6.4.15. Arbitrary-rank polymorphism

RankNTypes

Implies:ExplicitForAll Since:6.8.1

Allow types of arbitrary rank.

Rank2Types

Since:6.8.1

A deprecated alias of RankNTypes.

GHC’s type system supports arbitrary-rank explicit universal quantification in types. For example, all the following types are legal:

f1 :: forall a b. a -> b -> a
g1 :: forall a b. (Ord a, Eq  b) => a -> b -> a

f2 :: (forall a. a->a) -> Int -> Int
g2 :: (forall a. Eq a => [a] -> a -> Bool) -> Int -> Int

f3 :: ((forall a. a->a) -> Int) -> Bool -> Bool

f4 :: Int -> (forall a. a -> a)

Here, f1 and g1 are rank-1 types, and can be written in standard Haskell (e.g. f1 :: a->b->a). The forall makes explicit the universal quantification that is implicitly added by Haskell.

The functions f2 and g2 have rank-2 types; the forall is on the left of a function arrow. As g2 shows, the polymorphic type on the left of the function arrow can be overloaded.

The function f3 has a rank-3 type; it has rank-2 types on the left of a function arrow.

The language option RankNTypes (which implies ExplicitForAll) enables higher-rank types. That is, you can nest foralls arbitrarily deep in function arrows. For example, a forall-type (also called a “type scheme”), including a type-class context, is legal:

• On the left or right (see f4, for example) of a function arrow
• As the argument of a constructor, or type of a field, in a data type declaration. For example, any of the f1, f2, f3, g1, g2 above would be valid field type signatures.
• As the type of an implicit parameter
• In a pattern type signature (see Lexically scoped type variables)

The RankNTypes option is also required for any type with a forall or context to the right of an arrow (e.g. f :: Int -> forall a. a->a, or g :: Int -> Ord a => a -> a). Such types are technically rank 1, but are clearly not Haskell-98, and an extra extension did not seem worth the bother.

In particular, in data and newtype declarations the constructor arguments may be polymorphic types of any rank; see examples in Examples. Note that the declared types are nevertheless always monomorphic. This is important because by default GHC will not instantiate type variables to a polymorphic type (Impredicative polymorphism).

The obsolete language option Rank2Types is a synonym for RankNTypes. They used to specify finer distinctions that GHC no longer makes.

6.4.15.1. Examples

These are examples of data and newtype declarations whose data constructors have polymorphic argument types:

data T a = T1 (forall b. b -> b -> b) a

data MonadT m = MkMonad { return :: forall a. a -> m a,
                          bind   :: forall a b. m a -> (a -> m b) -> m b
                        }

newtype Swizzle = MkSwizzle (forall a. Ord a => [a] -> [a])

The constructors have rank-2 types:

T1 :: forall a. (forall b. b -> b -> b) -> a -> T a

MkMonad :: forall m. (forall a. a -> m a)
                  -> (forall a b. m a -> (a -> m b) -> m b)
                  -> MonadT m

MkSwizzle :: (forall a. Ord a => [a] -> [a]) -> Swizzle

In earlier versions of GHC, it was possible to omit the forall in the type of the constructor if there was an explicit context. For example:

newtype Swizzle' = MkSwizzle' (Ord a => [a] -> [a])

Since GHC 8.0 declarations such as MkSwizzle' will cause an out-of-scope error.

You construct values of types T1, MonadT, Swizzle by applying the constructor to suitable values, just as usual. For example,

a1 :: T Int
a1 = T1 (\x y->x) 3

a2, a3 :: Swizzle
a2 = MkSwizzle sort
a3 = MkSwizzle reverse

a4 :: MonadT Maybe
a4 = let r x = Just x
     b m k = case m of
           Just y -> k y
           Nothing -> Nothing
     in
     MkMonad r b

mkTs :: (forall b. b -> b -> b) -> a -> a -> [T a]
mkTs f x y = [T1 f x, T1 f y]

The type of the argument can, as usual, be more general than the type required, as (MkSwizzle reverse) shows. (reverse does not need the Ord constraint.)

When you use pattern matching, the bound variables may now have polymorphic types. For example:

f :: T a -> a -> (a, Char)
f (T1 w k) x = (w k x, w 'c' 'd')

g :: (Ord a, Ord b) => Swizzle -> [a] -> (a -> b) -> [b]
g (MkSwizzle s) xs f = s (map f (s xs))

h :: MonadT m -> [m a] -> m [a]
h m [] = return m []
h m (x:xs) = bind m x          $ \y ->
             bind m (h m xs)   $ \ys ->
             return m (y:ys)

In the function h we use the record selectors return and bind to extract the polymorphic bind and return functions from the MonadT data structure, rather than using pattern matching.

6.4.15.2. Subsumption

Suppose:

f1 :: (forall a b. Int -> a -> b -> b) -> Bool
g1 :: forall x y. Int -> y -> x -> x

f2 :: (forall a. (Eq a, Show a) => a -> a) -> Bool
g2 :: forall x. (Show x, Eq x) => Int -> a -> b -> b

then f1 g1 and f2 g2 are both well typed, despite the different order of type variables and constraints. What happens is that the argument is instantiated, and then re-generalised to match the type expected by the function.

But this instantiation and re-generalisation happens only at the top level of a type. In particular, none of this happens if the foralls are underneath an arrow. For example:

f3 :: (Int -> forall a b. a -> b -> b) -> Bool
g3a :: Int -> forall x y. x -> y -> y
g3b :: forall x. Int -> forall y. x -> y -> y
g3c :: Int -> forall x y. y -> x -> x

f4 :: (Int -> forall a. (Eq a, Show a) => a -> a) -> Bool
g4 ::  Int -> forall x. (Show x, Eq x) => x -> x) -> Bool

Then the application f3 g3a is well-typed, because g3a has a type that matches the type expected by f3. But f3 g3b is not well typed, because the foralls are in different places. Nor is f3 g3c, where the foralls are in the same place but the variables are in a different order. Similarly f4 g4 is not well typed, because the constraints appear in a different order.

These examples can be made to typecheck by eta-expansion. For example f3 (\x -> g3b x) is well typed, and similarly f3 (\x -> g3c x) and f4 (\x -> g4 x).

Historical note. Earlier versions of GHC allowed these now-rejected applications, by inserting automatic eta-expansions, as described in Section 4.6 of Practical type inference for arbitrary-aank types, where it is called “deep skolemisation”. But these automatic eta-expansions may silently change the semantics of the user’s program, and deep skolemisation was removed from the language by GHC Proposal #287. This proposal has many more examples.

6.4.15.3. Type inference

In general, type inference for arbitrary-rank types is undecidable. GHC uses an algorithm proposed by Odersky and Laufer (“Putting type annotations to work”, POPL‘96) to get a decidable algorithm by requiring some help from the programmer. We do not yet have a formal specification of “some help” but the rule is this:

For a lambda-bound or case-bound variable, x, either the programmer provides an explicit polymorphic type for x, or GHC’s type inference will assume that x’s type has no foralls in it.

What does it mean to “provide” an explicit type for x? You can do that by giving a type signature for x directly, using a pattern type signature (Lexically scoped type variables), thus:

\ f :: (forall a. a->a) -> (f True, f 'c')

Alternatively, you can give a type signature to the enclosing context, which GHC can “push down” to find the type for the variable:

(\ f -> (f True, f 'c')) :: (forall a. a->a) -> (Bool,Char)

Here the type signature on the expression can be pushed inwards to give a type signature for f. Similarly, and more commonly, one can give a type signature for the function itself:

h :: (forall a. a->a) -> (Bool,Char)
h f = (f True, f 'c')

You don’t need to give a type signature if the lambda bound variable is a constructor argument. Here is an example we saw earlier:

f :: T a -> a -> (a, Char)
f (T1 w k) x = (w k x, w 'c' 'd')

Here we do not need to give a type signature to w, because it is an argument of constructor T1 and that tells GHC all it needs to know.

6.4.15.4. Implicit quantification

GHC performs implicit quantification as follows. At the outermost level (only) of user-written types, if and only if there is no explicit forall, GHC finds all the type variables mentioned in the type that are not already in scope, and universally quantifies them. For example, the following pairs are equivalent:

f :: a -> a
f :: forall a. a -> a

g (x::a) = let
              h :: a -> b -> b
              h x y = y
           in ...
g (x::a) = let
              h :: forall b. a -> b -> b
              h x y = y
           in ...

Notice that GHC always adds implicit quantifiers at the outermost level of a user-written type; it does not find the inner-most possible quantification point. For example:

f :: (a -> a) -> Int
         -- MEANS
f :: forall a. (a -> a) -> Int
         -- NOT
f :: (forall a. a -> a) -> Int


g :: (Ord a => a -> a) -> Int
         -- MEANS
g :: forall a. (Ord a => a -> a) -> Int
         -- NOT
g :: (forall a. Ord a => a -> a) -> Int

If you want the latter type, you can write your foralls explicitly. Indeed, doing so is strongly advised for rank-2 types.

Sometimes there is no “outermost level”, in which case no implicit quantification happens:

data PackMap a b s t = PackMap (Monad f => (a -> f b) -> s -> f t)

This is rejected because there is no “outermost level” for the types on the RHS (it would obviously be terrible to add extra parameters to PackMap), so no implicit quantification happens, and the declaration is rejected (with “f is out of scope”). Solution: use an explicit forall:

data PackMap a b s t = PackMap (forall f. Monad f => (a -> f b) -> s -> f t)