4.1. Higher-Order Functions

Consider these functions double and square on integers:

let double x = 2 * x
let square x = x * x
val double : int -> int = <fun>
val square : int -> int = <fun>

Let’s use these functions to write other functions that quadruple and raise a number to the fourth power:

let quad x = double (double x)
let fourth x = square (square x)
val quad : int -> int = <fun>
val fourth : int -> int = <fun>

There is an obvious similarity between these two functions: what they do is apply a given function twice to a value. By passing in the function to another function twice as an argument, we can abstract this functionality:

let twice f x = f (f x)
val twice : ('a -> 'a) -> 'a -> 'a = <fun>

The function twice is higher-order: its input f is a function. And—recalling that all OCaml functions really take only a single argument—its output is technically fun x -> f (f x), so twice returns a function hence is also higher-order in that way.

Using twice, we can implement quad and fourth in a uniform way:

let quad x = twice double x
let fourth x = twice square x
val quad : int -> int = <fun>
val fourth : int -> int = <fun>

4.1.1. The Abstraction Principle

Above, we have exploited the structural similarity between quad and fourth to save work. Admittedly, in this toy example it might not seem like much work. But imagine that twice were actually some much more complicated function. Then if someone comes up with a more efficient version of it, every function written in terms of it (like quad and fourth) could benefit from that improvement in efficiency, without needing to be recoded.

Part of being an excellent programmer is recognizing such similarities and abstracting them by creating functions (or other units of code) that implement them. Bruce MacLennan names this the Abstraction Principle in his textbook Functional Programming: Theory and Practice (1990). The Abstraction Principle says to avoid requiring something to be stated more than once; instead, factor out the recurring pattern. Higher-order functions enable such refactoring, because they allow us to factor out functions and parameterize functions on other functions.

Besides twice, here are some more relatively simple examples, indebted also to MacLennan:

Apply. We can write a function that applies its first input to its second input:

let apply f x = f x
val apply : ('a -> 'b) -> 'a -> 'b = <fun>

Of course, writing apply f is a lot more work than just writing f.

Pipeline. The pipeline operator, which we’ve previously seen, is a higher-order function:

let pipeline x f = f x
let (|>) = pipeline
let x = 5 |> double
val pipeline : 'a -> ('a -> 'b) -> 'b = <fun>
val ( |> ) : 'a -> ('a -> 'b) -> 'b = <fun>
val x : int = 10

Compose. We can write a function that composes two other functions:

let compose f g x = f (g x)
val compose : ('a -> 'b) -> ('c -> 'a) -> 'c -> 'b = <fun>

This function would let us create a new function that can be applied many times, such as the following:

let square_then_double = compose double square
let x = square_then_double 1
let y = square_then_double 2
val square_then_double : int -> int = <fun>
val x : int = 2
val y : int = 8

Both. We can write a function that applies two functions to the same argument and returns a pair of the result:

let both f g x = (f x, g x)
let ds = both double square
let p = ds 3
val both : ('a -> 'b) -> ('a -> 'c) -> 'a -> 'b * 'c = <fun>
val ds : int -> int * int = <fun>
val p : int * int = (6, 9)

Cond. We can write a function that conditionally chooses which of two functions to apply based on a predicate:

let cond p f g x =
  if p x then f x else g x
val cond : ('a -> bool) -> ('a -> 'b) -> ('a -> 'b) -> 'a -> 'b = <fun>

4.1.2. The Meaning of “Higher Order”

The phrase “higher order” is used throughout logic and computer science, though not necessarily with a precise or consistent meaning in all cases.

In logic, first-order quantification refers primarily to the universal and existential (\(\forall\) and \(\exists\)) quantifiers. These let you quantify over some domain of interest, such as the natural numbers. But for any given quantification, say \(\forall x\), the variable being quantified represents an individual element of that domain, say the natural number 42.

Second-order quantification lets you do something strictly more powerful, which is to quantify over properties of the domain. Properties are assertions about individual elements, for example, that a natural number is even, or that it is prime. In some logics we can equate properties with sets of individual, for example the set of all even naturals. So second-order quantification is often thought of as quantification over sets. You can also think of properties as being functions that take in an element and return a Boolean indicating whether the element satisfies the property; this is called the characteristic function of the property.

Third-order logic would allow quantification over properties of properties, and fourth-order over properties of properties of properties, and so forth. Higher-order logic refers to all these logics that are more powerful than first-order logic; though one interesting result in this area is that all higher-order logics can be expressed in second-order logic.

In programming languages, first-order functions similarly refer to functions that operate on individual data elements (e.g., strings, ints, records, variants, etc.). Whereas higher-order function can operate on functions, much like higher-order logics can quantify over over properties (which are like functions).

4.1.3. Famous Higher-order Functions

In the next few sections we’ll dive into three of the most famous higher-order functions: map, filter, and fold. These are functions that can be defined for many data structures, including lists and trees. The basic idea of each is that:

  • map transforms elements,

  • filter eliminates elements, and

  • fold combines elements.