Re: Motion P1788/M0013.04 - Comparisons - Overflow / Infinity
On 20 Sep 2010, at 19:00, Arnold Neumaier wrote:
> Ian McIntosh wrote:
>> NH> Keep in mind, Ian McKintosh has suggested replacing unbounded closed
>> NH> intervals with "overflown" compact intervals. In practice, there is not
>> NH> really a difference between the two. For example, +INF and -INF are
>> NH> replaced by +OVR and -OVR in all compuations, where OVR is some unknown
>> NH> finite real number larger than the magnitude of the largest
>> NH> floating-point number. In this way, intervals such as [1,+OVR] remain
>> NH> compact intervals and are not unbounded like the interval [1,+INF].
>> NH> Arithmetic operations on the endpoints of compact overflown intervals
>> is
>> NH> the same as in Motion 5, i.e., (-OVR) + (-OVR) = -OVR, 0 * OVR = 0,
>> etc.
>> AN> But then OVR has a different meaning in different places, which is
>> AN> unacceptable from a mathematical point of view. It is just INF in
>> AN> duisguise, and it hides mathematical propoerties of the limit under
>> AN> the carpet. One cannot use it in a mathematical argument....
>> AN>
>> AN> To argue for existence, one _needs_ in certain case the unbounded case.
>> AN> Mathematical programming had once a tradition for using big M (which is
>> AN> about the same as your OVR), and it is now deprecated (though still
>> used
>> AN> by a few who don't like unbounded variables) since it behaves poorly
>> not
>> AN> only mathematically but also algorithmically.
>> I did suggest Overflow (OVR) as a better floating point value than Infinity
>> to represent an overflowed value, and that it would be very often more
>> useful as an interval endpoint.
>> In doing that I did not mean it to totally replace Infinity, and I did not
>> mean they would have the same semantics. The whole point is to distinguish
>> their semantics.
>> In IEEE floating point, you can get Infinity two ways. One is by dividing
>> a nonzero by zero. The other is by overflow. The following mostly ignores
>> signs for brevity:
>> IEEE nonzero / 0 = Infinity
>> IEEE overflow on any operation = Infinity
>
> We introduced in Motion 8 decorations to be able to distinguish the two where necessary inninterval bounds by having the decoration IsBounded.
I have several objections to the idea e.g.:
(1) What is, for instance, sqrt( [1,Overflow] ) or log( [1,Overflow] )?
(2) [a,Overflow] doesn't really represent one interval, but an infinite family of intervals: all [a,b] with b > MAXREAL. Hence how does one evaluate [1,Overflow] \subseteq [1,Overflow]? It seems to need a trit, or a bool_set, result.
Siegfried Rump spent some time about 2 years ago trying to make a provably consistent theory of this, and then went very quiet on it. If he couldn't do it, I suspect no such theory exists.
John