RE: Promotion of bare decorations & comparisons
John Pryce wrote:
> Nate, Jürgen, Michel, P1788
>
> Nate: on what you say (3 Jan, below) about Empty: P1788 agreed in Motion 3
that
> Empty *shall* be an interval, and we are not likely to change that now.
Hmm.
Then it appears Motion 42 should be entirely tossed out, too, since it also
invalidates many things P1788 previously agreed on in Motion 8:
"2.7. P1788 provides other nonarithmetic operations which are handled in an
analogous way, i.e., consistent with the idea that one can use the
operations as usual with or without any decoration, and with the forgetful
mechanism, and otherwise preserve as much information as possible."
Elaborated in more detail, the Rationale of Motion 8 says (I've made some
edits to use modern decoration names for sake of clarity):
"The framework presented in this motion also unifies the concept of NaI and
Empty in a semantically correct way. For example, given any non-empty bare
interval X, one probably wants to have
X \union Empty = X.
However, some applications may also need
X \union NaI = NaI,
which cannot be done naturally with flavors of Empty. But it can be neatly
handled by decorations, e.g., if
-- DAC is the decoration of a nonempty, bounded new interval
-- DAC is the decoration of an empty set created by intersecting two
decorated intervals with DAC decoration
-- NDF is the decoration of an interval created by an invalid
conversion or undefined operation
we may have
(X,DAC) \union (Empty,DAC) = (X,DAC)
as an unambiguous version of X \union Empty = X, and
(X,DAC) \union (Empty,NDF) = (X,NDF)
as an unambiguous version of X \union NaI = NaI."
Critical aspects of this example illustrate two concepts which Motion 42
currently prevents or forbids:
-- Decorated intersection and union (convex hull) operations
-- Empty may have "good" (DAC) and "bad" (NDF)
decorations/interpretations
> I have said enough to point out the problem. Please give some solutions to
get the New
> Year off to a good start.
IMO, the best solution I've seen so far is the decoration system which
includes EIN, since it gives all these features:
-- Decorated intersection and convex hull operations
-- Empty may have "good" and "bad" decorations/interpretations
-- Bare decorations may be safely defined as compressed decorated
empty sets
-- All bare decorations can promote to Empty
-- If Empty is not an interval, compressed interval arithmetic
avoids the false positives/negatives
-- Comparison operation on a bare decoration may be defined as the
equivalent comparison operation on Empty
This leaves open the questions about i) comparisons on decorated intervals
and ii) nonarithmetic operations (NAO) such as midpoint. I don't have a
fixed view about i) and for ii) I'd suggest its ok to return bare
decorations, for example, if NAO is undefined, etc.
Nate