Cumulativity Tailored for Nonmonotonic Reasoning

  • Tomi Janhunen
  • Ilkka Niemelä
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9060)


In nonmonotonic reasoning, conclusions can be retracted when new pieces of information are incorporated into premises. This contrasts with classical reasoning which is monotonic, i.e., new premises can only increase the set of conclusions that can be drawn. Slightly weaker properties, such as cumulativity and rationality, seem reasonable counterparts of such a monotonicity property for nonmonotonic reasoning but intriguingly it turned out that some major nonmonotonic logics failed to be cumulative. These observations led to the study of variants in hope of restoring cumulativity but not losing other essential properties. In this paper, we take a fresh view on cumulativity by starting from a notion of rule entailment in the context of answer set programs. It turns out that cumulativity can be revived if the expressive precision of rules subject to answer set semantics is fully exploited when new premises are being incorporated. Even stronger properties can be established and we illustrate how the approach can be generalized for major nonmonotonic logics.


Logic Program Stable Model Integrity Constraint Choice Rule Default Theory 
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.


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  1. 1.
    Brewka, G.: Cumulative default logic: In defense of nonmonotonic inference rules. Artificial Intelligence 50(2), 183–205 (1991)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Brewka, G., Makinson, D., Schlechta, K.: Cumulative inference relations for JTMS and logic programming. In: Dix, J., Schmitt, P.H., Jantke, K.P. (eds.) NIL 1990. LNCS, vol. 543, pp. 1–12. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  3. 3.
    Dix, J.: Default theories of poole-type and a method for constructing cumulative versions of default logic. In: Proceedings of ECAI 1992, pp. 289–293 (1992)Google Scholar
  4. 4.
    Dix, J.: Cumulativity and rationality in semantics of normal logic programs. In: Dix, J., Schmitt, P.H., Jantke, K.P. (eds.) NIL 1990. LNCS, vol. 543, pp. 13–37. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  5. 5.
    Dix, J.: Classifying semantics of disjunctive logic programs. In: Proceedings of JICSLP 1992, pp. 798–812. MIT Press (1992)Google Scholar
  6. 6.
    Gebser, M., Schaub, T.: Tableau calculi for answer set programming. In: Etalle, S., Truszczyński, M. (eds.) ICLP 2006. LNCS, vol. 4079, pp. 11–25. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Gebser, M., Schaub, T.: Generic tableaux for answer set programming. In: Dahl, V., Niemelä, I. (eds.) ICLP 2007. LNCS, vol. 4670, pp. 119–133. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Gelder, A.V., Ross, K., Schlipf, J.: The well-founded semantics for general logic programs. Journal of the ACM 38(3), 620–650 (1991)MathSciNetMATHGoogle Scholar
  9. 9.
    Gelfond, M., Lifschitz, V.: The stable model semantics for logic programming. In: Proceedings of ICLP 1988, pp. 1070–1080 (1988)Google Scholar
  10. 10.
    Gelfond, M., Lifschitz, V.: Classical negation in logic programs and disjunctive databases. New Generation Computing 9, 365–385 (1991)CrossRefMATHGoogle Scholar
  11. 11.
    Gelfond, M., Przymusinska, H., Lifschitz, V., Truszczynski, M.: Disjunctive defaults. In: Proceedings of KR 1991, pp. 230–237. Morgan Kaufmann (1991)Google Scholar
  12. 12.
    Gottlob, G., Mingyi, Z.: Cumulative default logic: Finite characterization, algorithms, and complexity. Artificial Intelligence 69(1-2), 329–345 (1994)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Janhunen, T.: Removing redundancy from answer set programs. In: Garcia de la Banda, M., Pontelli, E. (eds.) ICLP 2008. LNCS, vol. 5366, pp. 729–733. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  14. 14.
    Janhunen, T., Niemelä, I.: A scheme for weakened negative introspection in autoepistemic reasoning. In: Mundici, D., Gottlob, G., Leitsch, A. (eds.) KGC 1993. LNCS, vol. 713, pp. 211–222. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  15. 15.
    Kraus, S., Lehmann, D., Magidor, M.: Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence 44(1-2), 167–207 (1990)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Lifschitz, V., Pearce, D., Valverde, A.: Strongly equivalent logic programs. ACM Transactions on Computational Logic 2(4), 526–541 (2001)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Lifschitz, V., Tang, L., Turner, H.: Nested expressions in logic programs. Annals of Mathematics and Artificial Intelligence 25(3-4), 369–389 (1999)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Marek, W., Truszczyński, M.: Nonmonotonic Logic: Context-Dependent Reasoning. Springer, Berlin (1993)CrossRefMATHGoogle Scholar
  19. 19.
    McCarthy, J.: Applications of circumscription to formalizing commonsense knowledge. Artificial Intelligence 28, 89–116 (1986)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mikitiuk, A., Truszczynski, M.: Constrained and rational default logics. In: Proceedings of IJCAI 1995, pp. 1509–1517. Morgan Kaufmann (1995)Google Scholar
  21. 21.
    Moore, R.: Semantical consideration on nonmonotonic logic. Artificial Intelligence 25(1), 234–252 (1985)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Pearce, D.: Equilibrium logic. Annals of Mathematics and Artificial Intelligence 47(1-2), 3–41 (2006)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Moniz Pereira, L., Pinto, A.M.: Revised stable models – A semantics for logic programs. In: Bento, C., Cardoso, A., Dias, G. (eds.) EPIA 2005. LNCS (LNAI), vol. 3808, pp. 29–42. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  24. 24.
    Reiter, R.: A logic for default reasoning. Artificial Intelligence 13, 81–132 (1980)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Simons, P.: Extending the stable model semantics with more expressive rules. In: Gelfond, M., Leone, N., Pfeifer, G. (eds.) LPNMR 1999. LNCS (LNAI), vol. 1730, pp. 305–316. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  26. 26.
    Simons, P., Niemelä, I., Soininen, T.: Extending and implementing the stable model semantics. Artificial Intelligence 138(1-2), 181–234 (2002)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Turner, H.: Strong equivalence for logic programs and default theories (Made easy). In: Eiter, T., Faber, W., Truszczyński, M. (eds.) LPNMR 2001. LNCS (LNAI), vol. 2173, pp. 81–92. Springer, Heidelberg (2001)Google Scholar
  28. 28.
    Turner, H.: Strong equivalence made easy: nested expressions and weight constraints. Theory and Practice of Logic Programming 3(4-5), 609–622 (2003)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Tomi Janhunen
    • 1
  • Ilkka Niemelä
    • 1
  1. 1.Helsinki Institute for Information Technology HIIT, Department of Information and Computer ScienceAalto UniversityAALTOFinland

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