Question

This question is taken from an exam. I don’t know how to do that. 🙁

Question: Give an example of a haskell or ml function whose type is

( a -> b ) -> ( c -> a ) -> c -> b

How to do that?

Answer 1

What meaningful function could have the type mystery :: ( a -> b ) -> ( c -> a ) -> c -> b?

Let’s see, what could

mystery f g x

be?

It has three things to work with,

• one value x of type c,
• one function g of type c -> a and
• one function f of type a -> b.

It shall produce a value of type b.

The only argument that has anything to do with b is the function f, so the result must be f ???.

What can be the argument of f here? It must have type a, and the only way to produce a value of that type from the given arguments (ignoring bottoms, undefined, error "foo") is applying g to something, so it must be

mystery f g x = f (g ??)

But what could g be applied to? That must be a value of type c. Apart from bottoms, the only value of type c that can be constructed from the arguments is x, so

mystery f g x = f (g x)

must be function composition (or undefined).