Re: Kaucher intervals (Was: Undefined behaviour (Was: ...))
Dear collegues,
I would like to clarify the difference between Kaucher intervals,
directed intervals and modal intervals.
A Kaucher interval is a pair of reals. For example, [-1,1] is a proper
Kaucher interval, [1,-1] is an improper Kaucher interval. Proper
intervals are of course identified to the corresponding sets of reals,
while Kaucher arithmetic, which is defined for any proper/improper
arguments, coincides with the set interval arithmetic when restricted
to proper arguments.
A directed interval is a pair made of an interval and a sign taken
among + or -, e.g. ([-1,1],+) or ([-1,1],-). A modal interval is very
similar to a directed interval, as it is a pair made of an interval
and a quantifier \forall or \exists, e.g. ([-1,1],\exists) or
([-1,1],\forall). Although this is not formally defined using
equivalence classes, directed (resp. modal) intervals involving
degenerated intervals and different signs (resp. quantifiers) are
identified, e.g. the directed intervals ([1,1],+) is identified to
([1,1],-) (resp. ([1,1],\exists) is identified to ([1,1],\forall),
which is compatible with the semantics of modal intervals). Then,
directed intervals (and modal intervals) are isomorphic to Kaucher
intervals:
Consider a<=b
([a,b],+) <-----------> [a,b]
([a,b],-) <-----------> [b,a]
Kind regards,
Alexandre Goldsztejn
On Tue, Mar 17, 2009 at 2:56 PM, Michel Hack <hack@xxxxxxxxxxxxxx> wrote:
> It would have helped if early on we had seen a concise, though perhaps
> incomplete, description of Kaucher intervals, as Baker just gave. Not
> even the paper that Nate pointed us to was that clear, one had to guess
> it from the way they were used.
>
> So: a Kaucher interval is an interval with Lower and Upper bounds,
> always non-empty, with an additional flag that directs how the bounds
> are to participate in monotonic arithmetic. This is useful in cases
> where backwards containment needs to be calculated. (The mode flag is
> encoded in the order in which the two bounds are given in the interval
> representation, because that leads naturally to the desired effect, at
> least in common cases -- or does it always work, obviating the need to
> test the "flag"?)
>
> Critical in the context of the Expression-Rearrangement subgroup is
> how much knowledge of applicability of Kaucher rules is needed when
> expressions are transformed.
>
> Also -- how is Empty handled in Kaucher Arithmetic?
>
> Finally -- is "Infinity as a Member" decided one way or the other in
> Kaucher Arithmetic, and if not, are the issues similar?
>
> Michel.
>
> P.S. I started using "KA" -- but that could be misinterpreted as
> "Kahan Arithmetic". Would "MA" for "Modal Arithmetic" work,
> leaving "IA" as the generic term for Interval Arithmetic?
> Sent: 2009-03-17 14:05:27 UTC
>
--
Dr. Alexandre Goldsztejn
CNRS - University of Nantes
Office : +33 2 51 12 58 37 Mobile : +33 6 78 04 94 87
Web: www.goldsztejn.com
Email: alexandre.goldsztejn@xxxxxxxxxxxxxx