Neuro-Symbolic Synthesis of Formal Proofs

This paper investigates the synthesis of formal mathematical proofs through the integration of neural and symbolic AI paradigms. The limitations of purely symbolic automated theorem provers, such as combinatorial explosion and the need for handcrafted heuristics, have historically hindered their efficacy. Conversely, large-scale neural models, while proficient at pattern recognition and generating human-like text, lack the rigorous logical consistency required for formal verification. This work proposes a neuro-symbolic framework that leverages the strengths of both approaches. A neural component, typically a transformer-based language model, is employed to guide the proof search, suggesting promising axioms, tactics, and intermediate lemmas. This intuition-driven guidance is then validated and executed by a symbolic engine, such as a proof assistant (e.g., Lean, Coq, Isabelle/HOL), which guarantees logical soundness at every step. We explore a verifier-in-the-loop methodology where the symbolic system provides structured feedback, enabling the neural model to refine its heuristics through reinforcement learning. This synergistic loop mitigates neural model hallucination and grounds the proof synthesis process in a formal, verifiable context. The proposed architecture demonstrates that combining the statistical inference of neural networks with the deductive power of symbolic reasoners can significantly advance the state of automated theorem proving, making formal methods more scalable and accessible.