(Chapter written by Jacques Garrigue)
This chapter gives an overview of the new features in
Objective Caml 3: labels, and polymorphic variants.
If you have a look at modules ending in Labels in the standard
library, you will see that function types have annotations you did not
have in the functions you defined yourself.
#ListLabels.map;;
 : f:('a > 'b) > 'a list > 'b list = <fun>
#StringLabels.sub;;
The files /usr/local/bin/ocaml and /usr/local/lib/ocaml/stringLabels.cmi
make inconsistent assumptions over interface StringLabels
Such annotations of the form name: are called labels. They are
meant to document the code, allow more checking, and give more
flexibility to function application.
You can give such names to arguments in your programs, by prefixing them
with a tilde ~.
#let f ~x ~y = x  y;;
val f : x:int > y:int > int = <fun>
#let x = 3 and y = 2 in f ~x ~y;;
 : int = 1
When you want to use distinct names for the variable and the label
appearing in the type, you can use a naming label of the form
~name:. This also applies when the argument is not a variable.
#let f ~x:x1 ~y:y1 = x1  y1;;
val f : x:int > y:int > int = <fun>
#f ~x:3 ~y:2;;
 : int = 1
Labels obey the same rules as other identifiers in Caml, that is you
cannot use a reserved keyword (like in or to) as label.
Formal parameters and arguments are matched according to their
respective labels^{1}, the absence of label
being interpreted as the empty label.
This allows commuting arguments in applications. One can also
partially apply a function on any argument, creating a new function of
the remaining parameters.
#let f ~x ~y = x  y;;
val f : x:int > y:int > int = <fun>
#f ~y:2 ~x:3;;
 : int = 1
#ListLabels.fold_left;;
 : f:('a > 'b > 'a) > init:'a > 'b list > 'a = <fun>
#ListLabels.fold_left [1;2;3] ~init:0 ~f:(+);;
 : int = 6
#ListLabels.fold_left ~init:0;;
 : f:(int > 'a > int) > 'a list > int = <fun>
If in a function several arguments bear the same label (or no label),
they will not commute among themselves, and order matters. But they
can still commute with other arguments.
#let hline ~x:x1 ~x:x2 ~y = (x1, x2, y);;
val hline : x:'a > x:'b > y:'c > 'a * 'b * 'c = <fun>
#hline ~x:3 ~y:2 ~x:5;;
 : int * int * int = (3, 5, 2)
As an exception to the above parameter matching rules, if an
application is total, labels may be omitted. In practice, most
applications are total, so that labels can be omitted in applications.
#f 3 2;;
 : int = 1
#ListLabels.map succ [1;2;3];;
 : int list = [2; 3; 4]
But beware that functions like ListLabels.fold_left whose result
type is a type variable will never be considered as totally applied.
#ListLabels.fold_left (+) 0 [1;2;3];;
This expression has type int > int > int but is here used with type 'a list
When a function is passed as an argument to an higherorder function,
labels must match in both types. Neither adding nor removing labels
are allowed.
#let h g = g ~x:3 ~y:2;;
val h : (x:int > y:int > 'a) > 'a = <fun>
#h f;;
 : int = 1
#h (+);;
This expression has type int > int > int but is here used with type
x:int > y:int > 'a
An interesting feature of labeled arguments is that they can be made
optional. For optional parameters, the question mark ? replaces the
tilde ~ of nonoptional ones, and the label is also prefixed by ?
in the function type.
Default values may be given for such optional parameters.
#let bump ?(step = 1) x = x + step;;
val bump : ?step:int > int > int = <fun>
#bump 2;;
 : int = 3
#bump ~step:3 2;;
 : int = 5
A function taking some optional arguments must also take at least one
nonlabeled argument. This is because the criterion for deciding
whether an optional has been omitted is the application on a
nonlabeled argument appearing after this optional argument in the
function type.
#let test ?(x = 0) ?(y = 0) () ?(z = 0) () = (x, y, z);;
val test : ?x:int > ?y:int > unit > ?z:int > unit > int * int * int =
<fun>
#test ();;
 : ?z:int > unit > int * int * int = <fun>
#test ~x:2 () ~z:3 ();;
 : int * int * int = (2, 0, 3)
Optional parameters may also commute with nonoptional or unlabelled
ones, as long as they are applied simultaneously. By nature, optional
arguments do not commute with unlabeled arguments applied
independently.
#test ~y:2 ~x:3 () ();;
 : int * int * int = (3, 2, 0)
#test () () ~z:1 ~y:2 ~x:3;;
 : int * int * int = (3, 2, 1)
#(test () ()) ~z:1;;
This expression is not a function, it cannot be applied
Here (test () ()) is already (0,0,0) and cannot be further
applied.
Optional arguments are actually implemented as option types. If
you do not give a default value, you have access to their internal
representation, type 'a option = None  Some of 'a. You can then
provide different behaviors when an argument is present or not.
#let bump ?step x =
match step with
 None > x * 2
 Some y > x + y
;;
val bump : ?step:int > int > int = <fun>
It may also be useful to relay an optional argument from a function
call to another. This can be done by prefixing the applied argument
with ?. This question mark disables the wrapping of optional
argument in an option type.
#let test2 ?x ?y () = test ?x ?y () ();;
val test2 : ?x:int > ?y:int > unit > int * int * int = <fun>
#test2 ?x:None;;
 : ?y:int > unit > int * int * int = <fun>
4.1.2 
Labels and type inference 

While they provide an increased comfort for writing function
applications, labels and optional arguments have the pitfall that they
cannot be inferred as completely as the rest of the language.
You can see it in the following two examples.
#let h' g = g ~y:2 ~x:3;;
val h' : (y:int > x:int > 'a) > 'a = <fun>
#h' f;;
This expression has type x:int > y:int > int but is here used with type
y:int > x:int > 'a
#let bump_it bump x =
bump ~step:2 x;;
val bump_it : (step:int > 'a > 'b) > 'a > 'b = <fun>
#bump_it bump 1;;
This expression has type ?step:int > int > int but is here used with type
step:int > 'a > 'b
The first case is simple: g is passed ~y and then ~x, but f
expects ~x and then ~y. This is correctly handled if we know the
type of g to be x:int > y:int > int in advance, but otherwise
this causes the above type clash. The simplest workaround is to apply
formal parameters in a standard order.
The second example is more subtle: while we intended the argument
bump to be of type ?step:int > int > int, it is inferred as
step:int > int > 'a.
These two types being incompatible (internally normal and optional
arguments are different), a type error occurs when applying bump_it
to the real bump.
We will not try here to explain in detail how type inference works.
One must just understand that there is not enough information in the
above program to deduce the correct type of g or bump. That is,
there is no way to know whether an argument is optional or not, or
which is the correct order, by looking only at how a function is
applied. The strategy used by the compiler is to assume that there are
no optional arguments, and that applications are done in the right
order.
The right way to solve this problem for optional parameters is to add
a type annotation to the argument bump.
#let bump_it (bump : ?step:int > int > int) x =
bump ~step:2 x;;
val bump_it : (?step:int > int > int) > int > int = <fun>
#bump_it bump 1;;
 : int = 3
In practive, such problems appear mostly when using objects whose
methods have optional arguments, so that writing the type of object
arguments is often a good idea.
Normally the compiler generates a type error if you attempt to pass to
a function a parameter whose type is different from the expected one.
However, in the specific case where the expected type is a nonlabeled
function type, and the argument is a function expecting optional
parameters, the compiler will attempt to transform the argument to
have it match the expected type, by passing None for all optional
parameters.
#let twice f (x : int) = f(f x);;
val twice : (int > int) > int > int = <fun>
#twice bump 2;;
 : int = 8
This transformation is coherent with the intended semantics,
including sideeffects. That is, if the application of optional
parameters shall produce sideeffects, these are delayed until the
received function is really applied to an argument.
4.1.3 
Suggestions for labeling 

Like for names, choosing labels for functions is not an easy task. A
good labeling is a labeling which

makes programs more readable,
 is easy to remember,
 when possible, allows useful partial applications.
We explain here the rules we applied when labeling Objective Caml
libraries.
To speak in an ``objectoriented'' way, one can consider that each
function has a main argument, its object, and other arguments
related with its action, the parameters. To permit the
combination of functions through functionals in commuting label mode, the
object will not be labeled. Its role is clear by the function
itself. The parameters are labeled with names reminding either of
their nature or role. Best labels combine in their meaning nature and
role. When this is not possible the role is to prefer, since the nature will
often be given by the type itself. Obscure abbreviations should be
avoided.
ListLabels.map : f:('a > 'b) > 'a list > 'b list
UnixLabels.write : file_descr > buf:string > pos:int > len:int > unit
When there are several objects of same nature and role, they are all
left unlabeled.
ListLabels.iter2 : f:('a > 'b > 'c) > 'a list > 'b list > unit
When there is no preferable object, all arguments are labeled.
StringLabels.blit :
src:string > src_pos:int > dst:string > dst_pos:int > len:int > unit
However, when there is only one argument, it is often left unlabeled.
StringLabels.create : int > string
This principle also applies to functions of several arguments whose
return type is a type variable, as long as the role of each argument
is not ambiguous. Labeling such functions may lead to awkward error
messages when one attempts to omit labels in an application, as we
have seen with ListLabels.fold_left.
Here are some of the label names you will find throughout the
libraries.
Label 
Meaning 
f: 
a function to be applied 
pos: 
a position in a string or array 
len: 
a length 
buf: 
a string used as buffer 
src: 
the source of an operation 
dst: 
the destination of an operation 
init: 
the initial value for an iterator 
cmp: 
a comparison function, e.g. Pervasives.compare 
mode: 
an operation mode or a flag list 
All these are only suggestions, but one shall keep in mind that the
choice of labels is essential for readability. Bizarre choices will
make the program harder to maintain.
In the ideal, the right function name with right labels shall be
enough to understand the function's meaning. Since one can get this
information with OCamlBrowser or the ocaml toplevel, the documentation
is only used when a more detailed specification is needed.
Variants as presented in section 1.4 are a
powerful tool to build data structures and algorithms. However they
sometimes lack flexibility when used in modular programming. This is
due to the fact every constructor reserves a name to be used with a
unique type. On cannot use the same name in another type, or consider
a value of some type to belong to some other type with more
constructors.
With polymorphic variants, this original assumption is removed. That
is, a variant tag does not belong to any type in particular, the type
system will just check that it is an admissible value according to its
use. You need not define a type before using a variant tag. A variant
type will be inferred independently for each of its uses.
In programs, polymorphic variants work like usual ones. You just have
to prefix their names with a backquote character `.
#[`On; `Off];;
 : [> `Off  `On ] list = [`On; `Off]
#`Number 1;;
 : [> `Number of int ] = `Number 1
#let f = function `On > 1  `Off > 0  `Number n > n;;
val f : [< `Number of int  `Off  `On ] > int = <fun>
#List.map f [`On; `Off];;
 : int list = [1; 0]
[>`Off`On] list means that to match this list, you should at
least be able to match `Off and `On, without argument.
[<`On`Off`Number of int] means that f may be applied to `Off,
`On (both without argument), or `Number n where
n is an integer.
The > and < inside the variant type shows that they may still be
refined, either by defining more tags or allowing less. As such they
contain an implicit type variable. Both variant types appearing only
once in the type, the implicit type variables they constrain are not
shown.
The above variant types were polymorphic, allowing further refinement.
When writing type annotations, one will most often describe fixed
variant types, that is types that can be no longer refined. This is
also the case for type abbreviations. Such types do not contain < or
>, but just an enumeration of the tags and their associated types,
just like in a normal datatype definition.
#type 'a vlist = [`Nil  `Cons of 'a * 'a vlist];;
type 'a vlist = [ `Cons of 'a * 'a vlist  `Nil ]
#let rec map f : 'a vlist > 'b vlist = function
 `Nil > `Nil
 `Cons(a, l) > `Cons(f a, map f l)
;;
val map : ('a > 'b) > 'a vlist > 'b vlist = <fun>
Typechecking polymorphic variants is a subtle thing, and some
expressions may result in more complex type information.
#let f = function `A > `C  `B > `D  x > x;;
val f : ([> `A  `B  `C  `D ] as 'a) > 'a = <fun>
#f `E;;
 : [> `A  `B  `C  `D  `E ] = `E
Here we are seeing two phenomena. First, since this matching is open
(the last case catches any tag), we obtain the type [> `A  `B]
rather than [< `A  `B] in a closed matching. Then, since x is
returned as is, input and return types are identical. The notation as 'a denotes such type sharing. If we apply f to yet another tag
`E, it gets added to the list.
#let f1 = function `A x > x = 1  `B > true  `C > false
let f2 = function `A x > x = "a"  `B > true ;;
val f1 : [< `A of int  `B  `C ] > bool = <fun>
val f2 : [< `A of string  `B ] > bool = <fun>
#let f x = f1 x && f2 x;;
val f : [< `A of string & int  `B ] > bool = <fun>
Here f1 and f2 both accept the variant tags `A and `B, but the
argument of `A is int for f1 and string for f2. In f's
type `C, only accepted by f1, disappears, but both argument types
appear for `A as int & string. This means that if we
pass the variant tag `A to f, its argument should be both
int and string. Since there is no such value, f cannot be
applied to `A, and `B is the only accepted input.
Even if a value has a fixed variant type, one can still give it a
larger type through coercions. Coercions are normally written with
both the source type and the destination type, but in simple cases the
source type may be omitted.
#type 'a wlist = [`Nil  `Cons of 'a * 'a wlist  `Snoc of 'a wlist * 'a];;
type 'a wlist = [ `Cons of 'a * 'a wlist  `Nil  `Snoc of 'a wlist * 'a ]
#let wlist_of_vlist l = (l : 'a vlist :> 'a wlist);;
val wlist_of_vlist : 'a vlist > 'a wlist = <fun>
#let open_vlist l = (l : 'a vlist :> [> 'a vlist]);;
val open_vlist : 'a vlist > [> 'a vlist ] = <fun>
#fun x > (x :> [`A`B`C]);;
 : [< `A  `B  `C ] > [ `A  `B  `C ] = <fun>
You may also selectively coerce values through pattern matching.
#let split_cases = function
 `Nil  `Cons _ as x > `A x
 `Snoc _ as x > `B x
;;
val split_cases :
[< `Cons of 'a  `Nil  `Snoc of 'b ] >
[> `A of [> `Cons of 'a  `Nil ]  `B of [> `Snoc of 'b ] ] = <fun>
When an orpattern composed of variant tags is wrapped inside an
aliaspattern, the alias is given a type containing only the tags
enumerated in the orpattern. This allows for many useful idioms, like
incremental definition of functions.
#let num x = `Num x
let eval1 eval (`Num x) = x
let rec eval x = eval1 eval x ;;
val num : 'a > [> `Num of 'a ] = <fun>
val eval1 : 'a > [ `Num of 'b ] > 'b = <fun>
val eval : [ `Num of 'a ] > 'a = <fun>
#let plus x y = `Plus(x,y)
let eval2 eval = function
 `Plus(x,y) > eval x + eval y
 `Num _ as x > eval1 eval x
let rec eval x = eval2 eval x ;;
val plus : 'a > 'b > [> `Plus of 'a * 'b ] = <fun>
val eval2 : ('a > int) > [< `Num of int  `Plus of 'a * 'a ] > int = <fun>
val eval : ([< `Num of int  `Plus of 'a * 'a ] as 'a) > int = <fun>
To make this even more confortable, you may use type definitions as
abbreviations for orpatterns. That is, if you have defined type myvariant = [`Tag1 int  `Tag2 bool], then the pattern #myvariant is
equivalent to writing (`Tag1(_ : int)  `Tag2(_ : bool)).
Such abbreviations may be used alone,
#let f = function
 #myvariant > "myvariant"
 `Tag3 > "Tag3";;
val f : [< `Tag1 of int  `Tag2 of bool  `Tag3 ] > string = <fun>
or combined with with aliases.
#let g1 = function `Tag1 _ > "Tag1"  `Tag2 _ > "Tag2";;
val g1 : [< `Tag1 of 'a  `Tag2 of 'b ] > string = <fun>
#let g = function
 #myvariant as x > g1 x
 `Tag3 > "Tag3";;
val g : [< `Tag1 of int  `Tag2 of bool  `Tag3 ] > string = <fun>
4.2.1 
Weaknesses of polymorphic variants 

After seeing the power of polymorphic variants, one may wonder why
they were added to core language variants, rather than replacing them.
The answer is two fold. One first aspect is that while being pretty
efficient, the lack of static type information allows for less
optimizations, and makes polymorphic variants slightly heavier than
core language ones. However noticeable differences would only
appear on huge data structures.
More important is the fact that polymorphic variants, while being
typesafe, result in a weaker type discipline. That is, core language
variants do actually much more than ensuring typesafety, they also
check that you use only declared constructors, that all constructors
present in a datastructure are compatible, and they enforce typing
constraints to their parameters.
For this reason, you must be more careful about making types explicit
when you use polymorphic variants. When you write a library, this is
easy since you can describe exact types in interfaces, but for simple
programs you are probably better off with core language variants.
Beware also that certain idioms make trivial errors very hard to find.
For instance, the following code is probably wrong but the compiler
has no way to see it.
#type abc = [`A  `B  `C] ;;
type abc = [ `A  `B  `C ]
#let f = function
 `As > "A"
 #abc > "other" ;;
val f : [< `A  `As  `B  `C ] > string = <fun>
#let f : abc > string = f ;;
val f : abc > string = <fun>
You can avoid such risks by annotating the definition itself.
#let f : abc > string = function
 `As > "A"
 #abc > "other" ;;
Warning: this match case is unused.
val f : abc > string = <fun>
 1
 This correspond to the commuting label mode
of Objective Caml 3.00 through 3.02, with some additional flexibility
on total applications. The socalled classic mode (nolabels
options) is now deprecated for normal use.