# Rank-N types

### From HaskellWiki

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== About == |
== About == |
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− | As best as I can tell, rank-N types are exactly like [[existential type]]s - except that they're completely different. |
+ | Normal Haskell '98 types are considered Rank-1 types. A Haskell '98 type signature such as |

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+ | <hask>a -> b -> a</hask> |
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+ | implies that the type variables are universally quantified like so: |
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+ | <hask>forall a b. a -> b -> a</hask> |
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+ | <hask>forall</hask> can be floated out of the right-hand side of <hask>(->)</hask> if it appears there, so: |
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+ | <hask>forall a. a -> (forall b. b -> a)</hask> |
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+ | is also a Rank-1 type because it is equivalent to the previous signature. |
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+ | However, a <hask>forall</hask> appearing within the left-hand side of <hask>(->)</hask> cannot be moved up, and therefore forms another level or rank. The type is labeled "Rank-N" where N is the number of <hask>forall</hask>s which are nested and cannot be merged with a previous one. For example: |
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+ | <hask>(forall a. a -> a) -> (forall b. b -> b)</hask> |
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+ | is a Rank-2 type because the latter <hask>forall</hask> can be moved to the start but the former one cannot. Therefore, there are two levels of universal quantification. |
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+ | Rank-N type reconstruction is undecidable in general, and some explicit type annotations are required in their presence. |
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+ | Rank-2 or Rank-N types may be specifically enabled by the language extensions |
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+ | <hask>{-# LANGUAGE Rank2Types #-}</hask> or <hask>{-# LANGUAGE RankNTypes #-}</hask>. |
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− | Rank-2 types are a special case of rank-N types, and normal Haskell 98 types are all rank-1 types. |
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== Also see == |
== Also see == |

## Revision as of 12:48, 26 August 2007

## 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

(->)

forall a. a -> (forall b. b -> a)

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

However, aforall

(->)

forall

(forall a. a -> a) -> (forall b. b -> b)

forall

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 Also see

Rank-N types on the Haskell' website.