Re: Do I have a second? Overflow, New Motion
I second.
I agree with Alex Goldsztejn that " this is a real scientific challenge, that
needs all experts to agree", and I congratulate Nate for his efforts. Like him
I beleive that the correct approach to intervals is to start with
generalized intervals (in Kaucher sense).
S Markov
On 5 May 2012 at 9:59, Nate Hayes wrote:
Send reply to: "Nate Hayes" <nh@xxxxxxxxxxxxxxxxx>
From: "Nate Hayes" <nh@xxxxxxxxxxxxxxxxx>
To: "stds-1788" <stds-1788@xxxxxxxxxxxxxxxxx>
Subject: Re: Do I have a second? Overflow, New Motion
Date sent: Sat, 5 May 2012 09:59:10 -0500
Organization: Sunfish Studio, LLC
> Folks,
>
> The motion says:
>
> "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."
>
> The definitions (4) and (5) in the position paper are the same as those
> given by John Pryce in the current standard draft text. So the motion is not
> "removing unbounded intervals from the standard."
>
> If X is an unbounded interval, the definition (6) in the paper defines
> omega(X)
> as a family of intervals parameterized by an overflow threshold; in
> Corollary 2 upsilon is the union of all intervals in this family and we have
> upsilon(omega(X)) = X.
> This illustrates an unbounded intreval may be interpreted as overflow, and
> vice-versa.
>
> Re-interpreting X as omega(X) allows a midpoint operation to be defined as a
> real number at all levels of the standard, and the motion requires this.
> This is good for P1788 regardless of Kaucher arithmetic. That the motion may
> also allow Kaucher arithmetic to be developed at Level 1 (and possibly then
> extended to unbounded intervals at Level 1a) in a matter compatible to P1788
> is also good, no? Or is the purpose of P1788 truly to create a permanent
> fracture within the interval community and user base by forcing each side to
> have a mutually incompatible standard?
>
> We believe we are presenting a "win win" situation to P1788 in this regard.
>
> Nate
>
>
> ----- Original Message -----
> From: "Alexandre Goldsztejn" <alexandre.goldsztejn@xxxxxxxxx>
> To: "Nate Hayes" <nh@xxxxxxxxxxxxxxxxx>
> Cc: <rbk@xxxxxxxxxxxxx>; "stds-1788" <stds-1788@xxxxxxxxxxxxxxxxx>
> Sent: Friday, May 04, 2012 2:11 AM
> Subject: Re: Do I have a second? Overflow, New Motion
>
>
> > 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
> >
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