COMMUNICATION FUNCTION

A central issue in concurrency theory is that of modelling communication. The method that we present here yields synchronous communication. That is, the occurrence of communication between two processes is the result of the simultaneous performance of corresponding actions. For instance, one process may perform a send action, whereas the other performs a receive action, and if these actions take place simultaneously the result is a communicate action. Later on, we will also consider examples of asynchronous communication in process algebra.

In synchronous communication, every communication action has two parts. This can be enoted as follows: suppose an atomic action comm(5) represents the communication between send(5) and receive(5), then we write

If two actions do not communicate, then their communication is not defined. For example:

This will be formalized in the next paragraph.

DEFINITION

Let A be the set of atomic actions. A communication function on A is a partial binary function γ on A satisfying the following conditions:

1. for every a,b∈ A: γ(a,b) = γ(b,a), that is, communication is commutative;

2. for every a,b,c∈ A: γ(γ(a,b),c) = γ(aγ(b,C)): communication is associative.

When we write such an equation for a partial function, we also imply that one side of the equation is defined exactly when the other side is. Also, γ is not defined when one of its arguments is not defined or is not in A.