10 - Behavioural Effects of i/o Types

Summary

In the first half of this chapter, comprising Sections 10.1-10.5, we discuss the consequences of i/o types for behavioural properties of processes. For this, we compare the polvadic π-calculus (π) with the polvadic π-calculus with i/o types (π). In the former calculus the only constraint that types impose is that arities of names must be respected; in the latter, i/o information is also taken into account. In the second half of the chapter, comprising Sections 10.6 to 10.8, we study techniques for proving behavioural equivalences that crucially rely on the i/o constraints.

10.1 Type coercion

We first show that if processes of π are equivalent when i/o information is forgotten, then they are also equivalent in π, where by ‘forgotten’ we mean that all i/o types are coerced to connection types. The result is useful because proving equivalences in the π-calculus with only the connection type is easier (for instance, the definition of typed bisimilaritv is simpler).

The reason why Theorem 10.1.1 holds is that in π the coercion of i/o to connection types preserves type judgments. This may however be false in calculi with a richer subtyping relation:

Exercise* 10.1.2 Show that Theorem 10.1.1(1) fails in π where tuple types are replaced by records.

Most important is the behavioural effect of the refinement of connection types to i/o types. Processes that are distinguishable in π may become equivalent when types are so refined. The equivalences so gained are often highly desirable. In Section 10.2 we present five concrete examples of such equivalences. Then, in Sections 10.3 and 10.5, we present two very useful results: a law about wire processes in the Asynchronous π-calculus, and the Sharpened Replication Theorems. The equivalences in the examples and the results will be proved in the second half of the chapter. In Section 10.4 we exploit the wire processes and their laws to model the delayed input construct, in the Asynchronous π-calculus. More examples of the use of i/o types for reasoning can be found in Parts V and VI; see Theorem 13.2.22 and discussion afterwards, and Remark 15.3.17.