Fuzzy Implication Research Articles

An Algorithm for Producing Fuzzy Implications via Conical Sections

In the present study, a novel parametric family of fuzzy implications is introduced and its properties are examined. The parametric family of implications is produced only via a fuzzy negation. This in turn enables the effortless production of a wide range of implications from which to select the one that best fits a given problem, for example in fuzzy inference systems or fuzzy neural networks. The fuzzy negations that have been selected as a basis for the proposed methodology are strong, i.e., involutions, thus leading, in general, to the generated fuzzy implications possessing many desirable additional properties. We have examined which of these properties hold for the implications produced by our algorithm and under which conditions. Finally, it is demonstrated that the family of implications generated via the proposed methodology generalizes other well-established implications, including the Łukasiewicz implication.

An Approach to a Priori Assessment of Fuzzy Classification Models in Monitoring Tasks

The article addresses the problems of using automation tools to perform monitoring and management tasks applicable to assessing the quality of fuzzy classification models, where the classification procedure is implemented on the basis of knowledge (rules) in the absence of the training set. An approach is proposed to obtain a priori assessments of the classification quality based on the study of the used model sensitivity to changes in the values of internal parameters during the corresponding modeling. The interpretation of the modeling results in the form of risk assessment caused by the self-imperfection of the classification models is obtained. The article provides an example of a fuzzy classification model based on a comparison of the current state of a monitoring object described using fuzzy features with a set of predefined typical states, which form corresponding fuzzy equal (close) states (monitoring situations). The comparison is carried out using the fuzzy implication operation provided that the required reliability is met. The example of this model demonstrates how the type of implication operation, as well as the internal features of the model, affect the results of classification, and appropriate indicators are proposed, which are both an interpretation of generally accepted indicators for assessing the classification quality, and unique, inherent in the considered model. Computational experiments were carried out, which made it possible to obtain graphs of changes in classification quality assessment indicators for the considered model and its modification and visualize the influence of internal parameters of the model on the results of its application. A number of indicators are proposed that allow for an a priori assessment of the risks arising from the application of the model before its actual application.

Fuzzy implications-based transformation approaches from semi-three-way decision spaces to three-way decision spaces and their applications

Three-way decision spaces, as an important component of three-way decisions, greatly enrich their theoretical development and application prospects. Meanwhile, fuzzy implications, as a vital class of fuzzy logic connectives, have made great contributions to the solution of practical problems, especially complex decision-making problems. This paper considers the collaborative effect of the two, which inject new vitality into the theoretical development and application prospects of fuzzy implications and three-way decision spaces. As a vital component of three-way decision spaces, (semi-)decision evaluation functions have been widely studied based on fuzzy logic connectives and become a research hotspot. Specifically, this paper focuses on fuzzy implications-based transformation approaches from semi-three-way decision spaces to three-way decision spaces and their applications. Firstly, we present some novel fuzzy implications-based transformation approaches from semi-decision evaluation functions to decision evaluation functions, and construction approaches of semi-decision evaluation functions involving the existing semi-decision evaluation functions, fuzzy sets, interval-valued fuzzy sets and fuzzy relations. Secondly, we discuss the relationship between our approaches and the known construction approaches of three-way decision spaces. Notably, our approaches cover all existing approaches except the uninorms-based approaches. Finally, by the experiment results, we obtain our approaches are feasible, effective, superior to the known three-way decision spaces approaches and have good anti-noise ability. And, the parameter ρ of our approaches is also effective and stable.

Innovative Methods of Constructing Strict and Strong Fuzzy Negations, Fuzzy Implications and New Classes of Copulas

This paper presents new classes of strong fuzzy negations, fuzzy implications and Copulas. It begins by presenting two theorems with function classes involving the construction of strong fuzzy negations. These classes are based on a well-known equilibrium point theorem. After that, a construction of fuzzy implication is presented, which is not based on any negation. Finally, moving on to the area concerning copulas, we present proof about the third property of copulas. To conclude, we will present two original constructions of copulas. All the above constructions are motivated by a specific formula. For some specific conditions of the variables x, y and other conditions for the function f(x), the formula presented produces strict and strong fuzzy negations, fuzzy implications and copulas.

Research on intuitionistic fuzzy implications. Part 4

Continuing the research from Parts 1, 2 [2, 3] where intuitionistic fuzzy implications, determined as implications with “good” properties, were investigated, here we correct the list of the implications that satisfy the Modus Ponens from Part 3, [4], and further select among them those implications that satisfy the Modus Tollens, as well. We discuss some applications of these implications and show the relationship between every two of them.

The effectiveness of aggregation functions used in fuzzy local contrast constructions

In this paper, we explore the concept of local contrast of a fuzzy relation, which can be perceived as a measure for distinguishing the degrees of membership of elements within a defined region of an image. We introduce four distinct methods for constructing fuzzy local contrast: one uses a similarity measure, the second relies on the aggregation of similarity, the third is based on the aggregation of restricted equivalence, and the fourth utilizes the notion of equivalence. We further divide the constructions using similarity measures into two categories based on the two known definitions of similarity: distance-based similarity and aggregation function-based similarity. These construction methods also incorporate fuzzy implications and negations. Aggregation functions, which can be manipulated to enhance the effectiveness of the constructed fuzzy local contrast, play a significant role in most of our proposed constructions. For each construction method, several examples of fuzzy local contrasts are provided. The usefulness of the new fuzzy local contrasts is examined by applying them in image processing for salient region detection.

A Distance Measure Between Fuzzy Implications

In this paper, we study a distance measure between fuzzy implications. The proposed distance measure is normalized and, therefore, gives rise to the corresponding similarity measure. The existence of a distance measure of fuzzy implication and the quantification of the similarity of these is very important in applications.

Hierarchical fuzzy inference based on Bandler-Kohout subproduct

Fuzzy inference with the Bandler-Kohout subproduct (BKS) has been successfully applied in many fields such as fuzzy control, artificial intelligence, image processing, data mining, decision-making, prediction, classification and so on. However, one has to face with the rule explosion in these applications. To deal with this problem, hierarchical fuzzy systems with the compositional rule of inference (CRI) method have been constructed by a series of low-dimensional sub fuzzy systems. And it has been proved that hierarchical fuzzy inference method can efficiently restrain the explosion of fuzzy rules. Therefore, in order to increase the computational efficiency of the fuzzy inference based on the BKS when multi-input-single-output (MISO) fuzzy rules are involved, this paper mainly constructs two hierarchical fuzzy inference methods based on the BKS in which the if-then rules are respectively interpreted by fuzzy implications and ML-implications. Moreover, the validity of the two BKS hierarchical fuzzy inferences is studied with the GMP rules. Finally, two examples are employed to illustrate the computational efficiency of our proposed BKS hierarchical inference methods.

Fuzzy Inference Quintuple Implication Method Based on Single Valued Neutrosophic t-representable t-norm

In this paper, we study the single valued neutrosophic fuzzy inference quintuple implication method. Firstly, single valued neutrosophic fuzzy inference quintuple implication principles for fuzzy modus ponens and fuzzy modus tollens are given. Then, single-valued neutrosophic R-type quintuple implication solutions for fuzzy modus ponens (FMP) and fuzzy modus tollens (FMT) are given. Finally, the robustness of the quintuple implication method based on a left-continuous single-valued neutrosophic t-representable t-norm is investigated. As a special case of the main results, the sensitivity of quintuple implication solutions based on special single-valued neutrosophic residual implications is given.

Approximate hierarchical fuzzy reasoning based on the law of importation

Fuzzy reasoning has been widely used in applied sciences such as fuzzy control, artificial intelligence, image processing, data mining, machine learning, decision-making, prediction, classification and so on. However, one has to face with the rule explosion in the application of fuzzy reasoning. As an efficient tool to deal with the rule explosion of fuzzy system, the law of importation equation I(A(x,y),z)=I(x,I(y,z)) with aggregation functions (LIA) can be utilized to construct the fuzzy hierarchical system. In order to enhance the ability of this fuzzy hierarchical system in the actual environments, this paper aims mainly to construct a hierarchical fuzzy system involved in the fuzzy implications that approximately but not strictly satisfy (LIA) to approximate a fuzzy system with arbitrary accuracy. Concretely, we first define Ulam stability of (LIA). And the Ulam stability of (LIA) is then studied, that is, the fuzzy implications are sought to approximately fulfill (LIA) for some given aggregation functions. Further, a hierarchical fuzzy system based on the Ulam stability of (LIA) is constructed to approximate a fuzzy system with arbitrary accuracy. Finally, an example is presented to substantiate our theoretical discussion.

Approximation of the functional equation I(x, I(x, y)) = I(x, y)

The well-known iterative boolean-like law a →(a → b) = a → b can be generalized to the functional equation I(x, I(x, y)) = I(x, y), where I is a fuzzy implication. In this paper, we discuss an approximation of the equation, I(x, I(x, y)) ≈ I(x, y), i.e., the law is approximately valid. Furthermore, we study the property of approximation preserving with respect to compositions of fuzzy implications. Finally, we give a necessary condition and a sufficient condition for the approximate equation of (S, N)-implications.

Interval-valued pre-(quasi-)grouping functions and its application in constructing interval-valued directional monotonic fuzzy implications

How to expand the variable domain and monotonicity of aggregation functions to generate new aggregation functions is an important research content in aggregation functions. In this work, the concept of interval-valued pre-(quasi-)grouping functions is given by relaxing the interval monotonicity of interval-valued (quasi-)grouping functions to interval directional monotonicity. Then, some basic properties of interval-valued pre-(quasi-)grouping functions and the relationship between interval-valued pre-(quasi-)grouping functions and pre-(quasi-)grouping functions are presented. Accordingly, several construction methods of interval-valued pre-(quasi-)grouping functions are proposed. Finally, the concept of ( I G , IN ) -interval-valued directional monotonic fuzzy implications and QL -interval-valued directional monotonic operations are introduced on the basis of interval-valued pre-(quasi-)grouping functions I G , interval-valued overlap functions IO and interval-valued fuzzy negations IN. In addition, related studies were conducted on the basic properties of ( I G , IN ) -interval-valued directional monotonic fuzzy implications and QL -interval-valued directional monotonic operations.

Fuzzy inference with fuzzy implications satisfying generalized hypothetical syllogism

Fuzzy inference with fuzzy implications satisfying generalized hypothetical syllogism

A unified framework of fuzzy implications and coimplications

Fuzzy implications and coimplications play important roles in both theoretic and applied communities of fuzzy set theory. In this paper, we provide a unified framework for fuzzy implications and coimplications. Specifically, firstly, we introduce the concept of uni-implications, which is the unification of fuzzy implications and coimplications, and describe the structure of uni-implications. Secondly, we discuss the relationships among uni-implications, fuzzy boundary weak implications and fuzzy (co)implications. Thirdly, we present two constructions and equivalent characterizations of non-trivial uni-implications, respectively. Finally, we propose two binary operations ♠,♡ on some subset of the set of non-trivial uni-implications, denoted by FSNT, which make (FSNT,♠) a semigroup and (FSNT,♡) a non-commutative and non-idempotent monoid.

Generating methods of some classes of fuzzy implications obtained by unary functions and algebraic structures

The existing generating methods of fuzzy implications are closed in the set of all fuzzy implications, but not when applied to some families of these operators. In this paper, some binary operations are defined on some well-established families of fuzzy implications. Namely, the families of (S,N)-implications with continuous t-conorms, R-implications obtained from continuous t-norms, Yager's f- and g-generated implications, h- and generalized (h,e)-implications and k-implications are considered and it is proved that with these operations, they are lattices. Moreover, some other lattice substructures are defined in some of the families when some restrictions on the underlying generators of the implications are imposed. Furthermore, it is emphasized that these generating methods preserve important properties like the exchange principle and the law of importation.

Two Extensions of the Sugeno Class and a Novel Constructed Method of Strong Fuzzy Negation for the Generation of Non-Symmetric Fuzzy Implications

In this paper, we present two new classes of fuzzy negations. They are an extension of a well-known class of fuzzy negations, the Sugeno Class. We use it as a base for our work for the first two construction methods. The first method generates rational fuzzy negations, where we use a second-degree polynomial with two parameters. We investigate which of these two conditions must be satisfied to be a fuzzy negation. In the second method, we use an increasing function instead of the parameter δ of the Sugeno class. In this method, using an arbitrary increasing function with specific conditions, fuzzy negations are produced, not just rational ones. Moreover, we compare the equilibrium points of the produced fuzzy negation of the first method and the Sugeno class. We use the equilibrium point to present a novel method which produces strong fuzzy negations by using two decreasing functions which satisfy specific conditions. We also investigate the convexity of the new fuzzy negation. We give some conditions that coefficients of fuzzy negation of the first method must satisfy in order to be convex. We present some examples of the new fuzzy negations, and we use them to generate new non-symmetric fuzzy implications by using well-known production methods of non-symmetric fuzzy implications. We use convex fuzzy negations as decreasing functions to construct an Archimedean copula. Finally, we investigate the quadratic form of the copula and the conditions that the coefficients of the first method and the increasing function of the second method must satisfy in order to generate new copulas of this form.

Fuzzy implications - A (dis)similarity perspective

Fuzzy implications continue to remain an important class among the fuzzy logic connectives, having found utility in contexts outside of logical inference too. Recently, fuzzy implications have been shown to be a fertile source for obtaining distance functions with very beneficial properties. In this work, we show that fuzzy implications can also be a good wellspring of fuzzy compatibility relations with myriad properties. Quite interestingly, the converse also holds good, i.e., we also show that we can represent any fuzzy implication through a pair (d,E), where d is a distance function - called a pseudo-monometric - and E is a mono-similarity fuzzy relation. This representation helps us to propose an equivalent formulation of the problem of characterisation of QL-implications, that has the potential to offer computational savings in verifying if a QL-operation can be a QL-implication and also throws open further avenues of fruitful theoretical exploration.

Monometrics on betweenness sets: A general construction

Recently, the study of monometrics on betweenness relations has gathered impetus, with their applications ranging from rationalisation of ranking rules to penalty-based aggregation. The main challenge herein is to find a monometric with respect to the considered betweenness relation. While methods to generate monometrics on specific betweenness relations are known, no research has been done to investigate whether monometrics can be produced on any betweenness relation. One of the twin contributions of this short paper shows that certain betweenness relations preclude the existence of a non-trivial monometric. Secondly, we show how the distance functions obtained using fuzzy implications are useful in both establishing the existence of and building a monometric on betweenness relations that satisfy a specific transitivity.

ANALYSIS OF FUZZY IMPLICATION

Fuzzy logic (fuzzy sets) possesses some widespread application in decision-making due to its ability to handle uncertainties in the implemented data and information. The application of fuzzy logic in decision-making usually involves the design of fuzzy rule-based systems. The functioning of the designed fuzzy systems can be compactly defined as the fuzzy inference process. One of the main steps of fuzzy inference is fuzzy implication, and, therefore, the selection of proper implication function possesses the great impact on the corresponding decision-making process. The currently reported research provides a comparative analysis on two fuzzy implications: Ali-2 and Zadeh. The reported research involves the application of an efficiency index and is performed on a real-world example. Keywords: Fuzzy implication, If-Then rules, membership function, fuzzy set, efficiency index

Fuzzy Inference Full Implication Method Based on Single Valued Neutrosophic t-representable t-norm: Purposes, Strategies, and a Proof-of-Principle Study


 As a generalization of intuitionistic fuzzy sets, single-valued neutrosophic sets have certain advantages in solving indeterminate and inconsistent information. In this paper, we study the fuzzy inference full implication method based on single-valued neutrosophic t-representable t-norm. Firstly, single-valued neutrosophic fuzzy inference triple I principles for fuzzy modus ponens and fuzzy modus tollens are given. Then, single-valued neutrosophic R-type triple I solutions for FMP and FMT are given. Finally, the robustness of the full implication triple I method based on the left-continuous single-valued neutrosophic t-representable t-norm is investigated. As a special case of the main results, the sensitivity of full implication triple I solutions based on three special single-valued neutrosophic t-representable t-norms are given.