Weak Polymorphism Can Be

The weak polymorphic type system of Standard ML of New Jersey (SML/NJ) (MacQueen, 1992) has only been presented as part of the implementation of the SML/NJ compiler, not as a formal type system. As a result, it is not well understood. And while numerous versions of the implementation have been shown unsound, the concept has not been proved sound or unsound. We present an explanation of weak polymorphism and show that a formalization of this is sound. We also relate this to the SML/NJ implementation of weak polymorphism through a series of type systems that incorporate elements of the SML/NJ type inference algorithm. 1 Background A reference cell is an assignable store location and is the primary imperative feature of Standard ML (SML) (Milner, et al., 1990). However, it is well known that its Hindley-Milner type system (Milner, 1978) is unsound if reference types are polymorphically generalized in the usual manner. For example, in the following expression, polymorphically generalizing the type of ref nil allows that reference cell to be used as both an int list ref and a bool list ref: let val a = ref nil in a := 1]; not (hd(!a)) end This code stores a list of integers in the cell, but then reads a list of booleans from it. Such inconsistencies must not be allowed by the type system. While it is unsound to polymorphically generalize reference types, it is not necessarily unsound to generalize function types involving reference types. A simple example is the following expression: let val ref' = fn x => ref x in ref' 1; ref' true end 2 John Greiner The function ref', which creates reference cells, can safely be used at both the types int-> int ref and bool-> bool ref. A number of type systems have been proposed to allow code such as the latter example while preserving soundness Of particular interest for this paper are those of Tofte (1988), which is used in the deenition of SML, and MacQueen (1992), which is used in the New Jersey implementation of SML (SML/NJ). In the standard Hindley-Milner type system, generalization is allowed on all free type variables not occurring in the variable type assumption, which is a mapping from variables to type schemes. As shown by Tofte, if references are added to the language, the type system must also have a location type assumption assigning a type to each …