Arnold, Nate
I like your proposed 5-value decoration scheme (extract from Arnold's 10th Oct email quoted below). A couple of queries.
A. I had forgotten that, when promoting a bare to a decorated interval in the context of, say,
(bare xx) op (decorated yy), returning (decorated zz)
it is good sense to allot the most *pessimistic* decoration consistent with the available information, not the most optimistic.
On 26 Oct 2010, at 16:55, Arnold Neumaier wrote:
In the 5-value decoration scheme I had suggested, this would mean
that a bare interval is promoted to a decorated interval with decoration 4 if it is Empty, and decoration 2 otherwise.
Why decoration 2, "possibly everywhere defined"? Consider the function g(x) defined by the expression sqrt(2*x-x). Evaluating its interval version on [-3,-1] we get
sqrt([-6,-2]-[-3,-1]) = sqrt([-5,1]) = [0,1]
though g is nowhere defined on [-3,-1].
Thus the nonempty interval [0,1] could have arisen from evaluating a function that is nowhere defined on its input box. I deduce from this that correct pessimistic promotion of a nonempty interval gives it decoration 3, not 2.