axiomatics for fuzzy rough sets abstract a fuzzy t rough set consists of a set x and a t similarity relation r on x where t is a lower semi continuous triangular norm we generalize the farinas prade definition for the upper approximation operator a i x i x of a fuzzy t rough set x r given originally for the special case t min to the case of arbitrary t we propose a new definition for the lower approximation operator a i x i x of x r our definition satisfies the two important identities a a a and a a a as well as a number of other interesting properties we provide axiomatics to fully characterize those upper and lower approximations

on fuzzy approximation operators in attribute reduction with fuzzy rough sets generally speaking there are four fuzzy approximation operators defined on a general triangular norm t norm framework in fuzzy rough sets different types of t norms specify various approximation operators one issue whether and how the different fuzzy approximation operators affect the result of attribute reduction is then arisen this paper addresses this issue from the theoretical viewpoint by reviewing attribute reduction with fuzzy rough sets and then describing and proving some theorems which demonstrate the effects of the fuzzy approximation operators on the results of attribute reduction first we review some notions of attribute reduction with fuzzy rough sets such as positive region dependency degree and attribute reduction we then present and prove some theorems which describe how and to what degree fuzzy approximation operators impact the performance of attribute reduction finally we report some experimental simulation results which demonstrate the effectiveness and correctness of the theoretical contributions one main contribution in this paper is that we have described and proven that each attribute reduction obtained using one type of fuzzy lower approximation operator always contains one reduction obtained using the other type of fuzzy lower approximation operator

rough approximation operators on two universes of discourse and their fuzzy extensions based on a crisp binary relation between two universes of discourse one crisp covering and three crisp binary relations on the single universe of discourse are induced in this paper based on these induced notions four pairs of rough approximation operators are formulated so that the approximating sets and the approximated sets are on the same universe of discourse furthermore basic properties of the new approximation operators are investigated and the relationships among the operators are also examined subsequently conditions under which some or all of these approximation operators are equivalent are obtained on the other hand the generalizations of these rough approximation operators are also made in a fuzzy approximation space via a fuzzy implicator i and a triangular norm t consequently four pairs of fuzzy rough approximation operators are constructed basic properties and comparison of the fuzzy rough approximation operators are discussed if i is an r implicator with respect to a t norm t then it is shown that three pairs of the fuzzy rough approximation operators are identical iff the induced fuzzy covering is a fuzzy t partition

set approximations in fuzzy formal concept analysis formal concept analysis and rough set theory are two important tools in knowledge representation and knowledge discovery in relational information systems the purpose of this paper is to study rough set approximations within formal concept analysis in fuzzy environment properties of existent fuzzy concept lattices derived from an adjoint pair of operations are first reviewed and examined based on both lattice theoretic and fuzzy set theoretic operators two new pairs of rough fuzzy set approximations within fuzzy formal contexts are then defined finally properties of the rough fuzzy set approximation operators are presented in detail

neighborhood systems and approximate retrieval an approximate retrieval model is proposed based on the notion of neighborhood systems the knowledge used in the model consists of an information table in which each object is represented by its values on a finite set of attributes and neighborhood systems on attribute values which provide semantic similarity or closeness of different values an information table can be used for exact retrieval with the introduction of neighborhood systems to information tables one is able to perform approximate retrieval operations on neighborhood systems are introduced based on power algebras an ordering relation representing the information of a neighborhood system is suggested and examined approximate retrieval is carried out by the relaxation of the original query using neighborhood systems and the combination of intermediate results using neighborhood system operations the final retrieval results are presented according to the proposed ordering relation in contrast to many existing systems a main advantage of the proposed model is that the retrieval results are a non linear ordering of objects

a comprehensive study of implicator conjunctor based and noise tolerant fuzzy rough sets definitions properties and robustness analysis abstract both rough and fuzzy set theories offer interesting tools for dealing with imperfect data while the former allows us to work with uncertain and incomplete information the latter provides a formal setting for vague concepts the two theories are highly compatible and since the late 1980s many researchers have studied their hybridization in this paper we critically evaluate most relevant fuzzy rough set models proposed in the literature to this end we establish a formally correct and unified mathematical framework for them both implicator conjunctor based definitions and noise tolerant models are studied we evaluate these models on two different fronts firstly we discuss which properties of the original rough set model can be maintained and secondly we examine how robust they are against both class and attribute noise by highlighting the benefits and drawbacks of the different fuzzy rough set models this study appears a necessary first step to propose and develop new models in future research

information granularity in fuzzy binary grc model zadeh s seminal work in theory of fuzzy information granulation in human reasoning is inspired by the ways in which humans granulate information and reason with it this has led to an interesting research topic granular computing grc although many excellent research contributions have been made there remains an important issue to be addressed what is the essence of measuring a fuzzy information granularity of a fuzzy granular structure what is needed to answer this question is an axiomatic constraint with a partial order relation that is defined in terms of the size of each fuzzy information granule from a fuzzy binary granular structure this viewpoint is demonstrated for fuzzy binary granular structure which is called the binary grc model by lin we study this viewpoint from from five aspects in this study which are fuzzy binary granular structure operators partial order relations measures for fuzzy information granularity an axiomatic approach to fuzzy information granularity and fuzzy information entropies

granular variable precision l fuzzy rough sets based on residuated lattices abstract firstly as for the beginning of this paper we introduce the granular variable precision l fuzzy rough sets based on residuated lattices with arbitrary l fuzzy relations and then the equivalent expressions for the lower and upper approximation operators are given with l fuzzy implications and l fuzzy coimplications on arbitrary l fuzzy relations which can calculate the lower and upper approximation operators efficiently finally some basic properties of the granular variable precision l fuzzy rough sets on residuated lattices are investigated and the conclusions on the granular variable precision fuzzy rough sets are generalized into the situations of the truth values set becoming residuated lattices

type 2 fuzzy rough sets based on extended t norms a type 2 fuzzy rough set based on extended t norms is investigated from constructive and axiomatic approaches an approximation algorithm of type 2 fuzzy rough sets is listed based on cut sets weak type 2 fuzzy topologies are proposed to investigate the topological properties of type 2 fuzzy rough sets an interval valued type 2 fuzzy rough set is proposed and investigated this paper is devoted to investigate type 2 fuzzy rough sets based on extended t norms with respect to type 2 fuzzy relations with convex normal fuzzy truth values type 2 fuzzy rough sets are proposed on two finite universes of discourse and characterized from both constructive and axiomatic approaches in the constructive approach different classes of type 2 fuzzy rough sets are investigated an approximate algorithm of type 2 fuzzy rough sets based on cut sets is given with numerical examples in the axiomatic approach the axiomatic characterizations of type 2 fuzzy rough approximation operators are studied moreover the topological properties of type 2 fuzzy rough sets are discussed in the end this paper introduces interval valued type 2 fuzzy rough sets and discusses their properties

roughness of a fuzzy set an integration between the theories of fuzzy sets and rough sets has been attempted by providing a measure of roughness of a fuzzy set several properties of this new measure are established some of the possible applications for handling uncertainties in the field of pattern recognition are mentioned

generalized discrete timed automata decidable approximations for safety verification we consider generalized discrete timed automata with general linear relations over clocks and parameterized constants as clock constraints and with parameterized durations we look at three approximation techniques i e the r reset bounded approximation the b bounded approximation and the b r crossing bounded approximation and derive automata theoretic characterizations of the binary reachability under these approximations the characterizations allow us to show that the safety analysis problem is decidable for generalized discrete timed automata with unit durations and for deterministic generalized discrete timed automata with parameterized durations an example specification written in astral is used to run a number of experiments using one of the approximation techniques

three way decision on two universes abstract rough set models on two universes are important generalizations of pawlak s classical model in most of the two universe models the upper and lower approximations are constructed based on inclusion relation we generalize inclusion relation to the general evaluation function and define the models of three way decision on two universes as an important class of evaluation functions subsethood measures are considered we compare our models with other five existing two universe models in rough set theory and point out that the model of three way decision on two universe unifies the five two universe models besides properties of the two universe model of three way decision are also given more importantly we propose an approach to computing the pair of thresholds and our approach is based on the maximum value of the accuracy measure with respect to tri partitions of a universe induced by all thresholds

a high order laminated plate theory with improved in plane responses in order to improve the accuracy of the in plane responses of the shear deformable laminated composite plate theories a new high order laminated plate theory was developed based upon reissner s new mixed variational principle int j num meth eng 20 1366 1984 to this end a zig zag shaped c0 function and legendre polynomials were introduced into the approximate in plane displacement distributions across the plate thickness the accuracy of the present theory was examined by applying it to the cylindrical bending problem of laminated plates which had been solved exactly by pagano j comp mat 3 398 1969 a comparison with the exact solutions obtained for several symmetric and asymmetric cross ply laminates indicates that the present theory accurately estimates in plane responses even for small span to thickness ratios

relational interpretations of neighborhood operators and rough set approximation operators this paper presents a framework for the formulation interpretation and comparison of neighborhood systems and rough set approximations using the more familiar notion of binary relations a special class of neighborhood systems called 1 neighborhood systems is introduced three extensions of pawlak approximation operators are analyzed properties of neighborhood and approximation operators are studied and their connections are examined

constructive and axiomatic approaches of fuzzy approximation operators this paper presents a general framework for the study of rough set approximation operators in fuzzy environment in which both constructive and axiomatic approaches are used in constructive approach a pair of lower and upper generalized fuzzy rough and rough fuzzy respectively approximation operators is first defined the representations of both fuzzy rough approximation operators and rough fuzzy approximation operators are then presented the connections between fuzzy and crisp respectively relations and fuzzy rough and rough fuzzy respectively approximation operators are further established in axiomatic approach various classes of fuzzy approximation operators are characterized by different sets of axioms the minimal axiom sets of fuzzy approximation operators guarantee the existence of certain types of fuzzy or crisp relations producing the same operators

fuzzy granular structure distance a fuzzy granular structure refers to a mathematical structure of the collection of fuzzy information granules granulated from a dataset while a fuzzy information granularity is used to measure its uncertainty however the existing forms of fuzzy information granularity have two limitations one is that when the fuzzy information granularity of one fuzzy granular structure equals that of the other one can say that these two fuzzy granular structures possess the same uncertainty but these two fuzzy granular structures may be not equivalent to each other the other limitation is that existing axiomatic approaches to fuzzy information granularity are still not complete under which when the partial order relation among fuzzy granular structures cannot be found their coarseness fineness relationships will not be revealed to address these issues a so called fuzzy granular structure distance is proposed in this study which can well discriminate the difference between any two fuzzy granular structures besides this advantage the fuzzy granular structure distance has another important benefit it can be used to establish a generalized axiomatic constraint for fuzzy information granularity by using the axiomatic constraint the coarseness fineness of any two fuzzy granular structures can be distinguished in addition through taking the fuzzy granular structure distances of a fuzzy granular structure to the finest one and the coarsest one into account we also can build a bridge between fuzzy information granularity and fuzzy information entropy the applicable analysis on 12 real world datasets shows that the fuzzy granular structure distance and the generalized fuzzy information granularity have much better performance than existing methods

granular variable precision fuzzy rough sets with general fuzzy relations the variable precision fuzzy rough sets were proposed to remedy the defects of preexisting fuzzy rough set models however the variable precision fuzzy rough sets were only defined and investigated on fuzzy similarity relations in this paper the granular variable precision fuzzy rough sets with general fuzzy relations are proposed on arbitrary fuzzy relations the equivalent expressions of the approximation operators are given with fuzzy co implications on arbitrary fuzzy relations which can calculate efficiently the approximation operators the granular variable precision fuzzy rough sets are characterized from the constructive approach which are investigated on different fuzzy relations the conclusions on the variable precision fuzzy rough sets are generalized into the granular variable precision fuzzy rough sets

generalized fuzzy rough sets this paper presents a general framework for the study of fuzzy rough sets in which both constructive and axiomatic approaches are used in constructive approach a pair of lower and upper generalized approximation operators is defined the connections between fuzzy relations and fuzzy rough approximation operators are examined in axiomatic approach various classes of fuzzy rough approximation operators are characterized by different sets of axioms axioms of fuzzy approximation operators guarantee the existence of certain types of fuzzy relations producing the same operators

topological characterizations of covering for special covering based upper approximation operators the relationships among properties of covering approximation operators and their corresponding coverings have attracted intensive research in recent years in particular those among topological properties have drawn special attention because of their important applications in rough set theory in this paper we give topological characterizations of covering c for covering based upper approximation operators fh sh th and rh to be closure operators we also give intuitive characterizations of covering c and describe covering based approximation space u c as certain types of information exchange systems when sh or rh is a closure operator by applying our new characterizations we give inequalities about the relationship between the number of members in c and the number of elements in u and discuss relationships among conditions for different covering based upper approximation operators to be closure operators to the best of our knowledge it is the first time that such characterizations descriptions inequalities and discussions are systematically considered in the literature of rough set theory furthermore in this paper we also give several characterizations of unary coverings an important type of coverings in studying relationships among basic concepts in covering based rough sets by the relationships among different types of covering base approximation operators

on fuzzy rough sets based on tolerance relations dubois and prade 1990 1 introduced the notion of fuzzy rough sets as a fuzzy generalization of rough sets which was originally proposed by pawlak 1982 8 later radzikowska and kerre introduced the so called i t fuzzy rough sets where i is an implication and t is a triangular norm in the present paper by using a pair of implications i j we define the so called i j fuzzy rough sets which generalize the concept of fuzzy rough sets in the sense of radzikowska and kerre and that of mi and zhang basic properties of i j fuzzy rough sets are investigated in detail

some mathematical structures for computational information abstract this paper is about the basic underlying mathematical structures of various types of imprecise information set theory and probability measures are basic ingredients in extracting some specific type of information and in reasoning with uncertain knowledge but in this era of information technology it is desirable to consider also more complex types of information for example information coming from human perception see e g l a zadeh ieee trans circuits and systems 45 1 1999 105 119 as a first approximation the classical ingredients could be generalized to adequately model new types of information as such we will describe in this paper four types of non classical sets namely random sets rough sets conditional sets and fuzzy sets as well as their associated non additive measures

reasoning within expressive fuzzy rough description logics it is generally accepted that the management of imprecision and vagueness will yield more intelligent and realistic knowledge based applications description logics dls are suitable well known logics for managing structured knowledge that have gained considerable attention the last decade the current research progress and the existing problems of uncertain or imprecise knowledge representation and reasoning in dls are analyzed in this paper an integration between the theories of fuzzy dls and rough dls has been attempted by providing fuzzy rough dls based on fuzzy rough set theory the syntax semantics and properties of fuzzy rough dls are given it is proved that the satisfiability subsumption entailment and abox consistency reasoning in fuzzy rough dls may be reduced to the abox consistency reasoning in the corresponding fuzzy dls

some properties of l fuzzy approximation spaces based on bounded integral residuated lattices in this paper we present a general framework for the study of l fuzzy rough sets based on bounded integral residuated lattices we introduce and discuss the notions of l fuzzy topological spaces left right lower and left right upper l fuzzy approximation operators based on a complete bounded integral residuated lattice investigate union intersection and composition operations of l fuzzy approximation spaces and study some properties of the l fuzzy topologies generated by these l fuzzy rough approximation operators and the l relations induced by an l fuzzy topology

two fuzzy covering rough set models and their generalizations over fuzzy lattices by introducing the new concepts of fuzzy covering and fuzzy neighborhood we define two new types of fuzzy covering rough set models which can be regarded as bridges linking covering rough set theory and fuzzy rough set theory we show the properties of the two models and reveal the relationships between the two models and some others moreover we present the matrix representations of the newly defined lower and upper approximation operators so that the calculation of lower and upper approximations of subsets can be converted into operations on matrices finally we generalize the models and their matrix representations to l fuzzy covering rough sets which are defined over fuzzy lattices

generalized rough sets based on relations rough set theory has been proposed by pawlak as a tool for dealing with the vagueness and granularity in information systems the core concepts of classical rough sets are lower and upper approximations based on equivalence relations this paper studies arbitrary binary relation based generalized rough sets in this setting a binary relation can generate a lower approximation operation and an upper approximation operation but some of common properties of classical lower and upper approximation operations are no longer satisfied we investigate conditions for a relation under which these properties hold for the relation based lower and upper approximation operations

quick attribute reduction with generalized indiscernibility models we present a generalized indiscernibility reduction model girm and a concept of the granular structure in girm we prove that girm is compatible with three typical reduction models we present a generalized attribute reduction algorithm and a generalized positive region computing algorithm based on girm we present acceleration policies on two generalized algorithms and fast positive region computing approaches for three typical reduction models the efficiency of attribute reduction is one of the important challenges being faced in the field of big data processing although many quick attribute reduction algorithms have been proposed they are tightly coupled with their corresponding indiscernibility relations and it is difficult to extend specific acceleration policies to other reduction models in this paper we propose a generalized indiscernibility reduction model girm and a concept of the granular structure in girm which is a quantitative measurement induced from multiple indiscernibility relations and which can be used to represent the computation cost of varied models then we prove that our girm is compatible with three typical reduction models based on the proposed girm we present a generalized attribute reduction algorithm and a generalized positive region computing algorithm we perform a quantitative analysis of the computation complexities of two algorithms using the granular structure for the generalized attribute reduction we present systematic acceleration policies that can reduce the computational domain and optimize the computation of the positive region based on the granular structure we propose acceleration policies for the computation of the generalized positive region and we also propose fast positive region computation approaches for three typical reduction models experimental results for various datasets prove the efficiency of our acceleration policies in those three typical reduction models

catoptrical rough set model on two universes using granule based definition and its variable precision extensions rough set models on two universes are valuable generalizations of classic rough set model these models can be applied into formal concept analysis and decision making system this paper focuses on catoptrical approach for rough set models over two universes using granule based definition in this paper we propose a new catoptrical rough set model over dual universes the properties of the proposed model are discussed consequently a variable precision extension of the model is presented some properties of the extended model are obtained finally we investigate a further extension of the proposed model i e variable precision catoptrical rough set model with two parameters

formal concept analysis based on the topology for attributes of a formal context formal concept analysis fca concerns the hierarchical structures induced by a binary relation between a pair of sets it is widely applied in data analysis information retrieval and knowledge discovery within the framework of fca computing all formal concepts is the main challenge due to its exponential complexity it has be proved that all formal concepts is a closure system on objects hence the closure operator on objects are general used to generate all formal concepts of a formal context in the lexicographic tree approach a base on objects is used to generate extensions of all formal concepts and construct the formal concept lattice inspired from the approach we concentrate a base on attributes and generate intensions of all formal concepts in this paper to this end we firstly analyze an example the lexicographic tree approach is used to generate all formal concepts from the time complexity point of view we present that it is not trivial to generate the base on attributes by a simple symmetrical way from the methods based on objects then we deduce a set valued mapping from attributes to the power set of attributes in a formal context and define a binary relation on attributes by the set valued mapping using the binary relation on attributes we construct an approximation space and a topology for attributes respectively and obtain a base for the topology we prove that intensions of all formal concepts are included in the topology for attributes this means that the base can be used to generate intensions of all formal concepts of the formal context and construct the formal concept lattice more general our results represent relationships and the hierarchical structures among attributes of the formal context we present some typical applications in which the topology for attributes and the base for the topology are applied for association rules discovery from a formal context and linguistic concept analysis

set valued ordered information systems set valued ordered information systems can be classified into two categories disjunctive and conjunctive systems through introducing two new dominance relations to set valued information systems we first introduce the conjunctive disjunctive set valued ordered information systems and develop an approach to queuing problems for objects in presence of multiple attributes and criteria then we present a dominance based rough set approach for these two types of set valued ordered information systems which is mainly based on substitution of the indiscernibility relation by a dominance relation through the lower upper approximation of a decision some certain possible decision rules from a so called set valued ordered decision table can be extracted finally we present attribute reduction also called criteria reduction in ordered information systems approaches to these two types of ordered information systems and ordered decision tables which can be used to simplify a set valued ordered information system and find decision rules directly from a set valued ordered decision table these criteria reduction approaches can eliminate those criteria that are not essential from the viewpoint of the ordering of objects or decision rules

on a ktb modal fuzzy logic abstract an extension of the classical ktb modal logic to the fuzzy case is considered a new notion of similarity is introduced and a fuzzy logic based on this relation is proposed also the validity of formulas is defined by introducing a notion of logical consequence then the decision procedure of this logic is described in the tableau method

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

approximation operators on complete completely distributive lattices rough set a tool for data mining deals with the vagueness and granularity in information systems in 2006 chen et al initiated the study of rough approximations on a complete completely distributive lattice ccd lattice for short and brought generalizations of rough sets into a unified framework in this paper we discuss the approximation operators on a ccd lattice based on the concept of neighborhood three kinds of upper approximation operators and a kind of lower approximation operator are constructed basic properties of lower and upper approximation operators are examined furthermore the relationships among these operators are analyzed

the model of fuzzy variable precision rough sets the fuzzy rough set frs model has been introduced to handle databases with real values however frs was sensitive to misclassification and perturbation here misclassification means error or missing values in classification and perturbation means small changes of numerical data the variable precision rough sets vprss model was introduced to handle databases with misclassification however it could not effectively handle the real valued datasets now it is valuable from theoretical and practical aspects to combine frs and vprs so that a powerful tool which not only can handle numerical data but also is less sensitive to misclassification and perturbation can be developed in this paper we set up a model named fuzzy vprss fvprss by combining frs and vprs with the goal of making frs a special case first we study the knowledge representation ways of frs and vprs and then propose the set approximation operators of fvprs second we employ the discernibility matrix approach to investigate the structure of attribute reductions in fvprs and develop an algorithm to find all reductions third in order to overcome the np complete problem of finding all reductions we develop some fast heuristic algorithms to obtain one near optimal attribute reduction finally we compare fvprs with rs frs and several flexible rs based approaches with respect to misclassification and perturbation the experimental comparisons show the feasibility and effectiveness of fvprs

constructive and algebraic methods of the theory of rough sets this paper reviews and compares constructive and algebraic approaches in the study of rough sets in the constructive approach one starts from a binary relation and defines a pair of lower and upper approximation operators using the binary relation different classes of rough set algebras are obtained from different types of binary relations in the algebraic approach one defines a pair of dual approximation operators and states axioms that must be satisfied by the operators various classes of rough set algebras are characterized by different sets of axioms axioms of approximation operators guarantee the existence of certain types of binary relations producing the same operators

fuzzy rough approximations for set valued data rough set theory is one of important tools of soft computing and rough approximations are the essential elements in rough set models however the existing fuzzy rough set model for set valued data which is directly constructed based on a kind of similarity relation fail to explicitly define fuzzy rough approximations to solve this issue in this paper we propose two types of fuzzy rough approximations and define two corresponding relative positive region reducts furthermore two discernibility matrices and two discernibility functions are introduced to acquire these new proposed reducts and the relationships among the new reducts and the existing reducts are also be provided theoretical analyses demonstrate that the new types of reducts have less redundancy and are more diverse no lower number of reducts than those obtained by means of the existing matrices and experimental results illustrate the new reducts found by our methods outperform those obtained by existing method

using one axiom to characterize l fuzzy rough approximation operators based on residuated lattices abstract axiomatic characterization of approximation operators plays an important role in the study of rough set theory different axiom sets of abstract operators can illustrate different classes of rough set systems in this paper we are devoted to searching for a single axiom to characterize l fuzzy rough approximation operators based on residuated lattices axioms of l fuzzy set theoretic operators make sure of the existence of certain types of l fuzzy relations which produce the same operators we demonstrate that the lower upper l fuzzy rough approximation operators generated by a generalized l fuzzy relation can be characterized by only one axiom furthermore we also use one axiom to characterize l fuzzy rough approximation operators produced by the l fuzzy serial reflexive symmetric and t transitive relations as well as any of their compositions

three types of fuzzy covering based rough set models abstract in this paper three new types of fuzzy covering based rough set models are proposed first the membership degree based fuzzy covering rough set is defined in the fuzzy covering approximation space by separately comparing the membership degree of each object belonging to the approximated fuzzy set with those of the component fuzzy sets in the fuzzy covering for the same object second the membership function based fuzzy covering rough set is obtained by globally comparing the membership function of the approximated fuzzy set with those of the component fuzzy sets in the fuzzy covering finally the membership function based fuzzy covering rough set model is generalized to the fuzzy covering approximation space the properties of these new rough set models are discussed furthermore the relationship between these new rough sets and the relationship between the new rough set and pawlak rough set are described

fuzzy preference based rough sets preference analysis is an important task in multi criteria decision making the rough set theory has been successfully extended to deal with preference analysis by replacing equivalence relations with dominance relations the existing studies involving preference relations cannot capture the uncertainty presented in numerical and fuzzy criteria in this paper we introduce a method to extract fuzzy preference relations from samples characterized by numerical criteria fuzzy preference relations are incorporated into a fuzzy rough set model which leads to a fuzzy preference based rough set model the measure of attribute dependency of the pawlak s rough set model is generalized to compute the relevance between criteria and decisions the definitions of upward dependency downward dependency and global dependency are introduced algorithms for computing attribute dependency and reducts are proposed and experimentally evaluated by using two publicly available data sets

a novel algorithm for finding reducts with fuzzy rough sets attribute reduction is one of the most meaningful research topics in the existing fuzzy rough sets and the approach of discernibility matrix is the mathematical foundation of computing reducts when computing reducts with discernibility matrix we find that only the minimal elements in a discernibility matrix are sufficient and necessary this fact motivates our idea in this paper to develop a novel algorithm to find reducts that are based on the minimal elements in the discernibility matrix relative discernibility relations of conditional attributes are defined and minimal elements in the fuzzy discernibility matrix are characterized by the relative discernibility relations then the algorithms to compute minimal elements and reducts are developed in the framework of fuzzy rough sets experimental comparison shows that the proposed algorithms are effective

rough set approach to incomplete numerical data rough set theory has been applied successfully in many fields however classical rough set model can only deal with complete and symbolic data sets some researchers have proposed extended rough set models to handle incomplete data while others proposed extensions of classical rough set models to deal with numerical data however there exist many data which contain numerical attributes and missing values simultaneously we propose in this paper an extended rough set model i e tolerance fuzzy rough set model to deal with this type of data characterized with numerical attributes and missing values that is incomplete numerical data discernibility matrices and discernibility functions for incomplete numerical information systems and incomplete numerical decision systems are defined to compute reducts or relative reducts meanwhile the relationship between the proposed tolerance fuzzy rough set model and the tolerance rough set model is also examined it is shown that the tolerance fuzzy rough set model is an extension of the tolerance rough set model finally uncertainty measurement is also investigated it is suggested that the proposed tolerance fuzzy rough set model provide an optional approach to incomplete numerical data

fuzzy rough set theory for the interval valued fuzzy information systems the concept of the rough set was originally proposed by pawlak as a formal tool for modelling and processing incomplete information in information systems then in 1990 dubois and prade first introduced the rough fuzzy sets and fuzzy rough sets as a fuzzy extension of the rough sets the aim of this paper is to present a new extension of the rough set theory by means of integrating the classical pawlak rough set theory with the interval valued fuzzy set theory i e the interval valued fuzzy rough set model is presented based on the interval valued fuzzy information systems which is defined in this paper by a binary interval valued fuzzy relations r f i uxu on the universe u several properties of the rough set model are given and the relationships of this model and the others rough set models are also examined furthermore we also discuss the knowledge reduction of the classical pawlak information systems and the interval valued fuzzy information systems respectively finally the knowledge reduction theorems of the interval valued fuzzy information systems are built

a fuzzy covering based rough set model and its generalization over fuzzy lattice in this paper we study a new type of fuzzy covering based rough set model by introducing the notion of fuzzy minimal description we mainly address the following issues in this paper first we present the definition of fuzzy minimal description and study its properties then we define a novel type of fuzzy covering based rough set model and investigate the properties of this model furthermore the axiomizations and matrix representations of the fuzzy covering lower and fuzzy covering upper approximations are the vital problems we investigate in this paper moreover we explore the conditions under which two fuzzy coverings generate the same fuzzy covering lower or fuzzy covering upper approximations finally we generalize the model to l fuzzy covering based rough set which is defined over fuzzy lattices similar to the fuzzy covering based rough set model we also address the issues mentioned above to the l fuzzy covering based rough set

fuzzy rough attribute reduction via mutual information with an application to cancer classification establishing a classification model for cancer recognition based on dna microarrays is useful for cancer diagnosis feature selection is a key step to perform cancer classification with dna microarrays for there is a large number of genes from which to predict classes and a relatively small number of samples automatic methods must be developed for extracting relevant genes which are essential for classification this paper proposes a novel approach for reducing data redundancy based on fuzzy rough set theory and information theory a mutual information based algorithm for attribute reduction is suggested the method is applied to the problem of gene selection for cancer classification experimental results show that the algorithm is more effective than conventional rough sets based approaches

a rst a generalization of rough set theory the paper presents a transition from the crisp rough set theory to a fuzzy one called alpha rough set theory or in short a rst all basic concepts or rough set theory are extended i e information system indiscernibility dependency reduction core definability approximations and boundary the resulted theory takes into account fuzzy data and allows the approximation of fuzzy concepts besides the control of knowledge granularity is natural in a rst which is based on a parameterized indiscernibility relation a rst is developed to recognize non deterministic relationships using notions as a dependency a reduct and so forth on the other hand we introduce a notion of relative dependency as an alternative of the absolute definibility presented in rough set theory the extension a rst leads naturally to the new concept of alpha rough sets which represents sets with fuzzy non empty boundaries 2000 elsevier science inc all rights reserved

rough set approach to incomplete information systems abstract in the paper we present rough set approach to reasoning in incomplete information systems we propose reduction of knowledge that eliminates only that information which is not essential from the point of view of classification or decision making in our approach we make only one assumption about unknown values the real value of a missing attribute is one from the attribute domain however we do not assume which one we show how to find decision rules directly from such an incomplete decision table which are as little non deterministic as possible and have minimal number of conditions

attributes reduction using fuzzy rough sets fuzzy rough sets are the generalization of traditional rough sets to deal with both fuzziness and vagueness in data the existing researches on fuzzy rough sets are mainly concentrated on the construction of approximation operators less effort has been put on the attributes reduction of databases with fuzzy rough sets this paper mainly focuses on the attributes reduction with fuzzy rough sets after analyzing the previous works on attributes reduction with fuzzy rough sets we introduce formal concepts of attributes reduction with fuzzy rough sets and completely study the structure of attributes reduction an algorithm using discernibility matrix to compute all the attributes reductions is developed based on these lines of thought we set up a solid mathematical foundation for attributes reduction with fuzzy rough sets the experimental results show that the idea in this paper is feasible and valid

generalized rough sets over fuzzy lattices this paper studies generalized rough sets over fuzzy lattices through both the constructive and axiomatic approaches from the viewpoint of the constructive approach the basic properties of generalized rough sets over fuzzy lattices are obtained the matrix representation of the lower and upper approximations is given according to this matrix view a simple algorithm is obtained for computing the lower and upper approximations as for the axiomatic approach a set of axioms is constructed to characterize the upper approximation of generalized rough sets over fuzzy lattices

rudiments of rough sets abstract worldwide there has been a rapid growth in interest in rough set theory and its applications in recent years evidence of this can be found in the increasing number of high quality articles on rough sets and related topics that have been published in a variety of international journals symposia workshops and international conferences in recent years in addition many international workshops and conferences have included special sessions on the theory and applications of rough sets in their programs rough set theory has led to many interesting applications and extensions it seems that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences especially in research areas such as machine learning intelligent systems inductive reasoning pattern recognition mereology knowledge discovery decision analysis and expert systems in the article we present the basic concepts of rough set theory and point out some rough set based research directions and applications

attribute reduction for heterogeneous data based on the combination of classical and fuzzy rough set models attribute reduction with rough sets aims to delete superfluous condition attributes from a decision system by considering the inconsistency between condition attributes and the decision labels however heterogeneous condition attributes including symbolic and real valued ones always coexist for most decision systems and different types of attributes induce different kinds of granular structures the existing rough set models do not have explicit mechanisms to address different kinds of granular structures reasonably and effectively in this paper we aim to perform attribute reduction for decision systems with symbolic and real valued condition attributes by composing classical rough set and fuzzy rough set models we first define a discernibility relation for every symbolic and real valued condition attribute to characterize its discernible ability related to the decision labels with these discernibility relations we can develop a dependence function to measure the inconsistency between heterogeneous condition attributes and decision labels and attribute reduction aims to keep this dependence function with a small perturbation the proposed attribute reduction deals with heterogeneous condition attributes from the viewpoint of discernible ability and can consider the mutual effects between two types of attributes without preprocessing into single typed ones an algorithm to find reducts is developed and experiments are performed to demonstrate that the proposed idea is effective

the necessary and sufficient conditions for a fuzzy relation being τ euclidean the connections between special types of fuzzy relations and properties of fuzzy rough approximation operators have been established in recent years but euclidean fuzzy relation has not been considered yet in the present paper the necessary and sufficient conditions for a fuzzy relation being euclidean are investigated in three different fuzzy rough approximation spaces meanwhile it is proved that in i fuzzy rough approximation space where i is an r implication the properties the euclidean i fuzzy rough approximation operators possess are just the same as those in rough fuzzy approximation space

on definable concepts of rough set models in various generalized rough set models definable concepts are used to approximate other concepts based on a unified abstract rough set theory this paper investigates the mathematical structure of the set of definable concepts in several generalized rough set models such as relation based models covering based models and fuzzy based models it is shown that there exist two kinds of interesting structures for the set of definable concepts which are often used in rough set models the complementary structure and the equational structure in addition relations between different rough set models are discussed

the minimization of axiom sets characterizing generalized approximation operators in the axiomatic approach of rough set theory rough approximation operators are characterized by a set of axioms that guarantees the existence of certain types of binary relations reproducing the operators thus axiomatic characterization of rough approximation operators is an important aspect in the study of rough set theory in this paper the independence of axioms of generalized crisp approximation operators is investigated and their minimal sets of axioms are presented

twofold fuzzy sets and rough sets some issues in knowledge representation abstract this paper deals with the representation of sets where the membership of some elements may be ill known rather than just a matter of degree as in a fuzzy set the notion of a twofold fuzzy set is introduced when the relevant information for determining the membership status is incomplete a twofold fuzzy set is made of a nested pair of fuzzy sets the one which gathers the elements which more or less necessarily belong and the one which gathers the elements which more or less possibly belong twofold fuzzy sets are compared from a frontal and from a semantical point of view with other proposals and particularly with the notion of a rough set recently introduced by pawlak set operations of twofold fuzzy sets are discussed and the cardinality of a twofold fuzzy set is defined twofold fuzzy relations are also introduced finally various applications of twofold fuzzy sets in knowledge representation are briefly discussed

the relationship between l fuzzy rough set and l topology various fuzzy generalizations of rough approximations have been proposed in the literature this paper is devoted to the discussion of the relationship between l fuzzy rough sets and l topologies on an arbitrary universe finally one to one correspondence between the set of all reflexive transitive l relations and the set of all alexandrov l topologies is obtained

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

formal concept analysis based on fuzzy granularity base for different granulations this paper introduces granular computing grc into formal concept analysis fca it provides a unified model for concept lattice building and rule extraction on a fuzzy granularity base for different granulations one of the strengths of grc is that larger granulations help to hide some specific details whereas fca in a grc context can prevent losses due to concept lattice complexity however the number of superfluous rules increases exponentially with the scale of the decision context to overcome this we present some inference rules and maximal rules and prove that the set of all these maximal rules is complete and nonredundant thus users who want to obtain decision rules should generate maximal rules examples demonstrate that application of the method is valid and practicable in summary this approach utilizes fca in a grc context and provides a practical basis for data analysis and processing

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

knowledge acquisition in incomplete information systems a rough set approach abstract this paper deals with knowledge acquisition in incomplete information systems using rough set theory the concept of similarity classes in incomplete information systems is first proposed two kinds of partitions lower and upper approximations are then formed for the mining of certain and association rules in incomplete decision tables one type of optimal certain and two types of optimal association decision rules are generated two new quantitative measures random certainty factor and random coverage factor associated with each decision rule are further proposed to explain relationships between the condition and decision parts of a rule in incomplete decision tables the reduction of descriptors and induction of optimal rules in such tables are also examined