In this paper, some important issues of granularity are discussed mainly in information systems (ISs) based on binary relation. Firstly, the vector representation method of knowledge granules is proposed in an infor-mation system based on binary relation to eliminate limitations of set representation method. Secondly, operators among knowledge granularity are introduced and some important properties of them are studied carefully. Thirdly, distance between two knowledge granules is established and granular space is constructed based on it. Fourthly, axiomatic definition of knowledge granularity is investigated, and one can find that some existed knowledge granularities are special cases under the definition. In addition, as an application of knowledge granular space, an example is employed to validate some results in our work.
KeywordsBinary RelationGranular SpaceInformation SystemRough Set
T. Beaubouef, F. E. Petry and G. Arora, “Information-theoretic Measures of Uncertainty for Rough Sets and Rough Relational Databases,” Information Society, Vol. 109, 1998, pp. 185-195. doi:10.1016/S0020-0255(98)00019-X
T. S. Blyth, “Lat-tices and Ordered Algebraic Structures,” Springer-Verlag, London, 2005.
Z. Bonikowski, E. Bryniarski and U. Wy-braniec, “Extensions and Intentions in The Rough Set Theory,” Information Society, Vol. 107, 1998, pp. 149-167. doi:10.1016/S0020-0255(97)10046-9
S. Greco, B. Mata-razzo and R. Slowingski, “Rough Approximation of a Prefer-ence Relation by Dominance Relation,” European Journal of Operation Research, Vol. 117, 1999, pp. 63-68. doi:10.1016/S0377-2217(98)00127-1
R. Hosbs, “Granu-larity,” Proceedings of the ninth international joint conference on artificial intelligence, Los Angeles, California, 1985, pp. 432-435.
Q. Hu and D. Yu, “Entropies of Fuzzy Indis-cernibility Relation and Its Operations,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol. 12, No. 5, 2004, pp. 575-589. doi:10.1142/S0218488504003089
Z. J. Jiang and S. L. Sun, “Functional Analysis,” Higher education press, Beijing, 2005.
Z. J. Jiang and Z. Q. Wu, “Theory of Real Variable Functions,” People's education press, Beijing, 1973.
G. J. Klir, “Basic Issues of Computing with Granular Probabilities,” Proceedings of 1998 IEEE International Conference on Fuzzy Systems, Alaska, USA, 1998, pp. 101-105.
M. Krysz-kiewicz, “Rough Set Approach to Incomplete Information Sys-tems,” Information Society, Vol. 112, 1998, pp. 39-49. doi:10.1016/S0020-0255(98)10019-1
J. Y. Liang, K. S. Chin, C. Y. Dang and C. M. Yam, “A new Method for Meas-uring Uncertainty and Fuzziness in Rough Set Theory,” Inter-national Journal of General Systems, Vol. 31, No. 4, 2002, pp. 331-342. doi:10.1080/0308107021000013635
J. Y. Liang and Y. H. Qian, “Axiomatic Approach of Knowledge Granulation in Information System,” In: A. Sattar and B. H. Kang (Eds. ), Lecture Notes in Computer Science, Springer, Berlin, 2006, pp. 1074-1078.
J. Y. Liang, Z. Z. Shi and D. Y. Li, “Informa-tion Entropy, Rough entropy and Knowledge Granulation in Incomplete Information Systems,” International Journal of General Systems, Vol. 35, No. 6, 2006, pp. 641-654. doi:10.1080/03081070600687668
T. Y. Lin, “From Rough Sets and Neighborhood Systems to Information Granu-lation and Computing with Words,” European Congress on Intelligent Techniques and Soft Computing, 1997, pp. 1602-1606.
T. Y. Lin, “Introduction to Special Issues on Data Mining and Granular Computing,” International Journal of Approximate Reasoning, Vol. 40, 2005, pp. 1-2. doi:10.1016/j.ijar.2004.11.010
Z. Pawlak, “Rough Sets,” International Journal of computer and Information Society, Vol. 11, No. 5, 1982, pp. 341-356.
Z. Pawlak, “Rough Sets,” Theoretical Aspects of Reasoning about Date, Kluwer Academic Publishers, Boston, 1991.
Z. Pawlak, “Rough Sets Approach to Multi-attribute Decision Analysis,” European Journal of operational Research, Vol. 72, 1994, pp. 443-459. doi:10.1016/0377-2217(94)90415-4
Y. H. Qian and J. Y. Liang, “Combination Entropy and Combination Granulation in RoughSet Theory,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol. 16, No. 2, 2008, pp. 179-193. doi:10.1142/S0218488508005121
C. E. Shannon, “The Mathematical Theory of Communication,” The Bell System Technical Journal, Vol. 27, No. 3-4, 1948, pp. 373-423.
A. Skowron and J. Stepaniuk, “Tolerance Ap-proximation Space,” Fundamental Information, Vol. 27, 1996, pp. 245-253.
W. Swiniarski and A. Skowron, “Rough Set Methods in Feature Selection and Recognition,” Pattern Rec-ognition Letters Vol. 24, No. 6, 2003, pp. 883-849.
W. H. Xu, X. Y. Zhang and W. X. Zhang, “Knowledge Granulation, Knowledge Entropy and Knowledge Uncertainty Measure in Information Systems,” Applied Soft Computing, Vol. 9, 2009, pp. 1244-1251. doi:10.1016/j.asoc.2009.03.007
W. H. Xu, J. M. Zhong, X. Y. Zhang and W. X. Zhang, “Attribute Reduction in Or-dered Information Systems Based on Evidence Theory,” Knowledge and Information Systems, Vol. 25, 2010, pp. 169-184. doi:10.1007/s10115-009-0248-5
W. H. Xu and W. X. Zhang, “Knowledge Reduction and Matrix Computation in Inconsistent Ordered Information Systems,” International Journal of Business Intelligence and Data Mining, Vol. 3, 2008, pp. 409-425. doi:10.1504/IJBIDM.2008.022737
W. H. Xu, M. W. Shao and W. X. Zhang, “Knowledge Reduction Based on Evi-dence Reasoning Theory in Ordered Information Systems,” Lecture Notes in Artificial Intelligence, 2006, pp. 535-547.
W. H. Xu and W. X. Zhang, “Methods for Knowledge Reduction in Inconsistent Ordered Information Systems,” Journal of Applied Mathematics and Computing, Vol. 26, No. 1-2, 2008, pp. 313-323. doi:10.1007/s12190-007-0014-3
Y. Y. Yao, “Relational Interpretations of Neighborhood Operators and Rough set Ap-proximation Operators,” Information Sciences, Vol. 101, 1998, pp. 239-259.
Y. Y. Yao, “Granular Computing on Basic Issues and Possible Solutions,” Proceedings of the Fifth Inter-national Conference on Computing and Information, Kuwait, 2000, pp. 186-189.
Y. Y. Yao, “Information Granulation and Rough Set Approximation,” International Journal of Intel-ligent Systems, Vol. 16, No. 1, 2001, pp. 87-104. doi:10.1002/1098-111X(200101)16:1 3.0.CO;2-S
Y. Y. Yao, “A partition Model of Granular Comput-ing,” LNCS Transactions on Rough Sets I, 2004, pp. 232-253. doi:10.1007/978-3-540-27794-1_11
Y. Y. Yao, “Three Perspectives of Granular Computing,” The Proceedings, Inter-national Forum on Theory of GrC from Rough Set Perspective Journal of Nanchang Institute of Technology, Vol. 25, No. 2, 2006, pp. 16-21.
L. A. Zadeh, “Fuzzy sets and Informa-tion Granularity, Advances in fuzzy set theory and applica-tion,” North Holland Publishing, Amsterdam, 1979.
L. A. Zadeh, “Towards a Theory of Fuzzy Information Granulation and Its Centrality in Human Reasoning and Fuzzy Logic,” Fuzzy sets and systems, Vol. 19, 1997, pp. 111-127. doi:10.1016/S0165-0114(97)00077-8
W. X. Zhang and Y. Leung, “Information System and Knowledge Discovery,” Science Press, Beijing, 2003.
W. X. Zhang, J. S. Mi and W. Z. Wu, “Knowledge Reductions in Inconsistent Information Systems,” International Journal of Intelligent Systems, Vol. 18, 2003, 989-1000. doi:10.1002/int.10128
W. X. Zhang and W. Z. Wu, “An Introduction and a Survey for the Studies of Rough Set Theory,” Fuzzy Systems and Mathematics, Vol. 14, No. 4, 2000, pp. 1-12.
W. X. Zhang, Y. Y. Yao and Y. Leung, “Rough Sets and Concept Lattices,” Xi'an Jiaotong University Press, Xi'an, 2006.