KI - Künstliche Intelligenz

, Volume 31, Issue 1, pp 41–52 | Cite as

A Practical Comparison of Qualitative Inferences with Preferred Ranking Models

  • Christoph Beierle
  • Christian Eichhorn
  • Steven Kutsch
Technical Contribution


When reasoning qualitatively from a conditional knowledge base, two established approaches are system Z and p-entailment. The latter infers skeptically over all ranking models of the knowledge base, while system Z uses the unique pareto-minimal ranking model for the inference relations. Between these two extremes of using all or just one ranking model, the approach of c-representations generates a subset of all ranking models with certain constraints. Recent work shows that skeptical inference over all c-representations of a knowledge base includes and extends p-entailment. In this paper, we follow the idea of using preferred models of the knowledge base instead of the set of all models as a base for the inference relation. We employ different minimality constraints for c-representations and demonstrate inference relations from sets of preferred c-representations with respect to these constraints. We present a practical tool for automatic c-inference that is based on a high-level, declarative constraint-logic programming approach. Using our implementation, we illustrate that different minimality constraints lead to inference relations that differ mutually as well as from system Z and p-entailment.


Conditional logic Qualitative conditional Default rule Ranking function C-representation C-inference System Z P-entailment 



We are grateful to all our students who have been involved in the implementation of the software systems described here, in particular to Karl Södler, Martin Austen, and Fadil Kallat. We also thank the anonymous reviewers of this article for their detailed and helpful comments.


  1. 1.
    Adams E (1965) The logic of conditionals. Inquiry 8(1–4):166–197CrossRefGoogle Scholar
  2. 2.
    Beierle C, Eichhorn C, Kern-Isberner G (2016) Skeptical inference based on c-representations and its characterization as a constraint satisfaction problem. In: Gyssens M, Simari GR (eds) Proceedings of 9th International Symposium on Foundations of Information and Knowledge Systems, FoIKS 2016, Linz, Austria, March 7–11, 2016. LNCS, vol 9616. Springer, New York, pp 65–82Google Scholar
  3. 3.
    Beierle C, Hermsen R, Kern-Isberner G (2014) Observations on the minimality of ranking functions for qualitative conditional knowledge bases and their computation. In: Proceedings of the 27th International FLAIRS Conference, FLAIRS-2014. AAAI Press, Menlo Park, CA, pp 480–485Google Scholar
  4. 4.
    Beierle C, Kern-Isberner G, Södler K (2013) A declarative approach for computing ordinal conditional functions using constraint logic programming. In: 19th International Conference on Applications of Declarative Programming and Knowledge Management, INAP 2011, and 25th Workshop on Logic Programming, WLP 2011, Wien, Austria, revised selected papers. LNAI, vol 7773. Springer, New York, pp 175–192Google Scholar
  5. 5.
    Benferhat S, Cayrol C, Dubois D, Lang J, Prade H (1993) Inconsistency management and prioritized syntax-based entailment. In: Proceedings of the Thirteenth International Joint Conference on Artificial Intelligence (IJCAI’93), vol 1. Morgan Kaufmann Publishers, San Francisco, pp 640–647Google Scholar
  6. 6.
    Benferhat S, Dubois D, Prade H (1992) Representing default rules in possibilistic logic. In: Proceedings 3rd International Conference on Principles of Knowledge Representation and Reasoning KR’92, pp 673–684Google Scholar
  7. 7.
    DeFinetti B (1974) Theory of probability, vol 1, 2. John Wiley & Sons, New YorkGoogle Scholar
  8. 8.
    Dubois D, Prade H (2015) Possibility theory and its applications: where do we stand? In: Kacprzyk J, Pedrycz W (eds) Springer handbook of computational intelligence. Springer, Berlin, pp 31–60CrossRefGoogle Scholar
  9. 9.
    Goldszmidt M, Pearl J (1991) On the consistency of defeasible databases. Artif Intell 52(2):121–149MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Goldszmidt M, Pearl J (1996) Qualitative probabilities for default reasoning, belief revision, and causal modeling. Artif Intell 84:57–112MathSciNetCrossRefGoogle Scholar
  11. 11.
    Halpern J (2005) Reasoning about uncertainty. MIT Press, CambridgezbMATHGoogle Scholar
  12. 12.
    Kern-Isberner G (2001) Conditionals in nonmonotonic reasoning and belief revision. LNAI, vol 2087. Springer, New YorkGoogle Scholar
  13. 13.
    Kern-Isberner G (2002) Handling conditionals adequately in uncertain reasoning and belief revision. J Appl Non Class Log 12(2):215–237MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kern-Isberner G, Eichhorn C (2014) Structural inference from conditional knowledge bases. In: Unterhuber M, Schurz G (eds) Logic and probability: reasoning in uncertain environments. Studia logica, vol, 102, no 4. Springer Science+Business Media, Dordrecht, pp 751–769Google Scholar
  15. 15.
    Kraus S, Lehmann DJ, Magidor M (1990) Nonmonotonic reasoning, preferential models and cumulative logics. Artif Intell 44(1–2):167–207MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lehmann D (1995) Another perspective on default reasoning. Ann Math Artif Intell 15(1):61–82MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lehmann DJ, Magidor M (1992) What does a conditional knowledge base entail? Artif Intell 55(1):1–60MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Makinson D (1994) General patterns in nonmonotonic reasoning. In: Gabbay DM, Hogger CJ, Robinson JA (eds) Handbook of logic in artificial intelligence and logic programming, vol 3. Oxford University Press, New York, pp 35–110Google Scholar
  19. 19.
    Carlsson M, Ottosson G, Carlson B (1997) An open-ended finite domain constraint solver. In: Glaser H, Hartel PH, Kuchen H (eds) Programming languages: implementations, logics, and programs, (PLILP’97). LNCS, vol 1292. Springer, New York, pp 191–206CrossRefGoogle Scholar
  20. 20.
    Pearl J (1988) Probabilistic reasoning in intelligent systems. Morgan Kaufmann Publishers Inc., San FranciscozbMATHGoogle Scholar
  21. 21.
    Pearl J (1990) System Z: a natural ordering of defaults with tractable applications to nonmonotonic reasoning. In: Parikh R (ed) Proceedings of the 3rd Conference on Theoretical Aspects of Reasoning about Knowledge (TARK1990). Morgan Kaufmann Publishers Inc, San Francisco, pp 121–135Google Scholar
  22. 22.
    Spohn W (1988) Ordinal conditional functions: a dynamic theory of epistemic states. In: Harper W, 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. Springer Science+Business Media, Dordrecht, pp 105–134CrossRefGoogle Scholar
  23. 23.
    Spohn W (2012) The laws of belief: ranking theory and its philosophical applications. Oxford University Press, OxfordCrossRefGoogle Scholar
  24. 24.
    Thorn PD, Eichhorn C, Kern-Isberner G, Schurz G (2015) Qualitative probabilistic inference with default inheritance for exceptional subclasses. In: Beierle C, Kern-Isberner G, Ragni M, Stolzenburg F (eds) Proceedings of the 5th Workshop on Dynamics of Knowledge and Belief (DKB-2015) and the 4th Workshop KI and Kognition (KIK-2015). CEUR Workshop Proceedings, vol 1444Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Christoph Beierle
    • 1
  • Christian Eichhorn
    • 2
  • Steven Kutsch
    • 1
  1. 1.Faculty of Mathematics and Computer ScienceFernUniversität in HagenHagenGermany
  2. 2.Department of Computer ScienceTechnische Universität DortmundDortmundGermany

Personalised recommendations