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

arXiv: https://arxiv.org/abs/2605.16385 · HTML: https://mlsanity.com/paper/2605.16385/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Spatial and embodied reasoning, one of 17 directions in LLM reasoning.
- Builds on 20 of its 43 references that are in the map.
- First posted on arXiv 11 May 2026.

## Placed in

- LLM reasoning / [Spatial and embodied reasoning](https://mlsanity.com/lr/spatial-and-embodied-reasoning/)

## Builds on (most cited first, 20 of 20)

- [Qwen2.5-VL Technical Report](https://mlsanity.com/paper/2502.13923/) (2025)
- [A Survey of Large Language Models](https://mlsanity.com/paper/2303.18223/) (2023)
- [MathVista: Evaluating Mathematical Reasoning of Foundation Models in Visual Contexts](https://mlsanity.com/paper/2310.02255/) (2023)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [A Survey on Large Language Models for Code Generation](https://mlsanity.com/paper/2406.00515/) (2024)
- [A Comprehensive Overview of Large Language Models](https://mlsanity.com/paper/2307.06435/) (2023)
- [MathVerse: Does Your Multi-modal LLM Truly See the Diagrams in Visual Math Problems?](https://mlsanity.com/paper/2403.14624/) (2024)
- [Inter-GPS: Interpretable Geometry Problem Solving with Formal Language and Symbolic Reasoning](https://mlsanity.com/paper/2105.04165/) (2021)
- [A Survey on Large Language Models for Recommendation](https://mlsanity.com/paper/2305.19860/) (2023)
- [GeoQA: A Geometric Question Answering Benchmark Towards Multimodal Numerical Reasoning](https://mlsanity.com/paper/2105.14517/) (2021)
- [Thoughts Are All Over the Place: On the Underthinking of o1-Like LLMs](https://mlsanity.com/paper/2501.18585/) (2025)
- [UniGeo: Unifying Geometry Logical Reasoning via Reformulating Mathematical Expression](https://mlsanity.com/paper/2212.02746/) (2022)
- [VidHalluc: Evaluating Temporal Hallucinations in Multimodal Large Language Models for Video Understanding](https://mlsanity.com/paper/2412.03735/) (2024)
- [A Survey on Deep Learning for Theorem Proving](https://mlsanity.com/paper/2404.09939/) (2024)
- [FormalGeo: An Extensible Formalized Framework for Olympiad Geometric Problem Solving](https://mlsanity.com/paper/2310.18021/) (2023)
- [GeoX: Geometric Problem Solving Through Unified Formalized Vision-Language Pre-training](https://mlsanity.com/paper/2412.11863/) (2024)
- [A Symbolic Character-Aware Model for Solving Geometry Problems](https://mlsanity.com/paper/2308.02823/) (2023)
- [SOLIDGEO: Measuring Multimodal Spatial Math Reasoning in Solid Geometry](https://mlsanity.com/paper/2505.21177/) (2025)
- [Pi-GPS: Enhancing Geometry Problem Solving by Unleashing the Power of Diagrammatic Information](https://mlsanity.com/paper/2503.05543/) (2025)
- [NeSyGeo: A Neuro-Symbolic Framework for Multimodal Geometric Reasoning Data Generation](https://mlsanity.com/paper/2505.17121/) (2025)

## Built on by (most cited first, 2 of 2)

- [FormalAnalyticGeo: A Neural-Symbolic Based Framework for Multimodal Analytic Geometry Problem Generation](https://mlsanity.com/paper/2607.12982/) (2026)
- [Dense Neuro-Symbolic Reasoning in a Unified Geometry State](https://mlsanity.com/paper/2610.04280/) (2026)

## Abstract (arXiv)

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 are released at https://github.com/PremiLab-Math/Hilbert-Geo.