Re: Motion 37, Kaucher Arithmetic
Michel,
Kaucher arithmetic is historically restricted to closed and bounded
intervals due to the algebraic definition used by Kaucher to construct the
interval system. Modal analysis, on the other hand, introduces the Semantic
Theorems and therefore gives a set-theoretic definition of Kaucher
arithmetic. The FTIA for classical (Moore/Sunaga) intervals is just a
special case of the Semantic Theorems.
In the same way FTIA for classical interval arithmetic can be extended to
(IR)bar, so too can Semantic Theorems for modal arithmetic be extended to
(I*R)bar. As you and Ulrich both mention, this extended (I*R)bar system
contains the FTIA for intervals in (IR)bar as a special case.
The system (I*R)bar is therefore the most general system that is a superset
of classical, Kaucher and (IR)bar arithmetics.
Some operations in (I*R)bar however are not defined, e.g.,
[-oo,3] + [+oo,7] = [(-oo)+(+oo),3+7]
which involves addition of opposite-signed infinities. This is to be handled
by decorations.
Ulrich's motion therefore presents the algebraic definitions of Kaucher
arithmetic before the extension to (I*R)bar, which requires decorations.
Nate