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Enriched coalgebraic modal logic

Enriched coalgebraic modal logic
Enriched coalgebraic modal logic
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, ?) of models for an L-algebra (A, ?), and show that the forgetful V-functor from Mod(A, ?) 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.
Wilkinson, Toby
c711a6aa-a538-4b99-9048-3812a14d27a4
Wilkinson, Toby
c711a6aa-a538-4b99-9048-3812a14d27a4
Cirstea, Corina
ce5b1cf1-5329-444f-9a76-0abcc47a54ea

(2013) Enriched coalgebraic modal logic. University of Southampton, faculty of Physical Sciences and Engineering, Doctoral Thesis, 228pp.

Record type: Thesis (Doctoral)

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, ?) of models for an L-algebra (A, ?), and show that the forgetful V-functor from Mod(A, ?) 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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Published date: May 2013
Organisations: University of Southampton, Electronic & Software Systems

Identifiers

Local EPrints ID: 354112
URI: http://eprints.soton.ac.uk/id/eprint/354112
PURE UUID: 0cedb4d4-aea5-43e2-972c-e8073aac86bf

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Date deposited: 02 Jul 2013 11:38
Last modified: 18 Jul 2017 03:59

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Contributors

Author: Toby Wilkinson
Thesis advisor: Corina Cirstea

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