Re: Comparisons and decorations
Dan Zuras Intervals wrote:
Next, you continue to assert that disjoint, subset,
& interior are the only (numerical) predicates that
should be in an interval arithmetic.
If this is so I understand why the edifices created
by motions 13 & 20 have no meaning for you. They are
unneeded in a world with only these 3 simple numerical
predicates.
(I qualify that with the word 'numerical' to distinguish
them from predicates involving decorations.)
Yes, Probably, the literature contains nowhere a nontrivial use of
the other comparisons.
Next, let me guess as to how the predicates disjoint,
subset, & interior are written. For aa = [alo,ahi] &
bb = [blo,bhi], I'm guessing they are something like
disjoint(aa,bb) = (ahi < blo) || (bhi < alo)
subset(aa,bb) = (blo <= alo) && (ahi <= bhi)
interior(aa,bb) = (blo < alo) && (ahi < bhi)
Do I have that right?
Yes for bounded, nonempty intervals. If aa or bb is empty,
all three have the value true. If aa or bb are unbounded,
the formula for interior must be modified, as discussed
elsewhere in the last few days.
Then the example I cited in my earlier note is one in
which iteration turns aa(0) & bb(0) from
disjoint(aa(0)),bb(0)) = true &
subset(aa(0)),bb(0)) = false
into aa(i'') & bb(i'') with
disjoint(aa(i'')),bb(i'')) = false & (1)
subset(aa(i'')),bb(i'')) = true. (2)
Are you OK with that?
yes, but it doesn't matter. (1) is a flase negative, hence harmless.
(ii0 is a false positive, but is obtained by augmenting bb.
But in the applications, bb is supposed to be a domain, not a
range enclosure, hence it does not make sense to discuss what
happens when it is altered.
I am concerned that in a different context the false
positive on subset may lead one to believe that a
solution exists where one does not.
The only comarisons in the assumptions of theorems I know that
produce existence statements are of the form
aa subset bb, or aa interior bb, (3)
and that produce nonexistence statements are of the form
aa disjoint bb (4)
where in all cases bb is a _chosen_ domain and aa a computed range
enclosure.
Thus the only false positives that must be avoided by any specification
of the standard are those for (3) or (4) when aa is overestimated
but bb is exact.
And false negatives don't matter in any of these cases.
Arnold Neumaier