# A Thorough Formalization of Conceptual Spaces

## TL;DR

This paper refines the conceptual spaces framework by introducing fuzzy star-shaped sets, enabling better representation of correlations and efficient operations for learning and reasoning in knowledge representation.

## Contribution

It formalizes conceptual spaces with fuzzy star-shaped sets, addressing convexity issues and enhancing the framework's ability to model domain correlations and support computational operations.

## Key findings

- Formalization using fuzzy star-shaped sets improves conceptual space modeling
- Defines efficient operations for intersection, union, and projection
- Supports learning and reasoning processes in knowledge representation

## Abstract

The highly influential framework of conceptual spaces provides a geometric way of representing knowledge. Instances are represented by points in a high-dimensional 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 computationally efficient operations on concepts (intersection, union, and projection onto a subspace) and show that these operations can support both learning and reasoning processes.