Interpretable Graph Networks Formulate Universal Algebra Conjectures
Francesco Giannini Stefano Fioravanti Oguzhan Keskin Alisia Maria Lupidi Lucie Charlotte Magister Pietro Lio Pietro Barbiero
The rise of Artificial Intelligence (AI) recently empowered researchers to
investigate hard mathematical problems which eluded traditional approaches for
decades. Yet, the use of AI in Universal Algebra (UA) -- one of the fields
laying the foundations of modern mathematics -- is still completely unexplored.
This work proposes the first use of AI to investigate UA's conjectures with an
equivalent equational and topological characterization. While topological
representations would enable the analysis of such properties using graph neural
networks, the limited transparency and brittle explainability of these models
hinder their straightforward use to empirically validate existing conjectures
or to formulate new ones. To bridge these gaps, we propose a general algorithm
generating AI-ready datasets based on UA's conjectures, and introduce a novel
neural layer to build fully interpretable graph networks. The results of our
experiments demonstrate that interpretable graph networks: (i) enhance
interpretability without sacrificing task accuracy, (ii) strongly generalize
when predicting universal algebra's properties, (iii) generate simple
explanations that empirically validate existing conjectures, and (iv) identify
subgraphs suggesting the formulation of novel conjectures.