zbMATH — the first resource for mathematics

Calculations of graded ill-known sets. (English) Zbl 1296.03030
Summary: To represent a set whose members are known partially, the graded ill-known set is proposed. In this paper, we investigate calculations of function values of graded ill-known sets. Because a graded ill-known set is characterized by a possibility distribution in the power set, the calculations of function values of graded ill-known sets are based on the extension principle but generally complex. To reduce the complexity, lower and upper approximations of a given graded ill-known set are used at the expense of precision. We give a necessary and sufficient condition that lower and upper approximations of function values of graded ill-known sets are obtained as function values of lower and upper approximations of graded ill-known sets.
03E72 Theory of fuzzy sets, etc.
68T37 Reasoning under uncertainty in the context of artificial intelligence
Full Text: Link
[1] Prade, D. Dubois abd H.: A set-theoretic view of belief functions: Logical operations and approximations by fuzzy sets. Internat. J. General Syst. 12 (1986), 3, 193-226. · doi:10.1080/03081078608934937
[2] Dubois, D., Prade, H.: Incomplete conjunctive information. Comput. Math. Appl. 15 (1988), 797-810. · Zbl 0709.94588 · doi:10.1016/0898-1221(88)90117-4
[3] Dubois, D., Prade, H.: Gradualness, uncertainty and bipolarity: Making sense of fuzzy sets. Fuzzy Sets and Systems 192 (2012), 3-24. · Zbl 1238.03044 · doi:10.1016/j.fss.2010.11.007
[4] Inuiguchi, M.: Rough representations of ill-known sets and their manipulations in low dimensional space. Rough Sets and Intelligent Systems: To the Memory of Professor Zdzisław Pawlak (A. Skowronm and Z. Suraj, Vol. 1, Springer, Heidelberg 2012, pp. 305-327.
[5] Ngyuen, H. T.: Introduction to Random Sets. Chapman and Hall/CRC, Boca Raton 2006.
[6] Pawlak, Z.: Rough sets. Int. J. of Comput. and Inform. Sci. 11 (1982), 341-356. · Zbl 1142.68551 · doi:10.1016/j.ins.2006.06.007
[7] Shafer, G.: A Mathematical Theory of Evidence. Princeton Univ. Press, Princeton 1976. · Zbl 0359.62002
[8] Tijms, H.: Understanding Probability: Chance Rules in Everyday Life. Cambridge Univ. Press., Cambridge 2004. · Zbl 1073.60001
[9] Yager, R. R.: On different classes of linguistic variables defined via fuzzy subsets. Kybernetes 13 (1984), 103-110. · Zbl 0544.03008 · doi:10.1108/eb005681
[10] Zadeh, L. A.: Fuzzy sets. Inform. and Control 8 (1965), 3, 338-353. · Zbl 0942.00007 · doi:10.1016/S0019-9958(65)90241-X
[11] Zadeh, L. A.: The concept of linguistic variable and its application to approximate reasoning. Inform. Sci. 8 (1975), 199-246. · Zbl 0397.68071 · doi:10.1016/0020-0255(75)90036-5
[12] Zadeh, L. A.: Fuzzy sets as the basis for a theory of possibility. Fuzzy Sets and Systems 1 (1978), 3-28. · Zbl 0377.04002 · doi:10.1016/0165-0114(78)90029-5
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.