形式分析几何：一种基于神经符号的多模态解析几何问题生成框架
FormalAnalyticGeo: A Neural-Symbolic Based Framework for Multimodal Analytic Geometry Problem Generation
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- Xi’an Jiaotong-Liverpool University（西交利物浦大学）
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中文总结 AI 辅助
研究解析几何问题生成，提出基于神经符号的FormalAnalyticGeo框架，利用CDL及四个大语言模型组件，无需人工注释自动生成问题，形成闭环，生成的AnalyticGeo7K数据集问题误差小，框架和数据集将公开。
中文摘要 AI 辅助
随着多模态大语言模型的快速发展，数学推理取得了显著进展，但解析几何在很大程度上仍未得到充分探索，主要原因是带注释样本的稀缺。现有图表生成方法在解析几何方面存在困难。我们提出了FormalAnalyticGeo，一个用于全自动生成多模态解析几何问题的可扩展框架。该框架利用形式语言的严谨性，围绕CDL设计，通过符号距离场引擎将自由形式的问题文本与精确的图表渲染联系起来。它依次使用四个专门的大语言模型组件，质量验证器的结构化反馈驱动自动重试，形成闭环。大规模应用该框架产生了AnalyticGeo7K数据集，生成的问题实现了0.70%的中位数地面真值相对误差，82.3%的答案落在精确符号解的5%以内。我们的框架和数据集将公开发布。
英文摘要
Math reasoning has achieved significant progress with the rapid advancement of Multimodal Large Language Models (MLLMs), however analytic geometry remains largely underexplored, primarily due to the scarcity of annotated samples. Existing diagram generation approaches struggle with analytic geometry: template methods cannot handle constraint-driven layouts, and generative models lack the geometric precision to render annotated conic curves correctly. We present FormalAnalyticGeo, a scalable framework for fully automatic generation of multimodal analytic geometry problems. Leveraging the rigor of formal languages, we design the framework around CDL (Condition Description Language), a formal intermediate representation that bridges free-form problem text with precise diagram rendering via a Signed Distance Field (SDF) engine. The framework employs four specialized LLM components in sequence: a Generator that produces diverse analytic geometry problems, a Formalizer that converts each problem into CDL for SDF-based rendering, a Measurer that extracts ground-truth answers through vision-based measurement on the rendered diagrams, and a Quality Verifier that checks outputs at three stages. Structured feedback from the Quality Verifier drives automatic retry, forming a closed loop that eliminates any need for human annotation. Applying FormalAnalyticGeo at scale yields AnalyticGeo7K, a dataset of over 7K verified multimodal problems, each with aligned text, diagram, formal annotation, and ground truth.Experiments show that the generated problems achieve a median ground-truth relative error of 0.70\%, with 82.3\% of answers falling within 5\% of the exact symbolic solution. Our framework and dataset will be publicly released.
