Specifying and analysing networks of processes in CSPt (or in search of associativity)

In proposing theories of how we should design and specify networks of processes it is necessary to show that the semantics of any language we use to write down the intended behaviours of a system has several qualities. First in that the meaning of what is written on the page reflects the intention of the designer; second that there are no unexpected behaviours that might arise in a specified system that are hidden from the unsuspecting specifier; and third that the intention for the design of the behaviour of a network of processes can be communicated clearly and intuitively to others. In order to achieve this we have developed a variant of CSP, called CSPt, designed to solve the problems of termination of parallel processes present in the original formulation of CSP. In CSPt we introduced three parallel operators, each with a different kind of termination semantics, which we call synchronous, asynchronous and race. These operators provide specifiers with an expressive and flexible tool kit to define the intended behaviour of a system in such a way that unexpected or unwanted behaviours are guaranteed not to take place. In this paper we extend out analysis of CSPt and introduce the notion of an alphabet diagram that illustrates the different categories of events that can arise in the parallel composition of processes. These alphabet diagrams are then used to analyse networks of three processes in parallel with the aim of identifying sufficient constraints to ensure associativity of their parallel composition. Having achieved this we then proceed to prove associativity laws for the three parallel operators of CSPt. Next, we illustrate how to design and construct a network of three processes that satisfy the associativity law, using the associativity theorem and alphabet diagrams. Finally, we outline how this could be achieved for more general networks of processes.