Re: (Fwd) Re: Motion on interval flavors
Dear Professor Markov,
Let me clarify my experience. I see two main applications of Kaucher intervals:
1- Quantified propositions.
Together with the semantic brought by modal intervals, and mainly by
Professor Shary's work in the context of linear systems, Kaucher
intervals are able to certify quantified propositions. In the linear
case, there is clearly a superiority of Kaucher intervals over
classical interval for handling quantified interval linear systems,
but no concrete application up to my knowledge. In the nonlinear case,
the very same type of quantified propositions can be certified using
parametric existence theorems:
- page 28 of http://hal.archives-ouvertes.fr/hal-00294222/en/, I show
that the centered Kaucher form has formally the same expression as the
Hansen-Sengupta operator after formal rewriting.
- In several other papers, I use classical intervals to certify inner
approximations of universally quantified constraints (see e.g.
http://dx.doi.org/10.1007/s10601-008-9053-0). For this application,
the classical approach is more efficient and clearer (in my opinion).
2- Solving interval equations.
By this, I mean really find an interval that solves an interval
equation. Let me provide two example for people not used to Kaucher
intervals: Find an interval x such that x+[0,1]=[-2,2]. The interval
solution is x=[-2,1], which can be computed automatically using the
additive group properties of Kaucher arthmetic:
x=[-2,2]-dual[0,1]=[-2,2]-[1,0]=[-2,1]. An other example: Find an
interval x such that x+[-2,2]=[0,1]. Using the same group operations
x=[0,1]-dual[-2,2]=[0,1]-[2,-2]=[2,-1]. This latter result proves that
this interval equation has no proper (nonnegative radius) solution,
but only an improper (negative radius) one.
This is quite attractive, but, on the one hand, this application is
not upon applications advertised for including Kaucher arithmetic in
the standard. On the other hand, the interpretation of the solutions
to interval equations follows the quantified interpretations presented
in the first point: Linear interval equations can be pretty well
solved, but lake applications. Nonlinear interval equations can be
solved nicely for some academic example, but not anymore in general
(like nonlinear real equations). Anyway, classical intervals
algorithms should be able to provide the very same interpretations.
Kind regards,
Alexandre
On Sun, Jul 8, 2012 at 2:15 PM, Svetoslav Markov <smarkov@xxxxxxxxxx> wrote:
>
> Dear George,
>
>> What are Kaucher/modal IA useful for?
>
> This is a strange question, but let me take it seriously.
>
> To answer this question the analogy with negative numbers
> comes again into work.
>
> Negative numbers open a whole new world of problems
> to be solved. They also open a world of new basic theories,
> like linear algebra, rings etc. From practical point of
> view these tools induce new semantic. Do I have to give
> examples? All sciences give examples of the interpretation
> of negative numbers.
>
> Similarly, Kaucher/modal intervals open a new world of problems
> and new basic tools. Also, new semantic comes into play, namely
> the so-called modal sematic. The latter allows to formulate
> problems involving inclusions of sets, by means of logical quantors
> (exists-for all). These problems are then translated into algebraic
> problems using kaucher intervals. Then kaucher arithmetic is
> applied to solve the problems.
>
> There exists already a vast literature on the application of
> Kaucher/modal arithmetic. Sergei Shary and Alexandre Goldsztejn
> are just two names to be mentioned.
>
> Best regards,
>
> Svetoslav
>
>
>
> On 8 Jul 2012 at 11:31, Corliss, George wrote:
>
> From: "Corliss, George" <george.corliss@xxxxxxxxxxxxx>
> To: Svetoslav Markov <smarkov@xxxxxxxxxx>
> Copies to: "Corliss, George" <george.corliss@xxxxxxxxxxxxx>,
> stds-1788 <stds-1788@xxxxxxxxxxxxxxxxx>
> Subject: Re: Motion on interval flavors
> Date sent: Sun, 8 Jul 2012 11:31:49 +0000
>
>> Negative numbers may be useful for describing the state of my bank account. Also in 18th century England, it was common to throw debtors into prison, so the authorities clearly understood the concept of negative net worth.
>>
>> What are Kaucher/modal IA useful for?
>>
>> George Corliss
>> George.Corliss@xxxxxxxxxxxxx
>>
>>
>>
>> On Jul 8, 2012, at 5:52 AM, Svetoslav Markov wrote:
>>
>> > Dear prof Kulisch,
>> >
>> > I am quite puzzled by your attitude to Kaucher/modal IA.
>> >
>> > It seems to me that you, and probably other
>> > participants in this forum, are not aware
>> > with the history of the acceptance of negative numbers!
>> >
>> > I make the analogy with negative numbers because
>> > improper (Kaucher/modal) intervals are just
>> > like negative numbers (as their radii are negative).
>> >
>> > There is a lot in internet about the history of the acceptance
>> > of negative numbers, see e. g.:
>> >
>> > http://en.wikipedia.org/wiki/Negative_number
>> > (section "history")
>> >
>> > This section finishes like this:
>> >
>> > In A.D. 1759, Francis Maseres, an English mathematician, wrote that negative
>> > numbers "darken the very whole doctrines of the equations and make dark of the
>> > things which are in their nature excessively obvious and simple". He came to
>> > the conclusion that negative numbers were nonsensical.[14]
>> >
>> > In the 18th century it was common practice to ignore any negative results
>> > derived from equations, on the assumption that they were meaningless.[15]
>> >
>> > Are we living in the 18th century? Indeed, negative numbers were not
>> > real numbers before mathematicians accustomed to use them.
>> >
>> > Regards,
>> >
>> > Svetoslav
>> > PS. Here are some other links to "The History of Negative Numbers":
>> >
>> > http://nrich.maths.org/5961
>> >
>> > http://logica.ugent.be/albrecht/thesis/HPM2008.pdf
>> >
>> > http://www.ma.utexas.edu/users/mks/326K/Negnos.html
>> >
>> > http://www.basic-mathematics.com/history-of-negative-numbers.html
>> >
>> > http://logica.ugent.be/centrum/preprints/Numberline-ScienceAndEducation-rev.pdf
>> >
>> >
>> >
>> > A BBC film:
>> >
>> > http://www.bbc.co.uk/iplayer/episode/p003hyd9/In_Our_Time_Negative_Numbers/
>> >
>> >
>> >
>> > On 6 Jul 2012 at 15:56, Ulrich Kulisch wrote:
>> >
>> > Date sent: Fri, 6 Jul 2012 15:56:50 +0200
>> > From: Ulrich Kulisch <ulrich.kulisch@xxxxxxx>
>> > To: John Pryce <prycejd1@xxxxxxxxxxxxx>
>> > Copies to: stds-1788 <stds-1788@xxxxxxxxxxxxxxxxx>
>> > Subject: Re: Motion on interval flavors
>> >
>> >> Am 20.06.2012 07:49, schrieb John Pryce:
>> >>> I'm not very happy with the name "set-based" intervals, but calling them
>> >>> "standard" intervals is no longer appropriate, as well as clashing with the
>> >>> main meaning we give to "standard" (as a noun and also an adjective). Any
>> >>> better name welcomed.
>> >>>
>> >>> Regards
>> >>>
>> >>> John Pryce
>> >>>
>> >>
>> >> I see two kinds of set-based intervals, the closed and bounded intervals
>> >> of IR and the closed intervals of \overline{IR}. What about calling the
>> >> first cb-intervals and the second c-intervals.
>> >>
>> >> Kaucher or modal intervals are not really intervals. They are abstract
>> >> entities. Why not calling them i-intervals for inverse or improper
>> >> intervals or m-intervals.
>> >>
>> >> Best regards
>> >> Ulrich
>> >>
>> >> --
>> >> Karlsruher Institut f"ur Technologie (KIT)
>> >> Institut f"ur Angewandte und Numerische Mathematik
>> >> D-76128 Karlsruhe, Germany
>> >> Prof. Ulrich Kulisch
>> >>
>> >> Telefon: +49 721 608-42680
>> >> Fax: +49 721 608-46679
>> >> E-Mail: ulrich.kulisch@xxxxxxx
>> >> www.kit.edu
>> >> www.math.kit.edu/ianm2/~kulisch/
>> >>
>> >> KIT - Universit"at des Landes Baden-W"urttemberg
>> >> und nationales Grossforschungszentrum in der
>> >> Helmholtz-Gesellschaft
>> >
>> >
--
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