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Craig Interpolation of Epistemic Logics with Distributed Knowledge

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Foundations of Information and Knowledge Systems (FoIKS 2020)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 12012))

Abstract

Distributed Knowledge among agents is an important topic in multi-agent systems. While semantic studies of distributed knowledge have been done by several authors in the context of epistemic logic, there are a few proof-theoretic studies. This paper provides cut-free Gentzen-style sequent calculi for epistemic logics with distributed knowledge and establishes Craig Interpolation Theorem for the logics by a constructive method, i.e., Maehara method.

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Notes

  1. 1.

    Fagin et al. [3, p. 3] state as intuitive description for distributed knowledge “a group has distributed knowledge of a fact \(\varphi \) if the knowledge of \(\varphi \) is distributed among its members, so that by pooling their knowledge together the members of the group can deduce \(\varphi \)”. This seems clearer, at first sight, than the explanation we give here. Ågotnes et al. [1] states, however, that the above intuitive description is inappropriate by an illustrative example given in [1, Section 1].

  2. 2.

    In case 4, we assume the condition for both rule applications, because if the one of the two rule applications does not satisfy the condition, the whole derivation should be categorized into one of the rest cases.

  3. 3.

    Note that the condition \(\bigcup _{i=1}^n G_i \cup \bigcup _{j=1}^m H_j\subseteq H\) in \(\mathcal {E}\) can be obtained by the conditions \(\bigcup _{i=1}^n G_i \subseteq G\) and \(G \cup \bigcup _{j=1}^m H_j\subseteq H\) in \(\mathcal {D}\) through “cutting” G.

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Acknowledgement

We thank three reviewers of FoIKS 2020 for their helpful comments. The work of both authors was partially supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (C) Grant Number 19K12113 and JSPS Core-to-Core Program (A. Advanced Research Networks). The second author was also partially supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (B) Grant Number 17H02258.

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Authors and Affiliations

  1. Graduate School of Humanities and Human Sciences, Hokkaido University, Sapporo, Japan

    Ryo Murai

  2. Faculty of Humanities and Human Sciences, Hokkaido University, Sapporo, Japan

    Katsuhiko Sano

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  1. Ryo Murai
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  2. Katsuhiko Sano
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Correspondence to Ryo Murai .

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Editors and Affiliations

  1. CNRS, University of Toulouse, Toulouse, France

    Andreas Herzig

  2. University of Helsinki, Helsinki, Finland

    Juha Kontinen

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Murai, R., Sano, K. (2020). Craig Interpolation of Epistemic Logics with Distributed Knowledge. In: Herzig, A., Kontinen, J. (eds) Foundations of Information and Knowledge Systems. FoIKS 2020. Lecture Notes in Computer Science(), vol 12012. Springer, Cham. https://doi.org/10.1007/978-3-030-39951-1_13

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  • DOI: https://doi.org/10.1007/978-3-030-39951-1_13

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-39950-4

  • Online ISBN: 978-3-030-39951-1

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