Abstract
Denotational mathematics is a category of expressive mathematical structures that deals with high-level mathematical entities beyond numbers and sets, such as abstract objects, complex relations, behavioral information, concepts, knowledge, processes, intelligence, and systems. New forms of mathematics are sought, collectively known as denotational mathematics, in order to deal with complex mathematical entities emerged in cognitive informatics, computational intelligence, software engineering, and knowledge engineering. The domain and architecture of denotational mathematics are presented in this paper. Three paradigms of denotational mathematics, known as concept algebra, system algebra, and Real-Time Process Algebra (RTPA), are introduced. Applications of denotational mathematics in cognitive informatics and computational intelligence are elaborated. A set of case studies is presented on the modeling of iterative and recursive systems architectures and behaviors by RTPA, the modeling of autonomic machine learning by concept algebra, and the modeling of granular computing by system algebra.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aho, A.V., Sethi, R., Ullman, J.D.: Compilers: Principles, Techniques, and Tools. Addison-Wesley Publication Co., New York (1985)
Anderson, J.R.: The Architecture of Cognition. Harvard Univ. Press, Cambridge (1983)
Ashby, W.R.: Principles of the Self-Organizing System. In: von Foerster, H., Zopf, G. (eds.) Principles of Self-Organization, pp. 255–278. Pergamon, Oxford (1962)
Ashby, W.R.: Requisite Variety and Implications for Control of Complex Systems. Cybernetica 1, 83–99 (1985)
Bender, E.A.: Mathematical Methods in Artificial Intelligence. IEEE CS Press, Los Alamitos (1996)
Chomski, N.: Three Models for the Description of Languages. I.R.E. Transactions on Information Theory 2(3), 113–124 (1956)
Codin, R., Missaoui, R., Alaoui, H.: Incremental Concept Formation Algorithms Based on Galois (Concept) Lattices. Computational Intelligence 11(2), 246–267 (1995)
Colins, A.M., Loftus, E.F.: A Spreading-Activation Theory of Semantic Memory. Psychological Review 82, 407–428 (1975)
Ellis, D.O., Fred, J.L.: Systems Philosophy. Prentice-Hall, Englewood Cliffs (1962)
Ford, J.: Chaos: Solving the Unsolvable. Dynamics and Fractals. Academic Press, London (1986)
Ganter, B., Wille, R.: Formal Concept Analysis, pp. 1–5. Springer, Heidelberg (1999)
Hampton, J.A.: Psychological Representation of Concepts of Memory, pp. 81–110. Psychology Press, Hove, England (1997)
Heylighen, F.: Self-Organization, Emergence and the Architecture of Complexity. In: Proc. 1st European Conf. on System Science (AFCET), Paris, pp. 23–32 (1989)
Higman, B.: A Comparative Study of Programming Languages, 2nd edn., MacDonald (1977)
Hoare, C.A.R.: Communicating Sequential Processes. Communications of the ACM 21(8), 666–677 (1978)
Hoare, C.A.R.: Communicating Sequential Processes. Prentice-Hall International, London (1985)
Hurley, P.J.: A Concise Introduction to Logic, 6th edn. Wadsworth Pub. Co., ITP (1997)
Klir, G.J.: Facets of Systems Science. Plenum, New York (1992)
Lewis, H.R., Papadimitriou, C.H.: Elements of the Theory of Computation, 2nd edn. Prentice-Hall International, Englewood Cliffs (1998)
Lin, T.Y.: Granular Computing on Binary Relations (I): Data Mining and Neighborhood Systems. In: Proc. Rough Sets in Knowledge Discovery, pp. 107–120. Physica-Verlag, Heidelberg (1998)
Louden, K.C.: Programming Languages: Principles and Practice. PWS-Kent Publishing Co., Boston (1993)
Matlin, M.W.: Cognition, 4th edn. Harcourt Brace College Pub., NY (1998)
Medin, D.L., Shoben, E.J.: Context and Structure in Conceptual Combination. Cognitive Psychology 20, 158–190 (1988)
Milner, R. (ed.): A Calculus of Communication Systems. LNCS, vol. 92. Springer, Heidelberg (1980)
Murphy, G.L.: Theories and Concept Formation. In: Mechelen, I.V., et al. (eds.) Categories and Concepts, Theoretical Views and Inductive Data Analysis, pp. 173–200. Academic Press, London (1993)
Pavel, M.: Fundamentals of Pattern Recognition, 2nd edn. Addision-Wesley, Reading (1993)
Pawlak, Z.: Rough Logic, Bulletin of the Polish Academy of Science. Technical Science 5(6), 253–258 (1987)
Pedrycz, W. (ed.): Granular Computing: An Emerging Paradigm. Physica-Verlag, Heidelberg (2001)
Pnueli, A.: The Temporal Logic of Programs. In: Proc. 18th IEEE Symposium on Foundations of Computer Science, pp. 46–57. IEEE, Los Alamitos (1977)
O’Grady, W., Archibald, J.: Contemporary Linguistic Analysis: An Introduction, 4th edn. Pearson Education Canada Inc., Toronto (2000)
Rapoport, A.: Mathematical Aspects of General Systems Theory. General Systems Yearbook 11, 3–11 (1962)
Skarda, C.A., Freeman, W.J.: How Brains Make Chaos into Order. Behavioral and Brain Sciences 10 (1987)
Skowron, A., Stepaniuk, J.: Information Granules: Towards Foundations of Granular Computing. International Journal of Intelligent Systems 16, 57–85 (2001)
Smith, E.E., Medin, D.L.: Categories and Concepts. Harvard Univ. Press, Cambridge (1981)
von Bertalanffy, L.: Problems of Life: An Evolution of Modern Biological and Scientific Thought. C.A. Watts, London (1952)
von Neumann, J.: The Principles of Large-Scale Computing Machines. Annals of History of Computers 3(3), 263–273 (reprinted, 1946)
Wang, Y.: The Real-Time Process Algebra (RTPA). Annals of Software Engineering: A International Journal, USA 14, 235–274 (2002)
Wang, Y.: Keynote: On Cognitive Informatics. In: Proc. 1st IEEE International Conference on Cognitive Informatics (ICCI 2002), Calgary, Canada, pp. 34–42. IEEE CS Press, Los Alamitos (2002)
Wang, Y.: On Cognitive Informatics. Brain and Mind: A Transdisciplinary Journal of Neuroscience and Neurophilosophy, USA 4(3), 151–167 (2003)
Wang, Y.: Using Process Algebra to Describe Human and Software System Behaviors. Brain and Mind 4(2), 199–213 (2003)
Wang, Y.: On the Informatics Laws and Deductive Semantics of Software. IEEE Transactions on Systems, Man, and Cybernetics (C) 36(2), 161–171 (2006)
Wang, Y.: Cognitive Informatics and Contemporary Mathematics for Knowledge Representation and Manipulation. In: Wang, G.-Y., Peters, J.F., Skowron, A., Yao, Y. (eds.) RSKT 2006. LNCS (LNAI), vol. 4062, pp. 69–78. Springer, Heidelberg (2006)
Wang, Y.: Software Engineering Foundations: A Software Science Perspective. CRC Series in Software Engineering, vol. II. Auerbach Publications, USA (2007)
Wang, Y.: The Theoretical Framework of Cognitive Informatics. International Journal of Cognitive Informatics and Natural Intelligence 1(1), 1–27 (2007)
Wang, Y.: The OAR Model of Neural Informatics for Internal Knowledge Representation in the Brain. International Journal of Cognitive Informatics and Natural Intelligence 1(3), 64–75 (2007)
Wang, Y.: Keynote: On Theoretical Foundations of Software Engineering and Denotational Mathematics. In: Proc. 5th Asian Workshop on Foundations of Software, Xiamen, China, pp. 99–102 (2007)
Wang, Y.: The Theoretical Framework and Cognitive Process of Learning. In: Proc. 6th International Conference on Cognitive Informatics (ICCI 2007), Lake Tahoe, CA, pp. 470–479. IEEE CS Press, Los Alamitos (2007)
Wang, Y.: On the Big-R Notation for Describing Iterative and Recursive Behaviors. International Journal of Cognitive Informatics and Natural Intelligence 2(1), 17–28 (2008)
Wang, Y.: On Concept Algebra: A Denotational Mathematical Structure for Knowledge and Software Modeling. International Journal of Cognitive Informatics and Natural Intelligence 2(2), 1–19 (2008)
Wang, Y.: On System Algebra: A Denotational Mathematical Structure for Abstract System Modeling. International Journal of Cognitive Informatics and Natural Intelligence 2(2), 20–42 (2008)
Wang, Y.: RTPA: A Denotational Mathematics for Manipulating Intelligent and Computational Behaviors. International Journal of Cognitive Informatics and Natural Intelligence 2(2), 44–62 (2008)
Wang, Y.: Deductive Semantics of RTPA. International Journal of Cognitive Informatics and Natural Intelligence 2(2), 95–121 (2008)
Wang, Y., Wang, Y., Patel, S., Patel, D.: A Layered Reference Model of the Brain (LRMB). IEEE Trans. on Systems, Man, and Cybernetics (C) 36(2), 124–133 (2006)
Wang, Y., Lotfi, A.: On the System Algebra Foundation for Granular Computing. International Journal of Software Science and Computational Intelligence 1(1) (December 2008)
Wille, R.: Restructuring Lattice Theory: An Approach Based on Hierarchies of Concepts. In: Rival, I. (ed.) Ordered Sets, pp. 445–470. Reidel, Dordrecht (1982)
Wilson, L.B., Clark, R.G.: Comparative Programming Language. Addison-Wesley Publishing Co, Reading (1988)
Woodcock, J., Davies, J.: Using Z: Specification, Refinement, and Proof. Prentice Hall International, London (1996)
Yao, Y.Y.: Information Granulation and Rough Set Approximation. International Journal of Intelligent Systems 16(1), 87–104 (2001)
Yao, Y.Y.: A Comparative Study of Formal Concept Analysis and Rough Set Theory in Data Analysis. In: Tsumoto, S., Słowiński, R., Komorowski, J., Grzymała-Busse, J.W. (eds.) RSCTC 2004. LNCS (LNAI), vol. 3066, pp. 59–68. Springer, Heidelberg (2004)
Zadeh, L.A.: Fuzzy Sets and Systems. In: Fox, J. (ed.) Systems Theory, pp. 29–37. Polytechnic Press, Brooklyn (1965)
Zadeh, L.A.: Outline of a New Approach to Analysis of Complex Systems. IEEE Trans. on Sys. Man and Cyb. 1(1), 28–44 (1973)
Zadeh, L.A.: Fuzzy Sets and Information Granularity. In: Gupta, M.M., Ragade, R., Yager, R. (eds.) Advances in Fuzzy Set Theory and Applications, pp. 3–18. North-Holland, Amsterdam (1979)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Wang, Y. (2008). On Contemporary Denotational Mathematics for Computational Intelligence. In: Gavrilova, M.L., Tan, C.J.K., Wang, Y., Yao, Y., Wang, G. (eds) Transactions on Computational Science II. Lecture Notes in Computer Science, vol 5150. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87563-5_2
Download citation
DOI: https://doi.org/10.1007/978-3-540-87563-5_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-87562-8
Online ISBN: 978-3-540-87563-5
eBook Packages: Computer ScienceComputer Science (R0)