Question

What is the difference(s) between them?

 (i)
 gen ::  (a -> a -> a ) -> a -> a 
 gen  f x   = f x

 (ii)
 gen ::  (a -> a ) -> a -> a 
 gen  f x   = f x

 (iii)
 gen ::  (a -> a -> a -> a ) -> a -> a 
 gen  f x   = f x

first one gives error : “cannot construct infinite … “

second one is work

third one not work

Answer 1

Let’s try to figure out the type of the first function:
Imagine a is Int and f is normal addition. The second argument would be an Int. If you call e.g. gen (+) 3, the result is a function equivalent to (+3), which takes an Int and returns an Int. But your signature says that you just give back an Int.

So basically the compiler complains because it expects an a, and you give it an a->a, and there is no way to unify these types.

To fix it, you could either correct the signature, which would be gen :: (a -> a -> a ) -> a -> (a -> a), or to change the definition, e.g. gen f x = f x x

The second one is just a specialization of the identity function id :: t -> t.

The third one is similar to the first one.