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The Completeness Problem for Modal Logic

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Logical Foundations of Computer Science (LFCS 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10703))

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Abstract

We introduce the completeness problem for Modal Logic and examine its complexity. For a definition of completeness for formulas, given a formula of a modal logic, the completeness problem asks whether the formula is complete for that logic. We discover that completeness and validity have the same complexity — with certain exceptions for which there are, in general, no complete formulas. To prove upper bounds, we present a non-deterministic polynomial-time procedure with an oracle from PSPACE that combines tableaux and a test for bisimulation, and determines whether a formula is complete.

This research was partly supported by the project “TheoFoMon: Theoretical Foundations for Monitorability” (grant number: 163406-051) of the Icelandic Research Fund.

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Notes

  1. 1.

    According to our definition, for a finite model \(\mathcal {M}=(W,R,V)\) and \(a \in W\), V(a) can be infinite. However, we are mainly interested in \((W,R,V_P)\) for finite sets of propositions P, which justifies calling \(\mathcal {M}\) finite.

  2. 2.

    Although for the purposes of this paper we only consider a specific set of modal logics, it is interesting to note that the corollary can be extended to a much larger class of logics.

  3. 3.

    This is also a corollary of Lemma 4, as these are extensions of D and T.

  4. 4.

    We note that US is different from UP; for UP, if T has an accepting path for x, then it is guaranteed that it has a unique accepting path for x.

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Acknowledgments

The author is grateful to Luca Aceto for valuable comments that helped improve the quality of this paper.

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Authors and Affiliations

  1. School of Computer Science, Reykjavik University, Reykjavik, Iceland

    Antonis Achilleos

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  1. Antonis Achilleos
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Correspondence to Antonis Achilleos .

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Editors and Affiliations

  1. City University of New York, New York, New York, USA

    Sergei Artemov

  2. Cornell University, Ithaca, New York, USA

    Anil Nerode

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Achilleos, A. (2018). The Completeness Problem for Modal Logic. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2018. Lecture Notes in Computer Science(), vol 10703. Springer, Cham. https://doi.org/10.1007/978-3-319-72056-2_1

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  • DOI: https://doi.org/10.1007/978-3-319-72056-2_1

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