Fuzzy Implication Operators Research Articles

Retracted: Interval Information Content of Fuzzy Relation and the Application in the Fuzzy Implication Operators

Retracted: Interval Information Content of Fuzzy Relation and the Application in the Fuzzy Implication Operators

Novel variable precision fuzzy rough sets and three-way decision model with three strategies

Variable precision (fuzzy) rough sets are interesting generalizations of Pawlak rough sets and can handle uncertain and imprecise information well due to their error tolerance capability. The comparable property (CP for short), i.e., the lower approximation is included in the upper approximation, is fundamental in Pawlak rough sets since many theories and applications rely on this property. However, the CP is lost in many existing variable precision (fuzzy) rough sets. Therefore, a novel variable precision fuzzy rough set model with CP is proposed and an associated three-way decision model is developed. Firstly, a pair of variable precision fuzzy upper and lower approximation operators are defined and studied through fuzzy implication and co-implication operators. We show that this pair of approximation operators satisfy various properties (including CP) that the existing variable precision fuzzy approximation operators do not possess. Secondly, new methods for constructing loss functions and conditional probabilities are given by using CP, and then a new three-way decision model with three strategies (optimistic, pessimistic, and compromise strategies) is established. Finally, an explanatory example is provided to examine validity, stability and sensitivity of the proposed three-way decision model.

Three-Way Fuzzy Sets and Their Applications (III)

Three-way fuzzy inference is the theoretical basis of three-way fuzzy control. The proposed TCRI method is based on a Mamdani three-way fuzzy implication operator and uses one inference and simple composition operation. In order to effectively improve the TCRI method, this paper proposes a full implication triple I algorithm for three-way fuzzy inference and gives the triple I solution to the TFMP problem. The emphasis of our research is R0 and Go¨del triple I solution, which is related to three-way residual implication, as well as Zadeh’s and Mamdani’s triple I solution, which is based on three-way fuzzy implication operator. Then the three-way fuzzy controller is constructed by the proposed Zadeh’s and R0 triple I algorithm. Finally, the proposed triple I algorithm is applied to the three-way fuzzy control system, and its advantage is illustrated by the three-dimensional surface diagram of the control variable.

An embedded deep fuzzy association model for learning and explanation

This paper explores the complementary benefits of embedding a deep learning model as a fully data-driven fuzzy implication operator of a five-layer neuro-fuzzy system for learning and explanations for the predictions of both steady-state and dynamically changing data. In traditional Mandani-type neuro-fuzzy systems, the entailment performed by the implication is realized using the fuzzy implication operator based on the fuzzy rules formed in the rule base during encoding and recall. Given the presence of a group of test data that are significantly different from the training data, the realization of entailments through the use of the implication operator based on the fuzzy rules formed in traditional neuro-fuzzy systems may not be adequate. This paper attempts to adopt a more direct approach by embedding a deep learning model in the neuro-fuzzy system to serve as a fuzzy implication operator, thereby allowing the data-driven learning of fuzzy implication using the deep structure to provide a close correspondence to the real-world entailment of data. In addition, embedding the neuro-fuzzy architecture within the deep learning model allows the comprehension of the learning and explanation of the reasoning of the deep network. The induced fuzzy association rules impart transparency to the deep learning based implication using a common set of semantic meanings, which are amenable to human interpretability. The effectiveness of the proposed model is evaluated on a continuously stirred tank reactor dataset and three financial stock prediction datasets. Experimental results showed that the proposed model outperformed other state-of-the-art techniques based on the four datasets, which contain high levels of uncertainties.

Reverse triple I method based on the Pythagorean fuzzy inference model and its application

This paper is devoted to discussing the reverse triple I method based on the Pythagorean fuzzy set (PFS). We first propose the concepts of Pythagorean t-norm, Pythagorean t-conorm, residual Pythagorean fuzzy implication operator (RPFIO) and Pythagorean fuzzy biresiduum. The reverse triple I methods for Pythagorean fuzzy modus ponens (PFMP) and Pythagorean fuzzy modus tollens (PFMT) are also established. In addition, some interesting properties of the reverse triple I method of PFMP and PFMT inference models are analysed, including the robustness, continuity and reversibility. Finally, a practical problem is provided to illustrate the effectiveness of the reverse triple I method for PFMP in decision-making problems. The advantages of the new method over existing methods are also expounded. Overall, compared with the existing methods, the proposed methods are based on logical reasoning rather than using aggregation operators, so the novel methods are more logical, can better deal with the uncertain problems in complex decision-making environments and can completely reflect the decision-making opinions of decision-makers.

Pythagorean Fuzzy Full Implication Triple I Method and Its Application in Medical Diagnosis

This paper is devoted to the research of full implication triple I method under Pythagorean fuzzy environment. We first propose the concepts of Pythagorean t-norm, Pythagorean t-conorm, residual Pythagorean fuzzy implication operator (RPFIO) and Pythagorean fuzzy biresiduum. The full implication triple I method for Pythagorean fuzzy modus ponens (PFMP) and Pythagorean fuzzy modus tollens (PFMT) are also established. In addition, the properties of full implication triple I method of PFMP and PFMT models including the robustness, continuity and reversibility are analyzed. Finally, a practical problem is discussed to demonstrate the effectiveness of the Pythagorean fuzzy full implication multiple I method in medical diagnosis. The advantages of the new method are also explained. Overall, compared with the existing methods, the proposed methods are based on logical reasoning rather than using aggregation operators, so they can more accurately and completely express decision information.

Pythagorean fuzzy full implication multiple I method and corresponding applications

The overwhelming majority of existing decision-making methods combined with the Pythagorean fuzzy set (PFS) are based on aggregation operators, and their logical foundation is imperfect. Therefore, we attempt to establish two decision-making methods based on the Pythagorean fuzzy multiple I method. This paper is devoted to the discussion of the full implication multiple I method based on the PFS. We first propose the concepts of Pythagorean t-norm, Pythagorean t-conorm, residual Pythagorean fuzzy implication operator (RPFIO), Pythagorean fuzzy biresiduum, and the degree of similarity between PFSs based on the Pythagorean fuzzy biresiduum. In addition, the full implication multiple I method for Pythagorean fuzzy modus ponens (PFMP) is established, and the reversibility and continuity properties of the full implication multiple I method of PFMP are analyzed. Finally, a practical problem is discussed to demonstrate the effectiveness of the Pythagorean fuzzy full implication multiple I method in a decision-making problem. The advantages of the new method over existing methods are also explained. Overall, the proposed methods are based on logical reasoning, so they can more accurately and completely express decision information.

Interval Information Content of Fuzzy Relation and the Application in the Fuzzy Implication Operators

In rule optimization, some rule characteristics were extracted to describe the uncertainty correlations of fuzzy relations, but the concrete numbers cannot express correlations with uncertainty, such as “at least 0.1 and up to 0.5.” To solve this problem, a novel definition concerning interval information content of fuzzy relation has been proposed in this manuscript to realize the fuzziness measurement of the fuzzy relation. Also, its definition and expressions have also been constructed. Meanwhile based on the interval information content, the issues of fuzzy implication ranking and clustering were analyzed. Finally, utilizing the combination of possibility’s interval comparison equations and interval value’s similarity measure, the classifications of implication operators were proved to be realizable. The achievements in the presented work will provide a reasonable index to measure the fuzzy implication operators and lay a solid foundation for further research.

Willmott Fuzzy Implication in Fuzzy Databases

The object of the research are fuzzy functional dependencies on given relation scheme, and the question of their obtaining using the classical and innovated techniques. The attributes of the universal set are associated to the elements of the unit interval, and are turned into fuzzy formulas in this way. We prove that the dependency (which is treated as a fuzzy formula with respect to appropriately chosen valuation) is valid whenever it agrees with the attached two-elements fuzzy relation instance. The opposite direction of the claim is proven to be incorrect in this setting. Generalizing things to sets of attributes, we prove that particular fuzzy functional dependency follows form a set of fuzzy dependencies (in both, the world of two-element and the world of arbitrary fuzzy relation instances) if and only if the dependency is valid with respect to valuation anytime the set of fuzzy formulas agrees with the valuation. The results derived in paper show that the classical techniques in the procedure for generating new fuzzy dependencies may be replaced by the resolution ones, and hence automated. The research is conducted with respect to Willmott fuzzy implication operator

Fuzzy Functional and Multivalued Dependencies for Frank’s Class of Additive Generators

In this paper we consider all possible dependencies that can be built upon similarity-based fuzzy relations, that is, fuzzy functional and fuzzy multivalued dependencies. Motivated by the fact that the classical obtaining of new dependencies via inference rules may be tedious and uncertain, we replace it by the automated one, where the key role is played by the resolution principle techniques and the fuzzy formulas in place of fuzzy dependencies. We prove that some fuzzy multivalued dependency is actively correct with respect to given fuzzy relation instance if and only if the corresponding fuzzy formula is in line with the attached interpretation. Additionally, we require the tuples of the instance to be conformant (up to some extent) on the leading set of attributes. The equivalence as well as the conclusion are generalized to sets of attributes. The research is conducted by representing the attributes and fuzzy dependencies in the form of fuzzy formulas, and the application of fuzzy implication operators derived from carefully selected Frank’s classes of additive generators

Hesitant Fuzzy Co-tautological Sets using →implication

In this paper, I study some properties of hesitant fuzzy Co-tautological sets using hesitant fuzzy implication operator.

Some properties of fuzzy implications based on copulas

In 2011 Grzegorzewski introduced two new families of fuzzy implication functions called probabilistic implications and probabilistic S-implications. They are based on copulas and make a bridge between probability theory and fuzzy logic. Another family of fuzzy conditional implication operators was proposed by Dolati et al. in 2013. In this paper we consider some properties of these three classes of fuzzy implications like the law of contrapositions and the law of importation. Moreover, we examine intersections of these families of implications with R-implications, (S, N)-implications, QL-operations and Yager’s f- and g-generated implications.

Similarity measures, penalty functions, and fuzzy entropy from new fuzzy subsethood measures

In this study, we discuss a new class of fuzzy subsethood measures between fuzzy sets. We propose a new definition of fuzzy subsethood measure as an intersection of other axiomatizations and provide two construction methods to obtain them. The advantage of this new approach is that we can construct fuzzy subsethood measures by aggregating fuzzy implication operators which may satisfy some properties widely studied in literature. We also obtain some of the classical measures such as the one defined by Goguen. The relationships with fuzzy distances, penalty functions, and similarity measures are also investigated. Finally, we provide an illustrative example which makes use of a fuzzy entropy defined by means of our fuzzy subsethood measures for choosing the best fuzzy technique for a specific problem.

Fuzzy [formula omitted]-covering based [formula omitted]-fuzzy rough set models and applications to multi-attribute decision-making

Multi-attribute decision-making (MADM) can be regarded as a process of selecting the optimal one from all alternatives. Traditional MADM problems with fuzzy information are mainly focused on a fundamental tool which is a fuzzy binary relation. However some complicated problems cannot be effectively solved by a fuzzy relation. For this reason, in order to solve these issues, we set forth two decision-making methods that are stated in terms of novel and flexible fuzzy rough set models. For the purpose of defining these models we employ a fuzzy implication operator I and a triangular norm T. With these adaptable tools we design four kinds of fuzzy β-coverings based (I,T)-fuzzy rough set models. The elements that make these models different are the combination of fuzzy β-neighborhoods that intervene in the definitions of the lower (upper) approximations. Then, we discuss the relationships among these given four types of models. Finally, we propose two novel methodologies to solve MADM problems with evaluation of fuzzy information, which rely on these models. Through the analysis of the ranking results of these two methods, we observe that the optimal selected alternative is the same, which means that these two decision-making methods are reasonable. In addition, by comparing the ranking results of these two methods and the existing traditional methods (WA operator and TOPSIS), we observe that our proposed methods can solve the ranking problems that traditional methods cannot solve, which means that our proposed methods are superior to traditional methods.

Hesitant fuzzy Lukasiewicz implication operation and its application to alternatives’ sorting and clustering analysis

Hesitant fuzzy set (HFS) takes several possible values as the membership degree of an element to a set to express the decision makers’ hesitance when making decisions. Since its appearance, the HFS has been widely used in many fields, such as decision making, clustering analysis. Lukasiewicz implication operator, an indispensable part of implication operators, can grasp more nuances compared with the others. In this paper, we shall combine the Lukasiewicz implication operator with HFSs to realize a direct clustering analysis algorithm and a novel alternative sorting method in decision making under hesitant fuzzy environment. To do that, we first apply the Lukasiewicz implication operator to deal with HFEs by getting a hesitant fuzzy Lukasiewicz implication operator, and then construct a hesitant fuzzy triangle product and a hesitant fuzzy square product based on the new implication operator. After that, the hesitant fuzzy square product is applied to define the similarity degree between HFSs, and based on which, we develop a direct clustering algorithm for hesitant fuzzy information. Meanwhile, the hesitant fuzzy triangle product is used to induce a new alternative sorting method. Finally, two numerical examples are given to illustrate the effectiveness and practicability of our method and algorithm, one of which involves the evaluation analysis of the Arctic development risk.

Constructive methods for intuitionistic fuzzy implication operators

In this paper, two novel constructive methods for fuzzy implication operators are proposed. And they can be used to resolve the questions about how to construct intuitionistic fuzzy implication operators to inference sentence “If x is A, then y is B,” where A and B are intuitionistic fuzzy sets over X and Y, respectively. In the first method, cut sets and representation theorem of intuitionistic fuzzy sets are utilized through triple valued fuzzy implications as well as fuzzy relations to obtain seven intuitionistic fuzzy implication operators. In the second method, the Lebesgue measure of the fuzzy implications with value one and zero is used, and one intuitionistic fuzzy implication operator can be obtained. Finally, properties of those operators are discussed.

Using One Axiom to Characterize Fuzzy Rough Approximation Operators Determined by a Fuzzy Implication Operator

Using One Axiom to Characterize Fuzzy Rough Approximation Operators Determined by a Fuzzy Implication Operator

A Fuzzy Trustworthiness System with Probability Presentation Based on Center-of-gravity Method

Fuzzy methods are widely used in the study of trustworthiness. Based on this fact, the paper researches the fuzzy trustworthiness system and probability presentation theory based on bounded product implication and Larsen square implication. Firstly, we convert a group of single-input and single-output data into fuzzy inference rules and generate fuzzy relation by selecting the appropriate fuzzy implication operator, then calculate joint probability density function of two-dimensional random variables by using of this fuzzy relation. Two specific probability density functions can be obtained by selecting the fuzzy implication as bounded product implication or Larsen square implication. Secondly, we study the marginal distribution and numerical characteristics of these two kinds probability distributions and point out that two of these probability distributions have the same mathematical expectation and nearly the same variance and covariance. Finally, we study of the center-of-gravity fuzzy trustworthiness system based on these two probability distributions. We gave the sufficient conditions of universal approximations for those fuzzy trustworthiness systems.

INTUITIONISTIC FUZZY CO-TAUTOLOGICAL MATRIX

In this paper, we study some properties of intuitionistic fuzzy co-tautological matrix using intuitionistic fuzzy implication operator.

Direct clustering analysis based on intuitionistic fuzzy implication

In this paper, we investigate the technique for clustering analysis under intuitionistic fuzzy environment. We first develop an intuitionistic fuzzy implication operator and extend the Lukasiewicz implication operator to intuitionistic fuzzy environment, and then we define an intuitionistic fuzzy triangle product and an intuitionistic fuzzy square product. Furthermore, we apply the intuitionistic fuzzy triangle product to compare the alternatives in multiple attribute decision making with intuitionistic fuzzy information, and use the intuitionistic fuzzy square product to construct an intuitionistic fuzzy similarity matrix, based on which we develop a direct method for intuitionistic fuzzy clustering analysis. Some numerical examples are also given to illustrate and verify our results.