This chapter brings together all the concepts we have learned so far. For an extended example, it presents a collection of modules to implement the λ-calculus as a primitive functional programming language. Terms of the λ-calculus can be parsed, evaluated and the result displayed. It is hardly a practical language. Trivial arithmetic calculations employ unary notation and take minutes. However, its implementation involves many fundamental techniques: parsing, representing bound variables and reducing expressions to normal form. These techniques can be applied to theorem proving and computer algebra.

Chapter outline

We consider parsing and two interpreters for λ-terms, with an overview of the λ-calculus. The chapter contains the following sections:

A functional parser. An ml functor implements top-down recursive descent parsing. Parsers can be combined using infix operators that resemble the symbols for combining grammatical phrases.

Introducing the λ-calculus. Terms of this calculus can express functional programs. They can be evaluated using either the call-by-value or the call-by-name mechanism. Substitution must be performed carefully, avoiding variable name clashes.

Representing λ-terms inml. Substitution, parsing and pretty printing are implemented as ml structures.

The λ-calculus as a programming language. Typical data structures of functional languages, including infinite lists, are encoded in the λ-calculus. The evaluation of recursive functions is demonstrated.

A functional parser

Before discussing the λ-calculus, let us consider how to write scanners and parsers in a functional style.