RE: [boost] Formal Review for Interval Library beginning

From: David Bergman (davidb_at_[hidden])
Date: 2002-09-04 12:42:48

Doug,

An interval is not necessarily the same as an uncertainty interval
(being arithmetic or not...)

So, interval arithmetic is not probabilistic arithmetics, although the
latter arithmetics would be both useful and fun.

As I wrote in a reply to Peter's mail, his suggestion is not even an
equivalence relation, since it lacks reflexivity, e.g., [1 3] == [1, 3]
does not hold... 

In a probabilistic arithmetic, one would need (as suggested by someone
earlier in this thread) a distribution description, such as
"interval<double, guassian_distribution>". In such a beast, with the
"extremely certain" philosophy, we could only state that an interval is
equivalent to another one if they contain one single coinciding point,
such as [1, 1] == [1, 1].

But, then again, that is not what we are dealing with here. We are
dealing with real tangible goods: intervals, quite easily defined, and
compared. [a, b] == [c, d] iff a == c and b == d.

/David

-----Original Message-----
From: boost-bounces_at_[hidden]
[mailto:boost-bounces_at_[hidden]] On Behalf Of Douglas Gregor
Sent: Wednesday, September 04, 2002 8:43 AM
To: boost_at_[hidden]
Subject: Re: [boost] Formal Review for Interval Library beginning

On Wednesday 04 September 2002 06:50 am, Peter Dimov wrote:
> Natural equivalence for intervals? I don't see it.
>
> What I see as "natural" (regarding interval arithmetic, and not 
> intervals as objects) is:
>
> I relop J :- foreach(x in I, y in J) x relop y.

This brings to mind a small question: how should I think of a variable
'x' 
like this?
  interval<T> x(a, b);

Is 'x' every value in the interval [a,b]? Or is 'x' some value that is
bounded 
by the interval [a,b] (but our information isn't exact enough to get the
true 
value of x)? I know that the latter is the more useful interpretation
for 
static analysis, but it also seems to be the interpretation used in
interval 
arithmetic as the library was intended for (the true result is some
value 
bounded by the interval, and the interval bounds assure that the true
value 
doesn't fall out of the interval because of rounding).

If we were to agree on the second interpretation, then Peter's semantics
seem 
to be the only semantics that make sense to us non-mathematicians :)

        Doug

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