The combination of verifiable languages and LLMs has significantly influenced both the mathematical and computer science communities because it provides a rigorous foundation for theorem proving. Recent advancements in the field provide foundation models and sophisticated agentic systems pushing the boundaries of formal mathematical reasoning to approach the natural language capability of LLMs. However, little attention has been given to the formal physics reasoning, which also heavily relies on similar problem-solving and theorem-proving frameworks. To solve this problem, this paper presents, to the best of our knowledge, the first approach to enhance formal theorem proving in the physics domain. We compose a dedicated dataset PhysLeanData for the task. It is composed of theorems sampled from PhysLean and data generated by a conjecture-based formal data generation pipeline. In the training pipeline, we leverage DeepSeek-Prover-V2-7B, a strong open-source mathematical theorem prover, and apply Reinforcement Learning with Verifiable Rewards (RLVR) to train our model PhysProver. Comprehensive experiments demonstrate that, using only about 5.5K training samples, PhysProver achieves an overall 2.4% improvement in multiple sub-domains. Furthermore, after formal physics training, we observe 1.3% gains on the MiniF2F-Test benchmark, which indicates non-trivial generalization beyond physics domains and enhancement for formal math capability as well. The results highlight the effectiveness and efficiency of our approach, which provides a paradigm for extending formal provers outside mathematical domains. To foster further research, we will release both our dataset and model to the community.

Enhancing the formal math reasoning capabilities of Large Language Models (LLMs) has become a key focus in both mathematical and computer science communities in recent years. While significant progress has been made in using state-of-the-art Auto-Regressive (AR) LLMs for formal theorem proving, these models suffer from inherent limitations. Their next-token prediction generation methods may yield suboptimal performance due to the challenges of long-range coherence and the compounding of errors over long sequences. Recent advancements in diffusion LLMs (dLLMs), which generate text through iterative denoising of a multi-token block, offer a promising alternative. However, the application of dLLMs to formal mathematics, where maintaining long-range coherence is critical, remains largely understudied. To address the challenges above, we propose Diffusion-Proof, to the best of our knowledge, the first framework to train and apply dLLMs for formal theorem proving. Our frameworks contain training and inference methods for two models. The first one is dLLM-Prover-7B, which performs whole-proof writing with long-range coherent tactic usage. The second one is dLLM-Corrector-7B, which is a novel large block diffusion-based correction model. It leverages the in-filling capabilities of dLLMs to perform local proof correction using bi-directional information. Extensive experiments demonstrate that Diffusion-Proof relatively significantly outperforms the AR LLM baseline trained under the same dataset. Diffusion-Proof achieves an absolute improvement of 1.61% on ProofNet-Test and 6.14% on MiniF2F-Test benchmarks compare to the baseline. Notably, Diffusion-Proof successfully resolves one IMO problem that more advanced thinking model DeepSeek-Prover-V2-7B could not solve, showcasing the unique advantage of dLLMs in formal theorem proving.

Enhancing the mathematical reasoning capabilities of LLMs has garnered significant attention in both the mathematical and computer science communities. Recent works have made substantial progress in both Natural Language (NL) reasoning and Formal Language (FL) reasoning by leveraging the potential of pure Reinforcement Learning (RL) methods on base models. However, RL approaches struggle to impart new capabilities not presented in the base model, highlighting the need to integrate more knowledge like FL into NL math reasoning effectively. Yet, this integration is challenging due to inherent disparities in problem structure and reasoning format between NL and FL. To address these challenges, we introduce NL-FL HybridReasoning (NFL-HR), an end-to-end framework designed to incorporate the FL expert into NL math problem-solving. To bridge the NL and FL input format gap, we propose the NL-FL Problem Alignment method, which reformulates the Question-Answering (QA) problems in NL as existence theorems in FL. Subsequently, the Mixed Problem Input technique we provide enables the FL reasoner to handle both QA and existence problems concurrently. Lastly, we mitigate the NL and FL output format gap in reasoning through an LLM-based Answer Extraction mechanism. Comprehensive experiments demonstrate that the NFL-HR framework achieves 89.80% and 84.34% accuracy rates on the MATH-500 and the AMC benchmarks, surpassing the NL baseline by 4.60% and 4.82%, respectively. Notably, some problems resolved by our framework remain unsolved by the NL baseline model even under a larger number of trials.

Large Language Models (LLMs) have displayed astonishing abilities in various tasks, especially in text generation, classification, question answering, etc. However, the reasoning ability of LLMs still faces many debates, especially in math reasoning. The inherent ambiguity of Natural Language (NL) limits LLMs’ ability to perform verifiable reasoning, making the answers lack coherence and trustworthy support. To tackle the above challenges, we propose a novel framework named FANS: Formal ANswer Selection for LLM Natural Language Math Reasoning Using Lean4. It is a pioneering framework that utilizes Lean4 to enhance LLMs’ NL math reasoning ability. In particular, given an NL math question and LLM-generated answers, FANS first translates it into Lean4 theorem statements. Then it invokes another Lean4 prover LLM to produce proofs, and finally verifies the proofs by Lean4 compiler. Answers are selected based on the verifications. It enhances LLMs’ NL math ability in providing a computer-verifiable solution for its correct answer and proposes an alternative method for answer selection beyond the reward model based ones. Our experiments demonstrate the effectiveness of FANS with an improvement of nearly 2% across several math benchmarks, and even higher further based on reward models or in subfields such as algebra and number theory that Lean4 is better at. The code is available in https://github.com/MaxwellJryao/FANS.

Proving mathematical theorems using computer-verifiable formal languages like Lean significantly impacts mathematical reasoning. One approach to formal theorem proving involves generating complete proofs using Large Language Models (LLMs) based on Natural Language (NL) proofs. However, due to the scarcity of aligned NL and Formal Language (FL) theorem-proving data most modern LLMs exhibit suboptimal performance.This scarcity results in a paucity of methodologies for training LLMs and techniques to fully utilize their capabilities in composing formal proofs. To address these challenges, this paper proposes TheoremLlama, an end-to-end framework that trains a general-purpose LLM to be a Lean4 expert. TheoremLlama includes NL-FL dataset generation and bootstrapping method to obtain aligned dataset, curriculum learning and block training techniques to train the model, and iterative proof writing method to write Lean4 proofs that work together synergistically.Using the dataset generation method in TheoremLlama, we provide Open Bootstrapped Theorems (OBT), an NL-FL aligned and bootstrapped dataset. Our novel NL-FL bootstrapping method, where NL proofs are integrated into Lean4 code for training datasets, leverages the NL reasoning ability of LLMs for formal reasoning. The TheoremLlama framework achieves cumulative accuracies of 36.48% and 33.61% on MiniF2F-Valid and Test datasets respectively, surpassing the GPT-4 baseline of 22.95% and 25.41%. Our code, model checkpoints, and the generated dataset is published in GitHub

Data Synthesis is a promising way to train a small model with very little labeled data. One approach for data synthesis is to leverage the rich knowledge from large language models to synthesize pseudo training examples for small models, making it possible to achieve both data and compute efficiency at the same time. However, a key challenge in data synthesis is that the synthesized dataset often suffers from a large distributional discrepancy from the real task data distribution. Thus, in this paper, we propose Synthesis Step by Step (S3), a data synthesis framework that shrinks this distribution gap by iteratively extrapolating the errors made by a small model trained on the synthesized dataset on a small real-world validation dataset using a large language model. Extensive experiments on multiple NLP tasks show that our approach improves the performance of a small model by reducing the gap between the synthetic dataset and the real data, resulting in significant improvement compared to several baselines: 9.48% improvement compared to ZeroGen and 2.73% compared to GoldGen, and at most 15.17% improvement compared to the small model trained on human-annotated data.