Disjoint, subset, & interior, or more...???
Folks,
I have been having an offline discussion with John about
these comparison issues. He has (indirectly) convinced
me that I should make my concerns more public.
This will be an expression of my concerns & questions,
only. I have no good solutions.
Disjoint, subset, & interior.
I am moved by Arnold's arguments. But I am not convinced
by them. And I am moved for some reasons I'm sure Arnold
never intended.
You see, my attraction to his limited set of comparisons
is partially pedegogical rather than technical. Sooner or
later we are going to have to deal with users of interval
software who are trained to think in floating-point rather
than interval terms. The natural tendency will be to
attempt to convert their floating-point algorithms to
interval ones by, more or less, declaring all their
floating-point variables to be intervals instead.
This would be, of course, a HUGE mistake.
And what better way to convince them that it is a mistake
than to hit them right away with the fact that you cannot
compare two intervals as you once compared two floating-
point values? Sure, some of them would try to hack around
it. But at least some of them would look into why this is
the case &, just perhaps, learn something about how to do
REAL interval calculations. Correct interval calculations.
Its worth a shot, if for no other reason than that.
But back to the technical issue, I am not convinced by
Arnold's big 3: disjoint, subset, & interior. I'm OK with
the first two but the 3rd bothers me.
John reminded me that I raised this concern in another
context so let me couch it in terms of what are necessary
comparisons.
It seems that Arnold's argument is that in a mathematical
context the condition sought for ODE solvers is interior.
That is, subset with a smidge at both ends when those ends
are finite.
But in a computational context, with closed intervals that
can be expanded by computation, isn't that too restrictive?
Isn't the correct condition proper subset? That is, subset
& not equal?
Therefore, shouldn't the 3 comparisons be disjoint, subset,
& equal? From which proper subset can be derived?
Then there is another aspect of all this that worries me.
As Nate is fond of pointing out, intervals are ordered by
both subset & less than. Or more properly, PARTIALLY
ordered by both subset & less than.
So, two intervals can be disjoint (empty intersection) or
not (something in common). They can be subsets or not.
And they can be equal if both are subsets of each other.
Otherwise they are unordered by subset (some sort of non-
trivial overlap).
But, are there never interval methods which need to know
whether or not two intervals are ordered by less than?
At least to know that all elements of one interval are
less than any element of another? That is, strict &
disjoint less than.
It seems to me, at the very least, we must consider that
intervals are ordered by strict less than or not. The
various overlapping forms of less than may or may not be
important but this one seems to be clearly so.
And an equal comparison applies to the world of less than
as much as it does to subset.
Some of the other forms of less than can be derived from
disjoint, subset, equal, & strict less than if necessary.
Perhaps not all.
So I can argue for these 4 comparisons: disjoint, subset,
equal, & strict less than.
I don't know if these arguments are compelling but I can
make no similarly compelling arguments for the others.
I believe Baker is correct that someone should make some
sort of unifying or simplifying motion on this matter.
And I feel eminently unqualified to do so.
I am asking for your help.
Dan