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 ofp
is dropped, -
\p
means that the result ofp
is taken, -
p ^ q
means that results ofp
andq
are taken as a product, and -
p @ a
means that the results ofp
are passed to the functiona
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.