[std-interval] Interval comparison operators

[Dr John Pryce]

Lee, and participants in this group

At 19:12 04/06/06, R. Baker Kearfott wrote:
>My view is that, if there is a hopeless lack of consensus, it may
>indicate that there isn't a single "right" way, but choosing
>a particular way and having a standard is better than having none.
>However, we should strive to maximize the utility of the language
>in the standard.

Like Baker, I feel optimistic that a consensus will emerge. I 
STRONGLY agree with Baker's paragraph above.

Lee, you have written eloquently of the Principal Value (PV) model 
that you have been working on. If it is as different in 
concept/philosophy/goals (I don't know enough about it to know which 
of those words are most appropriate) from the "classical interval 
model" as you say, then certainly it needs a different standard from 
one that supports the classical model. Therefore, I do not see you as 
in conflict with the classical intervallers, but as complementary.

However I speak as the principal author of containment set (cset) 
theory for the last few years. Csets are NOT the most widely held 
view of classical intervals, and when I started working on them for 
Bill Walster under contract to Sun around 2001 they were generally 
seen as an eccentricity that would soon die out.

Evaluation of cset theory by the interval community was delayed 
slightly by commercial confidentiality. That prohibited putting some 
aspects of csets in the public domain - mainly Sun's implementation 
detail. The main reason for delay was that Bill and I had serious 
disagreement, not over what a cset-based interval implementation 
should do, but about what the results *meant*. I believed, and still 
do, that Bill's philosophy of csets at that time was mathematically 
inconsistent. As a result we could not agree on a paper to submit to 
the open literature. We spoke at conferences etc, but some of the 
assertions we made were incompatible, which was not good for credibility.

Maybe it was a blessing in disguise, because then I joined the ISL 
project steering committee, and working with 3 guys who have jointly 
over 50 years interval programming experience, plus a professional 
software engineer, widened my outlook so that the paper I eventually 
wrote, Pryce and Corliss "Interval arithmetic with containment sets" 
(2006), submitted to "Computing", see
     http://homepage.ntlworld.com/j.d.pryce/isloct05/IntvlArithCsets.pdf
was far clearer and more concise than it would otherwise have been.

And, to my surprise, my ISL colleagues decided of their own accord 
that a cset-based interval system was the one they preferred, though 
the well-known packages they had up to then used, and in one case 
written, were based on what I call the Simple System.

How did this come about? I believe it was because I had been able to 
write an account that showed such a system is
- rock-solid mathematically,
- actually useful.

Both are necessary conditions for acceptance in this corner of 
scientific computing (sadly, not sufficient). I urge you to move the 
task of writing up what PV intervals are all about to near the top of 
your priorities, if not the actual top. And get it into the open 
literature so that people can pull your ideas to pieces (as my cset 
ideas have been for several years), force you to refine them, and 
eventually be convinced you are making sense.

I find many things you write very weird or downright false (e.g. 
distinctions between line segments and sets, and tests on the empty 
set). That is because I am matching it against my model, which is 
firmly based on classical set theory. You have work to do to convince 
me, but my mind is open. As I am a mathematician, the "rock-solid 
mathematically" interests me more, initially, than the "actually 
useful". I think many people with more interval experience and/or 
more influence in the scientific world than I would agree with me.

PV intervals must be mathematically consistent, or they will die. But 
there is more than that. You must have an underlying purpose for 
them, or philosophy about them, that explains why this consistent bit 
of mathematics actually helps humans solve real life problems. From 
what you write, you are a fair way towards having both, but you won't 
convince the world unless you WRITE IT UP for public evaluation.

Best wishes for that project.

John Pryce