Formalized Conceptual Spaces with a Geometric Representation of Correlations
This work provides a theoretical extension to the conceptual spaces framework, potentially benefiting knowledge representation and cognitive modeling research, though it appears incremental with illustrative examples rather than empirical validation.
The paper addresses limitations in the conceptual spaces framework's convexity requirement by proposing a formalization based on fuzzy star-shaped sets, which enables geometric representation of correlations between domains and defines operations for concept manipulation and relation measurement.
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.