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International Workshop on Logic, Language, Information, and Computation

WoLLIC 2016: Logic, Language, Information, and Computation pp 109–124Cite as

Expressivity of Many-Valued Modal Logics, Coalgebraically

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

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 \(\diamondsuit \).

Keywords

  • Many-valued Modal Logics
  • Predicate Liftings
  • Hennessy-Milner Property
  • Bisimilarity
  • Coalgebraic Logic

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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

Authors
  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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