Singularities in Qualitative Reasoning - Artificial Intelligence and Symbolic Computation - page 276

Singularities in Qualitative Reasoning

Bjorn Gottfried

Centre for Computing Technologies
University of Bremen, Germany
[e-mail]

Abstract. Qualitative Reasoning is characterised by making knowledge
explicit in order to arrive at ecient reasoning techniques. It contrasts
with often intractable quantitative models. Whereas quantitative mod-
els require computations on continuous spaces, qualitative models work
on discrete spaces. A problem arises in discrete spaces concerning transi-
tions between neighbouring qualitative concepts. A given arrangement of
objects may comprise relations which correspond to such transitions, e.g.
an object may be neither left of nor right of another object but precisely
aligned with it. Such singularities are sometimes undesirable and influ-
ence underlying reasoning mechanisms. We shall show how to deal with
singular relations in a way that is more closely related to commonsense
reasoning than treating singularities as basic qualitative concepts.

1 Introduction

In this essay we shall discuss problems arising by describing arrangements of ob-
jects qualitatively. We are concerned with the relations depicted in Fig. 1, which
have been introduced in [3]. We refer to the set of these relations as BA .The

relations in describe arrangements of intervals in the two-dimensional plane

qualitatively; they can be considered as the two-dimensional analogue of Allen's
one-dimensional interval relations [1].

BA

is distinguished from other qualitative
representations (cf. [2])inthatitcomprisesonly disconnection relations .Rela-
tions between disconnected objects are of interest in a number of areas, mainly
when spatiotemporal interactions between objects are to be described. It could
be argued that connection relations are equally important. But there are no
connections, for example, between road-users in tra c, pedestrians walking in a
market square, sportsmen playing on a pitch, or generally between objects form-
ing patterns of spatiotemporal interactions. Sometimes the distances between
objects become very small, but they still remain detached from one another and
can generally change their orientation and position independently of other ob-
jects. We are simply interested in possible relations between objects that are not
connected.
It is less a question of motivating the necessity of different disconnection
relations, than of restricting the relations to those in general positions. In

BA

BA
the endpoints of all intervals are in general positions. The examples on the right
of Fig. 1 show relations in singular positions. These correspond to special cases