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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
forall
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
forall
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
forall
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
forall
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 #-}
or
{-# LANGUAGE RankNTypes #-}
.


2 Also see

Rank-N types on the Haskell' website.