Formal Mathematical Reasoning: A New Frontier in AI

TL;DR

<3-5 sentence high-level summary> Formal mathematical reasoning via proof assistants is proposed as a pivotal complement to traditional math LLMs in AI4Math, addressing data scarcity and verifiability by grounding reasoning in formal environments like Lean. The paper surveys autoformalization, neural theorem proving, verified reasoning in natural language, and formal-system verification, and outlines a roadmap of milestones and benchmarks to measure progress. It discusses data, algorithmic, and tooling challenges, advocates a hybrid of general-purpose and domain-specific systems, and emphasizes human-centered evaluation and collaborative tooling. The envisioned impact spans rigorous mathematical discovery, more reliable software/hardware verification, and scalable collaboration in formal mathematics.</3-5 sentence high-level summary>

Abstract

AI for Mathematics (AI4Math) is not only intriguing intellectually but also crucial for AI-driven discovery in science, engineering, and beyond. Extensive efforts on AI4Math have mirrored techniques in NLP, in particular, training large language models on carefully curated math datasets in text form. As a complementary yet less explored avenue, formal mathematical reasoning is grounded in formal systems such as proof assistants, which can verify the correctness of reasoning and provide automatic feedback. In this position paper, we advocate for formal mathematical reasoning and argue that it is indispensable for advancing AI4Math to the next level. In recent years, we have seen steady progress in using AI to perform formal reasoning, including core tasks such as theorem proving and autoformalization, as well as emerging applications such as verifiable generation of code and hardware designs. However, significant challenges remain to be solved for AI to truly master mathematics and achieve broader impact. We summarize existing progress, discuss open challenges, and envision critical milestones to measure future success. At this inflection point for formal mathematical reasoning, we call on the research community to come together to drive transformative advancements in this field.

Figures (5)

• Figure 1: State-of-the-art math LLMs such as NuminaMath numina typically undergo three stages: math pretraining, finetuning on step-by-step solutions, and further finetuning on tool-integrated solutions that interleave natural language reasoning with Python tool invocation.
• Figure 2: Formalizing mathematics using the Lean proof assistant de2015leanmoura2021lean.
• Figure 3: Common tasks using AI for formal mathematical reasoning in proof assistants.
• Figure 4: A neural theorem prover that combines tactic generation and proof search. This architecture is adopted by the majority of existing methods, with only a handful of exceptions first2023baldurxin2024deepseek.
• Figure 5: Autoformalization translates informal math to formal theorems and/or proofs automatically.