# Difference between revisions of "Rank-N types"

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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. |
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+ | 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

## 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 #-}`

.

## Also see

Rank-N types on the Haskell' website.