Decompose, Structure, and Repair: A Neuro-Symbolic Framework for Autoformalization via Operator Trees

Statement autoformalization acts as a critical bridge between human mathematics and formal mathematics by translating natural language problems into formal language. While prior works have focused on data synthesis and diverse training paradigms to optimize end-to-end Large Language Models (LLMs), they typically treat formal code as flat sequences, neglecting the hierarchical logic inherent in mathematical statements. In this work, we introduce Decompose, Structure, and Repair (DSR), a neuro-symbolic framework that restructures autoformalization into a modular pipeline. DSR decomposes statements into logical components and maps them to structured operator trees, leveraging this topological blueprint to precisely localize and repair errors via sub-tree refinement. Furthermore, we introduce PRIME, a benchmark of 156 undergraduate and graduate-level theorems selected from canonical textbooks and expertly annotated in Lean 4. Experimental results demonstrate that DSR establishes a new state-of-the-art, consistently outperforming baselines under equivalent computational budgets. The datasets, model, and code are available at https://github.com/XiaoyangLiu-sjtu/DSR.

Teaching computers to understand and verify advanced human mathematics is incredibly difficult. Current AI models try to translate math problems step-by-step like a regular spoken language, which often leads to cascading logical mistakes. We created a new system called DSR that fundamentally changes how AI approaches this translation. Instead of guessing word-by-word, DSR acts like a human mathematician: it first breaks the problem down and draws a structural "map" of the underlying logic. If the AI makes a mistake, this map allows it to pinpoint and surgically fix the exact error without having to rewrite everything from scratch. To test this, we also built PRIME, a rigorous new testing ground composed of graduate-level math problems. Our approach proves that giving AI a structured way to reason is far more effective than just throwing more computing power at it, bringing us a significant step closer to highly reliable tools for verifying real-world mathematical discoveries.