Dear Michel,
On Mon, Sep 23, 2013 at 7:22 PM, Michel Hack <
mhack@xxxxxxx> wrote:
>
> I'm glad I prompted Nate to resurface for a moment. Here are my very
> cursory notes on his document MIAstandard.pdf of Feb 4, 2013:
>
> Very nice exposition. Bounded and semibounded supported, but Empty is
> not an interval, anc comparisons with Empty are unordered. Very nice
> explanation of Modal vs Kaucher vs (what we would call) Common, unified
> by the Set(interval) operator: { x: min(l,r) <= x <= max(l,r) for [l,r] }
> Then proper([l,r]) means Ex(x in Set([l,r])) (l <= r)
> improp([l,r]) means Ax(x in Set([l,r])) (l >= r)
>
This is a common misunderstanding of the modal intervals theory. Consider the Kaucher interval operation
[-1,1]+[12,8]=[11,9]
It has two interpretations, using the so called * and ** modal interpretations:
forall x in [-1,1], forall z in [9,11], exists y in [8,12], z=x+y
and
forall y in [8,12], exists z in [9,11], exists x in [-1,1], z=x+y
None of these * or ** interpretations are obtained using the rules
> proper([l,r]) means Ex(x in Set([l,r])) (l <= r)
> improp([l,r]) means Ax(x in Set([l,r])) (l >= r)
you mentioned.
Regards,
Alexandre