When are type signatures necessary in Haskell?

Question

Many introductory texts will tell you that in Haskell type signatures are \"almost always\" optional. Can anybody quantify the \"almost\" part?

As far as I can tell,

Answer 1

Polymorphic recursion needs type annotations, in general.

f :: (a -> a) -> (a -> b) -> Int -> a -> b
f f1 g n x = 
    if n == (0 :: Int)
    then g x
    else f f1 (\z h -> g (h z)) (n-1) x f1

(Credit: Patrick Cousot)

Note how the recursive call looks badly typed (!): it calls itself with five arguments, despite f having only four! Then remember that b can be instantiated with c -> d, which causes an extra argument to appear.

The above contrived example computes

f f1 g n x = g (f1 (f1 (f1 ... (f1 x))))

where f1 is applied n times. Of course, there is a much simpler way to write an equivalent program.