Question

I have a foo function defined as follows

fun foo x y = x (x (x y));

How to deduce the function type?

The answer is:

val foo = fn : ('a -> 'a) -> 'a -> 'a

Answer 1

This process is called type inference which is done in SML by Hindley–Milner algorithm.

First let’s start with a generic type signature for foo:

val foo = fn : 'c -> 'b -> 'a

where x has type 'c and y has type 'b.

We have the following steps:

• x y is function application, x should have signature 'b -> 'd and we have an equation 'c = 'b -> 'd and x y has type 'd.
• x (x y) means we apply a function type 'b -> 'd on an argument type 'd so 'd = 'b and 'c = 'b -> 'b.
• x (x (x y)) means we apply a function type 'b -> 'b on an argument type 'b and return 'a (type of foo); it suffices 'b = 'a.
• after unification, we have 'c = 'a -> 'a and 'b = 'a and foo has a generic type: val foo = fn : ('a -> 'a) -> 'a -> 'a