Elementary structures in process theory (1): Sets with renaming

Abstract

We study a general algebraic framework that underlies a wide range of computational formalisms that use the notion of names, notably process calculi. The algebraic framework gives a rigorous basis for describing and reasoning about processes semantically, as well as offering new insights into existing constructions. The formal status of the theory is elucidated by introducing its alternative presentation, which is geometric in nature and is based on explicit manipulation of connections among nameless processes. Nameless processes and their relational theory form a coherent universe in their own right, which underlies existing graphical formalisms such as proof nets. We establish the formal equivalence between these two presentations, and illustrate how they can be used complementarily for the precise and effective description of diverse algebras and the dynamics of processes through examples.