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

2026-05-11

AI Summary

This work addresses the limitations of existing methods in solving solid geometry problems, which often struggle with 3D diagram understanding and complex reasoning due to the absence of a unified formal framework. The authors propose Hilbert-Geo, the first comprehensive formal language system tailored for solid geometry, comprising a Condition Description Language (CDL), a predicate library, and a theorem library. They further introduce Parse2Reason, a two-stage neuro-symbolic reasoning paradigm that first parses multimodal inputs into formal conditions and then performs relational reasoning and algebraic computation using the theorem library to generate rigorous, verifiable, and human-readable proofs. Evaluated on SolidFGeo2k and MathVerse-Solid, Hilbert-Geo achieves accuracies of 77.3% and 84.1%, respectively, substantially outperforming Gemini-2.5-Pro and GPT-5, and demonstrates strong generalization by attaining 80.2% accuracy on the plane geometry benchmark PlaneFGeo3k.

Abstract

Geometric problem solving, as a typical multimodal reasoning problem, has attracted much attention and made great progress recently, however most of works focus on plane geometry while usually fail in solid geometry due to 3D spatial diagrams and complex reasoning. To bridge this gap, we introduce Hilbert-Geo, the first unified formal language framework for solid geometry, including an extensive predicate library and a dedicated theorem bank. Based on this framework, we propose a Parse2Reason method containing two steps of first parsing then reasoning. In the parsing step, we utilize conditional description language (CDL), a formalized language composed of predicates specifically designed to construct geometric conditions, to represent both problem description (natural text) and solid diagrams (visual image). In the reasoning step, we leverage those formal CDL and the theorem bank to perform relational inference and algebraic computation, generating strictly correct, verifiable, and human-readable reasoning processes. Notably, our proposed Hilbert-Geo is also applicable to plane geometry. To advance geometric reasoning, we curate two expert-annotated dataset SolidFGeo2k and PlaneFGeo3k, which are furnished with geometric formal language annotations, solutions and answers. Extensive experiments show that our proposed method achieves the state-of-the-art (SOTA) performance 77.3% in SolidFGeo2k and 84.1% in MathVerse-Solid (one small subset in MathVerse dedicated to solid geometry), substantially outperforming leading MLLMs, such as Gemini-2.5-pro (54.2% on SolidFGeo2k) and GPT-5 (62.9% on MathVerse-Solid). In addition, our method achieves the SOTA accuracy 80.2% in PlaneFGeo3k, demonstrating the generality of the Hilbert-Geo in geometric reasoning. Our code and datasets will be publicly available.

Problem

• solid geometry
• geometric reasoning
• multimodal reasoning
• 3D spatial diagrams
• neural-symbolic reasoning

Innovation

• neural-symbolic reasoning
• formal language framework
• solid geometry
• conditional description language
• geometric problem solving