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.     

Roughness based on fuzzy ideals

The theory of rough set, proposed by Pawlak and the theory of fuzzy set, proposed by Zadeh are complementary generalizations of classical set theory. Many sets are naturally endowed with two binary operations: addition and multiplication. One concept which does this is a ring. This paper concerns a relationship between rough sets, fuzzy sets and ring theory. It is a continuation of ideas presented by Kuroki and Wang [N. Kuroki, P.P. Wang, The lower and upper approximations in a fuzzy group, Inform. Sci. 90 (1996) 203-220]. We consider a ring as a universal set and we assume that the knowledge about objects is restricted by a fuzzy ideal. In fact, we apply the notion of fuzzy ideal of a ring for definitions of the lower and upper approximations in a ring. Some characterizations of the above approximations are made and some examples are presented.     

Rough approximations on a complete completely distributive lattice with applications to generalized rough sets

The generalizations of rough sets considered with respect to similarity relation, covers and fuzzy relations, are main research topics of rough set theory. However, these generalizations have shown less connection among each other and have not been brought into a unified framework, which has limited the in-depth research and application of rough set theory. In this paper the complete completely distributive (CCD) lattice is selected as the mathematical foundation on which definitions of lower and upper approximations that form the basic concepts of rough set theory are proposed. These definitions result from the concept of cover introduced on a CCD lattice and improve the approximations of the existing crisp generalizations of rough sets with respect to similarity relation and covers. When T-similarity relation is considered, the existing fuzzy rough sets are the special cases of our proposed approximations on a CCD lattice. Thus these generalizations of rough sets are brought into a unified framework, and a wider mathematical foundation for rough set theory is established.     

积模糊粗集模型及其模糊知识粒的表示和分解

为处理人工智能中不精确和不确定的数据和知识,Pawlak提出了粗集理论。之后粗集理论被推广,其方法主要有二:一是减弱对等价关系的依赖;二是把研究问题的论域从一个拓展到多个。结合这两种思想,研究基于两个模糊近似空间的积模糊粗集模型及其模糊粗糙集的表示和分解。根据这种思想,可以从论域分解的角度探索降低高维模糊粗糙集计算的复杂度问题。先对模糊近似空间的分层递阶结构———λ-截近似空间进行研究,得到不同层次知识粒的相互关系;然后定义模糊等价关系的积,并研究其性质及算法;最后构建基于积模糊等价关系的积模糊粗集模型,并讨论了该模型中模糊粗糙集的表示及分解问题,分别从λ-截近似空间和一维模糊近似空间的角度去处理,给出了可分解集的上(下)近似的一个刻画,及模糊可分解集的上(下)近似的λ-截集分解算法。     

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.     

Granular computing based on fuzzy similarity relations

Rough sets and fuzzy rough sets serve as important approaches to granular computing, but the granular structure of fuzzy rough sets is not as clear as that of classical rough sets since lower and upper approximations in fuzzy rough sets are defined in terms of membership functions, while lower and upper approximations in classical rough sets are defined in terms of union of some basic granules. This limits further investigation of the existing fuzzy rough sets. To bring to light the innate granular structure of fuzzy rough sets, we develop a theory of granular computing based on fuzzy relations in this paper. We propose the concept of granular fuzzy sets based on fuzzy similarity relations, investigate the properties of the proposed granular fuzzy sets using constructive and axiomatic approaches, and study the relationship between granular fuzzy sets and fuzzy relations. We then use the granular fuzzy sets to describe the granular structures of lower and upper approximations of a fuzzy set within the framework of granular computing. Finally, we characterize the structure of attribute reduction in terms of granular fuzzy sets, and two examples are also employed to illustrate our idea in this paper.     

Fuzzy reasoning based on a new fuzzy rough set and its application to scheduling problems

In this paper, we study the fuzzy reasoning based on a new fuzzy rough set. First, we define a broad family of new lower and upper approximation operators of fuzzy sets between different universes using a set of axioms. Then, based on the approximation operators above, we propose the fuzzy reasoning based on the new fuzzy rough set. By means of the above fuzzy reasoning based on the new fuzzy rough set, for a given premise, we can obtain the fuzzy reasoning consequence expressed by the fuzzy interval constructed by the above two approximations of fuzzy sets. Furthermore, through the defuzzification of the lower and upper approximations, we can get the corresponding two values constructing the interval used as the fuzzy reasoning consequence after defuzzification. Then, from the above interval, a suitable value can be selected as the final reasoning consequence so that some special constraints are satisfied as possibly. At last, we apply the fuzzy reasoning based on the new fuzzy rough set to the scheduling problems, and numerical computational results show that the fuzzy reasoning based on the new fuzzy rough set is more suitable for the scheduling problems compared with the fuzzy reasoning based on the CRI method and the III method.     

Invertible approximation operators of generalized rough sets and fuzzy rough sets

This paper studies the classes of rough sets and fuzzy rough sets. We discuss the invertible lower and upper approximations and present the necessary and sufficient conditions for the lower approximation to coincide with the upper approximation in both rough sets and fuzzy rough sets. We also study the mathematical properties of a fuzzy rough set induced by a cyclic fuzzy relation.     

Learning fuzzy rules from fuzzy samples based on rough set technique

Although the traditional rough set theory has been a powerful mathematical tool for modeling incompleteness and vagueness, its performance in dealing with initial fuzzy data is usually poor. This paper makes an attempt to improve its performance by extending the traditional rough set approach to the fuzzy environment. The extension is twofold. One is knowledge representation and the other is knowledge reduction. First, we provide new definitions of fuzzy lower and upper approximations by considering the similarity between the two objects. Second, we extend a number of underlying concepts of knowledge reduction (such as the reduct and core) to the fuzzy environment and use these extensions to propose a heuristic algorithm to learn fuzzy rules from initial fuzzy data. Finally, we provide some numerical experiments to demonstrate the feasibility of the proposed algorithm. One of the main contributions of this paper is that the fundamental relationship between the reducts and core of rough sets is still pertinent after the proposed extension.     

多粒化模糊软粗糙集模型

为了扩大粗糙集理论的应用,特别是在模糊环境中的应用,基于模糊软集和模糊蕴涵算子,主要研究基于软模糊近似空间的乐观多粒化模糊软粗糙集模型。该模型将参数集根据客户的不同要求或目标进行重组,只选择若干相关参数集参与计算上、下近似,这样定义的上、下近似不再由整个属性集决定,而是根据重组后的多个属性集一并生成,从而使结果更加符合实际需求。另外,还定义了乐观多粒化模糊软粗糙集模型的截集并讨论了其相关性质。最后给出了算例。     

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.     

Rough approximations based on intersection of indiscernibility,similarity and outranking relations

Rough sets theory has proved to be a useful mathematical tool for dealing with the vagueness and granularity in information tables. Classical definitions of lower and upper approximations were originally introduced with reference to an indiscernibility relation. However, indiscernibility relation is still restrictive for many applications. Many real-world problems deal with assignment of some objects to some preference-ordered decision classes. And, the objects are described by a finite set of qualitative attributes and quantitative attributes. In this paper, we construct the indiscernibility relation for the subset of nominal attributes, the outranking relation for the subset of ordinal attributes, and the similarity relation for the subset of quantitative attributes. Then the global binary relation is generated by the intersection of indiscernibility relation, outranking relation and similarity relation. New definitions of lower and upper approximations of the upward and downward unions of decision classes are proposed based on the global relation. We also prove that the lower and upper approximation operations satisfy the properties of rough inclusion, complementarity, identity of boundaries, and monotonicity.     

铅锌密闭鼓风炉故障诊断知识获取的粗糙集方法

从粗糙集等价类概念出发，提出从不完整数据集中获取故障诊断知识的密闭鼓风炉故障诊断方法，将不完整数据集的训练事例划分为下近似和上近似两类，首先假设属性的未知特征值为任意可能值，然后根据从驯练事例中得到的上下近似进行提炼，最后从事例与近似互相作用以推导出确定的和可能的规则，得出规则概率，并估计出合适的属性的未知特征值，结合密闭鼓风炉悬料规则库的知识获取及其在故障诊断中的应用过程说明了该方法的有效性和实用性。     

Soft fuzzy rough sets and its application in decision making

Recently, the theory and applications of soft set has brought the attention by many scholars in various areas. Especially, the researches of the theory for combining the soft set with the other mathematical theory have been developed by many authors. In this paper, we propose a new concept of soft fuzzy rough set by combining the fuzzy soft set with the traditional fuzzy rough set. The soft fuzzy rough lower and upper approximation operators of any fuzzy subset in the parameter set were defined by the concept of the pseudo fuzzy binary relation (or pseudo fuzzy soft set) established in this paper. Meanwhile, several deformations of the soft fuzzy rough lower and upper approximations are also presented. Furthermore, we also discuss some basic properties of the approximation operators in detail. Subsequently, we give an approach to decision making problem based on soft fuzzy rough set model by analyzing the limitations and advantages in the existing literatures. The decision steps and the algorithm of the decision method were also given. The proposed approach can obtain a object decision result with the data information owned by the decision problem only. Finally, the validity of the decision methods is tested by an applied example.     

双论域下多粒度模糊粗糙集上下近似的包含关系

针对双论域上集合的多粒度乐观与悲观上下近似不具有包含关系的问题,本文给出了双论域上集合的多粒度上下近似具有包含关系的一个充分条件,进而采用标准化的方法将不具有包含关系的上下近似转化为具有包含关系的上下近似。通过实例验证,该方法能有效解决双论域下多粒度模糊粗糙集上下近似具有包含关系的问题。     

A roughness measure of fuzzy sets from the perspective of distance

Covering generalized rough set theory is an important extension of classical rough set theory. To characterize a fuzzy set in a given covering approximation space, a pair of fuzzy sets, called covering rough fuzzy lower and upper approximations, were introduced, but they do not describe well how much uncertainty is induced by the granularity of knowledge. In this paper, we first discuss the relationship between uncertainty and granularity of knowledge. Then we examine several commonly used distance measures, and indicate that some of them exhibit some limitations. Next we propose a roughness measure based on Minkowski distance, and examine some important properties of this measure. Finally, an illustrative example is provided to demonstrate the application of the roughness measure to incomplete information systems with fuzzy decision.     

On intuitionistic fuzzy rough sets and their topological structures

In this paper, lower and upper approximations of intuitionistic fuzzy sets with respect to an intuitionistic fuzzy approximation space are first defined. Properties of intuitionistic fuzzy approximation operators are examined. Relationships between intuitionistic fuzzy rough set approximations and intuitionistic fuzzy topologies are then discussed. It is proved that the set of all lower approximation sets based on an intuitionistic fuzzy reflexive and transitive approximation space forms an intuitionistic fuzzy topology; and conversely, for an intuitionistic fuzzy rough topological space, there exists an intuitionistic fuzzy reflexive and transitive approximation space such that the topology in the intuitionistic fuzzy rough topological space is just the set of all lower approximation sets in the intuitionistic fuzzy reflexive and transitive approximation space. That is to say, there exists an one-to-one correspondence between the set of all intuitionistic fuzzy reflexive and transitive approximation spaces and the set of all intuitionistic fuzzy rough topological spaces. Finally, intuitionistic fuzzy pseudo-closure operators in the framework of intuitionistic fuzzy rough approximations are investigated.     

Approximations and reducts with covering generalized rough sets

The covering generalized rough sets are an improvement of traditional rough set model to deal with more complex practical problems which the traditional one cannot handle. It is well known that any generalization of traditional rough set theory should first have practical applied background and two important theoretical issues must be addressed. The first one is to present reasonable definitions of set approximations, and the second one is to develop reasonable algorithms for attributes reduct. The existing covering generalized rough sets, however, mainly pay attention to constructing approximation operators. The ideas of constructing lower approximations are similar but the ideas of constructing upper approximations are different and they all seem to be unreasonable. Furthermore, less effort has been put on the discussion of the applied background and the attributes reduct of covering generalized rough sets. In this paper we concentrate our discussion on the above two issues. We first discuss the applied background of covering generalized rough sets by proposing three kinds of datasets which the traditional rough sets cannot handle and improve the definition of upper approximation for covering generalized rough sets to make it more reasonable than the existing ones. Then we study the attributes reduct with covering generalized rough sets and present an algorithm by using discernibility matrix to compute all the attributes reducts with covering generalized rough sets. With these discussions we can set up a basic foundation of the covering generalized rough set theory and broaden its applications.     

Reduction method based on a new fuzzy rough set in fuzzy information system and its applications to scheduling problems

In this paper, we present the concept of fuzzy information granule based on a relatively weaker fuzzy similarity relation called fuzzy T_L-similarity relation for the first time. Then, according to the fuzzy information granule, we define the lower and upper approximations of fuzzy sets and a corresponding new fuzzy rough set. Furthermore, we construct a kind of new fuzzy information system based on the fuzzy T_L-similarity relation and study its reduction using the fuzzy rough set. At last, we apply the reduction method based on the defined fuzzy rough set in the above fuzzy information system to the reduction of the redundant multiple fuzzy rule in the scheduling problems, and numerical computational results show that the reduction method based on the new fuzzy rough set is more suitable for the reduction of multiple fuzzy rules in the scheduling problems compared with the reduction methods based on the existing fuzzy rough set.     

Rough集定义的拓广

对Rough集定义进行了拓广,通过将拓广的定义与传统的Rough集定义的比较分析证明了拓广是有意义、实用的.