Expressivity in chain-based modal logics

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.

[1]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[2]  D. Mundici,et al.  Algebraic Foundations of Many-Valued Reasoning , 1999 .

[3]  Melvin Fitting,et al.  Many-valued modal logics II , 1992 .

[4]  Lluís Godo,et al.  A Fuzzy Modal Logic for Similarity Reasoning , 1999 .

[5]  George Metcalfe,et al.  A Hennessy-Milner Property for Many-Valued Modal Logics , 2014, Advances in Modal Logic.

[6]  Petr Hájek,et al.  A qualitative fuzzy possibilistic logic , 1995, Int. J. Approx. Reason..

[7]  Xavier Caicedo,et al.  Standard Gödel Modal Logics , 2010, Stud Logica.

[8]  P. Aglianò,et al.  Varieties of BL- algebras I: General properties. , 2003 .

[9]  M. de Rijke,et al.  Modal Logic , 2001, Cambridge Tracts in Theoretical Computer Science.

[10]  Martine De Cock,et al.  Spatial reasoning in a fuzzy region connection calculus , 2009, Artif. Intell..

[11]  George Metcalfe,et al.  A Real-Valued Modal Logic , 2018, Log. Methods Comput. Sci..

[12]  Francesco Paoli,et al.  Ordered Algebras and Logic , 2010 .

[13]  Lluis Godo,et al.  On the Minimum Many-Valued Modal Logic over a Finite Residuated Lattice , 2008, J. Log. Comput..

[14]  Xavier Caicedo,et al.  Bi-modal Gödel logic over [0, 1]-valued Kripke frames , 2011, J. Log. Comput..

[15]  Bruno Teheux,et al.  Extending Łukasiewicz Logics with a Modality: Algebraic Approach to Relational Semantics , 2013, Stud Logica.

[16]  Petr Hájek,et al.  Metamathematics of Fuzzy Logic , 1998, Trends in Logic.

[17]  Xavier Caicedo,et al.  Decidability of Order-Based Modal Logics , 2016 .

[18]  Petr Hájek,et al.  A Fuzzy Modal Logic for Belief Functions , 2001, Fundam. Informaticae.

[19]  Marta Bílková,et al.  Expressivity of Many-Valued Modal Logics, Coalgebraically , 2016, WoLLIC.

[20]  Robert McNaughton,et al.  A Theorem About Infinite-Valued Sentential Logic , 1951, J. Symb. Log..

[21]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[22]  Graham Priest,et al.  MANY-VALUED MODAL LOGICS: A SIMPLE APPROACH , 2008, The Review of Symbolic Logic.

[23]  George Georgescu,et al.  Tense Operators on MV-Algebras and Lukasiewicz-Moisil Algebras , 2007, Fundam. Informaticae.

[24]  DiaconescuDenisa,et al.  Tense Operators on MV-Algebras and ukasiewicz-Moisil Algebras , 2007 .

[25]  Umberto Straccia,et al.  Reasoning within Fuzzy Description Logics , 2011, J. Artif. Intell. Res..

[26]  Rafael Peñaloza,et al.  The limits of decidability in fuzzy description logics with general concept inclusions , 2015, Artif. Intell..

[27]  Lluis Godo,et al.  On modal extensions of Product fuzzy logic , 2017, J. Log. Comput..

[28]  D. Gabbay,et al.  Proof Theory for Fuzzy Logics , 2008 .

[29]  Melvin Fitting,et al.  Many-valued modal logics , 1991, Fundam. Informaticae.

[30]  Nicola Olivetti,et al.  Towards a Proof Theory of Gödel Modal Logics , 2011, Log. Methods Comput. Sci..

[31]  Nick Bezhanishvili,et al.  Finitely generated free Heyting algebras via Birkhoff duality and coalgebra , 2011, Log. Methods Comput. Sci..

[32]  Petr Hájek,et al.  Making fuzzy description logic more general , 2005, Fuzzy Sets Syst..

[33]  Christos Nomikos,et al.  Notions of Bisimulation for Heyting-Valued Modal Languages , 2012, J. Log. Comput..