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

2026-05-19 · Source: cs.CV updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

Hilbert-Geo is a novel neural-symbolic reasoning framework designed to solve solid geometric problems, an area where existing multimodal large language models (MLLMs) typically struggle due to complex 3D spatial diagrams and intricate reasoning. This framework introduces a unified formal language for solid geometry, including an extensive predicate library and a dedicated theorem bank. It employs a two-step "Parse2Reason" method: first, a Multimodal Formalization Parser (M2FP) converts natural language and visual diagrams into a formal Conditional Description Language (CDL); second, a Solid Geometry Reasoning Engine (SGRE) performs relational inference and algebraic computation using the CDL and theorem bank. Hilbert-Geo achieves state-of-the-art performance, with 77.3% accuracy on the SolidFGeo2k dataset and 84.1% on MathVerse-Solid, significantly outperforming MLLMs like Gemini-2.5-pro (54.2% on SolidFGeo2k) and GPT-5 (62.9% on MathVerse-Solid). The framework also demonstrates generality by achieving 80.2% accuracy on the PlaneFGeo3k dataset.

Key takeaway

Research Scientists developing AI for complex spatial reasoning should investigate neural-symbolic approaches like Hilbert-Geo. Its structured formalization framework and explicit reasoning engine effectively address the limitations of MLLMs in solid geometry, particularly concerning visual perception errors and logical inconsistencies. Consider adopting a similar Parse2Reason pipeline to enhance accuracy and verifiability in your own multimodal reasoning systems, especially for domains requiring rigorous, traceable solutions.

Key insights

Hilbert-Geo combines neural perception with symbolic reasoning to solve complex solid geometry problems, outperforming MLLMs.

Principles

• Formal language unifies multimodal geometric representations.
• Explicit theorem banks enable verifiable, human-readable reasoning.
• Parsing quality directly impacts downstream reasoning performance.

Method

The Parse2Reason method first translates multimodal inputs into Conditional Description Language (CDL) using a parser, then applies a reasoning engine with a theorem bank for symbolic deduction.

In practice

• Curate expert-annotated datasets for formal language training.
• Use fuzzy Jaccard similarity for robust parsing evaluation.
• Integrate formal logic to mitigate MLLM hallucinations in spatial reasoning.

Topics

• Hilbert-Geo
• Solid Geometry Problem Solving
• Neural-Symbolic Reasoning
• Formal Language Framework
• Multimodal Large Language Models

Best for: Research Scientist, AI Scientist, Machine Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by cs.CV updates on arXiv.org.