Currying

What is currying?

Currying is the technique of expressing a function taking multiple arguments as nested functions, each taking a single argument. A curried function returns another function as its result.

Consider the string concatenation operator (^):

(^)
(* : string -> string -> string = <fun> *)

The type string -> string -> string means “a function that takes a string and returns a function from string to string”. Compare with an uncurried version using pairs:

Curried	Uncurried
string -> string -> string	string * string -> string
Takes one string, returns a function	Takes a pair of strings, returns a string
Allows partial application	Must supply both arguments together

Defining curried functions

A curried function uses nested fun-notation:

let prefix = fun a -> fun b -> a ^ b
(* val prefix : string -> string -> string = <fun> *)

Applying this function to one argument yields another function:

let promote = prefix "Senior "
(* val promote : string -> string = <fun> *)

promote "Professor"
(* : string = "Senior Professor" *)

Promotion via partial application

Supplying the first argument of a curried function is called partial application. The resulting function has the first argument “fixed” and awaits the remaining arguments. This is sometimes called promotion: the function prefix applied to "Senior " promotes, yielding promote.

Mathematical intuition: the function x^y with y = 2 gives squaring; with y = 3 gives cubing; with y = 1 gives the identity function. Each is obtained by fixing the second argument.

Nesting of fun

The curried form nests fun expressions:

fun a -> fun b -> E

is read as: a function that takes a, and returns a function that takes b and computes E. In OCaml, -> associates to the right, so the type string -> string -> string means string -> (string -> string). Application associates to the left, so prefix "Senior " "Professor" means (prefix "Senior ") "Professor".