Coalgebras and modal expansions of logics

Alexander Kurz; Alessandra Palmigiano

doi:10.1016/j.entcs.2004.05.010

Coalgebras and modal expansions of logics

Alexander Kurz^*
, Alessandra Palmigiano

^*Corresponding author for this work

Research output: Contribution to Journal › Article › Academic › peer-review

Abstract

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 the logic that arises from the V-coalgebras. As an example, we also show how the Vietoris endofunctor on referential algebras extends the Vietoris endofunctor on Stone spaces. From another point of view, we examine when a category of 'spaces' (X, double-struck A sign), ie sets X equipped with an algebra double-struck A sign of subsets of X, allows for the definition of powerspaces V (and hence transition systems (X, double-struck A sign) → V(X, double-struck A sign)).

Original language	English
Pages (from-to)	243-259
Number of pages	17
Journal	Electronic Notes in Theoretical Computer Science
Volume	107
Issue number	1-4 SPEC. ISS.
DOIs	https://doi.org/10.1016/j.entcs.2004.05.010
Publication status	Published - 12 Dec 2004
Externally published	Yes

Funding

This paper was written during a visiting period of the second author to the Department of Computer Science of the University of Leicester. The second author wishes to thank the research group of Computer Science of the University of Leicester for their hospitality, and for partially funding her stay. We would like to thank Ramon Jansana for reading preliminary versions of this work and for many useful comments.

Keywords

Coalgebra
Modal expansion
Modal logic
Stone space
Vietoris endofunctor

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10.1016/j.entcs.2004.05.010

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title = "Coalgebras and modal expansions of logics",

abstract = "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 the logic that arises from the V-coalgebras. As an example, we also show how the Vietoris endofunctor on referential algebras extends the Vietoris endofunctor on Stone spaces. From another point of view, we examine when a category of 'spaces' (X, double-struck A sign), ie sets X equipped with an algebra double-struck A sign of subsets of X, allows for the definition of powerspaces V (and hence transition systems (X, double-struck A sign) → V(X, double-struck A sign)).",

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T1 - Coalgebras and modal expansions of logics

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AU - Palmigiano, Alessandra

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AB - 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 the logic that arises from the V-coalgebras. As an example, we also show how the Vietoris endofunctor on referential algebras extends the Vietoris endofunctor on Stone spaces. From another point of view, we examine when a category of 'spaces' (X, double-struck A sign), ie sets X equipped with an algebra double-struck A sign of subsets of X, allows for the definition of powerspaces V (and hence transition systems (X, double-struck A sign) → V(X, double-struck A sign)).

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KW - Modal expansion

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KW - Stone space

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Coalgebras and modal expansions of logics

Abstract

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Keywords

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