Re: [IEEE P-1788] Motion P1788/M0013.01:ComparisonOperations up for discussion
Dear George,
I think we need the intersection and the following operation
isSingleton(\x) :<=> x_inf == x_sup
together with glb and lub as defined in the Motion to realize the
required "certainly" and "possibly" relations.
Corliss, George schrieb:
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?
Then we can get the "certainly" and "possibly" relations by applying the
following rules:
We might want
\a certainly_less_than \b (\a < \b)
ok
\a certainly_less_than_or_equal_to \b
:<=> \a < \b || ( isSingleton(\a intersect \b)
&& ( (\a intersect \b) <= \b ) && ( \a <= (\a intersect \b) ) )
\a certainly_equals \b (\a == \b)
I think that is wrong. In my opinion it should be:
\a certainly_equals \b :<=> \a == \b && isSingleton(\a)
\a certainly_greater_than_or_equal_to \b
:<=> \b certainly_less_than_or_equal_to \a
\a certainly_greater_than \b (\b < \a)
ok
\a possibly_less_than \b
:<=> not(\b certainly_less_than_or_equal \a)
\a possibly_less_than_or_equal_to \b (not(\b < \a))
ok
\a possibly_equals \b
:<=> not(\a < \b) && not(\b < \a) (or: notEmpty(\a intersect \b))
\a possibly_greater_than_or_equal_to \b (not(\a < \b))
ok
\a possibly_greater_than \b
:<=> \b possibly_less_than \a
Since the proposed minimal set includes containment, we might add
\a certainly_contained_in \b (\a \contained_in \b)
ok
\a possibly_contained_in \b
\a possibly_contained_in \b :<=> \a possibly_equals \b (?)
Perhaps ..._contained_in_interior?
\a certainly_contained_in_interior \b :<=> \a \contained_in \b && not(
glb(\a,\b) == \a || glb(\a,\b) == \b || lub(\a,\b) == \a || lub(\a,\b)
== \b)
\a possibly_contained_in_interior \b :<=>
not(\a certainly_less_than_or_equal_to \b) &&
not(\b certainly_less_than_or_equal_to \a)
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
Best regards
Marco
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