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Default Logic and Bounded Treewidth

  • Johannes K. Fichte
  • Markus Hecher
  • Irina Schindler
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10792)

Abstract

In this paper, we study Reiter’s propositional default logic when the treewidth of a certain graph representation (semi-primal graph) of the input theory is bounded. We establish a dynamic programming algorithm on tree decompositions that decides whether a theory has a consistent stable extension (Ext). Our algorithm can even be used to enumerate all generating defaults (EnumSE) that lead to stable extensions. We show that our algorithm decides Ext in linear time in the input theory and triple exponential time in the treewidth (so-called fixed-parameter linear algorithm). Further, our algorithm solves EnumSE with a pre-computation step that is linear in the input theory and triple exponential in the treewidth followed by a linear delay to output solutions.

Keywords

Parameterized algorithms Tree decompositions Dynamic programming Reiter’s default logic Propositional logic 

References

  1. 1.
    Reiter, R.: A logic for default reasoning. AIJ 13, 81–132 (1980)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Marek, V.W., Truszczyński, M.: Nonmonotonic Logic: Context-dependent Reasoning. Artificial Intelligence. Springer, Berlin (1993).  https://doi.org/10.1007/978-3-662-02906-0 CrossRefzbMATHGoogle Scholar
  3. 3.
    Gottlob, G.: Complexity results for nonmonotonic logics. JLC 2(3), 397–425 (1992)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-21275-3 CrossRefzbMATHGoogle Scholar
  5. 5.
    Fichte, J.K., Meier, A., Schindler, I.: Strong backdoors for default logic. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 45–59. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-40970-2_4 Google Scholar
  6. 6.
    Meier, A., Schindler, I., Schmidt, J., Thomas, M., Vollmer, H.: On the parameterized complexity of non-monotonic logics. Arch. Math. Logic 54(5–6), 685–710 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Courcelle, B.: Graph rewriting: an algebraic and logic approach. In: van Leeuwen, J. (ed.) Handbook of theoretical computer science. Volume Formal Models and Semantics, vol. B, pp. 193–242. Elsevier Science Publishers, North-Holland (1990)Google Scholar
  8. 8.
    Kneis, J., Langer, A.: A practical approach to Courcelle’s theorem. Electron. Notes Theor. Comput. Sci. 251, 65–81 (2009)CrossRefzbMATHGoogle Scholar
  9. 9.
    Charwat, G., Woltran, S.: Dynamic programming-based QBF solving. In: Proceedings of the 4th International Workshop on Quantified Boolean Formulas (QBF 2016), pp. 27–40 (2016)Google Scholar
  10. 10.
    Fichte, J.K., Hecher, M., Morak, M., Woltran, S.: Answer set solving with bounded treewidth revisited. In: Balduccini, M., Janhunen, T. (eds.) LPNMR 2017. LNCS (LNAI), vol. 10377, pp. 132–145. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-61660-5_13 CrossRefGoogle Scholar
  11. 11.
    Fichte, J.K., Hecher, M., Morak, M., Woltran, S.: DynASP2.5: dynamic programming on tree decompositions in action. In: Proceedings of the 12th IPEC (2017)Google Scholar
  12. 12.
    Creignou, N., Meier, A., Müller, J.S., Schmidt, J., Vollmer, H.: Paradigms for parameterized enumeration. Th. Comput. Syst. 60(4), 737–758 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bodlaender, H., Koster, A.M.C.A.: Combinatorial optimization on graphs of bounded treewidth. Comput. J. 51(3), 255–269 (2008)CrossRefGoogle Scholar
  14. 14.
    Bliem, B., Charwat, G., Hecher, M., Woltran, S.: D-FLAT\(^\wedge \)2 subset minimization in dynamic programming on tree decompositions made easy. FI 147, 27–34 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Abiteboul, S., Hull, R., Vianu, V.: Foundations of Databases: The Logical Level, 1st edn. Addison-Wesley, Boston (1995)Google Scholar
  16. 16.
    Gelfond, M., Lifschitz, V., Przymusinska, H., Truszczynski, M.: Disjunctive Defaults, pp. 230–237. Morgan Kaufmann, Burlington (1991)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Technische Universität WienViennaAustria
  2. 2.Leibniz Universität HannoverHannoverGermany

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