Stable Modal Logics

Bezhanishvili, Guram; Bezhanishvili, Nick; Ilin, Julia

doi:10.1017/s1755020317000375

Cited by 10 publications

(11 citation statements)

References 29 publications

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“…For example, in the special case of topological spaces they simply correspond to continuous maps. They also generalize the stable morphisms studied in [BBI16,BBI18] to the setting of (descriptive) neighborhood frames. We plan to investigate this topic further in a sequel to this paper.…”

Section: Discussionmentioning

confidence: 79%

A Coalgebraic Approach to Dualities for Neighborhood Frames

Bezhanishvili¹,

Bezhanishvili

Groot

2022

Logical Methods in Computer Science

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We develop a uniform coalgebraic approach to J\'onsson-Tarski and Thomason type dualities for various classes of neighborhood frames and neighborhood algebras. In the first part of the paper we construct an endofunctor on the category of complete and atomic Boolean algebras that is dual to the double powerset functor on $\mathsf{Set}$. This allows us to show that Thomason duality for neighborhood frames can be viewed as an algebra-coalgebra duality. We generalize this approach to any class of algebras for an endofunctor presented by one-step axioms in the language of infinitary modal logic. As a consequence, we obtain a uniform approach to dualities for various classes of neighborhood frames, including monotone neighborhood frames, pretopological spaces, and topological spaces. In the second part of the paper we develop a coalgebraic approach to J\'{o}nsson-Tarski duality for neighborhood algebras and descriptive neighborhood frames. We introduce an analogue of the Vietoris endofunctor on the category of Stone spaces and show that descriptive neighborhood frames are isomorphic to coalgebras for this endofunctor. This allows us to obtain a coalgebraic proof of the duality between descriptive neighborhood frames and neighborhood algebras. 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 algebras. We conclude the paper by connecting the two types of dualities via canonical extensions, and discuss when these extensions are functorial.

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Section: Discussionmentioning

confidence: 79%

A Coalgebraic Approach to Dualities for Neighborhood Frames

Bezhanishvili¹,

Bezhanishvili

Groot

2022

Logical Methods in Computer Science

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“…For example, in the special case of topological spaces they simply correspond to continuous maps. They also generalize the stable morphisms studied in [8,9] to the setting of (descriptive) neighborhood frames. We plan to investigate this topic further in a sequel to this paper.…”

Section: Discussionmentioning

confidence: 79%

A Coalgebraic Approach to Dualities for Neighborhood Frames

Bezhanishvili¹,

Bezhanishvili²,

Groot³

2021

Preprint

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We develop a uniform coalgebraic approach to Thomason and Jónsson-Tarski type dualities for various classes of neighborhood frames and neighborhood algebras. In the first part of the paper we construct an endofunctor on the category of complete and atomic Boolean algebras that is dual to the double powerset functor on Set. This allows us to show that Thomason duality for neighborhood frames can be viewed as an algebra-coalgebra duality. We generalize this approach to any class of algebras for an endofunctor presented by one-step axioms in the language of infinitary modal logic. As a consequence, we obtain a uniform approach to dualities for various classes of neighborhood frames, including monotone neighborhood frames, pretopological spaces, and topological spaces.In the second part of the paper we develop a coalgebraic approach to Jónsson-Tarski duality for neighborhood algebras and descriptive neighborhood frames. We introduce an analogue of the Vietoris endofunctor on the category of Stone spaces and show that descriptive neighborhood frames are isomorphic to coalgebras for this endofunctor. This allows us to obtain a coalgebraic proof of the duality between descriptive neighborhood frames and neighborhood algebras. 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 algebras.We conclude the paper by connecting the two types of dualities via canonical extensions, and discuss when these extensions are functorial.

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“…Remark 1. We note that the above corollary can be proved for a larger class of stable and transitive stable logics of [5,6]. These are logics that are sound and complete with respect to classes of rooted frames closed under graph homomorphisms.…”

Section: Mmentioning

confidence: 86%

“…where r is the (only) basic program 6 and ϕ is a formula in the language L E . The program 1 stands for the universal program.…”

Section: Quotient-taking As a Model Transformermentioning

confidence: 99%