runST and function composition

Question

Why does this typecheck:

runST $ return $ True

While the following does not:

runST . return $ True

GHCI complains:

Couldn't match expected type `forall s. ST s c0'
            with actual type `m0 a0'
Expected type: a0 -> forall s. ST s c0
  Actual type: a0 -> m0 a0
In the second argument of `(.)', namely `return'
In the expression: runST . return

Answer 1

The short answer is that type inference doesn't always work with higher-rank types. In this case, it is unable to infer the type of (.), but it type checks if we add an explicit type annotation:

> :m + Control.Monad.ST
> :set -XRankNTypes
> :t (((.) :: ((forall s0. ST s0 a) -> a) -> (a -> forall s1. ST s1 a) -> a -> a) runST return) $ True
(((.) :: ((forall s0. ST s0 a) -> a) -> (a -> forall s1. ST s1 a) -> a -> a) runST return) $ True :: Bool

The same problem also happens with your first example, if we replace ($) with our own version:

> let app f x = f x
> :t runST `app` (return `app` True)
<interactive>:1:14:
    Couldn't match expected type `forall s. ST s t0'
                with actual type `m0 t10'
    Expected type: t10 -> forall s. ST s t0
      Actual type: t10 -> m0 t10
    In the first argument of `app', namely `return'
    In the second argument of `app', namely `(return `app` True)'

Again, this can be solved by adding type annotations:

> :t (app :: ((forall s0. ST s0 a) -> a) -> (forall s1. ST s1 a) -> a) runST (return `app` True)
(app :: ((forall s0. ST s0 a) -> a) -> (forall s1. ST s1 a) -> a) runST (return `app` True) :: Bool

What is happening here is that there is a special typing rule in GHC 7 which only applies to the standard ($) operator. Simon Peyton-Jones explains this behavior in a reply on the GHC users mailing list:

This is a motivating example for type inference that can deal with impredicative types. Consider the type of ($):

($) :: forall p q. (p -> q) -> p -> q

In the example we need to instantiate p with (forall s. ST s a), and that's what impredicative polymorphism means: instantiating a type variable with a polymorphic type.

Sadly, I know of no system of reasonable complexity that can typecheck [this] unaided. There are plenty of complicated systems, and I have been a co-author on papers on at least two, but they are all Too Jolly Complicated to live in GHC. We did have an implementation of boxy types, but I took it out when implementing the new typechecker. Nobody understood it.

However, people so often write

runST $ do ... 

that in GHC 7 I implemented a special typing rule, just for infix uses of ($). Just think of (f $ x) as a new syntactic form, with the obvious typing rule, and away you go.

Your second example fails because there is no such rule for (.).