This chapter introduces the module system of Objective Caml.
A primary motivation for modules is to package together related
definitions (such as the definitions of a data type and associated
operations over that type) and enforce a consistent naming scheme for
these definitions. This avoids running out of names or accidentally
confusing names. Such a package is called a structure and
is introduced by the struct...end construct, which contains an
arbitrary sequence of definitions. The structure is usually given a
name with the module binding. Here is for instance a structure
packaging together a type of priority queues and their operations:
#module PrioQueue =
struct
type priority = int
type 'a queue = Empty  Node of priority * 'a * 'a queue * 'a queue
let empty = Empty
let rec insert queue prio elt =
match queue with
Empty > Node(prio, elt, Empty, Empty)
 Node(p, e, left, right) >
if prio <= p
then Node(prio, elt, insert right p e, left)
else Node(p, e, insert right prio elt, left)
exception Queue_is_empty
let rec remove_top = function
Empty > raise Queue_is_empty
 Node(prio, elt, left, Empty) > left
 Node(prio, elt, Empty, right) > right
 Node(prio, elt, (Node(lprio, lelt, _, _) as left),
(Node(rprio, relt, _, _) as right)) >
if lprio <= rprio
then Node(lprio, lelt, remove_top left, right)
else Node(rprio, relt, left, remove_top right)
let extract = function
Empty > raise Queue_is_empty
 Node(prio, elt, _, _) as queue > (prio, elt, remove_top queue)
end;;
module PrioQueue :
sig
type priority = int
type 'a queue = Empty  Node of priority * 'a * 'a queue * 'a queue
val empty : 'a queue
val insert : 'a queue > priority > 'a > 'a queue
exception Queue_is_empty
val remove_top : 'a queue > 'a queue
val extract : 'a queue > priority * 'a * 'a queue
end
Outside the structure, its components can be referred to using the
``dot notation'', that is, identifiers qualified by a structure name.
For instance, PrioQueue.insert in a value context is
the function insert defined inside the structure
PrioQueue. Similarly, PrioQueue.queue in a type context is the
type queue defined in PrioQueue.
#PrioQueue.insert PrioQueue.empty 1 "hello";;
 : string PrioQueue.queue =
PrioQueue.Node (1, "hello", PrioQueue.Empty, PrioQueue.Empty)
Signatures are interfaces for structures. A signature specifies
which components of a structure are accessible from the outside, and
with which type. It can be used to hide some components of a structure
(e.g. local function definitions) or export some components with a
restricted type. For instance, the signature below specifies the three
priority queue operations empty, insert and extract, but not the
auxiliary function remove_top. Similarly, it makes the queue type
abstract (by not providing its actual representation as a concrete type).
#module type PRIOQUEUE =
sig
type priority = int (* still concrete *)
type 'a queue (* now abstract *)
val empty : 'a queue
val insert : 'a queue > int > 'a > 'a queue
val extract : 'a queue > int * 'a * 'a queue
exception Queue_is_empty
end;;
module type PRIOQUEUE =
sig
type priority = int
type 'a queue
val empty : 'a queue
val insert : 'a queue > int > 'a > 'a queue
val extract : 'a queue > int * 'a * 'a queue
exception Queue_is_empty
end
Restricting the PrioQueue structure by this signature results in
another view of the PrioQueue structure where the remove_top
function is not accessible and the actual representation of priority
queues is hidden:
#module AbstractPrioQueue = (PrioQueue : PRIOQUEUE);;
module AbstractPrioQueue : PRIOQUEUE
#AbstractPrioQueue.remove_top;;
Unbound value AbstractPrioQueue.remove_top
#AbstractPrioQueue.insert AbstractPrioQueue.empty 1 "hello";;
 : string AbstractPrioQueue.queue = <abstr>
The restriction can also be performed during the definition of the
structure, as in
module PrioQueue = (struct ... end : PRIOQUEUE);;
An alternate syntax is provided for the above:
module PrioQueue : PRIOQUEUE = struct ... end;;
Functors are ``functions'' from structures to structures. They are used to
express parameterized structures: a structure A parameterized by a
structure B is simply a functor F with a formal parameter
B (along with the expected signature for B) which returns
the actual structure A itself. The functor F can then be
applied to one or several implementations B_{1} ...B_{n}
of B, yielding the corresponding structures
A_{1} ...A_{n}.
For instance, here is a structure implementing sets as sorted lists,
parameterized by a structure providing the type of the set elements
and an ordering function over this type (used to keep the sets
sorted):
#type comparison = Less  Equal  Greater;;
type comparison = Less  Equal  Greater
#module type ORDERED_TYPE =
sig
type t
val compare: t > t > comparison
end;;
module type ORDERED_TYPE = sig type t val compare : t > t > comparison end
#module Set =
functor (Elt: ORDERED_TYPE) >
struct
type element = Elt.t
type set = element list
let empty = []
let rec add x s =
match s with
[] > [x]
 hd::tl >
match Elt.compare x hd with
Equal > s (* x is already in s *)
 Less > x :: s (* x is smaller than all elements of s *)
 Greater > hd :: add x tl
let rec member x s =
match s with
[] > false
 hd::tl >
match Elt.compare x hd with
Equal > true (* x belongs to s *)
 Less > false (* x is smaller than all elements of s *)
 Greater > member x tl
end;;
module Set :
functor (Elt : ORDERED_TYPE) >
sig
type element = Elt.t
type set = element list
val empty : 'a list
val add : Elt.t > Elt.t list > Elt.t list
val member : Elt.t > Elt.t list > bool
end
By applying the Set functor to a structure implementing an ordered
type, we obtain set operations for this type:
#module OrderedString =
struct
type t = string
let compare x y = if x = y then Equal else if x < y then Less else Greater
end;;
module OrderedString :
sig type t = string val compare : 'a > 'a > comparison end
#module StringSet = Set(OrderedString);;
module StringSet :
sig
type element = OrderedString.t
type set = element list
val empty : 'a list
val add : OrderedString.t > OrderedString.t list > OrderedString.t list
val member : OrderedString.t > OrderedString.t list > bool
end
#StringSet.member "bar" (StringSet.add "foo" StringSet.empty);;
 : bool = false
2.4 
Functors and type abstraction 

As in the PrioQueue example, it would be good style to hide the
actual implementation of the type set, so that users of the
structure will not rely on sets being lists, and we can switch later
to another, more efficient representation of sets without breaking
their code. This can be achieved by restricting Set by a suitable
functor signature:
#module type SETFUNCTOR =
functor (Elt: ORDERED_TYPE) >
sig
type element = Elt.t (* concrete *)
type set (* abstract *)
val empty : set
val add : element > set > set
val member : element > set > bool
end;;
module type SETFUNCTOR =
functor (Elt : ORDERED_TYPE) >
sig
type element = Elt.t
type set
val empty : set
val add : element > set > set
val member : element > set > bool
end
#module AbstractSet = (Set : SETFUNCTOR);;
module AbstractSet : SETFUNCTOR
#module AbstractStringSet = AbstractSet(OrderedString);;
module AbstractStringSet :
sig
type element = OrderedString.t
type set = AbstractSet(OrderedString).set
val empty : set
val add : element > set > set
val member : element > set > bool
end
#AbstractStringSet.add "gee" AbstractStringSet.empty;;
 : AbstractStringSet.set = <abstr>
In an attempt to write the type constraint above more elegantly,
one may wish to name the signature of the structure
returned by the functor, then use that signature in the constraint:
#module type SET =
sig
type element
type set
val empty : set
val add : element > set > set
val member : element > set > bool
end;;
module type SET =
sig
type element
type set
val empty : set
val add : element > set > set
val member : element > set > bool
end
#module WrongSet = (Set : functor(Elt: ORDERED_TYPE) > SET);;
module WrongSet : functor (Elt : ORDERED_TYPE) > SET
#module WrongStringSet = WrongSet(OrderedString);;
module WrongStringSet :
sig
type element = WrongSet(OrderedString).element
type set = WrongSet(OrderedString).set
val empty : set
val add : element > set > set
val member : element > set > bool
end
#WrongStringSet.add "gee" WrongStringSet.empty;;
This expression has type string but is here used with type
WrongStringSet.element = WrongSet(OrderedString).element
The problem here is that SET specifies the type element
abstractly, so that the type equality between element in the result
of the functor and t in its argument is forgotten. Consequently,
WrongStringSet.element is not the same type as string, and the
operations of WrongStringSet cannot be applied to strings.
As demonstrated above, it is important that the type element in the
signature SET be declared equal to Elt.t; unfortunately, this is
impossible above since SET is defined in a context where Elt does
not exist. To overcome this difficulty, Objective Caml provides a
with type construct over signatures that allows to enrich a signature
with extra type equalities:
#module AbstractSet =
(Set : functor(Elt: ORDERED_TYPE) > (SET with type element = Elt.t));;
module AbstractSet :
functor (Elt : ORDERED_TYPE) >
sig
type element = Elt.t
type set
val empty : set
val add : element > set > set
val member : element > set > bool
end
As in the case of simple structures, an alternate syntax is provided
for defining functors and restricting their result:
module AbstractSet(Elt: ORDERED_TYPE) : (SET with type element = Elt.t) =
struct ... end;;
Abstracting a type component in a functor result is a powerful
technique that provides a high degree of type safety, as we now
illustrate. Consider an ordering over character strings that is
different from the standard ordering implemented in the
OrderedString structure. For instance, we compare strings without
distinguishing upper and lower case.
#module NoCaseString =
struct
type t = string
let compare s1 s2 =
OrderedString.compare (String.lowercase s1) (String.lowercase s2)
end;;
module NoCaseString :
sig type t = string val compare : string > string > comparison end
#module NoCaseStringSet = AbstractSet(NoCaseString);;
module NoCaseStringSet :
sig
type element = NoCaseString.t
type set = AbstractSet(NoCaseString).set
val empty : set
val add : element > set > set
val member : element > set > bool
end
#NoCaseStringSet.add "FOO" AbstractStringSet.empty;;
This expression has type
AbstractStringSet.set = AbstractSet(OrderedString).set
but is here used with type
NoCaseStringSet.set = AbstractSet(NoCaseString).set
Notice that the two types AbstractStringSet.set and
NoCaseStringSet.set are not compatible, and values of these
two types do not match. This is the correct behavior: even though both
set types contain elements of the same type (strings), both are built
upon different orderings of that type, and different invariants need
to be maintained by the operations (being strictly increasing for the
standard ordering and for the caseinsensitive ordering). Applying
operations from AbstractStringSet to values of type
NoCaseStringSet.set could give incorrect results, or build
lists that violate the invariants of NoCaseStringSet.
2.5 
Modules and separate compilation 

All examples of modules so far have been given in the context of the
interactive system. However, modules are most useful for large,
batchcompiled programs. For these programs, it is a practical
necessity to split the source into several files, called compilation
units, that can be compiled separately, thus minimizing recompilation
after changes.
In Objective Caml, compilation units are special cases of structures
and signatures, and the relationship between the units can be
explained easily in terms of the module system. A compilation unit A
comprises two files:

the implementation file A.ml, which contains a sequence
of definitions, analogous to the inside of a struct...end
construct;
 the interface file A.mli, which contains a sequence of
specifications, analogous to the inside of a sig...end
construct.
Both files define a structure named A as if
the following definition was entered at toplevel:
module A: sig (* contents of file A.mli *) end
= struct (* contents of file A.ml *) end;;
The files defining the compilation units can be compiled separately
using the ocamlc c command (the c option means ``compile only, do
not try to link''); this produces compiled interface files (with
extension .cmi) and compiled object code files (with extension
.cmo). When all units have been compiled, their .cmo files are
linked together using the ocaml command. For instance, the following
commands compile and link a program composed of two compilation units
Aux and Main:
$ ocamlc c Aux.mli # produces aux.cmi
$ ocamlc c Aux.ml # produces aux.cmo
$ ocamlc c Main.mli # produces main.cmi
$ ocamlc c Main.ml # produces main.cmo
$ ocamlc o theprogram Aux.cmo Main.cmo
The program behaves exactly as if the following phrases were entered
at toplevel:
module Aux: sig (* contents of Aux.mli *) end
= struct (* contents of Aux.ml *) end;;
module Main: sig (* contents of Main.mli *) end
= struct (* contents of Main.ml *) end;;
In particular, Main can refer to Aux: the definitions and
declarations contained in Main.ml and Main.mli can refer to
definition in Aux.ml, using the Aux.ident notation, provided
these definitions are exported in Aux.mli.
The order in which the .cmo files are given to ocaml during the
linking phase determines the order in which the module definitions
occur. Hence, in the example above, Aux appears first and Main can
refer to it, but Aux cannot refer to Main.
Notice that only toplevel structures can be mapped to
separatelycompiled files, but not functors nor module types.
However, all moduleclass objects can appear as components of a
structure, so the solution is to put the functor or module type
inside a structure, which can then be mapped to a file.