adaptive neighborhood granularity selection and combination based on margin distribution optimization abstract granular computing aims to develop a granular view for interpreting and solving problems the model of neighborhood rough sets is one of effective tools for granular computing this model can deal with complex tasks of classification learning despite the success of the neighborhood model in attribute reduction and rule learning it still suffers from the issue of granularity selection namely it is an open problem to select a proper granularity of neighborhood for a specific task in this work we explore ensemble learning techniques for adaptively evaluating and combine the models derived from multiple granularity in the proposed framework base classifiers are trained in different granular spaces the importance of base classifiers is then learned by optimizing the margin distribution of the combined system experimental analysis shows that the proposed method can adaptively select a proper granularity and combining the models trained in multi granularity spaces leads to competent performance

rules in incomplete information systems abstract a new method of computing all optimal certain rules from an incomplete information system is presented and proved the method does not require changing the size of the original incomplete system additionally several existing rough set methods of computing decision rules from incomplete information systems are analyzed and compared we show which of these methods are capable of generating all optimal certain rules or a class of optimal certain rules and which methods may lead to generation of false rules

classification of coverings in the finite approximation spaces based on the notions of neighborhood and complementary neighborhood we consider the classification of coverings in the covering rough set theory we present a classification rule under which the coverings on a universe u are classified such that for any given pair of neighborhood or complementary neighborhood based lower and upper approximation operators on u all the different coverings in the same class generate the same pair of lower and upper approximations of each x u we show that there is a one to one correspondence between the equivalence classes of coverings and the topologies of the universe thus the number of the equivalence classes of coverings is equal to the number of topologies of the universe and therefore this number is much smaller than that of the coverings we also give an illustrative example to show how we can classify some given coverings by calculating the topologies moreover based on the relationship between the neighborhood and the complementary neighborhood we find that each class of coverings has a dual class and each pair of approximation operators has a dual pair of approximation operators for a finite universe considering one pair of approximation operators under one equivalence class of coverings is equivalent to considering its dual approximation operator pair under the dual class of coverings finally we also present a sufficient condition under which for any given pair of neighborhood based lower and upper approximation operators two coverings of the universe u generate the same pair of approximations of each x u if and only if they belong to the same equivalence class

a new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough sets traditional rough set theory is mainly used to extract rules from and reduce attributes in databases in which attributes are characterized by partitions while the covering rough set theory a generalization of traditional rough set theory does the same yet characterizes attributes by covers in this paper we propose a way to reduce the attributes of covering decision systems which are databases characterized by covers first we define consistent and inconsistent covering decision systems and their attribute reductions then we state the sufficient and the necessary conditions for reduction finally we use a discernibility matrix to design algorithms that compute all the reducts of consistent and inconsistent covering decision systems numerical tests on four public data sets show that the proposed attribute reductions of covering decision systems accomplish better classification performance than those of traditional rough sets

relationships among generalized rough sets in six coverings and pure reflexive neighborhood system rough set is a useful tool to deal with partition related uncertainty granularity and incompleteness of knowledge although the classical rough set is constructed on the basis of an indiscernibility relation it can also be generalized by using some weaker binary relations in this paper a systematic approach is used to study the generalized rough sets in six coverings and pure reflexive neighborhood system after two steps relationships among generalized rough sets in six coverings and pure reflexive neighborhood system are obtained the first step is to study the generalized rough sets in six coverings and to get relationships between every two covering rough set models the second step is to study the relationships between generalized rough sets in each covering and in pure reflexive neighborhood system the inclusion relations or equivalence relations among the seven upper lower approximations could be acquired finally the accuracy measures of generalized rough sets in six coverings and that in pure reflexive neighborhood system are compared the relationships among seven accuracy measures are also obtained some illustrative examples are employed to demonstrate our arguments

covering based multigranulation i t fuzzy rough set models and applications in multi attribute group decision making abstract by means of a fuzzy logical implicator and a t norm respectively denoted i and t we introduce covering based multigranulation i t fuzzy rough set models from fuzzy neighborhoods by using different implicators and t norms the corresponding axiomatic characterizations of covering based optimistic pessimistic and variable precision multigranulation i t fuzzy rough set models are investigated connections among these kinds of coverings based models are examined based on the theoretical analysis for the covering based multigranulation i t fuzzy rough set models solutions to problems in multi attribute group decision making by means of two kinds of decision making methods are respectively established an effective example is fully developed to illustrate these methodologies comparative analysis shows that the two ranking results obtained by means of two different decision making methods have a high consensus

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

attribute reduction of data with error ranges and test costs in data mining applications we have a number of measurement methods to obtain a data item with different test costs and different error ranges test costs refer to time money or other resources spent in obtaining data items related to some object observational errors correspond to differences in measured and true value of a data item in supervised learning we need to decide which data items to obtain and which measurement methods to employ so as to minimize the total test cost and help in constructing classifiers this paper studies this problem in four steps first data models are built to address error ranges and test costs second error range based covering rough set is constructed to define lower and upper approximations positive regions and relative reducts a closely related theory deals with neighborhood rough set which has been successfully applied to heterogeneous attribute reduction the major difference between the two theories is the definition of neighborhood third the minimal test cost attribute reduction problem is redefined in the new theory fourth both backtrack and heuristic algorithms are proposed to deal with the new problem the algorithms are tested on ten uci university of california irvine datasets experimental results show that the backtrack algorithm is efficient on rational sized datasets the weighting mechanism for the heuristic information is effective and the competition approach can improve the quality of the result significantly this study suggests new research trends concerning attribute reduction and covering rough set

roughness in mv algebras in this paper by considering the notion of an mv algebra we consider a relationship between rough sets and mv algebra theory we introduce the notion of rough ideal with respect to an ideal of an mv algebra which is an extended notion of ideal in an mv algebra and we give some properties of the lower and the upper approximations in an mv algebra

on minimization of axiom sets characterizing covering based approximation operators rough set theory was proposed by pawlak to deal with the vagueness and granularity in information systems the classical relation based pawlak rough set theory has been extended to covering based generalized rough set theory the rough set axiom system is the foundation of the covering based generalized rough set theory because the axiomatic characterizations of covering based approximation operators guarantee the existence of coverings reproducing the operators in this paper the equivalent characterizations for the independent axiom sets of four types of covering based generalized rough sets are investigated and more refined axiom sets are presented

estimates of heat kernel of fractional laplacian perturbed by gradient operators we construct a continuous transition density of the semigroup generated by delta alpha 2 b x cdot nabla for 1 alpha 2 d ge 1 and b in the kato class mathcal k d alpha 1 on mathbb r d for small time the transition density is comparable with that of the fractional laplacian

on fundamental solutions for non local parabolic equations with divergence free drift abstract we are concerned with non local parabolic equations in the presence of a divergence free drift term by using the classical nash approach we show the existence of fundamental solutions together with continuity estimates under weak regularity assumptions on the kernel of the non local term and the velocity of the drift term as an application we give an alternative proof of global regularity for the two dimensional dissipative quasi geostrophic equations in the critical case

a study of rough sets based on 1 neighborhood systems abstract a special class of neighborhood systems called 1 neighborhood systems are important in rough set theory by using a concept core originated in general topology we define two types of rough sets based on 1 neighborhood systems in this paper we discuss properties of these rough sets from the perspective of both common 1 neighborhood systems and several special classes of 1 neighborhood systems such as reflexive symmetric transitive or euclidean 1 neighborhood systems by using these properties we discuss the relationship among several classes of 1 neighborhood systems with various special properties we give a necessary and sufficient condition for a reflexive and symmetric 1 neighborhood system being a unary we also prove that a reflexive and transitive 1 neighborhood system is representative the proofs of these results show that the rough sets we defined in this paper not only have application background but also have theoretic importance

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

a novel method for attribute reduction of covering decision systems attribute reduction has become an important step in pattern recognition and machine learning tasks covering rough sets as a generalization of classical rough sets have attracted wide attention in both theory and application this paper provides a novel method for attribute reduction based on covering rough sets we review the concepts of consistent and inconsistent covering decision systems and their reducts and we develop a judgment theorem and a discernibility matrix for each type of covering decision system furthermore we present some basic structural properties of attribute reduction with covering rough sets based on a discernibility matrix we develop a heuristic algorithm to find a subset of attributes that approximate a minimal reduct finally the experimental results for uci data sets show that the proposed reduction approach is an effective technique for addressing numerical and categorical data and is more efficient than the method presented in the paper d g chen c z wang q h hu a new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough sets information sciences 177 17 2007 3500 3518

quantitative information architecture granular computing and rough set models in the double quantitative approximation space of precision and grade because precision and grade act as fundamental quantitative information in approximation space they are used in relative and absolute quantifications respectively at present the double quantification regarding precision and grade is a novel and valuable subject but quantitative information fusion has become a key problem thus this paper constructs the double quantitative approximation space of precision and grade pg approx space and tackles the fusion problem using normal logical operations it further conducts double quantification studies on granular computing and rough set models 1 first for quantitative information organization and storage we construct space and plane forms of pg approx space using the cartesian product and for quantitative information extraction and fusion we establish semantics construction and semantics granules of pg approx space 2 second by granular computing we investigate three primary granular issues quantitative semantics complete system and optimal calculation accordingly six types of fundamental granules are proposed based on the semantic microscopic and macroscopic descriptions their semantics forms structures calculations and relationships are studied and the granular hierarchical structure is achieved 3 finally we investigate rough set models in pg approx space accordingly model regions are proposed by developing the classical regions model expansion is systematically analyzed some models are constructed as their structures are obtained and a concrete model is provided based on the quantitative information architecture this paper systematically conducts and investigates double quantification and establishes a fundamental and general exploration framework

an entropy based uncertainty measurement approach in neighborhood systems abstract uncertainty measures can provide us with principled methodologies to analyze uncertain data and unveil the substantive characteristics of the data sets accuracy and roughness proposed by pawlak are two main tools to deal with uncertainty measurement issue in rough set theory many uncertainty measure methodologies for discrete valued information systems or discrete valued decision systems have been developed however there are only limited on the uncertainty measurement for neighborhood systems in this paper we address the issues of uncertainty of a neighborhood system and extend the traditional accuracy and roughness measures to deal with neighborhood systems in particular a concept called neighborhood entropy is first introduced to evaluate the uncertainty of a neighborhood information system consequently the entropy based roughness and approximation roughness measures of neighborhood system are presented theoretical analysis indicates that entropy based measures can be used to evaluate the uncertainty in neighborhood systems experiments are conducted on artificial data sets and standard uci data sets to test our proposed methodologies results demonstrate that the entropy based measures are effective and valid for evaluating the uncertainty of neighborhood systems

a systematic study on attribute reduction with rough sets based on general binary relations attribute reduction is considered as an important preprocessing step for pattern recognition machine learning and data mining this paper provides a systematic study on attribute reduction with rough sets based on general binary relations we define a relation information system a consistent relation decision system and a relation decision system and their attribute reductions furthermore we present a judgment theorem and a discernibility matrix associated with attribute reduction in each type of system based on the discernibility matrix we can compute all the reducts finally the experimental results with uci data sets show that the proposed reduction methods are an effective technique to deal with complex data sets

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 3 valued algebras rough set theory is an important tool for dealing with granularity and vagueness in information systems this paper studies a kind of rough set algebra the collection of all the rough sets of an approximation space can be made into a 3 valued lukasiewicz algebra we call the algebra a rough 3 valued lukasiewicz algebra in this paper we focus on the rough 3 valued lukasiewicz algebras which are a special kind of 3 valued lukasiewicz algebras firstly we examine whether the rough 3 valued lukasiewicz algebra is an axled 3 valued lukasiewicz algebra secondly we present the condition under which the rough 3 valued lukasiewicz algebra is also a 3 valued post algebra then we investigate the 3 valued post subalgebra problem of the rough 3 valued lukasiewicz algebra finally this paper studies the relationship between the rough 3 valued lukasiewicz algebra and the boolean algebra constructed by all the exact sets of the corresponding approximation space

a fast approach to attribute reduction in incomplete decision systems with tolerance relation based rough sets efficient attribute reduction in large incomplete decision systems is a challenging problem existing approaches have time complexities no less than o c 2 u 2 this paper derives some important properties of incomplete information systems then constructs a positive region based algorithm to solve the attribute reduction problem with a time complexity no more than o c 2 u log u furthermore our approach does not change the size of the original incomplete system numerical experiments show that the proposed approach is indeed efficient and therefore of practical value to many real world problems the proposed algorithm can be applied to both consistent and inconsistent incomplete decision systems

topological solution of missing attribute values problem in incomplete information tables in this paper we present a new method of data decomposition to avoid the necessity of reasoning from data with missing attribute values we define firstly a general binary relation on the original incomplete dataset this binary relation generates data subsets without missing values these data subsets are used to generate a topological base relation which decomposes datasets we investigate a new approach to find the missing values in incomplete datasets new pre topological approximations are initiated and some of their properties are proved also pre topological measures are defined and studied finally the reducts and the core of incomplete information system are determined

upper bounds for fundamental solutions to non local diffusion equations with divergence free drift we consider some non local diffusion equations in the presence of a divergence free drift term where the diffusion operators are related to certain dirichlet forms of jump type we derive pointwise upper bounds for fundamental solutions of the equations under weak assumptions for the velocity of the drift term our class of the velocity includes functions with the scale critical regularity and some growing functions at spatial infinity

dirichlet heat kernel estimates for fractional laplacian with gradient perturbation suppose that d 2 and 1 2 let d be a bounded c1 1 open set in rd and b an rd valued function on rd whose components are in a certain kato class of the rotationally symmetric stable process in this paper we derive sharp two sided heat kernel estimates for lb 2 b in d with zero exterior condition we also obtain the boundary harnack principle for lb in d with explicit decay rate

fuzzy rough sets based on generalized residuated lattices this paper is devoted to propose generalized l fuzzy rough sets as a further generalization of the notion of l fuzzy rough sets a quadruple of approximation operators are defined to suit the situation when generalized residuated lattices are non commutative generalized l fuzzy rough sets are characterized from both constructive and axiomatic approaches in the constructive approach various classes of generalized l fuzzy rough sets are investigated moreover the relationship between generalized l fuzzy rough sets and l topologies on an arbitrary universe is discussed with generalized lower and upper sets as an application of generalized l fuzzy rough sets fuzzy rough sets are proposed and studied on the unit interval 0 1 which are based on generalized residuated lattices induced by left continuous pseudo t norms

pointwise estimates for solutions of fractal burgers equation abstract in this paper we provide two sided estimates and uniform asymptotics for the solution of d dimensional critical fractal burgers equation u t 2 u b u u q 0 1 2 b r d for q 1 d and u 0 l 1 r d we consider also q 1 d under additional condition u 0 l r d in both cases we assume u 0 0 which implies that the solution is non negative the estimates are given in the terms of the function p t u 0 where p t denotes the semigroup for the operator t 2

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

estimates of gradient perturbation series abstract we give upper and lower bounds of perturbation series for transition densities corresponding to additive gradient perturbations satisfying certain space time integrability conditions

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

the algebraic structures of generalized rough set theory rough set theory is an important technique for knowledge discovery in databases and its algebraic structure is part of the foundation of rough set theory in this paper we present the structures of the lower and upper approximations based on arbitrary binary relations some existing results concerning the interpretation of belief functions in rough set backgrounds are also extended based on the concepts of definable sets in rough set theory two important boolean subalgebras in the generalized rough sets are investigated an algorithm to compute atoms for these two boolean algebras is presented

covering rough set structures for a locally finite covering approximation space abstract the purpose of this paper is to introduce the notion of locally finite covering approximation lfc for brevity space u c which is a generalization of a finite covering approximation space furthermore for a subset x of the universe u we develop two kinds of neighborhood systems derived from the covering c specifically k1 x and k2 x which are not comparable with each other next the paper establishes two topologies on u alexandroff and quasi discrete topologies generated by systems k1 x and k2 x as bases respectively after suggesting many examples explaining the above notions and systems we develop two new types of rough set structures an h rough set operator h h and a khalimsky k for short topological rough set operator d k d k the present paper investigates various properties of these operators and refers to certain advantages of them which can be used in applied science including digital geometry information geometry computer vision pattern recognition image processing and so on in order to accomplish this we use some tools such as granulations neighborhood systems typical pawlak s tools related topologies rough set structures associated with the k topology and so on furthermore through this work we can obtain strong connections between covering rough set theory and applied geometry in the present paper all sets u c and x u need not be finite

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

on lower and upper intension order relations by different cover concepts in this paper the concept of intension is used to introduce two types of ordering relations based on information that generates a cover for the universal set these types of ordering relations are distinct from the well known ordering relation based on set inclusion for these ordering relations we consider the algebraic structures that arise in various types of covers we show that in the case of a representative cover the algebraic structure resulting from the lower intension inclusion is a double stone algebra while in the case of a reduced cover it is a boolean algebra in addition the algebraic structure resulting from the upper intension inclusion in the case of a representative cover is a boolean algebra and in the case of a reduced cover the two boolean algebraic structures from lower and upper intension inclusions are isomorphic

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

related family a new method for attribute reduction of covering information systems in terms of attribute reduction of covering based rough sets the discernibility matrix is used as a conventional method to compute all attribute reducts however it is inapplicable to attribute reduction in certain circumstances in this article a new method referred to as the related family is introduced to compute all attribute reducts and relative attribute reducts for covering rough sets its core idea is to remove superfluous attributes while keeping the approximation space of covering information system unchanged the related family method is more powerful than the discernibility matrix method since the former can handle complicated cases that could not be handled by the latter in addition a simplified version of the related family and its corresponding heuristic algorithm are also presented

nullity based matroid of rough sets and its application to attribute reduction rough sets were proposed to deal with vagueness and incompleteness of knowledge in information systems in this field there are many optimization issues such as attribute reduction matroids generalized from matrices have been widely used in many fields particularly greedy algorithm design which plays an important role in attribute reduction therefore it is meaningful to combine matroids with rough sets to solve the optimization problems in this paper we construct a type of matroid of rough sets based on the concept of nullity and apply it to attribute reduction first we propose a nullity operator for rough sets to induce a matroid and then we present a specific type of matroid called a nullity based matroid second given the relationship between nullities and matrices we present two types of matrices to characterize this type of matroid and its nullity operator third the dual of this type of matroid is induced by the second type of matrix finally we apply the obtained matroids to attribute reduction issues in information systems in summary this paper provides a new approach to studying rough sets

on measurements of covering rough sets based on granules and evidence theory covering rough sets generalize classical rough sets by relaxing partitions of the universe to coverings the existing research of covering rough sets mainly concentrates on comparing of different kinds of approximation operators and designing algorithms for attribute reductions of decision systems less efforts have been made in measurements of covering rough sets to enrich this field in this paper we investigate the measurements of covering rough sets based on granules and evidence theory which can be used to solve more complex practical problems firstly it is shown that the most common sixteen pairs of covering approximation operators can be grouped into five pairs of dual covering approximation operators and six pairs of non dual covering approximation operators according to definitions based on element granule and subsystem secondly we define the rough degree and precision for the purpose of characterizing the uncertainty of these summarized covering approximation operators under a unified framework the underlying uncertainty of covering rough sets results from the roughness of granulation of coverings finally we investigate the relationships between the covering approximation operators and evidence theory it shows that some kinds of covering approximation operators can be characterized by belief and plausibility functions while others do not share this property the sufficient conditions under which covering approximation operators can be measured by belief and plausibility functions are further explored and the corresponding counterexamples are given to illustrate those that cannot be measured based on above discussions we develop a basic framework of numerical characterization on measurements of covering rough sets with granules and evidence theory

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

relationships between generalized rough sets based on covering and reflexive neighborhood system rough set theory is an important tool to deal with uncertainty granularity and incompleteness of knowledge the covering based rough sets is one of the most important extensions of the classical pawlak rough sets in covering based rough set theory many types of rough sets were already defined in the literature to find out the relationships among different types of covering based rough sets and give an axiomatic definition is a very important issue the generalized rough sets in neighborhood system is another important extension of the classical pawlak rough sets in this paper we investigate relationships between covering based rough sets and generalized rough sets in neighborhood system then twenty three types of the element based the granule based and the subsystem based definitions of covering approximation operators are unified under the framework of generalized approximation operators in neighborhood systems moreover properties of the generalized approximation operators in neighborhood systems are presented

on some types of fuzzy covering based rough sets fuzzy coverings are a natural extension of the coverings by replacing crisp sets with fuzzy sets recently an excellent introduction to the definition of a fuzzy covering is due to ma and two fuzzy covering based rough set models are presented in this paper by introducing some new definitions of fuzzy covering approximation spaces the properties of fuzzy covering approximation spaces and ma s fuzzy covering based rough set models are studied furthermore three new types of fuzzy covering based rough set models as generalizations of ma s models are first proposed in this paper first some properties of fuzzy covering and its fuzzy neighborhood family are proposed we present a necessary and sufficient condition for fuzzy neighborhood family induced by a fuzzy covering to be equal to the fuzzy covering itself then we study the characterizations of ma s fuzzy covering based rough set models and give a necessary and sufficient condition for two fuzzy coverings to generate the same fuzzy covering lower approximation or the same fuzzy covering upper approximation finally this paper proposes three new types of fuzzy covering based rough set models by introducing a new notion of a fuzzy complementary neighborhood

quantitative analysis for covering based rough sets through the upper approximation number coverings are a useful form of data while covering based rough sets provide an effective tool for dealing with this data covering based rough sets have been widely used in attribute reduction and rule extraction however few quantitative analyses for covering based rough sets have been conducted while many advances for classical rough sets have been obtained through quantitative tools in this paper the upper approximation number is defined as a measurement to quantify covering based rough sets and a pair of upper and lower approximation operators are constructed using the approximation number the operators not only inherit some important properties of existing approximation operators but also exhibit some new quantitative characteristics it is interesting to note that the upper approximation number of a covering approximation space is similar to the dimension of a vector space or the rank of a matrix

relationship between generalized rough sets based on binary relation and covering rough set theory is a powerful tool for dealing with uncertainty granularity and incompleteness of knowledge in information systems this paper systematically studies a type of generalized rough sets based on covering and the relationship between this type of covering based rough sets and the generalized rough sets based on binary relation firstly we present basic concepts and properties of this kind of rough sets then we investigate the relationships between this type of generalized rough sets and other five types of covering based rough sets the major contribution in this paper is that we establish the equivalency between this type of covering based rough sets and a type of binary relation based rough sets through existing results in binary relation based rough sets we present axiomatic systems for this type of covering based lower and upper approximation operations in addition we explore the relationships among several important concepts such as minimal description reduction representative covering exact covering and unary covering in covering based rough sets investigation of this type of covering based will benefit to our understanding of other types of rough sets based on covering and binary relation

a short note on algebraic t rough sets our aim in this paper is to introduce the concept of a set valued homomorphism for groups which is a generalization of ordinary homomorphism then using the definitions of lower inverse and upper inverse we introduce the definition of uniform set valued homomorphism and prove that every set valued homomorphism is uniform finally the concept of probabilistic t rough set is defined

axiomatization and conditions for neighborhoods in a covering to form a partition in this paper we study the axiomatic issue of a type of covering upper approximation operations this issue was proposed as an open problem we also further some known results by using only a single covering approximation operator to characterize the conditions for neighborhood n x x u to form a partition of universe u

stable estimates for source solution of critical fractal burgers equation abstract in this paper we provide two sided estimates for the source solution of d dimensional critical fractal burgers equation u t 2 b u u q 0 q 1 d 1 2 b r d by the density function of the isotropic stable process

multigranulation rough sets from partition to covering the classical multigranulation rough set mgrs theory offers a formal theoretical framework for solving the complex problem under multigranulation environment however it is noticeable that mgrs theory cannot be applied in multi source information systems with a covering environment in the real world to address this issue we firstly present in this paper three types of covering based multigranulation rough sets in which set approximations are defined by different covering approximation operators then by using two different approximation strategies i e seeking common reserving difference and seeking common rejecting difference two kinds of covering based multigranulation rough set are presented namely a covering based optimistic multigranulation rough set and a covering based pessimistic multigranulation rough sets finally we develop some properties and several uncertainty measures of the covering based multigranulation rough sets these results will enrich the mgrs theory and enlarge its application scope

object similarity measures and pawlak s indiscernibility on decision tables abstract in this paper we investigate the mathematical foundations of the notion of similarity between objects in relation to the granulations on a decision table d first of all we compare the endogenous granulation induced by pawlak s indiscernibility with the exogenous granulation induced by a similarity measure defined on pairs of objects and assuming values in the unit interval to this aim the starting point of our analysis is the introduction of the notion of refinement of the granulation induced by an attribute subset a through the object similarity measure more in detail we say that refines the granulation induced by a if assumes value 1 on a pair of objects if and only if they are a indiscernible next starting from two given families and of numerical maps defined on pairs of admissible values of d we determine a broad class of potential similarity measures on the objects of d refining sometimes under some specific additional hypotheses the a granulation on the object set of d with regard to a such class of similarity measures we establish several mathematical properties finally we focus our attention to the analysis of specific pairs of numerical maps and that have been classically studied in literature and for each of them we exhibit the main properties with respect to the aforementioned refinement of granulation

nested structure in parameterized rough reduction abstract in this paper by strict mathematical reasoning we discover the relationship between the parameters and the reducts in parameterized rough reduction this relationship named the nested reduction shows that the reducts act as a nested structure with the monotonically increasing parameter we present a systematic theoretical framework that provides some basic principles for constructing the nested structure in parameterized rough reduction some specific parameterized rough set models in which the nested reduction can be constructed are pointed out by strict mathematical reasoning based on the nested reduction we design several quick algorithms to find a different reduct when one reduct is already given here different refers to the reducts obtained on the different parameters all these algorithms are helpful for quickly finding a proper reduct in the parameterized rough set models the numerical experiments demonstrate the feasibility and the effectiveness of the nested reduction approach

communication between information systems with covering based rough sets communication between information systems is considered as an important issue in granular computing the concept of homomorphism is an effective mathematical tool to study information exchange between information systems this paper provides a study on some basic properties of covering information systems and decision systems under homomorphisms first we define consistent functions related to coverings and covering mappings between two universes and study their properties then we introduce the notions of homomorphisms of covering information systems and point out that a homomorphism is a special covering mapping between information systems furthermore we investigate some important properties of homomorphisms in covering information systems and decision systems it is proved that some basic properties of original systems such as set approximations attribute reductions can be reserved under the condition of homomorphisms in both covering information systems and covering decision systems

the lattice and matroid representations of definable sets in generalized rough sets based on relations abstract the definable set is a core concept in rough set theory it plays an important role in the characterizations of rough sets in this paper we study the lattice and matroid representations of definable sets in generalized rough sets based on relations first we propose the lower definable lattice consisting of all lower definable sets with set inclusion order then we give some conditions under which the lower definable lattice is distributive or geometric or boolean furthermore we discuss the relationship between distributive lattices and the lower definable lattices in generalized approximation spaces based on reflexive and transitive relations on the one hand we show that the lower definable lattice in a generalized approximation space based on reflexive and transitive relation is distributive on the other hand we obtain the result that a distributive lattice can induce a lower definable lattice in a generalized approximation space based on a reflexive and transitive relation finally we investigate the combination of generalized rough sets and matroids in terms of the lower definable lattice we show that if a lower definable lattice is a lattice of closed sets of a matroid then it must be an open closed set lattice of a matroid in addition we prove that some lower definable lattices are not the lattices of closed sets of matroids these results of this paper will benefit to our understanding of the relationship between matroids and generalized rough sets based on relations

relations of reduction between covering generalized rough sets and concept lattices the reduction theory plays an important role in data analysis this paper studies the relation between the reduction of a covering and the attribute reduction of a concept lattice the reduction of a covering from the perspective of concept lattices is investigated conversely the attribute reduction of a formal context is studied in the framework of covering generalized rough sets the results in this paper show that the reduction of a covering can be viewed as the attribute reduction of a derivative formal context moreover every reduct of a given formal context can be seen as the reduct of an induced covering as an application of the theoretical results an approach to the attribute reduction of concept lattices based on covering generalized rough sets is proposed furthermore experiments are given to show the effectiveness of the proposed method

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

the fourth type of covering based rough sets as a technique for granular computing rough sets deal with the vagueness and granularity in information systems covering based rough sets have been proposed to generalize this theory for wider application three types of covering based rough sets have been studied for different situations to make the theory more complete this paper proposes a fourth type of covering based rough sets compared with the existing ones the new type shows its special characteristic in the interdependency between its lower and upper approximations we carry out a systematical study of this new theory first we discuss basic properties such as normality contraction and monotone then we investigate the conditions for this type of covering based rough sets to satisfy the properties of pawlak s rough sets and study the interdependency between the lower and upper approximation operations in addition axiomatic systems for the lower and upper approximation operations are established lastly we address the relationships between this type of covering based rough sets and the three existing ones

topological and lattice structures of l fuzzy rough sets determined by lower and upper sets this paper builds the topological and lattice structures of l fuzzy rough sets by introducing lower and upper sets in particular it is shown that when the l relation is reflexive the upper resp lower set is equivalent to the lower resp upper l fuzzy approximation set then by the upper resp lower set it is indicated that an l preorder is the equivalence condition under which the set of all the lower resp upper l fuzzy approximation sets and the alexandrov l topology are identical however associating with an l preorder the equivalence condition that l interior resp closure operator accords with the lower resp upper l fuzzy approximation operator is investigated at last it is proven that the set of all the lower resp upper l fuzzy approximation sets forms a complete lattice when the l relation is reflexive

fractal burgers equations abstract the paper studies local and global in time solutions to a class of multidimensional generalized burgers type equations with a fractional power of the laplacian in the principal part and with general algebraic nonlinearity such equations naturally appear in continuum mechanics our results include existence uniqueness regularity and asymptotic behavior of solutions to the cauchy problem as well as a construction of self similar solutions the role of critical exponents is also explained

rough matroids based on relations rough sets provide an efficient tool for attribute reduction and rule extraction however many important problems in rough set theory including attribute reduction are np hard and therefore the algorithms for solving them are usually greedy as a generalization of linear independence in vector spaces matroids have wide applications in diverse fields particularly in greedy algorithm design in this paper we propose an integration of rough sets and matroids to exploit the advantages of both theories specifically we present definitions of lower and upper rough matroids based on relations from the viewpoint of approximation operators it is interesting that lower rough matroids based on partial orders coincide with poset matroids which are a well known generalization of matroids a matroid is represented by the lower rough matroid based on both a partial order and an equivalence relation lower and upper rough matroids based on equivalence relations coincide with each other finally we present some special types of examples of rough matroids from the viewpoints of approximations and graphs these interesting results demonstrate the potential for the combination between rough sets and matroids

multiset topologies induced by multiset relations multiset topology is a collection of multisets which satisfies the axioms of topology in this paper multiset topologies are obtained using binary multiset relations the relationship between multiset topologies is investigated and some of the properties of these multiset topologies are proved the quasi discrete multiset topology is obtained from a symmetric multiset relation instead of an equivalence multiset relation and several examples are given to indicate counter connections finally the rough multiset is introduced and the multiset topology induced by a multiset relation is used to generalize the rough multiset concept

rough sets and matroids from a lattice theoretic viewpoint this paper studies rough sets via matroidal approaches from a lattice theoretic viewpoint we firstly give a new interpretation of definable sets of pawlak rough set model i e the set of definable sets defines uniquely a matroid in which it is the family of open and closed sets then we induce two equivalence relations on a given universe based on a matroid defined on this universe one of the equivalence relations actually is defined on the set of all atoms of a geometric lattice corresponding to the matroid another is based on the transitivity of circuits properties of these two equivalence relations are then studied besides we also investigate the connections between relation based rough sets and matroids finally we point out that a geometric lattice can induce a series of coverings of a universe on which the corresponding matroid is defined and further relations of approximations based on the induced coverings are studied

using one axiom to characterize rough set and fuzzy rough set approximations axiomatic approaches are important for understanding the concepts of rough set theory the properties of the approximation operators constructed in rough set theory are determined by a binary relation a linkage in the form of a necessary and sufficient condition between the constructive approach and an axiomatic system is established in this paper various classes of rough set algebras are characterized by different sets of axioms with no restriction on the cardinality of the universal set we use only one axiom to describe the approximation generated by the reflexive symmetric and transitive relations respectively in addition we characterize lower and upper approximations of the pawlak rough set with only one axiom we also study a similar problem in the context of fuzzy sets

a comparison of two kinds of definitions of rough approximations based on a similarity relation in the present paper we investigate the three types of yao s lower and upper approximations of any set with respect to any similarity relation these types based on a right neighborhood also we define and investigate other three types of approximations for any similarity relations these new types based on the intersection of the right neighborhoods moreover we give a comparison between these types lastly the relationship between these definitions is introduced

roughness in cayley graphs in this paper rough approximations of cayley graphs are studied and rough edge cayley graphs are introduced furthermore a new algebraic definition for pseudo cayley graphs containing cayley graphs is proposed and a rough approximation is expanded to pseudo cayley graphs in addition rough vertex pseudo cayley graphs and rough pseudo cayley graphs are introduced some theorems are provided from which properties such as connectivity and optimal connectivity are derived this approach opens new research fields such as data networks

on axiomatic characterizations of three pairs of covering based approximation operators rough set theory is a useful tool for dealing with inexact uncertain or vague knowledge in information systems the classical rough set theory is based on equivalence relations and has been extended to covering based generalized rough set theory this paper investigates three types of covering generalized rough sets within an axiomatic approach concepts and basic properties of each type of covering based approximation operators are first reviewed axiomatic systems of the covering based approximation operators are then established the independence of axiom set for characterizing each type of covering based approximation operators is also examined as a result two open problems about axiomatic characterizations of covering based approximation operators proposed by zhu and wang in ieee transactions on knowledge and data engineering 19 8 2007 1131 1144 proceedings of the third ieee international conference on intelligent systems 2006 pp 444 449 are solved

on the topological properties of generalized rough sets in this paper we consider some topological properties of generalized rough sets induced by binary relations and show that1 any serial binary relation can induce a topology 2 let r be a binary relation on a universe u t r and e r denote the transitive closure and the equivalence closure of r respectively if r is a reflexive relation on u then r and t r induce the same topology i e t r t t r the interior and closure operators of the topology t r induced by r are the lower and upper approximation operators t r and t r respectively moreover r t r t r where r t r is the relation induced by the topology t r 3 when r is a reflexive and symmetric relation r and e r induce the same topology i e t r t e r the interior and closure operators of the topology t r induced by r are the lower and upper approximation operators e r and e r respectively moreover r t r e r 4 based on the above conclusions the notion of topological reduction of incomplete information systems is proposed and characterizations of reduction of consistent incomplete decision tables are obtained

categorical properties of m indiscernibility spaces this paper discusses categorical aspect of the pawlak rough set theory it is proved that the category of all m indiscernibility spaces and m equivalence relation preserving mappings between them is both a topological construct and a topos as an application of these results the notions of product m indiscernibility space sum m indiscernibility space quotient m indiscernibility space m indiscernibility subspace quotient mapping and isomorphism mapping are defined and structures of these m indiscernibility spaces and mappings are also given

an application of covering approximation spaces on network security based on the pawlak rough set theory this paper investigates separations in covering approximation spaces give some characterizations of these separations and some relations among these separations as an application of these results investigations on network security are converted into investigations on separations in covering approximation spaces by taking covering approximation spaces as mathematical models of networks results of this paper give further applications of the pawlak rough set theory in pattern recognition and artificial intelligence which makes it possible to research network security by logical methods and mathematical methods in computer science this contributes to giving risk assessments of securities and to raise grades of securities for networks

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

topological properties of generalized approximation spaces rough set theory is a powerful mathematical tool for dealing with inexact uncertain or vague information the core concepts of rough set theory are information systems and approximation operators of approximation spaces approximation operators draw close links between rough set theory and topology this paper concerns generalized approximation spaces via topological methods and studies topological properties of rough sets classical separation axioms compactness and connectedness for topological spaces are extended to generalized approximation spaces relationships among separation axioms for generalized approximation spaces and relationships between topological spaces and their induced generalized approximation spaces are investigated an example is given to illustrate a new approach to recover missing values for incomplete information systems by regularity of generalized 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

topological approaches to covering rough sets rough sets a tool for data mining deal with the vagueness and granularity in information systems this paper studies covering based rough sets from the topological view we explore the topological properties of this type of rough sets study the interdependency between the lower and the upper approximation operations and establish the conditions under which two coverings generate the same lower approximation operation and the same upper approximation operation lastly axiomatic systems for the lower approximation operation and the upper approximation operation are constructed

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

reduction and axiomization of covering generalized rough sets this paper investigates some basic properties of covering generalized rough sets and their comparison with the corresponding ones of pawlak s rough sets a tool for data mining the focus here is on the concepts and conditions for two coverings to generate the same covering lower approximation or the same covering upper approximation the concept of reducts of coverings is introduced and the procedure to find a reduct for a covering is given it has been proved that the reduct of a covering is the minimal covering that generates the same covering lower approximation or the same covering upper approximation so this concept is also a technique to get rid of redundancy in data mining furthermore it has been shown that covering lower and upper approximations determine each other finally a set of axioms is constructed to characterize the covering lower approximation operation

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

a characteristic of local existence for nonlinear fractional heat equations in lebesgue spaces abstract in this paper we consider the fractional heat equation u t 2 u f u with dirichlet conditions on the ball b r r d where 2 is the fractional laplacian f 0 0 is continuous and non decreasing we present the characterisations of f to ensure the equation has a local solution in l q b r provided that the non negative initial data u 0 l q b r for q 1 and 1 2 we show that the equation has a local solution in l q b r if and only if lim s sup s 1 q d f s and for q 1 and 1 2 if and only if 1 s 1 d f s d s where f s sup 1 t s f t t when lim s 0 f s s the same characterisations holds for the fractional heat equation on the whole space r d

approximation of sets based on partial covering in this current paper we reveal a mathematical tool that helps us to comprehend certain natural phenomena the main idea of this tool is a possible generalization of approximations of sets relying on the partial covering of the universe of discourse our starting point will be an arbitrary nonempty family b of subsets of an arbitrary nonempty universe u on the analogy of the definition of pawlak s type s algebra s u e over a finite universe let d b denote the family of subsets of u which contains the empty set and every set in b and it is closed under unions however d bneither covers the universe nor is closed under intersections in general our notions of lower and upper approximations are straightforward point free generalizations of pawlak s same approximations which are imitations of the e equivalence class based formulations both of them belong to d b our discussion will be within an overall approximation framework along which the common features of rough set theory and our approach can be treated uniformly to demonstrate the relationship of our approach with natural computing we will show an example relying on the so called meta program which is a recognition and evaluation program of the actual state of the natural and semi natural vegetation heritage of hungary

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

covering based rough set approximations we propose a framework for the study of covering based rough set approximations three equivalent formulations of the classical rough sets are examined by using equivalence relations partitions and s algebras respectively they suggest the element based the granule based and the subsystem based definitions of approximation operators covering based rough sets are systematically investigated by generalizing these formulations and definitions a covering of universe of objects is used to generate different neighborhood operators neighborhood systems coverings and subsystems of the power set of the universe they are in turn used to define different types of generalized approximation operators within the proposed framework we review and discuss covering based approximation operators according to the element granule and subsystem based definitions

relationships between covering based rough sets and relation based rough sets rough set theory is an important technique to deal with vagueness and granularity in information systems in rough set theory relation based rough sets and covering based rough sets are two important extensions of the classical rough sets this paper investigates relationships between relation based rough sets and covering based rough sets firstly we unify five types of covering based rough sets under a framework of a type of relation based rough sets and establish the equivalences between four types of covering based rough sets and the type of relation based rough sets respectively then within the proposed framework conditions required for two different coverings to generate the same approximation operators are investigated we also present some sufficient and necessary conditions for different types of covering based rough sets to be equal furthermore two examples are taken to illustrate applications of the discussion about relationships between covering based rough sets and relation based rough sets

on noise robustness of the mexican hat and hyperbolic wavelets with an experiment on chaos detection this paper studies the noise robustness of laplace of hyperbolic loh and laplace of gaussian log wavelets a typical example of using the loh and log to study duffing oscillation under normal and noisy conditions is given relative merits of each method are discussed

approaches to knowledge reduction of covering decision systems based on information theory in this paper we propose some new approaches for attribute reduction in covering decision systems from the viewpoint of information theory firstly we introduce information entropy and conditional entropy of the covering and define attribute reduction by means of conditional entropy in consistent covering decision systems secondly in inconsistent covering decision systems the limitary conditional entropy of the covering is proposed and attribute reductions are defined and finally by the significance of the covering some algorithms are designed to compute all the reducts of consistent and inconsistent covering decision systems we prove that their computational complexity are polynomial numerical tests show that the proposed attribute reductions accomplish better classification performance than those of traditional rough sets in addition in traditional rough set theory mibark algorithm g y wang h hu d yang decision table reduction based on conditional information entropy chinese j comput 25 2002 1 8 cannot ensure the reduct is the minimal attribute subset which keeps the decision rule invariant in inconsistent decision systems here we solve this problem in inconsistent covering decision systems