Re: Motion 31 draft text V04.4, extra notes
On 2012-04-04 10:51:24 -0500, Nate Hayes wrote:
> 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.
By "overflown" interval, I suppose you meant interval as an element
of some datatype, because mathematically, it is not an interval,
but a family of intervals (with the semantics being that the "true"
interval is an element of this family).
Anyway P1788 deals with "unbounded" intervals, not with "overflown"
intervals (which would have been much more complex, IMHO, e.g. for
\interior below).
> 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.
There could be several ways to specify the latter case (you have two
families of intervals, several ways to choose the quantifiers...);
"false" is not a canonical answer.
> So if one interprets the comparision operations in Motion 13 in the light of
> overflown intervals, then those endpoint formulas are correct.
P1788 should stick to a single choice: "unbounded" intervals or
"overflown" intervals. Mixing choices is a very bad idea.
> 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.
I don't see the discussion on midpoint changing anything about such
intervals.
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
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