Herald: A Natural Language Annotated Lean 4 Dataset
Herald: Translates Mathlib4 to natural language using dual augmentation, enhancing LLM performance in mathematical reasoning.
Key Findings
Methodology
The paper introduces a novel framework to translate Mathlib4 into natural language using the Lean-jixia system. It employs dual augmentation strategies: tactic-based and informal-based, resulting in the Herald dataset and a fine-tuned Herald translator.
Key Results
- The Herald translator achieved 93.2% accuracy on the miniF2F-test, significantly outperforming InternLM2-Math-Plus-7B at 74.0% and TheoremLlama at 50.1%.
- On the internal graduate-level textbook dataset, Herald achieved 22.5% accuracy, far exceeding InternLM2-Math-Plus-7B's 7.5%.
- The section-level translation framework was successfully applied to a template section in the Stack project.
Significance
The development of the Herald dataset and translator significantly enhances LLMs' automation capabilities in mathematical reasoning, addressing the scarcity of aligned natural-formal language datasets and providing a new path for automatic formalization of mathematical literature.
Technical Contribution
Herald introduces a structural-information-aware augmentation pipeline, offering a method to expand NL-FL datasets from any Lean project, improving LLM performance and enabling project-level formalization.
Novelty
Herald is the first to achieve large-scale translation of Mathlib4 into natural language, enhancing dataset quality and coverage through dual augmentation strategies.
Limitations
- Herald may struggle with translating complex mathematical concepts accurately.
- The translation order of dependencies is crucial, potentially affecting efficiency.
Future Work
Future work could explore autoformalization in more complex mathematical domains and applications in other formal languages.
AI Executive Summary
In mathematical reasoning, formal languages like Lean significantly reduce human errors but require extensive expertise to write. The scarcity of aligned natural-formal language datasets limits the training of large language models. To address this, the paper introduces the Herald framework, which translates Mathlib4 into natural language using dual augmentation strategies. The Herald translator excels across multiple datasets, particularly on the miniF2F-test and internal graduate-level textbook datasets. The research's significance lies in providing a new path for the automatic formalization of mathematical literature, advancing the automation of mathematical reasoning. Although Herald has limitations in handling complex concepts, its open-source nature provides a foundation for further research.
Deep Analysis
Background
Formal languages like Lean are widely used in mathematical reasoning, enabling automated proof verification and reducing human errors. However, writing these languages requires extensive expertise, and the scarcity of aligned natural-formal language datasets limits the training of large language models.
Core Problem
The core problem is the lack of aligned natural-formal language datasets, limiting the application of large language models in mathematical reasoning. Solving this issue is crucial for enhancing the automatic formalization of mathematical literature.
Innovation
The paper introduces the Herald framework, which translates Mathlib4 into natural language using dual augmentation strategies. Innovations include using the Lean-jixia system for analysis and employing tactic-based and informal-based strategies to enhance dataset quality and coverage.
Methodology
- �� Analyze Mathlib4 using the Lean-jixia system.
- �� Employ dual augmentation strategies: tactic-based and informal-based.
- �� Generate the Herald dataset and fine-tune the Herald translator.
Experiments
The experimental design includes testing the Herald translator on the miniF2F-test and internal graduate-level textbook datasets, comparing its performance with InternLM2-Math-Plus-7B and TheoremLlama.
Results
The Herald translator achieved 93.2% accuracy on the miniF2F-test and 22.5% on the internal graduate-level textbook dataset, significantly outperforming other models.
Applications
The Herald translator can be used for the automatic formalization of mathematical literature, particularly in graduate-level textbooks requiring high-precision translation.
Limitations & Outlook
Herald may struggle with translating complex mathematical concepts accurately, and the translation order of dependencies is crucial, potentially affecting efficiency.
Plain Language Accessible to non-experts
Imagine you're in a kitchen cooking. Herald acts like a smart assistant that helps you translate complex recipes into simple steps. You just tell it what dish you want to make, and it automatically generates a detailed cooking guide based on the available ingredients and steps. This way, even if you're not familiar with cooking, you can easily prepare a meal. Herald plays a similar role in mathematics, translating complex proofs into natural language, making them accessible to more people.
ELI14 Explained like you're 14
Hey there! Imagine you're playing a super complex game with lots of levels and quests. Herald is like an awesome guide that helps you break down these complex tasks into simple steps. This way, you can easily pass each level and become a game master! Herald does the same in math, making complex proofs easy to understand so you can become a math whiz too!
Glossary
Herald
A dataset and translator that converts Mathlib4 into natural language.
Used to enhance large language models' performance in mathematical reasoning.
Lean 4
A formal language used for mathematical reasoning.
Used in the paper to verify mathematical proofs.
Mathlib4
The mathematical library for Lean 4.
Serves as the source for the Herald dataset.
LLM
Large language model used for natural language processing tasks.
Used in the paper for translation and reasoning.
Autoformalization
The process of translating natural language mathematical reasoning into formal language.
Core function of Herald.
Open Questions Unanswered questions from this research
- 1 How to achieve autoformalization in more complex mathematical domains?
- 2 How to improve Herald's accuracy in handling complex concepts?
Applications
Immediate Applications
Mathematics Textbook Translation
Herald can be used to translate graduate-level mathematics textbooks, enhancing learning efficiency.
Long-term Vision
Mathematical Research Automation
Herald has the potential to automate more complex mathematical research in the future.
Abstract
Verifiable formal languages like Lean have profoundly impacted mathematical reasoning, particularly through the use of large language models (LLMs) for automated reasoning. A significant challenge in training LLMs for these formal languages is the lack of parallel datasets that align natural language with formal language proofs. To address this challenge, this paper introduces a novel framework for translating the Mathlib4 corpus (a unified library of mathematics in formal language Lean 4) into natural language. Building upon this, we employ a dual augmentation strategy that combines tactic-based and informal-based approaches, leveraging the Lean-jixia system, a Lean 4 analyzer. We present the results of this pipeline on Mathlib4 as Herald (Hierarchy and Retrieval-based Translated Lean Dataset). We also propose the Herald Translator, which is fine-tuned on Herald. Herald translator achieves a 93.2% accuracy (Pass@128) on formalizing statements in the miniF2F-test and a 22.5% accuracy on our internal graduate-level textbook dataset, outperforming InternLM2-Math-Plus-7B (74.0% and 7.5%) and TheoremLlama (50.1% and 4.0%). Furthermore, we propose a section-level translation framework for real-world applications. As a direct application of Herald translator, we have successfully translated a template section in the Stack project, marking a notable progress in the automatic formalization of graduate-level mathematical literature. Our model, along with the datasets, are open-sourced to the public.