Higher order function

Definition

A higher-order function is a function that takes other functions as arguments or returns a function as result.

Discussion

The major use is to abstract common behaviour into one place.

Examples

In the libraries

Many functions in the libraries are higher-order. The (probably) most commonly given examples are map and fold.

Two other common ones are curry, uncurry. A possible implementation of these is:

curry :: ((a,b)->c) -> a->b->c
curry f a b = f (a,b)

uncurry :: (a->b->c) -> ((a,b)->c)
uncurry f (a,b)= f a b

curry's first argument must be a function which accepts a pair. It applies that function to its next two arguments.

uncurry is the inverse of curry. Its first argument must be a function taking two values. uncurry then applies that function to the components of the pair which is the second argument.

Simple code examples

Rather than writing

doubleList []     = []
doubleList (x:xs) = 2*x : doubleList xs

and

tripleList []     = []
tripleList (x:xs) = 3*x : tripleList xs

we can parameterize out the difference

multList n [] = []
multList n (x:xs) = n*x : multList n xs

and define

tripleList = multList 3
doubleList = multList 2

leading to a less error prone definition of each.

But now, if we had the function

we could parameterize the difference again

operlist n bop [] = []
operlist n bop (x:xs) = bop n x : operlist n bop xs

and define doubleList as

doubleList = operList 2 (*)

but this ties us into a constant parameters

and we could redefine things as

mapList f [] = []
mapList f (x:xs) = f x : mapList f xs

and define doubleList as

doubleList = mapList (2*)

This higher-order function "mapList" can be used in a wide range of areas to simplify code. It is called map in Haskell's Prelude.

Mathematical examples

In mathematics the counterpart to higher-order functions are functionals (mapping functions to scalars) and function operators (mapping functions to functions). Typical functionals are the limit of a sequence, or the integral of an interval of a function.

limit :: [Double] -> Double
definiteIntegral :: (Double, Double) -> (Double -> Double) -> Double

Typical operators are the indefinite integral, the derivative, the function inverse.

indefiniteIntegral :: Double -> (Double -> Double) -> (Double -> Double)
derive :: (Double -> Double) -> (Double -> Double)
inverse :: (Double -> Double) -> (Double -> Double)

Here is a numerical approximation:

derive :: Double -> (Double -> Double) -> (Double -> Double)
derive eps f x = (f(x+eps) - f(x-eps)) / (2*eps)