Editing Higher-order function

{{Short description|Function that takes one or more functions as an input or that outputs a function}}{{More sources|date=November 2024}}{{Distinguish|Functor{{!}}Functor (category theory)}}In [[mathematics]] and [[computer science]], a '''higher-order function''' ('''HOF''') is a [[function (mathematics)|function]] that does at least one of the following:
* takes one or more functions as arguments (i.e. a [[procedural parameter]], which is a [[Parameter (computer science)|parameter]] of a [[Subroutine|procedure]] that is itself a procedure),
* returns a function as its result.
All other functions are ''first-order functions''. In mathematics higher-order functions are also termed ''[[operator (mathematics)|operators]]'' or ''[[functional (mathematics)|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 <math>(\tau_1\to\tau_2)\to\tau_3</math>.

==General examples==
* <code>[[map (higher-order function)|map]]</code> function, found in many functional programming languages, is one example of a higher-order function. It takes arguments as 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 (programming language)|C]] standard [[function (computer science)|function]] <code>qsort</code> is an example of this.
* [[Filter (higher-order function) | filter]]
* [[fold (higher-order function)|fold]]
* [[Prefix sum|scan]]
* [[apply]]
* [[Function composition (computer science)|Function composition]]
* [[Integral|Integration]]
* [[Callback (computer programming)| Callback]]
* [[Tree traversal]]
* [[Montague grammar]], a semantic theory of natural language, uses higher-order functions

==Support in programming languages==<!-- see also Category:Programming language comparisons -->

===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 {{code|twice}} takes a function, and applies the function to some value twice. If {{code|twice}} has to be applied several times for the same {{code|f}} it preferably should return a function rather than a value. This is in line with the "[[don't repeat yourself]]" principle.

====APL====
{{further information|APL (programming language)}}

<syntaxhighlight lang="apl">
      twice←{⍺⍺ ⍺⍺ ⍵}

      plusthree←{⍵+3}

      g←{plusthree twice ⍵}
    
      g 7
13
</syntaxhighlight>

Or in a tacit manner:

<syntaxhighlight lang="apl">
      twice←⍣2

      plusthree←+∘3

      g←plusthree twice
    
      g 7
13
</syntaxhighlight>

====C++====
{{further information|C++}}

Using {{code|std::function}} in [[C++11]]:

<syntaxhighlight lang="c++">
#include <iostream>
#include <functional>

auto twice = [](const std::function<int(int)>& f)
{
    return [f](int x) {
        return f(f(x));
    };
};

auto plus_three = [](int i)
{
    return i + 3;
};

int main()
{
    auto g = twice(plus_three);

    std::cout << g(7) << '\n'; // 13
}
</syntaxhighlight>

Or, with generic lambdas provided by C++14:

<syntaxhighlight lang="c++">
#include <iostream>

auto twice = [](const auto& f)
{
    return [f](int x) {
        return f(f(x));
    };
};

auto plus_three = [](int i)
{
    return i + 3;
};

int main()
{
    auto g = twice(plus_three);

    std::cout << g(7) << '\n'; // 13
}
</syntaxhighlight>

====C#====
{{further information|C Sharp (programming language)}}

Using just delegates:

<syntaxhighlight lang="csharp">
using System;

public class Program
{
    public static void Main(string[] args)
    {
        Func<Func<int, int>, Func<int, int>> twice = f => x => f(f(x));

        Func<int, int> plusThree = i => i + 3;

        var g = twice(plusThree);

        Console.WriteLine(g(7)); // 13
    }
}
</syntaxhighlight>

Or equivalently, with static methods:

<syntaxhighlight lang="csharp">
using System;

public class Program
{
    private static Func<int, int> Twice(Func<int, int> f)
    {
        return x => f(f(x));
    }

    private static int PlusThree(int i) => i + 3;

    public static void Main(string[] args)
    {
        var g = Twice(PlusThree);

        Console.WriteLine(g(7)); // 13
    }
}
</syntaxhighlight>

====Clojure====
{{further information|Clojure}}

<syntaxhighlight lang="clojure">
(defn twice [f]
  (fn [x] (f (f x))))

(defn plus-three [i]
  (+ i 3))

(def g (twice plus-three))

(println (g 7)) ; 13
</syntaxhighlight>

====ColdFusion Markup Language (CFML)====
{{further information|ColdFusion Markup Language}}

<syntaxhighlight lang="cfs">
twice = function(f) {
    return function(x) {
        return f(f(x));
    };
};

plusThree = function(i) {
    return i + 3;
};

g = twice(plusThree);

writeOutput(g(7)); // 13
</syntaxhighlight>

====Common Lisp====
{{further information|Common Lisp}}

<syntaxhighlight lang="lisp">
(defun twice (f)                                                                
  (lambda (x) (funcall f (funcall f x))))                                       
                                                                                
(defun plus-three (i)                                                           
  (+ i 3))                                                                      
                                                                                
(defvar g (twice #'plus-three))                                                 
                                                                                
(print (funcall g 7))                                                           
</syntaxhighlight>

====D====
{{further information|D (programming language)}}

<syntaxhighlight lang="d">
import std.stdio : writeln;

alias twice = (f) => (int x) => f(f(x));

alias plusThree = (int i) => i + 3;

void main()
{
    auto g = twice(plusThree);

    writeln(g(7)); // 13
}
</syntaxhighlight>

====Dart====
{{further information|Dart (programming language)}}

<syntaxhighlight lang="dart">
int Function(int) twice(int Function(int) f) {
    return (x) {
        return f(f(x));
    };
}

int plusThree(int i) {
    return i + 3;
}

void main() {
    final g = twice(plusThree);
    
    print(g(7)); // 13
}
</syntaxhighlight>

====Elixir====
{{further information|Elixir (programming language)}}

In Elixir, you can mix module definitions and [[anonymous function]]s

<syntaxhighlight lang="elixir">
defmodule Hof do
    def twice(f) do
        fn(x) -> f.(f.(x)) end
    end
end

plus_three = fn(i) -> i + 3 end

g = Hof.twice(plus_three)

IO.puts g.(7) # 13
</syntaxhighlight>

Alternatively, we can also compose using pure anonymous functions.

<syntaxhighlight lang="elixir">
twice = fn(f) ->
    fn(x) -> f.(f.(x)) end
end

plus_three = fn(i) -> i + 3 end

g = twice.(plus_three)

IO.puts g.(7) # 13
</syntaxhighlight>

====Erlang====
{{further information|Erlang (programming language)}}

<syntaxhighlight lang="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, _, {false, Y}) -> 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).
</syntaxhighlight>

In this Erlang example, the higher-order function {{code|or_else/2}} takes a list of functions ({{code|Fs}}) and argument ({{code|X}}). It evaluates the function {{code|F}} with the argument {{code|X}} as argument. If the function {{code|F}} returns false then the next function in {{code|Fs}} will be evaluated. If the function {{code|F}} returns {{code|{false, Y} }} then the next function in {{code|Fs}} with argument {{code|Y}} will be evaluated. If the function {{code|F}} returns {{code|R}} the higher-order function {{code|or_else/2}} will return {{code|R}}. Note that {{code|X}}, {{code|Y}}, and {{code|R}} can be functions. The example returns {{code|false}}.

====F#====
{{further information|F Sharp (programming language)}}

<syntaxhighlight lang="fsharp">
let twice f = f >> f

let plus_three = (+) 3

let g = twice plus_three

g 7 |> printf "%A" // 13
</syntaxhighlight>

====Go====
{{further information|Go (programming language)}}

<syntaxhighlight lang="go">
package main

import "fmt"

func twice(f func(int) int) func(int) int {
	return func(x int) int {
		return f(f(x))
	}
}

func main() {
	plusThree := func(i int) int {
		return i + 3
	}

	g := twice(plusThree)

	fmt.Println(g(7)) // 13
}
</syntaxhighlight>

Notice a function literal can be defined either with an identifier ({{code|twice}}) or anonymously (assigned to variable {{code|plusThree}}).

====Groovy====
{{further information|Groovy (programming language)}}

<syntaxhighlight lang="groovy">def twice = { f, x -> f(f(x)) }
def plusThree = { it + 3 }
def g = twice.curry(plusThree) 
println g(7) // 13

</syntaxhighlight>

====Haskell====
{{further information|Haskell}}

<syntaxhighlight lang="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
</syntaxhighlight>

====J====
{{further information|J (programming language)}}

Explicitly,

<syntaxhighlight lang="J">
   twice=.     adverb : 'u u y'

   plusthree=. verb   : 'y + 3'
   
   g=. plusthree twice
   
   g 7
13
</syntaxhighlight>

or tacitly,

<syntaxhighlight lang="J">
   twice=. ^:2

   plusthree=. +&3
   
   g=. plusthree twice
   
   g 7
13
</syntaxhighlight>

====Java (1.8+)====
{{further information|Java (programming language)|Java version history}}

Using just functional interfaces:

<syntaxhighlight lang="java">
import java.util.function.*;

class Main {
    public static void main(String[] args) {
        Function<IntUnaryOperator, IntUnaryOperator> twice = f -> f.andThen(f);

        IntUnaryOperator plusThree = i -> i + 3;

        var g = twice.apply(plusThree);

        System.out.println(g.applyAsInt(7)); // 13
    }
}
</syntaxhighlight>

Or equivalently, with static methods:

<syntaxhighlight lang="java">
import java.util.function.*;

class Main {
    private static IntUnaryOperator twice(IntUnaryOperator f) {
        return f.andThen(f);
    }

    private static int plusThree(int i) {
        return i + 3;
    }

    public static void main(String[] args) {
        var g = twice(Main::plusThree);

        System.out.println(g.applyAsInt(7)); // 13
    }
}
</syntaxhighlight>

====JavaScript====
{{further information|JavaScript}}

With arrow functions:

<syntaxhighlight lang="javascript">
"use strict";

const twice = f => x => f(f(x));

const plusThree = i => i + 3;

const g = twice(plusThree);

console.log(g(7)); // 13
</syntaxhighlight>

Or with classical syntax:

<syntaxhighlight lang="javascript">
"use strict";

function twice(f) {
  return function (x) {
    return f(f(x));
  };
}

function plusThree(i) {
  return i + 3;
}

const g = twice(plusThree);

console.log(g(7)); // 13
</syntaxhighlight>

====Julia====
{{further information|Julia (programming language)}}

<syntaxhighlight lang="jlcon">
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"{typeof(plusthree)}) (generic function with 1 method)

julia> g(7)
13
</syntaxhighlight>

====Kotlin====
{{further information|Kotlin (programming language)}}

<syntaxhighlight lang="kotlin">
fun twice(f: (Int) -> Int): (Int) -> Int {
    return { f(f(it)) }
}

fun plusThree(i: Int) = i + 3

fun main() {
    val g = twice(::plusThree)

    println(g(7)) // 13
}
</syntaxhighlight>

==== Lua ====
{{further information|Lua (programming language)}}

<syntaxhighlight lang="lua">
function twice(f)
  return function (x)
    return f(f(x))
  end
end

function plusThree(i)
  return i + 3
end

local g = twice(plusThree)

print(g(7)) -- 13
</syntaxhighlight>

==== MATLAB ====
{{further information|MATLAB}}

<syntaxhighlight lang="matlab">
function result = twice(f)
result = @(x) f(f(x));
end

plusthree = @(i) i + 3;

g = twice(plusthree)

disp(g(7)); % 13
</syntaxhighlight>

==== OCaml ====
{{further information|OCaml}}

<syntaxhighlight lang="ocaml" start="1">
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 ()
</syntaxhighlight>

====PHP====
{{further information|PHP}}

<syntaxhighlight lang="php">
<?php

declare(strict_types=1);

function twice(callable $f): Closure {
    return function (int $x) use ($f): int {
        return $f($f($x));
    };
}

function plusThree(int $i): int {
    return $i + 3;
}

$g = twice('plusThree');

echo $g(7), "\n"; // 13
</syntaxhighlight>

or with all functions in variables:

<syntaxhighlight lang="php">
<?php

declare(strict_types=1);

$twice = fn(callable $f): Closure => fn(int $x): int => $f($f($x));

$plusThree = fn(int $i): int => $i + 3;

$g = $twice($plusThree);

echo $g(7), "\n"; // 13
</syntaxhighlight>

Note that arrow functions implicitly capture any variables that come from the parent scope,<ref>{{Cite web|title=PHP: Arrow Functions - Manual|url=https://www.php.net/manual/en/functions.arrow.php|access-date=2021-03-01|website=www.php.net}}</ref> whereas anonymous functions require the {{code|use}} keyword to do the same.

====Perl====
{{further information|Perl}}

<syntaxhighlight lang="perl">
use strict;
use warnings;

sub twice {
    my ($f) = @_;
    sub {
        $f->($f->(@_));
    };
}

sub plusThree {
    my ($i) = @_;
    $i + 3;
}

my $g = twice(\&plusThree);

print $g->(7), "\n"; # 13
</syntaxhighlight>

or with all functions in variables:

<syntaxhighlight lang="perl">
use strict;
use warnings;

my $twice = sub {
    my ($f) = @_;
    sub {
        $f->($f->(@_));
    };
};

my $plusThree = sub {
    my ($i) = @_;
    $i + 3;
};

my $g = $twice->($plusThree);

print $g->(7), "\n"; # 13
</syntaxhighlight>

====Python====
{{further information|Python (programming language)}}

<syntaxhighlight lang="pycon">
>>> def twice(f):
...     def result(x):
...         return f(f(x))
...     return result

>>> plus_three = lambda i: i + 3

>>> g = twice(plus_three)
    
>>> g(7)
13
</syntaxhighlight>

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 {{code|g}} could be implemented equivalently:

<syntaxhighlight lang="pycon">
>>> @twice
... def g(i):
...     return i + 3

>>> g(7)
13
</syntaxhighlight>

====R====
{{further information|R (programming language)}}

<syntaxhighlight lang="R">
twice <- \(f) \(x) f(f(x))

plusThree <- function(i) i + 3

g <- twice(plusThree)

> g(7)
[1] 13
</syntaxhighlight>

====Raku====
{{further information|Raku (programming language)}}

<syntaxhighlight lang="perl6">
sub twice(Callable:D $f) {
    return sub { $f($f($^x)) };
}

sub plusThree(Int:D $i) {
    return $i + 3;
}

my $g = twice(&plusThree);

say $g(7); # 13
</syntaxhighlight>

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====
{{further information|Ruby (programming language)}}

<syntaxhighlight lang="ruby">
def twice(f)
  ->(x) { f.call(f.call(x)) }
end

plus_three = ->(i) { i + 3 }

g = twice(plus_three)

puts g.call(7) # 13
</syntaxhighlight>

====Rust====
{{further information|Rust (programming language)}}

<syntaxhighlight lang="rust">
fn twice(f: impl Fn(i32) -> i32) -> impl Fn(i32) -> i32 {
    move |x| f(f(x))
}

fn plus_three(i: i32) -> i32 {
    i + 3
}

fn main() {
    let g = twice(plus_three);

    println!("{}", g(7)) // 13
}
</syntaxhighlight>

====Scala====
{{further information|Scala (programming language)}}

<syntaxhighlight lang="scala">
object Main {
  def twice(f: Int => Int): Int => Int =
    f compose f

  def plusThree(i: Int): Int =
    i + 3

  def main(args: Array[String]): Unit = {
    val g = twice(plusThree)

    print(g(7)) // 13
  }
}
</syntaxhighlight>

====Scheme====
{{further information|Scheme (programming language)}}

<syntaxhighlight lang="scheme">
(define (compose f g) 
  (lambda (x) (f (g x))))

(define (twice f) 
  (compose f f))

(define (plus-three i)
  (+ i 3))

(define g (twice plus-three))

(display (g 7)) ; 13
(display "\n")
</syntaxhighlight>

====Swift====
{{further information|Swift (programming language)}}

<syntaxhighlight lang="swift">
func twice(_ f: @escaping (Int) -> Int) -> (Int) -> Int {
    return { f(f($0)) }
}

let plusThree = { $0 + 3 }

let g = twice(plusThree)

print(g(7)) // 13
</syntaxhighlight>

====Tcl====
{{further information|Tcl}}

<syntaxhighlight lang="tcl">
set twice {{f x} {apply $f [apply $f $x]}}
set plusThree {{i} {return [expr $i + 3]}}

# result: 13
puts [apply $twice $plusThree 7]
</syntaxhighlight>

Tcl uses apply command to apply an anonymous function (since 8.6).

====XACML====
{{further information|XACML}}

The XACML standard defines higher-order functions in the standard to apply a function to multiple values of attribute bags.

<syntaxhighlight lang="xquery">
rule allowEntry{
    permit
    condition anyOfAny(function[stringEqual], citizenships, allowedCitizenships)
}
</syntaxhighlight>

The list of higher-order functions in XACML can be found [[XACML#Higher order functions|here]].

====XQuery====
{{further information|XQuery}}

<syntaxhighlight lang="xquery">
declare function local:twice($f, $x) {
  $f($f($x))
};

declare function local:plusthree($i) {
  $i + 3
};

local:twice(local:plusthree#1, 7) (: 13 :)
</syntaxhighlight>

=== Alternatives ===
====Function pointers====
[[Function pointer]]s in languages such as [[C (programming language)|C]], [[C++]], [[Fortran]], and [[Pascal (programming language)|Pascal]] allow programmers to pass around references to functions. The following C code computes an approximation of the integral of an arbitrary function:

<syntaxhighlight lang="c">
#include <stdio.h>

double square(double x)
{
    return x * x;
}

double cube(double x)
{
    return x * x * x;
}

/* Compute the integral of f() within the interval [a,b] */
double integral(double f(double x), double a, double b, int n)
{
    int i;
    double sum = 0;
    double dt = (b - a) / n;
    for (i = 0;  i < n;  ++i) {
        sum += f(a + (i + 0.5) * dt);
    }
    return sum * dt;
}

int main()
{
    printf("%g\n", integral(square, 0, 1, 100));
    printf("%g\n", integral(cube, 0, 1, 100));
    return 0;
}
</syntaxhighlight>

The [[qsort]] function from the C standard library uses a function pointer to emulate the behavior of a higher-order function.

====Macros====
[[Macro (computer science)|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 [[type system#Static typing|statically typed]]; these languages generally rely on [[type system#Dynamic typing|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 [[interpreter (computing)|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, [[object (computer science)|objects]] can be an effective substitute. An object's [[method (computer science)|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 memory allocation|stack]]-based (versus [[dynamic memory allocation|heap]]-based) objects or [[Record (computer science)|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:

<syntaxhighlight lang="pascal">
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.
</syntaxhighlight>

The function <code>a()</code> takes a <code>Txy</code> record as input and returns the integer value of the sum of the record's <code>x</code> and <code>y</code> fields (3 + 7).

====Defunctionalization====
[[Defunctionalization]] can be used to implement higher-order functions in languages that lack  [[first class function|first-class functions]]:

<syntaxhighlight lang="cpp">
// Defunctionalized function data structures
template<typename T> struct Add { T value; };
template<typename T> struct DivBy { T value; };
template<typename F, typename G> struct Composition { F f; G g; };

// Defunctionalized function application implementations
template<typename F, typename G, typename X>
auto apply(Composition<F, G> f, X arg) {
    return apply(f.f, apply(f.g, arg));
}

template<typename T, typename X>
auto apply(Add<T> f, X arg) {
    return arg  + f.value;
}

template<typename T, typename X>
auto apply(DivBy<T> f, X arg) {
    return arg / f.value;
}

// Higher-order compose function
template<typename F, typename G>
Composition<F, G> compose(F f, G g) {
    return Composition<F, G> {f, g};
}

int main(int argc, const char* argv[]) {
    auto f = compose(DivBy<float>{ 2.0f }, Add<int>{ 5 });
    apply(f, 3); // 4.0f
    apply(f, 9); // 7.0f
    return 0;
}
</syntaxhighlight>

In this case, different types are used to trigger different functions via [[function overloading]]. The overloaded function in this example has the signature <code>auto apply</code>.

==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 message]]s

==References==
{{Reflist}}
{{Functions navbox}}

[[Category:Functional programming]]
[[Category:Lambda calculus]]
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[[Category:Subroutines]]
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[[Category:Articles with example Julia code]]
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