Partial application

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 support 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:

papply

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

curry

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

uncurry

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

Note that curry papply = curry.

• Currying
• η-conversion
• POP-2

• Simon Marlow and Simon Peyton Jones (2004, 2006). "Making a Fast Curry: Push/Enter vs. Eval/Apply for Higher-order Languages". ICFP '04 Proceedings of the ninth ACM SIGPLAN international conference on Functional programming.
• Benjamin C. Pierce et al. "Partial Application", "Digression: Currying". Software Foundations.

• Partial function application on Rosetta code.
• Partial application at Haskell Wiki
• Constant applicative form at Haskell Wiki
• The dangers of being too partial