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Properties of skeptical c-inference for conditional knowledge bases and its realization as a constraint satisfaction problem

  • Christoph Beierle
  • Christian Eichhorn
  • Gabriele Kern-Isberner
  • Steven Kutsch
Article

Abstract

While the axiomatic system P is an important standard for plausible, nonmonotonic inferences from conditional knowledge bases, it is known to be too weak to solve benchmark problems like Irrelevance or Subclass Inheritance. Ordinal conditional functions provide a semantic base for system P and 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 consider the skeptical inference relation, called c-inference, that is obtained by taking all c-representations of a given knowledge base into account. We study properties of c-inference and show in particular that it preserves the properties of solving Irrelevance and Subclass Inheritance. Based on a characterization of c-representations as solutions of a Constraint Satisfaction Problem (CSP), we also model skeptical c-inference as a CSP and prove soundness and completeness of the modelling, ensuring that constraint solvers can be used for implementing c-inference.

Keywords

Conditional Conditional knowledge base System P System Z C-representation C-inference Constraint satisfaction problem 

Mathematics Subject Classification (2010)

68T27 68T30 68T37 

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Notes

Acknowledgments

This work was supported by DFG-Grant KI1413/5-1 of Prof. Dr. Gabriele Kern-Isberner as part of the priority program “New Frameworks of Rationality” (SPP1516). Christian Eichhorn is supported by this Grant. This work benefitted very much from discussions led during Dagstuhl Seminar 15221 “Multi-disciplinary approaches to reasoning with imperfect information and knowledge - a synthesis and a roadmap of challenges”.

We thank all our students who have been involved in the implementation of the software system InfOCF, in particular Karl Södler, Martin Austen, Matthias Wirths, and Fadil Kallat. We are also very grateful to the anonymous referees of this article for their detailed and helpful comments.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Christoph Beierle
    • 1
  • Christian Eichhorn
    • 2
  • Gabriele Kern-Isberner
    • 2
  • Steven Kutsch
    • 1
  1. 1.Department of Mathematics and Computer ScienceUniversity of HagenHagenGermany
  2. 2.Department of Computer ScienceTU Dortmund UniversityDortmundGermany

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