Formalizing Complex Mathematical Statements with LLMs: A Study on Mathematical Definitions (2502.12065v2)

Abstract: Thanks to their linguistic capabilities, LLMs offer an opportunity to bridge the gap between informal mathematics and formal languages through autoformalization. However, it is still unclear how well LLMs generalize to sophisticated and naturally occurring mathematical statements. To address this gap, we investigate the task of autoformalizing real-world mathematical definitions -- a critical component of mathematical discourse. Specifically, we introduce two novel resources for autoformalisation, collecting definitions from Wikipedia (Def_Wiki) and arXiv papers (Def_ArXiv). We then systematically evaluate a range of LLMs, analyzing their ability to formalize definitions into Isabelle/HOL. Furthermore, we investigate strategies to enhance LLMs' performance including refinement through external feedback from Proof Assistants, and formal definition grounding, where we guide LLMs through relevant contextual elements from formal mathematical libraries. Our findings reveal that definitions present a greater challenge compared to existing benchmarks, such as miniF2F. In particular, we found that LLMs still struggle with self-correction, and aligning with relevant mathematical libraries. At the same time, structured refinement methods and definition grounding strategies yield notable improvements of up to 16% on self-correction capabilities and 43% on the reduction of undefined errors, highlighting promising directions for enhancing LLM-based autoformalization in real-world scenarios.

• The paper introduces novel datasets, Def_Wiki and Def_ArXiv, to benchmark LLMs in translating informal mathematical definitions into formal languages.
• The study evaluates models such as GPT-4o using Isabelle/HOL, revealing significant performance differences and challenges in autoformalization.
• The research proposes feedback-driven refinement and definition grounding strategies that improve self-correction by 16% and reduce undefined errors by 43%.

Formalizing Complex Mathematical Statements with LLMs: A Study on Mathematical Definitions

The paper Formalizing Complex Mathematical Statements with LLMs: A Study on Mathematical Definitions addresses the challenge of translating informal mathematical definitions into formal language using LLMs. This research is significant as it extends the autoformalization landscape beyond basic mathematical problems to more complex and real-world mathematical statements, offering insights into how well LLMs can bridge the gap between informal mathematical discourse and formal verification processes.

Key Contributions

1. Introduction of New Datasets: The authors develop two novel datasets — Def_Wiki and Def_ArXiv. Def_Wiki comprises definitions extracted from Wikipedia articles, while Def_ArXiv includes definitions from machine learning research papers. These datasets are specifically designed to test LLMs on their capacity to formalize sophisticated mathematical definitions, which are more intricate and context-dependent than those found in existing benchmarks.
2. Evaluation of LLMs: The paper conducts a systematic evaluation of various LLMs, including DeepSeekMath-7B, Llama3-8B, and GPT-4o, by assessing their ability to autoformalize definitions into Isabelle/HOL—a proof assistant language. The findings denote a marked difference in the performance of these models across different datasets, with GPT-4o showing relatively better success rates.
3. Challenges in Autoformalization: The evaluations reveal several challenges LLMs face, such as alignment with mathematical libraries and self-correction capabilities. Specifically, LLMs frequently generate incomplete formalizations and have difficulty incorporating relevant formal mathematical contexts.
4. Refinement Strategies: Two primary strategies are proposed to enhance LLM performance — (i) structured refinement through detailed feedback mechanisms from proof assistants, and (ii) formal definition grounding where LLMs are guided by introducing contextual elements from formal mathematical libraries. These strategies achieve up to 16% improvement in self-correction and 43% reduction in undefined errors.
5. Error Analysis and Categorical Refinement: An error analysis is conducted to identify typical failure points, leading to the development of categorical refinement techniques that guide LLMs to correct specific error types using structured instructions.

Implications and Future Directions

The paper has practical implications for improving the accuracy and reliability of autoformalization processes in areas requiring rigorous formal verification. The research highlights that while LLMs like GPT-4o can significantly enhance autoformalization, there is a need for further advancements in handling complex mathematical contexts and improving alignment with formal languages.

Future research could explore the integration of more sophisticated feedback mechanisms and adaptive techniques that allow LLMs to learn from varied and complex contexts automatically. Moreover, extending the paper to other proof assistants and formal languages could further validate the findings and lead to broader applications in AI-assisted formal verification processes.

In conclusion, the paper is a pivotal step toward leveraging LLMs for formalizing complex mathematical discourse, emphasizing the need for innovative strategies to address the unique challenges posed by real-world mathematical statements.