Currying

In mathematics and computer science, currying (named after Haskell Curry) is the technique of translating a function that takes multiple arguments into a sequence of families of functions, each taking a single argument.

In the prototypical example, one begins with a function

f

:

(

X

×

Y

)

→

Z

{\displaystyle f:(X\times Y)\to Z}

that takes two arguments, one from

X

{\displaystyle X}

and one from

Y

,

{\displaystyle Y,}

and produces objects in

Z

.

{\displaystyle Z.}

The curried form of this function treats the first argument as a parameter, so as to create a family of functions

f

x

:

Y

→

Z

.

{\displaystyle f_{x}:Y\to Z.}

The family is arranged so that for each object

x

{\displaystyle x}

in

X

,

{\displaystyle X,}

there is exactly one function

f

x

{\displaystyle f_{x}}

, such that for any

y

{\displaystyle y}

in

Y

{\displaystyle Y}

,

f

x

(

y

)

=

f

(

x

,

y

)

{\displaystyle f_{x}(y)=f(x,y)}

.

In this example,

curry

{\displaystyle {\mbox{curry}}}

itself becomes a function that takes

f

{\displaystyle f}

as an argument, and returns a function that maps each

x

{\displaystyle x}

to

f

x

.

{\displaystyle f_{x}.}

The proper notation for expressing this is verbose. The function

f

{\displaystyle f}

belongs to the set of functions

(

X

×

Y

)

→

Z

.

{\displaystyle (X\times Y)\to Z.}

Meanwhile,

f

x

{\displaystyle f_{x}}

belongs to the set of functions

Y

→

Z

.

{\displaystyle Y\to Z.}

Thus, something that maps

x

{\displaystyle x}

to

f

x

{\displaystyle f_{x}}

will be of the type

X

→

[

Y

→

Z

]

.

{\displaystyle X\to [Y\to Z].}

With this notation,

curry

{\displaystyle {\mbox{curry}}}

is a function that takes objects from the first set, and returns objects in the second set, and so one writes

curry

:

[

(

X

×

Y

)

→

Z

]

→

(

X

→

[

Y

→

Z

]

)

.