On the other hand, generalizations of **rough** sets to the **fuzzy** environment have also been made in the lit- erature [12,16,18,23,27] . By introducing the lower and upper approximations in **fuzzy** set theory, Dubois and Prade [11] , and Chakrabarty [5] formulated **rough** **fuzzy** sets and **fuzzy** **rough** sets. Using **fuzzy** set arguments and **fuzzy** logic **operators**, one can construct a variety of **fuzzy** **rough** set models. For examples, **fuzzy** simi- larity relations (or **fuzzy** T-similarity relations) are used in [26,31] , general **fuzzy** binary relations are used in [24,35,36,38,42] , and **fuzzy** **coverings** are employed in [10,20] to construct **fuzzy** **rough** sets. Alternatively, **fuzzy** **rough** sets can be obtained by extending the basic structure [0, 1] to the abstract algebraic structure. For examples, Radzikowska and Kerre [32] deﬁned L-**fuzzy** **rough** sets by the use of residuated lattice, and Deng [10] constructed a **fuzzy** **rough** set by the use of complete lattice. As for the applications of **fuzzy** **rough** sets, many signiﬁcant works have been done in recent years. Jensen and Shen [17] applied **fuzzy** **rough** set to feature selection. **Based** on an inclusion function of **fuzzy** sets, Hu et al. [14] constructed a type of **generalized** **fuzzy** **rough** set model by which a simple and eﬃcient hybrid attribute reduction algorithm was developed. Mitra [25] integrated **fuzzy** sets and **rough** sets into clustering techniques. By introducing a neighborhood **rough** set model, Hu et al. [15] established neighborhood classiﬁers for classiﬁcation learning. However, few literatures are focused on the model analysis of **fuzzy** **rough** sets **based** on **fuzzy** covering. As a few excep- tions, De Cock et al. [8] deﬁned **fuzzy** **rough** sets **based** on the R-foresets of all objects in a universe of dis- course with respect to (w.r.t.) a **fuzzy** binary relation. When R is a **fuzzy** serial relation, the family of all R- foresets forms a **fuzzy** covering of the universe of discourse. Analogously, Deng [10] examined the issue with **fuzzy** relations induced by a **fuzzy** covering. Li and Ma [20] , on the other hand, constructed two pairs of **fuzzy** **rough** **approximation** **operators** **based** on **fuzzy** **coverings**, the standard min operator T M , and the Kleene-

Show more
21 Read more

In this paper, we have developed a general framework for the study of ðI; T Þ-IVF **rough** sets on two universes of dis- course. By employing an IVF implicator I and an IVF t-norm T , we have introduced ðI ; T Þ-IVF **rough** **approximation** **operators** with reference to an IVF **approximation** space. Basic properties of IVF implicators and construction approach of IVF similarity relations have been presented as the preliminaries for the study of IVF **approximation** **operators**. We have examined basic properties of ðI; T Þ IVF **rough** lower and upper **approximation** **operators**. The connections between special types of IVF relations and IVF **rough** **approximation** **operators** have been established. At the same time, an oper- ator-oriented characterization of IVF **rough** sets has been proposed. We have shown that different axiom sets of I -lower and T -upper IVF set-theoretic **operators** guarantee the existence of different types of IVF relations which produce the same **operators**. We believe that the constructive approaches proposed in this paper will be more useful for practical applications of **rough** set theory in the IVF environment and the axiomatic approach will be helpful in studying the mathematical structures of IVF **rough** set algebras.

Show more
15 Read more

many researchers applied this theory to algebraic structures, such as [4], [5], [20], [6], [1], [13]. In 2016, Wang and Zhan [17] investigated **rough** semigroups and **rough** **fuzzy** semigroups **based** on **fuzzy** ideals. In particular, Davvaz [5] constructed a t-level relation **based** on a **fuzzy** ideal and showed that U(µ, t) is a congruence relation on rings, and the author investigated roughness in rings **based** on **fuzzy** ideals. Zhan et al. [25] investigated roughness in n-ary semigroups **based** on **fuzzy** ideals. In 2017, Zhan et al. [26], [27] inves- tigated roughness in hemirings **based** on strong h-ideals and roughness in non-associative po-semihypergroups **based** on pseudohyperorder relations, respectively. In addition, Zhan et al. [28] also studied **rough** soft n-ary semigroups **based** on a novel congruence relation and corresponding decision making. Wang et al. [18] studied soft **rough** semigroups and corresponding decision making applications. In 2018, Yang and Hu [22] discussed the communication between **fuzzy** information systems by using **fuzzy** covering mappings and **fuzzy** covering-**based** **rough** sets. Shao et al. [15] discussed the connections between two-universe **rough** sets and formal concepts. Wang and Zhan [19] investigated Z-soft **rough** **fuzzy** semigroups and its decision making. Rehman [14] studied **generalized** roughness in LA-semigroups. Shao et al. [16] investigated multi-granlation **rough** filters and **rough** **fuzzy** filters in pseudo-BCI algebras. Prasertpong and Sirip- itukdet [11] discussed **rough** sets induced by **fuzzy** relations approach in semigroups. Zhang and Zhan [32] combined **rough** sets with soft sets, they introduced the concept of **rough** soft BCK-algebras. In 2019, Hussain et al. [9] studied **rough** pythagorean **fuzzy** ideals in semigroups. Prasertpong and Siripitukdet [12] presented **generalized** **rough** sets in **approximation** spaces **based** on portions of successor classes induced by arbitrary binary relations between two universes. Yu et al. [23] investigated decision-theoretic **rough** set in lattice-valued decision information system. Zhan et al. [29] combined intuitionistic **fuzzy** sets with **rough** sets, they introduced intuitionistic **fuzzy** **rough** graphs. Zhang and Zhan [33] explored the relationships among several types of **fuzzy** soft β -covrings **based** **fuzzy** **rough** sets.

Show more
In this paper we review four **fuzzy** extensions of the so- called tight pair of covering **based** **rough** set approxima- tion **operators**. Furthermore, we propose two new ex- tensions of the tight pair: for the first model, we apply the technique of representation by levels to define the **approximation** **operators**, while the second model is an intuitive extension of the crisp **operators**. For the six models, we study which theoretical properties they sat- isfy. Moreover, we discuss interrelationships between the models.

Now, we would like to investigate under which condi- tions we have that all properties are satisfied. Prop- erties which require additional assumptions on **fuzzy** logic connectives are duality and exact approxima- tion. First, we want to check if exact **approximation** property holds under the assumptions of the duality property and vice versa. More precisely, we will con- struct counterexamples that it does not hold. First let us solve the doubt if we may use (qua ∀ , qua ∃ ) = (T, S) for the exact **approximation** property when we take S-implicator instead of R-implicator, which is the assumption for duality property. The answer is negative since provided Lukasiewicz implicator is both S-implicator and R-implicator so we have the same counterexample as before. Further on we take (qua ∀ , qua ∃ ) = (inf, sup). Now, let us show that un- der the assumptions of duality property we do not have the exact **approximation** property. We take de- Morgan triplet: T (x, y) = min(x, y), N (x) = 1 − x and S(x, y) = max(x, y) with I as S-implicator, i.e. I(x, y) = max(1 − x, y). Assume that

Show more
As is well known, there are at least two approaches to the study of **rough** set theory, namely the constructive and axiomatic approaches. In [ 34 ], the notion of **fuzzy** **rough** sets was **generalized** by taking an arbitrary residuated lattice as a basic algebraic structure and several classes of L-**fuzzy** **rough** sets have been considered and their properties have been investigated as well. Hence, it seems that the study of L-**rough** sets in [ 34 ] has been mainly concentrated on the constructive approach. In this paper, more efforts have been made on the axiomatic approach, and the axiomatic characterizations of L- **fuzzy** **rough** sets have been obtained. Moreover, certain kinds of L-**fuzzy** **approximation** **operators** have been characterized by corresponding axioms. The axiomatization of L-**fuzzy** **approximation** **operators** guarantees the existence of corresponding L-**fuzzy** relations which produce the **operators**. Finally, the relationship between L-**fuzzy** **rough** sets and L-**fuzzy** topological spaces has been proposed. We hope that the axiomatic approaches presented in this paper can be used to help us to gain much more insights into the mathematical structures of L-**fuzzy** **approximation** **operators**.

Show more
13 Read more

Abstract—This paper extends and generalizes the approximations of **fuzzy** **rough** sets dealing with **fuzzy** **coverings** of the universe induced by a weak **fuzzy** similarity relation. The weak **fuzzy** similarity relation is considered as a generalization of **fuzzy** similarity relation in representing a more realistic relationship between two objects in which it has weaker symmetric and transitive properties. Since the conditional symmetry in the weak **fuzzy** similarity relation is an asymmetric property, there are two distinct **fuzzy** similarity classes that provide two different **fuzzy** **coverings**. The generalization of **fuzzy** **rough** sets approximations is discussed **based** on two interpretations: object-oriented generalization and class- oriented generalization. More concepts of **generalized** **fuzzy** **rough** set approximations are introduced and defined, representing more alternatives to provide level-2 interval- valued **fuzzy** sets. Moreover, through combining several pairs of proposed approximations of the **generalized** **fuzzy** **rough** sets, it is possible to provide the level-2 type-2 **fuzzy** sets as an extension of the level-2 interval valued **fuzzy** sets. Some properties of the concepts are examined.

Show more
we take into consideration the combination of the vague set theory, **rough** set theory and **fuzzy** set theory. Here, we not only to deﬁne the **rough** approximations of a vague set in **fuzzy** **approximation** space, but also give roughness measure of a vague set in **fuzzy** **approximation** space.
The rest of this paper falls into ﬁve parts. In Section 2 , we review the basic concepts of the vague sets, b-operator and the **fuzzy** **approximation** space, etc. Meanwhile, we quote and prove several new properties of the b-operator. And the concept of b-complement operator is introduced. This establishes a basis for the deﬁnition of the **approximation** **operators** of the vague sets in **fuzzy** **approximation** space and the discussion of their properties partially. In Section 3 , we construct the lower and upper approximations of a vague set in **fuzzy** **approximation** space **based** on the b-operator and b-comple- ment operator. Simultaneously, we prove that the expressions of **approximation** **operators** meet the demand of vague sets. Following, we bring forward the concepts of the k-lower and k-upper approximations of a vague set using the k-cut rela- tion on **fuzzy** **approximation** space. Later on several properties of two types of **approximation** **operators** are studied. In Section 4 , two roughness measures of a vague set for the different **approximation** **operators** are posed, and a correspond- ing property is examined. In Section 5 , an example is given to illustrate the proposed concepts. Finally, we conclude in Section 6 .

Show more
16 Read more

From an application perspective, working with equivalence relations is often a too strong assumption in order to obtain useful results. Therefore, Pawlak’s model has been **generalized** along each of the three mentioned formulations. Indeed, by replacing the equivalence relation in the element-**based** definition by a general binary relation, or equiva- lently by a neighborhood operator, a first generalization is obtained [21, 23, 26, 27]. In this case, the binary relation or the neighborhood operator determines collections of sets which no longer form a partition of U . A second general- ization is obtained when we replace, in the granule-**based** definition, a partition by a covering, i.e., by a collection of non-empty sets such that its union is equal to U [18, 27, 31]. Finally, a third kind of **generalized** models is obtained when in the subsystem-**based** definition we replace the σ -algebra of subsets by a pair of systems: a closure system over U , i.e., a family of subsets of U that contains U and is closed under set-intersection, and its dual system [28]. However, these different generalizations are no longer equivalent, yielding different **generalized** **rough** set models.

Show more
19 Read more

[] studied (ν, T )-**fuzzy** **rough** **approximation** **operators** with respect to a T L-**fuzzy** ideal of the ring. Li and Yin [] **generalized** lower and upper approximations to ν-lower and T -upper **fuzzy** **rough** approximations with respect to T -congruence L-**fuzzy** relation on a semigroup. In order to have a more ﬂexible tool for analysis of an information system, recently, Davvaz has studied the concept of **generalized** **rough** sets called by him T -**rough** sets []. This is another generalization of **rough** sets. In this type of **generalized** **rough** sets, instead of equivalence relations, we require set-valued maps. This technique is use- ful, where it is diﬃcult to ﬁnd an equivalence relation among the elements of the universe set. In this **generalized** **rough** sets, a set-valued map gives rise to lower and upper gener- alized **approximation** **operators**. Ali et al. [] studied some topological properties of the sets which are ﬁxed by these **operators**. They also studied the degree of accuracy (DAG) for **generalized** **rough** sets and some properties of **fuzzy** sets which are induced by DAG. Davvaz, in [], also introduced the concept of set-valued homomorphisms for groups, which is a generalization of an ordinary homomorphism. Yamak et al. [, ] investigated some properties of **rough** approximations with respect to set-valued homomorphisms of rings and modules in the perspective of set-valued homomorphisms. In [], Ali et al. initiated the study of roughness in hemirings with respect to the Pawlak **approximation** space and also with respect to the **generalized** **approximation** space.

Show more
16 Read more

Ó 2007 Elsevier Inc. All rights reserved.
Keywords: Control theory; Decision model; **Fuzzy** attributes; **Fuzzy** **rough** sets; Variable precision **rough** sets
1. Introduction
Data obtained from decision processes constitute a valuable source of information, which can be used in knowledge engineering and for design of control systems. Modelling the human operator’s controlling behav- ior is an important issue. **Based** on process data, the classical control theory tries to create a mathematical model of the human operator. This approach assumes that the human operator can be treated like an addi- tional controller in a closed loop system. In contrast to the classical approach of control theory, a new par- adigm in the form of **fuzzy** set theory was elaborated in the recent decades. This approach turned out to be suitable for modelling the expert’s controlling behavior.

Show more
15 Read more

Many authors who studied axiomatic system of **rough** sets were **based** on set-theoretical methods
[5,7,8,10,11,25–32,34] . However, according to the literature, none of them used characteristic function as
we do in this paper. They either studied axiomatic system for Pawlak’s **rough** sets or studied axiomatic system for **fuzzy** **rough** sets, but not both. In this paper we use the inner and outer product methods to discuss the axiomatic systems of **rough** sets. Furthermore, we can use the same expression to characterize the lower and upper approximations of **rough** sets both in case of crisp sets and in case of **fuzzy** sets. As we know, the outer product method is ﬁrst used in **rough** sets. By using this approach, we obtain a uniﬁed axiomatic characterization of the lower and upper approximations for Pawlak’s **rough** sets and **fuzzy** **rough** sets without any restriction on the cardinality of universe.

Show more
11 Read more

The term approximate reasoning (AR) refers to methods and methodologies that enable reasoning with imprecise inputs to obtain meaningful outputs [?]. AR schemes involving **fuzzy** sets are one of the best known applications of **fuzzy** logic in the wider sense. **Fuzzy** Inference Systems (FIS) have many de- grees of freedom, viz., the underlying **fuzzy** partition of the input and output spaces, the **fuzzy** logic operations employed, the fuzzification and defuzzifi- cation mechanism used, etc. This freedom gives rise to a variety of FIS with differing capabilities. One of the important factors considered while employ- ing an FIS is its **approximation** capability. Many studies have appeared on this topic and due to space constraints, we only refer the readers to the following exceptional review on this topic [?] and the references therein.

Show more
15 Read more

This **Fuzzy** Nearest Neighbour (FNN) method improves the KNN method substantially, but in [4] it was noted that this classifier cannot adequately handle imperfect knowledge. In particular, when every training pattern is far removed from the test object, and hence there are no suitable neighbours, the algorithm is still forced to make clear-cut predictions. This is because the predicted membership degrees to the various decision classes always need to sum up to 1. To address this problem, Sarkar [4] introduced a so-called **fuzzy** **rough** ownership function that, when plugged into the conventional

Show more
4.2 Experimental Setup
The value of K is initialised as 30 and then decremented by 1 each time, resulting in 30 experiments for each dataset. For each choice of parameter K, 10× 10-fold cross-validation is performed. Note that this parameter is es- sential only for FNN. For FNN and FRNN-O, m is set to 2. For the new approaches, the **fuzzy** relation given in equation (10) was cho- sen. In the FRNN approach, the min t-norm and the Kleene-Dienes implicator I (defined by I(x, y) = max(1 − x, y)) were used. The VQNN approach was implemented using Q l = Q (0.1,0.6)

Over the past ten years, **rough** set theory (RST) has become a topic of great interest to re- searchers and has been applied to many domains [9, 4]. Given a dataset with discretized attribute values, it is possible to find a subset (termed a reduct) of the original attributes using **rough** set theory that are the most informative; all other attributes can be removed from the dataset with minimal information loss.

In this section, we first show how the decision tables can be used to explain the concept of granulation by partitions and **fuzzy** **rough** set approximations **based** on a **fuzzy** similarity relation. **Based** on this principle, the initial weights of the FRGNN are then determined; thereby providing a knowledge-**based** network. Such a network is found to be more efficient than other similar types of granular neural network [ 13 , 36 ] and the conventional MLP [ 9 ]. During training, these networks search for a set of connection weights that corresponds to some local minima. Note that there may be a large number of such minima corresponding to various good solutions. Therefore, if we can initially set weights of the network so as to correspond nearby one such solution, the searching space may be reduced and learning thereby becomes faster. Further, the architecture of the network can be made simpler by fixing the number of nodes in the hidden layer **based** on the class information. These are the characteristics that the proposed FRGNN is capable of achieving. The knowledge encoding procedure is described in the next section using the concepts of **fuzzy** **rough** sets and a **fuzzy** similarity relation ( Fig. 3 ).

Show more
20 Read more

We investigate L-**fuzzy** closure **operators** and L-**fuzzy** cotopologies in a complete residuated lattice. Also, we study the categorical relationship between L-**fuzzy** closure spaces and L- **fuzzy** cotopological space. Moreover, there exists the Galois correspondence between L-**fuzzy** cotopological spaces and L-**fuzzy** closure spaces. In particular, we give their examples.

15 Read more

b Department of Computing, School of Informatics, City University London,
Northampton Square, London EC1V 0HB, England, UK.
Abstract
Feature selection refers to the problem of selecting those input features that are most predictive of a given outcome; a problem encountered in many areas such as machine learning, pattern recognition and signal processing. In particular, solution to this has found successful application in tasks that involve datasets containing huge numbers of features (in the order of tens of thousands), which would otherwise be impossible to process further. Recent examples include text processing and web content classification. **Rough** set theory has been used as such a dataset pre-processor with much success, but current methods are inadequate at finding globally minimal reductions, the smallest sets of features possible. This paper proposes a technique that considers this problem from a propositional satisfiability perspective. In this framework, globally minimal subsets can be located and verified.

Show more
59 Read more

In our experiments, we perform to classify each of the five classes (normal, probe, denial of service (DoS), user to super-user, and remote to local) of patterns in the KDDCup’99 data. It is shown that using **fuzzy** **rough** c-means for clustering. The (training and testing) data set contains 1,011 randomly generated points from the five classes. The distribution of attacks in the KDD Cup dataset is extremely unbalanced. Some attacks are represented with only a few examples, e.g. the phf and ftp_write attacks, whereas the smurf and neptune attacks cover millions of records. In general, the distribution of attacks is dominated by probes and denial-of-service (DoS) attacks; the most interesting and dangerous attacks, such as compromises, are grossly under represented[18].

Show more