Functional Programming

What Is Functional Programming?

We've got quite far into the tutorial, yet we haven't really considered functional programming. All the features given so far (rich data types, pattern matching, type inference, nested functions) could exist in a "super C" language. These are cool features that make your code concise, easy to read, and have fewer bugs; however, they actually have very little to do with functional programming. In fact, the reason functional languages are so great is not because of functional programming but because we've been stuck with C-like languages for years, and in the meantime, cutting-edge programming has progressed considerably. So while we were writing struct { int type; union { ... } } for the umpteenth time, ML and Haskell programmers had safe variants and pattern matching on datatypes. While we were being careful to free all our mallocs, there have been languages with garbage collectors able to outperform hand coding since the 80s.

Let's explore functional programming.

The basic, and not very enlightening, distinction in functional programming is that functions are first-class citizens.

(Remember that OCaml runs code when it sees the double semicolon ;;. Throughout this tutorial, executable code examples begin with the command prompt #, end in ;;, and show the expected output. If it doesn't, it's showcasing the code pattern rather than an executable code block.)

Let's look at an example to illustrate:

# let double x = x * 2 in
    List.map double [1; 2; 3];;
- : int list = [2; 4; 6]

In this example, we first defined a nested function called double that took the argument x and returned x * 2. Then map calls double on each element of the given list ([1; 2; 3]) to produce the result: a list with each number doubled.

map is known as a higher-order function (HOF). Higher-order functions are just a fancy way of saying that the function takes a function as one of its arguments. Simple so far. If you're familiar with C/C++ then this looks like passing a function pointer around.

Closures are functions which carry around some of the "environment" in which they were defined. In particular, a closure can reference variables which were available at the point of its definition. Let's generalise the function above so that now we can take any list of integers and multiply each element by an arbitrary value n:

# let multiply n list =
    let f x = n * x in
      List.map f list;;
val multiply : int -> int list -> int list = <fun>

Hence:

# multiply 2 [1; 2; 3];;
- : int list = [2; 4; 6]
# multiply 5 [1; 2; 3];;
- : int list = [5; 10; 15]

The important point to note about the multiply function is the nested function f. This is a closure. Look at how f uses the value of n which isn't actually passed as an explicit argument to f. Instead f picks it up from its environment; it's an argument to the multiply function and hence available within this function.

This might sound straightforward, but let's look a bit closer at that call to map: List.map f list.

map is defined in the List module, far away from the current code. In other words, we're passing f into a module defined "a long time ago, in a galaxy far far away." For all we know, that code might pass f to other modules or save a reference to f somewhere and call it later. Whether it does or not, the closure will ensure that f always has access back to its parental environment, and to n.

Here's the code pattern (skeleton) extracted from a real example from LabGtk. This is actually a method on a class (we haven't talked about classes and objects yet, but just think of it as a function definition for now).

class html_skel obj = object (self)
  ...
  ...
  method save_to_channel chan =
    let receiver_fn content =
      output_string chan content;
      true
    in
      save obj receiver_fn
  ...
end

First of all you need to know that the save function called at the end of the method takes a function as its second argument: (receiver_fn). It repeatedly calls receiver_fn with snippets of text from the widget that it's trying to save.

Now look at the definition of receiver_fn. This function is a closure alright because it keeps a reference to chan from its environment.

Partial Function Applications

Let's define a plus function, which just adds two integers:

# let plus a b =
    a + b;;
val plus : int -> int -> int = <fun>

Some questions for people asleep at the back of the class.

1. What is plus?
2. What is plus 2 3?
3. What is plus 2?

Question 1 is easy: plus is a function. It takes two arguments that are integers and returns an integer. We write its type like this:

# plus;;
- : int -> int -> int = <fun>

Question 2 is even easier: plus 2 3 is a number, the integer 5. We write its value and type like this:

# 5;;
- : int = 5

But what about question 3? It looks like plus 2 is a typo or an incomplete copy-paste, but it isn't. If we type this into the OCaml toplevel, it returns:

# plus 2;;
- : int -> int = <fun>

This isn't an error. It's telling us that plus 2 is in fact a function, which takes an int and returns an int.

What sort of function is this? Let's experiment by first giving this mysterious function a name (f) and then trying it out on a few integers to see what it does:

# let f = plus 2;;
val f : int -> int = <fun>
# f 10;;
- : int = 12
# f 15;;
- : int = 17
# f 99;;
- : int = 101

In engineering, this is sufficient proof to state f (i.e., plus 2) is the function that adds 2 to its parameter.

Looking at the types of these expressions, we may be able to see some rationale for the strange -> arrow notation used for function types:

# plus;;
- : int -> int -> int = <fun>
# plus 2;;
- : int -> int = <fun>
# plus 2 3;;
- : int = 5

This process is called partial application of a function.

Remember our double and multiply functions from earlier on? multiply was defined as this:

# let multiply n list =
  let f x = n * x in
    List.map f list;;
val multiply : int -> int list -> int list = <fun>

We can now define double and triple functions very easily, like this:

# let double = multiply 2;;
val double : int list -> int list = <fun>
# let triple = multiply 3;;
val triple : int list -> int list = <fun>

They really are functions, look:

# double [1; 2; 3];;
- : int list = [2; 4; 6]
# triple [1; 2; 3];;
- : int list = [3; 6; 9]

You can also use partial application directly (without the intermediate f function) like this:

# let multiply n = List.map (( * ) n);;
val multiply : int -> int list -> int list = <fun>
# let double = multiply 2;;
val double : int list -> int list = <fun>
# let triple = multiply 3;;
val triple : int list -> int list = <fun>
# double [1; 2; 3];;
- : int list = [2; 4; 6]
# triple [1; 2; 3];;
- : int list = [3; 6; 9]

In the example above, (( * ) n) is the partial application of the ( * ) (times) function. Note the extra spaces needed so that OCaml doesn't think (* starts a comment.

You can also put infix operators into brackets to make functions. Here's an identical definition of the plus function as before:

# let plus = ( + );;
val plus : int -> int -> int = <fun>
# plus 2 3;;
- : int = 5

Here's some more examples of partial application:

# List.map (plus 2) [1; 2; 3];;
- : int list = [3; 4; 5]
# let list_of_functions = List.map plus [1; 2; 3];;
val list_of_functions : (int -> int) list = [<fun>; <fun>; <fun>]

What Is Functional Programming Good For?

Functional programming, like any good programming technique, is a useful tool for solving some classes of problems. It's very good for callbacks, which have multiple uses from GUIs to event-driven loops. It's great for expressing generic algorithms. List.map is really a generic algorithm for applying functions over any type of list. Similarly you can define generic functions over trees. Certain types of numerical problems can be solved more quickly with functional programming (for example, numerically calculating the derivative of a mathematical function).

Pure and Impure Functional Programming

A pure function is one without any side effects. A side effect means that the function keeps some sort of hidden state inside it. strlen is a good example of a pure function in C. If you call strlen with the same string, it always returns the same length. The output of strlen (the length) only depends on the inputs (the string) and nothing else. Many functions in C are, unfortunately, impure. For example, malloc. If you call it with the same number, it certainly won't return the same pointer to you. malloc, of course, relies on a huge amount of hidden internal state (objects allocated on the heap, the allocation method in use, grabbing pages from the operating system, etc.).

ML-derived languages like OCaml are "mostly pure," meaning they allow side effects through things like references and arrays. But most of the code you'll write will be "pure functional" because functional languages encourage this thinking. Haskell, another functional language, is pure functional. OCaml is therefore more practical because writing impure functions is sometimes useful.

There are various theoretical advantages of having pure functions. One advantage is that if a function is pure, it is called several times with the same arguments. The compiler only needs to actually call the function once. A good example in C is:

for (i = 0; i < strlen (s); ++i)
  {
    // Do something which doesn't affect s.
  }

If naively compiled, this loop is O(n²) because strlen (s) is called each time and strlen needs to iterate over the whole of s. If the compiler is smart enough to work out that strlen is pure functional and that s is not updated in the loop, then it can remove the redundant extra calls to strlen and make the loop O(n). Do compilers really do this? In the case of strlen yes, in other cases, probably not.

Concentrating on writing small pure functions allows you to construct reusable code using a bottom-up approach, testing each small function as you go along. The current fashion is to carefully plan your programs using a top-down approach, but in this author's opinion, this often results in projects failing.

Strictness vs. Laziness

C-derived and ML-derived languages are strict. Haskell and Miranda are non-strict, or lazy, languages. OCaml is strict by default, but it allows a lazy style of programming when needed.

In a strict language, arguments to functions are always evaluated first, and the results are then passed to the function. For example, in a strict language, the call give_me_a_three (1/0) is always going to result in a divide-by-zero error:

# let give_me_a_three _ = 3;;
val give_me_a_three : 'a -> int = <fun>
# give_me_a_three (1 / 0);;
Exception: Division_by_zero.

If you've programmed in any conventional language, this is just how things work. You might be surprised that things could work any other way.

In a lazy language, something stranger happens. Arguments to functions are only evaluated if the function actually uses them. Remember that the give_me_a_three function throws away its argument and always returns a 3? In a lazy language, the above call would not fail because give_me_a_three never looks at its first argument, so the first argument is never evaluated. Thus, the division by zero doesn't happen.

Lazy languages also let you do really odd things like defining an infinitely long list, provided you don't actually try to iterate over the whole list this works (say, instead, that you just try to fetch the first 10 elements).

OCaml is a strict language, but has a lazy module that lets you write lazy expressions. Here's an example. First we create a lazy expression for 1/0:

# let lazy_expr = lazy (1 / 0);;
val lazy_expr : int lazy_t = <lazy>

Notice the type of this lazy expression is int lazy_t.

Because give_me_a_three takes 'a (any type) we can pass this lazy expression into the function:

# give_me_a_three lazy_expr;;
- : int = 3

To evaluate a lazy expression, you must use the Lazy.force function:

# Lazy.force lazy_expr;;
Exception: Division_by_zero.

Boxed vs. Unboxed Types

"Boxes" is a term you'll hear often when discussing functional languages. It can be confusing, but in fact the distinction between boxed and unboxed types is quite simple if you've used C, C++, or Java before (in Perl, everything is boxed).

The way to understand a boxed object is to think of an object which has been allocated on the heap using malloc in C (or equivalently new in C++) and/or is referred to through a pointer. Take a look at this example C program:

#include <stdio.h>

void
printit (int *ptr)
{
  printf ("the number is %d\n", *ptr);
}

void
main ()
{
  int a = 3;
  int *p = &a;

  printit (p);
}

The variable a is allocated on the stack, and it is quite definitely unboxed.

The function printit takes a boxed integer and prints it.

The diagram below shows an array of unboxed (top) vs. boxed (below) integers:

(No prizes for guessing that the array of unboxed integers is much faster than the array of boxed integers.)

In addition, because there are fewer separate allocations, garbage collection is much faster and simpler on the array of unboxed objects.

In C or C++, you shouldn't have problems constructing either of the two array types above. In Java, you have two types, int which is unboxed and Integer which is boxed, hence considerably less efficient. In OCaml, the basic types are all unboxed.

Aliases for Function Names and Arguments

It's possible to use this as a neat trick to save typing: aliasing function names and function arguments.

Although we haven't looked at object-oriented programming (that's the subject for the "Objects" section), here's an example code pattern of an aliased function call that can be found in OCamlNet. All you need to know is that cgi # output # output_string "string" is a method call, similar to cgi.output().output_string ("string") in Java.

let begin_page cgi title =
  let out = cgi # output # output_string in
  out "<html>\n";
  out "<head>\n";
  out ("<title>" ^ text title ^ "</title>\n");
  out ("<style type=\"text/css\">\n");
  out "body { background: white; color: black; }\n";
  out "</style>\n";
  out "</head>\n";
  out "<body>\n";
  out ("<h1>" ^ text title ^ "</h1>\n")

The let out = ... is a partial function application for that method call (partial, because the string parameter hasn't been applied). out is therefore a function, which takes a string parameter.

out "<html>\n";

is equivalent to:

cgi # output # output_string "<html>\n";

By altering the code pattern, we saved ourselves a lot of typing there.

We can also add arguments. This alternative definition of print_string can be thought of as a kind of alias for a function name plus arguments:

let print_string = output_string stdout;;

output_string takes two arguments (a channel and a string), but since we have only supplied one, it is partially applied. So print_string is a function that expects one string argument.