On the Law of Importation $(x wedge y) longrightarrow z equiv (x longrightarrow (y longrightarrow z))$ in Fuzzy Logic

The law of importation, given by the equivalence (x Lambda y) rarr z equiv (xrarr (y rarr z)), is a tautology in classical logic. In A-implications defined by Turksen et aL, the above equivalence is taken as an axiom. In this paper, we investigate the general form of the law of importation J(T(x, y), z) = J(x, J(y, z)), where T is a t-norm and J is a fuzzy implication, for the three main classes of fuzzy implications, i.e., R-, S- and QL-implications and also for the recently proposed Yager's classes of fuzzy implications, i.e., f- and g-implications. We give necessary and sufficient conditions under which the law of importation holds for R-, S-, f- and g-implications. In the case of QL-implications, we investigate some specific families of QL-implications. Also, we investigate the general form of the law of importation in the more general setting of uninorms and t-operators for the above classes of fuzzy implications. Following this, we propose a novel modified scheme of compositional rule of inference (CRI) inferencing called the hierarchical CRI, which has some advantages over the classical CRI. Following this, we give some sufficient conditions on the operators employed under which the inference obtained from the classical CRI and the hierarchical CRI become identical, highlighting the significant role played by the law of importation.     

A general inequality of Chebyshev type for semi(co)normed fuzzy integrals

Generalization of the Chebyshev inequality for semi(co)normed fuzzy integrals on an abstract fuzzy measure space based on a binary operation is given. Also, Minkowski’s and Hölder’s inequalities for semi(co)normed fuzzy integrals are studied in a rather general form. The main results of this paper generalize some previous results. Finally, a conclusion is drawn and an open problem for further investigations is given.     

Balanced fuzzy sets

The paper presents a new approach to fuzzy sets and uncertain information based on an observation of asymmetry of classical fuzzy operators. Parallel is drawn between symmetry and negativity of uncertain information. The hypothesis is raised that classical theory of fuzzy sets concentrates the whole negative information in the value 0 of membership function, what makes fuzzy operators asymmetrical. This hypothesis could be seen as a contribution to a broad range discussion on unification of aggregating operators and uncertain information processing rather than an opposition to other approaches. The new approach “spreads” negative information from the point 0 into the interval [−1, 0] making scale and operators symmetrical. The balanced counterparts of classical operators are introduced. Relations between classical and balanced operators are discussed and then developed to the hierarchies of balanced operators of higher ranks. The relation between balanced norms, on one hand, and uninorms and nullnorms, on the other, are quite close: balanced norms are related to equivalence classes of some equivalence relation build on linear dependency in the spaces of uninorms and nullnorms. It is worth to stress that this similarity is raised by two entirely different approaches to generalization of fuzzy operators. This observation validates the generalized hierarchy of fuzzy operators to which both approaches converge. The discussion in this paper is aimed at presenting the idea and does not aspire to detailed exploration of all related aspects of uncertainty and information processing.     

基于模糊等价的毕达哥拉斯模糊集相似度构造方法

毕达哥拉斯模糊集是Zadeh模糊集的一种推广形式,其相似度刻画方法是毕达哥拉斯模糊集理论的重要研究内容。现有的毕达哥拉斯模糊集相似度大多针对具体问题而提出。为推广毕达哥拉斯模糊集理论的应用范围,文中基于模糊等价研究毕达哥拉斯模糊集相似度的一般构造方法。将模糊等价概念推广至毕达哥拉斯模糊数,提出了PFN(Pythagorean Fuzzy Number)模糊等价的概念,并给出了PFN模糊等价的构造方法。进一步,通过聚合算子给出了基于PFN模糊等价的毕达哥拉斯模糊集相似度的一般构造方法。通过实例说明了现有的一些相似度是文中构造的相似度的特例。     

Robustness of fuzzy reasoning via logically equivalence measure

In this paper, we discuss robustness of fuzzy reasoning. After proposing the definition of perturbation of fuzzy sets based on some logic-oriented equivalence measure, we present robustness results for various fuzzy logic connectives, fuzzy implication operators, inference rules and fuzzy reasoning machines, and discuss the relations between the robustness of fuzzy reasoning and that of fuzzy conjunction and implication operators. The robustness results are presented in terms of δ-equalities of fuzzy sets based on some logic-oriented equivalence measure, and the maximum of δ (which ensures the corresponding δ-equality holds) is derived.     

Simplified interval-valued intuitionistic fuzzy integrals and their use in park siting

The interval-valued intuitionistic fuzzy sets have received great attention of researchers because they can comprehensively depict the characters of things. In the past few years, some scholars have investigated the calculus of intuitionistic fuzzy information, but yet there is no research on the integrals in interval-valued intuitionistic fuzzy circumstance. To fill this vacancy, in this paper, we shall focus on investigating the integrals of simplified interval-valued intuitionistic fuzzy functions (SIVIFFs) and give their application in group decision making. We first develop the indefinite and definite integrals of SIVIFFs, and study their characteristics in detail. Then we establish the relationship between these two classes of integrals by giving two Newton–Leibniz formulas for SIVIFFs. Finally, a practical example concerning the park siting problem is given to illustrate the application of simplified interval-valued intuitionistic fuzzy integrals.     

多粒化的模糊粗糙集代数

众所周知,一个粗糙集代数是由一个集合代数加上一对近似算子构成的。首先利用公理化的方法探讨经典的多粒化模糊粗糙集代数系统,可知经典的多粒化模糊粗糙集代数没有很好的性质;其次,引入 具有最小(大)元的等价关系的定义,并给出了基于具有最小(大)元等价关系的多粒化模糊近似算子的概念,在此基础上讨论了模糊粗糙集代数的性质,并得到了诸多结果。     

On minimum-risk problems in fuzzy random decision systems

In a decision-making process, we may face a hybrid environment where linguistic and frequent imprecision nature coexists. The problem of frequent imprecision can be solved by probability theory, while the problem of linguistic imprecision can be tackled by possibility theory. Therefore, to solve this hybrid decision-making problem, it is necessary to combine both theories effectively. In this paper, we restrict our attention to this hybrid decision-making problem, where the input data are imprecise and described by fuzzy random variables. Fuzzy random variable is a mapping from a probability space to a collection of fuzzy variables, it is an appropriate tool to deal with twofold uncertainty with fuzziness and randomness in an optimization framework. The purpose of this paper is to present reasonable chances of a fuzzy random event characterized by fuzzy random variables so that they can connect with the expected value operators of a fuzzy random variable via Choquet integrals, just like the relation between the probability of a random event and the mathematical expectation of a random variable, and that between the credibility of a fuzzy event and the expected value operator of a fuzzy variable. Toward that end, we take fuzzy measure and fuzzy integral theory as our research tool, and present three kinds of mean chances of a fuzzy random event via Choquet integrals. After discussing the duality of the mean chances, we use the mean chances to define the expected value operators of a fuzzy random variable via Choquet integrals. To show the reasonableness of the mean chance approach, we prove the expected value operators defined in this paper coincide with those presented in our previous work. Using the mean chances, we present a new class of fuzzy random minimum-risk problems, where the objective and the constraints are all defined by the mean chances. To solve general fuzzy random minimum-risk optimization problems, a hybrid intelligent algorithm, which integrates fuzzy random simulations, genetic algorithm and neural network, is designed, and its feasibility and effectiveness are illustrated by numerical examples.     

Exact canonical decomposition of two-qubit operators in terms of CNOT

The canonical decomposition for two-qubit operators has proven very useful for applications in quantum computing. This decomposition generates equivalence classes up to local quantum gates. We provide a variety of complete, explicit decompositions of given two-qubit operators in terms of single, double, and triple controlled-NOT (CNOT) gates. By analytically addressing the needed pre- and post-tensor product factors, we demonstrate that exact results are possible, even when a parameter is included. The examples given are of interest to superconducting qubit, spin-based, dipolar molecule, and other quantum information processing systems.     

An axiomatic characterization of a fuzzy generalization of rough sets

In rough set theory, the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties. Several authors have proposed various fuzzy generalizations of rough approximations. In this paper, we introduce the definitions for generalized fuzzy lower and upper approximation operators determined by a residual implication. Then we find the assumptions which permit a given fuzzy set-theoretic operator to represent a upper (or lower) approximation derived from a special fuzzy relation. Different classes of fuzzy rough set algebras are obtained from different types of fuzzy relations. And different sets of axioms of fuzzy set-theoretic operator guarantee the existence of different types of fuzzy relations which produce the same operator. Finally, we study the composition of two approximation spaces. It is proved that the approximation operators in the composition space are just the composition of the approximation operators in the two fuzzy approximation spaces.     

Fuzzy controllers: synthesis and equivalences

It has been proved that fuzzy controllers are capable of approximating any real continuous control function on a compact set to arbitrary accuracy. In particular, any given linear control can be achieved with a fuzzy controller for a given accuracy. The aim of this paper is to show how to automatically build this fuzzy controller. The proposed design methodology is detailed for the synthesis of a Sugeno or Mamdani type fuzzy controller precisely equivalent to a given PI controller. The main idea is to equate the output of the fuzzy controller with the output of the PI controller at some particular input values, called modal values. The rule base and the distribution of the membership functions can thus be deduced. The analytic expression of the output of the generated fuzzy controller is then established. For Sugeno-type fuzzy controllers, precise equivalence is directly obtained. For Mamdani-type fuzzy controllers, the defuzzification strategy and the inference operators have to be correctly chosen to provide linear interpolation between modal values. The usual inference operators satisfying the linearity requirement when using the center of gravity defuzzification method are proposed     

基于ε-修正的直觉模糊信息集成方法及其在决策中的应用

已有的一些直觉模糊集成算子在处理一些特殊直觉模糊数时会出现反直觉现象。首先介绍了两个直觉模糊集成算子和直觉模糊数的比较方法。接着,举例说明了这些集成算子在某些情况下出现的反直觉现象。然后提出了基于ε-修正的直觉模糊集成算子,并讨论了ε取值对此算子结果的影响。之后建立了一种基于ε-修正的直觉模糊集成算子的决策方法。最后通过一个实例比较了原集成算子和本文提出的修正集成算子的集成结果,验证基于ε-修正的直觉模糊集成算子可以修正这些反直觉现象,这也拓宽了原集成算子的使用范围。     

Network-based decision making via generalized fuzzy integral operators

Recent advances in network-based decision making methods have given rise to computationally efficient solution methodologies for intelligent systems. One type of hierarchical network implementation, the fuzzy integral operator approach, is investigated. In this approach, we generalized the Choquet fuzzy integral as an excellent component for decision analysis and making. This involves extending the standard operators in information aggregation with generalized operators, resulting in increased flexibility. The characteristics of the Choquet fuzzy integrals and their generalizations are addressed and network-based decision making frameworks are then proposed. The trainable hierarchical networks are able to perceive and interpret complex decisions by using those processing elements called neurons. We also present a decision making experiment using the proposed network to learn appropriate functional relationships in the defective numeric fields detection domain. ©1999 John Wiley & Sons, Inc.     

Fuzzy interior operators

We study interior operators from the point of view of fuzzy set theory. The present approach generalizes the particular cases studied previously in the literature in two aspects. First, we use complete residuated lattices as structures of truth values thus generalizing several important cases like the classical Boolean case, (left-)continuous t-norms, MV-algebras, BL-algebras, etc. Second, and more importantly, we pay attention to graded subsethood of fuzzy sets, which turns out to play an important role. In the first part, we define, illustrate by examples and study general fuzzy interior operators. The second part is devoted to fuzzy interior operators induced by fuzzy equivalence relations (similarities).     

General type-2 fuzzy rough sets based on \alpha -plane Representation theory

Rough sets theory and fuzzy sets theory are mathematical tools to deal with uncertainty, imprecision in data analysis. Traditional rough set theory is restricted to crisp environments. Since theories of fuzzy sets and rough sets are distinct and complementary on dealing with uncertainty, the concept of fuzzy rough sets has been proposed. Type-2 fuzzy set provides additional degree of freedom, which makes it possible to directly handle highly uncertainties. Some researchers proposed interval type-2 fuzzy rough sets by combining interval type-2 fuzzy sets and rough sets. However, there are no reports about combining general type-2 fuzzy sets and rough sets. In addition, the $\alpha $ -plane representation method of general type-2 fuzzy sets has been extensively studied, and can reduce the computational workload. Motivated by the aforementioned accomplishments, in this paper, from the viewpoint of constructive approach, we first present definitions of upper and lower approximation operators of general type-2 fuzzy sets by using $\alpha $ -plane representation theory and study some basic properties of them. Furthermore, the connections between special general type-2 fuzzy relations and general type-2 fuzzy rough upper and lower approximation operators are also examined. Finally, in axiomatic approach, various classes of general type-2 fuzzy rough approximation operators are characterized by different sets of axioms.     

Grouping fuzzy sets by similarity

The paper presents results on factorization of systems of fuzzy sets. The factorization consists in grouping those fuzzy sets which are pairwise similar at least to a prescribed degree a. An obstacle to such factorization, well known in fuzzy set theory, is the fact that “being similar at least to degree a” is not an equivalence relation because, in general, it is not transitive. As a result, ordinary factorization using equivalence classes cannot be used. This obstacle can be overcome by considering maximal blocks of fuzzy sets which are pairwise similar at least to degree a. We show that one can introduce a natural complete lattice structure on the set of all such maximal blocks and study this lattice. This lattice plays the role of a factor structure for the original system of fuzzy sets. Particular examples of our approach include factorization of fuzzy concept lattices and factorization of residuated lattices.     

Fuzzy additive hybrid operators for network-based decision making

Evidence Aggregation Networks based on multiplicative fuzzy hybrid operators were introduced by Krishnapuram and Lee. They have been used for image segmentation, pattern recognition, and general multicriteria decision making. One of the drawbacks to these networks is that the training is complex and quite time consuming. In this article, we modify these aggregation networks to implement additive fuzzy hybrid connectives. We study the theoretical properties of two classes of such aggregation operators, one where the union and intersection components are based on multiplication, and the other where these components are derived from Yager connectives. These new networks have similar excellent properties such as backpropagation training and node interpretability for decision making under uncertainty as do their multiplicative precursors. They also have the advantage that training is easier since the derivatives of the additive hybrid operators are not as complex in form. the appropriate training algorithms are derived, and several examples given to illustrate the properties of the networks. © 1994 John Wiley & Sons, Inc.     

Identification of overlapping and non-overlapping community structure by fuzzy clustering in complex networks

This paper proposes a novel method based on fuzzy clustering to detect community structure in complex networks. In contrast to previous studies, our method does not focus on a graph model, but rather on a fuzzy relation model, which uses the operations of fuzzy relation to replace a traversal search of the graph for identifying community structure. In our method, we first use a fuzzy relation to describe the relation between vertices as well as the similarity in network topology to determine the membership grade of the relation. Then, we transform this fuzzy relation into a fuzzy equivalence relation. Finally, we map the non-overlapping communities as equivalence classes that satisfy a certain equivalence relation. Because most real-world networks are made of overlapping communities (e.g., in social networks, people may belong to multiple communities), we can consider the equivalence classes above as the skeletons of overlapping communities and extend our method by adding vertices to the skeletons to identify overlapping communities. We evaluated our method on artificial networks with built-in communities and real-world networks with known and unknown communities. The experimental results show that our method works well for detecting these communities and gives a new understanding of network division and community formation.     

逻辑等价度量下的模糊推理系统的鲁棒性

给出了基于模糊逻辑等价度量的模糊集的扰动的定义,讨论了模糊集扰动与模糊连接词及蕴涵算子扰动之间的关系,针对若干特殊的模糊连接词及蕴涵算子的扰动情形,给出了模糊推理系统的扰动的估计,并讨论了模糊推理系统的鲁棒性。