TY - JOUR
T1 - Complex fuzzy logic
AU - Ramot, Daniel
AU - Friedman, Menahem
AU - Langholz, Gideon
AU - Kandel, Abraham
N1 - Funding Information:
Manuscript received May 20, 2002; revised October 11, 2002 and December 9, 2002. This work was supported in part by the University of South Florida Software Testing Center (SOFTEC) under Grant 2108-004, and in part by the Space and Naval Warfare Systems Command (SPAWWR) under Grant N00039-01-1-2248.
PY - 2003/8
Y1 - 2003/8
N2 - A novel framework for logical reasoning, termed complex fuzzy logic, is presented in this paper. Complex fuzzy logic is a generalization of traditional fuzzy logic, based on complex fuzzy sets. In complex fuzzy logic, inference rules are constructed and "fired" in a manner that closely parallels traditional fuzzy logic. The novelty of complex fuzzy logic is that the sets used in the reasoning process are complex fuzzy sets, characterized by complex-valued membership functions. The range of these membership functions is extended from the traditional fuzzy range of [0,1] to the unit circle in the complex plane, thus providing a method for describing membership in a set in terms of a complex number. Several mathematical properties of complex fuzzy sets, which serve as a basis for the derivation of complex fuzzy logic, are reviewed in this paper. These properties include basic set theoretic operations on complex fuzzy sets - namely complex fuzzy union and intersection, complex fuzzy relations and their composition, and a novel form of set aggregation - vector aggregation. Complex fuzzy logic is designed to maintain the advantages of traditional fuzzy logic, while benefiting from the properties of complex numbers and complex fuzzy sets. The introduction of complex-valued grades of membership to the realm of fuzzy logic generates a framework with unique mathematical properties, and considerable potential for further research and application.
AB - A novel framework for logical reasoning, termed complex fuzzy logic, is presented in this paper. Complex fuzzy logic is a generalization of traditional fuzzy logic, based on complex fuzzy sets. In complex fuzzy logic, inference rules are constructed and "fired" in a manner that closely parallels traditional fuzzy logic. The novelty of complex fuzzy logic is that the sets used in the reasoning process are complex fuzzy sets, characterized by complex-valued membership functions. The range of these membership functions is extended from the traditional fuzzy range of [0,1] to the unit circle in the complex plane, thus providing a method for describing membership in a set in terms of a complex number. Several mathematical properties of complex fuzzy sets, which serve as a basis for the derivation of complex fuzzy logic, are reviewed in this paper. These properties include basic set theoretic operations on complex fuzzy sets - namely complex fuzzy union and intersection, complex fuzzy relations and their composition, and a novel form of set aggregation - vector aggregation. Complex fuzzy logic is designed to maintain the advantages of traditional fuzzy logic, while benefiting from the properties of complex numbers and complex fuzzy sets. The introduction of complex-valued grades of membership to the realm of fuzzy logic generates a framework with unique mathematical properties, and considerable potential for further research and application.
KW - Complex fuzzy logic
KW - Complex fuzzy relations
KW - Complex fuzzy sets
KW - Complex-valued grades of membership
UR - http://www.scopus.com/inward/record.url?scp=0041377015&partnerID=8YFLogxK
U2 - 10.1109/TFUZZ.2003.814832
DO - 10.1109/TFUZZ.2003.814832
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AN - SCOPUS:0041377015
SN - 1063-6706
VL - 11
SP - 450
EP - 461
JO - IEEE Transactions on Fuzzy Systems
JF - IEEE Transactions on Fuzzy Systems
IS - 4
ER -