Computing ML Equality Kinds Using Abstract Interpretation

The de nition of Standard ML provides a form of generic equality which is inferred for certain types, called equality types, on which it is possible to de ne an equality relation in ML. However, the standard de nition is incomplete in the sense that there are interesting and useful types which are not inferred to be equality types but for which an equality relation can be de ned in ML in a uniform manner. In this paper, a re nement of the Standard ML system of equality types is introduced and is proven sound and complete with respect to the existence of a de nable equality. The technique used here is based on an abstract interpretation of ML operators as monotone functions over a three point lattice. It is shown how the equality relation can be de ned (as an ML program) from the de nition of a type with our equality property. Finally, a sound, e cient algorithm for inferring the equality property which corrects the limitations of the standard de nition in all cases of practical interest is demonstrated. 1 Equality Types in Standard ML The ML language provides an extensible algebra of type constructions. The Standard ML dialect divides types into two classes, those which admit equality (also called equality types) and those which do not. This distinction is based on the structure of types. Primitive types like int and string have a prede ned equality operation, while equality can be de ned over compound types built up from primitive types using \concrete" constructions like product and disjoint union in the usual componentwise manner. Function types on the other hand do not posses a de nable equality relation (the existence of such a relation would solve the halting problem), nor do user-de ned abstract types (the compiler cannot determine when two concrete representations correspond to the same abstract value). As a rst approximation, therefore, the types admitting equality can be identi ed with the \hereditarily concrete" types built from primitive types using concrete type Information and Computation, vol. 2 (1993), pp. 303{323.