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Impredicative types

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Impredicative types are to be contrasted with [[rank-N types]].
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Impredicative types are an advanced form of polymorphism, to be contrasted with [[rank-N types]].
   
A standard Haskell type is universally quantified by default, and quantifiers can only appear at the top level of a type or to the right of function arrows.
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Standard Haskell allows polymorphic types via the use of type variables, which are understood to be ''universally quantified'': <tt>id :: a -> a</tt> means "''for all'' types <tt>a</tt>, <tt>id</tt> can take an argument and return a result of that type". All universal quantifiers ("for all"s) must appear at the beginning of a type.
   
A higher-rank polymorphic type allows universal quantifiers to appear to the left of function arrows as well, so that function arguments can be functions that are themselves polymorphic.
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Higher-rank polymorphism (e.g. [[rank-N types]]) allows universal quantifiers to appear inside function types as well. It turns out that appearing to the right of function arrows is not interesting: <tt>Int -> forall a. a -> [a]</tt> is actually the same as <tt>forall a. Int -> a -> [a]</tt>. However, higher-rank polymorphism allows quantifiers to the ''left'' of function arrows, too, and <tt>(forall a. [a] -> Int) -> Int</tt> really ''is'' different from <tt>forall a. ([a] -> Int) -> Int</tt>.
   
An impredicative type, on the other hand, allows universal quantifiers anywhere: in particular, may contain ordinary datatypes with polymorphic components. The GHC User's Guide gives the following example:
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Impredicative types take this idea to its natural conclusion: universal quantifiers are allowed ''anywhere'' in a type, even inside normal datatypes like lists or <tt>Maybe</tt>. The GHC User's Guide gives the following example:
   
 
<haskell>
 
<haskell>
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</haskell>
 
</haskell>
   
Impredicative types are enabled in GHC with the <hask>{-# LANGUAGE ImpredicativeTypes #-}</hask> pragma. They are among the less well-used and well-tested language extensions, and so some caution is advised in their use. They are rarely needed in practice.
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However, impredicative types do not mix very well with Haskell's type inference, so to actually use the above function with latest GHC you need to specify the full (unpleasant) type signature for the <tt>Just</tt> constructor:
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<haskell>
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ghci> f ((Just :: (forall a. [a] -> [a]) -> Maybe (forall a. [a] -> [a])) reverse)
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Just ([3],"olleh")
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</haskell>
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Other examples are more successful: see below.
   
 
=== See also ===
 
=== See also ===
   
 
* [http://www.haskell.org/ghc/docs/latest/html/users_guide/other-type-extensions.html#impredicative-polymorphism The GHC User's Guide on impredicative polymorphism].
 
* [http://www.haskell.org/ghc/docs/latest/html/users_guide/other-type-extensions.html#impredicative-polymorphism The GHC User's Guide on impredicative polymorphism].
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* [http://augustss.blogspot.co.uk/2011/07/impredicative-polymorphism-use-case-in.html A Pythonesque EDSL that makes use of impredicative polymorphism]
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* [http://stackoverflow.com/a/14065493/812053 A writeup of where ImpredicativePolymorphism is used in a GHC plugin to store a lookup table of strings to polymorphic functions]
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[[Category:Glossary]]

Revision as of 16:43, 4 January 2013

Impredicative types are an advanced form of polymorphism, to be contrasted with rank-N types.

Standard Haskell allows polymorphic types via the use of type variables, which are understood to be universally quantified: id :: a -> a means "for all types a, id can take an argument and return a result of that type". All universal quantifiers ("for all"s) must appear at the beginning of a type.

Higher-rank polymorphism (e.g. rank-N types) allows universal quantifiers to appear inside function types as well. It turns out that appearing to the right of function arrows is not interesting: Int -> forall a. a -> [a] is actually the same as forall a. Int -> a -> [a]. However, higher-rank polymorphism allows quantifiers to the left of function arrows, too, and (forall a. [a] -> Int) -> Int really is different from forall a. ([a] -> Int) -> Int.

Impredicative types take this idea to its natural conclusion: universal quantifiers are allowed anywhere in a type, even inside normal datatypes like lists or Maybe. The GHC User's Guide gives the following example:

f :: Maybe (forall a. [a] -> [a]) -> Maybe ([Int], [Char])
f (Just g) = Just (g [3], g "hello")
f Nothing  = Nothing

However, impredicative types do not mix very well with Haskell's type inference, so to actually use the above function with latest GHC you need to specify the full (unpleasant) type signature for the Just constructor:

ghci> f ((Just :: (forall a. [a] -> [a]) -> Maybe (forall a. [a] -> [a])) reverse)
Just ([3],"olleh")

Other examples are more successful: see below.

See also