Introduction to Part II

Summary

Part I presented a main line in the theory of the π-calculus, examining processes and behavioural equivalence in some depth. Part II continues the study of the calculus.

Its first segment addresses two of the book's central concerns: how to use π-calculus, and the question of its expressiveness. These two concerns are intimately connected, since using the calculus involves expressing things in it. Among the things whose expression is explored are computational data: simple data such as numbers and strings, and complex data such as records, lists, and graphs. Representing data is an attractive first exercise in using π-calculus, because it is simple and yet allows many common idioms to be demonstrated. The activity has a similar flavour to expressing entities such as numbers and partial recursive functions as terms of λ-calculus. As we will see, however, there are many striking points of contrast. We show by example how values of arbitrary datatypes and operations on them can be expressed as name-passing processes, and how evaluation of data is captured by reduction of those processes.

The calculus of Part I is sometimes referred to as the monadic π-calculus, because an interaction involves the communication of a single name between processes. The representations of data as processes make use of two additions to the monadic calculus. The first is polyadicity: the passing of tuples of names in communications. Polyadicity is perhaps the simplest addition to the monadic calculus that retains names as the only atomic data. As we will see, however, polyadicity forces consideration of types, an issue that can be left aside in the monadic calculus. The second addition to the monadic calculus used in representing data is recursive definition of processes. Polyadicity and recursion can be convenient when using π-calculus. Anything that can be expressed with their help can also be expressed without it, however. To demonstrate some of the reasoning techniques developed in Part I in alliance with polyadicity and recursion, we give an extended worked example. It establishes the equivalence of two implementations of a priority-queue datatype.