This chapter describes an ML-like language (i.e. implicitly typed λ-calculus with local definitions) and extends the framework of (Milner, 1978; Damas and Milner, 1982) with support for overloading using qualified types and an arbitrary system of predicates of the form described in the previous chapter. The resulting system retains the flexibility of the ML type system, while allowing more accurate descriptions of the types of objects. Furthermore, we show that this approach is suitable for use in a language based on type inference, in contrast for example with more powerful languages such as the polymorphic λ-calculus that require explicit type annotations.

Section 3.1 introduces the basic type system and Section 3.2 describes an ordering on types, used to determine when one type is more general than another. This is used to investigate the properties of polymorphic types in the system.

The development of a type inference algorithm is complicated by the fact that there are many ways in which the typing rules in our original system can be applied to a single term, and it is not clear which of these (if any!) will result in an optimal typing. As an intermediate step, Section 3.3 describes a syntax-directed system in which the choice of typing rules is completely determined by the syntactic structure of the term involved, and investigates its relationship to the original system. Exploiting this relationship, Section 3.4 presents a type inference algorithm for the syntax-directed system which can then be used to infer typings in the original system.