A Temporal Calculus of Communicating Systems

In this paper, we introduce a calculus of communicating systems which allows for the expression and analysis of timing constraints, for example as is important for real-time processes. We present the language, along with its formal semantics, and derive algebraic laws for reasoning about processes in the language. Though the core language is simple, we show that the language has several powerful derived operators which we demonstrate to be useful in several examples. 1 I n t r o d u c t i o n The analysis of the temporal properties of concurrent processes gives insight into some interesting aspects of concurrent programming, such as t imeout in fault-tolerant systems (eg, protocols), and durat ion control in critical real-time systems (eg, radiation exposure). There have been several studies made to provide a formalism within which these concepts can be expressed (e.g., [Bae89], [Gro90], [Jef89], [Koy83], [Ree86]). Many of these assume synchrony, which results in some of the more interesting temporal properties of processes being inexpressible, or they assume a global clock and require actions to happen at precise moments measured by that global dock, an approach which also has its drawbacks. The earlier report [Tof88] provided an extension to CCS, Milner's Calculus of Communicating Processes, which admit ted a notion of timing. That approach was unsatisfactory in that the account it gave of t ime was somewhat eccentric. Processes could only evolve simultaneously via communication, whilst t ime and action were otherwise interleaved. In order to give a fuller account of t ime for asynchronous processes, we rewrite the theory to derive our language TCCS, a Temporal Calculus of Communicating Systems, in which time is allowed to pass independent of the functional aspects of a processes. In order to give this fuller account, the process state transit ion system upon which the intuit ion of the calculus is built has semantically been split into two orthogonal parts, one describing the functional aspect of the process, and the other describing its temporal aspect. This allows for the separation of functional and temporal concerns in analysing process behaviour, but also there are sound physical motivations for imposing this separation. Computat ion involves energy changes, and by a result of quantum mechanics (cf., [Dir58], [Sch82]) energy changes and t ime cannot be measured simultaneously. Thus it seems reasonable when producing an observation-based model of t ime and computat ion not to permit the simultaneous observation of these two activities. We therefore assume that actions have no duration, although we could *Research supported by ESPRIT BRA Grant No. 3006 CONCUR