Modularity of Behaviours for Mathematical Operational Semantics

Some years ago, Turi and Plotkin gave a precise mathematical formulation of a notion of structural operational semantics: their formulation is equivalent to a distributive law of the free monad on a signature over the cofree copointed endofunctor on a behaviour endofunctor. From such a distributive law, one can readily induce a distributive law of the monad over the cofree comonad on the behaviour endofunctor, and much of their analysis can be carried out in the latter terms, adding a little more generality that proves to be vital here. Here, largely at the latter level of generality, we investigate the situation in which one has two sorts of behaviours, with operational semantics possibly interacting with each other. Our leading examples are given by combining action and timing, with a modular account of the operational semantics for the combination induced by that of each of the two components. Our study necessitates investigation and new results about products of comonads and liftings of monads to categories of coalgebras for the product of comonads, providing constructions with which one can readily calculate.