An Algebraic Framework for Defining Behaviours of Concurrent Systems. Part 2: The Axiomatic Presentation

Abstract

The paper is the second part of two-part paper that contributeswith a concept of a process, 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. In the first part a universal model of a process have been introduced and operations on processes have been defined. A process is viewed as a model of a run of a system (discrete, continuous, or of a mixed type). 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 is a continuation of the other can be composed sequentially. Independent processes, can be composed in parallel. Processes may be prefixes, i.e. independent components of initial segments of other processes. It has been shown that processes in a universe of objects and operations on such processes form a partial algebra, called algebra of processes, that is a specific partial category with respect to the sequential composition, and a specific partial monoid with respect to the parallel composition. 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 behaviour-oriented algebras, and properties of such algebras are investigated. In particular, it is shown how some of the behaviour-oriented algebras can be represented as algebras of processes, and how to use them to describe phenomena with states and processes provided with specific structures.