A Coalgebraic presentation of structured transition systems

Abstract

This paper relates labelled transition systems and coalgebras with the motivation of comparing and combining their complementary contributions to the theory of concurrent systems. The well-known mismatch between these two notions concerning the morphisms is resolved by extending the coalgebraic framework by lax cohomomorphisms. Enriching both labelled transition systems and coalgebras with algebraic structure for an algebraic specification, the correspondence is lost again. This motivates the introduction of lax coalgebras, where the coalgebra structure is given by a lax homomorphism. The resulting category of lax coalgebras and lax cohomomorphisms for a suitable endofunctor is shown to be isomorphic to the category of structured transition systems, where both states and transitions form algebras. The framework is also presented on a more abstract categorical level using monads and comonads, extending the bialgebraic approach introduced by Turi and Plotkin.