Interpretable Graph Networks Formulate Universal Algebra Conjectures

TL;DR

This paper introduces interpretable graph neural networks to analyze and validate conjectures in Universal Algebra, enabling better understanding and discovery of mathematical properties through AI.

Contribution

It presents the first AI approach for Universal Algebra conjectures using interpretable graph networks and a new dataset generation algorithm.

Findings

Enhanced interpretability without loss of accuracy

Strong generalization in property prediction

Empirical validation and formulation of conjectures

Abstract

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.