Subtyping arithmetical types

Abstract

We consider the type system formed by a finite set of primitive types such as integer, character, real, etc., and three type construction operators: (i) Cartesian product, (ii) disjoint sum, and (iii) recursive type definitions. Type equivalence is defined to obey the arithmetical rules: commutativity and associativity of product and sum and distributivity of product over sum. We offer a compact representation of the types in this system as multivariate algebraic functions. This type system admits two natural notions of subtyping: "multiplicative", which roughly corresponds to the notion of object-oriented subtyping, and "additive", which seems to be more appropriate in our context. Both kinds of subtyping can be efficiently computed if no recursive definitions are allowed. Our main result is that additive subtyping is undecidable in the general case. Perhaps surprisingly, this undecidability result is by reduction from Hilbert's Tenth Problem (H10): the solution of Diophantine equations. © 2001 ACM.