Knowledge Representation and Reasoning is at the heart of the great challenge of Artificial Intelligence, especially intelligent problem solvers (IPSs). Applications such as the intelligent problem solver in plane geometry and linear algebra have knowledge bases containing a complicated system of concepts, relations, operators, functions, and rules. Therefore, designing of the knowledge bases and the inference engines of those systems requires knowledge representations in the form of ontologies. Ontology COKB (Computational Object Knowledge Base) is suitable for these requirements. COKB model and reasoning algorithms for solving problems on it are essential parts of the ontology. Previous results of COKB model together with reasoning methods have not been complete, and it is needed to develop the knowledge representation model and reasoning algorithms. The perfect COKB model helps to represent knowledge domains and problems more adequately; reasoning techniques with new methods of reasoning and heuristics produce inference engines that solve more kinds of problems, more efficiently and more naturally. They have been used to design and to implement IPSs in plane geometry, analytic geometry, discrete mathematics and linear algebra.
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