Re: John, Arnold? Re: Amendment to property tracking
Baker, Jürgen, Dominique, P1788
On 5 Jun 2011, at 21:51, Ralph Baker Kearfott wrote:
> John, Arnold,
> Please give your opinions...
> On 06/03/2011 09:58 AM, Nate Hayes wrote:
>>
>> It is also important to remember that both FTIA and FTDIA are theories.
>> As such, all they do is make *predictions* about the result of an
>> interval expression. Neither actually *compute* anything. So even though
>> they are necessary, they are not enough for a standard.
>>
>> For FTIA, this is why P1788 passed Motion 5, which provides interval
>> arithmetic operations to compute range enclosures consistent with FTIA.
>>
>> For FTDIA, this is why P1788 should pass Motion 25, which provides the
>> necessary operations to compute decorations consistent with FTDIA.
>> ...
>> Dominique Lohez wrote:
>> Jürgen Wolff von Gudenberg a écrit :
>>>> I would like to raise the question:
>>>> "Do we really need an FTDIA ?"
>>>> or is an FTIA sufficient?
>>> Yes, we need FTDIA The meaning of the theorem is to fulfill the precept
>>> Thou shalt not lie
>>> For the interval width , this leads to FTIA
>>> For decorations this leads yo FTDIA
>>> IMHO Interval arithmetic and Decoration Calculations must not be dealt
>>> with separately.
This is how I look at it, though no doubt only a partial view.
Initially we thought of decorations as separate pieces of information that "tag along" while evaluating a function and are sort of subsidiary to the computed yy=f(xx) which, the FTIA says, encloses Range(f,xx).
It was Arnold's insight (and I think Nate was thinking on similar lines) to consider various combinations of the D+, D- and C bits as forming useful predicates p_d(f,xx) about pairs (function f, box xx). As a predicate can be regarded as "the set of things for which it is true", one could think of p_saf, p_def, etc. as being sets saf, def, etc.
If we can prove a given (f,xx) is in the set "saf", that is a smaller set and tells us more than if we can only prove it is in "def" or "con"; just as the smaller yy we can find, the more it tells us about Range(f,xx).
So, regarding decorated evaluation as a "joint enclosure", componentwise:
(computed range, computed decoration) \supseteq (true range, true decoration)
of f over xx, seemed sort of neat. That's what the FTDIA is.
Jürgen asks "Do we really need an FTDIA ?" Well, we need a decoration system in addition to the FTIA. Dominique is right that the 2 parts "must not be dealt with separately"; indeed CANNOT be dealt with separately.
But
- We don't HAVE to formulate it in the form of a FTDIA.
- There may yet be some flaw in the theory that makes such a formulation so messy that it doesn't help prospective users to understand the concepts.
- Or P1788, on studying a "final" FTDIA formulation, may just say "No thanks", and vote to go back to the original D+, D- and C bits.
Personally I continue to like the hierarchy of saf, def, etc. and the "joint enclosure" viewpoint, but some work is still needed to verify that it really can be made both tidy and watertight.
John