Formalized Conceptual Spaces with a Geometric Representation of Correlations

TL;DR

This paper enhances the conceptual spaces framework by formalizing it with fuzzy star-shaped sets, enabling geometric representation of correlations and new operations for concept manipulation.

Contribution

It introduces a formalization using fuzzy star-shaped sets, extending the framework to represent correlations and define operations for concept formation.

Findings

Formalization with fuzzy star-shaped sets improves the representation of correlations.

New operations enable creation and comparison of concepts.

Illustrative example demonstrates practical application.

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.