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

[Vincent Lefevre <vincent@xxxxxxxxxx>]

On 2012-04-11 10:03:43 -0500, Nate Hayes wrote:
> Vincent Lefevre wrote:
> >On 2012-04-11 08:46:42 -0500, Nate Hayes wrote:
[...]
> >>We don't actually perform arithmetic in a computer at Level 1, and at
> >>Level 2 with overflow one has
> >>   1 / [0,1] = [1,+OVR]
> >
> >This is wrong. At Level 1, 1 / [0,1] is defined as the smallest
> >(closed) interval that contains { 1 / x | x in [0,1] and x <> 0 },
> >and this is [1,+oo]. At Level 2, the interval must contain [1,+oo]
> >entirely. If [1,+OVR] means some interval [1,K] with K finite (but
> >we don't which one), the result is incorrect, because for any K,
> >[1,K] does not contain [1,+oo].
> 
> In my recent e-mails I've said that the interval [1,+OVR] should be
> considered as a family of intervals: the number of elements in the
> family is inifinite, but each element is closed and bounded.

So, basically, you get something that is equivalent to an unbounded
interval, but with a more complex definition!

> >>When restricted to bounded intervals, the Level 1 arithmetic is
> >>closed and cancellative for addition, subtraction, multiplication
> >>and division with 0 not in the denominator.
> >
> >This is not the definition of a closed arithmetic. If you can get
> >an interval as a result, say [0,1], you mustn't remove it from the
> >possible inputs of an operation.
> 
> At Level 2 its not.

Do you mean that you have an operation at Level 2 with no
corresponding operation at Level 1? This doesn't make sense.

> >>This is the oldest interval arithmetic of Ramon Moore, etc. and has
> >>withstood the test of time already. IMO this is better than the
> >>current P1788 model with unbounded intervals.
> >
> >P1788 needs a closed arithmetic.
> 
> The practice of interval arithmetic has thrived for over 50 years based on
> Moore's Level 1 model. Where is the evidence to support your claim to the
> contrary?
> 
> BTW, where is your answer to my original question about a concrete example?

I've given an example in another mail.

> All your points seem really academic to me.

No, really, I need to compute ranges of functions (that can be defined
by arbitrary expressions), and I need a correct answer.

> The arithmetic can be closed at Level 2, anyhow, with overflown intervals.

My point is that it must be closed at Level 1 too.

> For me, it is more important that the Level 1 arithmetic is cancellative,

It is cancellative on all the instances allowed by your model.
So, you do not miss anything.

> and the current P1788 model with unbounded intervals does not have this
> property.

Whatever model you choose, there are properties that will not be
satisfied.

> Unbounded intervals are unnecessary.

for you.

-- 
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
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