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

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.

[1]  H. Chertkow,et al.  Semantic memory , 2002, Current neurology and neuroscience reports.

[2]  Jordi Vallverd,et al.  Thinking Machines and the Philosophy of Computer Science: Concepts and Principles , 2010 .

[3]  Ismael Rodríguez,et al.  Testing the Behaviour of Entities in a Cognitive Language , 2008, Int. J. Cogn. Informatics Nat. Intell..

[4]  Yingxu Wang,et al.  RTPA: A Denotational Mathematics for Manipulating Intelligent and Computational Behaviors , 2008, Int. J. Cogn. Informatics Nat. Intell..

[5]  Yingxu Wang,et al.  Deductive Semantics of RTPA , 2008, Int. J. Cogn. Informatics Nat. Intell..

[6]  Yingxu Wang,et al.  The OAR Model of Neural Informatics for Internal Knowledge Representation in the Brain , 2007, Int. J. Cogn. Informatics Nat. Intell..

[7]  Yingxu Wang,et al.  The Theoretical Framework of Cognitive Informatics , 2007, Int. J. Cogn. Informatics Nat. Intell..

[8]  Witold Kinsner,et al.  A Relative Fractal Dimension Spectrum for a Perceptual Complexity Measure , 2008, Int. J. Cogn. Informatics Nat. Intell..

[9]  Yingxu Wang,et al.  Cognitive informatics models of the brain , 2006, IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews).

[10]  L. Beran,et al.  [Formal concept analysis]. , 1996, Casopis lekaru ceskych.

[11]  Yingxu Wang,et al.  The Cognitive Processes of Formal Inferences , 2007, Int. J. Cogn. Informatics Nat. Intell..

[12]  H. R. Quillian.  In semantic information processing , 1968 .

[13]  Yiyu Yao,et al.  Formal concept analysis based on hierarchical class analysis , 2005, Fourth IEEE Conference on Cognitive Informatics, 2005. (ICCI 2005)..

[14]  Yingxu Wang.  On the Big-R Notation for Describing Iterative and Recursive Behaviors , 2006, 2006 5th IEEE International Conference on Cognitive Informatics.

[15]  John R. Anderson.  Is human cognition adaptive? , 1991, Behavioral and Brain Sciences.

[16]  Yingxu Wang,et al.  Using Process Algebra to Describe Human and Software Behaviors , 2003 .

[17]  Yingxu Wang,et al.  On System Algebra: A Denotational Mathematical Structure for Abstract System Modeling , 2008, Int. J. Cogn. Informatics Nat. Intell..

[18]  Gregory L. Murphy,et al.  Theories and concept formation. , 1993 .

[19]  Xudong Guan,et al.  Main retina information processing pathways modeling , 2010, IEEE ICCI.

[20]  Yingxu Wang,et al.  On Laws of Work Organization in Human Cooperation , 2007, Int. J. Cogn. Informatics Nat. Intell..

[21]  Yingxu Wang.  On the informatics laws and deductive semantics of software , 2006, IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews).

[22]  Yingxu Wang,et al.  The Real-Time Process Algebra (RTPA) , 2002, Ann. Softw. Eng..

[23]  Chen Yan,et al.  Adaptive Narration in Multiplayer Ubiquitous Games , 2009, Int. J. Cogn. Informatics Nat. Intell..

[24]  Yiyu Yao.  Concept formation and learning: a cognitive informatics perspective , 2004, Proceedings of the Third IEEE International Conference on Cognitive Informatics, 2004..

[25]  Rokia Missaoui,et al.  INCREMENTAL CONCEPT FORMATION ALGORITHMS BASED ON GALOIS (CONCEPT) LATTICES , 1995, Comput. Intell..

[26]  Rudolf Wille,et al.  Restructuring Lattice Theory: An Approach Based on Hierarchies of Concepts , 2009, ICFCA.

[27]  A. Grafstein.  MIT Encyclopedia of the Cognitive Sciences , 2000 .

[28]  Yingxu Wang,et al.  Cognitive informatics models of the brain , 2006, IEEE Trans. Syst. Man Cybern. Syst..

[29]  Yingxu Wang,et al.  On Concept Algebra and Knowledge Representation , 2006, 2006 5th IEEE International Conference on Cognitive Informatics.

[30]  Allan Collins,et al.  A spreading-activation theory of semantic processing , 1975 .

[31]  Yingxu Wang,et al.  Software Engineering Foundations: A Software Science Perspective , 2007 .

[32]  W.,et al.  Categories and Concepts · Theoretical Views and Inductive Data Analysis , 2009 .

[33]  D. Medin,et al.  Context and structure in conceptual combination , 1988, Cognitive Psychology.

[34]  Yingxu Wang,et al.  On Abstract Systems and System Algebra , 2006, 2006 5th IEEE International Conference on Cognitive Informatics.

[35]  Yingxu Wang.  Discoveries and Breakthroughs in Cognitive Informatics and Natural Intelligence (Advances in Cognitive Informatics and Natural Intelligence (Acini) Book Series) , 2009 .

[36]  A. Diller.  Why AI and Robotics are Going Nowhere Fast , 2010 .