Cognitive Informatics and Contemporary Mathematics for Knowledge Manipulation

Although there are various ways to express entities, notions, relations, actions, and behaviors in natural languages, it is found in Cognitive Informatics (CI) that human and system behaviors may be classified into three basic categories known as to be, to have, and to do. All mathematical means and forms, in general, are an abstract and formal description of these three categories of system behaviors and their common rules. Taking this view, mathematical logic may be perceived as the abstract means for describing ‘to be,’ set theory for describing ‘to have,’ and algebras, particularly the process algebra, for describing ‘to do.’ This paper presents the latest development in a new transdisciplinary field known as CI. Three types of new mathematical structures, Concept Algebra (CA), System Algebra (SA), and Real-Time Process Algebra (RTPA), are created to enable rigorous treatment of knowledge representation and manipulation in terms of to be / to have / to do in a formal and coherent framework. A wide range of applications of the three knowledge algebras in the framework of CI has been identified in knowledge and software engineering.