Re: Motion P1788/M0005.02_Table_of_operations vote : NO

[Ulrich Kulisch <Ulrich.Kulisch@xxxxxxxxxxx>]

Dear Dominique LOHEZ:

To 1): I attach a short note entitled: "Mathematical Concepts of 
Computer Arithmetic". With respect to set inclusion as an order relation 
\overline(IR) is a complete lattice. The least element is the empty set 
and the greatest element is the set R = (-oo, +oo). According to the 
attached note \overline(IF) is a raster of \overline(IR). The least and 
greatest elements of \overline(IF) are the same as in \overline(IR), 
i.e., the empty set is the least element and the set R = (-oo, +oo) is 
the greatest element of \overline(IF) also.
I agree with you that everybody who is familiar with these concepts does 
not need mentioning the empty set in the definition of \overline(IR) and 
\overline(IF) in the denotations of Motion 5. But it is not wrong and 
for one or the other it may be helpful.

To 2): I have a certain sympathy with this. If you look into my position 
paper "Complete Interval Arithmetic and its Implementation on the 
Computer", published in the Dagstuhl proceedings  (on page 19) the 
operations are defined as you require. However, after discussing the 
matter with many colleagues I got convinced that it is better to define 
the operations as done in Motion 5. Then the property of conventional 
interval arithmetic:  A/B =  A×(1/B) is generally preserved for 
intervals. If B = [0, 0] then 1/B = empty set and A×(1/B) = empty set also.
Note that on the lower part of page 3 of Motion 5 (line 14 from below) a 
second division is defined which delivers the result (-oo, +oo) as you 
require.

I very much hope that you can change your vote into YES now. We still 
need a number of YES votes.

With best regards
Ulrich Kulisch



Dominique Lohez schrieb:
> My vote is NO
>
> I would vote YES  if
>          1)   In the denotation List  of denotations
>                  The reference to the empty set as an element of the 
> extended set of intervals is removed
>          2) In the division table  for  when 0 is an element of the 
> interval b
>               The second colun  has the following content
>                             header : The interval     [0,0] is 
> replaced by any interval  including 0
>                             result: On must always be (-infinity, 
> +infinity)
>                    Rationale
>
> While the function f :R-> R f(x)=1/x is not defined for x =0 , and no 
> extension is possible
> When working with intervals   such extensions are possible leading to 
> the results indicated above
>
>
> Dominique

Attachment: BasicMathConc.pdf
Description: Adobe PDF document