A Categorical Semantics of Fuzzy Concepts in Conceptual Spaces
This work provides a formal categorical semantics for fuzzy conceptual reasoning, which is incremental in extending existing frameworks to handle noise and compositionality.
The paper tackles the problem of modeling fuzzy concepts and reasoning within Gärdenfors' conceptual spaces framework by defining a symmetric monoidal category using log-concave functions, and extends this to probabilistic channels for noisy inputs, establishing a novel Markov category.
We define a symmetric monoidal category modelling fuzzy concepts and fuzzy conceptual reasoning within Gärdenfors' 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 Gärdenfors 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.