Axiomatic Theory of Complex Fuzzy Logic and Complex Fuzzy Classes

Abstract

Complex fuzzy sets, classes, and logic have an important role in applications, such as prediction of periodic events and advanced control systems, where several fuzzy variables interact with each other in a multifaceted way that cannot be represented effectively via simple fuzzy operations such as union, intersection, complement, negation, conjunction and disjunction. The initial formulation of these terms stems from the definition of complex fuzzy grade of membership. The problem, however, with these definitions are twofold: 1) the complex fuzzy membership is limited to polar representation with only one fuzzy component.2) The definition is based on grade of membership and is lacking the rigor of axiomatic formulation. A new interpretation of complex fuzzy membership enables polar and Cartesian representation of the membership function where the two function components carry uncertain information. Moreover, the new interpretation is used to define complex fuzzy classes and develop an axiomatic based theory of complex propositional fuzzy logic. Additionally, the generalization of the theory to multidimensional fuzzy grades of membership has been demonstrated. In this paper we propose an axiomatic framework for first order predicate complex fuzzy logic and use this framework for axiomatic definition of complex fuzzy classes. We use these rigorous definitions to exemplify inference in complex economic systems. The new framework overcomes the main limitations of current theory and provides several advantages. First, the derivation of the new theory is based on axiomatic approach and does not assume the existence of complex fuzzy sets or complex fuzzy classes. Second, the new form significantly improves the expressive power and inference capability of complex fuzzy logic and class theory. The paper surveys the current state of complex fuzzy sets, complex fuzzy classes, and complex fuzzy logic; and provides an axiomatic basis for first order predicate complex fuzzy logic and complex class theory.

The Orthogonality between Complex Fuzzy Sets and Its Application to Signal Detection

“…However, a complex fuzzy union using one of above three methods in the phase term does not preserve orthogonality. Moreover, a complex fuzzy union using Sum method in the phase term is orthogonality preserving (see Theorem 4) but does not have the property of function (18). Let us consider the following example.…”

“…Thus a complex fuzzy set is a set of vectors in the complex plane. This idea of membership vectors greatly inspired how definition of complex fuzzy sets operations [14,15] and complex fuzzy logic systems [16][17][18][19][20][21] and how to measures the difference between two complex fuzzy sets [22][23][24].…”

Control System Architecture for a Cement Mill Based on Fuzzy Logic

“…The second step is to define the fuzzy subsets of each input and output variable and create membership functions. The third step is to define fuzzy rules that relate each input membership function to each output membership function [5]. Upon the completion of a fuzzy system, the fuzzy process will fuzzify an input, check each rule to find a degree of truth, and then defuzzify the result into an output value.…”

Multiple Instance Mamdani Fuzzy Inference

“…Other extensions of fuzzy sets could be adapted to the context of multiple instance fuzzy logic. For example, complex fuzzy sets [40] or complex fuzzy classes [45] are based on fuzzy sets characterized by complex-valued membership functions. Because of the two dimensionality nature of a complex fuzzy set, one can think of using it to carry reasoning with bags containing two instances at most.…”