P1788 response to Motion 13
I had a private discussion with Nate about the post in which he submitted
the attachment Motion13.proposed.pdf, before I was able to look at that
attachment, because I was thoroughly confused by the following:
[Nate Hayes]
> So I fully agree with Prof. Markov that the fourth relation in Motion 13
> should be modified slightly and given the name "precedes or equals", i.e.,
> A \preceq B iff a2 <= b1.
I would call this "precedes or touches", not "precedes or equals".
> This leads to the family of relations depicted in Fig. 2, and also gives:
> A \prec B iff A \preceq B and not ( A = B ).
This makes no sense to me, as \prec would differ from \preceq
in only one degenerate case, namely when A and B are the same
singleton. If A and B are equal but not singleton then
NOT (A \preceq B)
because (a1 < a2 <= b1 < b2) is incompatible with a1=b1 and a2=b2.
The outcome of the discussion was:
[me writing to Nate]
If all you're trying to say is that precedes-or-touches is
a better primitive than strictly-precedes, I'm with you.
But bringing in (A=B) just confuses the issue, as does
the mention of "algebraic rules". The "least astonishing"
relationship between "<", "<=" and "=" only holds for
totally ordered domains, not partially-ordered domains
such as intervals (or vectors).
Others brought up the case of vectors, where (A < B) and
(A <= B) are defined componentwise, and it is possible
to have (A<=B) AND not(A<B) AND not(A=B) in that domain.
To define "strictly precedes" from "precedes-or-touches"
we need to stipulate an empty intersection. The other
direction is not so easy, which is why "precedes-or-touches"
is the better primitive. Note that a non-empty intersection
is necessarily a singleton here!
I've now seen the attachment, and have the following comments:
In Table 1, middle entry: I really dislike "precedes or is equal to",
and would prefer "precedes or touches". I agree that it could be read as
"each element of A precedes or is equal to some element of B, and
also each element of B follows or is equal to some element of A"
but that is pretty cumbersome.
The glyph used to denote "precedes or touches" can be as shown, as the
curved parallel lines don't necessarily suggest "equals", at least not
in the same way as the horizontal line of >= does.
In Table 2, I would add a note to "certainlyIsEqual(A,B)" pointing
out that (for standard intervals at least) this is equivalent to A
and B being the same singleton (a1=a2=b1=b2). Is that really what
the Fortran95 standard defines?
In Figures 1 and 2 I would also put "E \subset A" into the top left
quadrant (but close to the origin), as for standard intervals the
half-plane below the "singletons" line does not exist. For the
same reason I would raise the "D greater-or-equal A" in Figure 1
and "D follows-or-touches A" in Figure 2 above the "singletons" line
(and point out that the diagonal line is the "singletons" line).
Michel.
---Sent: 2010-04-12 18:22:12 UTC