This Part is concerned with the relationship between λ-calculus and π-calculus. The λ-calculus is the prototypical functional language. The λ-calculus talks about functions and their applicative behaviour. This contrasts with the π-calculus, which talks about processes and their interactive behaviour. Application is a special form of interaction, and therefore functions can be seen as a special kind of processes. In this Part we study how the functions of the λ-calculus (the computable functions) can be represented as π-calculus processes. The π-calculus semantics of a language induces a notion of equality on the terms of that language. We therefore also analyse the equality among functions that is induced by their representation as π-calculus processes.

A deep study of representations of functions as π-calculus processes is of interest for several reasons. From the π-calculus point of view, the representation is a significant test of expressiveness, and is helpful in getting deeper insight into the theory. From the λ-calculus point of view, the representation makes it possible to apply process-calculus techniques to λ-calculus, and also to analyse λ-terms in contexts that are not purely sequential. This study may be useful for providing a semantic foundation for languages having constructs for functions and for concurrency, and techniques for reasoning about them. (Behavioural equalities between functions preserved in sequential contexts may not be preserved in nonsequential contexts; we will see examples of this in Chapter 17.) The study may also be helpful in developing parallel implementations of functional languages and in the design of programming languages based on process calculi.

Structure of the Part The Part is composed of five chapters. The first, Chapter 14, is about the λ-calculus. The second, Chapter 15, is about the encoding of the untyped λ-calculus into π-calculus. Chapter 16 does the same for the typed λ-calculus. Chapters 17 and 18 are about the full-abstraction problem for the simplest of the π-calculus encodings, namely that of the untyped call-by-name λ-calculus. A more detailed summary follows.

In Section 14.1 we review the syntax and reduction rules of the untyped λ-calculus. In Section 14.2, we look at some properties of the λ-calculus that make it strikingly different from the π-calculus: sequentiality and confluence. We also touch on the differences and the similarities between the basic computational rules of the two calculi.