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In this paper, we explore the degree of reflexivity of fuzzy relations. We mainly focus on exploring two new types of the degree of reflexivity of fuzzy relations (we briefly call them as the type-II degree of reflexivity and the type-III degree of reflexivity) that are different from the existing degree of reflexivity of fuzzy relations in the literature (we briefly call it as the type-I degree of reflexivity). The notion of the type-II degree of reflexivity is introduced through the use of the type-I degree of reflexivity and R-implications derived from left-continuous t-norms, while the notion of the type-III degree of reflexivity is introduced through the use of fuzzy equalities based on R-implications derived from left-continuous t-norms. We also illustrate that these three types of the degree of reflexivity are not equal to each other in general, and further characterize when they are equal to each other. Moreover, it is shown that the type-I degree of reflexivity and the type-III degree of reflexivity preserve fuzzy equalities, while the type-II degree of reflexivity does not preserve fuzzy equalities in general. We also characterize when the type-II degree of reflexivity preserves fuzzy equalities. Finally, we show that for continuous t-norms or the nilpotent minimum t-norm, and for a given fuzzy relation R on a set X, there exists a sub- ε -reflexive (ε ∈ [ 0 , 1 ]) fuzzy relation S on X, which is close enough to R with respect to fuzzy equalities, such that the degree of the fuzzy equality between S and R is the type-III degree of reflexivity of R. [ABSTRACT FROM AUTHOR]

A fuzzy relation is a basic concept of fuzzy set theory. Fuzzy entropy as information entropy for a fuzzy relation is to measure the uncertainty of a fuzzy relation. Some measures on the uncertainty of a fuzzy relation have been presented in recent years by generalizing information entropy. However, a fuzzy relation can induce to a family of upper-fuzzy sets and a family of lower-fuzzy sets, respectively; the existing methods consider only the upper-fuzzy sets in a fuzzy relation and do not address the lower-fuzzy sets. Moreover, these methods sometimes need to use the equality between fuzzy sets, but the equality between fuzzy sets is actually very difficult to achieve. To solve the above problem, this paper proposes new fuzzy entropy based on class-consistent technology and considers its application in attribute reduction for heterogeneous data. Class-consistent technology is a technique for dealing with approximate equality of values. It replaces equality with approximate equality between two values in unit close interval. First of all, based on this technology, two equivalence relations are put forward by means of upper-fuzzy and the lower-fuzzy sets in a fuzzy relation. Then, upper-fuzzy entropy and lower-fuzzy entropy are presented by using these two equivalence relations. Next, new fuzzy entropy is constructed to measure the uncertainty of a fuzzy relation, and the constructed fuzzy entropy overcomes the weakness of the existing methods. Moreover, new fuzzy conditional entropy is defined. Finally, new fuzzy conditional entropy is applied to carry out attribute reduction for heterogeneous data. The experiment results confirm that the given reduction method is more effective than other methods. [ABSTRACT FROM AUTHOR]

Pythagorean fuzzy sets are an advanced tool of two-dimensional membership functions that are characterized by a membership degree and a non- membership degree satisfying the condition that the square sum of its membership degree and non- membership degree is equal to or less than one In this paper, it is studied a new approach via proximal spaces for Pythagorean fuzzy sets. To do this, the Pythagorean fuzzy proximity axioms are defined on proximal relator spaces. [ABSTRACT FROM AUTHOR]

Abstract Objective The study of total fuzzy graphs in all cases is crucial for the development of both theories and applications of the graph theory. Without theory the application will not be developed. Hence this manuscript attempted to theorize the conception of partially total fuzzy graphs. Results The article introduced the partially total fuzzy graph by keeping all the conditions of fuzziness as it is. From these definitions, it is endeavored to get the partial total fuzzy graph of a given fuzzy graph which is supported by illustrations. Also, some propositions and theorems related to this concept were developed and proved.

The traditional adaptive weighted fusion algorithm ignores the spatial correlation between the network sensing data and has high fusion bias, significantly reducing network data transmission quality. The network perception data fusion algorithm based on fuzzy time series is proposed, and the network perception data prediction algorithm based on fuzzy time series is used. The network data value is calculated using first and two-order fuzzy relations during the training stage, and the trend value of the network data is obtained. The trend value in the prediction stage dynamically obtains the network sense. Based on the relationship, the fuzzy relation of knowledge data is used to predict the network perception data for the next time series. The network sensing data fusion algorithm based on the confidence matrix is used to fuse the data by predicting the spatial correlation between the data and filtering the noise of the abnormal data to the fusion results. The proposed algorithm’s high fusion accuracy and improved quality of network data transmission are demonstrated by experimental results.

Fuzzy information granulation theory is based on the way humans granulate and reason about information, and it is essential to the remarkable ability of people to act logically in ambiguous and uncertain situations. In the study of fuzzy information granulation, instead of discussing single fuzzy granules, it is common to consider a fuzzy granular structure arising from a set of fuzzy information granules. Different approaches and perspectives may generate different fuzzy granular structures in the same universe by dividing the object into a number of meaningful fuzzy information granules. However, a specific task usually requires only a selection of representative fuzzy granular structures. Therefore, the main aim of this paper is to group fuzzy granular structures efficiently and accurately. To this end, we first introduce the distances between two fuzzy granular structures and illustrate the relevant properties. Subsequently, k-means and fuzzy c-means clustering algorithms are designed for clustering fuzzy granular structures, and their convergence is demonstrated. In this way, similar fuzzy granular structures can be grouped into the same class. In addition, two evaluation indicators, dispersion and separation, are constructed to evaluate the effect of clustering fuzzy granular structures. Experiments on 12 publicly available datasets demonstrate the feasibility and effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we characterize some properties of fuzzy congruence relations and obtain a fuzzy congruence relation generated by a fuzzy relation in residuated lattices. For this purpose, two various types of fuzzy relations (regular and irregular) are introduced. In order to obtain a fuzzy congruence relation generated by an irregular fuzzy relation it must convert to a regular fuzzy relation. [ABSTRACT FROM AUTHOR]

Spherical fuzzy sets are an advanced tool of three-dimensional membership functions which consist of membership, non-membership and hesitancy degrees. In this paper, it is introduced a new approach via proximal spaces for spherical fuzzy sets. To do this, the spherical fuzzy proximity axioms are defined on proximal relator spaces. Also, spherical fuzzy spatial Lodato proximity relation is studied. By using spherical fuzzy proximity relation, it is defined that descriptive proximity relation. An example is given how people are proximal(near) to each other via their description features. [ABSTRACT FROM AUTHOR]

The main goal behind mathematical cryptography is to keep messages and multimedia messages secret at a time when modern means of communication have spread and become very diverse. Fuzzy matrices as strong tools which was defined to deal with incomplete and uncertain data and many relationships in real life problems especially those which are related to images and graphs, may considered as important subjects for secret information and communication. The aim of this research paper is to present a new model and method for encrypting 2×2 fuzzy matrices using the basic concepts in neutrosophic number theory and El Gamal algorithm in cryptography, where we generalize El Gamal algorithm to become applicable to the ring of neutrosophic integer numbers that represents the studied fuzzy matrices. On the other hand, we study the applications of the novel algorithm to the encryption and decryption of some fuzzy relations represented in terms of fuzzy functions. 20 Mohammad Abobala, Ali Allouf, On A Novel Security Scheme for The Encryption and Decryption Of 2×2 Fuzzy Matrices with Rational Entries Based on The Algebra of Neutrosophic Integers and El-Gamal Crypto-System In addition, we illustrate many examples to clarify the validity of the new algorithm.

Abstract Artificial neural network is revolutionizing business and everyday life, bringing us to the next level in artificial intelligence. It has a unique ability to extract meaning from complex data to find patterns and detect trends that are too convoluted for the human brain. This paper analyzes the artificial neural network impact on different computational organizations by using the innovative structure of bipolar complex spherical fuzzy relation which is any subset of the Cartesian product of two bipolar complex spherical fuzzy sets. This notion has a comprehensive structure that consists of membership grade, abstinence grade, and non-membership grade. Furthermore, various kinds of bipolar complex spherical fuzzy relation with suitable examples are given and some authentic results also have been proved. These newly defined structures are used to investigate the impact of artificial neural network work on a variety of organizations. The innovative framework is also compared with the existing structure in the field of fuzzy set theory to prove its superiority.

In this paper, we study the concept of fuzzy metrics from the perspective of fuzzy relations. Specifically, we analyze the commonly used definitions of fuzzy metrics. We begin by noting that crisp metrics can be uniquely characterized by linear order relations. Further, we explore the criteria that crisp relations must satisfy in order to determine a crisp metric. Subsequently, we extend these conditions to obtain a fuzzy metric and investigate the additional axioms involved. Additionally, we introduce the definition of an extensional fuzzy metric or E-d-metric, which is a fuzzification of the expression d (x , y) = t . Thus, we examine fuzzy metrics from both the linear order and from the equivalence relation perspectives, where one argument is a value d (x , y) and the other is a number within the range [ 0 , + ∞) . [ABSTRACT FROM AUTHOR]

Objective: The study of total fuzzy graphs in all cases is crucial for the development of both theories and applications of the graph theory. Without theory the application will not be developed. Hence this manuscript attempted to theorize the conception of partially total fuzzy graphs. Results: The article introduced the partially total fuzzy graph by keeping all the conditions of fuzziness as it is. From these definitions, it is endeavored to get the partial total fuzzy graph of a given fuzzy graph which is supported by illustrations. Also, some propositions and theorems related to this concept were developed and proved. [ABSTRACT FROM AUTHOR]

Artificial neural network is revolutionizing business and everyday life, bringing us to the next level in artificial intelligence. It has a unique ability to extract meaning from complex data to find patterns and detect trends that are too convoluted for the human brain. This paper analyzes the artificial neural network impact on different computational organizations by using the innovative structure of bipolar complex spherical fuzzy relation which is any subset of the Cartesian product of two bipolar complex spherical fuzzy sets. This notion has a comprehensive structure that consists of membership grade, abstinence grade, and non-membership grade. Furthermore, various kinds of bipolar complex spherical fuzzy relation with suitable examples are given and some authentic results also have been proved. These newly defined structures are used to investigate the impact of artificial neural network work on a variety of organizations. The innovative framework is also compared with the existing structure in the field of fuzzy set theory to prove its superiority. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a concept of type-2 L-preorders for L being a complete residuated lattice. It can be considered as a common framework of L-preorders and hemimetrics, and also contains various kinds of fuzzy metrics, including Morsi fuzzy metrics, KM-fuzzy metrics and modular metrics, as natural examples. It is shown that the category of L-preordered sets can be reflectively and coreflectively embedded in that of type-2 L-preordered sets. A type-2 L-preorder can be supplied as different models for further study. [ABSTRACT FROM AUTHOR]

A novel image similarity index based on the greatest and smallest fuzzy set solutions of the max–min and min–max compositions of fuzzy relations, respectively, is proposed. The greatest and smallest fuzzy sets are found symmetrically as the min–max and max–min solutions, respectively, to a fuzzy relation equation. The original image is partitioned into squared blocks and the pixels in each block are normalized to [0, 1] in order to have a fuzzy relation. The greatest and smallest fuzzy sets, found for each block, are used to measure the similarity between the original image and the image reconstructed by joining the squared blocks. Comparison tests with other well-known image metrics are then carried out where source images are noised by applying Gaussian filters. The results show that the proposed image similarity measure is more effective and robust to noise than the PSNR and SSIM-based measures. [ABSTRACT FROM AUTHOR]

The challenging issues of computer networks and databases are not only the intrusion detection but also the reduction of false positives and increase of detection rate. In any intrusion detection system, anomaly detection mainly focuses on modeling the normal behavior of the users and detecting the deviations from normal behavior, which are assumed to be potential intrusions or threats. Several techniques have already been successfully tried for this purpose. However, the normal and suspicious behaviors are hard to predict as there is no precise boundary differentiating one from another. Here, rough set theory and fuzzy set theory come into the picture. In this article, a hybrid approach consisting of rough set theory and intuitionistic fuzzy set theory is proposed for the detection of anomaly. The proposed approach is a classification approach which takes the advantages of both rough set and intuitionistic fuzzy set to deal with inherent uncertainty, vagueness, and indiscernibility in the dataset. The algorithm classifies the data instances in such a way that they can be expressed using natural language. A data instance can possibly or certainly belong to a class with degrees of membership and non-membership. The empirical study with a real-world and a synthetic dataset demonstrates that the proposed algorithm has normal true positive rates of 91.989% and 96.99% and attack true positive rates of 91.289% and 96.29%, respectively. [ABSTRACT FROM AUTHOR]

Mathematical cryptography applies special properties of integers and number theory in encrypting and decrypting messages in our world which is increasingly in demand for information security, especially for social media and multimedia. The objective of this work is to generalize El-Gamal crypto algorithm to be applied by using symbolic 2-plithogenic integers instead of classical integers. Also, we apply the novel algorithm to the encryption and the decryption of square 3 × 3 fuzzy matrices with rational entries, and fuzzy relations which can be represented as fuzzy matrices with rational entries. In addition, many examples with numerical data will be presented. [ABSTRACT FROM AUTHOR]

This paper presents the idea of Alexandrov L-quasi-G-filter space and examines its relationship with L-fuzzy relations. It is proved that every Alexandrov L-quasi-G-filter induces an L-fuzzy relation and vice-versa. By identifying certain functors, the study has brought out the connections between the categories of Alexandrov L-quasi-G-filter spaces and Alexandrov L-fuzzy pre-uniform spaces. Further, the study has explored and thereby establishes the scope of Alexandrov L-quasi-G-filter spaces in mathematical modeling and decision-making processes. [ABSTRACT FROM AUTHOR]

The main goal behind mathematical cryptography is to keep messages and multimedia messages secret at a time when modern means of communication have spread and become very diverse. Fuzzy matrices as strong tools which was defined to deal with incomplete and uncertain data and many relationships in real life problems especially those which are related to images and graphs, may considered as important subjects for secret information and communication. The aim of this research paper is to present a new model and method for encrypting 2×2 fuzzy matrices using the basic concepts in neutrosophic number theory and El Gamal algorithm in cryptography, where we generalize El Gamal algorithm to become applicable to the ring of neutrosophic integer numbers that represents the studied fuzzy matrices. On the other hand, we study the applications of the novel algorithm to the encryption and decryption of some fuzzy relations represented in terms of fuzzy functions. In addition, we illustrate many examples to clarify the validity of the new algorithm. [ABSTRACT FROM AUTHOR]

In this article the relationship between fuzzy consequence and fuzzy interior operators with fuzzy relations is analyzed. While the situation has been studied in depth if the fuzzy relation is a ⁎-preorder it is not the case for arbitrary fuzzy relations. If R is an arbitrary fuzzy relation, a fuzzy consequence γ R and a fuzzy interior δ R operators can be derived from R. Using the Representation Theorem for ⁎-preorders, from the columns and rows of R two ⁎-preorders, R c o l and R r o w , can be generated that coincide with the ⁎-preorders generated by γ R and δ R respectively. A duality result between them and the ⁎-coherence of γ R are also shown. [ABSTRACT FROM AUTHOR]

The concept of closure operator is key in several branches of mathematics. In this paper, closure operators are extended to relational structures, more specifically to fuzzy relations in the framework of complete fuzzy lattices. The core of the work is the search for a suitable definition of (strong) fuzzy closure relation, that is, a fuzzy relation whose relation with fuzzy closure systems is one-to-one. The study of the properties of fuzzy closure systems and fuzzy relations helps narrow down this exploration until an appropriate definition is settled. [ABSTRACT FROM AUTHOR]

Introduction Fuzzy relational equations have played an important role in fuzzy set theory and fuzzy logic systems, and many researchers have discussed fuzzy relation equations based on different compositions of fuzzy relations, including max–product fuzzy relational equations. In this article, we study a class of posynomial geometric programming problem (PGPF), with the purpose of minimizing a posynomial subject to fuzzy relational equations with max–product composition. With the help of auxiliary variables, it is converted convert the PGPF into an equivalent programming problem whose objective function is a non-decreasing function with an auxiliary variable. Some preliminary definitions are introduced. Since the feasible solutions are not convex the basic programming techniques are not applicable. It is shown that an optimal solution consists of a maximum feasible solution and finite number of minimal feasible solutions by an equivalent programming problem. In fact, there are a lot minimal solutions and to obtain all them is tedious steps, boring and time consuming. Furthermore, we propose some rules for full simplifying the problem. Then by using a branch and bound approach and fuzzy relational equations (FRE) path, it is presented an algorithm to achieve an optimal solution to the PGPF. Finally, a numerical experiment is given to illustrate the steps of the algorithm. Material and methods In this paper, we develop an algorithm to deal with the PGPF based on a special structure of a solution set to the fuzzy relation equations. The material is arranged as summarizing some concepts and important properties related to max–product fuzzy relational equations and more,to present some main conclusions for developing the algorithm in order to solve the PGPF. Results and discussion We introduce three rules to simplify the PGPF. We give a step-by-step algorithm to the PGPF and then we use some examples to illustrate how the algorithm works. Also, four important concluding remarks are given. Conclusion The following conclusions were drawn from this research. It is proved that the feasible solutions are not convex so the basic programming techniques are not applicable. It is shown that an optimal solution consists of a maximum feasible solution and the finite number of minimal feasible solutions by an equivalent programming problem. By using a branch and bound approach and fuzzy relational equations (FRE) path, it is presented an algorithm to achieve an optimal solution to the PGPF. [ABSTRACT FROM AUTHOR]

Rough sets, as a powerful tool to deal with uncertainties and inaccuracies in data analysis, have been continuously concerned and studied by many scholars since it was put forward, especially the research on various rough set models. On the other hand, overlap and grouping functions, as two newly aggregation operators and mathematical model to handle the problems involving in information fusion, have been successfully applied in many real-life problems. In this paper, based on overlap and grouping functions, we propose a new fuzzy rough set model named (GO, O)-fuzzy rough sets and consider its characterizations along with topological properties. Properly speaking, firstly, we utilize QL-operators (and also QL-implications) constructed from overlap and grouping functions and fuzzy negations to define the lower approximation operator in (GO, O)-fuzzy rough set model named GO-lower fuzzy rough approximation operator and the upper approximation operator in (GO, O)-fuzzy rough set model is considered as the O-upper fuzzy rough approximation operator in (IO, O)-fuzzy rough set model proposed by Qiao recently. Secondly, we discuss lots of basic properties of (GO, O)-fuzzy rough sets, especially for the properties of GO-lower fuzzy rough approximation operator. Thirdly, we focus on the relationship between (GO, O)-fuzzy rough sets and concrete fuzzy relations. Finally, we give the topological properties of the upper and lower approximation operators in (GO, O)-fuzzy rough set model. [ABSTRACT FROM AUTHOR]

Considering a family { T c } c ∈ [ 0 , 1 ] generated from a t-norm T , the degree of T -transitivity of a fuzzy relation R is revisited and proved to coincide with the greatest c for which R is T c -transitive. This fact gives rise to the study of new families of t-norms to generate different degrees of transitivity with respect to them. The mappings transforming fuzzy relations into transitive fuzzy relations smaller than or equal to the given ones are studied. [ABSTRACT FROM AUTHOR]

This study builds a new forecasting model for time series based on some important improvements. First, we choose the universal set to be the percentage variation of the series. This universal set is divided to clusters by the automatic algorithm. The suitable number of cluster depends on the similar level of elements in the universal set. Second, a principle to find the relationship of each element in the series to the found clusters is established. Finally, we propose the forecasting rule from the established fuzzy relationships. The proposed model is illustrated in detail by the numerical examples, and can be quickly applied to real data by the established Matlab procedure. Comparing many series with the differences about the number of elements, fields, and characteristics, the proposed model has shown the outstanding advantages. Using the proposed model, we forecast the salty peak for a coastal province in Vietnam to illustrate for application of this study. [ABSTRACT FROM AUTHOR]

Attribute reduction in an information system (IS) is an important research topic in rough set theory (RST). This paper investigates attribute reduction for incomplete heterogeneous data based on information entropy. Information entropy in an incomplete IS with heterogeneous data (IISH) is first defined. Then, some derived notions of information entropy, such as joint information entropy, conditional information entropy, mutual information entropy, gain and gain ratio in an incomplete decision IS with heterogeneous data (IDISH), are presented. Next, information entropy is applied to perform attribute reduction in an IDISH. Two attribute reduction algorithms, based on gain and gain ratio, respectively, are proposed. Finally, in order to illustrate the feasibility and efficiency of the proposed algorithms, experimental analysis is carried out and comparisons are done. It is worth mentioning that the incomplete rate is used to deal with incomplete heterogeneous data. [ABSTRACT FROM AUTHOR]

The data readability, complexity reduction of learning algorithms, and enhancing predictability are the most important reasons for using feature selection methods, especially when there exist lots of features. In recent years, unsupervised feature selection techniques are well explored. Among the various methods of feature selection, algorithms based on information theory are effective in selecting useful attributes and eliminating useless ones. Mutual information and symmetric uncertainty based methods can effectively estimate feature relevancy, but the use of bivariate measures ignores possible dependencies among more than two features. To address this limitation, this research introduces a novel unsupervised feature selection method, called Fuzzy Multivariate Symmetric Uncertainty-Feature Selection (FMSU-FS). The proposed method also overcomes the need for discretization for continuous features that causes information loss. To evaluate the effectiveness of the proposed method, FMSU is compared with conventional methods based on information theory. The experimental results on benchmark datasets show improvement of our approach in measuring clustering performance. [ABSTRACT FROM AUTHOR]

The properties of fuzzy relations have been extensively studied, and the preservation of their properties plays a fundamental role in the various applications. However, either sufficient or necessity conditions for the preservation requires the aggregated functions of fuzzy relations to dominate or to be dominated by the corresponding operations, which constructs a significant limitation on applicable functions. This work concentrates on the preservation of transitivities and Ferrers property for the aggregation of comonotone or countermonotone fuzzy relations. Firstly, definitions of comonotonicity and countermonotonicity for binary functions are initially proposed. On the foundation of that, the relations of commuting and bisymmetry between min/max and commonly used increasing/decreasing functions are found. Afterwards, with the condition that underlying fuzzy relations are pair-wisely comonotone or countermonotone, theorems on the aggregation functions which can preserve the transitivities and the Ferrers property are proposed. Moreover, an interesting conclusion that the equivalent condition for the min-Ferrers property of fuzzy relations is clarified. [ABSTRACT FROM AUTHOR]

Fuzzy relation is one of the main research contents of fuzzy set theory. This paper obtains some results on fuzzy relations by studying relationships between fuzzy relations and their uncertainty measurement. The concepts of equality, dependence, partial dependence and independence between fuzzy relations are first introduced. Then, uncertainty measurement for a fuzzy relation is investigated by using dependence between fuzzy relations. Moreover, the basic properties of uncertainty measurement are obtained. Next, effectiveness analysis is carried out. Finally, an application of the proposed measures in attribute reduction for heterogeneous data is given. These results will be helpful for understanding the essence of a fuzzy relation. [ABSTRACT FROM AUTHOR]

An approximation space (A-space) is the base of rough set theory and a fuzzy approximation space (FA-space) can be seen as an A-space under the fuzzy environment. A fuzzy probability approximation space (FPA-space) is obtained by putting probability distribution into an FA-space. In this way, it combines three types of uncertainty (i.e., fuzziness, probability and roughness). This article is devoted to measuring the uncertainty for an FPA-space. A fuzzy relation matrix is first proposed by introducing the probability into a given fuzzy relation matrix, and on this basis, it is expanded to an FA-space. Then, granularity measurement for an FPA-space is investigated. Next, information entropy measurement and rough entropy measurement for an FPA-space are proposed. Moreover, information amount in an FPA-space is considered. Finally, a numerical example is given to verify the feasibility of the proposed measures, and the effectiveness analysis is carried out from the point of view of statistics. Since three types of important theories (i.e., fuzzy set theory, probability theory and rough set theory) are clustered in an FPA-space, the obtained results may be useful for dealing with practice problems with a sort of uncertainty. [ABSTRACT FROM AUTHOR]

The challenging issues of computer networks and databases are not only the intrusion detection but also the reduction of false positives and increase of detection rate. In any intrusion detection system, anomaly detection mainly focuses on modeling the normal behavior of the users and detecting the deviations from normal behavior, which are assumed to be potential intrusions or threats. Several techniques have already been successfully tried for this purpose. However, the normal and suspicious behaviors are hard to predict as there is no precise boundary differentiating one from another. Here, rough set theory and fuzzy set theory come into the picture. In this article, a hybrid approach consisting of rough set theory and intuitionistic fuzzy set theory is proposed for the detection of anomaly. The proposed approach is a classification approach which takes the advantages of both rough set and intuitionistic fuzzy set to deal with inherent uncertainty, vagueness, and indiscernibility in the dataset. The algorithm classifies the data instances in such a way that they can be expressed using natural language. A data instance can possibly or certainly belong to a class with degrees of membership and non-membership. The empirical study with a real-world and a synthetic dataset demonstrates that the proposed algorithm has normal true positive rates of 91.989% and 96.99% and attack true positive rates of 91.289% and 96.29%, respectively.