Higher-order function

Not to be confused with Functor (category theory).

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

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 list of elements, and as the result, returns a new list with f applied to each element from the list.
• 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.
• fold
• Function composition
• Integration
• Callback
• Tree traversal

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

OCAML

Further information: OCaml (programming language)

Explicitly

let add3 x = x + 3

let twice f x = f (f x)

print_int (twice add3 7) (* 13 *)

One-Line

print_int ((fun f x -> f (f x)) ((+)3) 7) (* 13 *)

APL

Further information: APL (programming language)

      twice←{⍺⍺ ⍺⍺ ⍵}

      plusthree←{⍵+3}

      g←{plusthree twice ⍵}
    
      g 7
13

Or in a tacit manner:

      twice←⍣2

      plusthree←+∘3

      g←plusthree twice
    
      g 7
13

J

Further information: J (programming language)

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

or point-free style,

   +&3(^:2) 7
13

Python

Further information: Python (programming language)

>>> def twice(f):
...   def result(a):
...     return f(f(a))
...   return result

>>> plusthree = lambda x: x+3

>>> g = twice(plusthree)
    
>>> g(7)
13

Pascal

Further information: Pascal (programming language)

 1 {$mode objfpc}
 2 
 3 type fun = function(x: Integer): Integer;
 4 
 5 function add3(x: Integer): Integer;
 6 begin
 7   result := x + 3;
 8 end;
 9 
10 function twice(func: fun; x: Integer): Integer;
11 begin
12   result := func(func(x));
13 end;
14 
15 begin
16   writeln(twice(@add3, 7)); { 13 }
17 end.

F#

Further information: F Sharp (programming language)

let twice f = f >> f

let f = (+) 3

twice f 7 |> printf "%A" // 13

D

Further information: D (programming language)

int delegate(int) twice(int delegate(int) f)
{
    int twiceApplied(int x)
    {
        return f(f(x));
    }
    return &twiceApplied;
}

import std.stdio;
int plusThree(int x)
{
    return x + 3;
}
writeln(twice(&plusThree)(7)); // 13

C#

Further information: C Sharp (programming language)

Func<Func<int,int>,Func<int,int>> twice = f => x => f(f(x));
Func<int,int> plusThree = x => x + 3;

Console.WriteLine(twice(plusThree)(7)); // 13

or more quickly

var twice = f => x => f(f(x));
var plusThree = x => x + 3;

Console.WriteLine(twice(plusThree)(7)); // 13

Haskell

Further information: Haskell (programming language)

twice :: (a -> a) -> (a -> a)
twice f = f . f

f :: Num a => a -> a
f = subtract 3

main :: IO ()
main = print (twice f 7) -- 1

Or more quickly:

twice f = f . f
main = print $ twice (+3) 7 -- 13

Clojure

Further information: Clojure

(defn twice [function x]
  (function (function x)))

(twice #(+ % 3) 7) ;13

In Clojure, "#" starts a lambda expression, and "%" refers to the next function argument.

Scheme

Further information: Scheme (programming language)

(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 (f x) is used to implement currying. It takes a single argument and returns a function. The evaluation of the expression ((f 3) 7) first returns a function after evaluating (f 3). The returned function is (lambda (y) (+ 3 y)). Then, it evaluates the returned function with 7 as the argument, returning 10. This is equivalent to the expression (add 3 7), since (f x) is equivalent to the curried form of (add x y).

Erlang

Further information: Erlang (programming language)

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).

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

Elixir

Further information: Elixir (programming language)

In Elixir, you can mix module definitions and anonymous functions

defmodule Hop do
    def twice(f, v) do
        f.(f.(v))
    end
end

add3 = fn(v) -> 3 + v end

IO.puts Hop.twice(add3, 7) #13

Alternatively, we can also compose using pure anonymous functions.

twice = fn(f, v) -> f.(f.(v)) end
add3 = fn(v) -> 3 + v end

IO.puts twice.(add3, 7) #13

JavaScript

Further information: JavaScript

const twice = (f, v) => f(f(v));
const add3 = v => v + 3;

twice(add3, 7); // 13

Go

Further information: Go (programming language)

func twice(f func(int) int, v int) int {
	return f(f(v))
}

func main() {
	f := func(v int) int {
		return v + 3
	}
	twice(f, 7) // returns 13
}

Notice a function literal can be defined either with an identifier (twice) or anonymously (assigned to variable f). Run full program on Go Playground!

Scala

Further information: Scala (programming language)

def twice(f:Int=>Int) = f compose f

twice(_+3)(7) // 13

Java (1.8+)

Further information: Java (programming language) and Java version history

Function<Function<Integer, Integer>, Function<Integer, Integer>> twice = f -> f.andThen(f);
twice.apply(x -> x + 3).apply(7); // 13

Kotlin

Further information: Kotlin (programming language)

fun <T> twice(f: (T)->T): (T)->T = {f(f(it))}
fun f(x:Int) = x + 3
println(twice(::f)(7)) // 13

Lua

Further information: Lua (programming language)

local twice = function(f,v)
    return f(f(v))
end

local f = function(v)
    return v + 3
end

print(twice(f,7)) -- 13

Swift

Further information: Swift (programming language)

// generic function
func twice<T>(_ v: @escaping (T) -> T) -> (T) -> T {
    return { v(v($0)) }
}

// inferred closure
let f = { $0 + 3 }

twice(f)(10) // 16

Rust

Further information: Rust (programming language)

// Take function f(x), return function f(f(x))
fn twice<A>(function: impl Fn(A) -> A) -> impl Fn(A) -> A
{
    move |a| function(function(a))
}

// Return x + 3
fn f(x: i32) -> i32 {
    x + 3
}

fn main() {
    let g = twice(f);
    println!("{}", g(7));
}

Ruby

Further information: Ruby (programming language)

def twice(f, x)
  f.call f.call(x)
end

add3 = ->(x) { x + 3 }
puts twice(add3, 7) #=> 13

C++

Further information: C++

With generic lambdas provided by C++14:

#include <iostream>

auto twice = [](auto f, int v)
{
    return f(f(v));
};
    
auto f = [](int i)
{
    return i + 3;
};
 
int main()
{   
    std::cout << twice(f, 7) << std::endl;
}

Or, using std::function in C++11 :

#include <iostream>
#include <functional>

auto twice = [](const std::function<int(int)>& f, int v)
{
    return f(f(v));
};
    
auto f = [](int i)
{
    return i + 3;
};
    
int main()
{
    std::cout << twice(f, 7) << std::endl;
}

D

Further information: D_(programming_language)

import std.stdio : writeln;

alias twice = (f, i) => f(f(i));
alias f = (int i) => i + 3;

void main()
{
    writeln(twice(f, 7));
}

ColdFusion Markup Language (CFML)

Further information: ColdFusion Markup Language

twice = function(f, v) {
    return f(f(v));
};

f = function(v) {
    return v + 3;
};

writeOutput(twice(f, 7)); // 13

PHP

Further information: PHP

$twice = function($f, $v) {
    return $f($f($v));
};

$f = function($v) {
    return $v + 3;
};

echo($twice($f, 7)); // 13

R

Further information: R (programming language)

twice <- function(func) {
  return(function(x) {
    func(func(x))
  })
}

f <- function(x) {
  return(x + 3)
}

g <- twice(f)

> print(g(7))
[1] 13

Perl 6

Further information: Perl 6

sub twice(Callable:D $c) {
    return sub { $c($c($^x)) };
}

sub f(Int:D $x) {
    return $x + 3;
}

my $g = twice(&f);

say $g(7); #OUTPUT: 13

In Perl 6, 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. Perl 6 also supports "pointy block" syntax for lambda expressions which can be assigned to a variable or invoked anonymously.

Tcl

Further information: Tcl

set twice {{f v} {apply $f [apply $f $v]}}
set f {{v} {return [expr $v + 3]}}

# result: 13
puts [apply $twice $f 7]

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

XQuery

Further information: XQuery

declare function local:twice($f, $x) {
  $f($f($x))
};

declare function local:f($x) {
  $x + 3
};

local:twice(local:f#1, 7) (: 13 :)

XACML

Further information: XACML

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

rule allowEntry{
    permit
    condition anyOfAny(function[stringEqual], citizenships, allowedCitizenships)
}

The list of higher-order functions is can be found here.

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 <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;
}

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<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;
}

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