This chapter introduces the module system of OCaml.

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 is the function insert defined inside the structure PrioQueue and PrioQueue.queue 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;; Error: 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

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;; Error: This expression has type string but an expression was expected of 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, OCaml provides a with type construct over signatures that allows enriching a signature with extra type equalities:

```
# module AbstractSet2 =
(Set : functor(Elt: ORDERED_TYPE) -> (SET with type element = Elt.t));;
module AbstractSet2 :
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 AbstractSet2(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_ascii s1) (String.lowercase_ascii 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;; Error: This expression has type AbstractStringSet.set = AbstractSet(OrderedString).set but an expression was expected of type NoCaseStringSet.set = AbstractSet(NoCaseString).set

Note 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), they 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 case-insensitive ordering). Applying operations from AbstractStringSet to values of type NoCaseStringSet.set could give incorrect results, or build lists that violate the invariants of NoCaseStringSet.

All examples of modules so far have been given in the context of the interactive system. However, modules are most useful for large, batch-compiled 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 OCaml, 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.

These two files together define a structure named A as if the following definition was entered at top-level:

module A: sig (* contents of file A.mli *) end = struct (* contents of file A.ml *) end;;

The files that define 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 ocamlc 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 top-level:

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 ocamlc 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.

Note that only top-level structures can be mapped to separately-compiled files, but neither functors nor module types. However, all module-class 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.