Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 513195, 16 pages
doi:10.1155/2011/513195
Research Article

The Uncertainty Measure of Hierarchical Quotient Space Structure

1College of Mathematics & Physics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
2Institute of Computer Science & Technology, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

Received 6 August 2010; Accepted 9 January 2011

Academic Editor: Muhammad R. Hajj

Copyright © 2011 Qinghua Zhang and Guoyin Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In the application of fuzzy reasoning, researchers usually choose the membership function optionally in some degree. Even though the membership functions may be different for the same concept, they can generally get the same (or approximate) results. The robustness of the membership function optionally chosen has brought many researchers' attention. At present, many researchers pay attention to the structural interpretation (definition) of a fuzzy concept, and find that a hierarchical quotient space structure may be a better tool than a fuzzy set for characterizing the essential of fuzzy concept in some degree. In this paper, first the uncertainty of a hierarchical quotient space structure is defined, the information entropy sequence of a hierarchical quotient space structure is proposed, the concept of isomorphism between two hierarchical quotient space structures is defined, and the sufficient condition of isomorphism between two hierarchical quotient space structures is discovered and proved also. Then, the relationships among information entropy sequence, hierarchical quotient space structure, fuzzy equivalence relation, and fuzzy similarity relation are analyzed. Finally, a fast method for constructing a hierarchical quotient space structure is presented.