Semantics of Fuzzy Sets in Rough Set Theory

The objective of this paper is to provide a semantic framework for fuzzy sets in the theory of rough sets. Rough membership functions are viewed as a special type of fuzzy membership functions interpretable using conditional probabilities. The relationships between fuzzy membership functions and rough membership functions, between core and support of fuzzy set theory and lower and upper approximation of rough set theory, are investigated. It is demonstrated that both theories share the same qualitative properties. Interpretations of fuzzy sets in rough set theory lead to constraints on membership values. Two types of constraints on membership values are studies, namely, constraints on membership values of related elements and constraints on membership values of related sets. The classical rough set model and generalized rough set models are discussed.

[1]  Y. Yao,et al.  Information Granulation and Approximation in a Decision-theoretic Model of Rough Sets , 2003 .

[2]  Anna Maria Radzikowska,et al.  Fuzzy Rough Sets Based on Residuated Lattices , 2004, Trans. Rough Sets.

[3]  Yiyu Yao,et al.  Information Granulation and Approximation in a Decision-Theoretical Model of Rough Sets , 2004 .

[4]  George J. Klir,et al.  Fuzzy sets and fuzzy logic - theory and applications , 1995 .

[5]  Yiyu Yao,et al.  Two views of the theory of rough sets in finite universes , 1996, Int. J. Approx. Reason..

[6]  Z. Pawlak,et al.  Rough membership functions , 1994 .

[7]  Yiyu Yao,et al.  Interpreting Fuzzy Membership Functions in the Theory of Rough Sets , 2000, Rough Sets and Current Trends in Computing.

[8]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[9]  D. Dubois,et al.  ROUGH FUZZY SETS AND FUZZY ROUGH SETS , 1990 .

[10]  Didier Dubois,et al.  The three semantics of fuzzy sets , 1997, Fuzzy Sets Syst..

[11]  S. K. Wong,et al.  Comparison of the probabilistic approximate classification and the fuzzy set model , 1987 .

[12]  Yiyu Yao,et al.  Generalization of Rough Sets using Modal Logics , 1996, Intell. Autom. Soft Comput..

[13]  Lech Polkowski,et al.  Rough Sets in Knowledge Discovery 2 , 1998 .

[14]  Jacek M. Zurada,et al.  Computational Intelligence: Imitating Life , 1994 .

[15]  Bo Zhang,et al.  Theory and Applications of Problem Solving , 1992 .

[16]  Janusz Zalewski,et al.  Rough sets: Theoretical aspects of reasoning about data , 1996 .

[17]  Yiyu Yao,et al.  A Decision Theoretic Framework for Approximating Concepts , 1992, Int. J. Man Mach. Stud..

[18]  Wei-Zhi Wu,et al.  Neighborhood operator systems and approximations , 2002, Inf. Sci..

[19]  Yiyu Yao,et al.  Relational Interpretations of Neigborhood Operators and Rough Set Approximation Operators , 1998, Inf. Sci..

[20]  Francesc Esteva,et al.  Review of Triangular norms by E. P. Klement, R. Mesiar and E. Pap. Kluwer Academic Publishers , 2003 .

[21]  Didier Dubois,et al.  A class of fuzzy measures based on triangular norms , 1982 .

[22]  Yiyu Yao,et al.  Constructive and Algebraic Methods of the Theory of Rough Sets , 1998, Inf. Sci..

[23]  Yiyu Yao,et al.  Rough sets and interval fuzzy sets , 2001, Proceedings Joint 9th IFSA World Congress and 20th NAFIPS International Conference (Cat. No. 01TH8569).

[24]  J. Kacprzyk,et al.  Advances in the Dempster-Shafer theory of evidence , 1994 .

[25]  Y. Yao,et al.  Generalized Rough Set Models , 1998 .

[26]  Sadaaki Miyamoto,et al.  Rough Sets and Current Trends in Computing , 2012, Lecture Notes in Computer Science.

[27]  Yiyu Yao,et al.  A Comparative Study of Fuzzy Sets and Rough Sets , 1998 .

[28]  W. Zakowski.  APPROXIMATIONS IN THE SPACE (U,π) , 1983 .