🤖 AI Summary
This work addresses the semantic gap and consistency challenges inherent in automatically translating informal mathematical texts into verifiable formal proofs. It proposes a two-stage approach: first, an intermediate representation called Relaxed Natural Formal Language (Relaxed NFL) is introduced to preserve the original reasoning structure while tolerating partially underspecified expressions; second, a combination of rule-driven mechanisms and large language model–guided heuristics refines this representation into a semantically precise Core Natural Formal Language (Core NFL), yielding verifiable proof obligations. The method enables end-to-end translation from natural mathematical discourse to formal proofs, maintaining linguistic naturalness while guaranteeing verifiability, and supports integration with domain-specific tactic languages and solvers.
📝 Abstract
Auto-formalization aims to translate informal mathematical content into formal languages that can be processed by theorem provers. However, directly targeting existing theorem provers requires LLMs to bridge a substantial representational gap between informal mathematical writing and formal proof languages. This gap also makes semantic consistency difficult to evaluate. We address these difficulties by introducing a Relaxed Natural Formal Language (Relaxed NFL) as an intermediate target for auto-formalization. The Relaxed NFL is designed to remain close to informal mathematical writing: it preserves the usual structure of informal reasoning and allows partially specified expressions and propositions, without requiring their precise interpretation to be fixed at the auto-formalization stage. The remaining ambiguity and implicitness inherited from informal reasoning are resolved during a later elaboration stage, which transforms Relaxed NFL proofs into Core Natural Formal Language (Core NFL) proofs with formally defined semantics. The elaboration procedure combines rule-based transformations with LLM-generated heuristics, while maintaining verifiability through explicit constraints on each transformation step. The Core NFL is then used to generate proof gaps, namely verification conditions that must hold for the formalized proof to be correct. These gaps are discharged by LLM-generated proof scripts written in a domain-specific tactic language, which provides commands for invoking theorem libraries and domain-specific solvers implemented as part of our system.