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Journal of Applied Non-Classical logics 2021年第2期31卷 108-129页

作者： Chalki, Aggeliki Koutras, Costas D. Zikos, Yorgos Computation Reasoning Laboratory (CoReLab) School of Electrical and Computer Engineering National Technical University of Athens Athens Greece Department of Informatics and Telecommunications University of Peloponnese Tripolis Greece Graduate Programme in Algorithms Logic Discrete Mathematics (ALMA) Department of Informatics Telecommunications National and Kapodistrian University of Athens Athens Greece

(Formula presented.) is the modal logic of directed partial pre-orders and/or the modal logic of reflexive and transitive relational frames with a final cluster. It holds a distinguished position in philosophical logic, where it has been advocated as the ‘correct’ logic of knowledge;it has also found interesting applications in the temporal logic of relativistic spacetime and the metamathematics of forcing in set theory. The satisfiability problem for (Formula presented.) is (Formula presented.) -complete: this is a result established in an AiML 2004 paper of Shapirovsky [(2004). On PSPACE-decidability in transitive modal logic. In R. A. Schmidt, I. Pratt-Hartmann, M. Reynolds, & H. Wansing (Eds.), Advances in modal logic 5 (pp. 269–287). King's College Publications] where the complexity classification of (Formula presented.) emerges as a consequence of a very general method for constructing (Formula presented.) decision procedures for transitive modal logics. We provide here a ‘classical’ proof in the standard Halpern-Moses style of Halpern and Moses [(1992). A guide to completeness and complexity for modal logics of knowledge and belief. Artificial Intelligence, 54(2), 319–379]. With little extra work, it is shown that the (Formula presented.) -completeness result extends to (Formula presented.), the multimodal version of (Formula presented.). We prove that the (Formula presented.) -completeness characterisation of monomodal (Formula presented.) persists even if we restrict ourselves to fragments with bounded modal depth, but the problem is (Formula presented.) -complete when it is restricted to formulae with modal depth at most one. The complexity of (Formula presented.) satisfiability for the fragment of the language with a finite number of propositional variables (but unbounded modal depth) remains (Formula presented.) -hard. For a finite language and bounded modal depth, (Formula presented.) -satisfiability can be checked in linear time. © 2021 Informa UK Limi

关键词： complexity of multimodal logics Modal logic S4.2 logic tableaux proof procedures

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A catalog ofweak many-valued modal axioms and their corresponding frame classes

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Journal of Applied Non-Classical logics 2003年第1期13卷 47-71页

作者： Koutras, Costas D. Graduate Programme in Logic Algorithms and Computation M.P.L.A. Dept. of Mathematics University of Athens Zographou 157 84 Panepistimioupolis Greece

In this paper we provide frame definability results for weak versions of classical modal axioms that can be expressed in Fittinǵs many-valued modal languages. These languages were introduced by M. Fitting in the early ́90s and are built on Heyting algebras which serve as the space of truth values. The possible-worlds frames interpreting these languages are directed graphs whose edges are labelled with an element of the underlying Heyting algebra, providing us a form of many-valued accessibility relation. Weak axioms of the form we treat here have been examined from the completeness perspective and further explored for applications in non-monotonic reasoning. Here, we introduce more weak many-valued modal axioms and prove a frame correspondence result for all of them. The classes of corresponding labelled frames possess algebraic properties which are strongly reminiscent of many classical ones, such as the Church-Rosser property, reflexivity, transitivity, partial functionality, etc.

关键词： Correspondence theory Many-valued logic Modal logic

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