Re: Kaucher intervals (Was: Undefined behaviour (Was: ...))
Dear Michel,
It would have helped if early on we had seen a concise, though perhaps
incomplete, description of Kaucher intervals, as Baker just gave. Not
even the paper that Nate pointed us to was that clear, one had to guess
it from the way they were used.
I am working hard on a tutorial for the modal intervals for this P1788 group. It has turned into bigger project than I expected, but finally I just have one section left. I hope to have it done soon.
So: a Kaucher interval is an interval with Lower and Upper bounds,
always non-empty, with an additional flag that directs how the bounds
are to participate in monotonic arithmetic. This is useful in cases
where backwards containment needs to be calculated. (The mode flag is
encoded in the order in which the two bounds are given in the interval
representation, because that leads naturally to the desired effect, at
least in common cases -- or does it always work, obviating the need to
test the "flag"?)
The Kaucher arithmetic always works... there is no need to test the flag. You can think of the flag as +/- sign: similar to how real numbers come in pairs of equal magnitude but opposite sign, the Kaucher intervals come in pairs of equal
interval set but opposite mode. The Kaucher flag functions much like the +/- sign of the real numbers.
Critical in the context of the Expression-Rearrangement subgroup is
how much knowledge of applicability of Kaucher rules is needed when
expressions are transformed.
Also -- how is Empty handled in Kaucher Arithmetic?
It is one of the properties of the Kaucher arithmetic that empty set is not needed. However, empty set can still be defined for other purposes.
Finally -- is "Infinity as a Member" decided one way or the other in
Kaucher Arithmetic, and if not, are the issues similar?
I don't believe it is decided. The issues are similar.
One particular view of the Kaucher arithmetic, i.e., the modal intervals by Gardenes, Sainz, et. al., require that infinities do not be member of the interval. There is a tutorial and explanation of why in my upcoming paper.
Sincerley,
Nate Hayes
Sunfish Studio, LLC