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Rank-N types

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1 About

Normal Haskell '98 types are considered Rank-1 types. A Haskell '98 type signature such as

a -> b -> a

implies that the type variables are universally quantified like so:

forall a b. a -> b -> a
can be floated out of the right-hand side of
if it appears there, so:
forall a. a -> (forall b. b -> a)

is also a Rank-1 type because it is equivalent to the previous signature.

However, a
appearing within the left-hand side of
cannot be moved up, and therefore forms another level or rank. The type is labeled "Rank-N" where N is the number of
s which are nested and cannot be merged with a previous one. For example:
(forall a. a -> a) -> (forall b. b -> b)
is a Rank-2 type because the latter
can be moved to the start but the former one cannot. Therefore, there are two levels of universal quantification.

Rank-N type reconstruction is undecidable in general, and some explicit type annotations are required in their presence.

Rank-2 or Rank-N types may be specifically enabled by the language extensions

{-# LANGUAGE Rank2Types #-}
{-# LANGUAGE RankNTypes #-}

2 Relation to Existentials

In order to unpack an existential type, you need a polymorphic function that works on any type that could be stored in the existential. This leads to a natural relation between higher-rank types and existentials; and an encoding of existentials in terms of higher rank types in continuation-passing style.

In general, you can replace

data T a1 .. ai = forall t1 .. tj. constraints => Constructor e1 .. ek
are types in terms of
Constructor exp1 .. expk -- application of the constructor
case e of (Constructor pat1 .. patk) -> res


data T' a1 .. ai = Constructor' (forall b. (forall constraints => e1 -> e2 -> ... -> ek -> b) -> b)
Constructor' (\f -> f exp1 .. expk)
case e of (Constructor' f) -> let k pat1 .. patk = res in f k

3 See also