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Advances in Knowledge Representation, Logic Programming, and Abstract Argumentation pp 96–111Cite as

Cumulativity Tailored for Nonmonotonic Reasoning

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9060))

Abstract

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.

The support from the Finnish Centre of Excellence in Computational Inference Research (COIN) funded by the Academy of Finland (under grant #251170) is gratefully acknowledged.

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References

  1. Brewka, G.: Cumulative default logic: In defense of nonmonotonic inference rules. Artificial Intelligence 50(2), 183–205 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  5. Dix, J.: Classifying semantics of disjunctive logic programs. In: Proceedings of JICSLP 1992, pp. 798–812. MIT Press (1992)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  9. Gelfond, M., Lifschitz, V.: The stable model semantics for logic programming. In: Proceedings of ICLP 1988, pp. 1070–1080 (1988)

    Google Scholar 

  10. Gelfond, M., Lifschitz, V.: Classical negation in logic programs and disjunctive databases. New Generation Computing 9, 365–385 (1991)

    Article  MATH  Google Scholar 

  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. Gottlob, G., Mingyi, Z.: Cumulative default logic: Finite characterization, algorithms, and complexity. Artificial Intelligence 69(1-2), 329–345 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  15. Kraus, S., Lehmann, D., Magidor, M.: Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence 44(1-2), 167–207 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lifschitz, V., Pearce, D., Valverde, A.: Strongly equivalent logic programs. ACM Transactions on Computational Logic 2(4), 526–541 (2001)

    Article  MathSciNet  Google Scholar 

  17. Lifschitz, V., Tang, L., Turner, H.: Nested expressions in logic programs. Annals of Mathematics and Artificial Intelligence 25(3-4), 369–389 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Marek, W., Truszczyński, M.: Nonmonotonic Logic: Context-Dependent Reasoning. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  19. McCarthy, J.: Applications of circumscription to formalizing commonsense knowledge. Artificial Intelligence 28, 89–116 (1986)

    Article  MathSciNet  Google Scholar 

  20. Mikitiuk, A., Truszczynski, M.: Constrained and rational default logics. In: Proceedings of IJCAI 1995, pp. 1509–1517. Morgan Kaufmann (1995)

    Google Scholar 

  21. Moore, R.: Semantical consideration on nonmonotonic logic. Artificial Intelligence 25(1), 234–252 (1985)

    Article  MathSciNet  Google Scholar 

  22. Pearce, D.: Equilibrium logic. Annals of Mathematics and Artificial Intelligence 47(1-2), 3–41 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  24. Reiter, R.: A logic for default reasoning. Artificial Intelligence 13, 81–132 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  26. Simons, P., Niemelä, I., Soininen, T.: Extending and implementing the stable model semantics. Artificial Intelligence 138(1-2), 181–234 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  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. Turner, H.: Strong equivalence made easy: nested expressions and weight constraints. Theory and Practice of Logic Programming 3(4-5), 609–622 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Helsinki Institute for Information Technology HIIT, Department of Information and Computer Science, Aalto University, 00076, AALTO, Finland

    Tomi Janhunen & Ilkka Niemelä

Authors
  1. Tomi Janhunen
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  2. Ilkka Niemelä
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Editors and Affiliations

  1. Institut für Informationssysteme, Technische Universität Wien, 184/3, Abteilung für wissensbasierte Systeme, Favoritenstraße 9-11, 1040, Wien, Austria

    Thomas Eiter

  2. Institut für Informatik, Abteilung Intelligente Systeme, Universität Leipzig, PF 100920, 04009, Leipzig, Germany

    Hannes Strass

  3. Department of Computer Science, University of Kentucky, 309 Davis Marksbury Building, 329 Rose Street, 40506-0633, Lexington, KY, USA

    Mirosław Truszczyński

  4. Institut für Informationssysteme, Technische Universität Wien, 184/2, Abteilung für Datenbanken und Artificial Intelligence, Favoritenstraße 9-11, 1040, Wien, Austria

    Stefan Woltran

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Janhunen, T., Niemelä, I. (2015). Cumulativity Tailored for Nonmonotonic Reasoning. In: Eiter, T., Strass, H., Truszczyński, M., Woltran, S. (eds) Advances in Knowledge Representation, Logic Programming, and Abstract Argumentation. Lecture Notes in Computer Science(), vol 9060. Springer, Cham. https://doi.org/10.1007/978-3-319-14726-0_7

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  • DOI: https://doi.org/10.1007/978-3-319-14726-0_7

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-14725-3

  • Online ISBN: 978-3-319-14726-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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