On intuitionistic fuzzy topologies based on intuitionistic fuzzy reflexive and transitive relations

Topologies and rough set theory are widely used in the research field of machine learning and cybernetics. An intuitionistic fuzzy rough set, which is the result of approximation of an intuitionistic fuzzy set with respect to an intuitionistic fuzzy approximation space, is an extension of fuzzy rough sets. For further studying the theories and applications of intuitionistic fuzzy rough sets, in this paper, we investigate the topological structures of intuitionistic fuzzy rough sets. We show that an intuitionistic fuzzy rough approximation space can induce an intuitionistic fuzzy topological space in the sense of Lowen if and only if the intuitionistic fuzzy relation in the approximation space is reflexive and transitive. We also examine the sufficient and necessary conditions that an intuitionistic fuzzy topological space can be associated with an intuitionistic fuzzy reflexive and transitive relation such that the induced lower and upper intuitionistic fuzzy rough approximation operators are, respectively, the intuitionistic fuzzy interior and closure operators of the given topology.     

Characterization of rough set approximations in Atanassov intuitionistic fuzzy set theory

The primitive notions in rough set theory are lower and upper approximation operators defined by a fixed binary relation and satisfying many interesting properties. Many types of generalized rough set models have been proposed in the literature. This paper discusses the rough approximations of Atanassov intuitionistic fuzzy sets in crisp and fuzzy approximation spaces in which both constructive and axiomatic approaches are used. In the constructive approach, concepts of rough intuitionistic fuzzy sets and intuitionistic fuzzy rough sets are defined, properties of rough intuitionistic fuzzy approximation operators and intuitionistic fuzzy rough approximation operators are examined. Different classes of rough intuitionistic fuzzy set algebras and intuitionistic fuzzy rough set algebras are obtained from different types of fuzzy relations. In the axiomatic approach, an operator-oriented characterization of rough sets is proposed, that is, rough intuitionistic fuzzy approximation operators and intuitionistic fuzzy rough approximation operators are defined by axioms. Different axiom sets of upper and lower intuitionistic fuzzy set-theoretic operators guarantee the existence of different types of crisp/fuzzy relations which produce the same operators.     

二型直觉模糊粗糙集

将二型直觉模糊集和粗糙集理论融合,建立二型直觉模糊粗糙集模型。首先,在二型直觉模糊近似空间中,定义了一对二型直觉模糊上、下近似算子,并讨论了二型直觉模糊关系退化为普通二型模糊关系和一般等价关系时,上、下近似算子的具体变化形式。然后,将普通二型模糊集之间包含关系的定义推广到了二型直觉模糊集,在此基础上研究了二型直觉模糊上、下近似算子的一些性质。最后,定义了自反的、对称的和传递的二型直觉模糊关系,并讨论了这3种特殊的二型直觉模糊关系与近似算子的特征之间的联系。该结论进一步丰富了二型模糊集理论和粗糙集理论,为二型直觉模糊信息系统的应用奠定了良好的理论基础。     

On generalized intuitionistic fuzzy rough approximation operators

In rough set theory, the lower and upper approximation operators defined by binary relations satisfy many interesting properties. Various generalizations of Pawlak’s rough approximations have been made in the literature over the years. This paper proposes a general framework for the study of relation-based intuitionistic fuzzy rough approximation operators within which both constructive and axiomatic approaches are used. In the constructive approach, a pair of lower and upper intuitionistic fuzzy rough approximation operators induced from an arbitrary intuitionistic fuzzy relation are defined. Basic properties of the intuitionistic fuzzy rough approximation operators are then examined. By introducing cut sets of intuitionistic fuzzy sets, classical representations of intuitionistic fuzzy rough approximation operators are presented. The connections between special intuitionistic fuzzy relations and intuitionistic fuzzy rough approximation operators are further established. Finally, an operator-oriented characterization of intuitionistic fuzzy rough sets is proposed, that is, intuitionistic fuzzy rough approximation operators are defined by axioms. Different axiom sets of lower and upper intuitionistic fuzzy set-theoretic operators guarantee the existence of different types of intuitionistic fuzzy relations which produce the same operators.     

On characterization of intuitionistic fuzzy rough sets based on intuitionistic fuzzy implicators

This paper presents a general framework for the study of relation-based (I,T)-intuitionistic fuzzy rough sets by using constructive and axiomatic approaches. In the constructive approach, by employing an intuitionistic fuzzy implicator I and an intuitionistic fuzzy triangle norm T, lower and upper approximations of intuitionistic fuzzy sets with respect to an intuitionistic fuzzy approximation space are first defined. Properties of (I,T)-intuitionistic fuzzy rough approximation operators are examined. The connections between special types of intuitionistic fuzzy relations and properties of intuitionistic fuzzy approximation operators are established. In the axiomatic approach, an operator-oriented characterization of (I,T)-intuitionistic fuzzy rough sets is proposed. Different axiom sets characterizing the essential properties of intuitionistic fuzzy approximation operators associated with various intuitionistic fuzzy relations are explored.     

粗糙support-intuitionistic模糊集及其聚类分析应用

粗糙集和模糊集理论已经被用于各种类型的不确定性建模中。Dubois和Prade研究了将模糊集和粗糙集结合的问题。提出了粗糙support-intuitionistic模糊集。介绍了粗糙集、粗糙直觉模糊集和support-intuitionistic模糊集等的概念;定义了在Pawlak近似空间中的support-intuitionistic模糊集的上下近似,讨论了一些粗糙support-intuitionistic模糊集近似算子的性质,给出了其相似度表达式;将其应用到聚类分析问题中,并通过一个实例验证其合理性。     

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.     

改进的粗糙直觉模糊集

在Pawlak近似空间中,针对直觉模糊目标集合,假设在信息粒度不变的情况下,试图寻求目标集合更好的近似集。在现有的粗糙直觉模糊集的基础之上,利用直觉模糊粗糙隶属函数,采用分段函数的形式建立直觉模糊集新的下近似与上近似算子,并讨论新模型的一些基本性质。与现有的粗糙直觉模糊集相比,改进后的模型无论在近似精度方面,还是与目标集合的相似度方面,都有了较大的改善和提高。最后通过数值算例验证了所给结论的正确性。     

Algorithm and axiomatization of rough fuzzy sets based finite dimensional fuzzy vectors

Rough sets, proposed by Pawlak and rough fuzzy sets proposed by Dubois and Prade were expressed with the different computing formulas that were more complex and not conducive to computer operations. In this paper, we use the composition of a fuzzy matrix and fuzzy vectors in a given non-empty finite universal, constitute an algebraic system composed of finite dimensional fuzzy vectors and discuss some properties of the algebraic system about a basis and operations. We give an effective calculation representation of rough fuzzy sets by the inner and outer products that unify computing of rough sets and rough fuzzy sets with a formula. The basis of the algebraic system play a key role in this paper. We give some essential properties of the lower and upper approximation operators generated by reflexive, symmetric, and transitive fuzzy relations. The reflexive, symmetric, and transitive fuzzy relations are characterized by the basis of the algebraic system. A set of axioms, as the axiomatic approach, has been constructed to characterize the upper approximation of fuzzy sets on the basis of the algebraic system.     

New topological approach of rough set generalizations

In this paper, we introduce a generalization of rough set theory using topological structures generated by any binary general relation. The class of all after-composed sets and the class of all fore-composed sets defined here are used to generate two topological spaces. These topologies are used to approximate sets topologically. Lower and upper approximations are defined topologically and some of their properties are studied using these topologies. Also, membership, equality, inclusion relations and power set notions in the generalized approximation space are studied. Many differences are shown between notions of ordinary set theory and notions of rough set theory. These notions introduced in this work are good future work to knowledge discovery and data mining.     

基于覆盖的区间直觉模糊粗糙集

在经典的覆盖近似空间中,定义了区间直觉模糊概念的粗糙近似。通过区间直觉模糊覆盖概念,给出了一种基于区间直觉模糊覆盖的区间直觉模糊粗糙集模型。讨论了两种模型的一些相关性质。     

基于蕴涵的区间值直觉模糊粗糙集

提出一种基于区间值直觉模糊蕴涵的区间值直觉模糊粗糙集模型.首先,介绍了区间值直觉模糊集、区间值直觉模糊关系和区间值直觉模糊逻辑算子的概念;然后,利用区间值直觉模糊三角模和区间值直觉模糊蕴涵,在区间值直觉模糊近似空间中定义了区间值直觉模糊集的上近似和下近似;最后,给出并证明了这些近似算子的一些性质.     

基于覆盖的直觉模糊粗糙集

通过直觉模糊覆盖概念将覆盖粗糙集模型进行推广,提出一种基于直觉模糊覆盖的直觉模糊粗糙集模型.首先,介绍了直觉模糊集、直觉模糊覆盖和直觉模糊逻辑算子等概念;然后,利用直觉模糊三角模和直觉模糊蕴涵,构建两对基于直觉模糊覆盖的下直觉模糊粗糙近似算子和上直觉模糊粗糙近似算子;最后,给出了这些算子的基本性质并研究了它们之间的对偶性.     

A review of: “ANTICIPATORY SYSTEMS”, by Robert Rosen,Pergamon Press,Oxford, 1985, X + 436pp.

In rough set theory there exists a pair of approximation operators, the upper and lower approximations, whereas in Dempster-Shafer theory of evidence there exists a dual pair of uncertainty measures, the plausibility and belief functions. It seems that there is some kind of natural connection between the two theories. The purpose of this paper is to establish the relationship between rough set theory and Dempster-Shafer theory of evidence. Various generalizations of the Dempster-Shafer belief structure and their induced uncertainty measures, the plausibility and belief functions, are first reviewed and examined. Generalizations of Pawlak approximation space and their induced approximation operators, the upper and lower approximations, are then summarized. Concepts of random rough sets, which include the mechanisms of numeric and non-numeric aspects of uncertain knowledge, are then proposed. Notions of the Dempster-Shafer theory of evidence within the framework of rough set theory are subsequently formed and interpreted. It is demonstrated that various belief structures are associated with various rough approximation spaces such that different dual pairs of upper and lower approximation operators induced by the rough approximation spaces may be used to interpret the corresponding dual pairs of plausibility and belief functions induced by the belief structures.     

Algorithm and axiomatization of rough fuzzy sets based finite dimensional fuzzy vectors

Rough sets, proposed by Pawlak and rough fuzzy sets proposed by Dubois and Prade were expressed with the different computing formulas that were more complex and not conducive to computer operations. In this paper, we use the composition of a fuzzy matrix and fuzzy vectors in a given non-empty finite universal, constitute an algebraic system composed of finite dimensional fuzzy vectors and discuss some properties of the algebraic system about a basis and operations. We give an effective calculation representation of rough fuzzy sets by the inner and outer products that unify computing of rough sets and rough fuzzy sets with a formula. The basis of the algebraic system play a key role in this paper. We give some essential properties of the lower and upper approximation operators generated by reflexive, symmetric, and transitive fuzzy relations. The reflexive, symmetric, and transitive fuzzy relations are characterized by the basis of the algebraic system. A set of axioms, as the axiomatic approach, has been constructed to characterize the upper approximation of fuzzy sets on the basis of the algebraic system.     

Generalized fuzzy rough sets determined by a triangular norm

The theory of rough sets has become well established as an approach for uncertainty management in a wide variety of applications. Various fuzzy generalizations of rough approximations have been made over the years. This paper presents a general framework for the study of T-fuzzy rough approximation operators in which both the constructive and axiomatic approaches are used. By using a pair of dual triangular norms in the constructive approach, some definitions of the upper and lower approximation operators of fuzzy sets are proposed and analyzed by means of arbitrary fuzzy relations. The connections between special fuzzy relations and the T-upper and T-lower approximation operators of fuzzy sets are also examined. In the axiomatic approach, an operator-oriented characterization of rough sets is proposed, that is, T-fuzzy approximation operators are defined by axioms. Different axiom sets of T-upper and T-lower fuzzy set-theoretic operators guarantee the existence of different types of fuzzy relations producing the same operators. The independence of axioms characterizing the T-fuzzy rough approximation operators is examined. Then the minimal sets of axioms for the characterization of the T-fuzzy approximation operators are presented. Based on information theory, the entropy of the generalized fuzzy approximation space, which is similar to Shannon’s entropy, is formulated. To measure uncertainty in T-generalized fuzzy rough sets, a notion of fuzziness is introduced. Some basic properties of this measure are examined. For a special triangular norm T = min, it is proved that the measure of fuzziness of the generalized fuzzy rough set is equal to zero if and only if the set is crisp and definable.     

覆盖模糊S-粗集模型

建立了基于覆盖理论的模糊S-粗糙集模型,并讨论其性质。在覆盖单向S-粗集x的最小描述的基础上,给出了x的最大描述的定义。给出了覆盖模糊S-粗集上 、下近似算子定义,讨论了算子的基本性质,证明了覆盖S-粗糙集模型下所有模糊集的下近似构成一个模糊拓扑,并得到模糊单向S-粗集X相对于覆盖单向S-粗集和覆盖约简单向S-粗集的上下近似分别相等。     

Approximation operators based on vague relations and roughness measures of vague sets

Rough set theory and vague set theory are powerful tools for managing uncertain, incomplete and imprecise information. This paper extends the rough vague set model based on equivalence relations and the rough fuzzy set model based on fuzzy relations to vague sets. We mainly focus on the lower and upper approximation operators of vague sets based on vague relations, and investigate the basic properties of approximation operators on vague sets. Specially, we give some essential characterizations of the lower and upper approximation operators generated by reflexive, symmetric, and transitive vague relations. Finally, we structure a parameterized roughness measure of vague sets and similarity measure methods between two rough vague sets, and obtain some properties of the roughness measure and similarity measures. We also give some valuable counterexamples and point out some false properties of the roughness measure in the paper of Wang et al.     

无限论域上粗糙集的拓扑结构

讨论了当论域不限制是有限集时满足自反、传递关系的广义近似空间中的近似算子的拓扑结构;证明了论域 上满足自反、传递关系的集合与其上所有的拓扑的集合是一一对应的;指出了该拓扑空间的拓扑基。     

对称传递关系的诱导拓扑及其可数性

粗糙集通过二元关系密切联系拓扑,并具有基于自反、自反传递、自反对称等关系的拓扑研究。采用对称传递关系构建拓扑并研究其可数性。基于对称传递关系,定义粗糙集近似集,由此建立拓扑及内部、闭包;针对构建拓扑,确立基与邻域基,得到第二可数性、第一可数性、可分性、林德洛夫性等可数性特征;提供实例分析。研究结果基于新二元关系揭示粗糙集与拓扑深入联系。