12.1 Simply-typed HOπ

We extend Base-π and move to higher order by allowing values built out of processes. We use Higher-Order π-calculus, briefly HOπ, as a general name for a higher-order extension of Base-π.

Passing a process is like passing a parameterless procedure. The recipient of a process can do nothing with it but execute it, possibly several times. Procedures gain great utility if they can be parametrized so that, when invoked, some arguments may be supplied. In the same way a higher-order process calculus gains power if the processes that are communicated may be parametrized (see also the discussion at the end of Section 13.1.1). A parametrized process, an abstraction, is an expression of the form (x).P. We have already met abstractions in previous Parts of the book - Definition 2.4.38, Section 7.4.1; in this Part and the next, they play a central role. We may also regard abstractions as components of input-prefixed processes, viewing a(x).P as an abstraction located at a. Indeed, the part (x).P of a(x).P behaves exactly like an abstraction. In (x).P as in a(x).P, the displayed occurrence of x is binding with scope P.

When an abstraction (x).P is applied to an argument w it yields the process P{w/x}. Application is the destructor for abstractions. The application of an abstraction v to a value v’ is written, where T is the type of v'. (The type annotation is to ensure that the typing derivation of a process is unique, which will facilitate the encoding of HOπ into π-calculus. An alternative is to annotate the binding occurrences of names in abstractions, as is common in λ-calculi. Our choice keeps the syntax of abstraction closer to that of input.) At the level of types, adding parametrization means adding function types; thus in the value v has type T → ⋄, where ⋄ is the behaviour type (the type of processes, Section 6.1).