.Institut für Informationssysteme 184/3Technische Universität WienViennaAustria

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