Introduction to OCaml, Part 2

In the previous chapter we've learnt how to use variables and arithmetic functions. Now it's time to learn how to make our own functions.

Anonymous functions

We will start with anonymous functions. The reason is that in OCaml, names functions are simply anonymous functions bound to name, and the special syntax for creating named functions is merely syntactic sugar.

The syntax of the anonymous functions is the following: fun <formal parameter> -> <expression>.

Let's write a simple program using an anonymous function:

let a = (fun x -> x * x) 3

let _ = print_int a

The fun x -> x * x function takes a single argument and squares it. Since we already know that the * operator works with integer, we can infer that this function has type int -> int. In the program it is applied to 3 and the result is bound to a, which is later printed.

Named functions

Now, before we learn of syntactic sugar for named functions, let's create a named function the hard way:

let square = fun x -> x * x

let a = square 3 (* 9 *)

As you can see, the syntax is exactly the same for variable and function bindings. In OCaml, they are both expressions. Like constants, functions (if not applied to any arguments), evaluate to themselves.

Bindings, scopes, and closures

Now let's examine how functions interacts with other bindings. Unlike other expressions, functions contain free variables. A free variable is a variable not bound to any value. In our square example, x is a free variable, and the expression in the right hand side contains no other variables. When that function is applied to an argument, x will be substituted with that argument in the x * x expression and that expression will be evaluated. We can visualize that process like this:

square 3 =
(fun x -> x * x) 3 =
(fun 3 -> 3 * 3) =
3 * 3 =
9

Of course, if we have previously created any bindings, we can refer to them in our functions.

let a = 2

let plus_two = fun x -> x + a

let b = plus_two 3 (* b = 5 *)

However, what happens is the name a is shadowed by another binding? Let's try and see what happens:

let a = 2

let plus_two = fun x -> x + a

let a = 3

let _ = print_int (plus_two a)

This program will print 5 (i.e. 2 + 3, not 3 + 3). The reason is that functions capture bound variables from the scope where they were created — forever. This is called lexical scoping. The alternative approach would be dynamic scoping where the value of a is be determined at the time when the plus_two function is applied. But in OCaml, values of bound variables are determined when functions are created, so we don't need to worry about it.

We can rewrite the plus_two function using a let ... in binding instead:

let plus_two =
  let a = 2 in
  fun x -> x + a

let b = plus_two 3

Here, the variable a doesn't even exist for the rest of the program because it was a local binding only visible to the fun x -> x + a expression, but it continues to exist for the plus_two function.

A function stored together with its environment is called a closure. Since you cannot prevent functions from capturing their environment in OCaml, there is no distinction between functions and closures, and for simplicity they are all called just functions. You should always remember about the effect though, and it has important practical uses.

For example, it can be used to create functions of multiple arguments using only functions of one argument. In fact, this is how functions of multiple arguments work in OCaml. Any function “of two arguments” really takes one argument and produces another function of one argument.

Functions of multiple arguments and currying

Let's write a function for calculating the average of two values.

let average = fun x -> (fun y -> (x +. y) /. 2.0)
  

As we can see, when applied to one argument, this function creates another function, where x is bound to the argument — a closure. So the expression let f = average 3.0 will be equivalent to let f = fun y -> (3.0 +. y) /. 2.0), because functions capture the bound variables from the scope where they are created, as we discussed already. The function f is thus a function that calculates the average of 3.0 and another arbitrary number, and can be applied to another argument. We could also avoid the intermediate binding and write it as (average 3.0) 4.0 is we wanted the average of 3.0 and 4.0. The type of the average function is therefore float -> (float -> float).

That's a lot of parentheses as you can see. To simplify it, OCaml and most other functional languages use a convention where arrows associate to the right. Therefore the type of the average function can be written as float -> float -> float, and it is assumed to mean float -> (float -> float). Likewise, you can apply it without any parentheses too: let a = average 3.0 4.0.

The process of creating a function “of multiple arguments” from functions of one argument is called currying, after Haskell Curry who was already mentioned in the history section, even though he wasn't the first to invent it, as it often happens with named laws.

Partial application

A big advantage of curried functions is that they make partial application especially easy. In many languages, partial application either needs a special syntax or isn't possible at all. In languages that use closures and currying, it is very easy: just use only the first argument(s) and omit the rest, and you get a function with first argument(s) fixed that can be applied to different remaining arguments as needed.

To see some real examples, let's introduce the OCaml's Printf.printf function that is used for formatted output. The reason we used print_int, print_string and similar in the first chapter is that we pretended that we know nothing about functions with more than one argument until we had a chance to learn about them systematically. In practice, people almost always use Printf.printf instead because it's much more powerful and convenient. Its name means that it belongs to the Printf module that comes with the standard library, but we will discuss modules later, for now let's consider it just an unusual name.

The type of that function is rather complicated and we will not discuss it right now. Let's just say that its first argument is a format string. Format string syntax is very similar to that of C and all languages inspired by its printf. The Hello World program could be written:

let () = Printf.printf "%s\n" "hello world"

When applied to a format string, that function will produce functions or one or more arguments depending on the format sting. For example, in let f = Printf.printf "%s %d", f will be a function of type string -> int -> unit.

Now let's write a simple program using Printf.printf and partial application of it:

let greet = Printf.printf "Hello %s!\n"

let () = greet "world"

The "Hello %s!\n" is stored in a closure together with the function that Printf.printf produced from it. A similar idiom in an object oriented language might have been greeter = new Formatter("Hello %s!\n"); greeter.format("world"). As Peter Norvig put it in one talk, objects are data with attached behaviour, while closures are behaviours with attached state data.

A danger of curried functions, however, is that failing to supply enough arguments is not a syntax error, but a valid expression, just not of the type that might be expected. Luckily, since OCaml is statically typed, this kind of errors rarely goes unnoticed and programs fail to compile.

Consider this program:

let add = fun x -> (fun y -> x + y)

let x = add 3 (* forgot second argument *)

let _ = Printf.printf "%d\n" x

It will fail to compile because add 3 expression has type int -> int, while Printf.printf "%d\n" is int -> unit.

The syntactic sugar for function definitions

Of course, creating functions by binding anonymous functions to names can quickly get cumbersome, especially when multiple arguments are involved, so OCaml provides syntactic sugar for it.

Let's rewrite the functions we've already written in a simpler way:

let plus_two x = x + 2

let average x y = (x +. y) /. 2.0

OCaml also supports multiple arguments to the fun keyword too, so anonymous functions of multiple arguments can be easily created to: let f = fun x y -> x + y.

Formal parameter is a pattern

We have already seen that the left hand side of a let-expression can be not only a name, but any valid pattern, including the wildcard or any valid constant. This also applied to the left hand side of function definitions.

For example, we can create a function that ignores its argument using the wildcard pattern:

let always_zero _ = 0

let always_one = fun _ -> 1

We can also use (), which is a constant of type unit, as a function argument. Since functions with no arguments cannot exist in OCaml, this is often done for functions used solely for their side effects:

let print_hello () = print_endline "hello world"

let _ = print_hello ()

The print_hello function in this example has type unit -> unit.

Operators are functions

It is actually not true that we haven't seen functions of multiple arguments in the first chapter, we just pretended that operators are special. While in many languages they are indeed special, in most functional languages operators are just functions that can be used in infix form.

In OCaml, every infix operator can also be used in prefix form if enclosed in parentheses:

let a = (+) 2 3
let b = (/.) 5 2
let c = (^) "hello " "world"

You can also define your own operators like any other functions using the same parentheses syntax:

let (^^) x y = x ^ x ^ y ^ y

let s = "foo" ^^ "bar" (* foofoobarbar *)

You can also define a prefix operator if you start it with the tilde:

let (~*) x = x * x

let a = ~* 2 (* 4 *)

The first character of the operator defines its associativity and precedence.

Named and optional arguments

OCaml supports named and optional arguments. Named arguments are preceded with the tilde symbol:

let greet ~greeting ~name = Printf.printf "%s %s!\n" greeting name

let _ = greet ~name:"world" ~greeting:"hello"

If you look at the inferred type of the greet function, you will see that argument labels are embedded in its type: greeting:string -> name:string -> unit. Named arguments can be used in any order as long as you specify the labels, but if argument come in the same order as they are defined in the function, you can omit the labels:

let greet ~greeting ~name = Printf.printf "%s %s!\n" greeting name

let _ = greet "hello" "world"

The syntax of optional argument is a bit more complicated. Suppose we want to write a function that takes a string and prints hello <string> by default, but allows us to use a different greeting, for example, “hi” This is how we can do it:

let greet ?(greeting="hello") name = Printf.printf "%s %s!\n" greeting name

let _ = greet "world" ~greeting:"hi"

The inferred type of this greet function will look like this: ?greeting:string -> string -> unit. It is recommended to put optional arguments first, because otherwise you will not be able to use your function with optional arguments omitted.

Exercises

Using the Printf.printf function, make a join_strings function that takes two strings and an optional separator argument that defaults to space and joins them. What is the type of that function?

Write a function that has type int -> float -> string using any functions we already encountered.

Part 1

Part 3