Motion 35 -- New levels 1 and 1a (amendment)

["Nate Hayes" <nh@xxxxxxxxxxxxxxxxx>]

P1788,

We submit the attached motion text and supporting position paper for
Motion 35 as a friendly amendment. Most of the position paper has been 
re-written for purposes of clarification.

Thanks to Alexandre, Arnold, Michel, John and others.

Nate

P.S. John, we answer your question about comparison operations in the
rationale.


As described in the accompanying position paper, P1788 shall change the
existing Level 1 and Level 2 model to the three-tiered level structure as
described in Section 2. Specifically, this means the following:

   -- The "mathematical intervals" at Level 1 are defined to be the
classic set of nonempty, closed and bounded intervals; this will be
called the level of "mathematical regularity" (MR) for interval
arithmetic. The FTIA, infimum, supremum, midpoint and radius are all
defined as in Section 2.1.

   -- In Level 1a, FTIA is extended to unbounded intervals and the empty
set according to (4) and (5); this will be called the level of "algebraic
closure" (AC) for interval arithmetic. More specifically, an unbounded
interval is interpreted as an overflow family parameterized (virtually)
by an overflow threshold, as defined and explained in Section 2.2.

   -- Level 2 is defined as in Section 2.3; this is the level of
"interval datums." The maximal real element of each interval datum format
defines the concrete value of each corresponding overflow threshold at
Level 1a.

   -- The midpoint operation is defined at Level 1a and Level 2 as a
real number for all overflow families except { empty }. We suggest
something similar to what is discussed in Section 3.2 and Table 1, but
we leave the actual definition to a future motion. The midpoint of
{ empty } is undefined. THIS MOTION DOES NOT DEFINE THE MIDPOINT OF
AN UNBOUNDED INTERVAL AT ANY LEVEL OF THE STANDARD.

Attachment: overflow.pdf
Description: Adobe PDF document