A Method for Knowledge Representation to Design Intelligent Problems Solver in Mathematics Based on Rela-Ops Model

Knowledge-base is a fundamental platform in the architecture of an intelligent system. Relations and operators are popular knowledge in practice knowledge domains. In this paper, we propose a method to represent the model by combining these kinds of knowledge, called the Rela-Ops model. This model includes foundation components consisting of concepts, relations, operators, and inference rules. It is built based on ontology and object-oriented approaches. Besides the structure, each concept of the Rela-Ops model is a class of objects which also have behaviors to solve problems on their own. The processing of algorithms for solving problems on the Rela-Ops model combines the knowledge of relations and operators in the reasoning. Furthermore, we also propose a knowledge model for multiple knowledge domains, in which each sub-domain has the form as the Rela-Ops model. These representation methods have been applied to build knowledge bases of Intelligent Problems Solver (IPS) in mathematics. The knowledge base of 2D-Analytical Geometry in a high-school is built by using the Rela-Ops model, and the knowledge base of Linear Algebra in university is designed by using the model for multiple knowledge domains. The IPS system can automatically solve basic and advanced exercises in respective courses. The reasoning of their solutions is done in a step-by-step approach. It is similar to the solving method by humans. The solutions are also pedagogical and suitable for the learner’s level and easy to be used by students studying 2D-Analytical Geometry in high-school and Linear Algebra in university.


I. INTRODUCTION
Science Technology Engineering and Math (STEM) is a modern approach for studying in the technological era [3]. Building the Intelligent Problem Solver (IPS) in STEM, especially for mathematical courses, is a grand challenge for artificial intelligence in education [1]- [3]. This system can automatically solve problems. Those problems are declared The associate editor coordinating the review of this manuscript and approving it for publication was Shouguang Wang . their hypothesis and goal by using a specification language [4], [5]. The system will automatically solve them or give some instructions to solve them. Besides the ability to solve common exercises, this system requires the pedagogy. Their proofs are suitable for the learner's level. They also help the learner understanding the method for solving problems. An IPS in education can solve many kinds of exercises from basic to advanced kinds in the course. It has to satisfy these requirements [4], [6], [36], [48]: • The program can solve conventional kinds of exercises in the course.
• The program is pedagogical.
• The program is useful for studying. For meeting these requirements, according to [4], [6], [7], [48], a method for building the knowledge base of IPS has to satisfy the following criteria: Universality [4], [6], [48]: This criterion provides the flexibility of the representation method. The method can be applied to representing various knowledge domains [4], [6]. It can be used directly or improved with some little things to use. A model has to represent the foundation of the knowledge domain, including concepts, relations between concepts, and inference rules [48]. That foundation is a knowledge kernel as ontology. The kernel can integrate with other knowledge to strengthen the ability to represent practical knowledge.
Usability [4], [7]: It has the simple specification language being adequately to represent knowledge domains. This language is also easy to employ for representing and updating knowledge domains. When it is applied, the knowledge base of the system gives a natural inference that can be understood by the user.
Practicality [6], [7]: The model must be applied to represent various real knowledge domains. This representation is similar to that of humans, and it can solve some practical problems in the knowledge domain.
Formality [7], [8]: The structure of the model is built based on the solid mathematical foundation. It is also used to construct the model of problems. The finiteness and effectiveness of algorithms for solving the problems are proven. The complexity of them is also evaluated.
Each criterion has four levels from 1 -4, respectively, from very weak -very strong as [6]. The meaning of each level is described in Fig. 1 as follows: Knowledge of relations and operators is a prevalent form of human knowledge. The model for this knowledge domain consists of concepts, relations, operators, and inference rules. Some methods for representing this knowledge have been studied, such as graph, logic, or ontology. Based on the mathematical structures about relations and operators, some models for representing the knowledge of relations [9], [11] and the knowledge of operators [10], [12] are studied. However, they cannot represent the knowledge domains that have both relations and operators. The result in [13] is using the syntax-semantics model to extract relations between the entities and the geometric attributes pattern; nonetheless, this method is not universality. It is based on the characteristic of geometric knowledge. Although the algebraic structures have also been used to represent this kind of knowledge, those methods have limitations and cannot satisfy the requirements of knowledge representation in intelligent systems [27], [28].
In this paper, a model for combining the knowledge of relations and operators, called the Rela-Ops model, is proposed. This model is built based on ontology and object-oriented approaches. Its foundation includes components: concepts, relations, operators, and inference rules, in which each concept in the Rela-Ops model is a class of objects. Based on its structure, this model also defines the sentence and problems on it. This model is useful in practical applications. In the inference processing, it combines the knowledge of relations and operators to solve current problems in the knowledge domain. Besides, the Rela-Ops model satisfies the criteria of a method for representing the knowledge base of IPS in education. More than that, we present a method to represent a knowledge domain, including multiple sub-domains; each sub-domain has the Rela-Ops model form. These proposed methods have been applied to build knowledge bases of IPS in mathematics. The knowledge base of 2D-Analytical Geometry in a high-school is built by using the Rela-Ops model, and the knowledge base of Linear Algebra in university is designed by using the model for multiple knowledge domains. These systems can solve the usual exercises of respective courses automatically. The reasoning of their solutions is clearly. It is similar to the solving method of humans. Solutions are also pedagogical and suitable for the learner's level.
The next section presents related works about methods constructing the knowledge base of IPS. Section 3 presents the knowledge model of relations and operators, which is called the Rela-Ops model. The syntax of the clauses and predicates on this model are also presented in that section. Section 4 presents problems with the Rela-Ops model and their models. Algorithms for solving them are designed based on their models and knowledge in the Rela-Ops model. Section 5 presents the knowledge model for multiple knowledge domains. Section 6 shows the applications of the proposed methods to build IPS for Linear Algebra in university and 2D-Analytical Geometry in high school. The last section concludes the paper.

II. RELATED WORK
Nowadays, there are many methods for representing knowledge, including formal logic, frames, networks, ontology, and algebraic approaches. However, they have some limitations to apply in practice, especially in designing knowledge bases of IPS in education.
Formal logic methods only can represent simple knowledge domains -many kinds of logic methods being studied for knowledge representation -for example, predicate methods, first-order logic, temporal logic, and description logic [14]- [16]. Logical methods are the formal representation of semantic [14]. In [15], [16], the authors represent the operators and relations in logic by matrices and tensors. Besides that, the study in [44] presents a logical method to describe the semantic and syntactical aspects of uncertain decision implications. This work is built based on a solid mathematical foundation, but those results are theoretical. The specification of inference rules in the real knowledge domain is not suitable for ordinary users, especially the students. When applying formal logic methods in knowledge domains about education, the representing is not natural; it is not similarly the way humans representing the knowledge of courses. Hence, those methods are not effective in designing the knowledge base of the system supporting learning.
Networks are suitable methods to represent concepts and their relationships in the knowledge domain [17]. However, they are not effective in representing the knowledge domain about computing. The knowledge graph performs a collection of interlinked descriptions of entities. The nodes of this graph represent items, and relations of nodes that interact with each other are represented by its edges [18]. This graph is a useful tool for semantic searching and describing the semantics of information. In [19], a knowledge graph embedding with the concepts model is proposed. This model represents the relationships between entities and their concepts. It can adjust a knowledge graph by the concept of information of entities from a concept graph. Besides that, there are some methods to extract relations from multiple knowledge graphs by considering the alignments between them [20]. Nonetheless, the knowledge graph is not sufficient in solving problems by reasoning methods, especially problems of an IPS in education.
Petri nets, which can be used to handle many problems [51], [52], are a modeling technique to construct knowledge-based systems in many fields [45]. They are useful for designing knowledge bases fuzzy reasoning of uncertain expert systems. The model combining Petri nets and information theory is sufficient to consider the development VOLUME 8, 2020 of a discrete event system. In [46], the authors propose a cloud reasoning Petri nets (CRPN) model to represent an uncertain knowledge domain. It used the operator of interval cloud hybrid averaging. However, the knowledge of mathematical courses cannot be represented by Petri nets, so this model cannot use to design the knowledge bases of IPS in mathematics.
Frames are a large part of knowledge representation schemes [21]. Some of IPS systems have been designed based on their knowledge bases as frames, such as symbolab [39], mathway [40]. By using frames, the systems only solve some kinds of problems which were set up, they cannot solve other kinds of problems, especially problems require the use of the depth knowledge or the combining knowledge. In [22], the authors proposed a sharing framework and algorithms for solving problems about explicit arithmetic and proving plane geometry theorems. They used the syntax-semantics model [13], [23] to extract relations for a math problem. That problem was solved through the set of extracted relations. However, this method does not support the learning; it does not show a pedagogical solution which has the reasoning similar to humans.
In intelligent tutoring systems (ITS), the content of a course is represented by ontology as a framework [24]. In robotics, ontology represents the knowledge about the environment, events, and actions, which make robots more autonomous, especially in automatic tutoring systems [25], [50]. Ontology COKB (Computational Object Knowledge Base) is an emerging method to build an IPS in education [26]. The mathematical foundation of the COKB model is not presented clearly. Some of its components, such as the components of operators and functions, have not yet had a solid structure.
In [47], a security knowledge representation artifact, called Domain Security Metamodel (DSM), is presented. This model includes a specific domain, contains information about security aspects. It is applied to the system integrating the solutions for evolution time naturally and directly. Nonetheless, this model only uses for security domains, such as the web service security domain; it does not use for knowledge domains of IPS in education.
The algebraic approach is a representation method based on mathematical structures. Some algebraic structures are used: groups, lattices, rings, fields, and integrating those structures [27]. By using this approach, the checking of information equivalence of knowledge is solved based on the symmetries of knowledge bases [28]. The knowledge base as logic is represented by the structure of matrices and tensors in linear algebra [29]. The structure of the knowledge base in [30] is created as a dynamic concept network mimicking human knowledge represented in the brain. This study is improved from the model of concept algebra in [31]. It is used to organize the knowledge bases of cognitive robots and machine learning systems. Nonetheless, those results are theoretical; they have not yet been applied in practical applications.
The knowledge base in [49] is represented by using the structure of relational algebra. This method is an analogical mechanism of inheritance in the Association-Oriented Database Metamodel. However, this method only can specify relationships of the knowledge base as data, and it cannot represent the real-world knowledge domain directly. Besides, there are many methods for integrating the knowledge bases in real life, such as image understanding [53], communication understanding [54]. However, those methods cannot solve problems in the learning knowledge domain. Hence, they are not sufficient to apply in IPS systems.
Methods for knowledge representation in education are compared based on criteria in Fig.1 as Table 1 [6].

III. STRUCTURE OF RELA-OPS MODEL
Rela-Ops model is a model for representing the combining knowledge of relations and operators. It is built based on ontology and object-oriented approach, in which an object in this model has behaviors to solve problems on its own. The components of this model are sets have their properties.
Definition 1: The structure of Rela-Ops model includes four components: In which, C is a set of concepts. Each concept c ∈ C is a class of objects which have their own structure and behaviors to solve problems on themselves. Each concept c also has an instance set I c . R is a set of relations on concepts. Each relation is a binary relation between concepts in C. This set represents the knowledge of relations in the knowledge domain K. The OPS-set is a set of operators. This set represents the knowledge of unary and binary operators on concepts in C. The RULES-set is a set of inference rules of the knowledge domain K.
In Fig. 2: Rela-Ops model consists of four components as part (I). This model is built based on ontology. Each concept in C has a particular structure, and objects of each concept also have behaviors to solve problems on them, as part (II). The structure of components in the Rela-Ops model is presented in sections 3.A and 3.C. The facts of this model are defined in section 3.B. Besides, the Rela-Ops model has general problems and an inference engine to solve them as part (III). The reasoning method for solving these problems will be presented in section 4. The structure of each component in the Rela-Ops model is presented in Table 2.  (c ∈ C, x is an object) VOLUME 8, 2020 x ∈ I c1 , y ∈ I c2 ) 5. Equal(e 1 , e 2 )::= (e 1 = e 2 ) (e 1 , e 2 are expressions) Let P K = {f |f is a predicate} α K : P K → N: assigning arities to a predicate.
c) if A is a sentence and x is an invidual variable:

Definition 9:
a) A fact in Rela-Ops model is a sentence as clause form (definition 3.3) or predicate form (definition 3.6). b) Let f and g be two facts. The unification of f and g, denoted f g, is defined as the following conditions: 1. f and g are clauses and have the same form k, with 1≤ k ≤ 5: use the unification of facts in [9], [10]. 2. OR if f = ¬f 1 and g = ¬g 1 , and f 1 , g 1 are facts: 3. OR if f = f 1 ∧ f 2 and g = g 1 ∧ g 2 , and f 1 , f 2 , g 1 , g 2 are facts: (f 1 g 1 and f 2 g 2 ) or (f 2 g 1 and f 2 g 1 ).

Definition 10 [10]: Relations and operations on a set of facts:
Let A and B be sets of facts, and f be a fact, the definition of relations and operators between them as follows:

C. STRUCTURE OF RULES-SET
A rule r ∈ RULES is one of the four kinds below: In the Rela-Ops model, the structure of each component has been built completely. The syntax of a clause and a predicate is defined clearly, thought that the facts and their unification are studied. Based on the structure of this model, the model of problems and algorithms for solving them will be presented in the next section. [facts] ENDRULE; kind::= deductive | generate | equivalent |equation. In practice, a knowledge domain K as the Rela-Ops model can be represented by a restriction model. These restriction models can reduce the complexity of an ordinary model. The knowledge of relations is a popular kind of knowledge. In this kind of knowledge domains, the computation is only the simple computing on real values, so the model for this knowledge does not need the component representing operators. A model of knowledge of relations is (C, R, RULES) as a restriction model as Rela-Ops model lacking OPS-set, this kind of model was studied in [9], [11]. Besides, the knowledge of operators is also used in the computation knowledge domains. In these kinds of knowledge domains, the relations between objects are often as computational relationships. The general problems in those knowledge domains are solving the equations, transforming the expression between objects. A model of knowledge of relations is (C, OPS, RULES) as a restriction model as Rela-Ops model lacking R-set, this kind of model was studied in [10], [12].

IV. MODEL OF PROBLEMS AND ALGORITHMS ON RELA-OPS MODEL
There are two kinds of problems on the Rela-Ops model: Problems on an object and general problems on the model. Problems on an object are its behaviors, and they are solved based on the reasoning on its structure. Model and the solving method of this kind of problem are presented in [9], [10]. In this section, we only mention the model and solving methods for general problems. Denote A problem P is solvable if and only if there exists a list S such that G.g S(E).  11. This algorithm will solve the problem P though these steps as follows: Input: The problem P = (O, Re, E)→G Output: The solution to problem P. The method of the following general algorithm uses forward chaining reasoning. It also uses heuristics rules in the reasoning process of searching for applied rules. Objects attend this process as active agents. We use the characteristic of the relations and operators to get new facts. This processing is done when it gets the goal. The algorithm for solving this kind of problem uses a forward chaining strategy to get new facts. The reasoning combines heuristic rules to make the deducing more effectively. This algorithm was studied and designed in [9], [10], [48]. It only needs to change about combining the facts of relations and equations in reasoning processing. 76998 VOLUME 8, 2020

Algorithm 1
Step Use the rules CONNECT the knowledge of K p and K q in Connect for generating the new objects only belonging to a knowledge domain (K p or K q ).
O pq : a set of new objects.
Step 5: Suppose that the objects in O pq belong to K q .
Use the objects in O pq and the facts in F pq and Known for transforming the problem P to the problem P' on K q .
Step 6: Solve the problem P' on the knowledge domain K q by using algorithm 1.
Update Known and Sol.

Step 7:
If problem P' is solvable Problem P has a solution Sol. Else There is no solution for problem P.

V. KNOWLEDGE MODEL FOR MULTIPLE KNOWLEDGE DOMAINS A. KNOWLEDGE MODEL FOR MULTIPLE KNOWLEDGE DOMAINS
In practice, a knowledge domain K can include multiple knowledge sub-domains K i (i = 1, 2, . . . , n). Each sub-domain has a known form, and they have certain relationships between them. In this paper, each sub-domain K i is modeled by using the Rela-Ops model or its reduced model, so we can establish a knowledge model M(K i ) of knowledge K i . There are also relationships on {K i }; thus, models {M(K i )} and their relations are also specified. We determine a knowledge model M(K) for the whole knowledge, K.
Definition 16: The model for multiple knowledge domains is a tube: In which, K = {K 1 , K 2 , . . . , K n } is a set of knowledge sub-domains, and each sub-domain has the form as the Rela-Ops model. Connect is a set of connective rules between sub-domains K i (i = 1, 2, . . . , n). In this paper, we only consider the rules for transforming the knowledge in K i into the knowledge in K j (j i; i, j = 1 . . . n).

B. PROBLEMS ON THE MODEL FOR MULTIPLE KNOWLEDGE DOMAINS
Let a knowledge domain K = (K, CONNECT), including knowledge sub-domains K i ∈ K (i = 1 . . . n). The general problem on the Rela-Ops model can solve problems that only use the knowledge of a knowledge domain K i . However, it cannot solve the problems which require the combination of multiple knowledge domains for solving. In this section, problems about combining the knowledge domains have been studied and solved. The model of these problems is also as: (O, F)→G. It is classified into two kinds: •Kind 1: The hypothesis of the problem has objects which belong to knowledge domains K p and K q (p q; p, q = 1 . . . n) •Kind 2: The hypothesis of the problem only has objects which belong to one knowledge domain K p (p = 1 . . . n) Both kinds of problems require using the knowledge in the other knowledge domains to solve them.
Algorithm 3: Solving a problem in kind 1 Given a knowledge domain K = (K, CONNECT) including knowledge sub-domains K i ∈ K (i = 1 . . . n), and P = (O, F) → G is a problem, P has objects which belong to knowledge domains K p and K q (p q; p, q = 1 . . . n) Input: The problem P = (O, F) → G such that: Output: The solution to problem P. 77000 VOLUME 8, 2020 There is no solution to problem P.

Algorithm 4: Solving a problem in kind 2
Given a knowledge domain K = (K, CONNECT) including knowledge sub-domains K i ∈ K (i = 1 . . . n), and P = (O, F) → G is a problem, P only has objects which belong to one knowledge domain K p (p = 1 . . . n).
Input: The problem P = (O, F) → G such that: Output: The solution to problem P.

VI. APPLICATION FOR DESIGNING INTELLIGENT PROBLEMS SOLVER SYSTEMS IN MATHEMATICS A. DESIGN AN INTELLIGENT SYSTEM FOR SOLVING PROBLEMS IN LINEAR ALGEBRA
Linear Algebra is a required course in the university. In this course, chapters about matrix, linear system, and vector space are the foundation of computational techniques. They help students improve their basic knowledge about mathematics, solving the problems. The IPS for Linear Algebra has to solve common exercises in this course, including basic and advanced kinds. Moreover, solutions to this system have to satisfy the requirements of an IPS. They have to ensure the criterion about pedagogy. They are readable, step-by-step, especially their reasoning simulates the method for solving of students. There are many current programs for solving exercises in Linear Algebra, but they have not yet tended to learn supporting systems, some requirements in education are missing. Symbolab [39] and Wolfram|Alpha [41] are websites for solving problems step-by-step; however, their knowledge bases are organized as frames, so they cannot be used to solve the problems that require in-depth knowledge. Maple [42] and Matlab [43] are computer algebra systems; their computation on linear algebra is fast; however, they do not have a reason to get solutions to problems similar to the solving method of learners.
In this section, we build an IPS system in Linear Algebra at university. In this course, chapters about Matrices, Linear equations system, and Vector space are essential for students. Our system can solve some kind of exercise in these chapters. Its knowledge base is represented by the knowledge model for multiple knowledge domains: (K, CONNECT)

1) K -SET OF SUB-DOMAINS
In this system, the knowledge domain about Linear Algebra is collected from [32], [33]. It is partitioned into three knowledge sub-domains: In which: •K 1 = (C 1 , R 1 , OPS 1 , RULES 1 ) is a knowledge domain about Matrices -Vectors.
The detail of these knowledge bases is represented in Appendix A.

2) CONNECT -SET OF CONNECTIVE RULES BETWEEN THE SUB-DOMAINS
This set includes the rules for transforming the knowledge in a sub-domain K i ∈ K into the other sub-domains K j (j i; i, j = 1, 2, 3). The connection of the knowledge K 1 and K 2 is the equivalent transforming between the set of roots of two linear equations systems and the augmented matrices of two corresponding linear equations systems. This transforming converts the problems in the knowledge K 2 into the problems in the knowledge K 1 by rule 2.2 in Appendix A.   we have the connection between the knowledge K 3 and K 2 through the finding a spanning-set and proving the linear independence of a set of vectors in the knowledge K 3 by solving a linear equations system in the knowledge K 2 . The transforming converts the problems in the knowledge K 3 into the problems in the knowledge K 2 by rules 3.2 and 3.3 in Appendix A. Fig. 3 represents the relations between the those knowledge sub-domains.

3) TESTING AND EXPERIMENTS
Our system can solve the typical kinds of basic and advanced problems for this course. Its solutions are clear, readable, and their reasoning is similarly the method for solving of students.
Firstly, the exercise is inputted to our program using the model of problems (O, F)→G as section 5.2. Our computer program has the specification language to represent the exercise. The inference engine of our system is designed based on algorithms for solving problems on the knowledge model for multiple knowledge domains as section 5.2. Then, the program solves this exercise automatically. Its solution trends to meet the requirements of an IPS in education [48]. Fig. 4 is the user-interface of our system: Symbolab is a website that can automatically solve mathematical problems. It gives step-by-step solutions [37]. It can solve some problems with matrices and equation systems. We compare our system and Symbolab about the ability to solve problems and the meeting requirements of an IPS in education.
Comparison of the ability to solve problems: The exercises are collected from books [32], [33]. They include three kinds: (1) Problems about Matrices -Vectors.
(2) Problems about the Linear equations system.
Before solving the problems, they have to be represented. The problems are specified based on the structure of the knowledge-based model. Rela-Ops model has the specification of facts, and it can represent many kinds of real facts. By using the Rela-Ops model, our program can represent the objects and facts of the problems more naturally, so the hypothesis of the problems is specified more appropriate for reality. The knowledge base in Symbolab is organized as frames, so it only can solve the kinds of exercises that were set up. Table 4 shows the comparison of some kinds of problems that can be solved by programs. The Solution of Our System: Step 1: We have: With this problem, Symbolab cannot solve problems of the knowledge domain about vector spaces. Here is the solution to our system. + The solution of this system: The solution to this problem is found by using algorithm 4. The current problem is in the knowledge domain K 3 (Vector Space). Firstly, using the rule ''The set of roots of a linear equations system with n variables is a vector subspace of R'', the problem will be changed to the problem about solving a linear equations system This problem is in the knowledge domain K 2 (Linear equations systems). Secondly, using the rule for transforming the set of roots to the augmented matrix, the problem in K 2 will be changed to the problem of finding the augmented matrix. This problem is in the knowledge domain K 1 (Matrices). It can be solved easily by using the transformations on a matrix. Finally, the solution of the original problem will be found by computing inversely.

FIGURE 5.
Results of testing in linear algebra course.
Step 4: Let Then B is a based-set of V.
The results of our system and Symbolab for solving problems in [33] are shown in Table 5 and Fig. 5: The knowledge base of Symbolab is designed as frames, so it only can solve the kinds of exercises that had been set up, especially it cannot solve exercises in a section about vector spaces. The knowledge base of our system is organized as a complete system. It can represent the knowledge liking the knowledge acquisition of the human, so it can solve some advanced problems. However, some exercises are too hard for the typical students, and they require the depth knowledge in linear algebra to solve them, such as combining the knowledge of calculus to compute the determinant of a general matrix; hence our system cannot solve it.
Comparison of the requirements of IPS in education: Beside solving the problems in linear algebra, the system has to support students to study this course, so it should meet the requirements of an IPS in education. Table 6 compares systems based on these criteria.

B. DESIGN AN INTELLIGENT SYSTEM FOR SOLVING PROBLEMS IN 2D-ANALYTIC GEOMETRY
2D-Analytic Geometry is an important knowledge domain in the high-school mathematical curriculum of Vietnam [34]. In this course, the student has to solve geometric plane problems by using the method of coordinate. In this section, we build an IPS for 2D-Analytic Geometry in high-school. The knowledge base of this system is represented by Rela-Ops model: The detail of this knowledge base is represented in Appendix B.

1) DESIGN THE INFERENCE ENGINE OF THE SYSTEM
For making inference processing of the system is faster and more effective, some heuristic rules are integrated into the inference engine for searching the proof of problems.

a: USE SAMPLE PROBLEMS
When solving a practical problem, we will consider a problem related to the current problem. If we had met the related problem, then we can use its results to get the solution to the practical problem more effectively. Sample Problems are related problems [35].
Example 19: Some sample problems have been used in our system: The solution of Sample problem: Step 1: A: Point, B: Point, A belongs d, B belongs d, Step 2: A: Point, n: Vector n is a normal vector of d, A belongs d, Solution of Sample problem: Step The algorithm for searching the sample problem in the knowledge domain about 2D-Analytic Geometry [35].
Algorithm 5: Give a problem P with the hypothesis (O, F) in the knowledge domain about 2D-Analytic Geometry. A sample problem for the problem P can be founded by these steps: Input: (O, F): hypothesis of the current problem P.
Sample: a set of sample problems. Output: Determine a problem S ∈ Sample is a sample problem of P. Step 2: if (Sample_found) then Return S is a sample problem of P.

b: GROUP OF RULES CAN BE APPLIED
When solving a problem, there are some rules that have more possibilities to apply for this problem. These rules belong to objects and facts in the practical problem. Hence, based on the hypothesis of the problem, we can show a group of rules that have the possibility to be used for solving a problem. Example 20: Some groups of rules: -Group of rules for problems about determining the equation of a unique line in a triangle, such as a bisector line, median line, and the height of a triangle. -Group of rules for problems about determining the distance between a point and a line in a circle.

2) TESTING AND EXPERIMENTAL RESULTS
Exercises of the course are collected from [34]. These exercises are classified into four kinds: Symbolab only can solve some fundamental problems in kind (i) and kind (ii), such as: determining an intersection point of two lines, the equation of a line through two given points. Hence, it is not an intelligent system for supporting to learn this course. Our system satisfies the requirements of an IPS in education. In the above reasoning steps, some heuristic rules have been used in the inference process: • The steps S1, S2, and S3 use the group of rules relating to individual lines in a triangle.
• The group of steps S5-S8 uses the Sample Problem SP2.
From those reasoning steps, the solution of this problem is as follows: Step   [34]. .area = 247/16 This solution is similar to the reasoning of the student for solving this problem. It uses the knowledge of 2D-Analytic Geometry in high-school. Using heuristics rules makes the inference process is more efficient in practice. Table 7 and Fig. 6 show the ability of our IPS system for solving problems in this course: The knowledge base of our system is organized as a complete system. It can solve many kinds of exercises in these courses. More than that, our system is pedagogical and suitable for the knowledge level of high-school students. This system is helpful in supporting the students in studying this course.  Our program has been tested and evaluated by 100 students of two high-schools in Ho Chi Minh City, Vietnam. All students had just graduated from high school in August 2018. Those students include 11 average students who have a GPA from 5.0 -7.0, 55 good students who have a GPA from 7.0 -8.0, and 34 outstanding students who have a GPA higher than 8.0.
This survey is interested in the requirements of an IPS in education: the sufficient of a knowledge base, the effectiveness of problem-solving, the pedagogy of the program, and the usefulness for studying this course. Firstly, each student selects 03 exercises from a set of solvable exercises (79 exercises). He/She checks their solutions. Secondly, the student inputs two other problems in four kinds of exercises. The program solves and shows solutions for those exercises. Finally, they assess each criterion with a level from 1 -5. Each level has the meaning as in [38], it is respectively very bad -very good.
The results of this survey are shown in Table 8 and Fig.7: Though the results of the survey, our program meets the requirements of IPS in education. It is also received excellent feedback from students. Our program is useful to support students studying this course. Its solutions are suitable for the knowledge level of students. The reasoning uses rules, theorems in the curriculum of this course. However, this program needs more research about its impact on developing students' solving-problem skills.

VII. CONCLUSION
In this paper, we propose a method to represent the Rela-Ops model, which combines the knowledge of relations and operators. This model includes components: concepts, relations, operators, and rules. The syntax of sentences and the unification of facts have also been defined. They make this model remaining flexible and more effective in practice. Based on the structure of the model, the problems have also been proposed and solved. Reasoning processing uses the characteristics of relations and operators to solve these problems. The algorithms have been proved the effectiveness Our proposed method can apply in many knowledge domains of STEM education. This model and some its restriction model were applied to build IPS systems in another mathematics courses, such as knowledge domains about Plane Geometry in middle school [11], Algebra in middle school [37], Solid Geometry in high school [9], Vector Algebra in high school [10], Discrete Mathematics in university [38]. In this paper, we presented the application of the Rela-Ops model in designing an IPS for 2D-Analytic Geometry in high-school. Table 9 compares the methods of knowledge representation based on the criteria of an IPS system in education.
Besides that, a knowledge model for multiple knowledge domains is studied, in which each sub-domain has the form as the Rela-Ops model. This model represents the connective relations between these domains. Besides, the problems for combining the knowledge of these domains are proposed and solved. In using this model, the knowledge base of linear algebra has been represented. This knowledge domain includes three domains: Matrices, Linear equations systems, and Vector spaces. This knowledge base is applied to design an intelligent system for solving problems in linear algebra. Its proofs are readable and human-alike solutions. This system can be used in supporting the learning of students.
In the future, problems with optimization of the searching rules on the Rela-Ops model will be studied. Some problems with the knowledge model for multiple knowledge domains will also be researched, such as the problems about the integration of knowledge domains for solving current problems. Our methods would be proved the merits when they could be applied in building the efficient, intelligent systems which have the knowledge from multiple domains, such as knowledge domains about mathematics, chemistry, and physics.

A. KNOWLEDGE BASE OF LINEAR ALGEBRA COURSE IN THE UNIVERSITY
The knowledge domain about Linear Algebra is partitioned into three knowledge sub-domains.  It also includes relations ''is-a'' between the kinds of matrixes, such as: SquareMatrix is-a Matrix, DiagnolMatrix is-a SquareMatrix... OPS 1 is referring to a set of operators between matrices and vectors concepts.  • Relations between a point and a line or a segment: a point belongs to a line (or a segment), a point is a midpoint of a segment.
• Relations between a line and a triangle: a line is a bisector (or height, or median) line of a triangle.
• Relations between two lines: parallel (//), perpendicular (⊥), intersection • Relations between a vector and a line: a vector is a normal (or direction) vector of a line.
• It also includes relations ''is-a'' between the kinds of triangles, such as Right Triangle is-a Triangle, Isosceles Triangle is-a Triangle, Equilateral Triangle is-a Isosceles Triangle. . .
OPS is referring to a set of operators between vectors.
RULES -set of inference rules + Some deductive rules in RULES-set: