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 that takes two arguments, one from and one from and produces objects in The curried form of this function treats the first argument as a parameter, so as to create a family of functions The family is arranged so that for each object in there is exactly one function , such that for any in , .
In this example, itself becomes a function that takes as an argument, and returns a function that maps each to The proper notation for expressing this is verbose. The function belongs to the set of functions Meanwhile, belongs to the set of functions Thus, something that maps to will be of the type With this notation, is a function that takes objects from the first set, and returns objects in the second set, and so one writes This is a somewhat informal example; more precise definitions of what is meant by "object" and "function" are given below. These definitions vary from context to context, and take different forms, depending on the theory that one is working in.
Currying is related to, but not the same as, partial application. The example above can be used to illustrate partial application; it is quite similar. Partial application is the function that takes the pair and together as arguments, and returns Using the same notation as above, partial application has the signature Written this way, application can be seen to be adjoint to currying.
The currying of a function with more than two arguments can be defined by induction.
Currying is useful in both practical and theoretical settings. In functional programming languages, and many others, it provides a way of automatically managing how arguments are passed to functions and exceptions. In theoretical computer science, it provides a way to study functions with multiple arguments in simpler theoretical models which provide only one argument. The most general setting for the strict notion of currying and uncurrying is in the closed monoidal categories, which underpins a vast generalization of the Curry–Howard correspondence of proofs and programs to a correspondence with many other structures, including quantum mechanics, cobordisms, and string theory.
The concept of currying was introduced by Gottlob Frege, developed by Moses Schönfinkel,Originally published as
Republished as and further developed by Haskell Curry.
Uncurrying is the dual transformation to currying, and can be seen as a form of defunctionalization. It takes a function whose return value is another function , and yields a new function that takes as parameters the arguments for both and , and returns, as a result, the application of and subsequently, , to those arguments. The process can be iterated.
Motivation Currying provides a way for working with functions that take multiple arguments, and using them in frameworks where functions might take only one argument. For example, some analytical techniques can only be applied to functions with a single argument. Practical functions frequently take more arguments than this. Frege showed that it was sufficient to provide solutions for the single argument case, as it was possible to transform a function with multiple arguments into a chain of single-argument functions instead. This transformation is the process now known as currying. All "ordinary" functions that might typically be encountered in mathematical analysis or in computer programming can be curried. However, there are categories in which currying is not possible; the most general categories which allow currying are the closed monoidal categories.
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. This property is inherited from lambda calculus, where multi-argument functions are usually represented in curried form.
Currying is related to, but not the same as partial application. In practice, the programming technique of closures can be used to perform partial application and a kind of currying, by hiding arguments in an environment that travels with the curried function.
History The "Curry" in "Currying" is a reference to logician Haskell Curry, who used the concept extensively, but Moses Schönfinkel had the idea six years before Curry. The alternative name "Schönfinkelisation" has been proposed. In the mathematical context, the principle can be traced back to work in 1893 by Frege.
The originator of the word "currying" is not clear. David Turner says the word was coined by Christopher Strachey in his 1967 lecture notes Fundamental Concepts in Programming Languages, but that source introduces the concept as "a device originated by Schönfinkel", and the term "currying" is not used, while Curry is mentioned later in the context of higher-order functions. John C. Reynolds defined "currying" in a 1972 paper, but did not claim to have coined the term.
Definition Currying is most easily understood by starting with an informal definition, which can then be molded to fit many different domains. First, there is some notation to be established. The notation denotes all functions from to . If is such a function, we write . Let denote the ordered pairs of the elements of and respectively, that is, the Cartesian product of and . Here, and may be sets, or they may be types, or they may be other kinds of objects, as explored below.
Given a function
currying constructs a new function
That is, takes an argument of type and returns a function of type . It is defined by
for of type and of type . We then also write
Uncurrying is the reverse transformation, and is most easily understood in terms of its right adjoint, the function
Set theory In set theory, the notation is used to denote the set of functions from the set to the set . Currying is the natural bijection between the set of functions from to , and the set of functions from to the set of functions from to . In symbols:
Indeed, it is this natural bijection that justifies the exponential notation for the set of functions. As is the case in all instances of currying, the formula above describes an adjoint pair of functors: for every fixed set , the functor is left adjoint to the functor .
In the category of sets, the object is called the exponential object.
Function spaces In the theory of function spaces, such as in functional analysis or homotopy theory, one is commonly interested in continuous functions between topological spaces. One writes (the Hom functor) for the set of all functions from to , and uses the notation to denote the subset of continuous functions. Here, is the bijection
while uncurrying is the inverse map. If the set of continuous functions from to is given the compact-open topology, and if the space is locally compact Hausdorff, then
is a homeomorphism. This is also the case when , and are compactly generated, although there are more cases.
One useful corollary is that a function is continuous if and only if its curried form is continuous. Another important result is that the application map, usually called "evaluation" in this context, is continuous (note that eval is a strictly different concept in computer science.) That is,
is continuous when is compact-open and locally compact Hausdorff. These two results are central for establishing the continuity of homotopy, i.e. when is the unit interval , so that can be thought of as either a homotopy of two functions from to , or, equivalently, a single (continuous) path in .
Algebraic topology In algebraic topology, currying serves as an example of Eckmann–Hilton duality, and, as such, plays an important role in a variety of different settings. For example, loop space is adjoint to reduced suspensions; this is commonly written as
where is the set of homotopy classes of maps , and is the suspension of A, and is the loop space of A. In essence, the suspension can be seen as the cartesian product of with the unit interval, modulo an equivalence relation to turn the interval into a loop. The curried form then maps the space to the space of functions from loops into , that is, from into . Then is the adjoint functor that maps suspensions to loop spaces, and uncurrying is the dual.
The duality between the mapping cone and the mapping fiber (cofibration and fibration) can be understood as a form of currying, which in turn leads to the duality of the long exact and coexact Puppe sequences.
In homological algebra, the relationship between currying and uncurrying is known as tensor-hom adjunction. Here, an interesting twist arises: the Hom functor and the tensor product functor might not lift to an exact sequence; this leads to the definition of the Ext functor and the Tor functor.
Domain theory In order theory, the theory of lattices of partially ordered sets, is a continuous function when the lattice is given the Scott topology. Scott-continuous functions were first investigated in the attempt to provide a semantics for lambda calculus (as ordinary set theory is inadequate to do this). More generally, Scott-continuous functions are now studied in domain theory, which encompasses the study of denotational semantics of computer algorithms. Note that the Scott topology is quite different than many common topologies one might encounter in the category of topological spaces; the Scott topology is typically finer, and is not sober.
The notion of continuity makes its appearance in homotopy type theory, where, roughly speaking, two computer programs can be considered to be homotopic, i.e. compute the same results, if they can be "continuously" refactored from one to the other.
Lambda calculi 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. Consider a function taking two arguments, and having the type , which should be understood to mean that x must have the type , y must have the type , and the function itself returns the type . The curried form of f is defined as
where is the abstractor of lambda calculus. Since curry takes, as input, functions with the type , one concludes that the type of curry itself is
The → operator is often considered right-associative, so the curried function type is often written as . Conversely, function application is considered to be left-associative, so that is equivalent to
That is, the parenthesis are not required to disambiguate the order of the application.
Curried functions may be used in any programming 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.
Type theory In type theory, the general idea of a type system in computer science is formalized into a specific algebra of types. For example, when writing , the intent is that and are types, while the arrow is a type constructor, specifically, the function type or arrow type. Similarly, the Cartesian product of types is constructed by the product type constructor .
The type-theoretical approach is expressed in programming languages such as ML and the languages derived from and inspired by it: Caml, Haskell, and F#.