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Approximations of System W Between c-Inference, System Z, and Lexicographic Inference

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Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2023)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 14294))

Abstract

Inductive inference operators have been introduced to formalize the process of completing a conditional belief base to a full inference relation. In this paper, we investigate the approximation of inductive inference operator system W with combinations of system Z (or equivalently rational closure) and c-inference, both of which are known to be extended by system W. We introduce general functions for generating inductive inference operators, the combination of two inductive inference operators by union, and the completion of an inductive inference operator by an arbitrary set of axioms. We construct the least inductive inference operator extending system Z and c-inference which, however, does not satisfy system P. We also construct the least inductive inference operator extending system Z and c-inference that also satisfies system P and show that it is strictly extended by system W. Furthermore, we develop approximations that extend system W and introduce an inductive inference operator that strictly extends system W and that is strictly extended by lexicographic inference. This leads to a map of inference relations between rational closure and c-inference on the one side and lexicographic inference on the other side.

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References

  1. Adams, E.W.: The Logic of Conditionals: An Application of Probability to Deductive Logic. Springer Science+Business Media, Dordrecht, NL, Synthese Library (1975)

    Google Scholar 

  2. Beierle, C., Eichhorn, C., Kern-Isberner, G.: Skeptical inference based on c-representations and its characterization as a constraint satisfaction problem. In: Gyssens, M., Simari, G. (eds.) FoIKS 2016. LNCS, vol. 9616, pp. 65–82. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-30024-5_4

    Chapter  MATH  Google Scholar 

  3. 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)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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, 103489 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  5. Beierle, C., Haldimann, J., Kollar, D., Sauerwald, K., Schwarzer, L.: An implementation of nonmonotonic reasoning with system W. In: Bergmann, R., Malburg, L., Rodermund, S.C., Timm, I.J. (eds.) KI 2022. LNCS, vol. 13404, pp. 1–8. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-15791-2_1

    Chapter  Google Scholar 

  6. Casini, G., Meyer, T., Moodley, K., Nortjé, R.: Relevant closure: a new form of defeasible reasoning for description logics. In: Fermé, E., Leite, J. (eds.) JELIA 2014. LNCS (LNAI), vol. 8761, pp. 92–106. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-11558-0_7

    Chapter  MATH  Google Scholar 

  7. Casini, G., Meyer, T., Varzinczak, I.: Taking defeasible entailment beyond rational closure. In: Calimeri, F., Leone, N., Manna, M. (eds.) JELIA 2019. LNCS (LNAI), vol. 11468, pp. 182–197. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-19570-0_12

    Chapter  Google Scholar 

  8. de Finetti, B.: La prévision, ses lois logiques et ses sources subjectives. Ann. Inst. H. Poincaré 7(1), 1–68 (1937). Engl. transl. Theory of Probability, J. Wiley & Sons (1974)

    Google Scholar 

  9. Giordano, L., Gliozzi, V.: Reasoning about multiple aspects in DLs: Semantics and closure construction. CoRR abs/1801.07161 (2018). http://arxiv.org/abs/1801.07161

  10. Giordano, L., Gliozzi, V.: Reasoning about exceptions in ontologies: from the lexicographic closure to the skeptical closure. Fundam. Inf. 176(3–4), 235–269 (2020). https://doi.org/10.3233/FI-2020-1973

    Article  MathSciNet  MATH  Google Scholar 

  11. Giordano, L., Gliozzi, V.: A reconstruction of multipreference closure. Artif. Intell. 290, 103398 (2021). https://doi.org/10.1016/j.artint.2020.103398

    Article  MathSciNet  MATH  Google Scholar 

  12. Goldszmidt, M., Pearl, J.: Qualitative probabilities for default reasoning, belief revision, and causal modeling. Artif. Intell. 84, 57–112 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Haldimann, J., Beierle, C.: Characterizing multipreference closure with system W. In: de Saint-Cyr, F.D., Öztürk-Escoffier, M., Potyka, N. (eds.) SUM 2022. LNCS, vol. 13562, pp. 79–91. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-18843-5_6

    Chapter  Google Scholar 

  14. Haldimann, J., Beierle, C.: Inference with system W satisfies syntax splitting. In: Kern-Isberner, G., Lakemeyer, G., Meyer, T. (eds.) Proceedings of the 19th International Conference on Principles of Knowledge Representation and Reasoning, KR 2022, Haifa, Israel. 31 July–5 August 2022, pp. 405–409 (2022)

    Google Scholar 

  15. Haldimann, J., Beierle, C.: Properties of system W and its relationships to other inductive inference operators. In: Varzinczak, I. (ed.) FoIKS 2022. LNCS, vol. 13388, pp. 206–225. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-11321-5_12

    Chapter  MATH  Google Scholar 

  16. Haldimann, J., Beierle, C., Kern-Isberner, G.: Extending c-representations and c-inference for reasoning with infeasible worlds. In: Sauerwald, K., Thimm, M. (eds.) 21st International Workshop on Non-Monotonic Reasoning, 2–4 September 2023, Rhodes, Greece. CEUR Workshop Proceedings, CEUR-WS.org (2023)

    Google Scholar 

  17. Haldimann, J., Beierle, C., Kern-Isberner, G., Meyer, T.: Conditionals, infeasible worlds, and reasoning with system W. In: The International FLAIRS Conference Proceedings, vol. 36, no. 1 (2023)

    Google Scholar 

  18. Heyninck, J., Kern-Isberner, G., Meyer, T.A., Haldimann, J., Beierle, C.: Conditional syntax splitting for non-monotonic inference operators. In: The 37th AAAI Conference on Artificial Intelligence (2023)

    Google Scholar 

  19. 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)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kern-Isberner, G.: Conditionals in nonmonotonic reasoning and belief revision, LNAI, vol. 2087. Springer (2001). https://doi.org/10.1007/3-540-44600-1

  21. Kern-Isberner, G., Beierle, C., Brewka, G.: Syntax splitting = relevance + independence: new postulates for nonmonotonic reasoning from conditional belief bases. In: Calvanese, D., Erdem, E., Thielscher, M. (eds.) Principles of Knowledge Representation and Reasoning: Proceedings of the 17th International Conference, KR 2020, pp. 560–571. IJCAI Organization (2020)

    Google Scholar 

  22. Komo, C., Beierle, C.: Nonmonotonic inferences with qualitative conditionals based on preferred structures on worlds. In: Schmid, U., Klügl, F., Wolter, D. (eds.) KI 2020. LNCS (LNAI), vol. 12325, pp. 102–115. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-58285-2_8

    Chapter  Google Scholar 

  23. Komo, C., Beierle, C.: Nonmonotonic reasoning from conditional knowledge bases with system W. Ann. Math. Artif. Intell. 90(1), 107–144 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kraus, S., Lehmann, D., Magidor, M.: Nonmonotonic reasoning, preferential models and cumulative logics. Artif. Intell. 44(1–2), 167–207 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lehmann, D., Magidor, M.: What does a conditional knowledge base entail? Artif. Intell. 55, 1–60 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lehmann, D.: Another perspective on default reasoning. Ann. Math. Artif. Intell. 15(1), 61–82 (1995). https://doi.org/10.1007/BF01535841

    Article  MathSciNet  MATH  Google Scholar 

  27. 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 (1994)

    Google Scholar 

  28. 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 (TARK 1990), pp. 121–135. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (1990)

    Google Scholar 

  29. 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 

  30. Tönnies, D.: Implementierung und empirische Untersuchung lexikographischer Inferenz für das nichtmonotone Schließen. Bachelor thesis, FernUniversität in Hagen, Germany (2022), (in German)

    Google Scholar 

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Acknowledgments

We are grateful to the anonymous reviewers of this paper for their valuable hints and comments. This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), grant BE 1700/10-1 awarded to Christoph Beierle as part of the priority program “Intentional Forgetting in Organizations” (SPP 1921). Jonas Haldimann was supported by this grant.

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Authors and Affiliations

  1. Knowledge-Based Systems, Faculty of Mathematics and Computer Science, FernUniversität in Hagen, 58084, Hagen, Germany

    Jonas Haldimann & Christoph Beierle

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  1. Jonas Haldimann
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  2. Christoph Beierle
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Correspondence to Jonas Haldimann .

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Editors and Affiliations

  1. CRIL CNRS University Artois, Lens Cedex, France

    Zied Bouraoui

  2. CRIL CNRS University Artois, Lens Cedex, France

    Srdjan Vesic

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Haldimann, J., Beierle, C. (2024). Approximations of System W Between c-Inference, System Z, and Lexicographic Inference. In: Bouraoui, Z., Vesic, S. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2023. Lecture Notes in Computer Science(), vol 14294. Springer, Cham. https://doi.org/10.1007/978-3-031-45608-4_15

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  • DOI: https://doi.org/10.1007/978-3-031-45608-4_15

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