Re: Revised version of Level 1 text (draft)

[Dominique Lohez <dominique.lohez@xxxxxxx>]

John Pryce a écrit :

Arnold, Nate, other discussers, and all of P1788

Note to the whole group:
I'm getting back to serious P1788 work after a break, rewriting my v03.1 draft text. Though P1788 as a whole has been relatively quiet, there has been much discussion among a group of about ten people, largely about the decoration system. I personally believe we have about 80% agreement on the main theoretical (Level 1) issues that underlie decorations. There are still important disagreements; only time will tell if full consensus can be found. 

Several of this gang of 10 have felt it is time to circulate to the whole group, so I am doing so, but slightly worried lest the issue discussed be seen as "angels dancing on the head of a pin" stuff. It is a technical issue, but a significant one, and such things have to be sorted out. Namely

   How should the intersection and union (= interval hull) operations handle decorated intervals?

The gist (see Arnold's 5 Dec 2010 paper, first few paras of 22 Jan revised version of §6, quoted below) is:
   These are not interval extensions of point functions, so one needs
  

This statement is clearly wrong for intersection, it is questionable for union.
Since  any real number   a is embedded into the set of interval as [a,a].
the intersection of real numbers can be defined by

intersection (x, y ) = x if x= y
Such a function is undefined if x# y exactly is the same way that sqrt(x) is undefined for x=-2
It can be useful   to follow    the notation  of    John's draft by explicitly introducing the NaN for the undefined real number
and furthermore replacing NaN  by the Empty interval which is obviously not a number.

intersection (x,y) =   {   x if x=y
                                 {   Empty otherwise

The problem with such a function lies in the fact it a function of two variables x and y  but it is only defined along the line x=y
so  for interval decoration(intersection(X, Y) ) is somewhere  undefined. It is only safe  or everywhere defined  if X and Y are real numbers.
So the decoration of the input interval are lost.


For union a similar discussion can be carried out. However NaN  ie  Empty must be considered as a valid input

We have

union(a,b )  {  Empty if a = Empty  and b = Empty
                       [a,a]  if    b= Empty
                       [b,b]   if  a = Empty
                      [min(a,b),max(a,b) ] otherwise

Now it must be noticed that the union of two real numbers is always an interval
Such a situation has already been encountered with the generalized power function ( Motion 10.02)

This mean that the union of two interval is a set of intervals
Such a set can be summarized as usual  by  a single interval in which all  the intervals of the set are included. However a big amount of information is then lost:
          Calculating A union    B  it is found that
                    If A and are not overlapping   all the interval in the set include the gap between   A and B
                   If A and are overlapping   all the interval in the set overlap the intersection of   A and B

At least this additional information should be captured by new decoration.
Where a gap exists the resulting function cannot be safe since otherwise the generalized power would be safe, and it is not.
Where a gap does not exist additional   discussion is required.


An alternative would be to consider that set of   intervals are   useful object   ...


Best regards

Dominique

           

-- 
Dr Dominique LOHEZ
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