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Re: KISS-decorations

Jurgen, Marco,

This is looking very good to me.

Here are a few thoughts about the new draft:

   -- I think it would be fine to remove section 1.3.5 from the current
motion, which has a primary focus on the most general case: decorated
intervals. It may be wiser to dedicate an entire separate motion that
considers the algebraic structure of bare intervals, bare decorations and
decorated intervals. See for example Tables 3 and 4 on p. 18 and Section 5.2
of my old Nov. 14 DRAFT position paper (which I attach for reference and
convenience).

   -- About FTDIA, Dominique has showed me in an offline discussion a
formulation that does not require a "containment order" of decorations. I
have some reason to believe his formulation may also be extended to include
the intersection and union (as presented in Motion 27), but I guess we shall
see.

   -- I agree that D4 and D3 in motion 25 are the finer shades of gray of
the decoration "safe" in motion 27; but still think its wiser and more
informative to keep the distiction and not lump them together. In terms of
KISS, I really don't see this would be "complicated".

So, except for the notes above, I believe we have converged. If imitation is
the most sincere form of flattery, I would otherwise plagarize motion 27
liberally as a friendly amendment to motion 25.

Sincerely,

Nate
----- Original Message ----- From: "J. Wolff von Gudenberg" <wolff@xxxxxxxxxxxxxxxxxxxxxxxxxxx> 
To: "John Pryce" <j.d.pryce@xxxxxxxxxxxx>; "Nate Hayes"
<nh@xxxxxxxxxxxxxxxxx>; "Dominique Lohez" <dominique.lohez@xxxxxxx>
Cc: "stds-1788@xxxxxxxx" <stds-1788@xxxxxxxx>
Sent: Friday, July 01, 2011 6:27 AM
Subject: Re: KISS-decorations



John, Nate, Dominique, P1788
Thank you for your helpful comments and corrections.
We have updated our motion again in two main aspects:
1. Theorem 1 has been corrected.
    John's remark on implemenntation of elementary functions is level 2
2. the treatment of empty sets has been changed such that f(empty) =
(empty, saf)
   That means we are joining Nate's decorations D4 and D3, so 4
decorations are enough.

    The following parts of the text have chaneged
        Definition 1
        remark 1 inserted
        theorem 1
        old remark 2 split into remark 3 and remark 4
        remark 5 inserted
        examples with empty set
        theoram 4

Concerning simplicity:
we think we should not go further in restrcting the concept of decorated
interval
the only concept which may be cut off i the use of bare decorations
On the other hand we have no objection against an OPTIONAL containment
order and a true FTDIA

we have changed the website

Juergen and Marco

Am 01.07.2011 09:21, schrieb John Pryce:

BTW Your second statement of theorem 1 is again wrong

               Since f(x)=x is continuous the John's counterexample is
again working.
               f(x) is a function defined by an expression in the John's
wording.
               F(X) is THE extension calculated by a interval
transcription of the expression.
               Then for F(X)   the isotonicity holds.

(a) I was about to point out the same two things. It's what I'm calling
the "natural interval extension" (Arnold asserts this is standard
terminology), aka what you call "straightforward interval computation"
that is isotonic.

And even that is only conditionally true.
IF the interval version of each elementary function is isotonic
THEN any function constructed from them by "straightforward interval
computation" is isotonic.

But if we define the interval version of EVERY elementary function
stupidly as in my counterexample (e.g. Entire if 0 not in xx, something
sensible otherwise) then obviously this applies to functions defined by
"straightforward interval computation" too, so isotonicity fails
spectacularly.

I seem to recall discussion by elementary-function experts that
isotonicity can fail "at the margins", i.e. by an ULP (unit in last
place) or two, if a point elementary function e(x), used in coding its
interval version, is "wobbly". E.g. if it is<1 ULP away from correct
value at one FP number, and>2 ULPs away at the next FP number. So there's
a nontrivial risk of isotonicity failing thanks to defects in library
functions.

(b) You say "the decoration systems of motions 25-26 are too
sophisticated...". I rather agree. However I think your, and Dominique's,
desire to remove the  "containment order" is more about presentation than
substance.

Like death and taxes, the "containment" order is a fact of life. It
simply expresses the fact that, because of interval widening due to
dependence, the computed decoration fact may not be the sharpest one that
actually holds. E.g. in your current scheme
   saf may be true but you actually compute def (which is valid but
weaker).
   saf may be true but you actually compute enc ...
   def may be true but you actually compute enc ...
   ndf may be true but you actually compute enc ...

Would you prefer it if the standard says something like the following?


The computed decoration always makes a true statement about the
continuity, definedness, etc., of the function f over the box xx.
However it may not be the strongest one available [give example] ... By
introducing a *containment order* and the notion of the *exact
decoration of f over xx *, one may express this in the form of a
*FTDIA*, which says that the computed interval yy and decoration d,
resulting from evaluating f(xx), _jointly_ enclose the exact range and
exact decoration, respectively, of f over xx. [followed by fuller
explanation]...

(* meaning bold introducing a new concept, _ meaning italic for
emphasis.)

This presents the FTDIA as an optional interpretation that one may find
enlightening, and the containment order as symbolically expressing an
unfortunate fact of life in the same way as
    (take-home pay T)<  (gross pay G)
symbolically expresses another unfortunate fact.

John


--
-                Prof. Dr. Juergen Wolff von Gudenberg
     o           Lehrstuhl fuer Informatik II
    / \          Universitaet Wuerzburg, Am Hubland, D-97074 Wuerzburg
InfoII o         Tel.: +49 931 / 31 86602 Fax ../31 86603
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