# Impredicative types

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− | Impredicative types are an advanced form of polymorphism, to be contrasted with [[rank-N types]]. | + | 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'': <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. | |

− | + | 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>. | |

− | + | 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> | ||

Line 13: | Line 13: | ||

</haskell> | </haskell> | ||

− | + | 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: | |

+ | |||

+ | <haskell> | ||

+ | ghci> f ((Just :: (forall a. [a] -> [a]) -> Maybe (forall a. [a] -> [a])) reverse) | ||

+ | Just ([3],"olleh") | ||

+ | </haskell> | ||

+ | |||

+ | Other examples are more successful: see below. | ||

=== See also === | === See also === | ||

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* [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]. | ||

* [http://augustss.blogspot.co.uk/2011/07/impredicative-polymorphism-use-case-in.html A Pythonesque EDSL that makes use of impredicative polymorphism] | * [http://augustss.blogspot.co.uk/2011/07/impredicative-polymorphism-use-case-in.html A Pythonesque EDSL that makes use of impredicative polymorphism] | ||

+ | * [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] | ||

[[Category:Glossary]] | [[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.