4 - Behavioural Equivalence, continued

Summary

4.1 Distinctions

The π-calculus makes no syntactic distinction between instantiable names and non-instantiable names, that is, between variables and constants. Sometimes, however, it is desirable to treat certain names as constants, that is, as not subject to instantiation. More generally, it is sometimes useful to work in a setting where certain names cannot be instantiated to the same name. This more general case is no more difficult technically than treating names as constants. This section shows how to treat the more general case, by means of a family of behavioural equivalences indexed by certain binary relations, distinctions, on names.

Suppose we wish to model some system as a π-calculus process, and that to achieve this, it is convenient to regard two names, s and r, as constants. For example, suppose s is to signify send and r receive in a process such as

In order that the occurrences of s and r in a process not be open to instantiation when it is placed in a context C, the hole in C must not be underneath an input prefix that binds s or r. Let us call the contexts with this property the (s, r)-faithful contexts.

Now suppose that two π-calculus processes describe the system in question, and that we wish to compare them. Then we would like an equivalence, that treats s and r appropriately, in that it satisfies the preservation property: if and is an (s, r)-faithful context, then. What is a suitable choice for ?

It is easy to see that neither bisimilaritv nor full bisimilaritv will do. The former is too weak, because in general, does not imply if the hole in C occurs under any input prefix. And the latter is too strong, because it demands bisimilaritv in all contexts, not just the (s, r)-faithful contexts.

We therefore need something in between, indeed a new class of equivalences, to handle all varieties of faithfulness. Generalizing slightly from the motivating example, we introduce a family of equivalences each of which demands bisimi-laritv under all substitutions that maintain the distinction between certain pairs of names. Each member of the family is indexed by a relation that prescribes the relevant pairs.