Motion 0035 -- New levels 1 and 1a
The proposed new Level 1 is clear: plain Moore intervals. That's not
what earlier motions decided to use, but then new motions can override
old ones. This new Level 1 may solve some issues (e.g. midpoint, and
mathematical exploitability of a regular monoid), but it does remove
unbounded intervals, and hence applicability to mathematical constraint
propagation problems. (I say "mathematical" to counter the argument
that physical constraints are never truly unbounded. I believe that
Interval Analysis *is* being used in its mathematical sense by people
such as Prof. Nicely, who gained fame for exposing the Pentium FDIV bug.)
Level 1a raises a different issue in my mind: its domain is a two-sorted
domain, where functions return either an interval or a set of intervals,
and the same applies to function arguments. This surely complicates the
definition of interval extension of a point function, even if we have
precise rules for the primitive arithmetic operations. I suppose the
domain could be made single-sorted by defining a non-owerflowed interval
to be a singleton family, consisting of exactly one bounded interval.
The empty interval could then be the empty family.
It really gets interesting when we consider Union and Intersection.
If we use the one-sorted convention mentioned above, the intersection
of a bounded and an overflowed interval would always be empty -- but
that is not what we want. If the lower bound of the overflowed interval
is interior to the bounded interval, we want a result that is a different
interval -- but Intersection does not create new elements, it selects a
subset of the given elements (in this case, family members, i.e. bounded
Level 1 intervals).
If on the other hand we use the two-sorted domain, we still get empty
intersections when one argument is a bounded interval (set of numbers)
and the other is an overflowed interval (set of bounded intervals).
Basically, to be useful, Intersection would have to be redefined in
a case-by-case manner, just as plain Union had to be replaced by the
Interval Hull -- though this too now needs a careful redefinition.
None of these complications have been addressed in the "Overflow"
paper, so I consider the motion to be essentially incomplete.
Michel.
P.S. The easiest way to get a one-sorted domain at Level 1a would be
to reduce an overflowed interval to the Union of its members --
but then we'd be right back with our present Level 1 that includes
unbounded intervals!
For Intersection a simple rule could be to replace an overflowed
argument with its smallest element. The point however is that
each operation will need a different way to interpret overflowed
arguments, which greatly complicates the presentation. Indeed,
this rule only applies to Intersection with a bounded interval.
So here we are: the rules for interpreting argument2 depend on
what argument1 is: not very nice!
---Sent: 2012-05-06 17:45:27 UTC