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P1788 membersIt's a pity that I didn't get to write this before the end of the official discussion period but too much to do...
Motion 3 says that 1788 should define an interval to be a closed connected subset of the reals R. Clearly, this excludes both cset models, and modal interval models, from being *directly* supported by the standard.
I have spent much time recently trying to understand modal and Kaucher intervals, via the papers of A. Neumaier "Computer graphics, linear interpolation, and nonstandard intervals", V. Kreinovich "Use of Modal Interval Analysis in Early Engineering Design" and N. Hayes "Introduction to Modal Intervals".
From comparing the claims made by the modal interval supporters with those made by myself and other cset interval supporters, I have come to the view that Motion 3 is right.
We should take seriously Dan Zuras' view (posting to ER subgroup 15 Mar 09):
If floating-point expertise has grown ... rare then expertise in interval methods is rarer still. Therefore, we must accept that part of our task in writing this standard is a pedagogical one. Interval methods are not just a funny flavor of floating-point. We must teach the world how to use intervals properly. We must teach them how to properly interpret their results. Further we must create a standard that is capable of being taught.
(Read the whole thing, for context.) We must not ignore this issue of making 1788 easy to grasp and to teach.Csets are totally natural -- for people with an analysis orientation like me. And the theory is so elegant! How can people not want a system that removes removable singularities and treats (x^2-1)/(x-1) as equivalent to (x+1), etc? Yet some take that view, and not just because they don't understand csets.
Modal intervals are totally natural -- for people with an algebra orientation. And the theory is so elegant! How can people not want a system that removes the need for an empty set and makes the intervals into a group under addition, etc? Yet some take that view, and not just because they don't understand modal theory.
Both approaches offer valuable ways of improving enclosures, as we saw with the Hayes-Neumaier challenge. But neither theory is "the natural completion" of classical intervals as proponents, I included, have sometimes thought. Their machinery is too heavy, and detracts from the pedagogical -- and evangelistic -- purpose of 1788 that Dan rightly identifies.
So I now believe that the KISS principle should trump the perceived virtues of either approach. I shall vote Yes to Motion 3.
Having come to this view, I think the Vienna proposal is close to what we need, in content if not entirely in presentation.
In view of the usefulness of these other systems, how should they be supported *on top* of a system that is close to Vienna? This seems quite easy for modals (as Nate has confirmed), and Vienna already provides low-level functions to aid a modal implementation.
It seems a lot harder for csets, largely because of the different underlying number system. Maybe one needs ways to switch in and out of a particular interval system, as I believe Van Snyder has recently suggested. Maybe this can only be satisfactorily addressed at the language level.
Best wishes John