Fuzzy Rough Approximation Operators Research Articles

Relational correspondences for L-fuzzy rough approximations defined on De Morgan Heyting algebras

We consider fuzzy rough sets defined on De Morgan Heyting algebras. We present a theorem that can be used to obtain several correspondence results between fuzzy rough sets and fuzzy relations defining them. We characterize fuzzy rough approximation operators corresponding to compositions of reflexive, transitive, mediate, Euclidean and adjoint fuzzy relations defined on De Morgan Heyting algebras using only a single axiom.

Fuzzy rough sets based on fuzzy quantification

Classical (fuzzy) rough sets exhibit sensitivity to noise, which is particularly undesirable for machine learning applications. One approach to solve this issue is by making use of fuzzy quantifiers, as done by the vaguely quantified fuzzy rough set (VQFRS) model. While this idea is intuitive, the VQFRS model suffers from both theoretical flaws as well as from suboptimal performance in applications. In this paper, we improve on VQFRS by introducing fuzzy quantifier-based fuzzy rough sets (FQFRS), which proposes an intuitive fuzzy rough approximation operator that utilizes general unary and binary quantification models. We show how several existing models fit inside FQFRS, as well as how it inspires novel ones. Additionally, we propose several binary quantification models to be used with FQFRS. Furthermore, we conduct a theoretical study of their properties, and investigate their potential by applying them to classification problems. In particular, we highlight the effectiveness of Yager's Weighted Implication-based (YWI) binary quantification model, which induces a fuzzy rough set model that is both a significant improvement on VQFRS, as well as a worthy competitor to the popular ordered weighted averaging based fuzzy rough set (OWAFRS) model.

Representations of L-fuzzy rough approximation operators

In recent years, there has been tremendous interest in developing lattice-valued rough set theory. In this framework, we primarily take into account the following two problems. Firstly, we include the well-known intuitionistic, interval-valued, neutrosophic and Pythagorean fuzzy rough sets into the framework of lattice-valued rough sets. Specifically, the four kinds of rough sets can be considered as special lattice-valued rough sets. Secondly, based on a completely distributive lattice L, we provide representations of the upper and lower L-fuzzy rough approximation operators by using four kinds of cut sets of an L-fuzzy set and an L-fuzzy relation, which generalizes the existing results in the case that L=[0,1]. In particular, we show that representations of intuitionistic and interval-valued fuzzy rough approximation operators provided by Zhou and Sun are special examples of our proposed representations.

Novel results on G -lower L -fuzzy rough approximation operator in ( O , G ) -fuzzy rough set model1

Lately, Jiang and Hu (H.B. Jiang, B.Q. Hu, On ( O , G ) -fuzzy rough sets based on overlap and grouping functions over complete lattices, Int. J. Approx. Reason. 144 (2022) 18-50.) put forward ( O , G ) -fuzzy rough sets via overlap and grouping functions over complete lattices. Meanwhile, they showed the characterizations of O -upper and G -lower L -fuzzy rough approximation operators in ( O , G ) -fuzzy rough set model based on some of specific L -fuzzy relations and studied the topological properties of the proposed model. Nevertheless, we discover that the partial results given by Jiang and Hu could be further optimized. So, as a replenish of the above article, in this paper, based on G -lower L -fuzzy rough approximation operator in ( O , G ) -fuzzy rough set model, we further explore several new conclusions on the relationship between G -lower L -fuzzy rough approximation operator and different L -fuzzy relations. In particular, the equivalent descriptions of relationship between G -lower L -fuzzy rough approximation operator and O -transitive ( O -Euclidean) L -fuzzy relations are investigated, which are not involved in above literature and can make the theoretical results of this newly fuzzy rough set model more perfect.

Axiomatic characterizations of (𝔾, 𝕆)-fuzzy rough approximation operators via overlap and grouping functions on a complete lattice

Recently, Jiang, H. B., and B. Q. Hu. [2022. “On (O,G)-Fuzzy Rough Sets Based on Overlap and Grouping Functions Over Complete Lattices.” International Journal of Approximate Reasoning 144: 18–50. doi:10.1016/j.ijar.2022.01.012] constructed a -fuzzy rough set model with the logical connectives–a grouping function and an overlap function on a complete lattice, which provided a new constructive approach to fuzzy rough set theory. The axiomatic approach is as important as the constructive approach in rough set theory. In this paper, we continue to study axiomatic characterizations of -fuzzy rough set. Traditionally, the associativity of the logical connectives plays a vital role in the axiomatic research of existing fuzzy rough set models. However, a grouping function and an overlap function lack the associativity. So we explore a novel axiomatic approach to -upper and -lower fuzzy rough approximation operators without associativity. Further, we provide single axioms to characterize -upper and -lower fuzzy rough approximation operators instead of sets of axioms. Finally, we use single axioms to characterize fuzzy rough approximation operators generated by various kinds of fuzzy relations including serial, reflexive, symmetric, -transitive, -transitive fuzzy relations as well as their compositions.

Enriched Fuzzy  -Algebra and Fuzzy RoughApproximation Operators

In this paper, we introduce the concept of enriched fuzzy -algebra and establish its relation to fuzzy rough approximation operators. Firstly, the concept of enriched fuzzy  -algebra is proposed as an extension of existing fuzzy -algebras, and several concrete examples are provided. Particularly, it has been proven that each - algebra can naturally generate an enriched fuzzy  -algebra. Secondly, the methods for constructing enriched fuzzy  -algebra using reflexive fuzzy rough approximation operators and serial rough fuzzy approximation operators are given, respectively. Finally, it is shown that enriched fuzzy  -algebras generated by -algebras must be induced by some rough fuzzy approximation operator.

A Further Study on the Single Axiom Characterization of (∇,▲)-Fuzzy Rough Approximation Operators

A Further Study on the Single Axiom Characterization of (∇,▲)-Fuzzy Rough Approximation Operators

On two new types of fuzzy rough sets via overlap functions and corresponding applications to three-way approximations

Fuzzy rough sets (FRSs) can be served as a useful technique to cope with inconsistent data by providing a pair of lower and upper fuzzy rough approximations of a fuzzy concept. Diversified FRS models have been proposed over the years. Meanwhile, t-norms and t-conorms, as two binary aggregation operators, play a critical part in the construction of fuzzy rough approximation operators. Nevertheless, one of common properties of t-norms and t-conorms is that they all fulfill associativity. Additionally, the associativity plays an important role in discussing properties of FRSs. Therefore, if the binary aggregation function does not satisfy associativity, how to define FRS models and discuss the relevant properties is a topic worthy of further investigation. Overlap functions, as a new kind of not necessary associative binary aggregation operator, have been continuously investigated by many scholars in theory and practical applications. Therefore, based on the advantages of overlap functions, and as the extension models for FRSs, we propose two new types of FRSs via overlap functions and their residual implications (or called IO-implications) in this paper. More specifically, we first give the definition of the 1st type of FRSs and study some essential properties. Subsequently, the definition of the 2nd type of FRSs is presented and the relevant properties are also investigated. Furthermore, the relationship between the 1st type of FRSs and the 2nd type of FRSs is described. That is, in a fuzzy approximation space, the 2nd type of lower approximation is contained in the 1st type of lower approximation, while the 1st type of upper approximation is included in the 2nd type of upper approximation when the overlap function has 1 as the identity element. At the end, a novel three-way decision (3WD) method is presented on an information table by using the new type of FRSs, and its practicality and superiority are verified via a numerical example and a series of experimental analyses.

On (GO, O)-fuzzy rough sets derived from overlap and grouping functions

Rough sets, as a powerful tool to deal with uncertainties and inaccuracies in data analysis, have been continuously concerned and studied by many scholars since it was put forward, especially the research on various rough set models. On the other hand, overlap and grouping functions, as two newly aggregation operators and mathematical model to handle the problems involving in information fusion, have been successfully applied in many real-life problems. In this paper, based on overlap and grouping functions, we propose a new fuzzy rough set model named (GO, O)-fuzzy rough sets and consider its characterizations along with topological properties. Properly speaking, firstly, we utilize QL-operators (and also QL-implications) constructed from overlap and grouping functions and fuzzy negations to define the lower approximation operator in (GO, O)-fuzzy rough set model named GO-lower fuzzy rough approximation operator and the upper approximation operator in (GO, O)-fuzzy rough set model is considered as the O-upper fuzzy rough approximation operator in (IO, O)-fuzzy rough set model proposed by Qiao recently. Secondly, we discuss lots of basic properties of (GO, O)-fuzzy rough sets, especially for the properties of GO-lower fuzzy rough approximation operator. Thirdly, we focus on the relationship between (GO, O)-fuzzy rough sets and concrete fuzzy relations. Finally, we give the topological properties of the upper and lower approximation operators in (GO, O)-fuzzy rough set model.

On (O,G)-fuzzy rough sets based on overlap and grouping functions over complete lattices

In rough set theory, upper and lower approximation operators are two crucial concepts. The study of these two approximation operators in the framework of lattice theory is an important generalization from the mathematical point of view. On the other hand, overlap and grouping functions, as two types of not necessarily associative binary aggregation functions different from the common binary aggregation functions triangular norms and triangular conorms, have not only rich theoretical results but also a wide range of practical applications. Therefore, based on overlap and grouping functions over complete lattices, this paper is devoted to proposing (O,G)-fuzzy rough sets as a further generalization of the notion of rough sets. Firstly, we define a pair of O-upper and G-lower L-fuzzy rough approximation operators and investigate basic properties of them. Then, the characterizations of (O,G)-fuzzy rough approximation operators are discussed by using different kinds of L-fuzzy relations. Meanwhile, we investigate the topological properties of (O,G)-fuzzy rough sets. Furthermore, we show a brief comparison of the (O,G)-fuzzy rough sets with other common rough set models. At the end of this paper, we further propose multigranulation (O,G)-fuzzy rough sets over complete lattices from the viewpoint of multigranulation structure.

Axiomatic characterizations of [formula omitted]-valued rough sets using a single axiom

Fuzzy rough approximation operators are the underlying concepts in fuzzy rough set theory. There are at least two approaches to develop these primary concepts, i.e., the constructive approach and the axiomatic approach. Single axiomatic characterizations of fuzzy rough approximation operators have got tons of attention. In this paper, considering L being a GL-quantale, we will develop the theory of L-valued rough sets with an L-set as the basic universe of defining L-valued rough approximation operators. Adopting the idea of single axiomatic characterizations of fuzzy rough sets, we will present the axiomatic characterizations of L-valued upper and lower rough approximation operators on an L-set with respect to reflexive, symmetric, transitive L-valued relations on an L-set as well as their compositions. Choosing an L-set as the universe will break the rules that adopting Zadeh’s fuzzy sets as the universe. By these results, we will further provide a new framework of axiomatic research of fuzzy rough set theory.

On (IO,O)-fuzzy rough sets based on overlap functions

In the last past years, as a class of continuous binary aggregation functions, there are many scholars keeping a watchful focus on overlap functions for their widely applicability in various actual problems. On the other side, after that rough sets were presented as a formal tool to handle indetermination and inaccuracy in the analysis of data, there arise many works concerning the extension study of rough approximation operators in rough set model. This paper mainly discusses the so-called (IO,O)-fuzzy rough set model obtained through extending the classical conjunction operator in rough approximation operator to overlap functions. Firstly, we investigate a number of essential properties of (IO,O)-fuzzy rough sets, especially for the connections between the obtained new fuzzy rough approximation operators and fuzzy relations. Secondly, we give the topological properties of (IO,O)-fuzzy rough sets. Finally, we show a brief comparison of the (IO,O)-fuzzy rough sets with other common fuzzy rough set models.

Single axioms for (S, T)-fuzzy rough approximation operators with fuzzy product operations

There are two different perspectives to study single axioms for (S, T)-fuzzy rough approximation operators, that is, ordinary fuzzy operations and fuzzy product operations. However, it is too complex and tedious to characterize (S, T)-fuzzy rough approximation operators with ordinary fuzzy operations, such as intersection, union and so on. To remedy these defects, this paper further investigates single axioms for (S, T)-fuzzy rough approximation operators with fuzzy product operations, where fuzzy relation is not limited into either a general fuzzy relation or a symmetric one. Considering a left-continuous t-norm T, we describe T-upper fuzzy rough approximation operators with fuzzy product operations by only one axiom. When t-conorm S is right-continuous and fuzzy negation N is strict, S-lower fuzzy rough approximation operators are characterized with fuzzy product operations by a single axiom.

Belief function of Pythagorean fuzzy rough approximation space and its applications

Rough set theory and evidence theory are two approaches to handle decision making and reduction problems of imprecise and uncertain knowledge. It is motivated that Pythagorean fuzzy set excels at describing the situation where the sum of non-membership degree and membership degree is greater than 1, and may have wider applications than intuitionistic fuzzy set. So in this paper we study probability measure of Pythagorean fuzzy sets and belief structure of Pythagorean fuzzy information systems based on rough set theory, and discuss the reduction of Pythagorean fuzzy information systems. First, we review the properties of Pythagorean fuzzy sets and the upper and lower Pythagorean fuzzy rough approximation operators on the level sets. Then using these properties, probability measure of Pythagorean fuzzy sets are constructed. And the belief and plausibility functions are studied by using the Pythagorean fuzzy rough upper and lower approximation operators. Finally, we apply the belief function to construct an attribute reduction algorithm, and an example is employed to illustrate the feasibility and validity of the algorithm.

Interval-valued pythagorean fuzzy rough approximation operators and its application

By combining interval-valued Pythagorean fuzzy sets with rough sets, the interval-valued Pythagorean fuzzy rough set model is first constructed in this paper. The connections between special interval-valued Pythagorean fuzzy relations and interval-valued Pythagorean fuzzy approximation operators are established subsequently. Then, we study the axiomatic characterizations of interval-valued Pythagorean fuzzy lower and upper approximation operators. Different axiom sets of interval-valued Pythagorean fuzzy set-theoretic operators ensure the existence of different types of interval-valued Pythagorean fuzzy relations producing the same operators. Finally, we give an example to illustrate the practical application of the newly proposed model.

A characterization of novel rough fuzzy sets of information systems and their application in decision making

AbstracVVt A novel concept of α-rough fuzzy sets is proposed as a generalization of rough fuzzy sets in information systems. A pair of rough fuzzy approximation operators is presented based on one type of correlation between attributes, namely, α-correlative relation. In addition, the properties of α-rough fuzzy sets are investigated. Finally, we develop an approach for decision-making problems based on lower and upper α-approximations and provide an actual example.

Labeled fuzzy approximations based on bisimulations

This paper focuses on the labeled fuzzy approximation space, which is considered as a relational structure consisting of a nonempty universal set and some fuzzy relations. To deduce knowledge hidden in the labeled fuzzy approximation space, based on the notion of bisimulations, lower and upper fuzzy rough approximation operators are constructed. Then basic properties of the fuzzy rough approximation operators are investigated. When the largest bisimulation is a trivial identity relation in some cases, the concept of simulations is proposed. Moreover, the lower and upper fuzzy rough relation approximation operators are first proposed and properties of the new operators are examined. Finally, the relationships between two kinds of approximations are discussed.

On (⊙,&)-fuzzy rough sets based on residuated and co-residuated lattices

Rough sets were introduced by Pawlak as a formal tool for dealing with imprecision and uncertainty in data analysis. After that time, varieties of fuzzy generalizations of rough approximations have been investigated in the literature. In particular, generalized fuzzy rough sets determined by a triangular norm have been considered. On the other hand, binary operations & and ⊙ on complete residuated and co-residuated lattice L can be regarded as two more extensive operations than left continuous triangular norms and right continuous triangular conorms, respectively. Therefore, as a further generalization of the notion of rough sets, this paper is devoted to proposing (⊙,&)-fuzzy rough sets based on residuated and co-residuated lattices from both constructive and axiomatic approaches. In the constructive approach, we define a pair of lower and upper L-fuzzy rough approximation operators determined by ⊙ and &, respectively. Meanwhile, various classes of (⊙,&)-fuzzy rough sets are investigated. In the axiomatic approach, the axiomatic characterizations of different (⊙,&)-fuzzy rough approximation operators are studied. Furthermore, the topological properties of (⊙,&)-fuzzy rough sets are discussed, especially from the categorical point of view. At the end of this paper, we give a brief introduction to a new model of fuzzy rough sets and discuss some properties of it.

Using single axioms to characterize (S, T)-intuitionistic fuzzy rough approximation operators

In this paper axiomatic characterizations of relation-based intuitionistic fuzzy rough approximation operators determined by an intuitionistic fuzzy triangular norm T and its dual intuitionistic fuzzy triangular conorm S on \([0, 1]\times [0, 1]\) are proposed. The constructive definitions and properties of S-lower and T-upper intuitionistic fuzzy rough approximation operators are first introduced. Operator-oriented characterizations of (S, T)-intuitionistic fuzzy rough approximation operators are then explored. Different sets of independent axioms for characterizing the essential properties of (S, T)-intuitionistic fuzzy rough approximation operators generated by various intuitionistic fuzzy relations are presented. Finally, it is examined that these sets of axioms can all be replaced by single axioms.

On Simplified Axiomatic Characterizations of (υ σ)-Fuzzy Rough Approximation Operators

The axiomatic approach is more appropriate than constructive approach for studying the algebraic structure of rough sets. In this paper, the more simple axiomatic characterizations of (υ σ)-fuzzy rough approximation operators are explored where υ is a residuated implicator and σis its dual implicator. Firstly, we review the existing independent axiomatic sets to characterize various types of υ-lower and σ-upper fuzzy rough approximation operators. Secondly, we present one-axiom characterizations of (υ σ)-fuzzy rough approximation operators constructed by a serial fuzzy relation on two universes. Furthermore, we show that (υ σ)-fuzzy rough approximation operators, corresponding to reexive, symmetric and T-transitive fuzzy relations, can be presented by only two axioms respectively. We conclude the paper by introducing some potential applications and future works.