On Concept Algebra: A Denotational Mathematical Structure for Knowledge and Software Modeling

Abstract

Concepts are the most fundamental unit of cognition that carries certain meanings in expression, thinking, reasoning, and system modeling. In denotational mathematics, a concept is formally modeled as an abstract and dynamic mathematical structure that encapsulates attributes, objects, and relations. The most important property of an abstract concept is its adaptive capability to autonomously interrelate itself to other concepts. This article presents a formal theory for abstract concepts and knowledge manipulation known as “concept algebra.” The mathematical models of concepts and knowledge are developed based on the object-attribute-relation (OAR) theory. The formal methodology for manipulating knowledge as a concept network is described. Case studies demonstrate that concept algebra provides a generic and formal knowledge manipulation means, which is capable to deal with complex knowledge and software structures as well as their algebraic operations.