Re: What intervals should be compared: a joint comment on motions 13.4, 20 and 21
Jürgen, Ulrich, p1788,
Please my comments in the text
Dominique
Jürgen Wolff von Gudenberg a écrit :
Dominique, P1788
I think the p1788 standard should be slim and easy to read.
Therefore I support motion 13.4 in defining 7 comparisons.
Arnold NEUMAÏER last position is far slimmer. It is
easy to read.
So there is another reason to your choice.
Motion 21 proposes the optional extension with the intervall
overlapping data type providing an interesting interface.
I understand that point.
The derivation of comparisons. like your BRA is an interesting
approach, but it has nothing to do with intervals.
Therefore, even when you change your "shall" into "should" I think to
vote against it
Intentionally the Motion 20 is not explicit on
intervals. The main interest of the framework defined is both
1) It provides a tool to prevent a wild use of
interval comparison
2) By defining the mathematical structure of
the set of all possible comparisons it make it necessary to clarify the
hypothesis
used
Nature of the interval set,
Properties of the comparisons
The choice of interval comparisons is not a part of motion
20 , is left to a separate motion.
On the other hand since the same framework can be applied
to all the propositions, the choice among the propositions can more
properly founded.
This can be illustrated on both the proposition of Arnold
NEUMAIER and your proposition.
I shall post to the list such an illustration as soon
as possible.
See my comments in the text
Juergen
Dominique Lohez schrieb:
Jûrgen, Ulrich, p1788
>From a reading of the three motions I would like to emphasize a
common feature (IMHO) and ask a question
The feature: The set of comparable intervals should be restricted
to I(R)\{\Emptyset}
I disagree
motion 8 tells us that there are bare intervals, hence the emptyset
is to be regarded
. The comparisons involving the empty set should be
worked out separately as this is
already the case for domain tetrit ( Motion 18)
The question: Given two bounded intervals A and B with width(A) =
0 and width(B) >0 should the A-interval and B-intervals be
considered as distinct mathematical objects with distinct rules for
comparison?
No, in mathematics a and the interval [a,a] is the same.
And we have decided in motion 3 that intervals are sets of real numbers
My wording was misleading .
The context is that A and A' are interval with width(A)=width(A') = 0
and B and B' with width(B)>0 and width(B') >0
The question can we find a relation R such that
R(A,A'), R(B, A) , R5B,B') are always false
and R(A,B) may be true.
IMHO this introduces two kinds of intervals.
And the properties of the A and B intervals are a special case of the
intervals properties. Thus A-interval and B-Interval are to be
considered as distinct mathematical objects.
DISCUSSION
1) The feature
In motion 13.4 , it is stated that
(I(R),\suseteq ) is a lattice
(I(R)\{\emptyset}, \le) is a lattice . thus the empty set
is irrelevant to \le
In that context the set of comparisons of intervals
(cf Motion 20) derived from \subseteq and \le can only be
calculated using the most restrictive set of intervals
that is I(R)\{\Emptyset}
that is a problem of motion 20
All the relations of this motion can then be retrieved
with the exception of the relation precedes or touches (\preceq)
Similarly all the overlapping states of motion 21 are
retrieved except meets and metBy
So we have to emphasize these relation or states ?
In motion 21 a similar result appears.
While the set of overlapping states are expected to
define an atomic basis for some Binary Relation (Motion n20)
the values given in motion 21 do not satisfy this
requirement
For example for a pair P =(\emptyset, X) for the
overlapping of P at least both the before and containedBy states are
true.
that is correct (remark 7) but that is overwritten by remark 10 where
now the atomic operations are considered as comparisons. then we have
before(P) = false containedBy(P) = true
This presentation looks like somewhat strange for me.
You start with the very formal definitions of Table 1.
You define your useful comparisons from the states.
And then you surreptitiously change the definitions of the state ( or of
the comparisons) within an anecdotal remark.
In my own taste, I had rather to face unpleasant problems at formal level.
A correction can be applied by redefining the states.
For example on can write
before'=before\wedge(\not contains]
\wedge (\not containedBy) \wedge (\not equal]
and similar expression for the other states.
Thus for any pair of intervals involving the
empty set only a single state among contains, equal or containedBy
can be true
that is remark 3 , see also table 2
However the new states do not correspond to atomic
relation of some BRA because the states contains, equal and
containedBy can be indefinitely be sliced into more specific states
That's why we don't use and need new states
The easiest solution is to decide that the empty
set is NOT a comparable interval.
except for containment
In this context the overlapping of two intervals A
and B
would produce one state true and only
one state true if both A and B are not empty
would produce no state true otherwise
we had a similar treatment of empty sets in previous version of the
postion paper
I have already noticed that.
Why do you change ?
The situation would be very similar to that
encountered in Motion 18
2° The question
Deriving the BRA defined the the \subseteq and
\le relations of Motion 13.4, it is found that
All the relation in motion 13.4 are retrieved but
the preceedes or touches relation
Similarly all the states in motion 21 are
retrieved but the meets and metBy states
In fact the overlaps and meets states are
merged into a single overlaps' state.
Similarly the overlappedBy ans metBy are
merged into a single overlappedBy' state.
In fact if X meets Y is true there exist a A-
interval such that A \subseteq X and A \subseteq Y.
In fact if X overlaps Y is true there exist a
B- interval such that B \subseteq X and B \subseteq Y.
In Allen's algebra only B-interval are
considered the distinction between metts and overlaps is founded
When bot A-intervals and B-intervals are allowed
this distinction is questionable. If it is retained the meets
relation ( or state) must be assumed as generator of the BRA.
A BRA with 26 atoms ( ir states) is produced.
Otherwise a BRA with only 11 atoms is used.
IMHO the second solution is clearly the best.
The main reason is that going from level 1 to level 2 the distinction
becomes anecdotal.
But i would like to hear arguments in favor of
the first solution.
see my statement
Bets regards
Dominique
--
Dr Dominique LOHEZ
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