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Formal Mathematical Reasoning: A New Frontier in AI

Kaiyu Yang, Gabriel Poesia, Jingxuan He, Wenda Li, Kristin Lauter, Swarat Chaudhuri, Dawn Song

arXiv:2412.16075v1cs.AIcs.LGcs.LO

TL;DR

Advanced mathematics is difficult for informal math-language-model approaches in part because high-quality training data is scarce. This position paper advocates formal mathematical reasoning and synthesizes progress, open challenges, and milestones. It describes progress in theorem proving and autoformalization, alongside applications such as formal verification.

  • Problem

    Informal math approaches are difficult to extend to advanced mathematics, where high-quality training data is difficult to obtain.

    Source: §2.1 · p. 4

  • Method

    The paper presents a position on formal mathematical reasoning and synthesizes progress, open challenges, and milestones for theorem proving, autoformalization, and related applications.

    Source: §1 · p. 3, §6 · p. 26

  • Results

    The paper reports progress in formal reasoning, including AlphaProof reaching silver-medal level at the IMO and applications in formal verification.

    Source: §1 · p. 3

  • Takeaways & Limitations

    Formal systems such as Lean can support rigorous verification of AI reasoning, with potential applications in mathematics and software verification.

    Source: §6 · p. 26

    The informal approach has been limited to domains where abundant high-quality data is available at low cost, making advanced mathematics difficult to address.

    Source: §2.1 · p. 4

Abstract

from arXiv · show

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.

1 Introduction

AI4Math has drawn on NLP-style training of math language models, but advanced mathematics exposes limits of that informal approach. The paper advocates formal mathematical reasoning as a complementary direction, reviews its progress, and identifies challenges and future milestones.

1 Introduction

  • The informal approach trains math LLMs on curated text datasets and uses inference-time techniques such as chain-of-thought prompting, self-consistency, and tool use.The passage reports success on GSM8K, MATH, and the AIMO Progress Prize, while noting that performance had mostly been limited to high-school math up to AIME level.Source: §1 · pp. 1–2
  • Advanced mathematics challenges the informal approach because high-quality training data is scarce and intricate proofs are difficult to evaluate, while scaling training data is difficult.The paper also describes inference-time search with neural verifiers as an emerging approach whose effectiveness on advanced problems remains open.Source: §1 · p. 2
  • Formal mathematical reasoning grounds models in formal systems that can verify reasoning and provide automatic feedback, enabling rigorous test-time checks against hallucination.The paper includes first- and higher-order logic, dependent type theory, and programs annotated with formal specifications among these systems.Source: §1 · p. 2
  • AlphaProof and AlphaGeometry use symbolic representations and proof-checking frameworks to execute neural reasoning steps and generate synthetic data, contributing to olympiad-level capabilities.AlphaProof combines autoformalization with reinforcement learning and reached silver-medal level at the IMO.Source: §1 · pp. 2–3
  • The paper positions formal reasoning as complementary to informal mathematics and maps open challenges, future directions, and milestones rather than aiming to be a comprehensive survey.It points to combining natural-language reasoning with autoformalization as a possible route to rigorously solve informal problems.Source: §1 · p. 3

2 AI for Mathematics (AI4Math) and the Formal Turn

The informal approach to AI4Math relies on increasingly curated training data, but advanced mathematics exposes limits in both data availability and proof evaluation. Formal systems offer soundness checks and automatic feedback, supporting theorem proving and autoformalization alongside complementary approaches to mathematical AI.

2.1 State-of-the-art Math LLMs and Their Limitations

  • Math LLMs are pretrained on math documents, finetuned on step-by-step solutions, then trained on tool-integrated trajectories that interleave natural-language reasoning with Python.NuminaMath’s example includes a 860K-problem dataset covering high-school and competition math.Source: §2.1 · pp. 3–4, Figure 1 · p. 4
  • The informal approach depends on abundant, high-quality data, making extension from pre-college math to data-scarce advanced mathematics difficult.Novel research problems are unlikely to have similar online solutions, and large-scale manual annotation is infeasible.Source: §2.1 · p. 4
  • Numeric-answer benchmarks do not directly evaluate advanced mathematics, where abstract conjectures and proofs require rigorous verification.Proof checking is difficult even for experienced mathematicians, and current LLMs struggle to verify their own outputs without external feedback.Source: §2.1 · pp. 4–5

2.2 AI for Formal Mathematical Reasoning

  • Formal systems encode mathematics through well-formed formulas and inference rules, enabling soundness checks and automatic feedback that can support learning and evaluation.Formal systems include axiomatic set theory, higher-order logic, and dependent type theory.Source: §2.2 · p. 5
  • In Lean, theorem proving proceeds interactively: tactics transform a goal into subgoals until none remain, and the assistant checks the resulting proof.Lean files can build on reusable libraries such as mathlib.Source: §2.2 · pp. 5–6
  • Autoformalization translates informal mathematics into formal statements or proofs, while neural theorem proving generates formal proofs from theorem statements, often using tactic generation and search.The prover’s search algorithm expands a tree by executing generated tactics until a proof is found.Source: §2.2 · pp. 6–7
  • Formal and informal reasoning can complement one another: autoformalization can expose natural-language reasoning to formal feedback that helps filter invalid steps.This integration is described as verified reasoning in natural language.Source: §2.2 · p. 7

2.3 Other Directions in AI for Mathematics

  • Other AI4Math directions include neural approximation of mathematical functions and domain-specific systems that use symbolic objects, search, or reinforcement learning on open problems.The cited examples include applications in cryptography, theoretical physics, control theory, graph theory, and combinatorics.Source: §2.3 · p. 8

3 Recent Progress in AI for Formal Mathematical Reasoning

The section surveys progress in autoformalization and theorem proving, then turns to verifiable reasoning applications in natural language and system verification. It highlights both useful mechanisms and open challenges.

3.1 Autoformalization

  • Rule-based systems use controlled grammars, while neural methods handle natural-language edge cases more flexibly but often rely on scarce aligned corpora.Autoformalization also supports prover training, informal-proof guidance, and validation of informal reasoning.Source: §3.1 · pp. 8–9
  • About 30% accuracy in formalizing competition-level statements and over 70% in informalizing more challenging undergraduate problems show the task asymmetry reported for LLMs.Fewer than five expert-crafted examples supported translation; synthetic aligned corpora and formal-environment signals were used in follow-up autoformalizers.Source: §3.1 · p. 9

3.2 Neural Theorem Proving

  • Expert iteration turns verifier-accepted proofs into training data, while proof-error messages can guide search; gains diminish after a few iterations and unproductive mistakes remain difficult to identify.Formal proving uses learned heuristics because proving in sufficiently expressive systems is undecidable.Source: §3.2 · pp. 9–10
  • Library learning explores reusable lemmas and higher-level abstractions, while tactic induction remains demonstrated only in simpler formalisms.LEGO-Prover proposes reusable lemmas; full-fledged Lean or Coq tactic languages remain undeveloped for tactic induction.Source: §3.2 · p. 10
  • Retrieval-augmented proving supplies relevant library lemmas, and the “new premises” split tests the need for premises unavailable during training.That split approximates formalizing a new domain that builds on recently proved lemmas.Source: §3.2 · p. 10

3.3 Verified Reasoning in Natural Language

  • Natural-language verifiers can improve performance without guaranteeing validity, while solver-based approaches have difficulty confirming that an ambiguous natural-language problem was formalized correctly.SAT/SMT systems may hide step-by-step reasoning inside the solver; LogicGuide instead constrains deductions while alternating formal and natural-language steps.Source: §3.3 · p. 11

3.4 Formal System Verification and Verified Generation

  • Formal system verification specifies correctness and security requirements before proving them or finding bugs, but specifications and proofs can demand substantial manual effort.The seL4 kernel verification involved more than 20 person-years; AI assistance can generate initial proofs, refine existing ones, and support SMT-based tasks.Source: §3.4 · p. 11
  • Coupling code generation with formal verification is proposed to address buggy or insecure generated code, while transpilation can use source code as a complete specification for equivalent target code.Other proposed routes develop verified artifacts before compilation or use verification-friendly languages such as Dafny and Verus.Source: §3.4 · pp. 11–12

4 Open Challenges and Future Directions

This section outlines open challenges and future directions for formal mathematical reasoning, spanning data, algorithms, AI tools, and verifiable code. It considers specialized algorithms and general models, more usable and collaborative proof assistants, and AI generation linked to formal verification, while highlighting challenges in human workflows, semantic correctness, and managing evolving systems’ specifications and proofs.

4.1 Data

  • Formal proof data remains scarce: Proof Pile contains only 500MB, and research-level mathematics has even less informal data, making scale alone an uncertain path to progress.The paper argues that data may remain scarce at the research frontier because abundant data could mean little novel research remains in that domain.Source: §4.1 · p. 12
  • Proposed data strategies include autoformalizing informal sources, generating synthetic problems and proofs, transferring across proof frameworks and modalities, and combining approaches as AlphaProof did.AlphaProof autoformalized one million IMO-like informal problems into one hundred million formal theorems, with synthetic proofs generated through expert iteration; generalizing this to research mathematics remains open.Source: §4.1 · pp. 12–13

4.2 Algorithms

  • Scaling autoformalization requires evaluation metrics aligned with human judgment, smaller reasoning steps, and interaction with proof systems to handle definitions and gaps in informal proofs.BLEU and compilation success do not correlate with human evaluations; logical-equivalence checks are one proposed metric.Source: §4.2 · pp. 13–14
  • Mathematical models must combine knowledge memorization with multi-step reasoning, long-context handling, abstraction, and planning, while Transformer reasoning generalization remains an open question.The paper proposes procedurally generated formal benchmarks with adjustable difficulty for controlled studies that can diagnose failures, including generalization to longer reasoning chains.Source: §4.2 · p. 14
  • Proof search has driven successes such as AlphaGeometry and AlphaProof, but systematic comparisons of search methods, models, decoding, and resource constraints are lacking.Value models could prioritize proof goals, yet goal provability is sensitive to small changes and negative training examples are difficult to obtain; theorem proving also has an unbounded action space that search cannot overcome without high-quality actions.Source: §4.2 · p. 15
  • Hierarchical proof decomposition and learned abstractions remain difficult: systems can sketch or split goals and synthesize lemmas, but realistic decomposition and new tactics in full-fledged provers remain open challenges.Mathematical abstractions include definitions, lemmas, and tactics that encode useful domain-specific strategies.Source: §4.2 · p. 16
  • Explicit retrieval can supply relevant definitions and lemmas missing from model knowledge, motivating formal-math-specific retrievers and methods for managing knowledge bases that grow over time.Existing systems use standard retrieval methods; one proposed direction uses structured neurosymbolic embeddings, while another caches useful proof subgoals as modules.Source: §4.2 · p. 17
  • Generalist systems can share knowledge across domains and identify connections, while specialists exploit domain-specific techniques; combining both approaches is a research direction for mathematical AI.Specialists can support domain-focused collaboration and offer controllable tasks for studying model reasoning.Source: §4.2 · pp. 17–18
  • AI math specialists and generalists are expected to coexist, with domain-specific algorithms used as tools by general LLMs and potentially learned universally.The paper identifies combining specialists and generalists as an avenue for progress.Source: §4.2 · p. 18

4.3 Tools for Assisting Human Mathematicians

  • Usability, rather than technical features, often bottlenecks adoption, motivating user-friendly engineering and ecological studies of mathematicians using proof assistants [226] [227].The paper suggests adapting methods from Human-Computer Interaction and Programming Languages research to understand users’ learning, conceptualization, confusion, and challenges.Source: §4.3 · p. 18
  • Proof assistants support modular, large-scale mathematical collaboration, with future human-computer projects potentially enabling AI agents to contribute autonomously through correctness-guaranteeing formal tools.Lean Blueprints exemplifies modular project organization, while distributed computing platforms such as Folding@Home offer a model for broader participation.Source: §4.3 · p. 18

4.4 Formal Verification and Verified Generation

  • Formal reasoning can assure correctness and security in AI-generated software and hardware, but semantic properties beyond syntax remain challenging despite constrained decoding [231] [232].Formal methods may also help programmers understand generated code, as demonstrated by runtime-value display [233].Source: §4.4 · pp. 18–19
  • Formal verification requires encoding bespoke system semantics and correctness requirements, while large, changing codebases make proof and specification management essential [142].The seL4 kernel verification comprises about 200,000 lines of specifications and proofs; the paper identifies AI-assisted proof engineering as underexplored.Source: §4.4 · p. 19
  • AI-based generation can jointly produce code or designs, specifications, and proofs for verifier checking [168] [234] [235] [236], but generated specifications must faithfully reflect developer intent.Specifications must be neither too strong nor too weak.Source: §4.4 · p. 19

5 Milestones and Success Measures

The paper proposes a capability-level framework across formal-mathematics tasks and argues that existing benchmarks miss higher-level capabilities, motivating new benchmarks and evaluation methods.

5 Milestones and Success Measures

  • The proposed framework defines capability levels for theorem proving and autoformalization, reviews existing benchmarks, and identifies gaps especially in emerging areas such as conjecturing.The authors call for new benchmarks and, in many cases, new evaluation methodologies.Source: §5 · p. 19

5.1 Theorem Proving

  • Current theorem-proving evaluations reach only Level 3, covering domain-general proof generation for relatively simple theorems.Higher levels envision autonomous formalization projects and mathematical discovery, for which meaningful progress measurement remains a challenge.Source: §5.1 · pp. 20–21

5.2 Verified Reasoning in Natural Language

  • Verified reasoning in natural language aims to combine formal and informal steps, selectively formalizing parts of a problem rather than requiring the whole problem to be formalized.The proposed evaluation should assess reasoning quality and include problems that resist complete formalization.Source: §5.2 · pp. 21–22

5.3 Autoformalization

  • Autoformalization milestones progress from human-viable encoding toward feedback-backed translation, reasoning over information gaps, self-correction, and novel definitions.Current pipelines lack a mature system for collecting and synchronizing informal-formal pairs, while automatic evaluation of translations remains difficult.Source: §5.3 · pp. 23–24

5.4 Conjecturing

  • Evaluating conjectures requires addressing what makes them interesting, useful, and valuable as mathematics changes.Novelty alone is insufficient because systematically generated conjectures may be uninteresting or unimportant.Source: §5.4 · pp. 24–25
  • Minimo’s conjecturing-proving loop generates increasingly difficult provable conjectures from domain axioms and improves performance on unseen theorems, but has been tested only in small domains.The approach remains untested at the scale of large libraries, and Minimo began to saturate after up to five iterations.Source: §5.4 · p. 25

5.5 Formal Verification and Verified Generation

  • Verification milestones progress from small code and design properties to project-scale verification, proof maintenance, and assistance with formal specifications.GPT-4o proves around 35% of miniCodeProps properties, and no high-quality benchmark covers generating verifiable code together with its specification even at Level 0.Source: §5.5 · pp. 25–26

6 Conclusion and Discussion

The paper argues that formal mathematical reasoning can support rigorously verifiable AI, while noting that benchmark evaluation may miss what mathematicians find useful.

6 Conclusion and Discussion

  • Formal systems such as Lean enable rigorous verification of AI reasoning, with potential applications in mathematics, software verification, and other rigorous reasoning tasks.The paper presents formal mathematical reasoning as a new frontier and calls for progress across theorem proving and autoformalization.Source: §6 · p. 26
  • The paper surveys advances and future directions in theorem proving and autoformalization, and proposes milestones for measuring progress.Source: §6 · p. 26
  • Benchmarks may not capture what mathematicians find meaningful or helpful, particularly on new and evolving formalization projects, motivating human-centered evaluation.The paper cites LeanDojo and PutnamBench as examples and points to formalizing Fermat’s Last Theorem as an evolving project.Source: §6 · p. 27
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