Vote on Motion 7
As acknowledged in Motion 7, no IA operation can have NaI as result.
Do we need NaI at all?
The only reason for considering Not An Interval is that intervals can have
one or more finite floating-point numbers as bounds and that there are bit
patterns in IEEE754 that do not denote any real or any of the infinities.
We should first have considered the question: should we inherit in this
way all aspects of IEEE754 in its current form?
Sometimes the idea is floated that ultimately the standards for
floating-point arithmetic and for interval arithmetic should merge. Such a
merger could be more than a smoothed-over version of the union of the two:
interval arithmetic can be used to clean up IEEE754.
The most valuable feature of interval arithmetic is that, as an algebraic
domain, it is closed under the arithmetic operations. The most valuable
contribution of interval arithmetic to IEEE754 can be to interpret
floating-point numbers as point intervals. The result of a floating-point
operation is then one of the bounds (which bound depends on the rounding
mode) of the interval resulting according to interval arithmetic. This is
worked out in a paper by Edmonson and myself
(http://arxiv.org/abs/0810.4196).
While this may appear outrageous to the floating-point community and seem
utopian to the hard-headed realists among us, we can at least leave open
this possibility and allow only floating-numbers in the representation of
intervals, whatever representations we will later decide to include in the
standard. There is plenty of scope for the use of non-Nans to represent
the empty interval.
According to the above considerations, my vote on motion 7 is NO. As for a
motion that I would support, I am at a loss how to word it.
"No interval shall be Not An Interval" seems tautologous. Perhaps the
existential quandary can be avoided by something more legalistic: "The
IEEE754 class of bit patterns named NaN shall not be used in any interval
representation."
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