Rethinking and Improving Autoformalization: Towards a Faithful Metric and a Dependency Retrieval-based Approach

As a central component in formal verification, statement autoformalization has been widely studied including the recent efforts from machine learning community, but still remains a widely-recognized difficult and open problem. In this paper, we delve into two critical yet under-explored gaps: 1) absence of faithful and universal automated evaluation for autoformalization results; 2) agnosia of contextual information, inducing severe hallucination of formal definitions and theorems. To address the first issue, we propose **BEq** (_**B**idirectional **E**xtended Definitional E**q**uivalence_), an automated neuro-symbolic method to determine the equivalence between two formal statements, which is formal-grounded and well-aligned with human intuition. For the second, we propose **RAutoformalizer** (_**R**etrieval-augmented **Autoformalizer**_), augmenting statement autoformalization by _Dependency Retrieval_, retrieving potentially dependent objects from formal libraries. We parse the dependencies of libraries and propose to _structurally informalise_ formal objects by the topological order of dependencies. To evaluate OOD generalization and research-level capabilities, we build a novel benchmark, _Con-NF_, consisting of 961 informal-formal statement pairs from frontier mathematical researches. Experiments validate the effectiveness of our approaches: BEq is evaluated on 200 diverse formal statement pairs with expert-annotated equivalence label, exhibiting significantly improved accuracy ($82.50\\% \mapsto 90.50\\%$) and precision ($70.59\\% \mapsto 100.0\\%$). For dependency retrieval, a strong baseline is devised. Our RAutoformalizer substantially outperforms SOTA baselines in both in-distribution ProofNet benchmark ($12.83\\% \mapsto 18.18\\%$, BEq@8) and OOD Con-NF scenario ($4.58\\%\mapsto 16.86\\%$, BEq@8).