# A Knowledge and Semantic Fusion Method for Automatic Geometry Problem Understanding
## 4 SGKO: Semantic-Enhanced Geometry Knowledge Ontology
### Model Scalability and Adaptability  
The SGKO model demonstrates good scalability, mainly due to two aspects: tree-structured modular management and atomic, modular, and componentized design. On the one hand, all entity and relationship types are categorized into corresponding branches through a tree structure, so when new types are added, they only need to be incorporated by adding nodes to the corresponding branches, without the need to restructure existing templates or algorithm frameworks. This design supports \"plug-and-play\" expansion, easily accommodating future knowledge evolution and model extension needs. For example, when extending the processing capability for the Circumcenter relationship, traditional rule/template-based methods and machine learning approaches incur tremendous updating costs. Specifically, traditional rule/template methods require designing and constructing an explosive number of rules or templates. Not only must the matching rule/template design for atomic relationships be considered (e.g., \"[Point] is the circumcenter of [Triangle]\" and \"[Triangle]'s circumcenter (is) [Point]\"), but the rule design for the conjunction of multiple relationships must also be addressed, leading to a vertical growth in the number of rules, such as the various conjunctive expressions involving the Circumcenter and PointOnLine relationships (e.g., \"[Triangle]'s circumcenter [Point] on  
Figure 6: Complete knowledge storage for Perpendicular. [Line]\", \"passing through [Triangle]'s circumcenter [Point] to draw [Line]\", \"[Line] passing through [Triangle]'s circumcenter [Point]\"), as well as horizontal expansion by considering possible combinations with various other relationship types and their different mention orders. For instance, beyond \"circumcenter\" and \"point on line\", additional relationships such as Parallel (e.g., \"through [Triangle]'s circumcenter [Point] draw [Line]\(||\)[Line]\") and Intersects (e.g., \"through [Triangle]'s circumcenter [Point] draw [Line] intersects [Line] at [Point]\") must also be incorporated, requiring rules/templates for various expression and orders (e.g., parallel-intersection or intersection-parallel). Machine-learning-based methods exhibit limitations in corpus collection and model adjustments. Specifically, relevant samples must be actively searched (e.g., collecting and selecting problems containing circumcenter relationships to construct a dataset), and retraining or fine-tuning already trained the model may face issues with imbalanced sample distribution or adversely affect the recognition of other already processed types, leading to significant expansion costs. In comparison, the proposed method only requires introducing a new subtree for Circumcenter in SGKO and filling in predefined knowledge. This includes defining the possible entity types (Triangle and Point), setting the relationship keyword (\"circumcenter\"), and selecting appropriate RSFs from the RSF library (e.g., \"\(e_{subject}\)\"\(e_{object}\)-kw\" and \"\(e_{object}\)-kw-\(e_{subject}\)\"). The expansion process requires neither extensive nor cumbersome work, nor does it depend on data, achieving an almost zero-cost and zero-impact expansion. A detailed comparison of expansion costs is provided in Table 4.  
On the other hand, the design philosophy of atomizing, modularizing, and compromising the management, maintenance, and application of knowledge also facilitates adaptation to other tasks within this domain, such as adding \"properties\" to polygons (e.g., \"opposite sides are parallel\" for Parallelogram) in automatic reasoning tasks. Furthermore, this design can be extended beyond geometry to other domains, ensuring broad applicability and scalability.