Title: Usefulness-driven Learning of Formal Mathematics
Primary area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)
Keywords: reasoning, formal mathematics, logic, conjecturing, theorem proving, reinforcement learning

Abstract: Creating an AI that can truly "do" mathematics requires more than just solving isolated problems. It must mimic the creative, progressive nature of human mathematicians, who build upon previous work to generate new knowledge. A crucial part of this process is proposing theorems that serve as useful building blocks for proving more advanced theorems. In this paper, we introduce UseForm, a novel framework that formalizes this notion of usefulness, and demonstrates how it can be used to train a usefulness-driven AI mathematician. UseForm determines a theorem's usefulness based on two criteria: its reusability in subsequent proofs and its contribution to increasing proof likelihood. We integrate UseForm into the self-play conjecturing-and-proving setting of Minimo (Poesia et al., 2024). That is, starting from only axioms, we iteratively train conjecturers to propose useful formal statements and provers that explicitly reuse them when generating formal proofs. We experimentally evaluate this usefulness-driven self-play approach across three mathematical domains: arithmetic, propositional logic, and group theory. Our evaluation considers two metrics: intrinsic usefulness, which measures how often our trained provers reuse theorems, and extrinsic usefulness, judged by a state-of-the-art large language model and external provers like SMT solvers. Our results demonstrate that our usefulness-trained model effectively generates a large number of intrinsically and extrinsically useful formal theorems. For instance, our approach outperforms the original Minimo by 2.9 times in extrinsic usefulness for arithmetic. Our work highlights the significant potential of integrating usefulness in AI-driven mathematical discovery.

TL;DR: We conjecture lemmas useful for automated theorem proving from base axioms through a usefulness-aware self-play loop.