Enriched coalgebraic modal logic

Abstract

We formalise the notion of enriched coalgebraic modal logic, and determine conditions on the category V (over which we enrich), that allow an enriched logical connection to be extended to a framework for enriched coalgebraic modal logic. Our framework uses V-functors L: A → A and T: X → X, where L determines the modalities of the resulting modal logics, and T determines the coalgebras that provide the semantics.

We introduce the V-category Mod(A,α) of models for an L-algebra (A,α), and show that the forgetful V-functor from Mod(A,α) to X creates conical colimits.

The concepts of bisimulation, simulation, and behavioural metrics (behavioural approximations), are generalised to a notion of behavioural questions that can be asked of pairs of states in a model. These behavioural questions are shown to arise through choosing the category V to be constructed through enrichment over a commutative unital quantale (Q, ?, I) in the style of Lawvere (1973).

Corresponding generalisations of logical equivalence and expressivity are also introduced,and expressivity of an L-algebra (A, ?) is shown to have an abstract category theoretic characterisation in terms of the existence of a so-called behavioural skeleton in the category Mod(A, ?).

In the resulting framework every model carries the means to compare the behaviour of its states, and we argue that this implies a class of systems is not fully defined until it is specified how states are to be compared or related.

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ORCID for Corina Cirstea: orcid.org/0000-0003-3165-5678

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