Introduction to Part IV

Summary

In this Part we study the use of types for reasoning about behavioural properties of mobile processes.

Although well-developed, the theory of the pure untyped π-calculus is often insufficient to prove ‘expected’ properties of processes. The reason is that when one uses π-calculus to describe a system, one normally follows a discipline that governs how names may be used; but this discipline is not explicit in π-terms, and therefore it cannot play a part in proofs. (The same happens for the λ-calculus, which is hardly ever used untyped, since each variable usually has an ‘intended’ functionality.) Types can be used to make such disciplines explicit. In Section 9.1, we illustrate this point with two examples that have to do with encapsulation.

The use of types affects contextuallv-defined behavioural equivalences such as barbed congruence, for in a typed calculus the processes being compared must obey the same typing, and the contexts in which they are tested must be compatible with this typing. Typically, in a typed calculus the class of legal contexts in which two processes may be tested is smaller than in the untyped calculus. As a consequence, more behavioural equivalences among processes hold. The behavioural effects of types are important for types such as i/o, linear, receptive, and polymorphic types, because such types express behavioural guarantees on the use of names. Types are important not only for behavioural equivalences, but more generally for behavioural properties; see for instance the security property discussed in Section 9.3.

Basic definitions, such as those of typed barbed congruence and typed labelled equivalences, will be parametric on the type system. However, most results will be presented on concrete π-calculi, usually extensions of π, the polvadic π-calculus. π is simple (for instance it has no basic types), it is an adequate testbed for the results, and it is a common core calculus in applications. The equivalences we present are also valid in other type systems, as long as the key conditions on the usage of names (such as i/o capabilities and receptiveness) are respected. We will point out when this is not the case. In examples we will often use also integer and boolean values.