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x y = {F l ,F m }
y z = F l
x z = x y ◦ y z
= {F l ,F m }◦F l
= F l ◦ F l ∪ F m ◦ F l
= {F l ,F m ,F r }∪{F r }
= {F l ,F m ,F r }
z
z
x y = F m
y z = F l
x z = x y ◦ y z
= F m ◦ F l
= F r
y
y
x
x
Fig. 3. Transitivity with a singular relation (left), and without any singularity (right)
composition result is less indeterminate. Note that we assume that
x z cannot be
perceived directly, as is actually the case in this figure.
3
Discussion
Hitherto qualitative representations have treated singular relations as being on
a par with general relations. This is useful in some areas, for example, in order
to distinguish whether an event happens before another event, or whether it
immediately follows ( meets )anotherone[1].Wehavearguedthatsingularre-
lations are not as important as general relations in some applications, and that
they form a different sort of relation since they do not accord with common-
sense reasoning. Characterising singularities on the basis of neighbourhoods, we
have treated them as relations of second order rather than basic relations. As
a consequence, the endpoints of basic relations always lie in general positions.
Indeed
forms a set of relations which covers all possible situations when
circumscribing singular relations by neighbourhoods of general relations -
BA
BA
leaves nothing undefined. This also holds for other qualitative representations.
To summarise, we have identified singularities as artefacts in qualitative rep-
resentations. They are problematic in some areas, in that they require precise
measurements whereas precision is normally avoided in qualitative reasoning.
We have outlined how to deal with singularities by means of sets of possible
relations, i.e. by defining singularities as sets of general relations.
References
1. J. F. Allen. Maintaining knowledge about temporal intervals. Communications of
the ACM , 26(11):832-843, 1983.
2. A. G. Cohn and S. M. Hazarika. Qualitative spatial representation and reasoning:
An overview. Fundamenta Informaticae , 43:2-32, 2001.
3. B. Gottfried. Reasoning about intervals in two dimensions. In
IEEE Int. Conference
on Systems, Man and Cybernetics , The Netherlands, 2004.

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