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Exceptions and hash tables

(unfinished)

Mutually recursive functions

Suppose I want to define two functions which call each other. This is actually not a very common thing to do, but it can be useful sometimes. Here's a contrived example (thanks to Ryan Tarpine): The number 0 is even. Other numbers greater than 0 are even if their predecessor is odd. Hence:

# let rec even n =
    match n with
    | 0 -> true
    | x -> odd (x-1);;
Error: Unbound value odd

The code above doesn't compile because we haven't defined the function odd yet! That's easy though. Zero is not odd, and other numbers greater than 0 are odd if their predecessor is even. So to make this complete we need that function too:

# let rec even n =
    match n with
    | 0 -> true
    | x -> odd (x-1)
    
  let rec odd n =
    match n with
    | 0 -> false
    | x -> even (x-1);;
Error: Unbound value odd

The only problem is... this program doesn't compile. In order to compile the even function, we already need the definition of odd, and to compile odd we need the definition of even. So swapping the two definitions around won't help either.

There are no "forward prototypes" (as seen in languages descended from C) in OCaml but there is a special syntax for defining a set of two or more mutually recursive functions, like odd and even:

# let rec even n =
    match n with
    | 0 -> true
    | x -> odd (x-1)
  and odd n =
    match n with
    | 0 -> false
    | x -> even (x-1);;
val even : int -> bool = <fun>
val odd : int -> bool = <fun>

You can also use similar syntax for writing mutually recursive class definitions and modules.

Aliases for function names and arguments

Recall that we talked about partial function application. 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 from OCamlNet of an aliased function call. 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";

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, expecting one string argument.

Labelled and optional arguments to functions

Labelled arguments

Python has a nice syntax for writing arguments to functions. Here's an example (from the Python tutorial, since I'm not a Python programmer):

def ask_ok(prompt, retries=4, complaint='Yes or no, please!'):
  # function definition omitted

Here are the ways we can call this Python function:

ask_ok ('Do you really want to quit?')
ask_ok ('Overwrite the file?', 2)
ask_ok (prompt='Are you sure?')
ask_ok (complaint='Please answer yes or no!', prompt='Are you sure?')

Notice that in Python we are allowed to name arguments when we call them, or use the usual function call syntax, and we can have optional arguments with default values.

You can do something similar in Perl:

sub ask_ok
{
  my %params = @_;
  
  my $prompt = $params{prompt};
  my $retries = exists $params{retries} ? $params{retries} : 4;
  
  # ... etc.
}
  
ask_ok (prompt => "Are you sure?", retries => 2);

OCaml also has a way to label arguments and have optional arguments with default values.

The basic syntax is:

# let rec range ~first:a ~last:b =
    if a > b then []
    else a :: range ~first:(a+1) ~last:b;;
val range : first:int -> last:int -> int list = <fun>

(Notice that both to and end are reserved words in OCaml, so they cannot be used as labels. So you cannot have ~from/~to or ~start/~end.)

The type of our previous range function was:

range : int -> int -> int list

And the type of our new range function with labelled arguments is:

range : first:int -> last:int -> int list

Confusingly, the ~ (tilde) is not shown in the type definition, but you need to use it everywhere else.

With labelled arguments, it doesn't matter which order you give the arguments anymore:

# range ~first:1 ~last:10;;
- : int list = [1; 2; 3; 4; 5; 6; 7; 8; 9; 10]
# range ~last:10 ~first:1;;
- : int list = [1; 2; 3; 4; 5; 6; 7; 8; 9; 10]

There is also a shorthand way to name the arguments, so that the label is the same as the variable in the function definition. Here is a function defined in lablgtk/gaux.ml (a library of useful oddities used in lablgtk):

# let may ~f x =
    match x with
    | None -> ()
    | Some x -> ignore(f x);;
val may : f:('a -> 'b) -> 'a option -> unit = <fun>

It's worth spending some time working out exactly what this function does, and also working out by hand its type signature. There's a lot going on. First of all, the parameter ~f is just shorthand for ~f:f (ie. the label is ~f and the variable used in the function is f). Secondly notice that the function takes two parameters. The second parameter (x) is unlabelled - it is permitted for a function to take a mixture of labelled and unlabelled arguments if you want.

What is the type of the labelled f parameter? Obviously it's a function of some sort.

What is the type of the unlabelled x parameter? The match clause gives us a clue. It's an 'a option.

This tells us that f takes an 'a parameter, and the return value of f is ignored, so it could be anything. The type of f is therefore 'a -> 'b.

The may function as a whole returns unit. Notice in each case of the match the result is ().

Thus the type of the may function is (and you can verify this in the OCaml interactive toplevel if you want):

may : f:('a -> 'b) -> 'a option -> unit

What does this function do? Running the function in the OCaml toplevel gives us some clues:

# may ~f:print_endline None;;
- : unit = ()
# may ~f:print_endline (Some "hello");;
hello
- : unit = ()

If the unlabelled argument is a “null pointer” then may does nothing. Otherwise may calls the f function on the argument. Why is this useful? We're just about to find out ...

Optional arguments

Optional arguments are like labelled arguments, but we use ? instead of ~ in front of them. Here is an example:

# let rec range ?(step=1) a b =
    if a > b then []
    else a :: range ~step (a+step) b;;
val range : ?step:int -> int -> int -> int list = <fun>

Note the somewhat confusing syntax, switching between ? and ~. We'll talk about that in the next section. Here is how you call this function:

# range 1 10;;
- : int list = [1; 2; 3; 4; 5; 6; 7; 8; 9; 10]
# range 1 10 ~step:2;;
- : int list = [1; 3; 5; 7; 9]

In this case, ?(step=1) fairly obviously means that ~step is an optional argument which defaults to 1. We can also omit the default value and just have an optional argument. This example is modified from lablgtk:

# type window = { mutable title: string;
                  mutable width: int;
                  mutable height: int }
    
  let create_window () =
    { title = "none"; width = 640; height = 480; }
    
  let set_title window title =
    window.title <- title
    
  let set_width window width =
    window.width <- width
    
  let set_height window height =
    window.height <- height
    
  let open_window ?title ?width ?height () =
    let window = create_window () in
    may ~f:(set_title window) title;
    may ~f:(set_width window) width;
    may ~f:(set_height window) height;
    window;;
type window = {
  mutable title : string;
  mutable width : int;
  mutable height : int;
}
val create_window : unit -> window = <fun>
val set_title : window -> string -> unit = <fun>
val set_width : window -> int -> unit = <fun>
val set_height : window -> int -> unit = <fun>
val open_window :
  ?title:string -> ?width:int -> ?height:int -> unit -> window = <fun>

This example is significantly complex and quite subtle, but the pattern used is very common in the lablgtk source code. Let's concentrate on the simple create_window function first. This function takes a unit and returns a window, initialized with default settings for title, width and height:

# create_window ();;
- : window = {title = "none"; width = 640; height = 480}

The set_title, set_width and set_height functions are impure functions which modify the window structure, in the obvious way. For example:

# let w = create_window () in
  set_title w "My Application";
  w;;
- : window = {title = "My Application"; width = 640; height = 480}

So far this is just the imperative "mutable records" which we talked about in the previous chapter. Now the complex part is the open_window function. This function takes 4 arguments, three of them optional, followed by a required, unlabelled unit. Let's first see this function in action:

# open_window ~title:"My Application" ();;
- : window = {title = "My Application"; width = 640; height = 480}
# open_window ~title:"Clock" ~width:128 ~height:128 ();;
- : window = {title = "Clock"; width = 128; height = 128}

It does what you expect, but how?! The secret is in the may function (see above) and the fact that the optional parameters don't have defaults.

When an optional parameter doesn't have a default, then it has type 'a option. The 'a would normally be inferred by type inference, so in the case of ?title above, this has type string option.

Remember the may function? It takes a function and an argument, and calls the function on the argument provided the argument isn't None. So:

may ~f:(set_title window) title;

If the optional title argument is not specified by the caller, then title = None, so may does nothing. But if we call the function with, for example,

open_window ~title:"My Application" ()

then title = Some "My Application", and may therefore calls set_title window "My Application".

You should make sure you fully understand this example before proceeding to the next section.

"Warning: This optional argument cannot be erased"

We've just touched upon labels and optional arguments, but even this brief explanation should have raised several questions. The first may be why the extra unit argument to open_window? Let's try defining this function without the extra unit:

# let open_window ?title ?width ?height =
    let window = create_window () in
    may ~f:(set_title window) title;
    may ~f:(set_width window) width;
    may ~f:(set_height window) height;
    window;;
Warning 16: this optional argument cannot be erased.
val open_window : ?title:string -> ?width:int -> ?height:int -> window =
  <fun>

Although OCaml has compiled the function, it has generated a somewhat infamous warning: "This optional argument cannot be erased", referring to the final ?height argument. To try to show what's going on here, let's call our modified open_window function:

# open_window;;
- : ?title:string -> ?width:int -> ?height:int -> window = <fun>
# open_window ~title:"My Application";;
- : ?width:int -> ?height:int -> window = <fun>

Did that work or not? No it didn't. In fact it didn't even run the open_window function at all. Instead it printed some strange type information. What's going on?

Recall currying and uncurrying, and partial application of functions. If we have a function plus defined as:

# let plus x y =
    x + y;;
val plus : int -> int -> int = <fun>

We can partially apply this, for example as plus 2 which is "the function that adds 2 to things":

# let f = plus 2;;
val f : int -> int = <fun>
# f 5;;
- : int = 7
# f 100;;
- : int = 102

In the plus example, the OCaml compiler can easily work out that plus 2 doesn't have enough arguments supplied yet. It needs another argument before the plus function itself can be executed. Therefore plus 2 is a function which is waiting for its extra argument to come along.

Things are not so clear when we add optional arguments into the mix. The call to open_window;; above is a case in point. Does the user mean "execute open_window now"? Or does the user mean to supply some or all of the optional arguments later? Is open_window;; waiting for extra arguments to come along like plus 2?

OCaml plays it safe and doesn't execute open_window. Instead it treats it as a partial function application. The expression open_window literally evaluates to a function value.

Let's go back to the original and working definition of open_window where we had the extra unlabelled unit argument at the end:

# let open_window ?title ?width ?height () =
    let window = create_window () in
    may ~f:(set_title window) title;
    may ~f:(set_width window) width;
    may ~f:(set_height window) height;
    window;;
val open_window :
  ?title:string -> ?width:int -> ?height:int -> unit -> window = <fun>

If you want to pass optional arguments to open_window you must do so before the final unit, so if you type:

# open_window ();;
- : window = {title = "none"; width = 640; height = 480}

you must mean "execute open_window now with all optional arguments unspecified". Whereas if you type:

# open_window;;
- : ?title:string -> ?width:int -> ?height:int -> unit -> window = <fun>

you mean "give me the functional value" or (more usually in the toplevel) "print out the type of open_window".

More ~shorthand

Let's rewrite the range function yet again, this time using as much shorthand as possible for the labels:

# let rec range ~first ~last =
    if first > last then []
    else first :: range ~first:(first+1) ~last;;
val range : first:int -> last:int -> int list = <fun>

Recall that ~foo on its own is short for ~foo:foo. This applies also when calling functions as well as declaring the arguments to functions, hence in the above the highlighted red ~last is short for ~last:last.

Using ?foo in a function call

There's another little wrinkle concerning optional arguments. Suppose we write a function around open_window to open up an application:

# let open_application ?width ?height () =
    open_window ~title:"My Application" ~width ~height;;
Error: This expression has type 'a option
       but an expression was expected of type int

Recall that ~width is shorthand for ~width:width. The type of width is 'a option, but open_window ~width: expects an int.

OCaml provides more syntactic sugar. Writing ?width in the function call is shorthand for writing ~width:(unwrap width) where unwrap would be a function which would remove the "option wrapper" around width (it's not actually possible to write an unwrap function like this, but conceptually that's the idea). So the correct way to write this function is:

# let open_application ?width ?height () =
    open_window ~title:"My Application" ?width ?height;;
val open_application : ?width:int -> ?height:int -> unit -> unit -> window =
  <fun>

When and when not to use ~ and ?

The syntax for labels and optional arguments is confusing, and you may often wonder when to use ~foo, when to use ?foo and when to use plain foo. It's something of a black art which takes practice to get right.

?foo is only used when declaring the arguments of a function, ie:

let f ?arg1 ... =

or when using the specialised "unwrap option wrapper" form for function calls:

# let open_application ?width ?height () =
    open_window ~title:"My Application" ?width ?height;;
val open_application : ?width:int -> ?height:int -> unit -> unit -> window =
  <fun>

The declaration ?foo creates a variable called foo, so if you need the value of ?foo, use just foo.

The same applies to labels. Only use the ~foo form when declaring arguments of a function, ie:

let f ~foo:foo ... =

The declaration ~foo:foo creates a variable called simply foo, so if you need the value just use plain foo.

Things, however, get complicated for two reasons: first, the shorthand form ~foo (equivalent to ~foo:foo), and second, when you call a function which takes a labelled or optional argument and you use the shorthand form.

Here is some apparently obscure code from lablgtk to demonstrate all of this:

let html ?border_width ?width ?height ?packing ?show () =  (* line 1 *)
  let w = create () in
  load_empty w;
  Container.set w ?border_width ?width ?height;            (* line 4 *)
  pack_return (new html w) ~packing ~show                  (* line 5 *)

On line 1 we have the function definition. Notice there are 5 optional arguments, and the mandatory unit 6th argument. Each of the optional arguments is going to define a variable, eg. border_width, of type 'a option.

On line 4 we use the special ?foo form for passing optional arguments to functions which take optional arguments. Container.set has the following type:

module Container = struct
  let set ?border_width ?(width = -2) ?(height = -2) w =
    (* ... *)

Line 5 uses the ~shorthand. Writing this in long form:

pack_return (new html w) ~packing:packing ~show:show

The pack_return function actually takes mandatory labelled arguments called ~packing and ~show, each of type 'a option. In other words, pack_return explicitly unwraps the option wrapper.

Addendum

If you think labels and optional arguments are complicated, that's because they are! Luckily, however, this is a relatively new feature in OCaml, and it's not yet widely used. In fact if you're not hacking on lablgtk, it's unlikely you'll see labels and optional arguments used at all (at the moment).

More variants (“polymorphic variants”)

Try compiling the following C code:

#include <stdio.h>
#include <fcntl.h>
#include <unistd.h>

enum lock { open, close };

main ()
{
  int fd, n;
  char buffer[256];

  fd = open ("/etc/motd", O_RDONLY);                     // line 12
  while ((n = read (fd, buffer, sizeof buffer)) > 0)
    write (1, buffer, n);
  close (fd);                                            // line 15
}

When I compile the code I get a whole bunch of errors including:

test.c: In function `main':
test.c:12: error: called object is not a function
test.c:15: error: called object is not a function

This illustrates one problem with enumerated types (enums) in C. In the example above, one enum statement reserves three symbols, namely lock, open and close. Here's another example:

enum lock { open, close };
enum door { open, close };

Compiling gives:

test.c:2: error: conflicting types for `open'
test.c:1: error: previous declaration of `open'
test.c:2: error: conflicting types for `close'
test.c:1: error: previous declaration of `close'

The first enum defines the symbol open as something of type enum lock. You cannot reuse that symbol in another enum.

This will be familiar to most C/C++ programmers, and they won't write naive code like that above. However the same issue happens with OCaml variants, but OCaml provides a way to work around it.

Here is some OCaml code, which actually does compile:

# type lock = Open | Close
  type door = Open | Close;;
type lock = Open | Close
type door = Open | Close

After running those two statements, what is the type of Open? We can find out easily enough in the toplevel:

# type lock = Open | Close;;
type lock = Open | Close
# type door = Open | Close;;
type door = Open | Close
# Open;;
- : door = Open

OCaml uses the most recent definition for Open, giving it the type door. This is actually not such a serious problem because if you accidentally tried to use Open in the type context of a lock, then OCaml's wonderful type inference would immediately spot the error and you wouldn't be able to compile the code.

So far, so much like C. Now I said that OCaml provides a way to work around the constraint that Open can only have one type. In other words, suppose I want to use Open to mean either "the Open of type lock" or "the Open of type door" and I want OCaml to work out which one I mean.

The syntax is slightly different, but here is how we do it:

# type lock = [ `Open | `Close ];;
type lock = [ `Close | `Open ]
# type door = [ `Open | `Close ];;
type door = [ `Close | `Open ]

Notice the syntactic differences:

  1. Each variant name is prefixed with ` (a back tick).
  2. You have to put square brackets ([]) around the alternatives.

The question naturally arises: What is the type of `Open?

# `Open;;
- : [> `Open ] = `Open

[> `Open] can be read as [ `Open | and some other possibilities which we don't know about ]. The “>” (greater than) sign indicates that the set of possibilities is bigger than those listed (open-ended).

There's nothing special about `Open. Any back-ticked word can be used as a type, even one which we haven't mentioned before:

# `Foo;;
- : [> `Foo ] = `Foo
# `Foo 42;;
- : [> `Foo of int ] = `Foo 42

Let's write a function to print the state of a lock:

# let print_lock st =
    match st with
    | `Open -> print_endline "The lock is open"
    | `Close -> print_endline "The lock is closed";;
val print_lock : [< `Close | `Open ] -> unit = <fun>

Take a careful look at the type of that function. Type inference has worked out that the st argument has type [< `Close | `Open]. The “<” (less than) sign means that this is a closed class. In other words, this function will only work on `Close or `Open and not on anything else.

# print_lock `Open;;
The lock is open
- : unit = ()

Notice that print_lock works just as well with a door as with a lock! We've deliberately given up some type safety, and type inference is now being used to help guess what we mean, rather than enforce correct coding.

This is only an introduction to polymorphic variants. Because of the reduction in type safety, it is recommended that you don't use these in your code. You will, however, see them in advanced OCaml code quite a lot precisely because advanced programmers will sometimes want to weaken the type system to write advanced idioms.