A Categorical Semantics of Fuzzy Concepts in Conceptual Spaces

TL;DR

This paper introduces a categorical framework for modeling fuzzy concepts and reasoning within conceptual spaces, utilizing log-concave functions to handle fuzzy reasoning with noisy inputs in a compositional manner.

Contribution

It develops a symmetric monoidal category for fuzzy concepts using log-concave functions, generalizes to probabilistic channels, and models fuzzy reasoning with noise within a novel Markov category.

Findings

Log-concave functions satisfy key criteria for fuzzy concepts.

The framework supports compositional fuzzy reasoning.

It introduces a Markov category for fuzzy probabilistic channels.

Abstract

We define a symmetric monoidal category modelling fuzzy concepts and fuzzy conceptual reasoning within G\"ardenfors' framework of conceptual (convex) spaces. We propose log-concave functions as models of fuzzy concepts, showing that these are the most general choice satisfying a criterion due to G\"ardenfors and which are well-behaved compositionally. We then generalise these to define the category of log-concave probabilistic channels between convex spaces, which allows one to model fuzzy reasoning with noisy inputs, and provides a novel example of a Markov category.