Re: Disjoint, subset, & interior, or more...???
Dan Zuras Intervals wrote:
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?
No, it isn't. For to be able to reach a positive conclusion,
one needs (in certain cases) the topological interior.
Thus if one only has a comparison that tests for proper subset,
one can never verify the required condition without having to
recode it oneself.
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.
Not in the version of Motion 0013.04. There Empty<=Empty is defined
to be false, which contradicts the definition of a partial order.
Moreover, <= is irrelvant in the context of ordinary interval
computations.
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 has some natural meaning (and some limited use) in the context
of Kaucher or modal intervals (where, instead, Empty is problematic).
Thus the introduction of <= type ordering relations should be left
an upgrade of P1788 ten years friom now that might standardize the
latter.