Proofs about a Folklore Let - Polymorphic TypeInference

The Hindley/Milner let-polymorphic type inference system has two diierent algorithms: one is the de facto standard Algorithm W that is bottom-up (or context-insensitive), and the other is a \folklore" algorithm that is top-down (or context-sensitive). Because the latter algorithm has not been formally presented with its soundness and completeness proofs, and its relation with the W algorithm has not been rigorously investigated, its use in place of (or in combination with) W is not well founded. In this article, we formally deene the context-sensitive, top-down type inference algorithm (named \M"), prove its soundness and completeness, and show a distinguishing property that M always stops earlier than W if the input program is ill typed. Our proofs can be seen as theoretical justiications for various type-checking strategies being used in practice. 1. INTRODUCTION Algorithm W , which is the standard presentation of the Hindley/Milner let-poly-morphic type inference system, fails late if the input program has a type error. Because the algorithm fails only at an application expression where its two subex-pressions (function and argument) have connicting types, an erroneous expression is often successfully type-checked long before its consequence collides at an application expression. This \bottom-up" Algorithm W thus reports the whole application expression as the problem area, implying some of its subexpressions are ill typed. Such a large type-error message does not help the programmer to nd the cause of the type problem. A diierent type inference algorithm, which has been used in an early ML compiler Leroy 1993], can cure this problem. This folklore algorithm carries a type