# Autoformalization

This review systematically surveys the progress, challenges, and prospects of AI for Mathematics (AI4Math), organizing the field into problem-specific modeling (guiding intuition, constructing counterexamples, formal reasoning in closed systems) and general-purpose modeling (natural language reasoning, formal reasoning, mathematical information retrieval), and reports the authors' own PKU undergraduate and PhD qualifying exam evaluations: GPT-4 averages below 60 while reasoning-enhanced models such as o1, DeepSeek-R1, o3-mini, and Gemini 2.5 Pro mostly exceed 90, with o3-mini averaging 84.4 on 58 PhD qualifying exam problems, while research-level mathematics remains an open challenge.

This review systematically surveys the progress, challenges, and prospects of AI for Mathematics (AI4Math), organizing the field into problem-specific modeling (guiding intuition, constructing counterexamples, formal reasoning in closed systems) and general-purpose modeling (natural language reasoning, formal reasoning, mathematical information retrieval), and reports the authors' own PKU undergraduate and PhD qualifying exam evaluations: GPT-4 averages below 60 while reasoning-enhanced models such as o1, DeepSeek-R1, o3-mini, and Gemini 2.5 Pro mostly exceed 90, with o3-mini averaging 84.4 on 58 PhD qualifying exam problems, while research-level mathematics remains an open challenge.

The work proposes Proof-Sketch-Guided Formal Model Synthesis (ProGS), an autoformalization method centered on model-based proof sketches: a sketch represents the proof structure of the target formal system as a tree, with internal nodes capturing case splits and inductive reasoning steps and leaf nodes corresponding to concrete state-transition events that realize individual subgoals; LLMs generate and repair these sketches, and verification failures are mapped back to specific nodes and subtrees to provide structured guidance for iterative repair; evaluation on a benchmark of 27 formal systems shows ProGS improves over state-of-the-art agentic formal modeling approaches in syntactic validity, deductive verifiability, and behavioral correctness.

The work proposes Proof-Sketch-Guided Formal Model Synthesis (ProGS), an autoformalization method centered on model-based proof sketches: a sketch represents the proof structure of the target formal system as a tree, with internal nodes capturing case splits and inductive reasoning steps and leaf nodes corresponding to concrete state-transition events that realize individual subgoals; LLMs generate and repair these sketches, and verification failures are mapped back to specific nodes and subtrees to provide structured guidance for iterative repair; evaluation on a benchmark of 27 formal systems shows ProGS improves over state-of-the-art agentic formal modeling approaches in syntactic validity, deductive verifiability, and behavioral correctness.