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A Duality for the Algebras of a Łukasiewicz n + 1-valued Modal System

Bruno Teheux
2007 Studia Logica: An International Journal for Symbolic Logic  
In this paper, we develop a duality for the varieties of a Lukasiewicz n + 1-valued modal system. This duality is an extension of Stone duality for modal algebras.  ...  Some logical consequences (such as completeness results, correspondence theory. . . ) are the derived and we propose some ideas for future research. 2000 Mathematics Subject Classification. 06D35, 03G25  ...  Introduction When one looks backwards in the history of modern logic, one can notice that many-valued logics and modal logics arose approximatively at the same time.  ... 
doi:10.1007/s11225-007-9074-5 fatcat:pykzs7vjhvgwnkzysgmiot34cu

Logic Column 15: Coalgebras and Their Logics [article]

Alexander Kurz
2006 arXiv   pre-print
By giving a suitably abstract description of transition systems as coalgebras, it is possible to derive logics for capturing properties of these transition systems in an elegant way.  ...  Lifting such a basic duality to a duality of coalgebras and modal algebras, as indicated in Diagram 3, will provide modal logics for coalgebras.  ...  Compared to [25] , we use the duality of algebras and coalgebras to lift the Stone duality from Boolean logic to modal logic.  ... 
arXiv:cs/0605128v1 fatcat:tkr62ke6rnaohapvpl4tz6elle

Duality for Logics of Transition Systems [chapter]

Marcello M. Bonsangue, Alexander Kurz
2005 Lecture Notes in Computer Science  
It has been used, for example, in the ground breaking work of Jónnson and Tarski [15] and Goldblatt [11] in modal logic and Abramsky [1, 2] in domain theory.  ...  The logical meaning of the duality is that the logic is sound and complete and expressive (or fully abstract) in the sense that non-bisimilar states are distinguished by some formula.  ...  Soundness and completeness of an infinitary modal logic for transition systems has been proved in [7] using a topological duality.  ... 
doi:10.1007/978-3-540-31982-5_29 fatcat:2o3zntlsnvffhep7kmjzkipusq

Coalgebras and their logics

Alexander Kurz
2006 ACM SIGACT News  
Some comments about the last Logic Column, on nominal logic.  ...  The critique of higherorder abstract syntax in that article prompted Karl Crary and Robert Harper to prepare a response to the leveled criticisms. The response should appear in an upcoming Column.  ...  Lifting such a basic duality to a duality of coalgebras and modal algebras, as indicated in Diagram 3, will provide modal logics for coalgebras.  ... 
doi:10.1145/1140612.1140628 fatcat:m72ui2imi5evdgnh5rlidvy6zi

Natural duality, modality, and coalgebra

Yoshihiro Maruyama
2012 Journal of Pure and Applied Algebra  
At the same time, however, natural duality theory has not subsumed important dualities in coalgebraic logic, including the Jónsson-Tarski topological duality and the Abramsky-Kupke-Kurz-Venema coalgebraic  ...  Such dualities play a fundamental role in recent developments of coalgebraic logic.  ...  This work was supported by the Nakajima foundation and by Grant-in-Aid for JSPS Fellows (236509).  ... 
doi:10.1016/j.jpaa.2011.07.002 fatcat:rla4ngz57ng7hlfl5qyxhxmkui

A Cook's tour of duality in logic: from quantifiers, through Vietoris, to measures [article]

Mai Gehrke, Tomas Jakl, Luca Reggio
2020 arXiv   pre-print
In particular, we focus on the use of (i) topological duality methods to solve problems in logic and computer science; (ii) category theory and, more particularly, free (and co-free) constructions; (iii  ...  ) these tools to unify the 'power' and 'structure' strands in computer science.  ...  In this paper we have discussed several examples of topological methods in logic and computer science, highlighting their duality theoretic nature.  ... 
arXiv:2007.15415v1 fatcat:d7fs2d66tffg3m7o5ob6xz6tt4

Topological duality for intuitionistic modal algebras

Barnaby P. Hilken
2000 Journal of Pure and Applied Algebra  
This paper describes a generalisation of the adjunction and duality between topological spaces and frames, inspired by the Kripke semantics of modal logic.  ...  This paper deÿnes an adjoint to this functor, and shows that the adjunction restricts to a duality between the subcategories in the image of the adjunction.  ...  duality for modal, temporal and program logics".  ... 
doi:10.1016/s0022-4049(98)00170-4 fatcat:bc6dy7pbnvhs7lmdyjettmav2e

Stone Coalgebras

Clemens Kupke, Alexander Kurz, Yde Venema
2003 Electronical Notes in Theoretical Computer Science  
This yields a duality between the category of modal algebras and that of coalgebras over the Vietoris functor.  ...  We prove that the so-called descriptive general frames, which play a fundamental role in the semantics of modal logics, can be seen as Stone coalgebras in a natural way.  ...  Acknowledgments We would like to thank the participants of the ACG-meetings at CWI, in particular, Marcello Bonsangue, Alessandra Palmigiano, and Jan Rutten.  ... 
doi:10.1016/s1571-0661(04)80638-8 fatcat:m2fs3hoggrcl7giex2amwykxim

Modal Logic and the Vietoris Functor [chapter]

Yde Venema, Jacob Vosmaer
2014 Leo Esakia on Duality in Modal and Intuitionistic Logics  
In [17] , Esakia uses the Vietoris topology to give a coalgebra-flavored definition of topological Kripke frames, thus relating the Vietoris topology, modal logic and coalgebra.  ...  Specifically, we look at Stone duality for the Vietoris hyperspace and the Vietoris powerlocale, and at recent work combining coalgebraic modal logic and the Vietoris functor.  ...  in §3.2.3.  ... 
doi:10.1007/978-94-017-8860-1_6 fatcat:awd6iv5xhbfjtk2hoh2h746kxm

Coalgebras and Modal Expansions of Logics

Alexander Kurz, Alessandra Palmigiano
2004 Electronical Notes in Theoretical Computer Science  
In this paper we construct a setting in which the question of when a logic supports a classical modal expansion can be made precise.  ...  Given a fully selfextensional logic S, we find sufficient conditions under which the Vietoris endofunctor V on S-referential algebras can be defined and we propose to define the modal expansions of S as  ...  In this paper we are focusing on expanding a logic with modal operators 2 and 3, that are normal and interrelated like in the modal logic K.  ... 
doi:10.1016/j.entcs.2004.05.010 fatcat:yoil4ppndzffvfdkxfzv7rl6em

A coalgebraic view on positive modal logic

Alessandra Palmigiano
2004 Theoretical Computer Science  
duality is established between the category of positive modal algebras and the category of K + -spaces in (J.  ...  The class of positive modal algebras is the one canonically associated with PML according to the theory of the algebrization of logics (Lecture Notes in Logic, Springer, Berlin, 1996).A Priestley-style  ...  Acknowledgements This work has been carried out during my stay as a visitor to the CWI in Amsterdam.  ... 
doi:10.1016/j.tcs.2004.07.026 fatcat:ysr6crxfnzc6zhp54wem4255gy

A Coalgebraic Approach to Dualities for Neighborhood Frames [article]

Guram Bezhanishvili, Nick Bezhanishvili, Jim de Groot
2022 arXiv   pre-print
Using one-step axioms in the language of finitary modal logic, we restrict this duality to other classes of neighborhood algebras studied in the literature, including monotone modal algebras and contingency  ...  We generalize this approach to any class of algebras for an endofunctor presented by one-step axioms in the language of infinitary modal logic.  ...  Acknowledgements 36 References 36 Acknowledgements We are grateful to the referees for careful reading and useful comments that have improved the presentation of the paper.  ... 
arXiv:2106.01628v2 fatcat:w4rxxjcj4ne5bn3wjanuoubn7e

Duality and the completeness of the modal μ-calculus

Simon Ambler, Marta Kwiatkowska, Nicholas Measor
1995 Theoretical Computer Science  
We consider the modal ~-calculus due to Kozen, which is a finitary modal logic with least and greatest fixed points of monotone operators.  ...  We extend the existing duality between the category of modal algebras with homomorphisms and the category of descriptive modal frames with contractions to the case of having fixed point operators.  ...  Acknowledgements The second author would like to acknowledge Samson Abramsky for many useful discussions, and the Theory and Formal Methods Section of Imperial College for providing an excellent research  ... 
doi:10.1016/0304-3975(95)00045-x fatcat:njjhubvhnbddxkdqc3ecmyvnkm

The Distributive Full Lambek Calculus with Modal Operators [article]

Daniel Rogozin
2020 arXiv   pre-print
In this paper, we study logics of bounded distributive residuated lattices with modal operators considering and in a noncommutative setting.  ...  After that, we establish a Priestley-style duality between residuated distributive modal algebras and topological Kripke structures based on Priestley spaces.  ...  The whole statement follows from the observation above and Theorem 4.6, Lemma 4.8, and Corollary 4.9. ✷ Topological duality In this section, we characterise a topological duality for residuated distributive  ... 
arXiv:2003.09975v3 fatcat:rbrz2afsw5fxrifkghudzr3fti

Stone coalgebras

Clemens Kupke, Alexander Kurz, Yde Venema
2004 Theoretical Computer Science  
This yields a duality between modal algebras and coalgebras for the Vietoris functor.  ...  We prove that the so-called descriptive general frames, which play a fundamental role in the semantics of modal logics, can be seen as Stone coalgebras in a natural way.  ...  Special thanks are due to one of the anonymous referees who provided us with very interesting comments, some of which have been incorporated in Remarks 4.5, 4.15 and 5.9.  ... 
doi:10.1016/j.tcs.2004.07.023 fatcat:st3xvxtfo5hw3pqlh3tst2bise
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