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Currying

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[[Category:Glossary]]
 
[[Category:Glossary]]
Currying is the process of transforming a function that takes multiple arguments into a function that takes just a single argument and returns another function if any arguments are still needed. In Haskell, ''all'' functions are considered curried: that is, ''all functions in Haskell take just single arguments.''
+
Currying is the process of transforming a function that takes multiple arguments into a function that takes just a single argument and returns another function if any arguments are still needed.
   
  +
<haskell>
  +
f :: a -> b -> c
  +
</haskell>
  +
is the '''curried''' form of
  +
<haskell>
  +
g :: (a, b) -> c
  +
</haskell>
  +
You can convert these two types in either directions with the Prelude functions <hask>curry</hask> and <hask>uncurry</hask>.
  +
<haskell>
  +
f = curry g
  +
g = uncurry f
  +
</haskell>
  +
Both forms are equally expressive.
  +
It holds
  +
<haskell>
  +
f x y = g (x,y) ,
  +
</haskell>
  +
however the curried form is usually more convenient because it allows [[partial application]].
  +
  +
In Haskell, ''all'' functions are considered curried: That is, ''all functions in Haskell take just single arguments.''
 
This is mostly hidden in notation, and so may not be apparent to a new Haskeller. Let's take the function <haskell>div :: Int -> Int -> Int</haskell> which performs integer division. The expression <hask>div 11 2</hask> unsurprisingly evaluates to <hask>5</hask>.
 
This is mostly hidden in notation, and so may not be apparent to a new Haskeller. Let's take the function <haskell>div :: Int -> Int -> Int</haskell> which performs integer division. The expression <hask>div 11 2</hask> unsurprisingly evaluates to <hask>5</hask>.
   
 
But there's more that's going on than immediately meets the untrained eye. It's a two-part process. First, <haskell>div 11</haskell> is evaluated and ''returns a function'' of type <haskell>Int -> Int</haskell> Then that resulting function is applied to the value <hask>2</hask>, and yields <hask>5</hask>.
 
But there's more that's going on than immediately meets the untrained eye. It's a two-part process. First, <haskell>div 11</haskell> is evaluated and ''returns a function'' of type <haskell>Int -> Int</haskell> Then that resulting function is applied to the value <hask>2</hask>, and yields <hask>5</hask>.
   
+
You'll notice that the notation for types reflects this: you can read <haskell>Int -> Int -> Int</haskell> incorrectly as "takes two <hask>Int</hask>s and returns an <hask>Int</hask>", but what it's ''really'' saying is "takes an <hask>Int</hask> and returns something of the type <hask>Int -> Int</hask>" -- that is, it returns a function that takes an <hask>Int</hask> and returns an <hask>Int</hask>. (One can write the type as <hask>Int x Int -> Int</hask> if you really mean the former -- but since all functions in Haskell are curried, that's not legal Haskell. Alternatively, using tuples, you can write <hask>(Int, Int) -> Int</hask>, but keep in mind that the tuple constructor <hask>(,)</hask> itself can be curried.)
You'll notice that the notation for types reflects this: you can read <haskell>Int -> Int -> Int</haskell> incorrectly as "takes two <hask>Int</hask>s and returns an <hask>Int</hask>", but what it's ''really'' saying is "takes an <hask>Int</hask> and returns something of the type <hask>Int -> Int</hask>--that is, it returns a function that takes an <hask>Int</hask> and returns an <hask>Int</hask>. (One can write the type as <hask>Int x Int -> Int</hask> if you really mean the former--but since all functions in Haskell are curried, that's not legal Haskell. Alternatively, using tuples, you can write <hask>(Int, Int) -> Int</hask>, but keep in mind that the tuple constructor <hask>(,)</hask> itself can be curried.)
 
   
 
Much of the time, currying can be ignored by the new programmer. The major advantage of considering all functions as curried is theoretical: formal proofs are easier when all functions are treated uniformly (one argument in, one result out). Having said that, there ''are'' Haskell idioms and techniques for which you need to understand currying.
 
Much of the time, currying can be ignored by the new programmer. The major advantage of considering all functions as curried is theoretical: formal proofs are easier when all functions are treated uniformly (one argument in, one result out). Having said that, there ''are'' Haskell idioms and techniques for which you need to understand currying.
   
Currying provides a convenient way of writing some functions without having to explicitly name them:
+
See
*<hask>(1+)</hask> (unsugared: <hask>(+) 1</hask>) is the "increment" function,
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* [[partial application]]
*<hask>(2*)</hask> is the "double" function,
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* [[Section of an infix operator]]
*<hask>("\t"++)</hask> is the "indent" function,
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* Sometimes it's valuable to think about functions abstractly without specifically giving all their arguments: this is the [[Pointfree]] style.
*<hask>(`elem` "AEIOU")</hask> is the "is-capital-vowel-in-English" function (ignoring the "sometimes Y").
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* Sometimes half the work of the function can be done looking only at the first argument (but there really ''is'' only one argument, remember?): see [[functional dispatch]].
These are examples of [[partial application]] (and of "section" notation).
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* Conversion between curried and uncurried style allows [[Composing functions with multiple values|composition of functions with multiple values]]
   
Sometimes it's valuable to think about functions abstractly without specifically giving all their arguments: this is the [[Pointfree]] style.
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== Exercises ==
   
Sometimes half the work of the function can be done looking only at the first argument (but there really ''is'' only one argument, remember?): see [[functional dispatch]].
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* Simplify <hask>curry id</hask> <!-- (,) -->
  +
* Simplify <hask>uncurry const</hask> <!-- fst -->
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* Express <hask>snd</hask> using <hask>curry</hask> or <hask>uncurry</hask> and other basic Prelude functions and without lambdas <!-- uncurry (flip const) -->
  +
* Write the function <hask>\(x,y) -> (y,x)</hask> without lambda and with only Prelude functions <!-- uncurry (flip (curry id)) -->

Latest revision as of 09:54, 3 January 2008

Currying is the process of transforming a function that takes multiple arguments into a function that takes just a single argument and returns another function if any arguments are still needed.

f :: a -> b -> c

is the curried form of

g :: (a, b) -> c
You can convert these two types in either directions with the Prelude functions
curry
and
uncurry
.
f = curry g
g = uncurry f

Both forms are equally expressive. It holds

f x y = g (x,y)    ,

however the curried form is usually more convenient because it allows partial application.

In Haskell, all functions are considered curried: That is, all functions in Haskell take just single arguments.

This is mostly hidden in notation, and so may not be apparent to a new Haskeller. Let's take the function
div :: Int -> Int -> Int
which performs integer division. The expression
div 11 2
unsurprisingly evaluates to
5
. But there's more that's going on than immediately meets the untrained eye. It's a two-part process. First,
div 11
is evaluated and returns a function of type
Int -> Int
Then that resulting function is applied to the value
2
, and yields
5
. You'll notice that the notation for types reflects this: you can read
Int -> Int -> Int
incorrectly as "takes two
Int
s and returns an
Int
", but what it's really saying is "takes an
Int
and returns something of the type
Int -> Int
" -- that is, it returns a function that takes an
Int
and returns an
Int
. (One can write the type as
Int x Int -> Int
if you really mean the former -- but since all functions in Haskell are curried, that's not legal Haskell. Alternatively, using tuples, you can write
(Int, Int) -> Int
, but keep in mind that the tuple constructor
(,)
itself can be curried.)

Much of the time, currying can be ignored by the new programmer. The major advantage of considering all functions as curried is theoretical: formal proofs are easier when all functions are treated uniformly (one argument in, one result out). Having said that, there are Haskell idioms and techniques for which you need to understand currying.

See

[edit] Exercises

  • Simplify
    curry id
  • Simplify
    uncurry const
  • Express
    snd
    using
    curry
    or
    uncurry
    and other basic Prelude functions and without lambdas
  • Write the function
    \(x,y) -> (y,x)
    without lambda and with only Prelude functions