Haskell Weird Kinds: Kind of (->) is ?? -> ? -> *

Question

When I was experimenting with Haskell kinds, and trying to get the kind of ->, and this showed up:

$ ghci
...
Prelude> :k (->)
(->)          


        

Answer 1

These are GHC-specific extensions of the Haskell kind system. The Haskell 98 report specifies only a simple kind system:

... type expressions are classified into different kinds, which take one of two possible forms:

The symbol * represents the kind of all nullary type constructors. If k1 and k2 are kinds, then k1->k2 is the kind of types that take a type of kind k1 and return a type of kind k2.

GHC extends this system with a form of kind subtyping, to allow unboxed types, and to allow the function construtor to be polymorphic over kinds. The kind lattice GHC supports is:

             ?
             /\
            /  \
          ??   (#)
          / \     
         *   #     

Where:       *   [LiftedTypeKind]   means boxed type
             #   [UnliftedTypeKind] means unboxed type
            (#)  [UbxTupleKind]     means unboxed tuple
            ??   [ArgTypeKind]      is the lub of {*, #}
            ?    [OpenTypeKind]     means any type at all

Defined in ghc/compiler/types/Type.lhs

In particular:

> error :: forall a:?. String -> a
> (->)  :: ?? -> ? -> *
> (\\(x::t) -> ...)

Where in the last example t :: ?? (i.e. is not an unboxed tuple). So, to quote GHC, "there is a little subtyping at the kind level".

For interested souls, GHC also supports coercion types and kinds ("type-level terms which act as evidence for type equalities", as needed by System Fc) used in GADTs, newtypes and type families.