Hilbert-Geo: Solving Solid Geometric Problems by Neural-Symbolic Reasoning

Researchers introduce Hilbert-Geo, a neural-symbolic AI framework for solving solid geometry problems by combining formal language representation with theorem-based reasoning. The system achieves 77.3% accuracy on solid geometry tasks, significantly outperforming leading AI models like GPT-4 and Gemini-2.5-pro, demonstrating advances in multimodal geometric reasoning.

Hilbert-Geo represents a meaningful advancement in AI's ability to tackle spatial reasoning problems that have historically challenged machine learning systems. The framework addresses a genuine gap in geometric problem-solving research, where most prior work focused on simpler 2D plane geometry while struggling with the complexity of 3D spatial relationships. By combining formal language representation (CDL) with theorem-based inference, the researchers created a hybrid approach that generates verifiable, human-readable solutions rather than black-box outputs.

The performance metrics are compelling: achieving 77.3% accuracy on solid geometry substantially exceeds GPT-4's 62.9% and Gemini-2.5-pro's 54.2%, suggesting that symbolic reasoning architectures outperform pure language models on mathematically-grounded problems. This aligns with a broader trend in AI research recognizing that end-to-end neural approaches have inherent limitations for reasoning tasks requiring logical consistency and verifiable correctness.

For the AI development community, this work demonstrates practical value in hybrid neural-symbolic systems that could improve performance in mathematics, physics, and engineering applications where formal correctness matters. The release of two curated datasets (SolidFGeo2k and PlaneFGeo3k) with formal annotations provides valuable resources for advancing geometric reasoning research.

The framework's applicability to both solid and plane geometry suggests scalability potential. Future developments might extend these principles to other formal reasoning domains beyond geometry, particularly in domains where domain-specific theorem banks and predicate libraries can be constructed.

• →Hilbert-Geo achieves 77.3% accuracy on solid geometry, substantially outperforming GPT-4 and Gemini-2.5-pro by leveraging neural-symbolic reasoning
• →The framework uses formal Conditional Description Language (CDL) to parse geometric problems and apply theorem-based inference for verifiable solutions
• →Hybrid neural-symbolic approaches demonstrate advantages over pure language models for problems requiring mathematical correctness and logical consistency
• →Two new expert-annotated datasets (SolidFGeo2k and PlaneFGeo3k) advance research infrastructure for geometric reasoning evaluation
• →The method's success on both 2D and 3D geometry suggests transferability potential to other formal reasoning domains