My understanding was that the implicitly polymorphic levity, did (->) not change because it's a type constructor?

Prelude> :info (->)

data (->) a b -- Defined in ‘GHC.Prim’

Prelude> :k (->)

(->) :: * -> * -> *

Basically I'm asking why ($) changed and (->) did not when (->) had similar properties WRT * and #.

Also does this encapsulate the implicit impredicativity of ($) for making runST $ work? I don't presently see how it would.

Worry not about the book, we already hand-wave FTP effectively. One more type shouldn't change much.

Thank you very much for answering, this has been very helpful already :)

--- Chris

On Thu, Feb 4, 2016 at 12:52 PM, Ryan Scott <ryan.gl.scott@gmail.com> wrote:

Hi Chris,

The change to ($)'s type is indeed intentional. The short answer is
that ($)'s type prior to GHC 8.0 was lying a little bit. If you
defined something like this:

    unwrapInt :: Int -> Int#
    unwrapInt (I# i) = i

You could write an expression like (unwrapInt $ 42), and it would
typecheck. But that technically shouldn't be happening, since ($) ::
(a -> b) -> a -> b, and we all know that polymorphic types have to
live in kind *. But if you look at unwrapInt :: Int -> Int#, the type
Int# certainly doesn't live in *. So why is this happening?

The long answer is that prior to GHC 8.0, in the type signature ($) ::
(a -> b) -> a -> b, b actually wasn't in kind *, but rather OpenKind.
OpenKind is an awful hack that allows both lifted (kind *) and
unlifted (kind #) types to inhabit it, which is why (unwrapInt $ 42)
typechecks. To get rid of the hackiness of OpenKind, Richard Eisenberg
extended the type system with levity polymorphism [1] to indicate in
the type signature where these kind of scenarios are happening.

So in the "new" type signature for ($):

    ($) :: forall (w :: Levity) a (b :: TYPE w). (a -> b) -> a -> b

The type b can either live in kind * (which is now a synonym for TYPE
'Lifted) or kind # (which is a synonym for TYPE 'Unlifted), which is
indicated by the fact that TYPE w is polymorphic in its levity type w.

Truth be told, there aren't that many Haskell functions that actually
levity polymorphic, since normally having an argument type that could
live in either * or # would wreak havoc with the RTS (otherwise, how
would it know if it's dealing with a pointer or a value on the
stack?). But as it turns out, it's perfectly okay to have a levity
polymorphic type in a non-argument position [2]. Indeed, in the few
levity polymorphic functions that I can think of:

    ($)        :: forall (w :: Levity) a (b :: TYPE w). (a -> b) -> a -> b
    error     :: forall (v :: Levity)  (a :: TYPE v). HasCallStack =>
[Char] -> a
    undefined :: forall (v :: Levity) (a :: TYPE v). HasCallStack => a

The levity polymorphic type never appears directly to the left of an arrow.

The downside of all this is, of course, that the type signature of ($)
might look a lot scarier to beginners. I'm not sure how you'd want to
deal with this, but for 99% of most use cases, it's okay to lie and
state that ($) :: (a -> b) -> a -> b. You might have to include a
disclaimer that if they type :t ($) into GHCi, they should be prepared
for some extra information!

Ryan S.
-----
[1] https://ghc.haskell.org/trac/ghc/wiki/NoSubKinds
[2] https://ghc.haskell.org/trac/ghc/ticket/11473
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--

Chris Allen

Currently working on http://haskellbook.com

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