Re: Motion 31 draft text V04.4, extra notes
John Pryce wrote:
First let me say I agree with criticisms I've seen of the Motion 13
Comparisons text in §5.6.9. They *need* definitions from first principles
in terms of sets, rather than endpoints. If we don't have that then how
they act on empty and unbounded inputs is likely to be ad-hoc and possibly
contradictory.
I remember the discussions surrounding this issue during the Motion 13
discussion period. Before too much time is invested rehashing the same
topic, I would recommend re-reading some of those threads.
In a nutshell, the issue as I recall boiled down to "unbounded" vs.
"overflown" intervals. To paraphrase Arnold Neumaier's distinction between
these two concepts:
-- The "unbounded" interval [7,+Inf] is a single interval (set of real
numbers) with an upper endpoint that is not bounded.
-- The "overflown" interval [7,+OVR] on the other hand is a family of
intervals. There are an infinite number of intervals in the family, but each
element of the family is closed and bounded.
The correct definitions and/or interpretations of the comparison operations
depend on these distinctions. For example, from a purely mathematical
perspective the comparison of unbounded intervals
[7,+Inf] \interior [1,+Inf]
is true; however as pointed out by Arnold the comparison of "overflown"
intervals
[7,+OVR] \interior [1,+OVR]
is necissarily false.
So if one interprets the comparision operations in Motion 13 in the light of
overflown intervals, then those endpoint formulas are correct.
There was a great deal of dissucssion at the time about what the endpoint
formula should be for unbounded intervals, instead, and this all led to very
complicated and slow implementations.
In my opinion, P1788 should consider restricting Level 1 to bounded
intervals and introduce "overflown" intervals at Level 2. After the recent
discussion on midpoint, it seems the committee is already leaning in this
direction anyways. It also means the formulas in Motion 13 which are very
simple and efficient could still be used for implementations.
Nate