What intervals should be compared: a joint comment on motions 13.4, 20 and 21
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} . 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?
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}
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
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.
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
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
The easiest solution is to decide that the empty set
is NOT a comparable interval.
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
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.
Bets regards
Dominique
--
Dr Dominique LOHEZ
ISEN
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France
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Email: Dominique.Lohez@xxxxxxx