For the past few classes we have been considering abstraction and
modular design, primarily through the use of the `module`

mechanism in OCaml. We have seen that good design principles include
writing clear specifications of interfaces, independent of the actual
implementation. We have also seen that writing good documentation of
the implementation is important. Today we will consider another means
of abstraction called *functors*, a construct that enables modules to be
combined by parameterizing a module in terms of other modules.

Consider the `SET`

data abstraction that we have looked at during the
past few classes:

module type SET = sig type 'a set val empty : 'a set val mem : 'a -> 'a set -> bool val add : 'a -> 'a set -> 'a set val rem : 'a -> 'a set -> 'a set val size: 'a set -> int val union: 'a set -> 'a set -> 'a set val inter: 'a set -> 'a set -> 'a set end

While this interface uses polymorphism to enable sets with
different types of elements to be created, any implementation of this
signature needs to use the built-in `=`

function in testing
whether an element is a member of such a set. Thus we cannot for
example have a set of strings where comparison of the elements is done
in a case-insensitive manner, or a set of integers where elements are
considered equal when their magnitudes (absolute values) are equal.
We could write two separate signatures, one for sets with
string elements and one for sets with integer elements, and then in
the implementation of each signature use an appropriate comparison
function. However this would yield a lot of nearly duplicated code,
both in the signatures and in the implementation. Such nearly
duplicated code is more work to write and maintain and more
importantly is often a source of bugs when things are changed in one
place and not another.

A *functor* is a mapping from modules to modules. It allows
the construction of a module parameterized by one or more other
modules. Functors allow us to create a set module that is
parameterized by another module that does the equality testing, thereby
allowing the same code to be used for different equality tests. To
make this concrete, we will consider an example using the following
signatures:

module type EQUAL = sig type t val equal : t -> t -> bool end

module type SETFUNCTOR = functor (Equal : EQUAL) -> sig type elt = Equal.t type set val empty : set val mem : elt -> set -> bool val add: elt -> set -> set val size: set -> int end

The signature `EQUAL`

describes the input type for the functor.
To implement `EQUAL`

, a module need only specify a type `t`

and a
comparison function `equal : t -> t -> bool`

, but these can be
anything.

The signature `SETFUNCTOR`

describes the type of the functor. This differs
from the `SET`

interface in several respects. First,
the keyword `functor`

indicates that it is a functor accepting a
parameter, which in this case is any module of type `EQUAL`

. Note how
the syntax is reminiscent of the notation for functions.
The parameter is referenced by the name `Equal`

in the body of `SETFUNCTOR`

,
but that does not have to be its actual name.

The body of `SETFUNCTOR`

describes the type of the module that will
be produced. In the body, instead of the polymorphic `'a`

of `SET`

,
the type of the elements is named `elt`

and is defined to be the same as
the type `t`

of the module `Equal`

, whatever that is.
There is also a fixed but unspecified type `set`

, along with some set
operations of the appropriate types, specified in terms of `elt`

and `set`

. (We have omitted a few of the operations
for simplicity of the presentation, although they could easily be added back in.)

Now we are ready to define a functor implementing
the `SETFUNCTOR`

signature.

module MakeSet : SETFUNCTOR = functor (Equal : EQUAL) -> struct open Equal type elt = t type set = elt list let empty = [] let mem x = List.exists (equal x) let add x s = if mem x s then s else x :: s let size = List.length end

First, the header

module MakeSet : SETFUNCTOR =

indicates that we are defining an implementation named `MakeSet`

of the functor type `SETFUNCTOR`

. The second line

functor (Equal : EQUAL) ->

indicates that we are defining a functor with parameter `Equal`

of type `EQUAL`

.
Again, the module implementing `EQUAL`

is referenced by
the name `Equal`

in the body of `MakeSet`

,
but that does not have to be its actual name.
In general there can be any number of parameter modules,
each of which must be specified with a name and signature.
Note that these parameters can only be modules, including other
parameterized modules—they cannot be
first-class objects of the language such as functions or other types.

Finally, the body of `MakeSet`

between `struct`

and `end`

describes the implementation of the output module. This module must satisfy the signature
described in the body of `SETFUNCTOR`

.