On Generalized Fuzzy Belief Functions in Infinite Spaces

Determined by a fuzzy implication operator, a general type of fuzzy belief structure and its induced dual pair of fuzzy belief and plausibility functions in infinite universes of discourse are first defined. Relationship between the belief-structure-based and the belief-space-based fuzzy Dempster-Shafer models is then established. It is shown that the lower and upper fuzzy probabilities induced by the fuzzy belief space yield a dual pair of fuzzy belief and plausibility functions. For any fuzzy belief structure, there must exist a fuzzy belief space such that the fuzzy belief and plausibility functions defined by the given fuzzy belief structure are just the lower and upper fuzzy probabilities induced by the fuzzy belief space, respectively. Essential properties of the fuzzy belief and plausibility functions are also examined. The fuzzy belief and plausibility functions are, respectively, a fuzzy monotone Choquet capacity and a fuzzy alternating Choquet capacity of infinite order.

[1]  D. Dubois,et al.  Fuzzy sets in approximate reasoning, part 1: inference with possibility distributions , 1999 .

[2]  Nehad N. Morsi,et al.  Axiomatics for fuzzy rough sets , 1998, Fuzzy Sets Syst..

[3]  Qinghua Hu,et al.  Fuzzy probabilistic approximation spaces and their information measures , 2006, IEEE Transactions on Fuzzy Systems.

[4]  Chris Cornelis,et al.  Fuzzy Rough Sets: The Forgotten Step , 2007, IEEE Transactions on Fuzzy Systems.

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

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

[7]  Wei-Zhi Wu,et al.  Knowledge acquisition in incomplete fuzzy information systems via the rough set approach , 2003, Expert Syst. J. Knowl. Eng..

[8]  King-Sun Fu,et al.  An Inexact Inference for Damage Assessment of Existing Structures , 1985, Int. J. Man Mach. Stud..

[9]  Andrzej Skowron,et al.  From the Rough Set Theory to the Evidence Theory , 1991 .

[10]  Wei-Zhi Wu,et al.  Attribute reduction based on evidence theory in incomplete decision systems , 2008, Inf. Sci..

[11]  Degang Chen,et al.  Measures of general fuzzy rough sets on a probabilistic space , 2008, Inf. Sci..

[12]  Ronald R. Yager,et al.  Generalized probabilities of fuzzy events from fuzzy belief structures , 1982, Inf. Sci..

[13]  Loredana Biacino,et al.  Fuzzy subsethood and belief functions of fuzzy events , 2007, Fuzzy Sets Syst..

[14]  L. Zadeh.  Fuzzy sets as a basis for a theory of possibility , 1999 .

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

[16]  Babak Nadjar Araabi,et al.  Generalization of the Dempster-Shafer theory: a fuzzy-valued measure , 1999, IEEE Trans. Fuzzy Syst..

[17]  Lotfi A. Zadeh,et al.  Fuzzy sets and information granularity , 1996 .

[18]  Philippe Smets,et al.  The degree of belief in a fuzzy event , 1981, Inf. Sci..

[19]  Alessandro Saffiotti,et al.  The Transferable Belief Model , 1991, ECSQARU.

[20]  John Yen,et al.  Generalizing the Dempster-Schafer theory to fuzzy sets , 1990, IEEE Trans. Syst. Man Cybern..

[21]  Didier Dubois,et al.  Fuzzy set connectives as combinations of belief structures , 1992, Inf. Sci..

[22]  Xizhao Wang,et al.  Learning fuzzy rules from fuzzy samples based on rough set technique , 2007, Inf. Sci..

[23]  Andrzej Skowron,et al.  The rough sets theory and evidence theory , 1990 .

[24]  Nehad N. Morsi,et al.  On the Relationship of Extended Necessity Measures To Implication Operators on the Unit Interval , 1995, Inf. Sci..

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

[26]  Yiyu Yao,et al.  Data mining using extensions of the rough set model , 1998, KDD 1998.

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

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

[29]  Yee Leung,et al.  Connections between rough set theory and Dempster-Shafer theory of evidence , 2002, Int. J. Gen. Syst..

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

[31]  Glenn Shafer,et al.  A Mathematical Theory of Evidence , 2020, A Mathematical Theory of Evidence.

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

[33]  Abraham Kandel,et al.  Constraints on belief functions imposed by fuzzy random variables , 1995, IEEE Trans. Syst. Man Cybern..

[34]  Fabio Cuzzolin,et al.  A Geometric Approach to the Theory of Evidence , 2008, IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews).

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

[36]  Philippe Smets,et al.  The Normative Representation of Quantified Beliefs by Belief Functions , 1997, Artif. Intell..

[37]  Arthur P. Dempster,et al.  Upper and Lower Probabilities Induced by a Multivalued Mapping , 1967, Classic Works of the Dempster-Shafer Theory of Belief Functions.

[38]  T. Denœux.  Modeling vague beliefs using fuzzy-valued belief structures , 2000 .

[39]  Wei-Zhi Wu,et al.  Knowledge reduction in random information systems via Dempster-Shafer theory of evidence , 2005, Inf. Sci..

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

[41]  Didier Dubois,et al.  Evidence measures based on fuzzy information , 1985, Autom..

[42]  John Yen,et al.  Computing generalized belief functions for continuous fuzzy sets , 1992, Int. J. Approx. Reason..

[43]  Abraham Kandel,et al.  Applicability analysis of fuzzy inference by means of generalized Dempster-Shafer theory , 1995, IEEE Trans. Fuzzy Syst..

[44]  Mei Zhang,et al.  A rough set approach to knowledge reduction based on inclusion degree and evidence reasoning theory , 2003, Expert Syst. J. Knowl. Eng..

[45]  Ivan Kramosil,et al.  Belief functions generated by fuzzy and randomized compatibility relations , 2003, Fuzzy Sets Syst..

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

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

[48]  D. Dubois,et al.  Properties of measures of information in evidence and possibility theories , 1987 .

[49]  G. Choquet.  Theory of capacities , 1954 .

[50]  L. Zadeh.  Probability measures of Fuzzy events , 1968 .

[51]  Etienne Kerre,et al.  Fuzzy implication operators and generalized fuzzy method of cases , 1993 .

[52]  George J. Klir,et al.  A principle of uncertainty and information invariance , 1990 .

[53]  Philippe Smets,et al.  The Combination of Evidence in the Transferable Belief Model , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[54]  Ronald R. Yager,et al.  On the normalization of fuzzy belief structures , 1996, Int. J. Approx. Reason..

[55]  Yiyu Yao,et al.  Interpretation of Belief Functions in The Theory of Rough Sets , 1998, Inf. Sci..

[56]  Thierry Denoeux,et al.  Modeling vague beliefs using fuzzy-valued belief structures , 2000, Fuzzy Sets Syst..

[57]  Mitsuru Ishizuka,et al.  Inference procedures under uncertainty for the problem-reduction method , 1982, Inf. Sci..