6.4.12. Levity 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 levity 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.

6.4.12.1. No levity-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 levity polymorphic. We thus forbid such constructions, via the following straightforward rule:

No variable may have a levity-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 levity polymorphic. There are no variables with a levity-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 levity-polymorphic.

6.4.12.2. Levity-polymorphic bottoms¶

We can use levity 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 levity-polymorphic variable, and so are accepted. Their polymorphism allows users to use these to conveniently stub out functions that return unboxed types.

6.4.12.3. Printing levity-polymorphic types¶

-fprint-explicit-runtime-reps¶: Print RuntimeRep parameters as they appear; otherwise, they are defaulted to LiftedRep.

Most GHC users will not need to worry about levity polymorphism or unboxed types. For these users, seeing the levity 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 levity 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