Reactivating Test-Time Scaling for Plane Geometry Problem Solving
2026-08-31 • Computation and Language
Computation and Language
AI summaryⓘ
The authors focus on solving plane geometry problems, which need both good picture understanding and step-by-step logical thinking. They found that current approaches struggle because they use strict symbolic programs and don't clearly link the picture to the logic steps. To fix this, the authors propose converting solutions into different types of reasoning paths and training models to better understand diagrams before solving. Their method improves accuracy on several geometry tests and works more efficiently than previous strategies when scaled up.
plane geometry problemsmultimodal reasoningsymbolic deductiontest-time scalingsymbolic programsreasoning traceschain-of-thought (CoT)self-consistencymulti-trace ensemblevisual grounding
Authors
Xiaoqiang Kang, Shengen Wu, Maizhen Ning, Xiaobo Jin, Kaizhu Huang, Yutao Yue, Xiaowei Huang, Qiufeng Wang
Abstract
Plane geometry problem (PGP) solving has become a critical benchmark for multimodal reasoning because it requires accurate visual perception and precise multi-step symbolic deduction. Although test-time scaling (TTS) has demonstrated remarkable success in general mathematical reasoning, it fails to scale effectively under the symbolic-program paradigm for plane geometry. We identify two key obstacles: limited reasoning diversity induced by rigid symbolic programs and insufficient explicit visual grounding before symbolic deduction. To address these issues, we propose Multi-Trace Synthesis (MTS), which converts each symbolic program into heterogeneous reasoning traces, including executable Python scripts and CoT-augmented variants. We further propose Perception-Augmented (PA) training, which parses diagrams into structured semantic clauses before deduction, and Consensus-Guided Multi-Trace Ensemble (CG-MTE) for efficient self-adaptive inference. Experiments on three geometry benchmarks show that our method consistently improves PGP-solving across model scales and achieves strong performance against both general-purpose MLLMs and specialized geometry solvers. Under test-time scaling, CG-MTE achieves comparable accuracy to high-budget self-consistency while reducing sampling cost by up to 8x. Code and data are publicly available at https://github.com/Jason8Kang/ReTTS-PGPS.