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

sal at dcs.st-and.ac.uk sal at dcs.st-and.ac.uk
Mon Apr 1 14:47:29 CEST 2002

```> I assume that JHD mean the interval from -1 to 1  by [-1,1].
> Then JHD asserts that if x= [-1,1],  then  x-x is zero, while
> [-1,1]- [-1,1]  is [-2,2].
> I think that x-x pretty much has to be [-2,2],  which of course includes
> one
> value of  0.
> Consider the extension of JHD's rule yielding 0:  p(x) where p is a
> program
> would not be the result of evaluating each operation as an interval
> operation,
> but the value [a,b] where a= min(p(x),x in [-1,1]), b = max ....
> Now this would be very neat, but might not be computable.
>

I think this tells us that there are two notions of interval needed here. One
trully represents the whole set, and the x= [-1,1], x-x  should be [-2,2]. The
other represents an unknown value in the set, and then x = [-1,1], x-x should
be 0.

Regarding sharing, I can see no possible problem with a  "syntactic" sharing,
wherte the OM tree is the unshared one, but repeated, syntactically identical
subtrees are merged for economy of space or bandwidth.  On the other hand,
gzip will more-or-less do this for you anyway.

A "sematic" sharing, creating an OM DAG is a little bit tricky, because
existing CDs have not been written with it in mind, so, for instance, we are
not clear which kind of intervals we have. Since we don't have a notion of
evaluation at OM level at all, the problem passes down to individual symbols,
and we would have to go through our whole collection of such checking for any
issues. That would not be a huge job. but it would be a job.

Steve

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