# A Method of Ontology Integration for Designing Intelligent Problem Solvers
## 4 Integration Between the Kernel and Other Knowledge
### Integration with the Knowledge of Functions -Functs -set
#### 4.3.2. Problems and Reasoning Methods on the Knowledge of Functions
Besides concepts in the kernel, \(\textbf{C}\) also includes concepts about the Linear Equations System, and Vector Space:

\[\textbf{C}\) = \(\textbf{C}_{\text{ kernel}}\cup\{\textit{Equation, EquationsSystem, CramerSystem, VectorSpace}\}.\)

\(\textbf{C}\) --set of relations. Besides the relations in the kernel, \(\textbf{C}\) also includes the relationships in the Linear Equations System and Vector Space:

\(\textbf{C}\) = R\({}_{\text{kernel}}\cup\{\textit{eigenvalue, eigenvector, sub-space, based-set, spanning-set, linearly independent,...}\}.\)

\(\textbf{Cps}\) is the set of operators between matrixes and vectors, operators in vector space.

\(\textbf{Cules}\) is the set of rules on the knowledge domain about linear algebra. It includes rules for generating a new object, equation rules, and equivalent rules in linear algebra.

The knowledge base of this knowledge domain is represented in Appendix A.

#### 5.2.2 Types of Exercises in Linear Algebra

Based on the knowledge base in Section 5.2.1 and the algorithms in Sections 3.2 and 4.2.2, an intelligent system for solving problems in linear algebra was designed. Some types of exercises in the curriculum were tested:

* Problems with the Matrix-Vector: Find the rank of a matrix, find the inverse matrix, find the value of an expression between matrixes.
* Problems with the Linear Equations System: Solve a linear equations system by the Gaussian method and Cramer method.
* Problems with the Vector Space: Determine the independence of vectors, determine the basis of a vector space.

**Example 5**.: _Given a set of vectors \(A=\{u=(x_{1},\,x_{2},\,x_{3},\,x_{4})\in\tilde{\mathbb{N}}^{d}|\,x_{1}=2x_{2}=x_ {3}\}\), and a subspace \(W=\textit{<A>}\), Determine the based-set of \(W\)._

\(+\) The model of this problem:

O: = [W: VectorSpace, x\({}_{1}\), x\({}_{2}\), x\({}_{3}\), x\({}_{4}\); \(\tilde{\mathbb{N}}\), A: 2\({}_{\text{kernel}}\)];

F: = [A = {(\{x_{1},x_{2},\,x_{3},\,x_{4}\})|\,x_{1}=2x_{2}}\)

, \(x_{1}=x_{3}\)], A is a spanning-set of W];

G: = [Determine: Based-set of W].

Design Knowledge Base of Graph Theory Using the Integrating Model (\(\mathbb{C}\), \(\mathbb{L}\), \(\mathbb{L}\)ules ) + Funcs

#### 5.3.1 Knowledge Base of Graph Theory

The knowledge about graph theory was collected from [25; 26; 27]. In this section, we only cover the knowledge about undirected graphs. The knowledge base of this knowledge domain can be represented by the method for integrating the kernel and knowledge of functions:

\[\textbf{(\mathbb{C},\mathbb{L},\mathbb{L}\text{\bfules})}+\textbf{Funcs}\.\]

\(\mathbb{C}\) --set of concepts. It includes concepts in the kernel and concepts about the knowledge of the graph:

\[\mathbb{C}=\mathbb{C}_{\text{kernel}}\cup\{\textit{Graph, Connected Graph, Simple Graph, \dots}\ \}.\]

\(\mathbb{L}\) --set of relationships. Besides relationships in the kernel, \(\mathbb{L}\) also includes relationships between graphs:

\[\text{F}_{1}=\ \Big{\{}\text{B}=\ (-2,\ \sqrt{3}),\text{C}=\ (2,\ \sqrt{3}) \Big{\}}\]0

**Funcs** --set of functions of the knowledge domain about graph theory:

**Funcs** = \(\{\textit{Degree, Path, Girth, EulerPath, HamiltonCycle}\dots\ \}.\)

**Funcs** --set of rules on the knowledge domain about graph theory. It includes deductive rules and equivalent rules. The knowledge base of this knowledge domain is represented in Appendix B. #### 5.3.2 Kinds of Exercises in Graph Theory

Based on the knowledge base and the inference engine in the above section, an intelligent system for solving problems in graph theory was designed. This system can solve many types of problems that are in the curriculum of this course. They are classified in the three following types [18]:

* Type 1: find the shortest paths between nodes in a graph, find a minimum spanning tree for a weighted undirected graph. These problems also have additional conditions;
* Type 2: Determine a graph from an expression of graphs;
* Type 3: Prove a property of a graph or a relationship between two graphs. **Example 6**.