How 5.5K Data Beats Gemini: Beihang’s Concise Symbolic Bridge for Plane Geometry Reasoning

Motivation

Plane geometry problem solving (PGPS) requires precise visual perception and rigorous logical reasoning. End‑to‑end joint fine‑tuning of multimodal large language models often fails because visual misinterpretation degrades the LLM’s reasoning ability.

Key Insight

Providing a perfect geometric description (ground‑truth CDL) to an LLM dramatically improves accuracy; Qwen‑3‑30B reaches 88.4 % when fed GT CDL, far above Claude‑Opus‑4.1, Gemini 2.5‑Pro and open‑source baselines.

Method

CDL Solver abandons end‑to‑end training and adopts a decoupled architecture:

MLLM Interpreter converts images into a structured Conditional Declaration Language (CDL).

LLM Solver solves the problem using the CDL output.

CDL defines three statement types:

ConsCDL – basic geometric skeletons (shapes, collinearity, concyclicity).

ImgCDL – numeric relations extracted from the diagram (lengths, angles, parallelism).

TextCDL – numeric relations extracted from the problem text.

The conciseness of CDL narrows the LLM’s search space and makes interpreter training more efficient.

Two‑Stage Training

Stage 1 – CoT‑enhanced supervised fine‑tuning : a Python parser generates chain‑of‑thought (CoT) reasoning steps. The model is trained to output

<think> … </think> <cdl> … </cdl>

, reasoning step‑by‑step before producing CDL.

Stage 2 – GRPO reinforcement learning with CDL‑matching reward : instead of a sparse solution‑based reward, a dense “CDL Matching Reward” is computed by greedy matching of generated CDL against ground‑truth GT CDL, measuring recall and precision for ConsCDL, ImgCDL and TextCDL. This high‑density signal accelerates convergence on tiny datasets.

Data Reconstruction

The Formalgeo7k dataset was rebuilt (v2) through double‑blind expert verification and enriched with high‑quality CoT annotations, yielding the Formalgeo7k‑Rec‑CoT benchmark.

Experimental Results

With only 5.5 k training examples, CDL Solver achieves:

85.7 % accuracy on FormalGeo (Qwen‑3‑VL 8B Interpreter + Qwen‑3 30B Solver).

3.9 % higher than Gemini 2.5 Pro (81.8 %) and 27.7 % higher than GPT‑4o (58.0 %).

Outperforms the previous state‑of‑the‑art DFE‑GPS model, which required 238 k data to reach 75.3 %.

Out‑of‑domain performance: 84.0 % on Unigeo and 80.8 % on MathVista.

Conclusion

Decoupling visual perception from logical reasoning via a concise, structured intermediate language preserves the LLM’s reasoning power while providing accurate visual grounding. High‑density process rewards such as CDL matching enable reinforcement learning to succeed with extremely limited data.

Code and models are released at https://github.com/dogehhh/CDL-Solver.