Re: Motion 31 draft text V04.4, extra notes

[Vincent Lefevre <vincent@xxxxxxxxxx>]

On 2012-04-16 10:45:06 -0500, Nate Hayes wrote:
> >>>>Who cares if I can do 1/[0,1]=[1,+Inf] at Level 1 if I can't assume
> >>>>   A + X = B + X
> >>>>means A = B!!
> >>>>
> >>>>That is not an arithmetic I want to work with.
> >>>
> >>>If you do not work with unbounded intervals, you still have
> >>> A + X = B + X ==> A = B
> >>>
> >>>I don't see any problem.
> >
> >And I would add the the above does not work with families of intervals
> >such as [1,+OVR]. So, you are not solving anything by replacing
> >unbounded intervals by families of intervals.
> It appears you are changing the subject, since I was speaking about
> Level 1 and now you are talking about Level 2.

I'm not changing the subject: Level 1 and Level 2 are closely related
(Level 1 exists to serve as a model for Level 2, for formalization),
Level 2 has been mentioned many times, and it is you who have said:

| So why include unbounded intervals in the Level 1 model? Especially
| when a similar treatment at Level 2 of "overflown" intervals can
| provide the same practical benefits?

Since you favor Level 2 for things related to unbounded intervals
(like 1 / [0,1]), it is quite natural to talk about this level.

> >>Well, how am I to not work with unbounded intervals at Level 1 if the
> >>standard requires they may be there?
> >
> >Since your system works only with bounded intervals, you must have
> >some hypotheses that guaranty that unbounded intervals will never
> >occur (e.g. if you consider 1 / X, you need to know that X doesn't
> >contain 0). So, you can just ignore unbounded intervals globally.
> It appears you are changing the subject again, for the same reasons
> mentioned above.
> 
> Anyhow, at Level 2 there is no practical difference between Inf/OVR in the
> interval arithmetics:
> Decorations and compressed intervals can make the distinction that you
> mention in both cases. For example, the full decorated result of 1 / [0,1]
> may be
>    ([1,+OVR],somewhereDefined)
> and with compressed intervals the result may be either the bare interval
>    [1,+OVR]
> or the bare decoration
>    somewhereDefined.
> So nothing is lost over the current P1788 model, i.e., what you're talking
> about isn't any argument in favor of unbounded intervals or against family
> of intervals (in fact, its not really relevant at all).

Since there is no practical difference, between Inf and OVR, I don't
see why you are complaining about unbounded intervals.

> >>I can't ignore anything the standard mandates unless I don't follow the
> >>standard.
> >
> >The standard mandates that unbounded intervals are supported by the
> >implementation, not that they will occur in the user application.
> It mandates they are in the Level 1 math equations, which are then no longer
> cancellative.

Level 1 will never occur in the implementation of the standard. On your
side, you can use whatever you like for math equations.

> >>Development of Kaucher arithmetic *requires* cancellation property!!!
> >
> >The standard is not about the Kaucher arithmetic anyway.
> That's news to me. The name of P1788 is "Standard for Interval
> Arithmetic"... and Kaucher arithmetic is "interval arithmetic".

It is a *particular* interval arithmetic, actually an extension of
conventional interval arithmetic. P1788 doesn't contain anything
about an implementation of Kaucher arithmetic (at least currently).

> >>>BTW, cancellation is invalid at Level 2.
> >>I already pointed that out just the other day.
> >
> >+ the fact that you do not want to work at Level 1, this makes it
> >useless.
> I have never said I don't want to work at Level 1. I've said I want to work
> in a Level 1 model that is cancellative. Please stop putting words in my
> mouth.

I recall what you've said:

| So why include unbounded intervals in the Level 1 model? Especially
| when a similar treatment at Level 2 of "overflown" intervals can
| provide the same practical benefits?

Basically this means: do not use Level 1, use Level 2.

-- 
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
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Work: CR INRIA - computer arithmetic / AriC project (LIP, ENS-Lyon)