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Abstract
Understanding the process of categorization is a
primary research goal in artificial intelligence. The
conceptual space framework provides a flexible approach
to modeling context-sensitive categorization
via a geometrical representation designed for
modeling and managing concepts.
In this paper we show how algorithms developed
in computational geometry, and the Region Connection
Calculus can be used to model important
aspects of categorization in conceptual spaces. In
particular, we demonstrate the feasibility of using
existing geometric algorithms to build and manage
categories in conceptual spaces, and we show how
the Region Connection Calculus can be used to reason
about categories and other conceptual regions.
1 Introduction
Categorization is a fundamental cognitive activity. The ability
to classify and identify objects with a high degree of exception
tolerance is a hallmark of intelligence, and an essential
skill for learning and communication. Understanding the
processes involved in constructing categories is a primary research
goal in artificial intelligence.
The conceptual space framework as developed by
G¨ardenfors [000] provides a flexible approach to modeling
context-sensitive categorization. Conceptual spaces are
based on a simple, yet powerful, geometrical representation
designed for modeling and managing concepts.
In this paper we show how algorithms developed in computational
geometry, and the Region Connection Calculus
(RCC) [Cohn et al., 17], a well known region-based spatial
reasoning framework, can be used to model important aspects
of categorization in conceptual spaces. In particular, we
demonstrate the feasibility of using existing geometric algorithms
to build and manage categories in conceptual spaces,
and we show how the RCC can be used to reason about categories
and other conceptual regions.
¤This paper appeared in the Proceedings of the Fourteenth International
Joint Conference of Artificial Intelligence, Morgan Kaufmann,
85 - , 001.
Conceptual Spaces
Conceptual spaces provide a framework for modeling the formation
and the evolution of concepts. They can be used to
explain psychological phenomena, and to design intelligent
agents [Chella et al., 18; G¨ardenfors, 000]. For the purposes
of this paper conceptual spaces provide the necessary
infrastructure for modeling the process of categorization.
Conceptual spaces are geometrical structures based on
quality dimensions. Quality dimensions correspond to the
ways in which stimuli are judged to be similar or different.
Judgments of similarity and difference typically generate an
ordering relation of stimuli, e.g. judgments of pitch generate
a natural ordering from "low" to "high" [G¨ardenfors,
000]. There have been extensive studies conducted over
the years that have explored psychological similarity judgments
by exposing human subjects to various physical stimuli.
Multi-dimensional scaling is a standard technique that
can be used to transform similarity judgments into a conceptual
space [Krusal and Wish, 178]. An interesting line of
inquiry is pursued by Balkenius [1] who attempts to explain
how quality dimensions in conceptual spaces could accrue
from psychobiological activity in the brain.
In conceptual spaces objects are characterized by a set of
attributes or qualities fq1; q; ; qng. Each quality qi takes
values in a domain Qi. For example, the quality of pitch (or
frequency) for musical tones could take values in the domain
of positive real numbers. Objects are identified with points in
the conceptual space C = Q1 x Q x Qn, and concepts are
regions in conceptual space.
In the definition above we use the standard mathematical
interpretation of "domain". In [G¨ardenfors, 000] however,
a domain is defined to be a set of integral dimensions, this
interpretation is consistent with its use in the psychology literature.
For example, pitch and volume constitute the integral
dimensions of sounds discernible by the human auditory perception
system. Integral dimensions are such that they cannot
be separated in the perceptual sense. The ability to bundle up
integral dimensions as a domain is an important part of the
conceptual spaces framework. Domains facilitate the sharing
and inheritance of integral dimensions across conceptual
spaces.
For the purpose of this paper, and without loss of generality,
we often identify a conceptual space C with Rn, but hasten
to note that conceptual spaces do not require the full rich-
ness of Rn. Domains can be continuous or discrete1. They
can also be based on a wide range of geometrical structures,
for example, according to psychological evidence the human
colour perception system is best represented using polar coordinates
[G¨ardenfors, 000].
For the purpose of problem solving, learning and communication,
agents adopt a range of conceptualizations using different
conceptual spaces depending on the cognitive task at
hand.
Similarity relations are fundamental to conceptual spaces.
They capture information about the similarity judgments. In
order to model some similarity relations we can endow a conceptual
space with a distance measure.
Definition 1 A distance measure d is a function from C x C
into T where C is a conceptual space and T is a totally ordered
set.
Distance measures lead to a natural model of similarity; the
smaller the distance between two objects in conceptual space,
the more similar they are. The relationship between distance
and similarity need not be linear, e.g. similarity may decay
exponentially with distance.
The properties of connectedness, star-shapedness and convexity
of regions in conceptual spaces will prove useful
throughout.
DefinitionA subset C of a conceptual space is
(i) connected if for every decomposition into the sum of two
nonempty sets C = C1 [ C, we have ¯ C1C [ C1¯ C 6= ; where ¯ C is the closure of C. In other words, C is
connected if it is not the disjoint union of two non-empty
closed sets.
(ii) star-shaped with respect to a point p (referred to as a
kernel point) if, for all points x in C, all points between
x and p are also in C.
(iii) convex if, for all points x and y in C, all points between
x and y are also in C.
DefinitionThe kernel of a star-shaped region C is the set
of all possible kernel points, and will be denoted kernel(C).
Connectedness is a topological notion, whilst starshapedness
and convexity rely only on a betweenness relation.
A qualitative betweenness relation can be specified in
terms of a similarity relation, S(a; b; c), which says that a is
more similar b than it is to c. Alternatively, a betweenness
relation can be used as primitive, and axioms introduced to
govern its behaviour [Borsuk and Szmielew, 160]. In the
special case where the distance measure is a metric, the betweenness
relation can be defined as "b is between a and c"
if and only if d(a; b) + d(b; c) = d(a; c).
Convex regions are star-shaped, and in many topological
settings star-shaped regions are connected. The kernel of a
convex region is the region itself, and under the Euclidean
metric kernels are convex.
1They can even be small and finite e.g. fmale, femaleg.
A scientific representation of colour would require a different
representation however, one that captures important scientific features
of the electromagnetic spectrum such that the wave properties
of wavelength and amplitude constitute integral dimensions.
Constraints like connectedness, star-shapedness and convexity
can be used to impose ontological structure on the categorization
of the conceptual space, i.e. not any old region
can serve as a category. In fact, there is compelling evidence
that natural properties correspond to convex regions in conceptual
space, and using the idea of a natural property in this
way G¨ardenfors [000] is able to sidestep the enigmatic problems
associated with induction.
In section 4 we show how categorization, the central theme
of this paper, occurs in conceptual spaces, but first we briefly
describe the RCC.
Region Connection Calculus
The RCC is a qualitative approach to spatial reasoning. It was
developed in an attempt to build a commonsense reasoning
model for space, and its remarkable utility has been illustrated
in numerous innovative applications [Cohn et al., 17].
The RCC approach is region-based where spatial regions
are identified with their closures. The RCC is based on a connection
relation, C(X; Y ), which stands for "region X connects
with region Y ". The connection relation, C, is reflexive
and symmetric. Despite the fact that the basic building blocks
in the RCC are regions, C can be given a topological interpretation,
namely C(X; Y ) holds when the topological closures
of regions X and Y share at least one point.
The RCC framework comprises several families of binary
topological relations. One family, the RCC5 fragment uses
the following Jointly Exhaustive and Pairwise Disjoint base
relations to describe the relationship between two regions (see
Figure 1); DR (discrete), EQ (identical), PP (proper part),
PP¡1(inverse PP), and PO (partial overlap).
[Aurenhammer, 187] Aurenhammer, F., Power Diagrams
Properties, Algorithms and Applications, SIAM Journal
of Computing Surveys, 16(1)78-6, 187.
[Aurenhammer, 11] Aurenhammer, F., Voronoi Diagrams
A Survey of a Fundamental Data Structure, ACM Computing
Surveys, (), 45 - 405, 11.
[Balkenius, 1] Balkenius, C., Are There Dimensions in
the Brain? in Spinning Ideas, Electronic Essays Dedicated
to Peter G¨ardenfors on His Fiftieth Birthday online
at http//www.lucs.lu.se/spinning.
[Borsuk and Szmielew, 160] Borsuk, K., and Szmielew, W,
Foundations of Geometry. Amsterdam North Holland,
160.
Vol 81 LNCS,- 44, Springer-Verlag, 15.
[Okabe et al., 000] Okabe, A., Boots, B., Sugihara, K., and
Chiu, S.N., Spatial Tessellations, nd Ed, Wiley, 000.
[Petitot, 18] Petitot, J., Morphodynamics and the Categorical
Perception of Phonological Units, Theoretical Linguistics,
15 5 -71, 18.
[Renz and Nebel, 1] Renz, J. and Nebel, B., On the Complexity
of Qualitative Spatial Reasoning A Maximal
Tractable Fragment of the Region Connection Calculus,
Artificial Intelligence, 108 (1-) 6 -1, 1.
[Rosch, 175] Rosch, E., Cognitive representations of semantic
categories, Journal of Experimental Psychology
General, 104 1 - , 175.
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