Knowledge about relations plays a crucial role in human’s knowledge. Different methods for representing this type of knowledge have been proposed. However, due to the lack of theoretical foundations, these methods cannot guarantee criteria in knowledge representation such as formality, universality, usability and practicality. They are not adequate to represent the knowledge domains in practice which have many components. Based on formal ontology approach, a knowledge model about relations, called Rela-model, is presented in this paper. It has the components such as concepts, relations between concepts, and rules. The concepts in this model consist of attributes, facts and rules of itself. Each object in a concept has also equipped its behavior to solve problems on it. The methods for solving problems based on Rela-model are also studied. The general problems on this model are the following: Given some objects and facts on them, determine the closure of set of attributes and facts on the objects or determine an object or consider a relation between the objects. The algorithms to solve problems are designed and their properties, such as finiteness, effectiveness, have also been proved. Besides the solid mathematical foundation, Rela-model also has a simple specification language which can effectively represent the knowledge, thus it can be used in many real situations. Our approach is also applied to build two systems: the intelligent problem solver about solid geometry in high school mathematics, and the expert system to diagnose diseases in diabetic microvascular complication.

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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. INDEX TERMS Knowledge representation, knowledge-based systems, intelligent problem-solver, knowledge engineering.

Nowadays, building intelligent systems for science, technology, engineering, and math (STEM) education is necessary to support the studying of learners. Intelligent problem solver (IPS) is a system that can be able to solve or tutor how to solve the problems automatically. Learners only declare hypothesis and goal of problems based on a sufficient specification language. They can request the program to solve it automatically or to give instructions that help them to solve it themselves. Knowledge representation plays a vital role in these kinds of intelligent systems. There are various methods for knowledge representation; however, they do not meet the requirements of an IPS in STEM education. In this paper, we propose the criteria of a knowledge model for an IPS in education. These criteria orient to develop a method for knowledge representation to meet actual requirements in practice, especially pedagogical requirements. For proving the effectiveness of these criteria, a knowledge model is also constructed. This model can satisfy these criteria and be applied to build IPS for courses, such as mathematics and physics.

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