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智能化定理证明技术与系统

Intelligent Theorem Proving Techniques and Systems

  • 摘要: 自动化定理证明一直是数学与计算机领域的核心难题之一,旨在让机器能够自动或半自动地完成形式化定理的证明,在数学证明、软件形式验证等领域有着重要的应用前景。近期的研究工作表明以大语言模型为代表的智能化技术可以有效生成部分形式证明,并在国际数学奥林匹克竞赛中展现出接近人类顶尖专家的水平。这为实现自动化定理证明提供了新的路径。本文旨在梳理近期基于智能化技术的定理证明发展现状,并展望未来的研究方向。具体而言,本文首先介绍了经典定理证明器的基本分类与思路,并从自动形式化、引理选择与证明生成3个角度回顾近期智能化定理证明技术的进展。随后,本文介绍了近期典型的数学定理证明系统,以及定理证明技术在软件形式验证领域的应用情况。最后,本文探讨了当前定理证明面临的主要挑战以及未来的研究方向。

     

    Abstract: Automating theorem proving has long been one of the core challenges in mathematics and computer science. It aims to enable machines to (semi-)automatically generate proofs of formal theorems, and has important application prospects in areas such as mathematical theorem proving and software verification. Recent progress has shown that intelligent techniques represented by large language models can effectively generate formal proofs in certain domains, and have demonstrated human-expert-level performance in the International Mathematical Olympiad. This provides a new pathway for addressing the challenge of theorem proving. This article aims to review the recent status of theorem proving based on intelligent techniques. Specifically, this article first introduces the basic classifications and approaches of classical theorem provers, and covers recent progress in intelligent theorem proving from three aspects: autoformalization, premise selection, and proof generation. Then, this article introduces several representative theorem proving systems, as well as the application of theorem proving techniques in the field of software verification. Finally, this article discusses the main challenges currently faced by theorem proving and outlines future research directions.

     

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