What is the meaning of parentheses in Haskell type signatures?

Question

Take the type signature of fmap (the Functor method) as an example:

(a -> b) -> f a -> f b

How is that different from the following type signature?

a -> b -> f a -> f b

Is there even a difference between those two type signatures?

Answer 1

Yes, there is a difference, because the -> type constructor is right-associative. In other words,

a -> b -> f a -> f b

is equivalent to

a -> (b -> (f a -> f b))

This type signature denotes a function that takes a parameter of type a and returns a function, which itself takes a parameter of type b and returns a function, which itself takes a parameter of type f a and returns a value of type f b.

On the other hand,

(a -> b) -> f a -> f b

denotes a function that takes a parameter of type a -> b (i.e. a function that takes a parameter of type a and returns a value of type b) and returns a function, which itself takes a parameter of type f a and returns a value of type f b.

Here is a contrived example that illustrates the difference between the two type signatures:

f :: (Int -> Bool) -> [Int] -> [Bool]
f = map

g :: Int -> Bool -> [Int] -> [Bool]
g n b = map (\n' -> (n' == n) == b)

λ> let ns = [42, 13, 42, 17]

λ> f (== 42) ns
[True,False,True,False]

λ> g 42 True ns
[True,False,True,False]

λ> g 42 False ns
[False,True,False,True]