Real-time calculi and expansion theorems

Abstract

Using process calculi such as CCS it is possible to specify concisely the desired observable behaviour of a distributed system. However, these specifications focus only on the qualitative aspects of a system and leaves import quantitative aspects such as timing considerations unspecified. A system may provide the desired services in the specified order and hence be correct with respect to a qualitative specification. However, if the services are not enabled at the required moments in time, the system is nevertheless incorrect from a practical point of view.

This has motivated a recent introduction of several process calculi with explicit inclusion of real-time aspects. Traditionally, axiomatization and decidability results for process calculi are based on an expansion theorem enabling parallel composition to be removed from process descriptions. In contrast to the traditional work on process calculi, the axiomatization and decidability problems associated with these real-time calculi have proven notoriously hard problems. For instance, in the early work by Wang [Wan90] a simple timed extension of CCS is proposed together with an erroneous expansion theorem. Later, in [Wan91], Wang's timed version of CCS is extended with time variables in order to have an expansion theorem. As the main result of this paper we demonstrate that for certain simple types of real-time calculi no such expansion theorem can exist, i.e. parallel composition can not be removed in general. Thus, in order to deal with the problems of axiomatization and decidability new techniques, such as the use of time variables, are needed.

The proof of the result utilizes a translation from the real-timed calculus to the model of timed graphs. We consider this translation to be of independent interest.

This work has been supported by the Danish Research Council through the DART project. The first author is supported by a scholarship from Aalborg University and is currently on leave from the Danish telecommunication research laboratory (TFL).