Lecture 10: Functors

In the previous lecture we considered abstraction and modular design, primarily through the use of the module mechanism in OCaml. 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.

The body of MakeSet is like the body of any other module. In this example the open directive is used so that the names t and equal can be used without qualifying them as Equal.t and Equal.equal.

It is also worth noting the partial evaluation of both equal and List.exists in:

  let mem x = List.exists (equal x)

To write it out in full, we might have written

  let mem x s = List.exists (fun y -> equal x y) s

but the shorter version is equivalent. In both cases, we are using the fact that fun z -> f z is equivalent to just f. For example, both fun y -> equal x y and equal x are functions that test whether a given element is equal to the value of x.

Now we show how to create modules using the functor MakeSet. To do this, we need an implementation of the EQUAL signature. Say, for example, we want to test equality of strings in a case-independent fashion. Here is a module that does this.

module StringNoCase =
  struct
    type t = string
    let equal s1 s2 =
      String.lowercase s1 = String.lowercase s2
  end

Now we can use MakeSet to create a string set module with case-insensitive equality by applying it to StringNoCase:

module SSet = MakeSet (StringNoCase)

Evaluating this expression, the interpreter prints out:

module SSet :
  sig
    type elt = StringNoCase.t
    type set = MakeSet(StringNoCase).set
    val empty : set
    val mem : elt -> set -> bool
    val add : elt -> set -> set
    val size : set -> int
  end

That is, the SSet module defines the types set and elt and the function mem, add, and size, but the actual implementation is hidden.

Now we can use this set abstraction to create and manipulate sets of strings with case-insensitive comparison.

# let s = SSet.add "I like CS 3110" SSet.empty;;
val s : SSet.set = <abstr>
# SSet.mem "i LiKe cs 3110" s;;
- : bool = true
# SSet.size s;;
- : int = 1

After doing this, creating a module for sets of integers using absolute value comparison involves almost no additional code. We only need to create another module implementing EQUAL and use it as the parameter to MakeSet:

module IntAbs = struct
  type t = int
  let equal i1 i2 = abs i1 = abs i2
end

module ISet = MakeSet (IntAbs)

Now we can use this set abstraction to create and manipulate sets of integers with absolute value comparison:

# let i = ISet.add 1 ISet.empty;;
val i : ISet.set = <abstr>
# ISet.mem 1 i;;
- : bool = true
# ISet.mem (-1) i;;
- : bool = true
# ISet.size i;;
- : int = 1
# let i = ISet.add (-1) i;;
val i : ISet.set = <abstr>
# ISet.size i;;
- : int = 1

Caveats

There are a few subtleties with functors that are worth mentioning. First, note that we did not specify the signature EQUAL when we defined StringNoCase. We might have written

module StringNoCase : EQUAL =
  struct
    type t = string
    let equal s1 s2 =
      String.lowercase s1 = String.lowercase s2
  end

but this would have been a bad idea:

# module SSet = MakeSet (StringNoCase);;
module SSet :
  sig
    type elt = StringNoCase.t
    type set = MakeSet(StringNoCase).set
    val empty : set
    val mem : elt -> set -> bool
    val add : elt -> set -> set
    val size : set -> int
  end
# let s = SSet.add "I like CS 3110" SSet.empty;;
Characters 17-33:
  let s = SSet.add "I like CS 3110" SSet.empty;;
                   ^^^^^^^^^^^^^^^^
Error: This expression has type string but is here used with type
         SSet.elt = StringNoCase.t

The issue here is that the signature EQUAL does not expose the type definition type t = string in the implementation StringNoCase, so the functor is not free to use that information. It may only deal with the module StringNoCase through its signature. This is consistent with the principle of information hiding through the use of signatures. Thus the module that is produced does not know that StringNoCase.t is really string. But if we omit the EQUAL, then the signature of StringNoCase is inferred from the implementation:

# module StringNoCase =
    struct
      type t = string
      let equal s1 s2 =
        String.lowercase s1 = String.lowercase s2
    end;;
module StringNoCase :
  sig
    type t = string
    val equal : string -> string -> bool
  end

(formatting inserted by hand for clarity). You can see that here the type definition type t = string is exposed, and the functor may now use that information.

Contravariance

Another good reason for not specifying the signature is that modules can implement lots of different signatures and can be used in different ways. For example, if we had defined

module StringNoCase =
  struct
    type t = string
    let compare s1 s2 =
      String.compare (String.lowercase s1) (String.lowercase s2)
    let equal s1 s2 = compare s1 s2 = 0
  end

then StringNoCase implements not only EQUAL, but also Map.OrderedType and Set.OrderedType from the OCaml library, so it can also be used as an argument to Map.Make and Set.Make. If we had specified EQUAL, then we would have precluded those uses.

We can always restrict it if we like:

# module StringNoCaseEq = (StringNoCase : EQUAL);;
module StringNoCaseEq : EQUAL

The modules StringNoCaseEq and StringNoCase have the same implementation, but their signatures are different. This is very much like a type upcast in Java. It is an upcast because it is going from a more specific specification (fewer instances) to a more general specification (more instances), which means it can be used as the argument to fewer functors. This inverse relationship is known as contravariance.

...with type...

For clarity, we may wish to define the result signature of a functor independently from the signature of the functor itself. So instead of the definition of SETFUNCTOR as given above, we may wish to write something like

module type SET =
  sig
    type elt
    type set
    val empty : set
    val mem : elt -> set -> bool
    val add: elt -> set -> set
    val size: set -> int
  end

module type SETFUNCTOR = functor (Equal : EQUAL) -> SET

The difficulty here is that we need a way to equate the type elt of SET with Equal.t. We could do that in the previous definition of SETFUNCTOR by writing type elt = Equal.t in the body, but here there is no parameter module Equal around when we define SET.

To handle this, OCaml allows you to write

module type SETFUNCTOR =
  functor (Equal : EQUAL) -> SET with type elt = Equal.t
Now it can link up the two types:
# module type SETFUNCTOR =
    functor (Equal : EQUAL) -> SET with type elt = Equal.t;;
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