# [om] A Proposal for extending OpenMath with structure sharing

jhd at cs.bath.ac.uk jhd at cs.bath.ac.uk
Fri Apr 5 14:22:02 CEST 2002

```On Thu, 4 Apr 2002, Bruce Miller wrote:
> It seems a couple of issues have become conflated here.
>
> 1) Richard may be right that there is only one
> official definition of "interval", namely
>
>     a) X=[a,b] : all x such that a<=x<=b  (for closed)
>
> But, "interval arithmetic" can mean a couple of different
> things. In a straightforward "Arithmetic on Intevals",
> I can't understand why X-X => 0; What should "identity"
> mean in such an arithmetic?
>
> OTOH, many (most?) people using interval arithmetic,
> eg. for error analysis, are interested in using
> an interval to represent
>    "An unknown x that is in the interval X".
> It's impossible to compute the tightest bounds on a
> result without knowing the history or correlations of
> the arguments.  So such systems overestimate the bounds,
> but may use some simple rules like
>     x-x => 0 or x/x => 1,
> where "identity" does mean something.  Those rules would
> seem inappropriate for the first kind of interval arithmetic,
> however.
>
> In summary, I would say that -- eventually -- OM's interval
> CD should make clear which model is intended, or provide
> CDs for both concepts (or alternatively, different CD's
> for different arithmetics on intervals, or ....
I wasn't referring particularly to the current CD, and I expect that a
second CD to support interval arithmetic is the way to go.
> Some clarification, somewhere).  The current cd just
> says "interval", and I'm suspecting we're not all thinking
I personally do not believe that the current interval CD is about interval
arithmetic in that sense.
> the same thing.
>
The best example I think is ABbott's, which Strotmann posted: in the
formula for a cubic, there are two occurrences of sqrt(-4a^3-27b^2). It
doesn't matter which you use AS LONG AS YOU'RE CONSISTENT.

James
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