Autoformalization aims to convert informal mathematical proofs into machine-verifiable formats, bridging the gap between natural and formal languages. However, ensuring semantic alignment between the informal and formalized statements remains challenging. Existing approaches heavily rely on manual verification, hindering scalability. To address this, we introduce \textsc{FormalAlign}, the first automated framework designed for evaluating the alignment between natural and formal languages in autoformalization. \textsc{FormalAlign} trains on both the autoformalization sequence generation task and the representational alignment between input and output, employing a dual loss that combines a pair of mutually enhancing autoformalization and alignment tasks. Evaluated across four benchmarks augmented by our proposed misalignment strategies, \textsc{FormalAlign} demonstrates superior performance. In our experiments, \textsc{FormalAlign} outperforms GPT-4, achieving an Alignment-Selection Score 11.58\% higher on \forml-Basic (99.21\% vs. 88.91\%) and 3.19\% higher on MiniF2F-Valid (66.39\% vs. 64.34\%). This effective alignment evaluation significantly reduces the need for manual verification. Both the dataset and code can be accessed via~\url{https://github.com/rookie-joe/FormalAlign}.

Large language models (LLMs) have exhibited their problem-solving ability in mathematical reasoning. Solving realistic optimization (OPT) problems in industrial application scenarios requires advanced and applied math ability. However, current OPT benchmarks that merely solve linear programming are far from complex realistic situations. In this work, we propose E-OPT, a benchmark for end-to-end optimization problem-solving with human-readable inputs and outputs. E-OPT contains rich optimization problems, including linear/nonlinear programming with/without table data, which can comprehensively evaluate LLMs' solving ability. In our benchmark, LLMs are required to correctly understand the problem in E-OPT and call code solver to get precise numerical answers. Furthermore, to alleviate the data scarcity for optimization problems, and to bridge the gap between open-source LLMs on a small scale (e.g., Llama-2-7b and Llama-3-8b) and closed-source LLMs (e.g., GPT-4), we further propose a novel data synthesis method namely ReSocratic. Unlike general data synthesis methods that proceed from questions to answers, ReSocratic first incrementally synthesizes optimization scenarios with mathematical formulations step by step and then back-translates the generated scenarios into questions. In such a way, we construct the ReSocratic-29k dataset from a small seed sample pool with the powerful open-source large model DeepSeek-V2. To demonstrate the effectiveness of ReSocratic, we conduct supervised fine-tuning with ReSocratic-29k on multiple open-source models. The results show that Llama3-8b is significantly improved from 13.6% to 51.7% on E-OPT, while DeepSeek-V2 reaches 61.0%, approaching 65.5% of GPT-4.

Formal verification (FV) has witnessed growing significance with current emerging program synthesis by the evolving large language models (LLMs). However, current formal verification mainly resorts to symbolic verifiers or hand-craft rules, resulting in limitations for extensive and flexible verification. On the other hand, formal languages for automated theorem proving, such as Isabelle, as another line of rigorous verification, are maintained with comprehensive rules and theorems. In this paper, we propose FVEL, an interactive Formal Verification Environment with LLMs. Specifically, FVEL transforms a given code to be verified into Isabelle, and then conducts verification via neural automated theorem proving with an LLM. The joined paradigm leverages the rigorous yet abundant formulated and organized rules in Isabelle and is also convenient for introducing and adjusting cutting-edge LLMs. To achieve this goal, we extract a large-scale FVELER3. The FVELER dataset includes code dependencies and verification processes that are formulated in Isabelle, containing 758 theories, 29,125 lemmas, and 200,646 proof steps in total with in-depth dependencies. We benchmark FVELER in the FVEL environment by first fine-tuning LLMs with FVELER and then evaluating them on Code2Inv and SV-COMP. The results show that FVEL with FVELER fine-tuned Llama3- 8B solves 17.39% (69 -> 81) more problems, and Mistral-7B 12% (75 -> 84) more problems in SV-COMP. And the proportion of proof errors is reduced. Project page: https://fveler.github.io/.

Autoformalization, the conversion of natural language mathematics into formal languages, offers significant potential for advancing mathematical reasoning. However, existing efforts are limited to formal languages with substantial online corpora and struggle to keep pace with rapidly evolving languages like Lean 4. To bridge this gap, we propose a new benchmark \textbf{Form}alization for \textbf{L}ean~\textbf{4} (\textbf{\name}) designed to evaluate the autoformalization capabilities of large language models (LLMs). This benchmark encompasses a comprehensive assessment of questions, answers, formal statements, and proofs. Additionally, we introduce a \textbf{P}rocess-\textbf{S}upervised \textbf{V}erifier (\textbf{PSV}) model that leverages the precise feedback from Lean 4 compilers to enhance autoformalization. Our experiments demonstrate that the PSV method improves autoformalization, enabling higher accuracy using less filtered training data. Furthermore, when fine-tuned with data containing detailed process information, PSV can leverage the data more effectively, leading to more significant improvements in autoformalization for Lean 4. Our dataset and code are available at \url{https://github.com/rookie-joe/PDA}.

In this work, we propose a novel method named \textbf{Auto}mated Process Labeling via \textbf{C}onfidence \textbf{V}ariation (\textbf{\textsc{AutoCV}}) to enhance the reasoning capabilities of large language models (LLMs) by automatically annotating the reasoning steps. Our approach begins by training a verification model on the correctness of final answers, enabling it to generate automatic process annotations. This verification model assigns a confidence score to each reasoning step, indicating the probability of arriving at the correct final answer from that point onward. We detect relative changes in the verification's confidence scores across reasoning steps to automatically annotate the reasoning process. This alleviates the need for numerous manual annotations or the high computational costs associated with model-induced annotation approaches. We experimentally validate that the confidence variations learned by the verification model trained on the final answer correctness can effectively identify errors in the reasoning steps. Subsequently, we demonstrate that the process annotations generated by \textsc{AutoCV} can improve the accuracy of the verification model in selecting the correct answer from multiple outputs generated by LLMs. Notably, we achieve substantial improvements across five datasets in mathematics and commonsense reasoning. The source code of \textsc{AutoCV} is available at \url{https://github.com/rookie-joe/AUTOCV}.

Recent advances in automated theorem proving leverages language models to explore expanded search spaces by step-by-step proof generation. However, such approaches are usually based on short-sighted heuristics (e.g., log probability or value function scores) that potentially lead to suboptimal or even distracting subgoals, preventing us from finding longer proofs. To address this challenge, we propose POETRY (PrOvE Theorems RecursivelY), which proves theorems in a recursive, level-by-level manner in the Isabelle theorem prover. Unlike previous step-by-step methods, POETRY searches for a verifiable sketch of the proof at each level and focuses on solving the current level's theorem or conjecture. Detailed proofs of intermediate conjectures within the sketch are temporarily replaced by a placeholder tactic called sorry, deferring their proofs to subsequent levels. This approach allows the theorem to be tackled incrementally by outlining the overall theorem at the first level and then solving the intermediate conjectures at deeper levels. Experiments are conducted on the miniF2F and PISA datasets and significant performance gains are observed in our POETRY approach over state-of-the-art methods. POETRY on miniF2F achieves an average proving success rate improvement of 5.1%. Moreover, we observe a substantial increase in the maximum proof length found by POETRY, from 10 to 26.

Recent large language models (LLMs) have witnessed significant advancement in various tasks, including mathematical reasoning and theorem proving. As these two tasks require strict and formal multi-step inference, they are appealing domains for exploring the reasoning ability of LLMs but still face important challenges. Previous studies such as Chain-of-Thought (CoT) have revealed the effectiveness of intermediate steps guidance. However, such step-wise annotation requires heavy labor, leading to insufficient training steps for current benchmarks. To fill this gap, this work introduces MUSTARD, a data generation framework that masters uniform synthesis of theorem and proof data of high quality and diversity. MUSTARD synthesizes data in three stages: (1) It samples a few mathematical concept seeds as the problem category. (2) Then, it prompts a generative language model with the sampled concepts to obtain both the problems and their step-wise formal solutions. (3) Lastly, the framework utilizes a proof assistant (e.g., Lean Prover) to filter the valid proofs. With the proposed MUSTARD, we present a theorem-and-proof benchmark MUSTARDSAUCE with 5,866 valid data points. Each data point contains an informal statement, an informal proof, and a translated formal proof that passes the prover validation. We perform extensive analysis and demonstrate that MUSTARD generates validated high-quality step-by-step data. We further apply the MUSTARDSAUCE for fine-tuning smaller language models. The fine-tuned Llama 2-7B achieves a 15.41% average relative performance gain in automated theorem proving, and 8.18% in math word problems. Codes and data are available at https://github.com/Eleanor-H/MUSTARD.

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 Figure 1: An example theorem generated by GPT-4 splitting the Metamath library into three sets: (OpenAI, 2023). GPT-4 wrongly refers to the intermediate axioms, library, and problem based on their theorem (A (B A) A A) as proving depth. We conduct extensive experiments ((A B) (A A)). In Step 4, it applies "ax-to investigate whether current LMs can 1" but obtains the wrong expression instead of correct generate theorems in the library and benefit (A (B A)) and can not derive (A A) even the problem theorems proving. The results with the incorrect Steps 4 and 5. demonstrate that high-quality ATG data facilitates models' performances on downstream

In-context learning (ICL) with large language models (LLMs) has become the modern tools of choice for many natural language processing tasks. However, how the text style of in-context examples influences the performance of LLMs still remains under-explored. This paper presents a novel and effective approach, named \textbf{AlignedCoT}, to improve the reasoning capability of LLMs by aligning the in-context examples with the native style of LLMs.''Native'' refers to the inherent characteristic of LLMs which can be probed by zero-shot scenarios.AlignedCoT is widely applicable to ICL methods, making it easy to combine with state-of-the-art techniques to further improve the LLMs' performance. We conduct extensive and comprehensive experiments on several benchmarks on mathematical question-answering, common-sense reasoning, and text understanding. The empirical results demonstrate that our AlignedCoT significantly improves performance over the carefully handcrafted demonstrations. Specifically, with AlignedCoT, we observe an average +3.2\% improvement for \texttt{gpt-3.5-turbo} compared to the carefully handcrafted CoT on multi-step reasoning benchmarks.Furthermore, we use AlignedCoT to rewrite the CoT text style in the training set, which improves the performance of Retrieval Augmented Generation by 3.6\%.The source code and dataset is available at https://github.com/yangzhch6/AlignedCoT

In our study, we delve into the realm of evaluating Despite large language models (Arkoudas, 2023; large language models' (LLMs) ability to generate OpenAI, 2022) perform strikingly in plenty of reasoning counterfactually coherent thoughts. Specifically, benchmarks (Cobbe et al., 2021; Hendrycks we proposed an innovative evaluation system et al., 2021a), late studies observe an internal inconsistency that quantitatively measures the evolution of information in their reasoning processes (Saparov and in statement pairs, ensuring that they adhere He, 2023; Arkoudas, 2023). The inconsistency is to a specified logical relationship. Our approach attributed to misunderstanding and misapplication includes designing a specialized task where models of logical relations. However, logical relations in are presented with mismatched argument-premise complex language reasoning are not yet properly pairs bound by a specific logical relation. The objective quantified and evaluated.