Personal tools

Rank-N types

From HaskellWiki

(Difference between revisions)
Jump to: navigation, search
m (+category)
(a real description of rank n types)
Line 4: Line 4:
 
== About ==
 
== About ==
   
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
  +
  +
<hask>a -> b -> a</hask>
  +
  +
implies that the type variables are universally quantified like so:
  +
  +
<hask>forall a b. a -> b -> a</hask>
  +
  +
<hask>forall</hask> can be floated out of the right-hand side of <hask>(->)</hask> if it appears there, so:
  +
  +
<hask>forall a. a -> (forall b. b -> a)</hask>
  +
  +
is also a Rank-1 type because it is equivalent to the previous signature.
  +
  +
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:
  +
  +
<hask>(forall a. a -> a) -> (forall b. b -> b)</hask>
  +
  +
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.
  +
  +
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
  +
<hask>{-# LANGUAGE Rank2Types #-}</hask> or <hask>{-# LANGUAGE RankNTypes #-}</hask>.
   
Rank-2 types are a special case of rank-N types, and normal Haskell 98 types are all rank-1 types.
 
   
 
== 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
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.