Higher-order function

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In mathematics and computer science, a higher-order function (also functional form, functional or functor) is a function that does at least one of the following:[1]

  • takes one or more functions as an input
  • outputs a function

All other functions are first-order functions. In mathematics higher-order functions are also known as operators or functionals. The derivative in calculus is a common example, since it maps a function to another function.

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 are values with types of the form (\tau_1\to\tau_2)\to\tau_3.[citation needed]

General examples[edit]

The 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. Another very common kind of higher-order function in those languages which support them are 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.

Other examples of higher-order functions include fold, function composition, integration, and the constant-function function λxy.x.

Support in programming languages[edit]

Direct support[edit]

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 some value, and applies the function to this value twice.

Python:

def twice(function, x):
    return function(function(x))
 
def f(x):
    return x + 3
 
print(twice(f, 7)) # 13

Haskell:

twice function x = (function . function) x
 
f = (-3)
 
main = print (twice f 7) -- 1

Javascript:

function twice(f, x){
    return f(f(x));
}
 
function f(x){
    return x*3;
}
 
twice(f, 7); // 63

Ruby:

def twice(f, x)
  f.call(f.call(x))
end
 
f1 = ->(x){ x / 3 }
 
print twice(f1, 9) # 1

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).

(define (add x y) (+ x y))
(define (f x)
  (lambda (y) (+ x y)))
(display ((f 3) 7))
(display (add 3 7))

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.

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 JavaScript example the higher-order function ArrayForEach takes an array and a method in as arguments and calls the method on every element in the array. Although the method may or may not return a value, there is not mapping involved since mapping requires saving returned values to a list.

function ArrayForEach(array, func) {
    for (var i = 0; i < array.length; i++) {
        if (i in array) {
            func(array[i]);
        }
    }
}
 
function log(msg) {
    console.log(msg);
}
 
ArrayForEach([1,2,3,4,5], log);

However, this could also be implemented using the map function, which in Javascript can accept functions with no return value.

function log(msg) {
    console.log(msg);
}
 
[1,2,3,4,5].map(log);

Alternatives[edit]

Function Pointers[edit]

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) {
    double sum = 0;
    double dt = (b - a) / n;
    for (int 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[edit]

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[edit]

In other imperative programming languages it is possible to achieve some of the same algorithmic results as are obtained through use of 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[edit]

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[edit]

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 through the use of overloading. The overloaded function in this example has the signature auto apply.

See also[edit]

References[edit]

External links[edit]