GOLD: Geometry Problem Solver with Natural Language Description

TL;DR

GOLD tackles automated geometry problem solving by transforming geometry diagrams into natural language descriptions and leveraging large language models to generate solution programs. By separately modeling symbols and geometric primitives, GOLD constructs sym2geo and geo2geo relations, converts them into NL, and uses LLMs to solve problems from UniGeo, PGPS9K, and Geometry3K data. The approach yields substantial accuracy gains over Geoformer and PGPSNet, and ablations confirm the value of NL descriptions and the embedded representations for reliable relation extraction. This diagram-to-language strategy enhances interpretability and enables seamless integration with LLM-based reasoning, offering a scalable path for multimodal geometry problem solving with practical impact on educational AI tools.

Abstract

Addressing the challenge of automated geometry math problem-solving in artificial intelligence (AI) involves understanding multi-modal information and mathematics. Current methods struggle with accurately interpreting geometry diagrams, which hinders effective problem-solving. To tackle this issue, we present the Geometry problem sOlver with natural Language Description (GOLD) model. GOLD enhances the extraction of geometric relations by separately processing symbols and geometric primitives within the diagram. Subsequently, it converts the extracted relations into natural language descriptions, efficiently utilizing large language models to solve geometry math problems. Experiments show that the GOLD model outperforms the Geoformer model, the previous best method on the UniGeo dataset, by achieving accuracy improvements of 12.7% and 42.1% in calculation and proving subsets. Additionally, it surpasses the former best model on the PGPS9K and Geometry3K datasets, PGPSNet, by obtaining accuracy enhancements of 1.8% and 3.2%, respectively.

Figures (4)

• Figure 1: The illustration of the GOLD Model. The diagram $\mathcal{D}$, problem text $\mathcal{T}$, and solution program $\mathcal{P}$ used in this illustration are sourced from the PGPS9K dataset pgps9k. The symbols and geometric primitives in the diagram are annotated using the notations from the Notation Table, which are consistent with the colours of extracted relations of sym2geo and geo2geo.
• Figure 2: Top-left: the performance of the GOLD (using T5-base) with (w) and without (w/o) the geo2geo. Top-right: Geometry math problem. Bottom: Predicted diagram description with and without the geo2geo. The same text between (w) and (w/o) is omitted for space consideration, where the red text is geo2geo relations.
• Figure 3: An example from the 111-th problem in the PGPS9K dataset. This case shows that models' natural language descriptions and solution programs outputs with and without spatial_embedding. The purple notations in the diagram are added by us. Note that the different parts of diagram descriptions between w/o and w are coloured red.
• Figure 4: The top-left bar chart compares GOLD (T5-base as the problem-solving module) accuracy in solving geometry math problems, with (w) and without (w/o) the use of geo_type_embedding. The top-right diagram is from the 375th problem in the PGPS9K dataset, while the bottom part shows the predicted diagram descriptions for two different cases. Purple notations in the diagram are added for better visual comprehension. The differences between the two diagram description texts are highlighted in red. It should be noted that the same texts in the w to the w/o section are omitted, which are represented by "...".