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Communicating Sequential Processes (CSP), is a theoretical framework for discussing concurrent phenomena [7]. In this note, we begin an investigation into the nature of sets of communicating sequential processes. Sets of sequential processes arise naturally when one considers processes contained within specified bounds which provides their approximate description. Adopting the trace formalism, we may express those bounds in a natural way by means of containment of traces. We endow families of process traces and a fortiori, families of processes, with a rough set topology. We show in this note that basic operators on processes preserve exact sets of processes and they are non-expansive (i.e., non-destructive in terminology of [7]) with respect to the metric D on rough sets [6], [5], restricted to exact sets of processes. [ABSTRACT FROM AUTHOR] more...

We present here an implementation relation intended to formalise the notion that a system built of communicating processes is an acceptable implementation of another base. or target, system in the event that the two systems have different interfaces. Such a treatment has clear applicability in the software development process, where (the interface of) an implementation component may be expressed at a different level of abstraction to (the interface of) the relevant specification component. Technically. processes are formalised using Hoare's CSP language, with its standard failures-divergences model. The implementation relation is formulated in terms of failures and divergences of the implementation and target processes. Interlace difference is modelled by endowing the implementation relation with parameters called extraction patterns. These are intended to interpret implementation behaviour as target behaviour, and suitably constrain the former in connection to well-formedness and deadlock properties. We extend the results of our previous work and replace implementation relations previously presented by a single, improved scheme. We also remove all the restrictions previously placed upon target processes. Two basic kinds of results are obtained: realisability and compositionality. The latter means that a target composed of several connected systems may be implemented by connecting their respective implementations. The former means that, if target and implementation in fact have the same interface, then the implementation relation they should satisfy collapses into standard implementation pre-order. We also show how to represent processes and extraction patterns in a manner amenable to computer implementation, and detail a graph-theoretic restatement of the conditions defining the implementation relation, from which algorithms for their automatic verification are easily derived. [ABSTRACT FROM AUTHOR] more...