Calculus of Communicating Systems: a synopsis

This article summarizes the essential ideas as presented in the book \Communication and Concurrency" by R.Milner [Mil89]. It presents intuitively the concepts of observation equivalence and observation congruence based on a fundamental notion of bisimulation. 1 Modelling Concurrent Systems There are many ways of specifying, designing and implementing a system. Different approaches may arrive at the same system. When are they (i.e., speci cations, designs and implementations) essentially the same? Basically, they are just abstractions of the same system but with di erent levels of details. At a given level of abstraction, di erent descriptions of the same system should be equivalent. For deterministic systems, we could say that two systems are equivalent if they produce identical outputs for the same input. That is, they satisfy the same input-output relation. Di erent algorithms of encoding a solution with varying degree of performance will be considered equivalent as long as they produce the same results. But, what about concurrent systems? They are highly non-deterministic, i.e., they may produce di erent outputs at di erent time for the same input. Some may end up with deadlocks, and others may not. How can they be compared and analyzed? We could consider that: As long as they exhibit the same ongoing behaviour, they are equivalent. In this sense, a system with deadlocks will be di erent from one without. So then, what are these ongoing behaviours of concurrent systems? To answer these questions, rst we need establish a model of concurrent systems, and then examine the essential properties that characterize their behaviours. An agent (or sometimes called a process) is any (concurrent) system whose behaviour consists of discrete actions. An action is either an interaction (a communication) with another agent, or it occurs independently; it is instantaneous