7.8. Kind polymorphism

This section describes kind polymorphism, and extension enabled by -XPolyKinds. It is described in more detail in the paper Giving Haskell a Promotion, which appeared at TLDI 2012.

7.8.1. Overview of kind polymorphism

Currently there is a lot of code duplication in the way Typeable is implemented (Section 7.5.4, “Deriving Typeable instances”):

class Typeable (t :: *) where
  typeOf :: t -> TypeRep

class Typeable1 (t :: * -> *) where
  typeOf1 :: t a -> TypeRep

class Typeable2 (t :: * -> * -> *) where
  typeOf2 :: t a b -> TypeRep

Kind polymorphism (with -XPolyKinds) allows us to merge all these classes into one:

data Proxy t = Proxy

class Typeable t where
  typeOf :: Proxy t -> TypeRep

instance Typeable Int  where typeOf _ = TypeRep
instance Typeable []   where typeOf _ = TypeRep

Note that the datatype Proxy has kind forall k. k -> * (inferred by GHC), and the new Typeable class has kind forall k. k -> Constraint.

7.8.2. Overview

Generally speaking, with -XPolyKinds, GHC will infer a polymorphic kind for un-decorated declarations, whenever possible. For example:

data T m a = MkT (m a)
-- GHC infers kind   T :: forall k. (k -> *) -> k -> *

Just as in the world of terms, you can restrict polymorphism using a kind signature (sometimes called a kind annotation) (-XPolyKinds implies -XKindSignatures):

data T m (a :: *) = MkT (m a)
-- GHC now infers kind   T :: (* -> *) -> * -> *

There is no "forall" for kind variables. Instead, when binding a type variable, you can simply mention a kind variable in a kind annotation for that type-variable binding, thus:

data T (m :: k -> *) a = MkT (m a)
-- GHC now infers kind   T :: forall k. (k -> *) -> k -> *

The kind "forall" is placed just outside the outermost type-variable binding whose kind annotation mentions the kind variable. For example

f1 :: (forall a m. m a -> Int) -> Int
         -- f1 :: forall (k:BOX).
         --       (forall (a:k) (m:k->*). m a -> Int)
         --       -> Int

f2 :: (forall (a::k) m. m a -> Int) -> Int
         -- f2 :: (forall (k:BOX) (a:k) (m:k->*). m a -> Int)
         --       -> Int

Here in f1 there is no kind annotation mentioning the polymorphic kind variable, so k is generalised at the top level of the signature for f1, making the signature for f1 is as polymorphic as possible. But in the case of of f2 we give a kind annotation in the forall (a:k) binding, and GHC therefore puts the kind forall right there too.

(Note: These rules are a bit indirect and clumsy. Perhaps GHC should allow explicit kind quantification. But the implicit quantification (e.g. in the declaration for data type T above) is certainly very convenient, and it is not clear what the syntax for explicit quantification should be.)

7.8.3. Polymorphic kind recursion and complete kind signatures

Just as in type inference, kind inference for recursive types can only use monomorphic recursion. Consider this (contrived) example:

data T m a = MkT (m a) (T Maybe (m a))
-- GHC infers kind  T :: (* -> *) -> * -> *

The recursive use of T forced the second argument to have kind *. However, just as in type inference, you can achieve polymorphic recursion by giving a complete kind signature for T. A complete kind signature is present when all argument kinds and the result kind are known, without any need for inference. For example:

data T (m :: k -> *) :: k -> * where
  MkT :: m a -> T Maybe (m a) -> T m a

The complete user-supplied kind signature specifies the polymorphic kind for T, and this signature is used for all the calls to T including the recursive ones. In particular, the recursive use of T is at kind *.

What exactly is considered to be a "complete user-supplied kind signature" for a type constructor? These are the forms:

  • For a datatype, every type variable must be annotated with a kind. In a GADT-style declaration, there may also be a kind signature (with a top-level :: in the header), but the presence or absence of this annotation does not affect whether or not the declaration has a complete signature.

    data T1 :: (k -> *) -> k -> *       where ...   -- Yes  T1 :: forall k. (k->*) -> k -> *
    data T2 (a :: k -> *) :: k -> *     where ...   -- Yes  T2 :: forall k. (k->*) -> k -> *
    data T3 (a :: k -> *) (b :: k) :: * where ...   -- Yes  T3 :: forall k. (k->*) -> k -> *
    data T4 (a :: k -> *) (b :: k)      where ...   -- Yes  T4 :: forall k. (k->*) -> k -> *
    data T5 a (b :: k)             :: * where ...   -- NO  kind is inferred
    data T6 a b                         where ...   -- NO  kind is inferred
  • For a class, every type variable must be annotated with a kind.

  • For a type synonym, every type variable and the result type must all be annotated with kinds.

    type S1 (a :: k) = (a :: k)    -- Yes   S1 :: forall k. k -> k
    type S2 (a :: k) = a           -- No    kind is inferred
    type S3 (a :: k) = Proxy a     -- No    kind is inferred

    Note that in S2 and S3, the kind of the right-hand side is rather apparent, but it is still not considered to have a complete signature -- no inference can be done before detecting the signature.

  • An open type or data family declaration always has a complete user-specified kind signature; un-annotated type variables default to kind *.

    data family D1 a           	-- D1 :: * -> *
    data family D2 (a :: k)    	-- D2 :: forall k. k -> *
    data family D3 (a :: k) :: *    -- D3 :: forall k. k -> *
    type family S1 a :: k -> *      -- S1 :: forall k. * -> k -> *
    class C a where                 -- C  :: k -> Constraint
      type AT a b                   -- AT :: k -> * -> *

    In the last example, the variable a has an implicit kind variable annotation from the class declaration. It keeps its polymorphic kind in the associated type declaration. The variable b, however, gets defaulted to *.

  • A closed type familey has a complete signature when all of its type variables are annotated and a return kind (with a top-level ::) is supplied.

7.8.4. Kind inference in closed type families

Although all open type families are considered to have a complete user-specified kind signature, we can relax this condition for closed type families, where we have equations on which to perform kind inference. GHC will infer kinds for the arguments and result types of a closed type family.

GHC supports kind-indexed type families, where the family matches both on the kind and type. GHC will not infer this behaviour without a complete user-supplied kind signature, as doing so would sometimes infer non-principal types.

For example:

type family F1 a where
  F1 True  = False
  F1 False = True
  F1 x     = x
-- F1 fails to compile: kind-indexing is not inferred

type family F2 (a :: k) where
  F2 True  = False
  F2 False = True
  F2 x     = x
-- F2 fails to compile: no complete signature

type family F3 (a :: k) :: k where
  F3 True  = False
  F3 False = True
  F3 x     = x
-- OK

7.8.5. Kind inference in class instance declarations

Consider the following example of a poly-kinded class and an instance for it:

class C a where
  type F a

instance C b where
  type F b = b -> b

In the class declaration, nothing constrains the kind of the type a, so it becomes a poly-kinded type variable (a :: k). Yet, in the instance declaration, the right-hand side of the associated type instance b -> b says that b must be of kind *. GHC could theoretically propagate this information back into the instance head, and make that instance declaration apply only to type of kind *, as opposed to types of any kind. However, GHC does not do this.

In short: GHC does not propagate kind information from the members of a class instance declaration into the instance declaration head.

This lack of kind inference is simply an engineering problem within GHC, but getting it to work would make a substantial change to the inference infrastructure, and it's not clear the payoff is worth it. If you want to restrict b's kind in the instance above, just use a kind signature in the instance head.