# Chapter 2 The module system

- Structures
- Signatures
- Functors
- Functors and type abstraction
- Modules and separate compilation

This chapter introduces the module system of OCaml.

## 2.1 Structures

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)

## 2.2 Signatures

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;;

## 2.3 Functors

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 B1 …Bn of B, yielding the corresponding structures A1 …An.

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;;
    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 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;;
    Error: This expression has type
             AbstractStringSet.set = AbstractSet/1139(OrderedString).set
           but an expression was expected of type
             NoCaseStringSet.set = AbstractSet/1171(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 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`.

## 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, 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.

Both files 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 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 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 `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 top-level structures can be mapped to separately-compiled files, but not 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.