Categorical Foundations of Explainable AI: A Unifying Formalism of Structures and Semantics

Explainable AI (XAI) aims to answer ethical and legal questions associated with the deployment of AI models. However, a considerable number of domain-specific reviews highlight the need of a mathematical foundation for the key notions in the field, considering that even the term "explanation" still lacks a precise definition. These reviews also advocate for a sound and unifying formalism for explainable AI, to avoid the emergence of ill-posed questions, and to help researchers navigate a rapidly growing body of knowledge. To the authors knowledge, this paper is the first attempt to fill this gap by formalizing a unifying theory of XAI. Employing the framework of category theory, and feedback monoidal categories in particular, we first provide formal definitions for all essential terms in explainable AI. Then we propose a taxonomy of the field following the proposed structure, showing how the introduced theory can be used to categorize all the main classes of XAI systems currently studied in literature. In summary, the foundation of XAI proposed in this paper represents a significant tool to properly frame future research lines, and a precious guidance for new researchers approaching the field.