MLton 20210117

# StaticSum

While SML makes it impossible to write functions whose types would depend on the values of their arguments, or so called dependently typed functions, it is possible, and arguably commonplace, to write functions whose types depend on the types of their arguments. Indeed, the types of parametrically polymorphic functions like map and foldl can be said to depend on the types of their arguments. What is less commonplace, however, is to write functions whose behavior would depend on the types of their arguments. Nevertheless, there are several techniques for writing such functions. Type-indexed values and fold are two such techniques. This page presents another such technique dubbed static sums.

## Ordinary Sums

Consider the sum type as defined below:

    structure Sum = struct
       datatype ('a, 'b) t = INL of 'a | INR of 'b
    end

While a generic sum type such as defined above is very useful, it has a number of limitations. As an example, we could write the function out to extract the value from a sum as follows:

    fun out (s : ('a, 'a) Sum.t) : 'a =
        case s
         of Sum.INL a => a
          | Sum.INR a => a

As can be seen from the type of out, it is limited in the sense that it requires both variants of the sum to have the same type. So, out cannot be used to extract the value of a sum of two different types, such as the type (int, real) Sum.t. As another example of a limitation, consider the following attempt at a succ function:

    fun succ (s : (int, real) Sum.t) : ??? =
        case s
         of Sum.INL i => i + 1
          | Sum.INR r => Real.nextAfter (r, Real.posInf)

The above definition of succ cannot be typed, because there is no type for the codomain within SML.

## Static Sums

Interestingly, it is possible to define values inL, inR, and match that satisfy the laws

    match (inL x) (f, g) = f x
    match (inR x) (f, g) = g x

and do not suffer from the same limitions. The definitions are actually quite trivial:

    structure StaticSum = struct
       fun inL x (f, _) = f x
       fun inR x (_, g) = g x
       fun match x = x
    end

Now, given the succ function defined as

    fun succ s =
        StaticSum.match s
           (fn i => i + 1,
            fn r => Real.nextAfter (r, Real.posInf))

we get

    succ (StaticSum.inL 1) = 2
    succ (StaticSum.inR Real.maxFinite) = Real.posInf

To better understand how this works, consider the following signature for static sums:

    structure StaticSum :> sig
       type ('dL, 'cL, 'dR, 'cR, 'c) t
       val inL : 'dL -> ('dL, 'cL, 'dR, 'cR, 'cL) t
       val inR : 'dR -> ('dL, 'cL, 'dR, 'cR, 'cR) t
       val match : ('dL, 'cL, 'dR, 'cR, 'c) t -> ('dL -> 'cL) * ('dR -> 'cR) -> 'c
    end = struct
       type ('dL, 'cL, 'dR, 'cR, 'c) t = ('dL -> 'cL) * ('dR -> 'cR) -> 'c
       open StaticSum
    end

Above, 'd stands for domain and 'c for codomain. The key difference between an ordinary sum type, like (int, real) Sum.t, and a static sum type, like (int, real, real, int, real) StaticSum.t, is that the ordinary sum type says nothing about the type of the result of deconstructing a sum while the static sum type specifies the type.

With the sealed static sum module, we get the type

    val succ : (int, int, real, real, 'a) StaticSum.t -> 'a

for the previously defined succ function. The type specifies that succ maps a left int to an int and a right real to a real. For example, the type of StaticSum.inL 1 is (int, 'cL, 'dR, 'cR, 'cL) StaticSum.t. Unifying this with the argument type of succ gives the type (int, int, real, real, int) StaticSum.t -> int.

The out function is quite useful on its own. Here is how it can be defined:

    structure StaticSum = struct
       open StaticSum
       val out : ('a, 'a, 'b, 'b, 'c) t -> 'c =
        fn s => match s (fn x => x, fn x => x)
    end

Due to the value restriction, lack of first class polymorphism and polymorphic recursion, the usefulness and convenience of static sums is somewhat limited in SML. So, don’t throw away the ordinary sum type just yet. Static sums can nevertheless be quite useful.

### Example: Send and Receive with Argument Type Dependent Result Types

In some situations it would seem useful to define functions whose result type would depend on some of the arguments. Traditionally such functions have been thought to be impossible in SML and the solution has been to define multiple functions. For example, the Socket structure of the Basis library defines 16 send and 16 recv functions. In contrast, the Net structure (net.sig) of the Basic library designed by Stephen Weeks defines only a single send and a single receive and the result types of the functions depend on their arguments. The implementation (net.sml) uses static sums (with a slighly different signature: static-sum.sig).

### Example: Picking Monad Results

Suppose that we need to write a parser that accepts a pair of integers and returns their sum given a monadic parsing combinator library. A part of the signature of such library could look like this

    signature PARSING = sig
       include MONAD
       val int : int t
       val lparen : unit t
       val rparen : unit t
       val comma : unit t
       (* ... *)
    end

where the MONAD signature could be defined as

    signature MONAD = sig
       type 'a t
       val return : 'a -> 'a t
       val >>= : 'a t * ('a -> 'b t) -> 'b t
    end
    infix >>=

The straightforward, but tedious, way to write the desired parser is:

    val p = lparen >>= (fn _ =>
            int    >>= (fn x =>
            comma  >>= (fn _ =>
            int    >>= (fn y =>
            rparen >>= (fn _ =>
            return (x + y))))))

In Haskell, the parser could be written using the do notation considerably less verbosely as:

    p = do { lparen ; x <- int ; comma ; y <- int ; rparen ; return $ x + y }

SML doesn’t provide a do notation, so we need another solution.

Suppose we would have a "pick" notation for monads that would allows us to write the parser as

    val p = `lparen ^ \int ^ `comma ^ \int ^ `rparen @ (fn x & y => x + y)

using four auxiliary combinators: `, \, ^, and @.

Roughly speaking

- `p means that the result of p is dropped,

- \p means that the result of p is taken,

- p ^ q means that results of p and q are taken as a product, and

- p @ a means that the results of p are passed to the function a and that result is returned.

The difficulty is in implementing the concatenation combinator ^. The type of the result of the concatenation depends on the types of the arguments.

Using static sums and the product type, the pick notation for monads can be implemented as follows:

    functor MkMonadPick (include MONAD) = let
       open StaticSum
    in
       struct
          fun `a = inL (a >>= (fn _ => return ()))
          val \ = inR
          fun a @ f = out a >>= (return o f)
          fun a ^ b =
              (match b o match a)
                 (fn a =>
                     (fn b => inL (a >>= (fn _ => b)),
                      fn b => inR (a >>= (fn _ => b))),
                  fn a =>
                     (fn b => inR (a >>= (fn a => b >>= (fn _ => return a))),
                      fn b => inR (a >>= (fn a => b >>= (fn b => return (a & b))))))
       end
    end

The above implementation is inefficient, however. It uses many more bind operations, >>=, than necessary. That can be solved with an additional level of abstraction:

    functor MkMonadPick (include MONAD) = let
       open StaticSum
    in
       struct
          fun `a = inL (fn b => a >>= (fn _ => b ()))
          fun \a = inR (fn b => a >>= b)
          fun a @ f = out a (return o f)
          fun a ^ b =
              (match b o match a)
                 (fn a => (fn b => inL (fn c => a (fn () => b c)),
                           fn b => inR (fn c => a (fn () => b c))),
                  fn a => (fn b => inR (fn c => a (fn a => b (fn () => c a))),
                           fn b => inR (fn c => a (fn a => b (fn b => c (a & b))))))
       end
    end

After instantiating and opening either of the above monad pick implementations, the previously given definition of p can be compiled and results in a parser whose result is of type int. Here is a functor to test the theory:

    functor Test (Arg : PARSING) = struct
       local
          structure Pick = MkMonadPick (Arg)
          open Pick Arg
       in
          val p : int t =
              `lparen ^ \int ^ `comma ^ \int ^ `rparen @ (fn x & y => x + y)
       end
    end

## Also see

There are a number of related techniques. Here are some of them.

- Fold

- TypeIndexedValues