Characterization of rough set approximations in Atanassov intuitionistic fuzzy set theory

Abstract The primitive notions in rough set theory are lower and upper approximation operators defined by a fixed binary relation and satisfying many interesting properties. Many types of generalized rough set models have been proposed in the literature. This paper discusses the rough approximations of Atanassov intuitionistic fuzzy sets in crisp and fuzzy approximation spaces in which both constructive and axiomatic approaches are used. In the constructive approach, concepts of rough intuitionistic fuzzy sets and intuitionistic fuzzy rough sets are defined, properties of rough intuitionistic fuzzy approximation operators and intuitionistic fuzzy rough approximation operators are examined. Different classes of rough intuitionistic fuzzy set algebras and intuitionistic fuzzy rough set algebras are obtained from different types of fuzzy relations. In the axiomatic approach, an operator-oriented characterization of rough sets is proposed, that is, rough intuitionistic fuzzy approximation operators and intuitionistic fuzzy rough approximation operators are defined by axioms. Different axiom sets of upper and lower intuitionistic fuzzy set-theoretic operators guarantee the existence of different types of crisp/fuzzy relations which produce the same operators.

[1]  Anna Maria Radzikowska,et al.  Rough Approximation Operations Based on IF Sets , 2006, ICAISC.

[2]  Daniel S. Yeung,et al.  Approximations and reducts with covering generalized rough sets , 2008, Comput. Math. Appl..

[3]  S. D. Comer,et al.  On connections between information systems, rough sets and algebraic logic , 1993 .

[4]  Zhang Zhi-ming.  Intuitionistic fuzzy rough sets based on intuitionistic fuzzy coverings , 2010 .

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

[6]  J. Recasens,et al.  UPPER AND LOWER APPROXIMATIONS OF FUZZY SETS , 2000 .

[7]  Przemyslaw Grzegorzewski,et al.  Some notes on (Atanassov's) intuitionistic fuzzy sets , 2005, Fuzzy Sets Syst..

[8]  Gianpiero Cattaneo,et al.  Basic intuitionistic principles in fuzzy set theories and its extensions (A terminological debate on Atanassov IFS) , 2006, Fuzzy Sets Syst..

[9]  Wei-Zhi Wu,et al.  Generalized fuzzy rough sets , 2003, Inf. Sci..

[10]  Javier Gutiérrez García,et al.  Order-theoretic, topological, categorical redundancies of interval-valued sets, grey sets, vague sets, interval-valued "intuitionistic" sets, "intuitionistic" fuzzy sets and topologies , 2005, Fuzzy Sets Syst..

[11]  S. Nanda,et al.  Fuzzy rough sets , 1992 .

[12]  Etienne E. Kerre,et al.  On the relationship between some extensions of fuzzy set theory , 2003, Fuzzy Sets Syst..

[13]  Andrzej Skowron,et al.  Rudiments of rough sets , 2007, Inf. Sci..

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

[15]  Chris Cornelis,et al.  Intuitionistic fuzzy rough sets: at the crossroads of imperfect knowledge , 2003 .

[16]  C. J. V. Rijsbergen,et al.  Rough Sets, Fuzzy Sets and Knowledge Discovery , 1994, Workshops in Computing.

[17]  Anna Maria Radzikowska,et al.  A comparative study of fuzzy rough sets , 2002, Fuzzy Sets Syst..

[18]  Anna Maria Radzikowska,et al.  Characterisation of main classes of fuzzy relations using fuzzy modal operators , 2005, Fuzzy Sets Syst..

[19]  Stephen D. Comer,et al.  An algebraic approach to the approximation of information , 1991, Fundamenta Informaticae.

[20]  Lei Zhou,et al.  On characterization of intuitionistic fuzzy rough sets based on intuitionistic fuzzy implicators , 2009, Inf. Sci..

[21]  K. Atanassov.  More on intuitionistic fuzzy sets , 1989 .

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

[23]  Yee Leung,et al.  On characterizations of (I, J)-fuzzy rough approximation operators , 2005, Fuzzy Sets Syst..

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

[25]  Didier Dubois,et al.  Terminological difficulties in fuzzy set theory - The case of "Intuitionistic Fuzzy Sets" , 2005, Fuzzy Sets Syst..

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

[27]  Krassimir T. Atanassov,et al.  Answer to D. Dubois, S. Gottwald, P. Hajek, J. Kacprzyk and H. Prade's paper "Terminological difficulties in fuzzy set theory - the case of "Intuitionistic Fuzzy Sets" , 2005, Fuzzy Sets Syst..

[28]  H. Thiele.  On axiomatic characterizations of fuzzy approximation operators. I. The fuzzy rough set based case , 2001 .

[29]  Helmut Thiele,et al.  On axiomatic characterization of fuzzy approximation operators. II. The rough fuzzy set based case , 2001, Proceedings 31st IEEE International Symposium on Multiple-Valued Logic.

[30]  Chris Cornelis,et al.  Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification, application , 2004, Int. J. Approx. Reason..

[31]  Helmut Thiele,et al.  On axiomatic characterisations of crisp approximation operators , 2000, Inf. Sci..

[32]  Danish Nadeem,et al.  Rough Intuitionistic Fuzzy Sets , 2002, JCIS.

[33]  Tsau Young Lin,et al.  Rough Approximate Operators: Axiomatic Rough Set Theory , 1993, RSKD.

[34]  Helmut Thiele,et al.  On Axiomatic Characterizations of Fuzzy Approximation Operators , 2000, Rough Sets and Current Trends in Computing.

[35]  Wei-Zhi Wu,et al.  Constructive and axiomatic approaches of fuzzy approximation operators , 2004, Inf. Sci..

[36]  Wen-Xiu Zhang,et al.  An axiomatic characterization of a fuzzy generalization of rough sets , 2004, Inf. Sci..

[37]  Andrzej Skowron,et al.  Rough sets: Some extensions , 2007, Inf. Sci..

[38]  Xizhao Wang,et al.  On the generalization of fuzzy rough sets , 2005, IEEE Transactions on Fuzzy Systems.

[39]  Yee Leung,et al.  Generalized fuzzy rough sets determined by a triangular norm , 2008, Inf. Sci..

[40]  Jiye Liang,et al.  Converse approximation and rule extraction from decision tables in rough set theory , 2008, Comput. Math. Appl..

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

[42]  Krassimir T. Atanassov,et al.  Intuitionistic fuzzy sets , 1986 .