Coalgebraic Geometric Logic: Basic Theory

Using the theory of coalgebra, we introduce a uniform framework for adding modalities to the language of propositional geometric logic. Models for this logic are based on coalgebras for an endofunctor on some full subcategory of the category of topological spaces and continuous functions. We investigate derivation systems, soundness and completeness for such geometric modal logics, and we we specify a method of lifting an endofunctor on Set, accompanied by a collection of predicate liftings, to an endofunctor on the category of topological spaces, again accompanied by a collection of (open) predicate liftings. Furthermore, we compare the notions of modal equivalence, behavioural equivalence and bisimulation on the resulting class of models, and we provide a final object for the corresponding category

Distributive Laws for Monotone Specifications

Turi and Plotkin introduced an elegant approach to structural operational semantics based on universal coalgebra, parametric in the type of syntax and the type of behaviour. Their framework includes abstract GSOS, a categorical generalisation of the classical GSOS rule format, as well as its categorical dual, coGSOS. Both formats are well behaved, in the sense that each specification has a unique model on which behavioural equivalence is a congruence. Unfortunately, the combination of the two formats does not feature these desirable properties. We show that monotone specifications - that disallow negative premises - do induce a canonical distributive law of a monad over a comonad, and therefore a unique, compositional interpretation.Comment: In Proceedings EXPRESS/SOS 2017, arXiv:1709.0004

Generic Trace Semantics via Coinduction

Trace semantics has been defined for various kinds of state-based systems, notably with different forms of branching such as non-determinism vs. probability. In this paper we claim to identify one underlying mathematical structure behind these "trace semantics," namely coinduction in a Kleisli category. This claim is based on our technical result that, under a suitably order-enriched setting, a final coalgebra in a Kleisli category is given by an initial algebra in the category Sets. Formerly the theory of coalgebras has been employed mostly in Sets where coinduction yields a finer process semantics of bisimilarity. Therefore this paper extends the application field of coalgebras, providing a new instance of the principle "process semantics via coinduction."Comment: To appear in Logical Methods in Computer Science. 36 page