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.More advanced theorem proving is if an agent (for instance, a generative LM) can leverage its creativity to generate new but also reasonable theorems that properly substitute part of a proof and also be saved as reusable knowledge for future theorem proving.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.

Self-Consistency samples diverse reasoning chains with answers and chooses the final answer by majority voting. It is based on forward reasoning and cannot further improve performance by sampling more reasoning chains when saturated. To further boost performance, we introduce backward reasoning to verify candidate answers. Specifically, for mathematical tasks, we mask a number in the question and ask the LLM to answer a backward question created by a simple template, i.e., to predict the masked number when a candidate answer is provided. Instead of using forward or backward reasoning alone, we propose **FOBAR** to combine **FO**rward and **BA**ckward **R**easoning for verification. Extensive experiments on six standard mathematical data sets and three LLMs show that FOBAR achieves state-of-the-art performance. In particular, FOBAR outperforms Self-Consistency, which uses forward reasoning alone, demonstrating that combining forward and backward reasoning is more accurate in verification. In addition, FOBAR achieves higher accuracy than existing verification methods, showing the effectiveness of the simple template used in backward reasoning and the proposed combination.

Recent advances in neural theorem-proving resort to large language models and tree searches. When proving a theorem, a language model advises single-step actions based on the current proving state and the tree search finds a sequence of correct steps using actions given by the language model. However, prior works often conduct constant computation efforts for each proving state while ignoring that the hard states often need more exploration than easy states. Moreover, they evaluate and guide the proof search solely depending on the current proof state instead of considering the whole proof trajectory as human reasoning does. Here, to accommodate general theorems, we propose a novel Dynamic-Tree Driven Theorem Solver (DT-Solver) by guiding the search procedure with state confidence and proof-level values. Specifically, DT-Solver introduces a dynamic-tree Monte-Carlo search algorithm, which dynamically allocates computing budgets for different state confidences, guided by a new proof-level value function to discover proof states that require substantial exploration. Experiments on two popular theorem-proving datasets, PISA and Mathlib, show significant performance gains by our DT-Solver over the state-of-the-art approaches, with a 6.65% improvement on average in terms of success rate. And especially under low computing resource settings (11.03% improvement on average).

Automated theorem proving (ATP) has become an appealing domain for exploring the reasoning ability of the recent successful generative language models. However, current ATP benchmarks are mainly focus on symbolic inference, but rarely involve the understanding of complex number combination reasoning. In this work, we propose TRIGO, an ATP benchmark that not only requires a model to reduce a trigonometric expression with step-by-step proof but also evaluates a generative LM’s reasoning ability on formulas and capability to manipulate, group, and factor number terms. We gather trigonometric expressions and their reduced forms from web, annotate the simplification process manually, and translate it into the “Lean” formal language system. We then automatically generate additional examples from the annotated samples to expand the dataset. Furthermore, we also create three automatically generated training and testing datasets of varying difficulty and distributions. Our extensive experiments show our proposed TRIGO poses a new challenge for advanced generative LM’s including GPT-4 which is pre-trained on a considerable amount of open-source formal theorem-proving language data, and provide a new tool to study the generative LM’s ability on both formal and mathematical reasoning.