Autoformalization: Bridging Informal and Formal Math

Updated 6 August 2025

Autoformalization is the process of automatically translating informal math content into machine-verifiable formal proofs, bridging human reasoning with rigorous formal systems.
It leverages neural machine translation, large language models, and neuro-symbolic pipelines to ensure syntactic correctness and semantic consistency.
Empirical evaluations demonstrate enhanced pass rates and accuracy, making autoformalization pivotal for scalable theorem proving, formal verification, and reliable mathematical library construction.

Autoformalization is the task of automatically translating informal mathematical content—typically written in human natural language or LaTeX—into formal statements or proofs in a machine-verifiable proof assistant language. This process aims to bridge the gap between human mathematical discourse and formal systems, undergirding advances in automated reasoning, formal verification, and large-scale mathematical knowledge management.

1. Problem Definition and Significance

Autoformalization can be formally characterized as a mapping $f\colon S \rightarrow F$ , where $S$ is the set of informal statements and $F$ is the space of corresponding formal representations, i.e., for $s \in S$ there exists $c \in F$ with $f(s) = c$ (Zhang et al., 17 Feb 2025). The process underpins applications in automated theorem proving, formal verification, and the construction of reliable mathematical libraries (Weng et al., 29 May 2025). Manually formalizing mathematics is labor-intensive and error-prone; automated approaches offer scalability and rigorous checks that single out subtleties often missed by informal folk mathematics (Patel et al., 2023, Zhang et al., 12 Jun 2025).

The importance of autoformalization is amplified by its role in:

Enabling high-confidence formal verification in mathematics, program synthesis, and system reliability (Wu et al., 2022, Weng et al., 29 May 2025).
Lowering the entry barrier to formal methods, making advanced formal verification approachable to non-experts (Lu et al., 4 Jun 2024).
Enhancing the trust and interpretability of LLM-generated quantitative reasoning through formal verification layers (Zhou et al., 26 Mar 2024).

2. Core Methodologies

Multiple paradigms have been pursued, combining neural, symbolic, and neuro-symbolic techniques. The primary approaches are as follows:

Neural Machine Translation Frameworks

Neural machine translation (NMT) architectures treat informal–formal translation as a sequence modeling problem, with both supervised (encoder–decoder with attention) and unsupervised (backtranslation, denoising, shared encoders) instantiations (Wang et al., 2019). Cross-lingual pretraining, using masked LLMing, is used to bolster unsupervised performance by enabling language-agnostic representations. For example, the decoder probability at each time step is

$P(y_{t}\mid y_{1}\ldots y_{t-1}, x) = \mathrm{softmax}(W\cdot h_t)$

where $h_t$ is the (attention-augmented) decoder state (Wang et al., 2019).

LLMs and Few-shot Learning

LLMs trained on web-scale text and code (e.g., Codex, GPT-4, PaLM) exhibit non-trivial ability to translate informal mathematical statements—even at competition-level difficulty—into formal languages like Isabelle/HOL, Lean, and Coq in a few-shot setting (Wu et al., 2022, Xie et al., 15 Jul 2025). Carefully engineered prompts and in-context retrieval methods further improve performance, especially when tailored to the mathematical domain at hand (Azerbayev et al., 2023).

Neuro-symbolic and Process-driven Pipelines

Hybrid frameworks combine LLMs for initial translation and symbolic verification engines (theorem provers, SMT solvers) for post-generation validation (Murphy et al., 27 May 2024, Zhou et al., 26 Mar 2024). In these frameworks, generated code is processed via a proof assistant’s typechecker and automated tactic engines to ensure internal consistency, followed by iterative error feedback to prompt model refinement. Feedback loops based on Lean REPL or Isabelle verification are integral for improving semantic quality, filtering invalid outputs, and providing fine-grained process supervision (Lu et al., 4 Jun 2024, Poiroux et al., 11 Jun 2024).

Consistency, Alignment, and Retrieval-Augmented Generation

Recent advances emphasize semantic and syntactic consistency, leveraging:

Symbolic equivalence via automated theorem proving and logical matching (Li et al., 28 Oct 2024);
Semantic alignment by informalizing formal candidates and comparing their embedding-based similarity to the original statement (Li et al., 28 Oct 2024, Lu et al., 14 Oct 2024);
Contextually rich few-shot prompts retrieved dynamically from large formal corpora (MS-RAG) to ensure terminology and style consistency (Zhang et al., 5 Oct 2024);
Iterative error-driven auto-correction with formal tool feedback (Auto-SEF) (Zhang et al., 5 Oct 2024).

Data Augmentation and Backtranslation

Synthetic data is generated by informalizing large-scale formal corpora (e.g., using GPT-4 to generate natural language from formal Lean/Isabelle) and backtranslating these into the formal domain—a strategy that enables scaling training (even multilingual) with minimal manual annotation (Jiang et al., 2023, Chan et al., 18 Feb 2025). High-fidelity, quality-controlled synthetic corpora have been found to be more effective than large, more heterogeneous datasets (Chan et al., 18 Feb 2025).

3. Evaluation Frameworks and Benchmarking

Rigorous assessment of autoformalization systems hinges on both syntactic and semantic metrics:

Syntactic correctness: Does the formal code typecheck and compile in the theorem prover? Pass rates and type error proportions are standard metrics (Poiroux et al., 11 Jun 2024).
Semantic equivalence: Does the formalization capture the intended meaning of the informal statement? Methods include BLEU, edit distance, manual annotation of semantic consistency, symbolic equivalence classes (using ATPs), and embedding-based measures (Wang et al., 2019, Li et al., 28 Oct 2024, Lu et al., 14 Oct 2024).

Tables of benchmark datasets and models:

Dataset	Language(s)	Domain Level
ProofNet	Lean 3/4	Undergraduate, diversified
miniF2F	Lean, Isabelle	Olympiad, high school
MMA	Lean4, Isabelle	Multilingual/multidomain
LeanEuclid	Lean	Euclidean geometry
FMC	Lean	Olympiad
arXiv2Formal	Lean 3	Research mathematics
Def_Wiki/ArXiv	Isabelle/HOL	Advanced ML definitions

Model/Framework	Main Features
GPT-4, DeepSeekMath	LLMs with few-shot, multilingual, pretraining
Codex, PaLM	Code-trained LLMs, few-shot
LeanDojo, DeepSeek-Prover	Proof-oriented autoformalization
FormalAlign	Automated alignment evaluation, dual loss
MS-RAG, Auto-SEF	Retrieval, denoising, error-driven correction

4. Empirical Results and Performance Analysis

Performance gaps between base LLMs and autoformalization-optimized systems are pronounced. Typical findings:

On miniF2F and ProofNet, vanilla LLMs achieve 0–16% pass rates, but with data augmentation, error filtering, and self-consistency selection, rates climb as high as 53.2% (Jiang et al., 2023, Poiroux et al., 11 Jun 2024).
Quality-first data generation (distilled backtranslation, line-wise proof state annotation) yields substantially better token efficiency and accuracy than training on large, diverse, but loosely aligned datasets (Chan et al., 18 Feb 2025).
Techniques such as symbolic equivalence-based reranking, semantic consistency (embedding-based), and alignment-evaluation methods (FormalAlign) deliver significant absolute accuracy gains—up to 1.35x improvement in pass@1 compared to log-probability selection (Li et al., 28 Oct 2024, Lu et al., 14 Oct 2024).

Robustness remains limited in highly abstract or specialized topics (e.g., category theory, advanced topology), and performance can drop sharply when provers’ libraries lack the definitions used in the informal text (Gulati et al., 1 Jun 2024, Zhang et al., 17 Feb 2025).

5. Challenges, Limitations, and Open Problems

Data Scarcity: The dearth of aligned informal–formal pairs hinders both model generalization and systematic evaluation (Jiang et al., 2023, Weng et al., 29 May 2025).
Syntactic and Semantic Failures: Many errors arise from type mismatches, undefined symbols (e.g., referencing non-existent predicates), and inconsistencies in referencing formal libraries (Zhang et al., 17 Feb 2025, Poiroux et al., 11 Jun 2024).
Handling Implicitness and Context: Research-level mathematics often omits critical definitions and types, making entity linking, context modeling, and type refinement essential (Patel et al., 2023).
Compositional and Cross-Domain Generalization: LLMs tend to underperform on long, compositionally complex inputs or less represented mathematical subdisciplines (Gulati et al., 1 Jun 2024).
Evaluation Scalability: Manual verification does not scale; automated, multi-granular evaluation techniques—such as “ensemble of LLM judges” decomposed into logical preservation, mathematical consistency, and formal quality—better approximate expert judgment (Zhang et al., 12 Jun 2025).

6. Future Directions and Prospects

Research priorities articulated across the literature include:

Construction and curation of larger, higher-fidelity, and more diverse autoformalization datasets spanning underrepresented domains and definition types (Zhang et al., 17 Feb 2025, Jiang et al., 2023, Xie et al., 15 Jul 2025).
Tighter integration of symbolic reasoning and LLMs, especially process-level feedback, stepwise refinement, and error correction using theorem prover APIs (Lu et al., 4 Jun 2024, Zhang et al., 5 Oct 2024).
Enhanced entity linking, implicit assumption detection, and context-aware grounding to better align informal text with formal library content (Patel et al., 2023, Zhang et al., 17 Feb 2025).
Development of hybrid evaluators that fuse automated symbolic/semantic checks with interpretable LLM-based judgment for scalable yet nuanced assessment (Lu et al., 14 Oct 2024, Zhang et al., 12 Jun 2025).
Methods improving self-correction, definition grounding, and generalization to research-grade mathematical writing through structured refinement and prompt engineering (Zhang et al., 17 Feb 2025, Li et al., 28 Oct 2024).
Broadened applications, including formalization of game-theoretic scenarios, program verification, and translation of complex scientific/technical protocols (Mensfelt et al., 18 Sep 2024).

7. Broader Impacts and Implications

Autoformalization enables automated theorem provers and LLMs to function as collaborative partners in mathematical discovery, proof auditing, and knowledge curation. Integration of autoformalization and rigorous formal verification holds promise for scientific reproducibility, programmable mathematics, and the reliable deployment of formal methods in software engineering, hardware verification, and beyond (Weng et al., 29 May 2025, Zhou et al., 26 Mar 2024).

The trend toward multidimensional evaluation, high-fidelity synthetic corpora, and process-driven symbolic verification signals a maturing ecosystem where human and machine faculties can be jointly leveraged to scale trustworthy mathematical reasoning.