Interpretable Graph Networks Formulate Universal Algebra Conjectures
This work addresses the lack of AI applications in Universal Algebra, enabling researchers to explore hard mathematical problems with improved transparency and empirical validation.
The authors tackled the problem of using AI to investigate conjectures in Universal Algebra, a foundational mathematical field, by developing interpretable graph networks that enhance interpretability without sacrificing accuracy, strongly generalize in predicting properties, validate existing conjectures, and suggest new ones.
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.