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2025 Volume 40
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RESEARCH ARTICLE   Open Access    

Inductive inference from weakly consistent belief bases

  • Institute for Logic and Computation, TU Wien, 1040 Vienna, Austria

  • University of Cape Town and CAIR, 7700 Cape Town, South Africa

  • Department of Mathematics and Computer Science, FernUniversität in Hagen, 58084 Hagen, Germany

  • Department of Computer Science, TU Dortmund University, 44227 Dortmund, Germany

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  • Abstract: We consider nonmonotonic inferences from belief bases that contain conditionals enforcing some of the possible worlds to be infeasible and thus completely implausible. In contrast to belief bases satisfying the strong notion of consistency requiring every world to be at least somewhat plausible, we call such belief bases weakly consistent. First, we review the treatment of weakly consistent belief bases by the seminal approaches of p-entailment, which coincides with system P, and of system Z, which coincides with rational closure. Then we focus on c-inference, an inductive inference operator that has been shown to exhibit many desirable properties put forward for nonmonotonic reasoning. It is based on c-representations, which are a special kind of ranking model ordering worlds according to their plausibility. While c-representation is defined for strongly consistent belief bases only, in this article, we extend the notions of c-representation and of c-inference to cover also weakly consistent belief bases. We adapt a constraint satisfaction problem (CSP) characterizing c-representations to capture extended c-representations, and we show how this extended CSP can be used to characterize extended c-inference, providing a basis for its implementation. We show various properties of extended c-inference and in particular, we prove that also the extended notion of c-inference fully satisfies syntax splitting. Furthermore, we extend and evaluate credulous and weakly skeptical c-inference to weakly consistent belief bases and provide characterizations for them as CSPs.
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  • Cite this article

    Jonas Philipp Haldimann, Christoph Beierle, Gabriele Kern-Isberner. 2025. Inductive inference from weakly consistent belief bases. The Knowledge Engineering Review 40(1), doi: 10.1017/S0269888925100088
    Jonas Philipp Haldimann, Christoph Beierle, Gabriele Kern-Isberner. 2025. Inductive inference from weakly consistent belief bases. The Knowledge Engineering Review 40(1), doi: 10.1017/S0269888925100088
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RESEARCH ARTICLE   Open Access    

Inductive inference from weakly consistent belief bases

  • Institute for Logic and Computation, TU Wien, 1040 Vienna, Austria

  • University of Cape Town and CAIR, 7700 Cape Town, South Africa

  • Department of Mathematics and Computer Science, FernUniversität in Hagen, 58084 Hagen, Germany

  • Department of Computer Science, TU Dortmund University, 44227 Dortmund, Germany

  • Received: 26 December 2024
  • Revised: 25 October 2025
  • Accepted: 29 October 2025
  • Published online: 03 December 2025
The Knowledge Engineering Review  40,  2025, 40(1)  |  Cite this article

Abstract: Abstract: We consider nonmonotonic inferences from belief bases that contain conditionals enforcing some of the possible worlds to be infeasible and thus completely implausible. In contrast to belief bases satisfying the strong notion of consistency requiring every world to be at least somewhat plausible, we call such belief bases weakly consistent. First, we review the treatment of weakly consistent belief bases by the seminal approaches of p-entailment, which coincides with system P, and of system Z, which coincides with rational closure. Then we focus on c-inference, an inductive inference operator that has been shown to exhibit many desirable properties put forward for nonmonotonic reasoning. It is based on c-representations, which are a special kind of ranking model ordering worlds according to their plausibility. While c-representation is defined for strongly consistent belief bases only, in this article, we extend the notions of c-representation and of c-inference to cover also weakly consistent belief bases. We adapt a constraint satisfaction problem (CSP) characterizing c-representations to capture extended c-representations, and we show how this extended CSP can be used to characterize extended c-inference, providing a basis for its implementation. We show various properties of extended c-inference and in particular, we prove that also the extended notion of c-inference fully satisfies syntax splitting. Furthermore, we extend and evaluate credulous and weakly skeptical c-inference to weakly consistent belief bases and provide characterizations for them as CSPs.

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    Acknowledgments

    • This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—512363537, grant BE 1700/12-1 awarded to Christoph Beierle.

    • This work was partially supported by the Austrian Science Fund (FWF) projects P30873, PIN8884924, and the FWF and netidee SCIENCE project T1349-N.

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    • This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
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    Jonas Philipp Haldimann, Christoph Beierle, Gabriele Kern-Isberner. 2025. Inductive inference from weakly consistent belief bases. The Knowledge Engineering Review 40(1), doi: 10.1017/S0269888925100088
    Jonas Philipp Haldimann, Christoph Beierle, Gabriele Kern-Isberner. 2025. Inductive inference from weakly consistent belief bases. The Knowledge Engineering Review 40(1), doi: 10.1017/S0269888925100088
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