CS 312 Recitation 8
Beyond signatures: Functors

Recall that the implementation of queue given in class made use of two lists. More generally, a functional queue can be implemented using two stacks, s1 and s2. Stack s1 is used for enqueuing, s2 for dequeuing. When dequeuing, if stack s2 was empty, reverse s1 and consider it the new s2. A linked list is a possible (but not the only) implementation for a stack. Here is an implementation using stacks: 

structure Queue :> QUEUE = 
      struct
      structure S = Stack

      type 'a queue = 'a S.stack * 'a S.stack
      exception EmptyQueue

      val empty : 'a queue = (S.empty, S.empty)

      fun isEmpty ((s1,s2):'a queue) = 
        S.isEmpty s1 andalso S.isEmpty s2 

      fun enqueue (x:'a, (s1,s2):'a queue) : 'a queue = 
        (S.push (x,s1), s2)

      fun rev (s: 'a S.stack): 'a S.stack = let
        fun loop (old: 'a S.stack, new: 'a S.stack): 'a S.stack = 
          if S.isEmpty old then new
          else loop (S.pop old, S.push(S.top old, new))
      in
        loop(s, S.empty)                                                                               loop (s, S.empty)
      end

      fun dequeue ((s1,s2): 'a queue) : 'a * 'a queue = 
        if S.isEmpty s2
          then dequeue(S.empty, S.pop (rev s1)) 
                    handle S.EmptyStack => raise EmptyQueue
        else (s1, S.pop s2)

     fun map (f: 'a -> 'b) ((s1,s2): 'a queue): 'b queue = 
                (S.map f s1, S.map f s2)
  end

Suppose now that you had another implementation of stacks, say Stack2. Suppose you wanted to have at your disposal two implementations of Queue, one using Stack and the other using  Stack2.  The only way to achieve this with the above code is to duplicate it, creating two different versions of Queue. This is clearly not ideal - for one, you now have to maintain twice the code. Furthermore, it is not clear exactly what Queue assumes about the functionality of Stack. How do we know that substituting Stack2 for Stack will not break any of the Queue functionality? After all, it's possible that Stack and Stack2 don't even have the same signature.

Functors

To solve this problem, ML provides a mechanism called functors. The intuition behind functors is quite simple. In the above example, we know that our implementation of Queue will use a stack implementation. Creating a specific Queue structure could involve the following steps:

1. Create a stack structure S in the desired implementation (Stack  or Stack2),
2. Create a Queue structure making use of S, by passing S in as a parameter to some sort of "function".

For step 2, we need a "function" that takes in a structure and returns another structure - in this case, takes in a stack and returns a queue. Such "functions" are precisely the ML functors.

Our Queue functor will also need to specify (and check that its argument satisfies) all the stack functionality expected by Queue. Fortunately, we already know how to do this; after all, specifying functionality is exactly what signatures do. The Queue functor will specify a signature for the stack structure it expects, and will only accept structures that instantiate that signature.

More concretely, suppose we have the following stack signature:

  signature STACK = 
    sig
      type 'a stack
      exception EmptyStack

      val empty : 'a stack
      val isEmpty : 'a stack -> bool

      val push : ('a * 'a stack) -> 'a stack
      val pop : 'a stack -> 'a stack
      val top : 'a stack -> 'a
      val map : ('a -> 'b) -> 'a stack -> 'b stack
    end
Suppose this is the functionality desired by our Queue. We can write a Queue functor as follows:
functor QueueFn (S:STACK) = 
    struct
      type 'a queue = 'a S.stack * 'a S.stack
      exception EmptyQueue

      val empty : 'a queue = (S.empty, S.empty)

      fun isEmpty ((s1,s2):'a queue) = 
        S.isEmpty s1 andalso S.isEmpty s2 

      fun enqueue (x:'a, (s1,s2):'a queue) : 'a queue = 
        (S.push (x,s1), s2)

      fun rev (s: 'a S.stack): 'a S.stack = let
        fun loop (old: 'a S.stack, new: 'a S.stack): 'a S.stack = 
          if S.isEmpty old then new
          else loop (S.pop old, S.push(S.top old, new))
      in
        loop(s, S.empty)                                                                               loop (s, S.empty)
      end

      fun dequeue ((s1,s2): 'a queue) : 'a * 'a queue = 
        if S.isEmpty s2
          then dequeue(S.empty, S.pop (rev s1)) 
                    handle S.EmptyStack => raise EmptyQueue
        else (s1, S.pop s2)

     fun map (f: 'a -> 'b) ((s1,s2): 'a queue): 'b queue = 
        (S.map f s1, S.map f s2)
  end
Suppose now that Stack and Stack2 both have the signature STACK. We create two stack structures, one using each implementation:
 structure s1 = Stack
                                                                                                                                                                                                                                                                                                                                                                                                structure s2 = Stack2

It's now easy to create two corresponding queue structures by applying the functor:

structure q1 = QueueFn(s1)
structure q2 = QueueFn(s2)

To complete the picture, note that we can also specify that we want the result of the functor to conform to the QUEUE signature. Recall that this signature was:

signature QUEUE =
   sig
      type 'a queue
      exception EmptyQueue
      val empty : 'a queue
      val isEmpty : 'a queue -> bool
      val enqueue : ('a * 'a queue) -> 'a queue
      val dequeue : 'a queue -> 'a queue
      val front : 'a queue -> 'a
      val map : ('a -> 'b) -> 'a queue -> 'b queue
   end
We can update our functor as follows:
functor QueueFn (S:STACK) : QUEUE = 
   struct
	(rest of code as before)
   end
As you see, functors provide a very powerful abstraction mechanism. Another example of their use is when we need to construct a structure that will have an ordering, such as a Binary Search Tree or a Dictionary. You can, of course, fix the ordering by explicitly passing in the ordering function to the ADT each time the ADT is constructed. However, we could also use functors to do this. Given the following signature:
signature ORDER = sig
   type element
   val compare: element * element -> order
end
We can create functors for Binary Search Trees and Dictionaries as follows:
functor BinarySearchTreeFn(O:ORDER)
   struct
      (BST code as before)
   end

functor DictionaryFn(O:ORDER) =
   struct
      (Dictionary code as before)
   end
Which simplifies life for us quite a bit.

More fun with functors

Suppose you have a structure that makes use of not one, but two or more other structures. This seems to call for a functor that can take in more than one structure as an argument. However, as you have seen with ordinary functions, we don't need to worry about multiple arguments, as long as we can "package" them into a single value somehow. In the case of functions, we could pass in a tuple containing all the arguments. In the case of functors, we can "wrap" our argument structures into another structures, and pass only this in.

As an example, suppose we want to pass in two independent structures A and B to a functor FooFn. We proceed as follows:
functor FooFn (Arg : sig
                        structure A : A_SIG
                        structure B : B_SIG
                     end) = 
	(* body using Arg.A and Arg.B *)
and we instantiate the functor as follows:
structure Foo = FooFn (struct
                          structure A = A
                          structure B = B
                       end)
Since this looks a bit clunky, ML provides a special abbreviation: we can forget about the surrounding structure and pass in the specification of the elements of the structure directly. Thus, the above example can be rewritten:
functor FooFn (structure A : A_SIG
               structure B : B_SIG) = ...

structure Foo = FooFn (structure A = A
                       structure B = B)
You may be slightly worried that this abbreviation loses the name for the wrapping structure (Arg). To reassure you, have a look at how the above abbreviation is actually implemented.
functor FooFn (structure A : A_SIG
               structure B : B_SIG) = ...
is implemented as
functor FooFn (uid : struct
                           structure A : A_SIG
                           structure B : B_SIG
                     end) =
 struct       
    local       
       open uid   
    in     
        <body>  
    end
 end
As you see, because of the "open", we don't need to worry about the name of the wrapping structure.

Footnote: the "local" keyword in ML is similar to "let", except it works at the declaration level rather than the expression level. Here's an example to see how it works:
- local
     val a = 3
     val b = 10
  in
         val x = a + b
     val y = a * b
  end;

val x = 13 : int
val y = 30 : int

- x;
val it = 13 : int

- a;
stdIn:139.1 Error: unbound variable or constructor: a


Notes adapted from Chapter 3 of Riccardo Pucella's SML Notes