A Transformation System for Unique Minimal Normal Forms of Conditional Knowledge Bases
Conditional knowledge bases consisting of sets of conditionals are used in inductive nonmonotonic reasoning and can represent the defeasible background knowledge of a reasoning agent. For the comparison of the knowledge of different agents, as well as of different approaches to nonmonotonic reasoning, it is beneficial if these knowledge bases are as compact and straightforward as possible. To enable the replacement of a knowledge base \(\mathcal R\) by a simpler, but equivalent knowledge base \(\mathcal R'\), we propose to use the notions of elementwise equivalence or model equivalence for conditional knowledge bases. For elementwise equivalence, we present a terminating and confluent transformation system on conditional knowledge bases yielding a unique normal form for every \(\mathcal R\). We show that an extended version of this transformation system takes model equivalence into account. For both transformation system, we prove that the obtained normal forms are minimal with respect to subset inclusion and the corresponding notion of equivalence.
This work was supported by DFG-Grant KI1413/5-1 to Gabriele Kern-Isberner as part of the priority program “New Frameworks of Rationality” (SPP 1516). Christian Eichhorn is supported by this Grant. We thank the anonymous reviewers for their valuable hints and comments.
- 2.Beierle, C., Eichhorn, C., Kern-Isberner, G.: On transformations and normal forms of conditional knowledge bases. In: Benferhat, S., Tabia, K., Ali, M. (eds.) Proceedings of the 30th International Conference on Industrial, Engineering, Other Applications of Applied Intelligent Systems (IEA/AIE-2017). LNAI, vol. 10350, pp. 488–494. Springer, Heidelberg (2017)Google Scholar
- 4.de Finetti, B.: La prévision, ses lois logiques et ses sources subjectives. Ann. Inst. H. Poincaré 7(1), 1–68 (1937). English translation in Kyburg, H., Smokler, H.E. (eds.): Studies in Subjective Probability, pp. 93–158. Wiley, New York (1974)Google Scholar
- 7.Knuth, D.E., Bendix, P.B.: Simple word problems in universal algebra. In: Leech, J. (ed.) Computational Problems in Abstract Algebra, pp. 263–297. Pergamon Press (1970)Google Scholar
- 8.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 (1990)Google Scholar