Impredicative types

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Impredicative types are an advanced form of polymorphism, to be contrasted with rank-N types.

Standard Haskell allows polymorphic types via the use of type variables, which are understood to be universally quantified: id :: a -> a means "for all types a, id can take an argument and return a result of that type". All universal quantifiers ("for all"s) must appear at the beginning of a type.

Higher-rank polymorphism (e.g. rank-N types) allows universal quantifiers to appear inside function types as well. It turns out that appearing to the right of function arrows is not interesting: Int -> forall a. a -> [a] is actually the same as forall a. Int -> a -> [a]. However, higher-rank polymorphism allows quantifiers to the left of function arrows, too, and (forall a. [a] -> Int) -> Int really is different from forall a. ([a] -> Int) -> Int.

Impredicative types take this idea to its natural conclusion: universal quantifiers are allowed anywhere in a type, even inside normal datatypes like lists or Maybe. The GHC User's Guide gives the following example:

f :: Maybe (forall a. [a] -> [a]) -> Maybe ([Int], [Char])
f (Just g) = Just (g [3], g "hello")
f Nothing  = Nothing

However, impredicative types do not mix very well with Haskell's type inference, so to actually use the above function with GHC 7.6.1 you need to specify the full (unpleasant) type signature for the Just constructor:

ghci> f ((Just :: (forall a. [a] -> [a]) -> Maybe (forall a. [a] -> [a])) reverse)
Just ([3],"olleh")

Other examples are more successful: see below.

See also