Formalized Conceptual Spaces with a Geometric Representation of Correlations

Abstract

The highly influential framework of conceptual spaces provides a geometric way of representing knowledge. Instances are represented by points in a similarity space and concepts are represented by convex regions in this space. After pointing out a problem with the convexity requirement, we propose a formalization of conceptual spaces based on fuzzy star-shaped sets. Our formalization uses a parametric definition of concepts and extends the original framework by adding means to represent correlations between different domains in a geometric way. Moreover, we define various operations for our formalization, both for creating new concepts from old ones and for measuring relations between concepts.We present an illustrative toy-example and sketch a research project on concept formation that is based on both our formalization and its implementation.

Criteria for naturalness in conceptual spaces

“…Within the literature on conceptual spaces, one also often encounters spaces that are not drawn from similarity data but composed from pre-defined dimensions. For instance, Bechberger and Kühnberger (2019) treat a combination of AGE and HEIGHT as a conceptual space (see Fig. 4 on p. 14).…”

“…The choice of this metric is justified by the intuitive assumption that these dimensions are perceived separately and are thus not integral. Furthermore, Bechberger and Kühnberger (2019) emphasise that only cuboids are convex in Manhattan spaces. As a consequence, the correlation between younger age and smaller size is not captured in a convex representation.…”

“…As a consequence, the correlation between younger age and smaller size is not captured in a convex representation. According to Bechberger and Kühnberger (2019), one should give up convexity requirements in favour of representing this correlation. They prefer the non-convex representation in Fig.…”

“…For these multi-domain concepts, correlations between dimensions become more salient. However, as noted by Bechberger and Kühnberger (2019), the representation of correlations can prima facie conflict with convexity requirements.…”

“…The dimensions HEIGHT and AGE covary within the category of children. Bechberger and Kühnberger (2019) convincingly argue that only a nonconvex representation of the concept child can capture this correlation. Moreover, they also claim that the non-convex representation is closer to an intuitive understanding of child.…”