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Expressivity in chain-based modal logics

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Abstract

We investigate the expressivity of many-valued modal logics based on an algebraic structure with a complete linearly ordered lattice reduct. Necessary and sufficient algebraic conditions for admitting a suitable Hennessy–Milner property are established for classes of image-finite and (appropriately defined) modally saturated models. Full characterizations are obtained for many-valued modal logics based on complete BL-chains that are finite or have the real unit interval [0, 1] as a lattice reduct, including Łukasiewicz, Gödel, and product modal logics.

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References

  1. Aglianò, P., Montagna, F.: Varieties of BL-algebras I: general properties. J. Pure Appl. Algebra 181, 105–129 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bílková, M., Dostal, M.: Expressivity of many-valued modal logics, coalgebraically. In: Proceedings of WoLLIC 2016, volume 9803 of LNCS, pp. 109–124. Springer (2016)

  3. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  4. Borgwardt, S., Distel, F., Peñaloza, R.: The limits of decidability in fuzzy description logics with general concept inclusions. Artif. Intell. 218, 23–55 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bou, F., Esteva, F., Godo, L., Rodríguez, R.: On the minimum many-valued logic over a finite residuated lattice. J. Logic Comput. 21(5), 739–790 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Caicedo, X., Metcalfe, G., Rodríguez, R., Rogger, J.: Decidability in order-based modal logics. J. Comput. Syst. Sci. 88, 53–73 (2017)

    Article  MATH  Google Scholar 

  7. Caicedo, X., Rodríguez, R.: Standard Gödel modal logics. Stud. Log. 94(2), 189–214 (2010)

    Article  MATH  Google Scholar 

  8. Caicedo, X., Rodríguez, R.: Bi-modal Gödel logic over [0,1]-valued Kripke frames. J. Logic Comput. 25(1), 37–55 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cignoli, R., D’Ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning. Trends in Logic, vol. 7. Kluwer, Dordrecht (1999)

    MATH  Google Scholar 

  10. Diaconescu, D., Georgescu, G.: Tense operators on MV-algebras and Łukasiewicz–Moisil algebras. Fundam. Inform. 81(4), 379–408 (2007)

    MATH  Google Scholar 

  11. Diaconescu, D., Metcalfe, G., Schnüriger, L.: A real-valued modal logic. In: Proceedings of AiML 2016, pp. 236–251. College Publications (2016)

  12. Eleftheriou, P.E., Koutras, C.D., Nomikos, C.: Notions of bisimulation for Heyting-valued modal languages. J. Logic Comput. 22(2), 213–235 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fitting, M.C.: Many-valued modal logics. Fundam. Inform. 15(3–4), 235–254 (1991)

    MathSciNet  MATH  Google Scholar 

  14. Fitting, M.C.: Many-valued modal logics II. Fundam. Inform. 17, 55–73 (1992)

    MathSciNet  MATH  Google Scholar 

  15. Godo, L., Hájek, P., Esteva, F.: A fuzzy modal logic for belief functions. Fundam. Inform. 57(2–4), 127–146 (2003)

    MathSciNet  MATH  Google Scholar 

  16. Godo, L., Rodríguez,R.: A fuzzy modal logic for similarity reasoning. In: Fuzzy Logic and Soft Computing, pp 33–48. Kluwer (1999)

  17. Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)

    Book  MATH  Google Scholar 

  18. Hájek, P.: Making fuzzy description logic more general. Fuzzy Sets Syst. 154(1), 1–15 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hájek, P., Harmancová, D., Verbrugge, R.: A qualitative fuzzy possibilistic logic. Int. J. Approx. Reason. 12, 1–19 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hansoul, G., Teheux, B.: Extending Łukasiewicz logics with a modality: Algebraic approach to relational semantics. Stud. Log. 101(3), 505–545 (2013)

    Article  MATH  Google Scholar 

  21. Marti, M., Metcalfe, G.: Hennessy–Milner properties for many-valued modal logics. In: Proceedings of AiML 2014, pp. 407–420. College Publications (2014)

  22. McNaughton, R.: A theorem about infinite-valued sentential logic. J. Symb. Logic 16(1), 1–13 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  23. Metcalfe, G., Olivetti, N.: Towards a proof theory of Gödel modal logics. Log. Methods Comput. Sci. 7(2), 1–27 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Metcalfe, G., Olivetti, N., Gabbay, D.: Proof Theory for Fuzzy Logics. Springer, Berlin (2008)

    MATH  Google Scholar 

  25. Metcalfe, G., Paoli, F., Tsinakis, C.: Ordered algebras and logic. In: Hosni, H., Montagna, F. (eds.) Uncertainty and Rationality, vol. 10, pp. 1–85. Publications of the Scuola Normale Superiore di Pisa, Pisa (2010)

    Google Scholar 

  26. Priest, G.: Many-valued modal logics: a simple approach. Rev. Symb. Log. 1, 190–203 (2008)

    MathSciNet  MATH  Google Scholar 

  27. Schockaert, S., De Cock, M., Kerre, E.: Spatial reasoning in a fuzzy region connection calculus. Artif. Intell. 173(2), 258–298 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Straccia, U.: Reasoning within fuzzy description logics. J. Artif. Intell. Res. 14, 137–166 (2001)

    MathSciNet  MATH  Google Scholar 

  29. Vidal, A., Godo, L., Esteva, F.: On modal extensions of product fuzzy logic. J. Log. Comput. 27(1), 299–336 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Computer Science, University of Bern, Neubrückstrasse 10, 3012, Bern, Switzerland

    Michel Marti

  2. Mathematical Institute, University of Bern, Sidlerstrasse 5, 3012, Bern, Switzerland

    George Metcalfe

Authors
  1. Michel Marti
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  2. George Metcalfe
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Corresponding author

Correspondence to George Metcalfe.

Additional information

Preliminary results from this work were reported in the proceedings of AiML 2014 [21].

Supported by Swiss National Science Foundation Grant 200021_165850.

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Marti, M., Metcalfe, G. Expressivity in chain-based modal logics. Arch. Math. Logic 57, 361–380 (2018). https://doi.org/10.1007/s00153-017-0573-4

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  • DOI: https://doi.org/10.1007/s00153-017-0573-4

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