# A Knowledge and Semantic Fusion Method for Automatic Geometry Problem Understanding
## 6 Experiment and Application
### Experiment and Analysis
#### 6.1.1 Self-Verification Test and Validity Verification  
Dataset (A): Aiming at the semantic understanding of geometry proof problems, this paper collects 130 plane geometry proof problem texts without circles from the latest years' math entrance examination questions, mainstream textbooks (People's Education Press), textbook companions, exercise books, and mock exams. These problems include complex relationship descriptions and form the foundation for method construction. The dataset was statistically analyzed from two perspectives: relationship distribution and RMTs. Detailed statistics are provided in Tables 8 and 9.  
Experimental Setup: A system implementing the proposed method was built using Python 3.8, and the final problem semantic KG was constructed using Neo4j 4.4.16. All problems in the dataset were input into the system for comprehension. Manual verification was performed on intermediate outputs, entity sets and relation triplets, and final semantic KGs to validate text-level ERE and knowledge-level ERE.  
Performance: The system successfully parsed all 130 geometry problems in the dataset, with entities and relationships accurately identified and extracted at both the text and knowledge levels. Notably, the relationship extraction at the text level achieved satisfactory performance: First was the accurate recognition of similar texts. For instance, the system identified \"AB's extension\" as an entity mention, \"AB's extension BE\" as a relationship description, \"AB's extension intersects CD\" as mentioning one relationship, and \"AB's  
\begin{table}
\begin{tabular}{l l l l} \hline
**Module** & **Traditional Method** & **Proposed Method** & **Optimization** \\ \hline \multirow{3}{*}{Template Complexity} & Time: (_Exponential_) & Time: (_Linear_) & \\  & \(\mathrm{O}(R\cdot V^{k})\) & \(O(R)\) & Hash table indexing, \\  & Space: (_Exponential_) & Space: (_Linear_) & Tree-structured SGKO, \\  & \(\mathrm{O}(R\cdot V^{k})\) & \(\mathrm{O}(R\cdot C+K)\) & Limited RSFs. \\ \hline \multirow{3}{*}{Text-Level ERE} & Time: (_Polynomial_) & Time: (_Linear_) & Hash table lookup \\  & \(\mathrm{O}(N\cdot K\cdot T)\) & \(\mathrm{O}(N\cdot K+E)\) & (\(O(1)\) per keyword), \\  & Space: (_Exponential_) & Space: (_Linear_) & Dynamic Template \\  & \(\mathrm{O}(N\cdot L+R\cdot V^{k})\) & \(\mathrm{O}(N\cdot L+E+R)\) & Matching. \\ \hline \multirow{3}{*}{Knowledge-Level ERE} & Time: (_Quadratic_) & Time: (_Linearithmic_) & \\  & \(\mathrm{O}(M^{2})\) & \(\mathrm{O}(M\cdot\log M)\) & Hash-based \\  & Space: (_Quadratic_) & Space: (_Linear_) & deduplication. \\ \cline{1-1}  & \(O(M^{2})\) & \(\mathrm{O}(M)\) & \\ \hline \multirow{3}{*}{Full Pipeline} & Time: (_Exponential_) & Time: (_Linearithmic_) & \\  & \(\mathrm{O}(R\cdot V^{k}+M^{2})\) & \(\mathrm{O}(N\cdot K+M\cdot\log M)\) & \\ \cline{1-1}  & Space: (_Exponential_) & Space: (_Linear_) & \\ \cline{1-1}  & \(\mathrm{O}(R\cdot V^{k}+M^{2})\) & \(\mathrm{O}(N\cdot L+R+M)\) & \\ \hline \end{tabular} Key Symbols: \(R\): Relationship types, \(V\): Variants per relationship, \(k\): Concurrency level, \(N\): Clauses, \(K\): Keywords per clause, \(E\): Entities, \(M\): Entities including both original and decomposed entities, \(L\): Clause length, C: Atomic components (constant).  
\end{table}
Table 7: Summary of complexity.  
extension BE intersects AC\" as containing two relationship descriptions. Second was the robust handling of complex relationships that span across sentences or share semantic roles, such as \"the bisector AG of \(\angle\)DAE intersects CD at point G, (and) intersects the extension line of BC at point F\" and \"through the midpoint O of AC, draw EF \(\perp\) AC (and) intersects BC at point E, intersecting AD at point F\". Third, the method exhibits fault tolerance when the key information (keywords and entity names) is correct. For example, it can correctly recognize the relationship mentioned in \"CF 'were' perpendicular to AC\" because only the core atomic unit \"perpendicular\" is set as the keyword.