Arnold, Mr. Hayes and colleagues,
Although I have worked on Kaucher intervals and their modal
interpretation for several years and do really appreciate the beauty
of this theory, I agree with Arnold's claim:
Therefore, it is wise if the standard does not commit itself to
modal arithmetic. It is far from being a mature subject matter.
This claim is supported by the following arguments, presented
informally because I believe a formal discussion about the inclusion
of the Kaucher intervals (and modal intervals arithmetic) in the
standard should wait for the output of the modal interval subgroup
leaded my Mr. Hayes, and for its arguments in favor of the inclusion
of Kaucher intervals.
The Kaucher intervals present three advantages:
1- They provide quantified interpretations through the modal intervals
interpretation.
2- They allow improving the range enclosure using some monotonicity
assumptions and the dual operation.
3- They allow solving algebraic interval equations.
However, points 1 and 2 are really well handled in the context of
classical intervals while Point 3 seems anecdotic to me (though useful
in some situations).
On the other hand, including the Kaucher intervals in the standard
presents several disadvantages:
1- Kaucher intervals, modal arithmetic and modal interpretations are
actually not completely compatible with classical set intervals. I
suppose that the main goal of the modal intervals subgroup will be to
provide everyone with arguments against the present claim. So a formal
discussion about this point shall wait for a formal output of the
modal intervals subgroup.
2- The reverse mode is central for the usage of the standard in the
context of constraint programming. Kaucher intervals are not
compatible with reverse mode. Again, a formal discussion about this
point shall wait for a formal output of the modal intervals subgroup.
3- Users of the standard will have to get used to interval
computations, which is not easy for people used to floating point
computations. Getting used to Kaucher intervals seems far more
difficult, and I foresee that we will loose a lot of potential users
as soon as they start reading a documentation that focuses on Kaucher
intervals and modal arithmetic.
Kaucher intervals and modal arithmetic could be included in the
standard through separated functions, with right conventions for their
interaction with standard (set) intervals. In this way, people who
really use it will be able to, while people who don't will just see a
section dedicated to this in the standard documentation toc, which is
also a nice opportunity for them to jump to this section and check
whether Kaucher intervals can help them.
Kind regards,
Alexandre Goldztejn