6.4.14. Representation polymorphism

In order to allow full flexibility in how kinds are used, it is necessary to use the kind system to differentiate between boxed, lifted types (normal, everyday types like Int and [Bool]) and unboxed, primitive types (Unboxed types and primitive operations) like Int#. We thus have so-called representation polymorphism.

Here are the key definitions, all available from GHC.Exts:

TYPE :: RuntimeRep -> Type   -- highly magical, built into GHC

data Levity = Lifted    -- for things like `Int`
            | Unlifted  -- for things like `Array#`

data RuntimeRep = BoxedRep Levity  -- for anything represented by a GC-managed pointer
                | IntRep           -- for `Int#`
                | TupleRep [RuntimeRep]  -- unboxed tuples, indexed by the representations of the elements
                | SumRep [RuntimeRep]    -- unboxed sums, indexed by the representations of the disjuncts
                | ...

type LiftedRep = BoxedRep Lifted

type Type = TYPE LiftedRep    -- Type is just an ordinary type synonym

The idea is that we have a new fundamental type constant TYPE, which is parameterised by a RuntimeRep. We thus get Int# :: TYPE IntRep and Bool :: TYPE LiftedRep. Anything with a type of the form TYPE x can appear to either side of a function arrow ->. We can thus say that -> has type TYPE r1 -> TYPE r2 -> TYPE LiftedRep. The result is always lifted because all functions are lifted in GHC. Levity polymorphism

A special case of representation polymorphism is levity polymorphism, where we abstract over a variable of kind Levity, such as:

example :: forall (l :: Levity) (a :: TYPE (BoxedRep l)). (Int -> a) -> a
example f = f 42

With UnliftedDatatypes, we can even declare levity-polymorphic data types:

type PEither :: Type -> Type -> TYPE (BoxedRep l)
data PEither l r = PLeft l | PRight r No representation-polymorphic variables or arguments

If GHC didn’t have to compile programs that run in the real world, that would be the end of the story. But representation polymorphism can cause quite a bit of trouble for GHC’s code generator. Consider

bad :: forall (r1 :: RuntimeRep) (r2 :: RuntimeRep)
              (a :: TYPE r1) (b :: TYPE r2).
       (a -> b) -> a -> b
bad f x = f x

This seems like a generalisation of the standard $ operator. If we think about compiling this to runnable code, though, problems appear. In particular, when we call bad, we must somehow pass x into bad. How wide (that is, how many bits) is x? Is it a pointer? What kind of register (floating-point or integral) should x go in? It’s all impossible to say, because x’s type, a :: TYPE r1 is representation-polymorphic. We thus forbid such constructions, via the following straightforward rule:

No variable may have a representation-polymorphic type.

This eliminates bad because the variable x would have a representation-polymorphic type.

However, not all is lost. We can still do this:

($) :: forall r (a :: Type) (b :: TYPE r).
       (a -> b) -> a -> b
f $ x = f x

Here, only b is representation-polymorphic. There are no variables with a representation-polymorphic type. And the code generator has no trouble with this. Indeed, this is the true type of GHC’s $ operator, slightly more general than the Haskell 98 version.

Because the code generator must store and move arguments as well as variables, the logic above applies equally well to function arguments, which may not be representation-polymorphic. Representation-polymorphic bottoms

We can use representation polymorphism to good effect with error and undefined, whose types are given here:

undefined :: forall (r :: RuntimeRep) (a :: TYPE r).
             HasCallStack => a
error :: forall (r :: RuntimeRep) (a :: TYPE r).
         HasCallStack => String -> a

These functions do not bind a representation-polymorphic variable, and so are accepted. Their polymorphism allows users to use these to conveniently stub out functions that return unboxed types. Inference and defaulting

GHC does not infer representation-polymorphic types. If the representation of a variable is not specified, it will be assumed to be LiftedRep. For example, if you write f a b = a b, the inferred type of f will be

f :: forall {a :: Type} {b :: Type}. (a -> b) -> a -> b

even though

f :: forall {rep} {a :: Type} {b :: TYPE rep}. (a -> b) -> a -> b

would also be legal, as described above.

Likewise, in a user-written signature f :: forall a b. (a -> b) -> a -> b GHC will assume that both a and b have kind Type. To use a different representation, you have to specify the kinds of a and b.

During type inference, GHC does not quantify over variables of kind RuntimeRep nor Levity. Instead, they are defaulted to LiftedRep and Lifted respectively. Likewise, Multiplicity variables (Linear types) are defaulted to Many. Printing representation-polymorphic types


Print RuntimeRep and Levity parameters as they appear; otherwise, they are defaulted to LiftedRep and Lifted, respectively.

Most GHC users will not need to worry about representation polymorphism or unboxed types. For these users, seeing the representation polymorphism in the type of $ is unhelpful. And thus, by default, it is suppressed, by supposing all type variables of type RuntimeRep to be LiftedRep when printing, and printing TYPE LiftedRep as Type (or * when StarIsType is on).

Should you wish to see representation polymorphism in your types, enable the flag -fprint-explicit-runtime-reps. For example,

ghci> :t ($)
($) :: (a -> b) -> a -> b
ghci> :set -fprint-explicit-runtime-reps
ghci> :t ($)
  :: forall (r :: GHC.Types.RuntimeRep) a (b :: TYPE r).
     (a -> b) -> a -> b