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Rank-N types

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Normal Haskell '98 types are considered Rank-1 types. A Haskell '98 type signature such as
 
Normal Haskell '98 types are considered Rank-1 types. A Haskell '98 type signature such as
+
<haskell>a -> b -> a</haskell>
<hask>a -> b -> a</hask>
 
 
 
implies that the type variables are universally quantified like so:
 
implies that the type variables are universally quantified like so:
+
<haskell>forall a b. a -> b -> a</haskell>
<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:
+
<haskell>forall a. a -> (forall b. b -> a)</haskell>
<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.
 
is also a Rank-1 type because it is equivalent to the previous signature.
   
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In general, you can replace
 
In general, you can replace
  +
<haskell>data T a1 .. ai = forall t1 .. tj. constraints => Constructor e1 .. ek</haskell>
  +
(where <hask>e1..ek</hask> are types in terms of <hask>a1..ai</hask> and <hask>t1..tj</hask>)
   
<hask>data T a1 .. ai = forall t1 .. tj. constraints => Constructor e1 .. ek</hask> (where <hask>e1..ek</hask> are types in terms of <hask>a1..ai</hask> and <hask>t1..tj</hask>)
+
<haskell>Constructor exp1 .. expk -- application of the constructor</haskell>
 
<hask>Constructor exp1 .. expk -- application of the constructor</hask>
 
   
<hask>case e of (Constructor pat1 .. patk) -> res</hask>
+
<haskell>case e of (Constructor pat1 .. patk) -> res</haskell>
   
 
with
 
with
   
<hask>data T' a1 .. ai = Constructor' (forall b. (forall t1..tj. constraints => e1 -> e2 -> ... -> ek -> b) -> b)</hask>
+
<haskell>data T' a1 .. ai = Constructor' (forall b. (forall t1..tj. constraints => e1 -> e2 -> ... -> ek -> b) -> b)</haskell>
   
<hask>Constructor' (\f -> f exp1 .. expk)</hask>
+
<haskell>Constructor' (\f -> f exp1 .. expk)</haskell>
   
<hask>case e of (Constructor' f) -> let k pat1 .. patk = res in f k</hask>
+
<haskell>case e of (Constructor' f) -> let k pat1 .. patk = res in f k</haskell>
   
== Also see ==
+
== See also ==
   
[http://hackage.haskell.org/trac/haskell-prime/wiki/RankNTypes Rank-N types] on the Haskell' website.
+
* [http://hackage.haskell.org/trac/haskell-prime/wiki/RankNTypes Rank-N types] on the Haskell' website.
  +
* [http://www.haskell.org/ghc/docs/latest/html/users_guide/other-type-extensions.html#universal-quantification The GHC User's Guide on higher-ranked polymorphism].

Revision as of 01:04, 6 September 2012


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 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
(where
e1..ek
are types in terms of
a1..ai
and
t1..tj
)
Constructor exp1 .. expk -- application of the constructor
case e of (Constructor pat1 .. patk) -> res

with

data T' a1 .. ai = Constructor' (forall b. (forall t1..tj. 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