An Algebraic Framework for Defining Behaviours of Concurrent Systems. Part 1: The Constructive Presentation

Abstract

The paper is the first part of a two-part paper that contributes with a concept of a process viewed as a model of a run of a phenomenon (discrete, continuous, or of a mixed type), with operations allowing to define complex processes in terms of their components, with the respective algebras, and with the idea of using the formal tools thus obtained to describe the behaviours of concurrent systems. A process may have an initial state (a source), a final state (a target), or both. A process can be represented by a partially ordered multiset. Processes of which one can be a continuation of the other can be composed sequentially. Independent processes, i.e. processes which do not disturb each other, can be composed in parallel. Processes may be prefixes, i.e. independent components of initial segments of other processes. Processes in a universe of objects and operations on such processes form a partial algebra, called algebra of processes. Algebras of processes are partial categories with respect to the sequential composition, and partial monoids with respect to the parallel composition. Algebras of processes can be used to define behaviours of concurrent systems. The behaviour of a system can be defined as the set of possible processes of this system with a structure on this set. Such a set is prefix-closed. The structure on this set reflects the prefix order and, possibly, specific features of the behaviour like observability, the relation to flow of real time, etc. Algebras of processes can also be used to define behaviours, to define operations on behaviours similar to those in the existing calculi of behaviours, and to define random behaviours. The first part of the whole paper investigates algebras of processes and their applications to describing behaviours of systems. In the second part the properties of algebras of processes described in the first part are regarded as axioms defining a class of abstract partial algebras, called behaviouroriented algebras, and they initiate a theory of such algebras.