Graph2Tac: Learning Hierarchical Representations of Math Concepts in Theorem proving

Concepts abound in mathematics and its applications. They vary greatly between subject areas, and new ones are introduced in each mathematical paper or application. A formal theory builds a hierarchy of definitions, theorems and proofs that reference each other. When an AI agent is proving a new theorem, most of the mathematical concepts and lemmas relevant to that theorem may have never been seen during training. This is especially true in the Coq proof assistant, which has a diverse library of Coq projects, each with its own definitions, lemmas, and even custom tactic procedures used to prove those lemmas. It is essential for agents to incorporate such new information into their knowledge base on the fly. We work towards this goal by utilizing a new, large-scale, graph-based dataset for machine learning in Coq. We leverage a faithful graph-representation of Coq terms that induces a directed graph of dependencies between definitions to create a novel graph neural network, Graph2Tac (G2T), that takes into account not only the current goal, but also the entire hierarchy of definitions that led to the current goal. G2T is an online model that is deeply integrated into the users' workflow and can adapt in real time to new Coq projects and their definitions. It complements well with other online models that learn in real time from new proof scripts. Our novel definition embedding task, which is trained to compute representations of mathematical concepts not seen during training, boosts the performance of the neural network to rival state-of-the-art k-nearest neighbor predictors.