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 will be released to the public soon.

ASSESS: A Semantic and Structural Evaluation Framework for Statement Similarity

Statement autoformalization, the automated translation of statements from natural language into formal languages, has seen significant advancements, yet the development of automated evaluation metrics remains limited. Existing metrics for formal statement similarity often fail to balance semantic and structural information. String-based approaches capture syntactic structure but ignore semantic meaning, whereas proof-based methods validate semantic equivalence but disregard structural nuances and, critically, provide no graded similarity score in the event of proof failure. To address these issues, we introduce ASSESS (A Semantic and Structural Evaluation Framework for Statement Similarity), which comprehensively integrates semantic and structural information to provide a continuous similarity score. Our framework first transforms formal statements into Operator Trees to capture their syntactic structure and then computes a similarity score using our novel TransTED (Transformation Tree Edit Distance) Similarity metric, which enhances traditional Tree Edit Distance by incorporating semantic awareness through transformations. For rigorous validation, we present EPLA (Evaluating Provability and Likeness for Autoformalization), a new benchmark of 524 expert-annotated formal statement pairs derived from miniF2F and ProofNet, with labels for both semantic provability and structural likeness. Experiments on EPLA demonstrate that TransTED Similarity outperforms existing methods, achieving state-of-the-art accuracy and the highest Kappa coefficient. The benchmark, and implementation code will be made public soon.