Re: Notations, new Motion
Ulrich and P1788
On 16 Apr 2012, at 16:13, Ulrich Kulisch wrote:
> For consistency the same scheme of denotations should be kept for the subsets
> representable on computers. This leads to the following denotations:
> R the set of real numbers.
> \overline{R} \overline{R} := R ∪ {−∞,+∞}.
> IR the set of nonempty, closed and bounded real intervals.
> \overline{IR} the set of closed real intervals, including unbounded intervals and the empty set.
> F the set of (finite) floating-point numbers representable in some floating-point format.
> \overline{F} \overline{F} := F ∪ {−∞,+∞}.
> IF the intervals of IR whose bounds are in F.
> \overline{IF} the intervals of \overline{IR} whose bounds are in \overline{F} and the empty set.
Changing notation does not change content -- though a good notation can make content much easier to express.
I agree with Ulrich that we should not depart more than necessary from well established notation. Maybe the notation described in the Level 1 text V04.4, in the Notes to §5.2, departs too much for too little benefit. It is easy to amend, and I am happy to do so if we get something better.
However, I think there are one or two things Ulrich's comments don't address, which I *was* trying to address by the notation chosen in v04.4. Points:
A.
At Level 2 we shall often be referring to the inf-sup type derived from some floating-point format F (including unbounded intervals and the empty set), namely \overline{IF} in Ulrich's text above.
I believe more and more that Level 2 will often need to refer to the classical set of (bounded, nonempty) intervals derived from F, namely IF, above.
Reason: I think the most straightforward way to reconcile the apparently conflicting demands of standard and modal interval arithmetic is *not* to force standard intervals to behave like modals, or the other way round. Instead require that at Level 2, standard arithmetic and modal arithmetic behave identically on the set IF of (standard) classical intervals in a given format. More on this shortly.
B.
I want these "I" symbols to behave syntactically as operators. It's all very well saying "Symbol F stands for a generic format, and IF [resp. \overline{IF}] is its derived inf-sup type". We have lots of specific, named, formats like binary64. I don't want to be forced to say "Let F be binary64, and let T be the type \overline{IF}." It's far more concise and expressive to say "Let T = \overline{I binary64}" or similar.
C.
As far as possible I don't want there to exist two nearly identical denotations of the same thing.
Here is one option I see. (Imagine names like binary64 being in \tt font, while R, I, F, ... and Rbar=\overline{R} are in \mathbb font.)
- binary64, and other specific denotations, stand for the set
of *finite* numbers represented in that format.
- To make the full set of numbers put a line on top, so
\overline{binary64} means the full set of numbers including
-oo and +oo. So \overline in this context denotes the
operator that maps *any* subset S of R, to the set
S U {-oo,+oo}.
- I denotes the operator that maps any subset S of Rbar to the
family of (nonempty) intervals whose bounds are in S.
- \overline{I} is as I, except it includes the empty set in
the family.
This is not consistent with 754 usage, in which binary64 stands for the set of numbers *including* -oo and +oo. But it seems a fair compromise.
- Ulrich's \overline{IF} is actually
\overline{I}\overline{F},
that is
"take F, form F U {-oo,+oo}, then form the family of
all intervals whose bounds are in the latter, plus
the empty set".
- Ulrich's IF is I acting on F.
This seems better than the notation in v04.4, which was complicated by introducing an \underline{I}. It reproduces Ulrich's notation cited above, but the "I"s are indeed operators. If it is agreed that any set of numbers *without* an overline shall be a subset of R (so doesn't include -oo,+oo) then requirement C above is probably satisfied.
Views please.
John Pryce