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Expressivity of Many-Valued Modal Logics, Coalgebraically

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Logic, Language, Information, and Computation (WoLLIC 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9803))

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Abstract

We apply methods developed to study coalgebraic logic to investigate expressivity of many-valued modal logics which we consider as coalgebraic languages interpreted over set-coalgebras with many-valued valuations. The languages are based on many-valued predicate liftings. We provide a characterization theorem for a language generated by a set of such modalities to be expressive for bisimilarity: in addition to the usual condition on the set of predicate liftings being separating, we indicate a sufficient and sometimes also necessary condition on the algebra of truth values which guarantees expressivity. Thus, adapting results of Schröder [16] concerning expressivity of boolean coalgebraic logics to many-valued setting, we generalize results of Metcalfe and Martí [13], concerning Hennessy-Milner property for many-valued modal logics based on \(\Box \) and \(\diamondsuit \).

M. Bílková—The work of the first author has been supported by the joint project of Austrian Science Fund (FWF) I1897-N25 and Czech Science Foundation (GACR) 15-34650L.

M. Dostál—The work of the second author has been supported by the project No. GA13-14654S of the Czech Science Foundation.

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Notes

  1. 1.

    We would like to stress that we do not include constants for elements of \({\mathscr {V}}\) in the language (cf. Examples 3 and 7).

  2. 2.

    This in fact says that B is a \(T\times {\mathscr {V}}^{{ At }}\)-bisimulation, where the second part of the functor encodes the valuations.

  3. 3.

    In case that \({\mathscr {V}}=2\) separability is in fact sufficient for expressivity. The reason is that the classical propositional logic is functionally complete and each boolean function \(\sigma :2^n\rightarrow 2\) is definable by a formula with n variables (cf. Definition 4).

  4. 4.

    Not to be confused with the double contravariant powerset functor whose coalgebras are neighbourhood frames.

  5. 5.

    cf. Examples 7 and 10. This does not entail expressivity.

  6. 6.

    Defined like this, using the multiplication of reals, the semantics of \(\diamondsuit \) is not expressed by a first-order formula of Łukasziewicz logic.

  7. 7.

    It is straightforward to generalize Theorem 3 to the polyadic setting, and in this particular example we will not need any expressible propositional formulas.

References

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Authors and Affiliations

  1. Institute of Computer Science, The Czech Academy of Sciences, Prague, Czech Republic

    Marta Bílková & Matěj Dostál

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  1. Marta Bílková
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  2. Matěj Dostál
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Correspondence to Matěj Dostál .

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Editors and Affiliations

  1. Dept of Mathematics & Statistics, Univ of Helsinki Dept of Mathematics & Statistics, Helsinki, Finland

    Jouko Väänänen

  2. Department of Mathematics and Statistics, University of Helsinki , Helsinki, Finland

    Åsa Hirvonen

  3. Centro de Informática , Recife, Pernambuco, Brazil

    Ruy de Queiroz

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Bílková, M., Dostál, M. (2016). Expressivity of Many-Valued Modal Logics, Coalgebraically. In: Väänänen, J., Hirvonen, Å., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2016. Lecture Notes in Computer Science(), vol 9803. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-52921-8_8

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