Dear Mr. Hayes,
I am really surprised to see how much time and energy people working
on the standard are loosing because you want to include modal
intervals and Kaucher arithmetic into the standard!
Your proposal to replace unbounded intervals by overflow is a typical
example of the reason why I argued some months ago that Kaucher
intervals should not be included in the standard: Kaucher arithmetic
is not defined for unbounded intervals, so there are only two ways to
include them. First, define the Kaucher arithmetic for unbounded
intervals, but this is a real scientific challenge, that needs all
experts to agree, and I understand very well you could not do that.
Second, remove unbounded intervals for the standard, what you
unbelievably propose now.
Removing unbounded intervals from the standard just makes no sense!
They appear naturally as domains in so many models! Your claim that
nothing can be proved for unbounded interval is outwardly false.
Consider for example
f(x)= 1+x^2 (1+sin(x))
and the unbounded interval (-oo,+oo). Then the simple interval
evaluation f(-oo,+oo)=[1,+oo] allows proving that the equation f(x)=0
has no real solution! Every interval library handling correctly
unbounded intervals will perform this easy computation. There are many
other situations where one can prove properties on unbounded
intervals.
In my opinion, unbounded intervals are necessary for any interval
arithmetic package. Removing them from the standard just to have a
chance to see modal intervals and Kaucher arithmetic included sounds
very poor to me. Of course, computing the midpoint of an unbounded
interval is a real issue, that needs to be discussed urgently.
However, removing unbounded intervals to fix this issue also sounds
very poor to me.
We now know that including Modal intervals and Kaucher arithmetic into
the standard will exclude unbounded intervals from the standard. Will
you also propose in the future that we exclude everything that is not
compatible with modal intervals, like reverse operations?!
Alexandre Goldsztejn
On Fri, May 4, 2012 at 2:00 AM, Nate Hayes <nh@xxxxxxxxxxxxxxxxx> wrote:
That's fine with me; or if you wish to continue this will be our motion:
-------------------------------------------
As described in the accompanying position paper, P1788 shall change the
existing Level 1 and Level 2 model to the three-tiered level structure as
described in Section 2. Specifically, this means the following:
-- The "mathematical intervals" at Level 1 are defined to be the classic
set of nonempty, closed and bounded intervals; this is the level of
"mathematical regularity" (MR) for interval arithmetic. The FTIA,
infimum,
supremum, midpoint and radius are all defined as in Section 2.1.
-- In Level 1a, FTIA is extended to unbounded intervals and the empty
set according to (4) and (5); this is the level of "algebraic closure"
(AC)
for interval arithmetic. More specifically, an unbounded interval is
interpreted as a family of intervals parameterized (virtually) by an
overflow threshold, as defined and explained in Section 2.2.
-- Level 2 is defined as in Section 2.3; this is the level of "interval
datums." The maximal real element of each interval datum format defines
the
concrete value of each corresponding overflow threshold at Level 1a.
-- All of the "infinities" in the current model are changed to
"overflow", i.e., lower-case omega.
-- The midpoint operation is defined at Level 1a and Level 2 for all
nonempty intervals as a real number (we suggest something similar to what
is
discussed in Section 3.2, but we leave the actual definition for a future
motion); the midpoint of an empty interval is left to a future motion.
-------------------------------------------
Sincerely,
Nate
P.S. One of our engineers found a type-o (the natural domain D_f of a
real
function should be a "subset of" R^n, not an "element of"), so I attach a
corrected version.
----- Original Message ----- From: "Ralph Baker Kearfott"
<rbk@xxxxxxxxxxxx>
To: "Michel Hack" <mhack@xxxxxxx>
Cc: "stds-1788" <stds-1788@xxxxxxxxxxxxxxxxx>
Sent: Thursday, May 03, 2012 3:47 PM
Subject: Re: Do I have a second? Overflow, New Motion
OK, I guess that's reasonable, if that's OK with Nate.
Baker
On 05/02/2012 01:18 PM, Michel Hack wrote:
As Baker's P.S. indicates, there is as yet no actual motion on
the table, so I don't understand what a "second" would be about!
Nate's position paper is well-written btw, and I will post a few
comments soon. But at this point it is just a position paper,
and we don't need a motion to put this into the public list of
position papers. (It may become the Rationale for an upcoming
motion, of course.)
Michel.
---Sent: 2012-05-02 18:23:52 UTC
--
---------------------------------------------------------------
Ralph Baker Kearfott, rbk@xxxxxxxxxxxxx (337) 482-5346 (fax)
(337) 482-5270 (work) (337) 993-1827 (home)
URL: http://interval.louisiana.edu/kearfott.html
Department of Mathematics, University of Louisiana at Lafayette
(Room 217 Maxim D. Doucet Hall, 1403 Johnston Street)
Box 4-1010, Lafayette, LA 70504-1010, USA
---------------------------------------------------------------
--
Dr. Alexandre Goldsztejn
CNRS - Laboratoire d'Informatique de Nantes Atlantique
Office : +33 2 51 12 58 37 Mobile : +33 6 78 04 94 87
Web: www.goldsztejn.com
Email: alexandre.goldsztejn@xxxxxxxxxxxxxx