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  , and Chakrabarty  formulated roughfuzzy sets and fuzzyrough sets. Using fuzzy set arguments and fuzzy logic operators, one can construct a variety of fuzzyrough 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 fuzzycoverings are employed in [10,20] to construct fuzzyrough sets. Alternatively, fuzzyrough sets can be obtained by extending the basic structure [0, 1] to the abstract algebraic structure. For examples, Radzikowska and Kerre  deﬁned L-fuzzyrough sets by the use of residuated lattice, and Deng  constructed a fuzzyrough set by the use of complete lattice. As for the applications of fuzzyrough sets, many signiﬁcant works have been done in recent years. Jensen and Shen  applied fuzzyrough set to feature selection. Based on an inclusion function of fuzzy sets, Hu et al.  constructed a type of generalizedfuzzyrough set model by which a simple and eﬃcient hybrid attribute reduction algorithm was developed. Mitra  integrated fuzzy sets and rough sets into clustering techniques. By introducing a neighborhood rough set model, Hu et al.  established neighborhood classiﬁers for classiﬁcation learning. However, few literatures are focused on the model analysis of fuzzyrough sets based on fuzzy covering. As a few excep- tions, De Cock et al.  deﬁned fuzzyrough 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  examined the issue with fuzzy relations induced by a fuzzy covering. Li and Ma  , on the other hand, constructed two pairs of fuzzyroughapproximationoperatorsbased on fuzzycoverings, the standard min operator T M , and the Kleene-
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 roughapproximationoperators 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 approximationoperators. We have examined basic properties of ðI; T Þ IVF rough lower and upper approximationoperators. The connections between special types of IVF relations and IVF roughapproximationoperators 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.
many researchers applied this theory to algebraic structures, such as , , , , , . In 2016, Wang and Zhan  investigated rough semigroups and roughfuzzy semigroups based on fuzzy ideals. In particular, Davvaz  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.  investigated roughness in n-ary semigroups based on fuzzy ideals. In 2017, Zhan et al. ,  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.  also studied rough soft n-ary semigroups based on a novel congruence relation and corresponding decision making. Wang et al.  studied soft rough semigroups and corresponding decision making applications. In 2018, Yang and Hu  discussed the communication between fuzzy information systems by using fuzzy covering mappings and fuzzy covering-basedrough sets. Shao et al.  discussed the connections between two-universe rough sets and formal concepts. Wang and Zhan  investigated Z-soft roughfuzzy semigroups and its decision making. Rehman  studied generalized roughness in LA-semigroups. Shao et al.  investigated multi-granlation rough filters and roughfuzzy filters in pseudo-BCI algebras. Prasertpong and Sirip- itukdet  discussed rough sets induced by fuzzy relations approach in semigroups. Zhang and Zhan  combined rough sets with soft sets, they introduced the concept of rough soft BCK-algebras. In 2019, Hussain et al.  studied rough pythagorean fuzzy ideals in semigroups. Prasertpong and Siripitukdet  presented generalizedrough sets in approximation spaces based on portions of successor classes induced by arbitrary binary relations between two universes. Yu et al.  investigated decision-theoretic rough set in lattice-valued decision information system. Zhan et al.  combined intuitionistic fuzzy sets with rough sets, they introduced intuitionistic fuzzyrough graphs. Zhang and Zhan  explored the relationships among several types of fuzzy soft β -covrings basedfuzzyrough sets.
In this paper we review four fuzzy extensions of the so- called tight pair of covering basedrough 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 approximationoperators, 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
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 fuzzyrough sets was generalized by taking an arbitrary residuated lattice as a basic algebraic structure and several classes of L-fuzzyrough 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- fuzzyrough sets have been obtained. Moreover, certain kinds of L-fuzzyapproximationoperators have been characterized by corresponding axioms. The axiomatization of L-fuzzyapproximationoperators guarantees the existence of corresponding L-fuzzy relations which produce the operators. Finally, the relationship between L-fuzzyrough 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-fuzzyapproximationoperators.
Abstract—This paper extends and generalizes the approximations of fuzzyrough sets dealing with fuzzycoverings 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 fuzzycoverings. The generalization of fuzzyrough sets approximations is discussed based on two interpretations: object-oriented generalization and class- oriented generalization. More concepts of generalizedfuzzyrough 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 generalizedfuzzyrough 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.
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 fuzzyapproximation space, but also give roughness measure of a vague set in fuzzyapproximation 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 fuzzyapproximation 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 approximationoperators of the vague sets in fuzzyapproximation space and the discussion of their properties partially. In Section 3 , we construct the lower and upper approximations of a vague set in fuzzyapproximation space based on the b-operator and b-comple- ment operator. Simultaneously, we prove that the expressions of approximationoperators 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 fuzzyapproximation space. Later on several properties of two types of approximationoperators are studied. In Section 4 , two roughness measures of a vague set for the different approximationoperators 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 .
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 . However, these different generalizations are no longer equivalent, yielding different generalizedrough set models.
 studied (ν, T )-fuzzyroughapproximationoperators 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 fuzzyrough 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 generalizedrough sets called by him T -rough sets . This is another generalization of rough sets. In this type of generalizedrough 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 generalizedrough sets, a set-valued map gives rise to lower and upper gener- alized approximationoperators. 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 generalizedrough 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 generalizedapproximation space.
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Keywords: Control theory; Decision model; Fuzzy attributes; Fuzzyrough sets; Variable precision rough sets
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.
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 fuzzyrough 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 fuzzyrough sets without any restriction on the cardinality of universe.
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.
This Fuzzy Nearest Neighbour (FNN) method improves the KNN method substantially, but in  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  introduced a so-called fuzzyrough ownership function that, when plugged into the conventional
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 fuzzyrough 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 fuzzyrough sets and a fuzzy similarity relation ( Fig. 3 ).
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.
b Department of Computing, School of Informatics, City University London,
Northampton Square, London EC1V 0HB, England, UK.
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.
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 fuzzyrough 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.