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Re: Motion 52: final "Expressions" text for vote

Bill, P-1788,

I'm confused at your objections.  Isn't this at the
heart of extended arithmetic, impacted by decisions such
as whether intervals are subsets of the reals or extended
reals, and guiding developments such as decorations?  Yes,
there have been various extended real systems developed,
and there has been underlying theory for each of them.  For instance,
the published C-set theory, actually commissioned by you
when you were at Sun, is applicable both to when the underlying
number system is the reals and the extended reals.  Developers
of the various systems have been active in the working group,
and their input has guided what we presently have as the
standards text.  If you look back several years, you will
see in the record discussion of precisely those functions
you list in your email.  I think the answers to your questions
are contained in those discussions, but I would welcome
clarification of your objections otherwise.

In short, my perception is that we have considered this, that
there is underlying theory (more than one), and that the
working group has taken this into account in formulating
the document.

Baker

On 11/23/2013 01:14 PM, G. William (Bill) Walster wrote:

Vladik,

The difficulty is that a given interval can contain values that are
outside the natural domain of definition of a given function or
operation.  For example what is the smallest set of intervals that must
be included in the result of the following operations on the interval X
= [-1, +1]:

1/X
1/(1/(x))
ln(X)
1/ln(X)
X/X
X/X - X/X

The answers have nothing whatever to do with implementation or a
standard for it.  The answers do depend on the definition of the
universal set of intervals and the real numbers contained therein and
the natural domain of any given expression and/or function. For example,
in addition to real numbers in the open interval (-oo, +oo), shall
intervals include {-oo, +oo} and/or oo = 1/0 and their reciprocals?  If
so, how?

Shall there be an empty interval that is not the same as the empty set
because the empty interval is not a subset of every interval? If so,
when must the empty interval be included in the result of an interval
expression evaluation to prevent a containment failure?

Are there any conditions under which X - X = 0 for given non-degenerate
interval variables X?

Cheers,

Bill


On 11/22/13 8:04 PM, Kreinovich, Vladik wrote:

For intervals proper, the defiition is the same always,
f([x1],...,[xn]) is the smallest interval that contains the range of
the given function f(x1,...,xn) on given intervals [x1], ..., [xn],
i.e., the set of all the values
{f(x1,...,xn):x1 is in [x1], ..., xn is in [xn]}.

The questions arise when we want to extend this to situations when
[xi] are not intervals

(there are also definitions for decorations)

________________________________________
From: Bill Walster [billwalster@xxxxxxxxx] on behalf of G. William
(Bill) Walster [bill@xxxxxxxxxxx]
Sent: Friday, November 22, 2013 2:31 PM
To: Kreinovich, Vladik; Michel Hack; stds-1788
Subject: Re: Motion 52: final "Expressions" text for vote

Vladik,

Then please point me to the purely mathematical foundation for computing
with "well defined" intervals.

Cheers,

Bill





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R. 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
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