Re: Supporting set- and modal-paradigms
Dear John and colleagues,
Before going so deep in details, we should all agree on what kind of
generalized intervals and arithmetic we discuss for potential
inclusion in the standard. I think this discussion should be conducted
in the modal intervals subgroup instead, but since it has moved to the
main list let me provide my point of view.
> - MOD has another set of abstract interval objects. Following Nate's paper
> they *are* ordered pairs (xx, Q) where xx is a nonempty set-interval and Q
> is either \E or \A (the existential or the universal quantifier,
> respectively). When Q is \E the interval is "proper", else "improper".
I don't think this definition is coherent, though being the original
modal intervals definition. Indeed it is required in every context
where modal intervals are used that for any real x both ([x,x],E) and
([x,x],A) are identified. Therefore, the only correct mathematical
definition for these objects is to consider the quotient set of
{ (xx,Q) | xx in IR , Q in {E,A} }
by the equivalence relation ~ defined by
(xx,Q) ~ (yy,R) iff (xx=yy and Q=R) or (xx=yy=[z,z] for some real number z)
Remark: The modal intervals theory proposes instead to define an
equality between modal intervals that satisfies ([x,x],E)=([x,x],A)
(cf. Lemma I.2.5 of main reference Modal intervals (basic tutorial)
presented at MISC1999) , which is different of the usual mathematical
equality between pairs. I believe equivalence classes is the right
mathematical formalization of this non standard equality.
This definition is rather complicated while not mandatory since the
original modal intervals theory intensively uses a bijection between
modal intervals and Kaucher intervals. Kaucher intervals can thus
safely replace modal intervals providing a more consistent
mathematical definition in the standard.
Several arithmetics can be defined on Kaucher intervals (see e.g.
Neumaier's paper assessing nonstandard interval arithmetic). On the
other hand, all these definitions coincide for every unary operations
(like exp or sin) and also for the binary operations +, -, x and /,
but some special care should be given to the other two variable
operations. I suggest we use in general the star modal intervals
arithmetic (denoted f* in the modal intervals theory, in opposition to
the f** arithmetic) since among the several arithmetics that can be
defined on Kaucher intervals, this is the one that provide nice
interpretations in terms of quantified propositions that generalize
the classical intervals arithmetic.
Kind regards,
Alexandre Goldsztejn
--
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