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In mathematics and computer science, a higher-order function is a function that does at least one of the following: * takes one or more functions as arguments (i.e. procedural parameters), * returns a function as its result. All other functions are ''first-order functions''. In mathematics higher-order functions are also termed ''operators'' or ''functionals''. The differential operator in calculus is a common example, since it maps a function to its derivative, also a function. Higher-order functions should not be confused with other uses of the word "functor" throughout mathematics, see Functor (disambiguation). In the untyped lambda calculus, all functions are higher-order; in a typed lambda calculus, from which most functional programming languages are derived, higher-order functions that take one function as argument are values with types of the form $\left(\tau_1\to\tau_2\right)\to\tau_3$.

General examples

* map function, found in many functional programming languages, is one example of a higher-order function. It takes as arguments a function ''f'' and a collection of elements, and as the result, returns a new collection with ''f'' applied to each element from the collection. * Sorting functions, which take a comparison function as a parameter, allowing the programmer to separate the sorting algorithm from the comparisons of the items being sorted. The C standard function qsort is an example of this. * filter * fold * apply * Function composition * Integration * Callback * Tree traversal * Montague grammar, a semantic theory of natural language, uses higher-order functions

Support in programming languages

Direct support

''The examples are not intended to compare and contrast programming languages, but to serve as examples of higher-order function syntax'' In the following examples, the higher-order function takes a function, and applies the function to some value twice. If has to be applied several times for the same it preferably should return a function rather than a value. This is in line with the "don't repeat yourself" principle.

APL

twice← plusthree← g← g 7 13 Or in a tacit manner: twice←⍣2 plusthree←+∘3 g←plusthree twice g 7 13

C++

Using in C++11: #include #include auto twice = [](const std::function& f) ; auto plus_three = [](int i) ; int main() Or, with generic lambdas provided by C++14: #include auto twice = [](const auto& f) ; auto plus_three = [](int i) ; int main()

C#

Using just delegates: using System; public class Program Or equivalently, with static methods: using System; public class Program

Clojure

(defn twice (fn (f (f x)))) (defn plus-three (+ i 3)) (def g (twice plus-three)) (println (g 7)) ; 13

ColdFusion Markup Language (CFML)

twice = function(f) ; plusThree = function(i) ; g = twice(plusThree); writeOutput(g(7)); // 13

D

import std.stdio : writeln; alias twice = (f) => (int x) => f(f(x)); alias plusThree = (int i) => i + 3; void main()

Elixir

In Elixir, you can mix module definitions and anonymous functions defmodule Hof do def twice(f) do fn(x) -> f.(f.(x)) end end end plus_three = fn(i) -> 3 + i end g = Hof.twice(plus_three) IO.puts g.(7) # 13 Alternatively, we can also compose using pure anonymous functions. twice = fn(f) -> fn(x) -> f.(f.(x)) end end plus_three = fn(i) -> 3 + i end g = twice.(plus_three) IO.puts g.(7) # 13

Erlang

or_else([], _) -> false; or_else([F | Fs], X) -> or_else(Fs, X, F(X)). or_else(Fs, X, false) -> or_else(Fs, X); or_else(Fs, _, ) -> or_else(Fs, Y); or_else(_, _, R) -> R. or_else([fun erlang:is_integer/1, fun erlang:is_atom/1, fun erlang:is_list/1], 3.23). In this Erlang example, the higher-order function takes a list of functions () and argument (). It evaluates the function with the argument as argument. If the function returns false then the next function in will be evaluated. If the function returns then the next function in with argument will be evaluated. If the function returns the higher-order function will return . Note that , , and can be functions. The example returns .

F#

let twice f = f >> f let plus_three = (+) 3 let g = twice plus_three g 7 |> printf "%A" // 13

Go

package main import "fmt" func twice(f func(int) int) func(int) int func main() Notice a function literal can be defined either with an identifier () or anonymously (assigned to variable ).

Haskell

twice :: (Int -> Int) -> (Int -> Int) twice f = f . f plusThree :: Int -> Int plusThree = (+3) main :: IO () main = print (g 7) -- 13 where g = twice plusThree

J

Explicitly, twice=. adverb : 'u u y' plusthree=. verb : 'y + 3' g=. plusthree twice g 7 13 or tacitly, twice=. ^:2 plusthree=. +&3 g=. plusthree twice g 7 13

Java (1.8+)

Using just functional interfaces: import java.util.function.*; class Main Or equivalently, with static methods: import java.util.function.*; class Main

JavaScript

"use strict"; const twice = f => x => f(f(x)); const plusThree = i => i + 3; const g = twice(plusThree); console.log(g(7)); // 13

Julia

julia> function twice(f) function result(x) return f(f(x)) end return result end twice (generic function with 1 method) julia> plusthree(i) = i + 3 plusthree (generic function with 1 method) julia> g = twice(plusthree) (::var"#result#3") (generic function with 1 method) julia> g(7) 13

Kotlin

fun twice(f: (Int) -> Int): (Int) -> Int fun plusThree(i: Int) = i + 3 fun main()

Lua

local function twice(f) return function (x) return f(f(x)) end end local function plusThree(i) return i + 3 end local g = twice(plusThree) print(g(7)) -- 13

MATLAB

function result = twice(f) result = @inner function val = inner(x) val = f(f(x)); end end plusthree = @(i) i + 3; g = twice(plusthree) disp(g(7)); % 13

OCaml

let twice f x = f (f x) let plus_three = (+) 3 let () = let g = twice plus_three in print_int (g 7); (* 13 *) print_newline ()

PHP

or with all functions in variables: fn(int $x): int =>$f($f($x)); $plusThree = fn(int$i): int => $i + 3;$g = $twice($plusThree); echo $g(7), "\n"; // 13 Note that arrow functions implicitly capture any variables that come from the parent scope, whereas anonymous functions require the keyword to do the same. Pascal type fun = function(x: Integer): Integer; function twice(f: fun; x: Integer): Integer; begin result := f(f(x)); end; function plusThree(i: Integer): Integer; begin result := i + 3; end; begin writeln(twice(@plusThree, 7)); end. Perl use strict; use warnings; sub twice sub plusThree my$g = twice(\&plusThree); print $g->(7), "\n"; # 13 or with all functions in variables: use strict; use warnings; my$twice = sub ; my $plusThree = sub ; my$g = $twice->($plusThree); print $g->(7), "\n"; # 13 Python >>> def twice(f): ... def result(x): ... return f(f(x)) ... return result >>> plusthree = lambda i: i + 3 >>> g = twice(plusthree) >>> g(7) 13 Python decorator syntax is often used to replace a function with the result of passing that function through a higher-order function. E.g., the function could be implemented equivalently: >>> @twice ... def g(i): ... return i + 3 >>> g(7) 13 R twice <- function(f) plusThree <- function(i) g <- twice(plusThree) > print(g(7)) 13 Raku sub twice(Callable:D$f) sub plusThree(Int:D $i) my$g = twice(&plusThree); say $g(7); # 13 In Raku, all code objects are closures and therefore can reference inner "lexical" variables from an outer scope because the lexical variable is "closed" inside of the function. Raku also supports "pointy block" syntax for lambda expressions which can be assigned to a variable or invoked anonymously. Ruby def twice(f) ->(x) end plus_three = ->(i) g = twice(plus_three) puts g.call(7) # 13 Rust fn twice(f: impl Fn(i32) -> i32) -> impl Fn(i32) -> i32 fn plus_three(i: i32) -> i32 fn main() Scala object Main Scheme (define (add x y) (+ x y)) (define (f x) (lambda (y) (+ x y))) (display ((f 3) 7)) (display (add 3 7)) In this Scheme example, the higher-order function is used to implement currying. It takes a single argument and returns a function. The evaluation of the expression first returns a function after evaluating . The returned function is . Then, it evaluates the returned function with 7 as the argument, returning 10. This is equivalent to the expression , since is equivalent to the curried form of . Swift func twice(_ f: @escaping (Int) -> Int) -> (Int) -> Int let plusThree = let g = twice(plusThree) print(g(7)) // 13 Tcl set twice set plusThree # result: 13 puts pply$twice $plusThree 7 Tcl uses apply command to apply an anonymous function (since 8.6). XACML The XACML standard defines higher-order functions in the standard to apply a function to multiple values of attribute bags. rule allowEntry The list of higher-order functions in XACML can be found here. XQuery declare function local:twice($f, $x) ; declare function local:plusthree($i) ; local:twice(local:plusthree#1, 7) (: 13 :)

Alternatives

Function pointers

Function pointers in languages such as C and C++ allow programmers to pass around references to functions. The following C code computes an approximation of the integral of an arbitrary function: #include double square(double x) double cube(double x) /* Compute the integral of f() within the interval ,b*/ double integral(double f(double x), double a, double b, int n) int main() The qsort function from the C standard library uses a function pointer to emulate the behavior of a higher-order function.

Macros

Macros can also be used to achieve some of the effects of higher-order functions. However, macros cannot easily avoid the problem of variable capture; they may also result in large amounts of duplicated code, which can be more difficult for a compiler to optimize. Macros are generally not strongly typed, although they may produce strongly typed code.

Dynamic code evaluation

In other imperative programming languages, it is possible to achieve some of the same algorithmic results as are obtained via higher-order functions by dynamically executing code (sometimes called ''Eval'' or ''Execute'' operations) in the scope of evaluation. There can be significant drawbacks to this approach: *The argument code to be executed is usually not statically typed; these languages generally rely on dynamic typing to determine the well-formedness and safety of the code to be executed. *The argument is usually provided as a string, the value of which may not be known until run-time. This string must either be compiled during program execution (using just-in-time compilation) or evaluated by interpretation, causing some added overhead at run-time, and usually generating less efficient code.

Objects

In object-oriented programming languages that do not support higher-order functions, objects can be an effective substitute. An object's methods act in essence like functions, and a method may accept objects as parameters and produce objects as return values. Objects often carry added run-time overhead compared to pure functions, however, and added boilerplate code for defining and instantiating an object and its method(s). Languages that permit stack-based (versus heap-based) objects or structs can provide more flexibility with this method. An example of using a simple stack based record in Free Pascal with a function that returns a function: program example; type int = integer; Txy = record x, y: int; end; Tf = function (xy: Txy): int; function f(xy: Txy): int; begin Result := xy.y + xy.x; end; function g(func: Tf): Tf; begin result := func; end; var a: Tf; xy: Txy = (x: 3; y: 7); begin a := g(@f); // return a function to "a" writeln(a(xy)); // prints 10 end. The function a() takes a Txy record as input and returns the integer value of the sum of the record's x and y fields (3 + 7).

Defunctionalization

Defunctionalization can be used to implement higher-order functions in languages that lack first-class functions: // Defunctionalized function data structures template struct Add ; template struct DivBy ; template struct Composition ; // Defunctionalized function application implementations template auto apply(Composition f, X arg) template auto apply(Add f, X arg) template auto apply(DivBy f, X arg) // Higher-order compose function template Composition compose(F f, G g) int main(int argc, const char* argv[]) In this case, different types are used to trigger different functions via [[function overloading]]. The overloaded function in this example has the signature auto apply.

See also

*First-class function *Combinatory logic *Function-level programming *Functional programming *Kappa calculus - a formalism for functions which ''excludes'' higher-order functions *Strategy pattern *Higher order messages

References

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