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From Simplified Kripke-Style Semantics to Simplified Analytic Tableaux for Some Normal Modal Logics

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AI*IA 2019 – Advances in Artificial Intelligence (AI*IA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11946))

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Abstract

Modal logics \(\mathsf{K45}\), \(\mathsf{KB4}\), \(\mathsf{KD45}\) and \(\mathsf{S5}\) are of particular interest in knowledge representation, especially in the context of knowledge and belief modelling. Pietruszczak showed that these logics are curious for another reason, namely for the fact that their Kripke-style semantics can be simplified. A simplified frame has the form \(\langle W,A\rangle \), where \(A\subseteq W\). A reachability relation R may be defined as \(R=W\times A\), which, however, makes it superfluous to explicitly refer to it. It is well-known that \(\mathsf{S5}\) is determined by Kripke frames with \(R=W\times W\), i.e., \(A=W\). Pietruszczak showed what classes of simplified frames determine \(\mathsf{K45}\), \(\mathsf{KD45}\), and \(\mathsf{KB4}\). These results were generalized to the extensions of these logics by Segerberg’s formulas. In this paper, we devise sound, complete and terminating prefixed tableau algorithms based on simplified semantics for these logics. Since no separate rules are needed to handle the reachability relation and prefixes do not store any extra information, the calculi are accessible and conceptually simple and the process of countermodel-construction out of an open tableau branch is straightforward. Moreover, we obtain a nice explanation of why these logics are computationally easier than most modal logics, in particular \(\textsc {NP}\)-complete.

Research reported in this paper is supported by the National Science Centre, Poland (grant number: DEC-2017/25/B/HS1/01268).

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Notes

  1. 1.

    The assumptions can be read in the following way: A1: John believes in whatever is (logically) true, A2: If John believes that \(\upvarphi \) is true, then he believes that he believes that \(\upvarphi \) is true, A3: If John disbelieves that \(\upvarphi \) is true, then he believes that he disbelieves that \(\upvarphi \) is true. The conclusion can be read as follows: C: John believes that whatever he believes is true.

  2. 2.

    For a thorough survey of prefixed tableau calculi for various modal logics between \(\mathsf{K}\) and \(\mathsf{S5}\) see, e.g., [8, 18].

  3. 3.

    Henceforth, we will refer to these rules as common rules.

  4. 4.

    Note that it is an analogous condition to Technique 9.1 from [18], however using it in the framework of simplified tableaux shows explicitly why it does not violate completeness.

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Acknowledgements

We would like to thank the anonymous reviewers whose comments helped substantially improve this paper.

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  1. Department of Logic, University of Łódź, Łódź, Poland

    Yaroslav Petrukhin & Michał Zawidzki

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  1. Yaroslav Petrukhin
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Correspondence to Michał Zawidzki .

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  1. University of Calabria, Rende, Italy

    Mario Alviano

  2. University of Calabria, Rende, Italy

    Gianluigi Greco

  3. University of Calabria, Rende, Italy

    Francesco Scarcello

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Petrukhin, Y., Zawidzki, M. (2019). From Simplified Kripke-Style Semantics to Simplified Analytic Tableaux for Some Normal Modal Logics. In: Alviano, M., Greco, G., Scarcello, F. (eds) AI*IA 2019 – Advances in Artificial Intelligence. AI*IA 2019. Lecture Notes in Computer Science(), vol 11946. Springer, Cham. https://doi.org/10.1007/978-3-030-35166-3_9

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  • DOI: https://doi.org/10.1007/978-3-030-35166-3_9

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