RE: Motion 52: final "Expressions" text for vote
Dear Vincent, Thanks for your comment. I agree that this is just one of the goals, not THE goal.
Yes, we need intersection, union, etc., but my point is our ulmitae goal is applications -- whether to computations in pure mathematical computing or to problems from engineering and science. While computing the range may not be the only reason, if we have a choice, we should select the one which is most appropriate from the viewpoint of the corresponding application problem.
Re x-x, I think we are in full agreement
________________________________________
From: stds-1788@xxxxxxxx [stds-1788@xxxxxxxx] on behalf of Vincent Lefevre [vincent@xxxxxxxxxx]
Sent: Sunday, November 24, 2013 6:40 PM
To: stds-1788@xxxxxxxxxxxxxxxxx
Subject: Re: Motion 52: final "Expressions" text for vote
On 2013-11-23 22:22:49 +0000, Kreinovich, Vladik wrote:
> The goal of interval computations is to provide enclosures for
> ranges of functions of real variables over intervals.
Well, I disagree. This is just *one* of the goals. Another goal is
to do pure computations on intervals; this includes set operations
like union (yielding hull on intervals) and intersection.
> Interpretation of expressions should be left to the implementation,
> it is not a question of mathematcial foundations.
>
> It is a question of mathematical foundations of what is the range of
> x-x when x is in [1,2]. Clearly, the range is 0. It is a question of
> implementation whether X-X, where X is an interval, would return
> [0,0] or a wider interval -- as long as what it returns is an
> enclosure for the exact range.
I disagree or don't think this should be said like that. Say, for
some language, one could have the following various levels (very
simplified):
1. Program text (a sequence of characters).
2. Some kind of program graph, after lexical and semantic analysis.
3. Another program graph, after some language-specific optimizations.
For instance, the language could regard for some types, x-x to be
equivalent to 0.
4. For the various "interval expressions", the interpretation in P1788
(via bindings).
IMHO, in P1788 (4), X-X must not be simplified to [0,0]. But such
a transformation could be done at the language level (3), before
regarding the expression as being an interval expression.
> It would be great if we could always return the exact range, but
> since the problem of computing the exact range is NP-hard, we will
> inevitably sometimes produce the enclosure with excess width.
One would also need to be careful with requirements on Empty.
For instance, the exact range of some expression could be Empty,
but a range computed in an efficient way could be non-Empty.
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
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Work: CR INRIA - computer arithmetic / AriC project (LIP, ENS-Lyon)