In computer science, partial application (or partial function application) refers to the process of fixing a number of arguments to a function, producing another function of smaller arity. Given a function , we might fix (or 'bind') the first argument, producing a function of type . Evaluation of this function might be represented as . Note that the result of partial function application in this case is a function that takes two arguments. Partial application is sometimes incorrectly called currying, which is a related, but distinct concept.


Intuitively, partial function application says "if you fix the first arguments of the function, you get a function of the remaining arguments". For example, if function div(x,y) = x/y, then div with the parameter x fixed at 1 is another function: div1(y) = div(1,y) = 1/y. This is the same as the function inv that returns the multiplicative inverse of its argument, defined by inv(y) = 1/y.

The practical motivation for partial application is that very often the functions obtained by supplying some but not all of the arguments to a function are useful; for example, many languages have a function or operator similar to plus_one. Partial application makes it easy to define these functions, for example by creating a function that represents the addition operator with 1 bound as its first argument.


In languages such as ML, Haskell and F#, functions are defined in curried form by default. Supplying fewer than the total number of arguments is referred to as partial application.

In languages with first-class functions, one can define curry, uncurry and papply to perform currying and partial application explicitly. This might incur a greater run-time overhead due to the creation of additional closures, while Haskell can use more efficient techniques.[1]

Scala implements optional partial application with placeholder, e.g. def add(x: Int, y: Int) = {x+y}; add(1, _: Int) returns an incrementing function. Scala also supports multiple parameter lists as currying, e.g. def add(x: Int)(y: Int) = {x+y}; add(1) _.

Clojure implements partial application using the partial function defined in its core library.[2]

The C++ standard library provides bind(function, args..) to return a function object that is the result of partial application of the given arguments to the given function. Alternatively, lambda expressions can be used:

int f(int a, int b);
auto f_partial = [](int a) { return f(a, 123); };
assert(f_partial(456) == f(456, 123) );

In Java, MethodHandle.bindTo partially applies a function to its first argument.[3] Alternatively, since Java 8, lambdas can be used:

public static <A, B, R> Function<B, R> partialApply(BiFunction<A, B, R> biFunc, A value) {
    return b -> biFunc.apply(value, b);

In Raku, the assuming method creates a new function with fewer parameters.[4]

The Python standard library module functools includes the partial function, allowing positional and named argument bindings, returning a new function.[5]

In XQuery, an argument placeholder (?) is used for each non-fixed argument in a partial function application.[6]


In the simply-typed lambda calculus with function and product types (λ→,×) partial application, currying and uncurrying can be defined as

(((a × b) → c) × a) → (bc) = λ(f, x). λy. f (x, y)
((a × b) → c) → (a → (bc)) = λf. λx. λy. f (x, y)
(a → (bc)) → ((a × b) → c) = λf. λ(x, y). f x y

Note that curry papply = curry.

Mathematical formulation and examples

Partial application can be a useful way to define several useful notions in mathematics.

Given sets and , and a function , one can define the function

where is the set of functions . The image of under this map is . This is the function which sends to . There are often structures on which mean that the image of restricts to some subset of functions , as illustrated in the following examples.

Group actions

A group action can be understood as a function . The partial evaluation restricts to the group of bijections from to itself. The group action axioms further ensure is a group homomorphism.

Inner-products and canonical map to the dual

An inner-product on a vector space over a field is a map . The partial evaluation provides a canonical map to the dual vector space, . If this is the inner-product of a Hilbert space, the Riesz representation theorem ensures this is an isomorphism.

Cross-products and the adjoint map for Lie algebras

The partial application of the cross product on is . The image of the vector is a linear map such that . The components of can be found to be .

This is closely related to the adjoint map for Lie algebras. Lie algebras are equipped with a bracket . The partial application gives a map . The axioms for the bracket ensure this map is a homomorphism of Lie algebras.

See also


  1. ^ Marlow & Peyton Jones 2004
  2. ^ "clojure/clojure, partial function". GitHub. Retrieved 2020-07-18.
  3. ^ "MethodHandle (Java Platform SE 7)". Retrieved 2018-09-12.
  4. ^ "Method assuming". Retrieved 2018-09-12.
  5. ^ "10.2. functools — Higher-order functions and operations on callable objects — Python 3.7.0 documentation". Retrieved 2018-09-12.
  6. ^ "XQuery 3.1: An XML Query Language". Retrieved 2018-09-12.

Further reading