5 - Subcalculi

Summary

5.1 The Asynchronous π-calculus

The π-calculus takes over from CCS a kind of synchronized communication, in which an interaction involves the joint participation of two processes. The prefix operator expresses temporal precedence: the sole capability of xy. P is to send via x, and that of x(z). Q to receive via x, and in each case it is only after the relevant capability has been exercised that the behaviour expressed by the term underneath the prefix can unfold. The combination of prefixing and synchronized communication is versatile and tractable.

On the other hand, some concurrent systems, especially distributed systems, use forms of asynchronous communication, in which the act of sending a datum and the act of receiving it are separate. Often, such communication involves an explicit medium of some kind, and sending and receiving data involve putting them into and taking them out of the medium. Media exhibit a variety of characteristics. For instance, they may have bounded or unbounded capacity, and they may or may not preserve some ordering among the data they contain. There may be an arbitrary delay between a datum being sent and it being received. If the communication medium has unbounded capacity, no process need wait to send a datum.

This section and the following four are about a subcalculus of the π-calculus in which communication can be understood as asynchronous. The key step in achieving this is the decree that, in the subcalculus, the only term that can appear underneath an output prefix is 0. Thus, the only output-prefixed terms are of the shape xy. 0, or, in abbreviated form, xy. In a term of the subcalculus, an unguarded occurrence of a particle xy can be thought of as a datum y in an implicit communication medium, tagged with x to indicate that it is available to any unguarded subterm of the form x(z).P. Thus, in the evolution of a term, the datum y can be considered to be sent when xy becomes unguarded, and to be received when xy disappears via an intraaction of the form

This reduction has two effects: the particle xy is consumed and removed from the communication medium; and the unguarded output particles in P{y/z} are liberated and put into the medium.