MLton 20210117

# ForLoops

A for-loop is typically used to iterate over a range of consecutive integers that denote indices of some sort. For example, in OCaml a for-loop takes either the form

    for <name> = <lower> to <upper> do <body> done

or the form

    for <name> = <upper> downto <lower> do <body> done

Some languages provide considerably more flexible for-loop or foreach-constructs.

A bit surprisingly, Standard ML provides special syntax for while-loops, but not for for-loops. Indeed, in SML, many uses of for-loops are better expressed using app, foldl/foldr, map and many other higher-order functions provided by the Basis Library for manipulating lists, vectors and arrays. However, the Basis Library does not provide a function for iterating over a range of integer values. Fortunately, it is very easy to write one.

## A fairly simple design

The following implementation imitates both the syntax and semantics of the OCaml for-loop.

    datatype for = to of int * int
                 | downto of int * int

    infix to downto

    val for =
        fn lo to up =>
           (fn f => let fun loop lo = if lo > up then ()
                                      else (f lo; loop (lo+1))
                    in loop lo end)
         | up downto lo =>
           (fn f => let fun loop up = if up < lo then ()
                                      else (f up; loop (up-1))
                    in loop up end)

For example,

    for (1 to 9)
        (fn i => print (Int.toString i))

would print 123456789 and

    for (9 downto 1)
        (fn i => print (Int.toString i))

would print 987654321.

Straightforward formatting of nested loops

    for (a to b)
        (fn i =>
            for (c to d)
                (fn j =>
                    ...))

is fairly readable, but tends to cause the body of the loop to be indented quite deeply.

## Off-by-one

The above design has an annoying feature. In practice, the upper bound of the iterated range is almost always excluded and most loops would subtract one from the upper bound:

    for (0 to n-1) ...
    for (n-1 downto 0) ...

It is probably better to break convention and exclude the upper bound by default, because it leads to more concise code and becomes idiomatic with very little practice. The iterator combinators described below exclude the upper bound by default.

## Iterator combinators

While the simple for-function described in the previous section is probably good enough for many uses, it is a bit cumbersome when one needs to iterate over a Cartesian product. One might also want to iterate over more than just consecutive integers. It turns out that one can provide a library of iterator combinators that allow one to implement iterators more flexibly.

Since the types of the combinators may be a bit difficult to infer from their implementations, let’s first take a look at a signature of the iterator combinator library:

    signature ITER =
      sig
        type 'a t = ('a -> unit) -> unit

        val return : 'a -> 'a t
        val >>= : 'a t * ('a -> 'b t) -> 'b t

        val none : 'a t

        val to : int * int -> int t
        val downto : int * int -> int t

        val inList : 'a list -> 'a t
        val inVector : 'a vector -> 'a t
        val inArray : 'a array -> 'a t

        val using : ('a, 'b) StringCvt.reader -> 'b -> 'a t

        val when : 'a t * ('a -> bool) -> 'a t
        val by : 'a t * ('a -> 'b) -> 'b t
        val @@ : 'a t * 'a t -> 'a t
        val ** : 'a t * 'b t -> ('a, 'b) product t

        val for : 'a -> 'a
      end

Several of the above combinators are meant to be used as infix operators. Here is a set of suitable infix declarations:

    infix 2 to downto
    infix 1 @@ when by
    infix 0 >>= **

A few notes are in order:

- The 'a t type constructor with the return and >>= operators forms a monad.

- The to and downto combinators will omit the upper bound of the range.

- for is the identity function. It is purely for syntactic sugar and is not strictly required.

- The @@ combinator produces an iterator for the concatenation of the given iterators.

- The ** combinator produces an iterator for the Cartesian product of the given iterators.

  - See ProductType for the type constructor ('a, 'b) product used in the type of the iterator produced by **.

- The using combinator allows one to iterate over slices, streams and many other kinds of sequences.

- when is the filtering combinator. The name when is inspired by OCaml's guard clauses.

- by is the mapping combinator.

The below implementation of the ITER-signature makes use of the following basic combinators:

    fun const x _ = x
    fun flip f x y = f y x
    fun id x = x
    fun opt fno fso = fn NONE => fno () | SOME ? => fso ?
    fun pass x f = f x

Here is an implementation the ITER-signature:

    structure Iter :> ITER =
      struct
        type 'a t = ('a -> unit) -> unit

        val return = pass
        fun (iA >>= a2iB) f = iA (flip a2iB f)

        val none = ignore

        fun (l to u) f = let fun `l = if l<u then (f l; `(l+1)) else () in `l end
        fun (u downto l) f = let fun `u = if u>l then (f (u-1); `(u-1)) else () in `u end

        fun inList ? = flip List.app ?
        fun inVector ? = flip Vector.app ?
        fun inArray ? = flip Array.app ?

        fun using get s f = let fun `s = opt (const ()) (fn (x, s) => (f x; `s)) (get s) in `s end

        fun (iA when p) f = iA (fn a => if p a then f a else ())
        fun (iA by g) f = iA (f o g)
        fun (iA @@ iB) f = (iA f : unit; iB f)
        fun (iA ** iB) f = iA (fn a => iB (fn b => f (a & b)))

        val for = id
      end

Note that some of the above combinators (e.g. **) could be expressed in terms of the other combinators, most notably return and >>=. Another implementation issue worth mentioning is that downto is written specifically to avoid computing l-1, which could cause an Overflow.

To use the above combinators the Iter-structure needs to be opened

    open Iter

and one usually also wants to declare the infix status of the operators as shown earlier.

Here is an example that illustrates some of the features:

    for (0 to 10 when (fn x => x mod 3 <> 0) ** inList ["a", "b"] ** 2 downto 1 by real)
        (fn x & y & z =>
           print ("("^Int.toString x^", \""^y^"\", "^Real.toString z^")\n"))

Using the Iter combinators one can easily produce more complicated iterators. For example, here is an iterator over a "triangle":

    fun triangle (l, u) = l to u >>= (fn i => i to u >>= (fn j => return (i, j)))