Abstract
For nonmonotonic reasoning in the context of a knowledge base \(\mathcal {R}\) containing conditionals of the form If A then usually B, system P provides generally accepted axioms. Inference solely based on system P, however, is inherently skeptical because it coincides with reasoning that takes all ranking models of \(\mathcal {R}\) into account. System Z uses only the unique minimal ranking model of \(\mathcal {R}\), and c-inference, realized via a complex constraint satisfaction problem, takes all c-representations of \(\mathcal {R}\) into account. C-representations constitute the subset of all ranking models of \(\mathcal {R}\) that are obtained by assigning non-negative integer impacts to each conditional in \(\mathcal {R}\) and summing up, for every world, the impacts of all conditionals falsified by that world. While system Z and c-inference license in general different sets of desirable entailments, the first major objective of this article is to present system W. System W fully captures and strictly extends both system Z and c-inference. Moreover, system W can be represented by a single strict partial order on the worlds over the signature of \(\mathcal {R}\). We show that system W exhibits further inference properties worthwhile for nonmonotonic reasoning, like satisfying the axioms of system P, respecting conditional indifference, and avoiding the drowning problem. The other main goal of this article is to provide results on our investigations, underlying the development of system W, of upper and lower bounds that can be used to restrict the set of c-representations that have to be taken into account for realizing c-inference. We show that the upper bound of n − 1 is sufficient for capturing c-inference with respect to \(\mathcal {R}\) having n conditionals if there is at least one world verifying all conditionals in \(\mathcal {R}\). In contrast to the previous conjecture that the number of conditionals in \(\mathcal {R}\) is always sufficient, we prove that there are knowledge bases requiring an upper bound of 2n− 1, implying that there is no polynomial upper bound of the impacts assigned to the conditionals in \(\mathcal {R}\) for fully capturing c-inference.
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Adams, E.W.: The logic of conditionals: an application of probability to deductive logic. Synthese library. Springer Science+Business media, dordrecht NL (1975)
Beierle, C., Eichhorn, C., Kern-Isberner, G.: Skeptical inference based on c-representations and its characterization as a constraint satisfaction problem. In: FoIKS-2016, volume 9616 of LNCS, pages 65–82. Springer (2016)
Beierle, C., Eichhorn, C., Kern-Isberner, G., Kutsch, S.: Skeptical, weakly skeptical, and credulous inference based on preferred ranking functions. In: Kaminka, G.A., Fox, M., Bouquet, P., Hüllermeier, E., Dignum, V., Dignum, F., van Harmelen, F. (eds.) Proceedings 22nd European Conference on Artificial Intelligence, ECAI-2016, volume 285 of Frontiers in Artificial Intelligence and Applications, pages 1149–1157. IOS Press (2016)
Beierle, C., Eichhorn, C., Kern-Isberner, G., Kutsch, S.: Properties of skeptical c-inference for conditional knowledge bases and its realization as a constraint satisfaction problem. Ann. Math. Artif. Intell. 83(3-4), 247–275 (2018)
Beierle, C., Eichhorn, C., Kern-Isberner, G., Kutsch, S.: Properties and interrelationships of skeptical, weakly skeptical, and credulous inference induced by classes of minimal models. Artif. Intell., 297 (2021)
Beierle, C., Eichhorn, C., Kutsch, S.: A practical comparison of qualitative inferences with preferred ranking models. KI – Künstliche Intelligenz 31 (1), 41–52 (2017)
Beierle, C., Kern-Isberner, G.: Semantical investigations into nonmonotonic and probabilistic logics. Ann. Math. Artif. Intell. 65(2-3), 123–158 (2012)
Beierle, C., Kutsch, S.: Regular and sufficient bounds of finite domain constraints for skeptical c-inference. In: Benferhat, S., Tabia, K., Ali, M. (eds.) Advances in Artificial Intelligence: From Theory to Practice, volume 10350 of LNCS, pages 477–487. Springer (2017)
Beierle, C., Kutsch, S.: Computation and comparison of nonmonotonic skeptical inference relations induced by sets of ranking models for the realization of intelligent agents. Appl. Intell. 49(1), 28–43 (2019)
Beierle, C., Kutsch, S., Sauerwald, K.: Compilation of static and evolving conditional knowledge bases for computing induced nonmonotonic inference relations. Ann. Math. Artif. Intell. 87(1-2), 5–41 (2019)
Benferhat, S., Cayrol, C., Dubois, D., Lang, J., Prade, H.: Inconsistency Management and Prioritized Syntax-Based Entailment. Inproceedings of the Thirteenth International Joint Conference on Artificial Intelligence (IJCAI’93), Volume 1, Pages 640–647, San Francisco, CA, USA. Morgan Kaufmann Publishers (1993)
Benferhat, S., Dubois, D., Prade, H.: Representing default rules in possibilistic logic. Inproceedings 3th International Conference on Principles of Knowledge Representation and Reasoning KR’92, pp 673–684 (1992)
de Finetti, B.: La prévision, ses lois logiques et ses sources subjectives. Ann. Inst. H. Poincaré 7(1), 1–68 (1937). English translation in Studies in Subjective Probability, ed. H. Kyburg and H.E. Smokler, 1974, 93–158. New York: Wiley & Sons
Dubois, D., Prade, H.: Conditional objects as nonmonotonic consequence relationships. Special Issue on Conditional Event Algebra. IEEE Transactions on Systems Man and Cybernetics 24(12), 1724–1740 (1994)
Dubois, D., Prade, H.: Possibility theory and its applications: Where do we stand?. In: Kacprzyk, J., Pedrycz, W. (eds.) Springer Handbook of Computational Intelligence, pages 31–60. Springer, Berlin (2015)
Goldszmidt, M., Pearl, J.: On the consistency of defeasible databases. Artif. Intell. 52(2), 121–149 (1991)
Goldszmidt, M., Pearl, J.: Qualitative probabilities for default reasoning, belief revision, and causal modeling. Artif. Intell. 84(1-2), 57–112 (1996)
Hawthorne, J., Makinson, D.: The quantitative/qualitative watershed for rules of uncertain inference. Stud. Logica. 86(2), 247–297 (2007)
Kern-Isberner, G.: Conditionals in nonmonotonic reasoning and belief revision, volume 2087 of LNAI Springer (2001)
Kern-Isberner, G.: A thorough axiomatization of a principle of conditional preservation in belief revision. Ann. Math. Artif. Intell. 40(1-2), 127–164 (2004)
Kern-Isberner, G., Beierle, C., Brewka, G.: Syntax splitting = relevance + independence: New postulates for nonmonotonic reasoning from conditional belief bases. In: KR-2020, pp 560–571 (2020)
Kern-Isberner, G., Brewka, G.: Strong syntax splitting for iterated belief revision. Inproceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence, IJCAI 2017, Melbourne, Australia, August 19-25, 2017, pp 1131–1137 (2017)
Kern-Isberner, G., Logica, C.: Eichhorn. Structural inference from conditional knowledge bases. Studia Special Issue Logic and Probability:, Reasoning in Uncertain Environments 102(4), 751–769 (2014)
Komo, C., Beierle, C.: Nonmonotonic inferences with qualitative conditionals based on preferred structures on worlds. In: Schmid, U., Klügl, F., Wolter, D (eds.) KI, Advances in Artificial Intelligence - 43rd German Conference on AI, Bamberg, Germany, September 21-25, 2020, Proceedings, volume 12325 of LNCS, pages 102–115, Springer, 2020 (2020)
Komo, C., Beierle, C.: Upper and lower bounds for finite domain constraints to realize skeptical c-inference over conditional knowledge bases. In: International Symposium on Artificial Intelligence and Mathematics (ISAIM), Fort lauderdale, FL, USA, January 6-8, 2020 (2020)
Kraus, S., Lehmann, D., Magidor, M.: Nonmonotonic reasoning, preferential models and cumulative logics. Artif. Intell. 44, 167–207 (1990)
Kutsch, S., Beierle, C.: InfOCF-Web: An online tool for nonmonotonic reasoning with conditionals and ranking functions. In: Zhou, Z. (ed.) Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence, IJCAI 2021, Virtual Event / Montreal, Canada, 19-27 August 2021, pages 4996–4999. ijcai.org (2021)
Kutsch, S., Beierle, C.: Semantic classification of qualitative conditionals and calculating closures of nonmonotonic inference relations. Int. J. Approx. Reason. 130, 297–313 (2021)
Lehmann, D., Magidor, M.: What does a conditional knowledge base entail Artif. Intell. 55, 1–60 (1992)
Lewis, D.: Counterfactuals. Harvard University Press, Cambridge (1973). Mass.
Makinson, D.: General patterns in nonmonotonic reasoning. In: Gabbay, D., Hogger, C., Robinson, J. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming, volume 3, pages 35–110. Oxford University Press (1994)
Parikh, R.: Beliefs, belief revision, and splitting languages. Logic, Language, and Computation 2, 266–278 (1999)
Pearl, J.: 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), pages 121–135, San Francisco, CA, USA. Morgan Kaufmann Publishers Inc (1990)
Peppas, P., Williams, M.-A., Chopra, S., Foo, N.Y.: Relevance in belief revision. Artif. Intell. 229(1-2), 126–138 (2015)
Spohn, W.: Ordinal Conditional Functions: A Dynamic Theory of Epistemic States. In: Causation in Decision, Belief Change and Statistics: Proceedings of the Irvine Conference on Probability and Causation, volume 42 of The Western Ontario Series in Philosophy of Science, pages 105–134, Dordrecht, NL. Springer Science+Business Media (1988)
Spohn, W.: The laws of belief: ranking theory and its philosophical applications. Oxford univ. press, Oxford UK (2012)
Thorn, P.D., Eichhorn, C., Kern-Isberner, G., Schurz, G.: 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 & Kognition (KIK-2015) co-located with 38th German Conference on Artificial Intelligence (KI-2015), volume 1444 of CEUR Workshop Proceedings (2015)
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Komo, C., Beierle, C. Nonmonotonic reasoning from conditional knowledge bases with system W. Ann Math Artif Intell 90, 107–144 (2022). https://doi.org/10.1007/s10472-021-09777-9
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DOI: https://doi.org/10.1007/s10472-021-09777-9
Keywords
- Conditional
- Conditional knowledge base
- Nonmonotonic inference
- System P
- System Z
- C-representation
- C-inference
- Constraint satisfaction problem
- System W