A new interpretation of complex membership grade

Tamir, Dan E.; Lü, Jin; Kandel, Abraham

doi:10.1002/int.20454

Cited by 108 publications

(54 citation statements)

References 37 publications

(30 reference statements)

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“…Their formalism, however, is limited due to the fact that they restrict the membership function to representation using polar coordinate system where only the magnitude carries fuzzy information. Motivated by similar considerations, Tamir et al, extends the formalism proposed by Ramot et al, and introduce complex class theory where both component of a complex fuzzy grade of membership carries fuzzy information [18]. This enables reasoning about processes with multi-dimensional components where each component is carrying fuzzy information and the interaction between the components cannot be decomposed and represented via primitive, one dimensional, fuzzy set theory and fuzzy logic operations such as union, intersection, conjunction, and disjunction.…”

Section: Current and Related Research On Complex Fuzzy Setsmentioning

confidence: 99%

“…Alternatively, polar representation where both the magnitude and the phase values of the complex membership function convey fuzzy information, can be utilized [18]. Furthermore, they provide a new interpretation of complex fuzzy grades of membership as a representation of a complex fuzzy class along with complex fuzzy class operations.…”

Section: Axiomatic Theory Of Complex Fuzzy Logic and Complex Fuzzy CLmentioning

confidence: 99%

“…The rest of the paper is organized in the following way: section 2 introduces current research based on the concepts of complex fuzzy sets [15]- [18]. Section 3 provides the axiomatic definition of first order complex fuzzy predicate logic [19].…”

Section: Axiomatic Theory Of Complex Fuzzy Logic and Complex Fuzzy CLmentioning

confidence: 99%

“…The current formulation of complex fuzzy sets, complex fuzzy classes, and propositional fuzzy logic is based on the definition of grade of membership. This section briefly reviews the first two formalisms [15]- [18]. The next section introduces axiomatic fuzzy logic and complex fuzzy logic theory and includes a discussion on propositional complex fuzzy logic [19].…”

Section: Current and Related Research On Complex Fuzzy Setsmentioning

confidence: 99%

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Axiomatic Theory of Complex Fuzzy Logic and Complex Fuzzy Classes

Tamir

Kandel

2011

INT J COMPUT COMMUN

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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.

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Section: Current and Related Research On Complex Fuzzy Setsmentioning

confidence: 99%

Section: Axiomatic Theory Of Complex Fuzzy Logic and Complex Fuzzy CLmentioning

confidence: 99%

Section: Axiomatic Theory Of Complex Fuzzy Logic and Complex Fuzzy CLmentioning

confidence: 99%

Section: Current and Related Research On Complex Fuzzy Setsmentioning

confidence: 99%

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Axiomatic Theory of Complex Fuzzy Logic and Complex Fuzzy Classes

Tamir

Kandel

2011

INT J COMPUT COMMUN

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“…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].…”

Section: Introductionmentioning

confidence: 99%

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

Dai

2017

Symmetry

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Abstract:A complex fuzzy set is a set whose membership values are vectors in the unit circle in the complex plane. This paper establishes the orthogonality relation of complex fuzzy sets. Two complex fuzzy sets are said to be orthogonal if their membership vectors are perpendicular. We present the basic properties of orthogonality of complex fuzzy sets and various results on orthogonality with respect to complex fuzzy complement, complex fuzzy union, complex fuzzy intersection, and complex fuzzy inference methods. Finally, an example application of signal detection demonstrates the utility of the orthogonality of complex fuzzy sets.

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Pythagorean Membership Grades, Complex Numbers, and Decision Making

Yager

Abbasov²

2013

Int. J. Intell. Syst.

1,161

675

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We describe the idea of Pythagorean membership grades and the related idea of Pythagorean fuzzy subsets. We focus on the negation and its relationship to the Pythagorean theorem. We look at the basic set operations for the case of Pythagorean fuzzy subsets. A relationship is shown between Pythagorean membership grades and complex numbers. We specifically show that Pythagorean membership grades are a subclass of complex numbers called Π‐i numbers. We investigate operations that are closed under Π‐i numbers. We consider the problem of multicriteria decision making with satisfactions expressed as Pythagorean membership grades, Π‐i numbers. We look at the use of the geometric mean and ordered weighted geometric operator for aggregating criteria satisfaction.

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A new interpretation of complex membership grade

Cited by 108 publications

References 37 publications

Axiomatic Theory of Complex Fuzzy Logic and Complex Fuzzy Classes

Axiomatic Theory of Complex Fuzzy Logic and Complex Fuzzy Classes

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

Pythagorean Membership Grades, Complex Numbers, and Decision Making

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