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Modal Logics with Hard Diamond-Free Fragments

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Logical Foundations of Computer Science (LFCS 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9537))

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Abstract

We investigate the complexity of modal satisfiability for certain combinations of modal logics. In particular we examine four examples of multimodal logics with dependencies and demonstrate that even if we restrict our inputs to diamond-free formulas (in negation normal form), these logics still have a high complexity. This result illustrates that having D as one or more of the combined logics, as well as the interdependencies among logics can be important sources of complexity even in the absence of diamonds and even when at the same time in our formulas we allow only one propositional variable. We then further investigate and characterize the complexity of the diamond-free, 1-variable fragments of multimodal logics in a general setting.

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Notes

  1. 1.

    There are numerous semantics for modal logic, but in this paper we only use Kripke semantics.

  2. 2.

    In general, in \(A\oplus _\subseteq B\), if A a bimodal (resp. unimodal) logic, the modalities 1 and 2 (resp. modality 1) come(s) from A and 3 (resp. 2) comes from logic B.

  3. 3.

    It may seem strange that we introduce languages with diamonds and then only consider their diamond-free fragments. When we discuss K, we consider the full language, so we introduce diamonds for \(L_1,L_2,L_3\) for uniformity.

  4. 4.

    We can consider more logics as well, but these ones are enough to make the points we need. Besides, it is not hard to extend the reasoning of this section to other logics (ex. B, S4, KD45 and due to the observation above, also K, K4), especially since the absence of diamonds makes the situation simpler.

  5. 5.

    Frames for D have serial accessibility relations; frames for T have reflexive accessibility relations; frames for D4 have serial and transitive accessibility relations; frames for S5 have accessibility relations that are equivalence relations (reflexive, symmetric, transitive).

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Acknowledgments

The author is grateful to anonymous reviewers, whose input has greatly enhanced the quality of this paper.

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Authors and Affiliations

  1. The Graduate Center of The City University of New York, New York, USA

    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. (2016). Modal Logics with Hard Diamond-Free Fragments. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2016. Lecture Notes in Computer Science(), vol 9537. Springer, Cham. https://doi.org/10.1007/978-3-319-27683-0_1

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  • DOI: https://doi.org/10.1007/978-3-319-27683-0_1

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