Higher-order function explained

In mathematics and computer science, a higher-order function (HOF) is a 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 of a 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 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

(\tau _{1\to\tau} _{2)\to\tau} _{3} .

General examples

• [[map (higher-order function)|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 713

Or in a tacit manner:

twice←⍣2

plusthree←+∘3

g←plusthree twice g 713

C++

Using in C++11:

1. include
2. include

auto twice = [](const std::function& f)

auto plus_three = [](int i)

int main

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

1. 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 [f] (fn [x] (f (f x))))

(defn plus-three [i] (+ 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

Common 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))

D

import std.stdio : writeln;

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

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

void main

Dart

int Function(int) twice(int Function(int) f)

int plusThree(int i)

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 endend

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

plus_three = fn(i) -> i + 3 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 -> IntplusThree = (+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 713

or tacitly,

twice=. ^:2

plusthree=. +&3 g=. plusthree twice g 713

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

With arrow functions:

"use strict";

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

const plusThree = i => i + 3;

const g = twice(plusThree);

console.log(g(7)); // 13

Or with classical syntax:

"use strict";

function twice(f)

function plusThree(i)

const g = twice(plusThree);

console.log(g(7)); // 13

Julia

julia> function twice(f) function result(x) return f(f(x)) end return result endtwice (generic function with 1 method)

julia> plusthree(i) = i + 3plusthree (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

function twice(f) return function (x) return f(f(x)) endend

function plusThree(i) return i + 3end

local g = twice(plusThree)

print(g(7)) -- 13

MATLAB

function result = twice(f)result = @(x) f(f(x));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

declare(strict_types=1);

function twice(callable $f): Closure

function plusThree(int $i): int

$g = twice('plusThree');

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

or with all functions in variables:

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

Note that arrow functions implicitly capture any variables that come from the parent scope,^[1] whereas anonymous functions require the keyword to do the same.

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

>>> plus_three = lambda i: i + 3

>>> g = twice(plus_three) >>> 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))[1] 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 (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")

Swift

func twice(_ f: @escaping (Int) -> Int) -> (Int) -> Int

let plusThree =

let g = twice(plusThree)

print(g(7)) // 13

Tcl

set twice set plusThree

1. result: 13

puts [apply $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, C++, Fortran, and Pascal allow programmers to pass around references to functions. The following C code computes an approximation of the integral of an arbitrary function:

1. include

double square(double x)

double cube(double x)

/* Compute the integral of f within the interval [a,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 10end.

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 structurestemplate struct Add ;template struct DivBy ;template struct Composition ;

// Defunctionalized function application implementationstemplateauto apply(Composition f, X arg)

templateauto apply(Add f, X arg)

templateauto apply(DivBy f, X arg)

// Higher-order compose functiontemplateComposition 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

Notes and References

1. Web site: PHP: Arrow Functions - Manual. 2021-03-01. www.php.net.