Currying

In computer science, currying, invented by Moses Schönfinkel and Gottlob Frege, is the technique of transforming a function that takes multiple arguments into a function that takes a single argument (the other arguments having been specified by the curry).

Nomenclature

The name "currying", coined by Christopher Strachey in 1967, is a reference to logician Haskell Curry. An alternative name, Schönfinkelisation, has been proposed. ^[1]

Definition

Given a function f of type Failed to parse (Missing texvc executable; please see math/README to configure.): f \colon (X \times Y) \to Z , then currying it makes a function Failed to parse (Missing texvc executable; please see math/README to configure.): \mbox{curry}(f) \colon X \to (Y \to Z) . That is, Failed to parse (Missing texvc executable; please see math/README to configure.): \mbox{curry}(f)

takes an argument of type Failed to parse (Missing texvc executable; please see math/README to configure.):  X 
and returns a function of type Failed to parse (Missing texvc executable; please see math/README to configure.):  Y \to Z 

. Uncurrying is the reverse transformation. In a different sense, the opposite of currying is Apply, in that Apply will restore the curried argument.

Intuitively, currying says "if you fix some arguments, you get a function of the remaining arguments". For example, if function div stands for the curried form of the division operation x / y, then div with the parameter x fixed at 1 is another function: the same as the function inv that returns the multiplicative inverse of its argument, defined by inv(y) = 1 / y.

The practical motivation for currying is that very often the functions obtained by supplying some but not all of the arguments to a curried function are useful; for example, many languages have a function or operator similar to plus_one. Currying makes it easy to define these functions.

Some programming languages have built-in syntactic support for currying, where certain multi-argument functions are expanded to their curried form; notable examples are ML and Haskell. Any language that supports closures can be used to write curried functions.

Mathematical view

In theoretical computer science, currying provides a way to study functions with multiple arguments in very simple theoretical models such as the lambda calculus in which functions only take a single argument.

When viewed in a set-theoretic light, currying becomes the theorem that the set Failed to parse (Missing texvc executable; please see math/README to configure.): A^{B\times C}

of functions from

Failed to parse (Missing texvc executable; please see math/README to configure.): B\times C

to Failed to parse (Missing texvc executable; please see math/README to configure.): A

, and the set Failed to parse (Missing texvc executable; please see math/README to configure.): (A^B)^C

of functions from

Failed to parse (Missing texvc executable; please see math/README to configure.): C

to the set of functions from Failed to parse (Missing texvc executable; please see math/README to configure.): B
to Failed to parse (Missing texvc executable; please see math/README to configure.): A

, are isomorphic.

In category theory, currying can be found in the universal property of an exponential object, which gives rise to the following adjunction in cartesian closed categories: There is a natural isomorphism between the morphisms from a binary product Failed to parse (Missing texvc executable; please see math/README to configure.): f \colon (X \times Y) \to Z

and the morphisms to an exponential object Failed to parse (Missing texvc executable; please see math/README to configure.):  g \colon X \to Z^Y 

. In other words, currying is the statement that product and Hom are adjoint functors; this is the key property of being a Cartesian closed category.

See also

• Lazy evaluation
• Closure (computer science)
• s_mn theorem

References

1. ^ I. Heim and A. Kratzer (1998). Semantics in Generative Grammar. Blackwell.

External links

Look up currying in Wiktionary, the free dictionary.

• Currying in Python
• Implicit currying in Scheme
• Currying in Ruby
• Currying in Smalltalk
• Currying in Algol68G
• Currying != Generalized Partial Application! - post at Lambda-the-Ultimate.org
• Currying in Scalade:Currying

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