A System Z-like Approach for First-Order Default Reasoning

  • Gabriele Kern-Isberner
  • Christoph Beierle
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9060)


Default rules of the form “If A then (usually, probably) B” can be represented conveniently by conditionals. To every consistent knowledge base \(\mathcal{R}\) with such qualitative conditionals over a propositional language, system Z assigns a unique minimal model that accepts every conditional in \(\mathcal{R}\) and that is therefore a model of \(\mathcal{R}\) inductively completing the explicitly given knowledge. In this paper, we propose a generalization of system Z for a first-order setting. For a first-order conditional knowledge base \(\mathcal{R}\) over unary predicates, we present the definition of a system Z-like ranking function, prove that it yields a model of \(\mathcal{R}\), and illustrate its construction by a detailed example.


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  1. 1.
    Adams, E.: Probability and the logic of conditionals. In: Hintikka, J., Suppes, P. (eds.) Aspects of Inductive Logic, pp. 265–316. North-Holland, Amsterdam (1966)CrossRefGoogle Scholar
  2. 2.
    Beierle, C., Kern-Isberner, G.: Methoden wissensbasierter Systeme - Grundlagen, Algorithmen, Anwendungen, 5th, revised and extended edn. Springer, Wiesbaden (2014)Google Scholar
  3. 3.
    Bonatti, P., Faella, M., Sauro, L.: Defeasible inclusions in low-complexity DLs. Journal of Artificial Intelligence Research 42, 719–764 (2011)MathSciNetMATHGoogle Scholar
  4. 4.
    Brewka, G.: Tweety - still flying: Some remarks on abnormal birds applicable rules and a default prover. In: Kehler, T. (ed.) Proceedings of the 5th National Conference on Artificial Intelligence, Philadelphia, PA, August 11-15. Science, vol. 1, pp. 8–12. Morgan Kaufmann (1986)Google Scholar
  5. 5.
    Britz, K., Heidema, J., Meyer, T.: Semantic preferential subsumption. In: Proceedings of KR-2008. pp. 476–484. AAAI Press/MIT Press (2008)Google Scholar
  6. 6.
    Delgrande, J.: On first-order conditional logics. Artificial Intelligence 105, 105–137 (1998)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    de Finetti, B.: La prévision, ses lois logiques et ses sources subjectives. Ann. Inst. H. Poincaré 7 (1937), English translation in Kyburg, H., Smokler, H.E.: Studies in Subjective Probability, pp. 93-158. Wiley, New York (1964)Google Scholar
  8. 8.
    Gelfond, M., Leone, N.: Logic programming and knowledge representation – the A-prolog perspective. Artificial Intelligence 138, 3–38 (2002)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Giordano, L., Gliozzi, V., Olivetti, N.: Minimal model semantics and rational closure in description logics. In: Proceedings of DL-2013 (2013)Google Scholar
  10. 10.
    Goldszmidt, M., Pearl, J.: Qualitative probabilities for default reasoning, belief revision, and causal modeling. Artificial Intelligence 84, 57–112 (1996)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kern-Isberner, G.: A thorough axiomatization of a principle of conditional preservation in belief revision. Annals of Mathematics and Artificial Intelligence 40(1-2), 127–164 (2004)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kern-Isberner, G., Eichhorn, C.: Structural inference from conditional knowledge bases. Studia Logica, Special Issue Logic and Probability: Reasoning in Uncertain Environments 102(4) (2014)Google Scholar
  13. 13.
    Kern-Isberner, G., Thimm, M.: A ranking semantics for first-order conditionals. In: De Raedt, L., Bessiere, C., Dubois, D., Doherty, P., Frasconi, P., Heintz, F., Lucas, P. (eds.) Proceedings 20th European Conference on Artificial Intelligence, ECAI-2012. Frontiers in Artificial Intelligence and Applications, vol. 242, pp. 456–461. IOS Press (2012)Google Scholar
  14. 14.
    Lehmann, D., Magidor, M.: What does a conditional knowledge base entail? Artificial Intelligence 55, 1–60 (1992)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Pearl, J.: System Z: A natural ordering of defaults with tractable applications to nonmonotonic reasoning. In: Proc. of the 3rd Conf. on Theor. Asp. of Reasoning about Knowledge, TARK 1990, pp. 121–135. Morgan Kaufmann Publishers Inc., San Francisco (1990)Google Scholar
  16. 16.
    Spohn, W.: Ordinal conditional functions: a dynamic theory of epistemic states. In: Harper, W., Skyrms, B. (eds.) Causation in Decision, Belief Change, and Statistics, II, pp. 105–134. Kluwer Academic Publishers (1988)Google Scholar
  17. 17.
    Spohn, W.: The Laws of Belief: Ranking Theory and Its Philosophical Applications. Oxford University Press (2012)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Gabriele Kern-Isberner
    • 1
  • Christoph Beierle
    • 2
  1. 1.Department of Computer ScienceTU DortmundGermany
  2. 2.Department of Computer ScienceUniversity of HagenGermany

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