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Archive for Mathematical Logic

, Volume 57, Issue 3–4, pp 361–380 | Cite as

Expressivity in chain-based modal logics

  • Michel Marti
  • George Metcalfe
Article

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.

Keywords

Modal logic Many-valued logic Bisimulation Modal equivalence Hennessy–Milner property 

Mathematics Subject Classification

03B45 03B50 03G10 

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References

  1. 1.
    Aglianò, P., Montagna, F.: Varieties of BL-algebras I: general properties. J. Pure Appl. Algebra 181, 105–129 (2003)MathSciNetCrossRefMATHGoogle Scholar
  2. 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)Google Scholar
  3. 3.
    Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2001)CrossRefMATHGoogle Scholar
  4. 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)MathSciNetCrossRefMATHGoogle Scholar
  5. 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)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Caicedo, X., Metcalfe, G., Rodríguez, R., Rogger, J.: Decidability in order-based modal logics. J. Comput. Syst. Sci. 88, 53–73 (2017)CrossRefMATHGoogle Scholar
  7. 7.
    Caicedo, X., Rodríguez, R.: Standard Gödel modal logics. Stud. Log. 94(2), 189–214 (2010)CrossRefMATHGoogle Scholar
  8. 8.
    Caicedo, X., Rodríguez, R.: Bi-modal Gödel logic over [0,1]-valued Kripke frames. J. Logic Comput. 25(1), 37–55 (2015)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cignoli, R., D’Ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning. Trends in Logic, vol. 7. Kluwer, Dordrecht (1999)MATHGoogle Scholar
  10. 10.
    Diaconescu, D., Georgescu, G.: Tense operators on MV-algebras and Łukasiewicz–Moisil algebras. Fundam. Inform. 81(4), 379–408 (2007)MATHGoogle Scholar
  11. 11.
    Diaconescu, D., Metcalfe, G., Schnüriger, L.: A real-valued modal logic. In: Proceedings of AiML 2016, pp. 236–251. College Publications (2016)Google Scholar
  12. 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)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Fitting, M.C.: Many-valued modal logics. Fundam. Inform. 15(3–4), 235–254 (1991)MathSciNetMATHGoogle Scholar
  14. 14.
    Fitting, M.C.: Many-valued modal logics II. Fundam. Inform. 17, 55–73 (1992)MathSciNetMATHGoogle Scholar
  15. 15.
    Godo, L., Hájek, P., Esteva, F.: A fuzzy modal logic for belief functions. Fundam. Inform. 57(2–4), 127–146 (2003)MathSciNetMATHGoogle Scholar
  16. 16.
    Godo, L., Rodríguez,R.: A fuzzy modal logic for similarity reasoning. In: Fuzzy Logic and Soft Computing, pp 33–48. Kluwer (1999)Google Scholar
  17. 17.
    Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)CrossRefMATHGoogle Scholar
  18. 18.
    Hájek, P.: Making fuzzy description logic more general. Fuzzy Sets Syst. 154(1), 1–15 (2005)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Hájek, P., Harmancová, D., Verbrugge, R.: A qualitative fuzzy possibilistic logic. Int. J. Approx. Reason. 12, 1–19 (1995)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Hansoul, G., Teheux, B.: Extending Łukasiewicz logics with a modality: Algebraic approach to relational semantics. Stud. Log. 101(3), 505–545 (2013)CrossRefMATHGoogle Scholar
  21. 21.
    Marti, M., Metcalfe, G.: Hennessy–Milner properties for many-valued modal logics. In: Proceedings of AiML 2014, pp. 407–420. College Publications (2014)Google Scholar
  22. 22.
    McNaughton, R.: A theorem about infinite-valued sentential logic. J. Symb. Logic 16(1), 1–13 (1951)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Metcalfe, G., Olivetti, N.: Towards a proof theory of Gödel modal logics. Log. Methods Comput. Sci. 7(2), 1–27 (2011)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Metcalfe, G., Olivetti, N., Gabbay, D.: Proof Theory for Fuzzy Logics. Springer, Berlin (2008)MATHGoogle Scholar
  25. 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. 26.
    Priest, G.: Many-valued modal logics: a simple approach. Rev. Symb. Log. 1, 190–203 (2008)MathSciNetMATHGoogle Scholar
  27. 27.
    Schockaert, S., De Cock, M., Kerre, E.: Spatial reasoning in a fuzzy region connection calculus. Artif. Intell. 173(2), 258–298 (2009)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Straccia, U.: Reasoning within fuzzy description logics. J. Artif. Intell. Res. 14, 137–166 (2001)MathSciNetMATHGoogle Scholar
  29. 29.
    Vidal, A., Godo, L., Esteva, F.: On modal extensions of product fuzzy logic. J. Log. Comput. 27(1), 299–336 (2017)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Institute of Computer ScienceUniversity of BernBernSwitzerland
  2. 2.Mathematical InstituteUniversity of BernBernSwitzerland

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