ATG: Benchmarking Automated Theorem Generation for Generative Language Models
This work addresses the challenge of automated theorem generation for generative language models, which is incremental as it builds on existing theorem-proving benchmarks by focusing on generation rather than just proving.
The paper tackles the problem of generative language models' limited ability to generate new or reusable theorems, which hinders proving harder theorems with exponentially growing search spaces, by proposing an Automated Theorem Generation (ATG) benchmark; results show that high-quality ATG data improves downstream automated theorem proving performance, but current models still have room for generating more advanced theorems.
Humans can develop new theorems to explore broader and more complex mathematical results. While current generative language models (LMs) have achieved significant improvement in automatically proving theorems, their ability to generate new or reusable theorems is still under-explored. Without the new theorems, current LMs struggle to prove harder theorems that are distant from the given hypotheses with the exponentially growing search space. Therefore, this paper proposes an Automated Theorem Generation (ATG) benchmark that evaluates whether an agent can automatically generate valuable (and possibly brand new) theorems that are applicable for downstream theorem proving as reusable knowledge. Specifically, we construct the ATG benchmark by splitting the Metamath library into three sets: axioms, library, and problem based on their proving depth. We conduct extensive experiments to investigate whether current LMs can generate theorems in the library and benefit the problem theorems proving. The results demonstrate that high-quality ATG data facilitates models' performances on downstream ATP. However, there is still room for current LMs to develop better ATG and generate more advanced and human-like theorems. We hope the new ATG challenge can shed some light on advanced complex theorem proving.