Re: Motion 31 draft text V04.4, extra notes
Michel Hack wrote:
Nate Hayes wrote:
What I meant is that at Level 1 there is no definition of midpoint for
unbounded intervals. So why include unbounded intervals in the Level 1
model? Especially when a similar treatment at Level 2 of "overflown"
intervals can provide the same practical benefits? IMO the Level 1
model is then cleaner and simpler.
If we were only talking about intervals representing uncertain numbers,
which is where midpoint notions are most significant, Nate would be right.
But I believe we also want to support intervals as ranges, e.g. for
constraint propagation. In that case it is essential to provide for
unbounded intervals at Level 1.
Folks, I apologize in advance for being pedantic but someone needs to
explain this to me:
Why is it "essential"?
Specifically: what is a concrete example of a computer algorithm that can
produce more meaningful results with unbounded Level 1 intervals as opposed
to "overflown" intervals at Level 2?
I'm not aware of any interval computer algorithm that operates at Level 1
with an infinite amount of magnitude and precision (save Computer Algebra
Systems, which IMHO is beyond the scope of P1788), and no branch-and-bound
algorithm, for example, can prove anything useful about a Level 1 domain
x >= 0
beyond the largest Level 2 number. In my experience such an algorithm is
ill-conditioned if one is trying to numerically prove something about the
domain of a function outside the numerical limits of the underlying system.
So why does [0,+OVR] not suffice as input to such an algorithm as opposed to
[0,+Inf]? I don't see how unbounded intervals fix or solve anything in this
respect, or that they even provide any additional meaning to the results
that can be computed by such an algorithm.
So can someone please give a very specific, concrete example of an interval
algorithm that demonstrates why unbounded intervals are "essential"?
Specifically, a concrete example where the results are more meanigful than
those computed by a similar algorithm that would use "overflown" Level 2
intervals instead? A reference or white-paper would be fine.
I thank Michel for some preliminary offline discussion about this. Between
the two of us it seems we couldn't come up with such a concrete example.
Nate
P.S. On the note of Computer Algebra Systems, anyhow, I would point out
again that working with bounded intervals gives stronger algebraic
capabilities.