On the Semantics of Message Passing Processes

Abstract

Let J be a shape in some category Shp for which there is a functor k: Shp → Cat. A categorical transition system (or system) is a pair (J, K(J)→C) consisting of a shape labelled by a functor in a category in C.Systems generalize conventional labelled transition systems. By choosing a suitable universe of shapes, systems can model concurrent and asynchronous computation. By labelling in a category, rather than an alphabet or term algebra, the actions of an algorithm or process can have structure.We study a class of systems called twisted systems having the form S =(J,FJ̃ → C) where J is a reflexive graph and : RGrph → RGrph is the twisted graph construction. The relevance of twisted systems lies in the relationship between twists and spans. A functor FJ → Sp(C) into a bicategory of spans is equivalent to a functor FJ̃ → C.The connection with spans means that when the target category C = Set, then following Burstall, a twisted system can be viewed as a generalized flow-chart. The theory extends to modeling interacting processes. If U is a system, then a process of type U is a system S and a morphism p: S → U. The system U represents the interface to the process. It describes what can be observed and what the process offers to the environment for interaction. The system S describes the internal behaviour of the process and the morphism p describes how S realizes observable behaviour. Processes compose by pullback over a common interface.