Skeptical Inference Based on C-Representations and Its Characterization as a Constraint Satisfaction Problem

  • Christoph Beierle
  • Christian Eichhorn
  • Gabriele Kern-Isberner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9616)


The axiomatic system P is an important standard for plausible, nonmonotonic inferences that is, however, known to be too weak to solve benchmark problems like irrelevance, or subclass inheritance (so-called Drowning Problem). Spohn’s ranking functions which provide a semantic base for system P have often been used to design stronger inference relations, like Pearl’s system Z, or c-representations. While each c-representation shows excellent inference properties and handles particularly irrelevance and subclass inheritance properly, it is still an open problem which c-representation is the best. In this paper, we focus on the generic properties of c-representations and consider the skeptical inference relation (c-inference) that is obtained by taking all c-representations of a given knowledge base into account. In particular, we show that c-inference preserves the properties of solving irrelevance and subclass inheritance which are met by every single c-representation. Moreover, we characterize skeptical c-inference as a constraint satisfaction problem so that constraint solvers can be used for its implementation.


  1. 1.
    Adams, E.W.: The Logic of Conditionals: An Application of Probability to Deductive Logic. Synthese Library. Springer, Dordrecht (1975)CrossRefMATHGoogle Scholar
  2. 2.
    Beierle, C., Kern-Isberner, G.: A declarative approach for computing ordinal conditional functions using constraint logic programming. In: Tompits, H., Abreu, S., Oetsch, J., Pührer, J., Seipel, D., Umeda, M., Wolf, A. (eds.) INAP/WLP 2011. LNCS (LNAI), vol. 7773, pp. 175–192. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  3. 3.
    Benferhat, S., Cayrol, C., Dubois, D., Lang, J., Prade, H.: Inconsistency management and prioritized syntax-based entailment. In: Proceedings of the Thirteenth International Joint Conference on Artificial Intelligence (IJCAI 1993), vol. 1, pp. 640–647. Morgan Kaufmann Publishers, San Francisco (1993)Google Scholar
  4. 4.
    Dubois, D., Prade, H.: Conditional objects as nonmonotonic consequence relations. In: Principles of Knowledge Representation and Reasoning: Proceedings of the Fourth International Conference (KR 1994), pp. 170–177. Morgan Kaufmann Publishers, San Francisco (1996)Google Scholar
  5. 5.
    Dubois, D., Prade, H.: Possibility theory and its applications: where do we stand? In: Kacprzyk, J., Pedrycz, W. (eds.) Springer Handbook of Computational Intelligence, pp. 31–60. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  6. 6.
    Finetti, B.D.: Theory of Probability, vol. 1,2. Wiley, New York (1974)MATHGoogle Scholar
  7. 7.
    Goldszmidt, M., Pearl, J.: On the consistency of defeasible databases. Artif. Intell. 52(2), 121–149 (1991)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Goldszmidt, M., Pearl, J.: Qualitative probabilities for default reasoning, belief revision, and causal modeling. Artif. Intell. 84(1–2), 57–112 (1996)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kern-Isberner, G., Eichhorn, C.: Structural inference from conditional knowledge bases. In: Unterhuber, M., Schurz, G. (eds.) Logic and Probability: Reasoning in Uncertain Environments, pp. 751–769 (2014). No. 102(4) in Studia Logica. Springer, Dordrecht (2014)Google Scholar
  10. 10.
    Kern-Isberner, G.: Conditionals in Nonmonotonic Reasoning and Belief Revision. LNCS (LNAI), vol. 2087. Springer, Heidelberg (2001)CrossRefMATHGoogle Scholar
  11. 11.
    Kern-Isberner, G.: A thorough axiomatization of a principle of conditional preservation in belief revision. Ann. Math. Artif. Intell. 40, 127–164 (2004)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Makinson, D.: General patterns in nonmonotonic reasoning. In: Gabbay, D.M., Hogger, C.J., Robinson, J.A. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 3, pp. 35–110. Oxford University Press, New York (1994)Google Scholar
  13. 13.
    Pearl, J.: System Z: a natural ordering of defaults with tractable applications to nonmonotonic reasoning. In: Proceedings of the 3rd Conference on Theoretical Aspects of Reasoning About Knowledge (TARK1990), pp. 121–135. Morgan Kaufmann Publishers Inc., San Francisco (1990)Google Scholar
  14. 14.
    Spohn, W.: Ordinal conditional functions: a dynamic theory of epistemic states. In: Harper, W.L., Skyrms, B. (eds.) Causation in Decision, Belief Change and Statistics: Proceedings of the Irvine Conference on Probability and Causation. The Western Ontario Series in Philosophy of Science, vol. 42, pp. 105–134. Springer, Dordrecht (1988)CrossRefGoogle Scholar
  15. 15.
    Spohn, W.: The Laws of Belief: Ranking Theory and Its Philosophical Applications. Oxford University Press, Oxford (2012)CrossRefGoogle Scholar
  16. 16.
    Thorn, P.D., Eichhorn, C., Kern-Isberner, G., Schurz, G.: Qualitative probabilistic inference with default inheritance for exceptional subclasses. In: PROGIC 2015: The Seventh Workshop on Combining Probability and Logic (2015)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Christoph Beierle
    • 1
  • Christian Eichhorn
    • 2
  • Gabriele Kern-Isberner
    • 2
  1. 1.Department of Computer ScienceUniversity of HagenHagenGermany
  2. 2.Department of Computer ScienceTU DortmundDortmundGermany

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