# A Knowledge and Semantic Fusion Method for Automatic Geometry Problem Understanding
## 4 SGKO: Semantic-Enhanced Geometry Knowledge Ontology  
### The Upper Knowledge System Layer  
According to the dataset resources, \"EntityType\" and \"RelationshipType\" are designated as top-level classes in the knowledge system, categorizing various entity and relationship types. Entity types are grouped into Geometry Shape, Expression, and Value. Geometry Shape includes subclasses like Point, Line, Angle, and Circle, with Polygons further subdivided into Triangle and Quadrilateral. Triangle is classified into Isosceles, Equilateral, and Right Triangle, while Quadrilateral includes Square, Parallelogram, Rhombus, and Trapezoid. Trapezoid is further detailed into Isoscele, Right, and Isoscele Right Trapezoid. Values distinguish between Numerical Value (e.g., 4, 3.2, 1/2) and Degree Value (e.g., 18 deg), while Expression is split into Operation Expression (e.g., 1/2(AB + CD), 2 AB\({}^{2}\) + CD\({}^{2}\)) and Non-operation Expression (e.g., 2 AB, 1/4 CF). To enhance text-level ERE, a new entity type \"ShapeLimit\" and a corresponding relationship type \"shapeLimits\" are introduced to describe specific polygon prefixes like \"isosceles\" and \"equilateral\", treated as independent ShapeLimit entities. The relationship type \"ShapeLimits\" defines constraints these prefixes impose on polygons, facilitating combinations of ShapeLimit, PolygonType, and Polygon entities in logical sequences of \"special shape description prefix\(\rightarrow\)polygon type\(\rightarrow\)polygon name\", as illustrated in Table 1.  
This approach reduces entity types and may enhance GPU accuracy by converting unary semantic relationships into binary relationships for more streamlined processing. Descriptions such as \"isosceles triangle\" and \"right trapezoid\" can be treated either as independent entities or combined with the Geometry Shape entity, impacting GPU accuracy when increasing the Bayesian denominator. The introduction of the ShapeLimit and shapeLimits relationship types aligns with the Subject-Predicate-Object (SPO) structure, facilitating uniform processing. This allows subclasses under ShapeLimit and Polygon categories to be processed separately, simplifying entity management while distinguishing between conceptual- and object-level information. Conceptual-level entities like ShapeLimit and Polygon categories are retained within the SGKO, whereas specific polygon names are processed and included in the problem semantic KGs. Relationship types are classified under \"Text-level RelationshipType\" and \"Knowledge-level RelationshipType\" in our knowledge system, differentiating relationships relevant to the text-level ERE and those pertinent to the knowledge-level ERE without overlap in their processing. Text-level relationships focus on geometry predicates, while knowledge-level relationships pertain to arithmetic operations. For text-level relationship types, two categories are defined: \begin{table}
\begin{tabular}{l l l} \hline
**No.** & **Combination** & **Example** \\ \hline
1 & ShapeLimit-PolygonType–Polygon & Isosceles Trapezoid \(ABCD\) \\ \hline
2 & ShapeLimit-Polygon & Equilateral Triangle \(\triangle ABC\), Isosceles \\  & Triangle \(\triangle ABC\) \\ \hline
3 & PolygonType–Polygon & Parallelogram \(EFGH\), Trapezoid \(ABCD\), \\  & Rectangle \(ABCD\) \\ \hline
4 & ShapeLimit–ShapeLimit–Polygon & Right Isosceles Triangle \(\triangle ABC\) \\ \hline
5 & Polygon & Triangle \(\triangle ABC\) \\ \hline \end{tabular}
\end{table} 
Table 1: Free combination of polygon descriptions. **Definition 1** (Autonomous Relationship Type and Dependent Relationship Type). : _Let \(U_{RT}\) represent the set of all text-level relationship types. \(RT_{Autonomous}\) and \(RT_{Dependent}\) denote the sets of Autonomous Relationship Type and Dependent Relationship Type, respectively. For any \(rt\in U_{RT}\), if its relationship has a complete SPO structure, which independently conveys complete semantics, then \(rt\in RT_{Autonomous}\); otherwise, \(rt\in RT_{Dependent}\).