Re: Decorations and Motion 22
Kreinovich, Vladik wrote:
Please check whether my understanding is correct.
Almost, but your explanations are still somewhat ambiguous.
Decorations are properties of the function whose interval extension
is being computed, not of the resulting interval.
Bounded is clear, that we have finite bounds.
It means that the underlying function is bounded on the input box.
So 2*[1,maxreal] = [2,inf] would be bounded, though the interval part
looks unbounded. Thus this decoration takes care of overflow issues.
Otherwise, one would not need a decoration to handle this.
Continuous, if I remember correctly, means that this interval came as a result of continuous operations, so it may be an enclosure of the actual range, but at least we know that it did not come from computing the interval hull of a discrete set such as sign([-1,1]).
... and that the input intervals also came from continuous evaluations.
About everywhere defined: if I understand correctly, this means that in all previous steps, we did not have possible un-defined values. For example, sqrt([0,4]) is [0,2] and everywhere defined while sqrt([-1.4]) is still [0,2] but this time NOT everywhere defined, only possible everywhere defined -- since it could be that [-1,4] is only an enclosure for the actual interval [0,4] for which the previous operation is everywhere defined.
Is this correct?
yes.
Re different between nowhere defined and not valid I am not so sure.
invalid arises upon the attempt to construct an interval in a way that
doesn't make sense, like conversion from '[1,-1]' or from '[1x,%]'
or from an already invalid interval.
From: stds-1788@xxxxxxxx [mailto:stds-1788@xxxxxxxx] On Behalf Of Arnold Neumaier
Let me write the presence of the decorations x.dec=0,...,4 as
d0: safe = everywhere defined, continuous, and bounded
d1: everywhere defined
d2: possibly everywhere defined
d3: nowhere defined
d4: not valid
The decorations are partially ordered according to information content
as d0>d1>d2<d3<d4. Thus d2 is the least informative decoration, where
nothing is claimed anymore; it is the bare decoration equivalent of
the bare interval Entire. d4 is sticky in all arithmetic operations.