the "set paradigm" is harmful
I agree with Arnold on this is one, but would like to elaborate
on the classification:
> What are meaningful definitions of approximate numbers?
>
> 1. A number known to lie between to known numbers.
> This gives traditional interval arithmetic.
>
> 2. A number known to deviate from a given number by
> at most a given amount. This is equivalent to a
> special case of 1., special since it does not
> cater for unbounded intervals.
>
>
> 3. Probabilistic versions of 1. or 2.
>
> It is clear that we should not consider option 3.
It seems to me that the concept of "approximate numbers"
does carry probabilistic connotations, and midpoint-radius
is an appropriate representation only if the distribution
of values is reasonably centred and can thus be approximated
by an implied Gaussian distribution. In that interpretation,
there is no containment guarantee -- it is sort of like a
quantum well. Some operations can skew the distribution so
badly that the concept breaks down, and that happens not only
when the "interval" contains a singularity, but also when it
is near a singularity. So "approximate number arithmetic"
should throw an exception when the result distribution is
excessively skewed -- or, in a practical implementation, when
the radius exceeds either an absolute or a relative threshold.
(Those thresholds would be part of the local environment.)
When those conditions are satisfied, it does indeed become
practical to use a different format for the radius than for
the midpoint.
All things considered, "approximate numbers" form a different
structure. They raise different concerns (e.g. there are two
forms of multiplication, and there can be arithmetic exceptions).
So they should not even be called "intervals". They can of course
be converted to intervals (but not necessarily in the obvious way,
because of the possible lack of an implied containment guarantee),
and narrow intervals can be converted to approximate numbers.
Plain Interval Arithmetic is probably a useful way to support an
"Approximate Number" application, and it might be interesting to
find out what assist functions might be useful -- e.g. an efficient
test for error bounds. The Vienna Proposal already supports the
necessary conversion operations.
Michel.
Sent: 2009-02-09 13:59:07 UTC