The Good, the Bad, and the Odd: Cycles in Answer-Set Programs

  • Johannes Klaus Fichte
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7415)

Abstract

Backdoors of answer-set programs are sets of atoms that represent “clever reasoning shortcuts” through the search space. Assignments to backdoor atoms reduce the given program to several programs that belong to a tractable target class. Previous research has considered target classes based on notions of acyclicity where various types of cycles (good and bad cycles) are excluded from graph representations of programs. We generalize the target classes by taking the parity of the number of negative edges on bad cycles into account and consider backdoors for such classes. We establish new hardness results and non-uniform polynomial-time tractability relative to directed or undirected cycles.

Keywords

Answer-Set Programming Non-monotonic Reasoning 

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References

  1. 1.
    Apt, K.R., Blair, H.A., Walker, A.: Towards a theory of declarative knowledge. In: Foundations of Deductive Databases and Logic Programming, pp. 89–148. Morgan Kaufmann (1988)Google Scholar
  2. 2.
    Arikati, S.R., Peled, U.N.: A polynomial algorithm for the parity path problem on perfectly orientable graphs. Discrete Applied Mathematics 65(1-3), 5–20 (1996)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer (1999)Google Scholar
  4. 4.
    Eiter, T., Gottlob, G.: On the computational cost of disjunctive logic programming: Propositional case. Annals of Mathematics and AI 15(3-4), 289–323 (1995)MathSciNetMATHGoogle Scholar
  5. 5.
    Fichte, J.K., Szeider, S.: Backdoors to tractable answer-set programming. Extended and updated version of a paper that appeared in IJCAI 2011, CoRR abs/1104.2788 (2012)Google Scholar
  6. 6.
    Gaspers, S., Szeider, S.: Backdoors to Satisfaction. CoRR abs/1110.6387 (2011)Google Scholar
  7. 7.
    Van Gelder, A.: The alternating fixpoint of logic programs with negation. In: Proceedings of the Ninth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, pp. 1–10. ACM (1989)Google Scholar
  8. 8.
    Van Gelder, A., Ross, K.A., Schlipf, J.S.: The well-founded semantics for general logic programs. Journal of the ACM 38(3), 620–650 (1991)MATHGoogle Scholar
  9. 9.
    Gelfond, M., Lifschitz, V.: The stable model semantics for logic programming. In: Proceedings of the Fifth International Conference and Symposium on Logic Programming (ICLP/SLP), vol. 2, pp. 1070–1080. MIT Press (1988)Google Scholar
  10. 10.
    Gelfond, M., Lifschitz, V.: Classical negation in logic programs and disjunctive databases. New Generation Computing 9(3/4), 365–386 (1991)CrossRefGoogle Scholar
  11. 11.
    Gottlob, G., Scarcello, F., Sideri, M.: Fixed-parameter complexity in AI and nonmonotonic reasoning. AI 138(1-2), 55–86 (2002)MathSciNetMATHGoogle Scholar
  12. 12.
    Lapaugh, A.S., Papadimitriou, C.H.: The even-path problem for graphs and digraphs. Networks 14(4), 507–513 (1984)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Lin, F., Zhao, X.: On odd and even cycles in normal logic programs. In: Proceedings of the Nineteenth National Conference on AI (AAAI), pp. 80–85. AAAI Press (2004)Google Scholar
  14. 14.
    Marek, V.W., Truszczynski, M.: Stable models and an alternative logic programming paradigm: a 25-Year Perspective. In: The Logic Programming Paradigm, pp. 375–398 (1999)Google Scholar
  15. 15.
    Montalva, M., Aracena, J., Gajardo, A.: On the complexity of feedback set problems in signed digraphs. Electronic Notes in Discrete Mathematics 30, 249–254 (2008)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Niemelä, I.: Logic programs with stable model semantics as a constraint programming paradigm. Annals of Mathematics and AI 25(3), 241–273 (1999)MATHGoogle Scholar
  17. 17.
    Robertson, N., Seymour, P., Thomas, R.: Permanents, Pfaffian orientations, and even directed circuits. Annals of Mathematics 150(3), 929–975 (1999)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Schaub, T.: Collection on answer set programming (ASP) and more. Tech. rep., University of Potsdam (2008), http://www.cs.uni-potsdam.de/~torsten/asp
  19. 19.
    Vazirani, V., Yannakakis, M.: Pfaffian Orientations, 0/1 Permanents, and Even Cycles in Directed Graphs. In: Lepistö, T., Salomaa, A. (eds.) ICALP 1988. LNCS, vol. 317, pp. 667–681. Springer, Heidelberg (1988)CrossRefGoogle Scholar
  20. 20.
    Williams, R., Gomes, C., Selman, B.: Backdoors to typical case complexity. In: Proceedings of the Eighteenth International Joint Conference on AI (IJCAI), pp. 1173–1178. Morgan Kaufmann (2003)Google Scholar
  21. 21.
    Williams, R., Gomes, C., Selman, B.: On the connections between backdoors, restarts, and heavy-tailedness in combinatorial search. In: Proceedings of the Sixth International Conference on Theory and Applications of Satisfiability Testing (SAT), pp. 222–230. Morgan Kaufmann (2003)Google Scholar
  22. 22.
    Yuster, R., Zwick, U.: Finding Even Cycles Even Faster. In: Shamir, E., Abiteboul, S. (eds.) ICALP 1994. LNCS, vol. 820, pp. 532–543. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  23. 23.
    Zhao, J.: A Study of Answer Set Programming. MPhil thesis, The Hong Kong University of Science and Technology, Dept. of Computer Science (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Johannes Klaus Fichte
    • 1
  1. 1.Institute of Information SystemsVienna University of TechnologyViennaAustria

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