1.1 Syntax

This section introduces the language of the π-calculus. Its terms, the processes, express mobile systems.

The meanings of processes are discussed informally, and name-binding and name-substitution, two notions needed for the formal account, are explained.

The simplest entities of the π-calculus are names. They can be thought of as names of communication links. Processes use names to interact, and pass names to one another by mentioning them in interactions. Names received by a process can be used and mentioned by it in further interactions. A countablv-infinite set of names is presupposed, ranged over by lower-case letters.

Processes evolve by performing actions. The capabilities for action are expressed via the prefixes, of which there are four kinds:

The first capability is to send the name y via the name x, and the second to receive any name via x - the role of z will be explained shortly. The third is a capability for unobservable action - this will also be explained soon. The fourth is a conditional capability: the capability π if x and y are the same name.

We now define the processes and a particular subclass of them.

Definition 1.1.1 (π-calculus) The processes and the summations of the π- calculus are given respectively by

We give a brief, informal account of the intended interpretation of processes, beginning with the summations.

• (1) 0 is inaction; it is a process that can do nothing.

• (2) The prefix it. P has a single capability, expressed by π; the process P cannot proceed until that capability has been exercised.

• The output prefix xy. P can send the name y via the name x and continue as P.

• The input prefix x(z).P can receive any name via x and continue as P with the received name substituted for z.