It was clear to me from the beginning that I shall vote No on Motion
45. But the motion was accepted. So reading the text of the motion
more carefully now a number of inconsistencies occur to me. I wonder
whether somebody of the Yes voters can help me staighten these out.
The product of two real numbers is a particular form of a dot
product. Now consider the interval product with a>0 and b<0:
[0, a] × [b, 0] = [a×b, +0] or [0, 0] × [c, d] = [+0, +0]
By motion 45 the zeros in the results have to carry a plus sign. This
looks very strange to me in particular since a×b<0. A signed zero does
not make sense in interval arithmetic. An interval with zero as one
bound has already a direction.
Similar for division by an interval where a bound is -oo or +oo:
If in the result of an interval division one bound is zero, like [a,
0] or [0, b], the other bound shows immediately whether the elements
are positive or negative. So there is absolutely no need for
indicating this by a minus or plus symbol in front of the zero.
A real interval is defined as a closed and connected set of real
numbers. In case of unbounded intervals -oo and +oo are just used as
bounds. They are themselves not real numbers! So in case of interval
division the rules for computing the bounds of the result do not
strictly follow the pattern of IEEE 754 arithmetic for extended real
numbers. For instance 0×(-oo) = (-oo)×0 = 0×(+oo) = (+oo)×0 = 0 in
interval arithmetic and not NaN like in IEEE 754 arithmetic.
Well defined interval arithmetic over the real numbers (or the subset
of floating-point numbers) including CA and the EDP leads to a
calculus that is free of exceptions. We must not make the mistake
defining interval arithmetic over all IEEE 754 datums!
The development of IEEE P1788 suffers from the fact that many members
of the group take IEEE 754 arithmetic as basic truth for scientific
computing. This may be true for floating-point arithmetic. But it is
not true for interval arithmetic. The two should be separated more
clearly. Within IEEE P1788 we have the unique chance of defining
interval arithmetic including CA and the EDP as a powerful but simple
calculus that is free of exceptions.
What in Motion 45 under 1. is called the "broader aim" actually is a
much narrower aim.
I shall comment on a number of misunderstandings concerning CA and the
EDP in another mail.
With best regards
Ulrich