Question

Functors in Standard ML are related to the module system and can generate structures based on other structures. An example of a functor generating list combinators for various types of lists is given below, but this example has a problem:

The various types of lists all have advantages — for example, lazy lists can be infinitely long, and concantenation lists have a O(1) concat operator. But when all of these list types conform to the same signature, the functor can only use their general properties.

My question is therefore: What is a good example of when functors are useful and the various generated structures don’t lose their special abilities?

signature MYLIST =
sig
  type 'a t
  val null : 'a t -> bool
  val empty : 'a t
  val cons : 'a * 'a t -> 'a t
  val hd : 'a t -> 'a
  val tl : 'a t -> 'a t
end

structure RegularList : MYLIST =
struct
  type 'a t = 'a list
  val null = List.null
  val empty = []
  val cons = op::
  val hd = List.hd
  val tl = List.tl
end

structure LazyList : MYLIST =
struct
  datatype 'a t = Nil | Cons of 'a * (unit -> 'a t)
   val empty = Nil
   fun null Nil = true
    | null _ = false
   fun cons (x, xs) = Cons (x, fn () => xs)
   fun hd Nil = raise Empty
    | hd (Cons (x, _)) = x
   fun tl Nil = raise Empty
    | tl (Cons (_, f)) = f ()
end

structure ConcatList : MYLIST =
struct
  datatype 'a t = Nil | Singleton of 'a | Concat of 'a t * 'a t
  val empty = Nil
  fun null Nil = true
    | null (Singleton _) = false
    | null (Concat (xs, ys)) = null xs andalso null ys
  fun cons (x, xs) = Concat (Singleton x, xs)
  fun hd Nil = raise Empty
    | hd (Singleton x) = x
    | hd (Concat (xs, ys)) = hd xs
  fun tl Nil = raise Empty
    | tl (Singleton x) = Nil
    | tl (Concat (xs, ys)) = (* exercise *)
end

signature MYLISTCOMB =
sig
  type 'a t
  val length : 'a liste -> int
  val map : ('a -> 'b) -> 'a liste -> 'b liste
  val foldl : ('a * 'b -> 'b) -> 'b -> 'a liste -> 'b
  val append : 'a liste * 'a liste -> 'a liste
  val concat : 'a liste liste -> 'a liste
  val sort : ('a * 'a -> order) -> 'a t -> 'a t
end

functor ListComb (X : MYLIST) : MYLISTCOMB =
struct
  type 'a t = 'a X.t
  open X

  fun length xs =
      if null xs then 0
      else 1 + length (tl xs)

  fun map f xs =
      if null xs then empty
      else cons(f (hd xs), map f (tl xs))

  fun foldl f e xs =
      if null xs then e
      else foldl f (f (hd xs, e)) (tl xs)

  fun append (xs, ys) =
      if null xs then ys
      else cons (hd xs, append (tl xs, ys))

  fun concat xs =
      if null xs then empty
      else append (hd xs, concat (tl xs))

  fun sort cmp xs = (* exercise *)
end

structure RegularListComb = ListComb (RegularList)
structure LazyListComb = ListComb (LazyList)
structure ConcatListComb = ListComb (ConcatList)

Answer 1

Not sure I fully understand your question. Obviously, functors are useful for defining modular abstractions that (1) are polymorphic, (2) require a whole set of operations over their type parameters, and (3) provide types as part of their result (in particular, abstract types), and (4) provide an entire set of operations.

Note that your example doesn’t make use of (3), which probably is the most interesting aspect of functors. Imagine, for example, implementing an abstract matrix type that you want to parameterise over the vector type it is based on.

One specific characteristic of ML functors — as well as of core-language polymorphic functions — is that they are parametric. Parametricity is a semantic property saying that evaluation (of polymorphic code) is oblivious to the concrete type(s) it is instantiated with. That is an important property, as it implies all kinds of semantic goodness. In particular, it provides very strong abstraction and reasoning principles (see e.g. Wadler’s "Theorem’s for free!", or the brief explanation I gave in reply to another question). It also is the basis for type-erasing compilation (i.e., no types are needed at runtime).

Parametricity implies that a single functor cannot have different implementations for different types — which seems to be what you are asking about. But of course, you are free to write multiple functors that make different semantic/complexity assumptions about their parameters.

Hope that kind of answers your question.