My thoughts on FTDIA
John, Arnold, P1788,
Thanks for sharing your opinions.
I'd like to re-iterate a few key points why I believe P1788 needs to pass
Motion 25.
As mentioned before, FTIA and FTDIA are both theories. All they do is
predict what are the valid results of an interval expression. Evaluation or
computation, by contrast, is the practice.
John Pryce wrote:
So, regarding decorated evaluation as a "joint enclosure", componentwise:
(computed range, computed decoration) \supseteq (true range, true
decoration)
of f over xx, seemed sort of neat. That's what the FTDIA is.
For the reasons mentioned above, this statement doesn't entirely make sense
because FTDIA does not compute or evaluate anything. It only predicts the
superset relation should be valid.
On the other hand, it is the *property tracking* which actually computes the
range enclosure and decoration on the left. In his own words, I believe John
therefore demonstrates both the need for property tracking and its relation
to FTDIA.
Another key point.
For the reasons Arnold would have Motion 25 rejected:
Arnold Neumaier wrote:
But I think Motion 25 is far too low level on arithmetic operations...
Thus I would recommend voting No to the current version of the motion.
we would then have to repeal Motion 5, as well... since the interval
operations in Motion 5 are low-level and do not give optimal
(non-pessimistic) bounds of complicated interval expressions otherwise
predicted by FTIA, either.
To my understanding, IEEE 1788 is *supposed* to be a standard that is
practically implementable at a low-level, notably at the hardware level. If
this is true, then John and Arnold need to show that FTDIA is practicable at
this level, and even more importantly that such practice can be done in a
standardized manner. Neither of them have accomplished this.
As illustrated very nicely in Vladik's paper posted recently to P1788,
evaluation of interval expressions via computation on intermediate steps is
an obvious solution to the problem. This is the interval arithmetic we all
know and love over the past 50+ years, and that can obvsiouly be implemented
in hardware very efficiently. For FTIA, this clearly leads to Motion 5; and
for FTDIA, this clearly leads to Motion 25.
Nate