Categories for Dynamic Epistemic Logic

Abstract

The primary goal of this paper is to recast the semantics of modal logic, and dynamic epistemic logic (DEL) in particular, in category-theoretic terms. We first review the category of relations and categories of Kripke frames, with particular emphasis on the duality between relations and adjoint homomorphisms. Using these categories, we then reformulate the semantics of DEL in a more categorical and algebraic form. Several virtues of the new formulation will be demonstrated: The DEL idea of updating a model into another is captured naturally by the categorical perspectivewhich emphasizes a family of objects and structural relationships among them, as opposed to a single object and structure on it. Also, the categorical semantics of DEL can be merged straightforwardly with a standard categorical semantics for first-order logic, providing a semantics for first-order DEL.

Unification of Modal Logic via Topological Categories

“…There are already several categorical reformulation and generalisation of DEL in the literature, e.g. see [17,12], but most of them are based on relational semantics, while our approach applies to arbitrary topological categories. We first define the notion of a product type, which generalises event models: Definition 5.2 (Product Type).…”

On Inverse Operators in Dynamic Epistemic Logic

“…Regarding the categorical reformulation of MTSs, in [9,13], Cîrstea and Sadrzadeh give a coalgebraic model of modal logic for actions and agents. They take T X = P κ (X) E × (1 + X) A × ℘(P), where κ is a regular cardinal, as the coalgebra functor (over Set).…”

“…4 focuses on capturing this indexing mechanism by Fam construction. Categorical semantics of DEL in [9,13,14] does not have such indexing systems. On the other hand, the first author extends index update functions (t α component of MTS) to relations in [16].…”