# A Knowledge and Semantic Fusion Method for Automatic Geometry Problem Understanding
## 6 Experiment and Application
### Experiment and Analysis
#### 6.1.2 Application Verification Test and Error Analysis
In the baseline methods, S2-Like enhances flexibility through POS tagging and fine-grained matching, but its reliance on lexical cues results in significant shortcomings in recognizing numerical and expression entities (e.g., in \"AM = 1/2(AB + BC)\"), adversely affecting the processing of related relationships. Its fundamental dependency on sentence templates for splitting complex expressions still limits recall. Moreover, it fails to effectively handle relationship directionality and nested expressions (e.g., misinterpreting \"the perpendicular bisector of AE with respect to BC\" as \"AE perpendicular bisects BC\"), which severely impacts precision. This hybrid method fails to balance flexibility and accuracy. In summary, the proposed method--through dynamic template construction, knowledge-guided extraction, and direction-aware semantic parsing--significantly out \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Precision (P)** & **Recall (R)** & **F1 Score (F1)** \\ \hline STM & 0.924 & 0.782 & 0.848 \\ TBC & 0.942 & 0.863 & 0.900 \\ BBC & 0.894 & 0.804 & 0.847 \\ S2-like & 0.877 & 0.729 & 0.796 \\ Proposed Method & 0.996 & 0.952 & 0.974 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Performance comparison of relationship extraction between the proposed method and the baseline methods. performs existing baseline methods, providing a more robust and efficient solution for the extraction of complex geometry relationships. Error Analysis: A further detailed analysis was conducted on the problems with text-level ERE failures. The errors were attributed to the following four factors: (1) unprocessed entity types, (2) processed entity types without set knowledge attributes, (3) unprocessed relationship types, and (4) processed relationship types without set knowledge attributes. Tables 13 and 14 present the extraction statistics of the proposed method from the perspectives of target relationships and RMTs. Based on further statistical analysis of the results, the shortcomings in handling unary relationships stem from unprocessed entity types, primarily due to extended datasets containing more diverse geometry shape types, while numerical relationships involve more specialized mathematical symbols (e.g., radicals) or complex numerical values and expressions. However, these two relationship types predominantly appear in simpler RMTs (e.g., SC-B or SC-T), meaning their improvement can be achieved through broader data exposure and enhancements in SGKO. In contrast, binary and ternary relationships, as the most frequent types in geometry problems, are heavily associated with complex RMTs. In particular, ternary relationships--frequently appearing in intricate forms (e.g., SI-T, MC-series, and MT) and often mentioned in concurrent descriptions--pose the greatest challenges for GPU. Analysis reveals that errors in binary relationships mainly arise from unprocessed entity types and unprocessed relationship types, while errors in ternary relationships are primarily caused by shared unprocessed entities in concurrent RMTs. In other words, the obstacles for binary relationships largely lie in the completeness of knowledge, whereas the difficulties for ternary relationships are more tied to their expression patterns. Further, the disparities in error distribution across the tables and the above analysis demonstrate that failures in entity and relationship extraction are not random but systematically linked to specific relationship types and the structural complexity of the text. For the four failure causes, a further examination of their detailed handling in failure cases is presented in the Table 15.  
Template construction is designed at the atomic relationship level and the processing unit for entity relationship extraction so that error propagation is confined within a single atomic relationship. Even if a particular relationship type is undefined or incompletely configured, it only affects the directly related text fragment without interfering with the