Type Classes are Signatures of Abstract Types

Abstract

We present an extension of Haskell’s type class concept in which a type class is identified with the signature of an abstract type As shown by Mitchell and Plotkin, abstract types can be expressed using existential quantification. Unlike in Mitchell and Plotkin’s work, an abstract type does not come with one — and only one — implementation. Rather, any concrete type can be declared to be an implementation by a clause that corresponds to an instance declaration in Haskell. We introduce F-bounded existential quantification, where an abstract type has the form:

$$\exists \alpha .C(\alpha ).\tau (\alpha ).$$

Here, C(α) is a set of constraints that restricts the range of the bound variable α, and τ(α) is a type constructed from α. The expression reads “some type τ(α), where α is some arbitrary fixed type satisfying constraints C(α)”. The constraint set C corresponds to a type class. Just like a type class, it contains declarations for overloaded identifiers as well as conformity clauses that declare one abstract type to be more specific than another.

The generalization of type classes to abstract types has the advantage of greater expressiveness: We can model polymorphic abstract types and heterogeneous data structures, concepts which cannot be expressed in Haskell. An example of a polymorphic abstract type is ∀α.Bag α, the abstract type of all bags with elements of type α. In Haskell, we would either have to fix the element type, or we would have to fix the implementation of Bag.

Our extension shares the desirable properties of the type class approach in that it is fully static and in that type reconstruction is feasible.

Supported by the Defense Advanced Research Projects Agency/Information Systems Technology Office under the Office of Naval Research contract N00014-90-J-1110