Verifiable Geometry Problem Solving: Solver-Driven Autoformalization and Theorem Proposing

2026-06-26

AI Summary

This work addresses the disconnect and rigidity in existing geometric problem-solving approaches, particularly in automatic formalization and theorem prediction. The authors propose SD-GPS, a novel solver-driven geometric problem-solving framework that integrates a symbolic solver as an execution oracle throughout both formalization and reasoning. By leveraging solvability-guided formalization and proof-state-aware auxiliary lemma generation, the method enables closed-loop, verifiable geometric reasoning. SD-GPS combines the QwenVL3-2B model with supervised formal language adaptation, solvability-guided reinforcement learning, and a symbolic verification filtering mechanism. Evaluated on the Geometry3K and PGPS9K benchmarks, the approach significantly outperforms current large language models and neuro-symbolic methods across standard solution generation, multiple-choice questions, and cross-modal reference tasks.

Abstract

Geometry Problem Solving have increasingly adopt the neuro-symbolic paradigm, combining neural intuition with symbolic rigor. However, current frameworks suffer from severe bottlenecks in two core stages: autoformalization, which treats multimodal translation as a static task decoupled from downstream solver compatibility, and theorem prediction, where solvers frequently hit a deductive impasse due to fixed rule libraries. To address these, we propose SD-GPS, a solver-driven framework that treats the symbolic solver as an execution oracle throughout both formalization and deduction. First, Solver-Driven Autoformalization unifies supervised formal-language adaptation and solvability-guided reinforcement learning into a single module built on QwenVL3-2B, making executability the central training signal. Second, Verified Theorem Proposing introduces an impasse-aware agent that proposes local auxiliary lemmas from current proof states, ensuring soundness by filtering all proposals through symbolic verification. Empirical evaluations on Geometry3K and PGPS9K demonstrate that SD-GPS consistently outperforms existing MLLM, neural, and neuro-symbolic methods across standard completion, multiple-choice, and cross-modal reference regimes, proving that closing the loop between multimodal perception and symbolic execution significantly improves geometric reasoning, offering profound insights into how neural agents can be grounded by formal systems to achieve verifiable problem-solving capabilities.

Problem

• autoformalization
• theorem proposing
• geometry problem solving
• neuro-symbolic reasoning
• solver compatibility

Innovation

• solver-driven autoformalization
• verified theorem proposing
• neuro-symbolic reasoning
• symbolic verification
• geometric problem solving