Re: Motion 3
Yes, Juergen, I understand, and I am satisfied
with your answer. I hope everybody understands
you correctly.
It should be made clear that: intervals (as closed
and connected sets of real numbers) form an additive
monoid which is naturally extended to an aditive
group by considering pairs of intervals (as new
"intervals". This is a standard procedure in mathematics,
in particular used in the definition of numbers.
The rest is a question of representation, as you write:
-- modal intervals (a set of reals and a predicate) or
-- directed intervals (a set of reals and a direction);
and I would add:
-- Kaucher intervals - in the narrow sense (a pair of
real numbers; both in infsup or midrad presentation).
My suggestion is that we accept the term "standard
interval" in this algebraic sense, arrising naturally
from the the basic notion (closed and connected sets
of real numbers). This is the simplest that can be done.
For a rationale, see my scan06 paper (not published).
Ofcourse, there are some problems to be setled
(like the one that narrow intervals in mid-rad presentation
are just finite sets, etc) but the important thing is to
decide what do we mean by "interval". That is why
IMO motion 3 is very important.
BTW, it is somewhat confusing to speak about
"semantic of intervals" as this semantics depends
on applications. We should concentrate on the
algebra (algebraic properties) of intervals.
Regards,
Svetoslav
On 30 Mar 2009 at 20:13, J"urgen Wolff v Gudenberg wrote:
> Dear Svetoslav,
> I do not intend to block extensions such as modal intervals (a set of
> reals and a predicate) or directed intervals (a set of reals and a
> direction). You may also define an extension where you use a pair of
> intervals.
> What I do want, however, is to keep the standards' (P1788) definition of
> a (standard) interval clear and simple.
> regards
> Juergen
>
> Svetoslav Markov schrieb:
> >
> > Dear Juergen,
> >
> > before voting Motion 3 I want to ask you:
> >
> > Does your interpretation/definition of "interval"
> > excludes the possibility to consider
> > a pair of two "intervals" again as an "interval"?
> >
> > I ask this question because as
> > we know the natural way to introduce
> > negative numbers in mathematics is
> > to define them as pairs of positive numbers.
> > This is natural because positive numbers
> > do not admit inverse additive.
> >
> > The situation is same with intervals, and the natural
> > way to interval arithmetic is to consider pairs of
> > intervals again as intervals. Will the standard
> > prohibit such a semantic?
> >
> > As far as I remember such is the semantic involved
> > in your paper:
> >
> > Wolff v. Gudenberg, J., Determination of Minimum Sets
> > of the Set of Zeros of a Function}, Computing 24, 1980, 203--212).
> >
> > Regards,
> >
> > Svetoslav
> >
> >
> > ====================
> >
> >
> > > Wolff v. Gudenberg :
> > > Dear members of the working group.
> > >Please note the following motion concerning the semantics of intervals
> > >So it is not about structure and procedure, not to fiX the final
> > > formulation, but about contents
> > >Comments welcome
> > >Juergen
> >
> >>>
> >>> ===Motion P1788/M0003.01_Set_of_reals===
> >>> Proposer: Juergen Wolff von Gudenberg
> >>> Seconder: required
> >>>
> >>> ===Motion text===
> >>> The P1788 Interval arithmetic standard defines intervals as closed
> >>> and connected sets of real numbers.
> >>> That means that +/- Infinity may be used to denote an unbounded
> >>> interval but are never considered as members of an interval.
> >>>
> >>> ===Rationale===
> >>> The Vienna Proposal section 1.2
> >>> Kulisch's position paper "Complete Interval Arithmetic"
> >>> The extensive discussion in the mailing list
> >>>
> >
>