Singularities in Qualitative Reasoning - Artificial Intelligence and Symbolic Computation - page 278

prises only 23 relations, there exist 226 relations when we additionally consider
singular relations [3] - a significant difference since all these relations need to be
distinguished when analysing and interpreting situations. More importantly, sin-
gular relations are somewhat misplaced in the context of qualitative reasoning.
We are not at all interested in whether objects are precisely aligned. We focus
on coarse relations between objects, which are simple to obtain and which allow
ecient commonsense reasoning. For instance, we want to know whether one
object is to the left of another one, whether it is moving in the same direction,
and the like. What distinguishes qualitative relations from metrical relations is
that they can be recognised easily by perception. However, this does not apply
to singular relations, which require precise measurements. We conclude that sin-
gular relations are not compatible with commonsense reasoning, although there
are exceptions in those fields where singular relations are as easy to obtain as
general relations. For example, in the case of events we often know whether one
event follows another one directly: after the performance a reception is held in
the foyer; there is no time to go shopping between the performance and the
reception - these events meet in time.

2.1 Representing Singular Relations

Having said that singular relations are incompatible with the idea of common-
sense reasoning - more so in two dimensions than in one - we must show how
to deal appropriately with singular relations. We cannot simply exclude them,
since we need to represent every conceivable arrangement of intervals. One way
of dealing with them consists in assigning singular relations to similar general
relations. The singular relation in the first example could be assigned to F l ,

since there is only one point that is not actually in relation F l ;thismaybean

appropriate solution in applications in which coarse reasoning is performed. But
when such a precise distinction matters we are outside the scope of qualitative
reasoning.
The second example, Fig. 1.(b), is more dicult to handle. If we regard this
arrangement as D l then we are heading for a problem. What about the con-

verse relation? If it is regarded as D r , then it holds for both intervals that each

is contained in the other one - a quite awkward situation. For this reason we
have to proceed as we do whenever we encounter indeterminate information in
any qualitative representation: by sets of possible relations. In Fig. 1.(b) we
would represent the singular relation by {D l ,C l ,FO l ,BO l } and its converse by

{D r ,C r ,FO r ,BO r } . In this way, we can deal with parallel intervals which are

equal in length. The representation does not seem to be very precise, but preci-
sion is exactly what we want to avoid in a qualitative representation. When can
we be sure whether parallel lines really are equal in length? Only when we have
precise measuring tools. Isn't there always a little uncertainty left when working
without such tools? At most we know that two lines in a given arrangement are
likely to be equal in length, but at the same time we also know that they may

be something else - something similar. Similar relations form a neighbourhood
in the BA -graph, and such neighbourhoods circumscribe the singular relations.