Spatial Cognition and Computation 1: 181–204, 1999.
© 2000 Kluwer Academic Publishers. Printed in the Netherlands.
First-order qualitative spatial representation
languages with convexity
Department of Computer Science, University of Manchester, UK
Abstract. In recent years, there has been considerable interest within the AI community in
qualitative descriptions of space. The idea is that a language in which we can say such things
as “region a is convex” or “region b is a part of region c” might be sufﬁcient for characterizing
useful properties of everyday spatial arrangements, while avoiding complex and error-sensitive
numerical coordinate descriptions. However, such qualitative representation languages are
inevitably balanced on a semantic knife-edge: too little expressiveness, and they are useless
for the everyday tasks we want them for; too much, and they exhibit the over-precision which
motivated qualitative representation languages in the ﬁrst place. The aim of this paper is to
demonstrate how sharp that knife-edge is, and thus to establish some limits on what such
qualitative spatial description languages might be like.
Key words: afﬁne geometry, convexity, logic, mereology, model theory, qualitative spatial
In recent years, there has been considerable interest within the AI community
in qualitative descriptions of space. The idea is roughly this. Suppose we
have a language in which we can say such things as “region a is convex”
or “region b isapartofregionc” and so on; then maybe such a language
would enable us to characterize the spatial properties of everyday objects
to an extent that sufﬁces for – say – object recognition or route planning
or commonsense mechanical reasoning. Thus – so the thought goes – we
could perform a range of useful tasks without the need to rely on numerical
(coordinate) descriptions, which are computationally complex, error-sensitive
and often simply unavailable. The hope is that, by choosing an appropriate
qualitative spatial description language, we might increase the effectiveness
of an artiﬁcially intelligent agent operating in or reasoning about space.
However, such qualitative spatial representation languages are inevitably
balanced on a semantic knife-edge: too little expressiveness, and they are