Currying

In mathematics and computer science, currying is the technique of transforming a function that takes multiple arguments (or an n-tuple of arguments) in such a way that it can be called as a chain of functions each with a single argument. It was invented by Moses Schönfinkel and later re-invented by Haskell Curry,^[1]^{[not specific enough to verify]}

Uncurrying is the dual transformation to currying, and can be seen as a form of defunctionalization. It takes a function f(x) which returns another function g(y) as a result, and yields a new function f'(x, y) which takes a number of additional parameters and applies them to the function returned by f. The process can be iterated if necessary.

Motivation

Currying is similar to the process of calculating a function of multiple variables for some given values on paper. For example, given the function $f(x,y)=y/x$ :

To evaluate

f(2,3)

, first replace

x

with 2.

Since the result is a function of

y

, this function

g(y)

can be defined as

g(y)=f(2,y)=y/2

.

Next, replace the

y

argument with 3, producing

g(3)=f(2,3)=3/2

.

On paper, using classical notation, this is usually done all in one step. However, each argument can be replaced sequentially as well. Each replacement results in a simpler function taking one fewer argument. This produces a chain of functions as in lambda calculus, where each function takes only one argument, and multi-argument functions are usually represented in curried form.

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 (often called partial application) are useful; for example, many languages have a function or operator similar to plus_one. Currying makes it easy to define these functions, for example by currying the addition operator with 1 as its first argument.

This is similar in computer code: If we let f be a function $f(x,y)=y/x$ , then the function $g_{x}(y)=(y\mapsto f(x,y))$ is a curried version of f. In particular, $g_{(2)}(y)=(y\mapsto f(2,y))$ is the curried equivalent of the example above.

Definition

Given a function f of type $f\colon (X\times Y)\to Z$ , then currying it makes a function ${\text{curry}}(f)\colon X\to (Y\to Z)$ . That is, ${\text{curry}}(f)$ takes an argument of type $X$ and returns a function of type $Y\to Z$ . Uncurrying is the reverse transformation, and is most easily understood in terms of its right adjoint, apply.

The → operator is often considered right-associative, so the curried function type $X\to (Y\to Z)$ is often written as $X\to Y\to Z$ . Conversely, function application is considered to be left-associative, so that $f\;\langle x,y\rangle$ is equivalent to ${\text{curry}}(f)\;x\;y$ .

Intuitively, currying says "if you fix the first arguments of the function, 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 (i.e. div 1) is another function: the same as the function inv that returns the multiplicative inverse of its argument, defined by inv(y) = 1 / y.

Some programming languages almost always use curried functions to achieve multiple arguments; notable examples are ML and Haskell, where in both cases all functions have exactly one argument.

Curried functions may be used in any language that supports closures; however, uncurried functions are generally preferred for efficiency reasons, since the overhead of partial application and closure creation can then be avoided for most function calls.

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 is the natural code between the set $A^{B\times C}$ of functions from $B\times C$ to $A$ , and the set $(A^{B})^{C}$ of functions from $C$ to the set of functions from $B$ to $A$ . 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 $f\colon (X\times Y)\to Z$ and the morphisms to an exponential object $g\colon X\to Z^{Y}$ . In other words, currying is the statement that product and Hom are adjoint functors; that is there is a natural transformation:

\hom(A\times B,C)\cong \hom(A,\hom(B,C)).

This is the key property of being a Cartesian closed category.

Under the Curry-Howard correspondence, the existence of currying and uncurrying is equivalent to the logical theorem $(A\land B)\to C\Leftrightarrow A\to (B\to C)$ , as tuples (product type) corresponds to conjunction in logic, and function type corresponds to implication.

Curry is a continuous function in the Scott topology.

Naming

The name "currying", coined by Christopher Strachey in 1967, is a reference to logician Haskell Curry. The alternative name "Schönfinkelisation", has been proposed as a reference to Moses Schönfinkel.^[2]

Notes

^ (Barendregt 2000, p. 8)
^ I. Heim and A. Kratzer (1998). Semantics in Generative Grammar. Blackwell.

References

Heim, Irene; Kratzer, Angelika (1998), Semantics in a Generative Grammar, Malden: Blackwall Publishers

External links

Currying != Generalized Partial Application! - post at Lambda-the-Ultimate.org

Language-Specific Implementations

[1] (Barendregt 2000, p. 8)

[2] I. Heim and A. Kratzer (1998). Semantics in Generative Grammar. Blackwell.

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