]>
Didier and Xavier write:
> > Using an object oriented style, you could consider the functions on sets as
> > methods, and pack them with each set (instead of the stamp).
>
> Please, no object-oriented nonsense. This is a matter of type
> abstraction. I strongly doubt objects will help clarify anything here.
>
> More to the point, the solution to which you finally arrive (storing
> the ordering function in each set, and give it as argument to
> set__empty), which is also the solution proposed by Damien Doligez,
> solves the problem for functions that take only one set argument, but
> fail on functions such as "union", which take two. In general you
> can't decide whether the two orderings are the same ....
While I agree with Xavier's other arguments about why an
object-oriented approach is probably to be avoided here, it's worth
remarking that a sufficiently powerful type system for objects can
also handle cases like set union.
Let Object be a type constructor that maps a description of an object
interface (a vector of method types) to a description of the set of
objects with this interface. For example
SetI = Fun(Rep) { addElement: Rep->Int->Rep,
member: Rep->Int->Bool }
describes the interface of simple set-of-integer objects. (The
reasons for the lambda-abstraction over the type variable Rep are
technical, but they don't matter here; what's important is just the
intuition that each of the methods implements a certain kind of
transformation on a set-object's internal state: addElelement
transforms an internal state and an integer into a new internal state
and member maps an internal state and an integer into a boolean.)
Then
Set = Object(SetI)
is the type of integer-set objects.
Now, a particular implementation of some set-objects with this type
may be based on a binary tree representation and may have --
internally -- a particular comparison function, but there is no way to
make use of these facts to implement functions like union, because
these aspects of the objects are not accessible through the public
interface. So let's make them accessible:
SetI' = Fun(Rep) { addElement: Rep->Int->Rep,
member: Rep->Int->Bool,
compare: Int->Int->Bool,
asTree: Rep->IntTree }
Now if
Set' = Object(SetI')
then we can easily implement functions like union on elements of Set'.
But there are two problems:
1) The asTree function is not somehing we want in the interface of
our sets, since we want to be free to reimplement them using some
different representation.
2) A function that takes two sets are arguments cannot be sure that
they were created with the same comparison function.
Fortunately, we can solve both of these problems at the same time
using a simple trick.
Informally,
SetI' < SetI
(the interface of Set' objects is richer than that of Set objects) and
Object(SetI') < Object(SetI).
[This is also true formally in a certain system, but that doesn't
concern us here].
Now, for each comparison function, we want to build a type of sets
based on that comparison function, where the only operations that show
in the interface of these sets are addElement and member. In other
words, we want a function
buildSetPackage : (Int->Int->Bool) -> SetPackage
where SetPackage is an existential type -- an implementation of an
abstract data type:
SetPackage = Some(SetI' < SetI) {
emptySet : Object(SetI'),
union : Object(SetI') -> Object(SetI') -> Object(SetI')
}
>From the fact that the hidden representation type SetI' is declared to
be smaller than SetI (this is a "partially abstract type," in Cardelli
and Wegner's terminology), it follows that we can send the messages
addElement and member to instances of Object(SetI') just as if they
were instances of Object(SetI). But the union operation must be
applied to two arguments with the same hidden SetI', and the only way
to obtain a new object with this hidden type is to use the emptySet
provided by this package, so the implementation of union is justified
in assuming that the comparison functions of its two arguments are
identical. (Indeed, the comparison function need not be stored with
the sets any more: it can be stored in the package itself, which
brings us essentially full-circle.)
Well, there it is. The punchline, in the end, is something like this:
1) for things like set union and comparison, what you want is a
powerful system of abstract data types -- a higher-order module
system -- as Xavier suggested, and
2) such a module system interacts smoothly with a view of sets as
objects. (In fact, you can even "inherit" things like union
functions, but that's a topic for another discussion!)
Cheers,
Benjamin
P.S. Details of this construction are in a forthcoming tech report by
David N. Turner and myself. If anybody's interested I can send you a
postscript file.