Re: Discussion paper: what are the level 2 datums?
Nate Hayes wrote:
Dan Zuas wrote:
Let us assume for the moment that we deal with them
entirely at level 2. At that level a bare interval
is not so much an interval without decorations as an
interval for which the decorations have no meaning.
No!
At Level 2, a bare interval is an element of overline-IR. Period. This
includes closed real intervals and the empty set.
Similarly, a bare decoration is something for which
the interval part has no meaning.
No. A bare decoration is just an abstract object representing the state
of various decoration attributes, e.g., the "domain" tetrit or a
"defined and continuous" bit, etc.
A bare decoration has no interval part.
This is also my reading of motion 8.
For if it were it would simply be something that
introduces the possibility that the thing that has
no meaning or the thing being ignored might admit
errors down the line. Or, at the very least, cause
problems for a correctness proof.
Exactly.
This is why in Motion 8
bare intervals
bare decorations
decorated intervals
are specifically singled-out and defined as different Level 2 objects,
each with its own meaning; and that precise rules about how these
objects may be used in arithmetic and non-arithmetic functions or
operations are also provided so that the result is a reliable Level 2
system of computing.
One really important example is that it should never be possible to
construct a decorated interval from just a bare decoration. Another
important example is that it should never be possible to convert a bare
decoration into a bare interval (e.g. the empty set) or vice-versa.
Yes. This was a deliberately designed feature to ensure that decorations
can take the role of NaI without corrupting interval computations.
The Level 2 model is the primary motive, for reasons described above.
However, the Level 2 model _does_ also anticipate that at Level 3 there
will eventually be issues of storage space and efficiency that will come
into play.
This is why, for example, there are both bare intervals and bare
decorations as opposed to just decorated intervals. The assumption is
that for certain implementations, always computing with decorated
intervals will be overkill (in fact I think this is actually true for
just about all practical interval implemenations I can think of).
Therefore, the Level 2 model anticipates that some interval
implementations and/or algorithms will choose to use only bare intervals
at some times and only bare decorations at others. But the Level 2 model
does not in any way dictate or require how users and/or implementors
make those decisions. It defers those decisions to implementors, users,
compiler designers, hardware engineers, language committes, library
vendors, etc., assuming that those people are the experts in thier
respective fields and can make the best decisions in that regard as to
when and/or how to use bare intervals vs. bare decorations etc.
But the Level 2 model does dictate the terms, rules, and conditions that
those implementation experts must follow to ensure their final product
conforms to the standard and is also a reliable system of computing. For
example, it decrees that no implementer shall allow a bare decoration
(or its moral equivalent) be converted to an empty set, etc.
I fully agree.
Arnold Neumaier