Dear George,
I said erroneously:
"The first and fourst order relations are of
different nature."
Sorry, I meant:
The second and fourth order relations are of
different nature.
Svetoslav
------- Forwarded message follows -------
From: "Svetoslav Markov" <smarkov@xxxxxxxxxx>
To: "Corliss, George" <george.corliss@xxxxxxxxxxxxx>,
P1788 <stds-1788@xxxxxxxxxxxxxxxxx>
Date sent: Wed, 07 Apr 2010 10:40:14 +0300
Subject: Re: [IEEE P-1788] Motion P1788/M0013.01:ComparisonOperations
up for discussion
Priority: normal
[ Double-click this line for list subscription options ]
Dear George,
before your question is correctly answered,
I would like to make the following notational
suggestions.
The first and fourst order relations are of
different nature.
To avoid notational confusions I would
suggest that the first order relation is denoted
differently than \le, for instance by
\preceq , as already done in several works.
I also suggest that for uniformity the last
notation is changed to \le (less_than_or_equal_to)
In algebra it is accepted that the strict analogues
of these two order relations are defined by the
respective relations and the equality relation.
For example, see:
Garrett Birkhoff. Lattice theory, Providence,
Rhode island, 1967.
There the order relation \le, that is:
\less_than_or_equal_to
is first defined by the properties reflexivity,
antisymmetry and transitivity. Then it says:
"If x \le y and x \neq y, then one writes x < y."
Of couse, it could be done reverse, see e. g.
B.L.Van der Waerden. Algebra, Springer, 1971.
There first the (strickt) order relation x < y is defined,
then it says:
"x \le y means that either x < y or x = y.
At any case one of the two order relations
x \le y
and
x < y
is always a consequence of the other one plus relation "=",
same with the order relation \preceq
I strongly support the inclusion of \preceq, which has
been often used in our group since 1970. G. Birkhoff
also advocates this relation in his paper :
Birkhoff, G.: The Role of Order in Computing, in: Moore, R. (ed.),
Reliability in Computing, Academic Press, 1988, pp. 357–378.
Regards,
Svetoslav
On 7 Apr 2010 at 2:41, Corliss, George wrote:
From: "Corliss, George" <george.corliss@xxxxxxxxxxxxx>
To: P1788 <stds-1788@xxxxxxxxxxxxxxxxx>
Copies to: "Corliss, George" <george.corliss@xxxxxxxxxxxxx>
Subject: Re: [IEEE P-1788] Motion P1788/M0013.01:ComparisonOperations
up for discussion
Date sent: Wed, 7 Apr 2010 02:41:34 +0000
P1788,
On Apr 6, 2010, at 6:58 AM, Ralph B Kearfott wrote:
Since Motion 13 (comparison operations) has been proposed
(by Bo Einarsson) and seconded (by Dan Zuras),
it is now up for discussion as a position paper. Discussion will proceed
until the end of April 27, after which the voting period will begin.
I try to argue consistently for simplicity. My interpretation of Motion 13 is to propose a minimal set of comparisons. That is, given the set described in the motion, which I think of as
\a == \b iff a1 = b1 && a2 = b2;
// \a: |--------|
// \b: |--------|
\a <= \b iff a1 <= b1 && a2 <= b2;
// \a: |--------|
// \b: |--------|
\a \contained_in \b iff b1 <= a1 && a2 <= b2;
// \a: |--------|
// \b: |--------------|
\a < \b iff a2 < b1;
// \a: |-----|
// \b: |-----|
Can we express any (most?) comparisons I might want to make between intervals in terms of these four comparisons without further recourse to extracting and testing endpoints?
We might want
\a certainly_less_than \b (\a < \b)
\a certainly_less_than_or_equal_to \b
\a certainly_equals \b (\a == \b)
\a certainly_greater_than_or_equal_to \b
\a certainly_greater_than \b (\b < \a)
\a possibly_less_than \b
\a possibly_less_than_or_equal_to \b (not(\b < \a))
\a possibly_equals \b
\a possibly_greater_than_or_equal_to \b (not(\a < \b))
\a possibly_greater_than \b
Since the proposed minimal set includes containment, we might add
\a certainly_contained_in \b (\a \contained_in \b)
\a possibly_contained_in \b
Perhaps ..._contained_in_interior?
and the negation of all the above. Perhaps others?
Can anyone code each of these in terms of the four proposed comparisons without further recourse to extracting and testing endpoints? Presumably also allowing not(), negation(), and perhaps some other operations?
John had posed that as a challenge to me. After an evening of thought, there are several I could not get. Can you?
If we can do that without TOO ugly expressions, I'm a strong supporter of Motion 13 for its simplicity.
Alternatively, is there a well-developed theory of a minimal set of comparisons?
Dr. George F. Corliss
Electrical and Computer Engineering
Marquette University
P.O. Box 1881
1515 W. Wisconsin Ave
Milwaukee WI 53201-1881 USA
414-288-6599; GasDay: 288-4400; Fax 288-5579
George.Corliss@xxxxxxxxxxxxx
www.eng.mu.edu/corlissg
Prof. Svetoslav Markov, DSci, PhD
Head, Dept. "Biomathematics", phone: +359-2-979-3704
Inst. of Mathematics and Informatics, fax: +359-2-971-3649
Bulgarian Academy of Sciences, e-mail: smarkov@xxxxxxxxxx
"Acad. G. Bonchev" st., block 8,
BG-1113 Sofia, BULGARIA mobile (gsm): 0885871584
URL: http://www.math.bas.bg/~bio/
------- End of forwarded message -------