Re: Motion on interval flavors
Dear Alexandre,
I agree with you. I also beleive that, in principle,
Kaucher/modal intervals can be avoided.
However, the analogy with negative numbers
comes again into play. Everything that is formulated
using negative numbers can be reformulated without
using negative numbers. Your arguments are similar
to the arguments of those prominent mathematicians
who up to the 18th century refused to work with negative
numbers. I was trying to explain this with the paper that I
attached a few days ago. One can work in an additive
monoid instead of a group, unless one uses the additional
operation |a-b| . For the case of interval arithmetic I developed
a so-called "extended interval arithmetic" which uses the same
basic idea. Instead of working in group structures one works
in monoid structure introducing and using new (so-called inner)
operations.
The historical lesson of the negative numbers teaches us
that the correct way is to work in group structures
whenever possible. Kaucher/modal intervals provide
such structures. I agree that these structures involve improper
elements that are dificult to be swallowed. However, the situation
is exactly as with negative numbers. We actually only use the
algebraic properties of the group system(s), such as existence of
opposite elements and simple distributivity law. The actual computation
and interpretation is based on positive numbers. In the case of numbers
we use the sign-magnitude form when computing and interprating
result, thereby giving various interpretation of the sign (like dept in
the example of George Corliss). As you know, improper Kaucher
intervals have meaningfull interpretation in terms of proper intervals.
Note also that the applications of negative numbers increased
only when mathematicians started to use them systematically and got
used to the (group-like) properties of real numbers. The history
teaches us that soon or later the same will happen with Kaucher intervals.
Now my conclusion and my question:
Conclusion: Kaucher (improper) intervals can be avoided,
real (negative) numbers also.
Question: what for are then real (negative) numbers?
Those who can answer this question can understand what for
are Kaucher intervals.
Best regards,
Svetoslav
On 8 Jul 2012 at 15:18, Alexandre Goldsztejn wrote:
Date sent: Sun, 8 Jul 2012 15:18:24 +0200
Subject: Re: (Fwd) Re: Motion on interval flavors
From: Alexandre Goldsztejn <alexandre.goldsztejn@xxxxxxxxx>
To: Svetoslav Markov <smarkov@xxxxxxxxxx>
Copies to: stds-1788 <stds-1788@xxxxxxxxxxxxxxxxx>
> 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