Explainable AI (XAI) aims to address the human need for safe and reliable AI systems. However, numerous surveys emphasize the absence of a sound mathematical formalization of key XAI notions -- remarkably including the term "explanation" which still lacks a precise definition. To bridge this gap, this paper presents the first mathematically rigorous definitions of key XAI notions and processes, using the well-funded formalism of Category theory. We show that our categorical framework allows to: (i) model existing learning schemes and architectures, (ii) formally define the term "explanation", (iii) establish a theoretical basis for XAI taxonomies, and (iv) analyze commonly overlooked aspects of explaining methods. As a consequence, our categorical framework promotes the ethical and secure deployment of AI technologies as it represents a significant step towards a sound theoretical foundation of explainable AI.
@article{arxiv.2304.14094,
title = {Categorical Foundations of Explainable AI: A Unifying Theory},
author = {Pietro Barbiero and Stefano Fioravanti and Francesco Giannini and Alberto Tonda and Pietro Lio and Elena Di Lavore},
journal= {arXiv preprint arXiv:2304.14094},
year = {2023}
}