core set analysis in inconsistent decision tables core set inconsistency always causes confusion regarding how to select the proper core set for data reduction in inconsistent decision tables in this paper partitions of knowledge granules are introduced to analyze this inconsistency and it is concluded that there are only three types of effective partitions those that focus on exact information those that focus on exact partial and negative information and those that focus on exact partial negative and probabilistic information all useful core sets are calculated systematically by converting the three types of partitions to corresponding discernibility matrices then we define three types of rules positive inexact and confidence rules based on the three types of partitions using these rules an intelligible rule based strategy is proposed to select the proper core set for a practical application which resolves the confusion caused by core set inconsistency and completes the process of data reduction experimental analysis and industrial results demonstrate the effectiveness of the selection strategy

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

defining rough sets using tolerances compatible with an equivalence abstract we consider tolerances t compatible with an equivalence e on u meaning that the relational product e t is included in t we present the essential properties of e compatible tolerances and study rough approximations defined by such e and t we consider rough set pairs xe xt where the lower approximation xe is defined as is customary in rough set theory but xt allows more elements to be possibly in x than xe motivating examples of e compatible tolerances are given and the essential lattice theoretical properties of the ordered set of rough sets xe xt x u are established

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

examples of smash nilpotent cycles on rationally connected varieties abstract voevodsky has conjectured that numerical and smash equivalence coincide on a smooth projective variety we prove this conjecture holds for uniruled 3 folds 4 folds whose mrcc quotient has dimension 2 and for smooth complete intersections with very small degree

relative reducts in consistent and inconsistent decision tables of the pawlak rough set model a relative reduct can be considered as a minimum set of attributes that preserves a certain classification property this paper investigates three different classification properties and suggests three distinct definitions accordingly in the pawlak rough set model while the three definitions yield the same set of relative reducts in consistent decision tables they may result in different sets in inconsistent tables relative reduct construction can be carried out based on a discernibility matrix the study explicitly stresses a fact that the definition of a discernibility matrix should be tied to a certain property regarding the three classification properties we can define three distinct definitions accordingly based on the common structure of the specific definitions of relative reducts and discernibility matrices general definitions of relative reducts and discernibility matrices are suggested

approaches to knowledge reduction based on variable precision rough set model this paper deals with approaches to knowledge reduction based on variable precision rough set model the concepts of lower distribution reduct and upper distribution reduct based on variable precision rough sets vprs are first introduced their equivalent definitions are then given and the relationships among lower and upper distribution reducts and alternative types of knowledge reduction in inconsistent systems are investigated it is proved that for some special thresholds lower distribution reduct is equivalent to the maximum distribution reduct whereas upper distribution reduct is equivalent to the possible reduct the judgement theorems and discernibility matrices associated with the lower and upper distribution reducts are also established from which we can obtain the approaches to knowledge reduction in vprs

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

attribute reduction based on evidence theory in incomplete decision systems attribute reduction is a basic issue in knowledge representation and data mining this paper deals with attribute reduction in incomplete information systems and incomplete decision systems based on dempster shafer theory of evidence the concepts of plausibility reduct and belief reduct in incomplete information systems as well as relative plausibility reduct and relative belief reduct in incomplete decision systems are introduced it is shown that in an incomplete information system an attribute set is a belief reduct if and only if it is a classical reduct and a plausibility consistent set must be a classical consistent set in a consistent incomplete decision system the concepts of relative reduct relative plausibility reduct and relative belief reduct are all equivalent in an inconsistent incomplete decision system an attribute set is a relative plausibility reduct if and only if it is a relative reduct a plausibility consistent set must be a belief consistent set and a belief consistent set is not a plausibility consistent set in general

characterizing reducts in multi adjoint concept lattices abstract the construction of reducts that is minimal sets of attributes containing the main information of a database is a fundamental task in different frameworks such as in formal concept analysis fca and rough set theory rst this paper will be focused on a general fuzzy extension of fca called multi adjoint concept lattice and we present a study about the attributes generating meet irreducible elements and on the reducts in this framework from this study we introduce interesting results on the cardinality of reducts and the consequences in the classical case

data analysis based on discernibility and indiscernibility rough set theory models similarities and difierences of objects based on the notions of indiscernibility and discernibility with respect to any subset of attributes one can deflne two pairs of dual relations the strong indiscernibility and weak discernibility relations and the weak indiscernibility and strong discernibility relations the similarities of objects are examined by the indiscernibility relations and the difierence by the discernibility relations respectively alternatively one can construct an indiscernibility matrix to represent the family of strong indiscernibility or weak discernibility relations one also can construct a discernibility matrix to represent the family of strong discernibility or weak indiscernibility relations the consideration of the matrix counterpart of relations and the relation counterpart of matrices brings more insights into rough set theory based on indiscernibility and discernibility three difierent types of reducts can be constructed keeping the indiscernibility discernibility and indiscernibility anddiscernibility relations respectively although the indiscernibility reducts have been intensively studied in the literature the other two types of reducts are relatively new and require more attention the existing methods for constructing the indiscernibility reducts also can be applied to construct the other two types of reducts an empirical experiment for letter recognition is reported for demonstrating the usefulness of the discussed relations and reducts

class specific attribute reducts in rough set theory abstract the concept of attribute reducts plays a fundamental role in rough set analysis there are at least two possibilities to define an attribute reduct a classification based or global attribute reduct is a minimal subset of condition attributes that preserves the positive region of the decision classification namely the positive regions of all decision classes in a decision table a class specific class dependent or local attribute reduct is a minimal subset of condition attributes that preserves the positive region of a particular decision class while a classification based reduct may not work equally well for every decision class a class specific attribute reduct is optimally tailored to a particular decision class however studies in rough set theory are dominated by classification based reducts there is very limited investigation on class specific reducts an objective of this paper is to draw attention to class specific reducts we systematically compare the two types of reducts and investigate their relationships with respect to both individual reducts and families of all reducts it is possible to derive a class specific reduct from a classification based reduct and to derive a classification based reduct from a family of class specific reducts the families of all class specific reducts provide a pair of lower and upper bounds of the family of all classification based reducts based on a three way classification of attributes into the pair wise disjoint sets of core marginal and nonuseful attributes we examine relationships between the corresponding classes of classification based and class specific attributes the union of the sets of class specific core attributes is the set of classification based core attributes it is only possible to obtain an upper bound for the set of classification based marginal attributes and a lower bound for the set of classification based nonuseful attributes from the family of the class specific corresponding sets of attributes

dynamic maintenance of approximations in set valued ordered decision systems under the attribute generalization set valued information systems are important type of data tables and generalized models of single valued information systems approximations are the focal point of approaches to knowledge discovery based on rough set theory which can be used to extract and represent the hidden knowledge in the form of decision rules attribute generalization refers to dynamic change of the attribute set in an information system with respect to the requirements of real life applications in this paper we focus on maintaining approximations dynamically in set valued ordered decision systems under the attribute generalization firstly a matrix based approach for computing approximations of upward and downward unions of decision classes is constructed by introducing the dominant and dominated matrices with respect to the dominance relation then incremental approaches for updating approximations are proposed which involves several modifications to relevant matrices without having to retrain from the start on all accumulated training data finally comparative experiments on data sets from uci as well as synthetic data sets show the proposed incremental updating methods are efficient and effective for dynamic attribute generalization

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

theory and applications of granular labelled partitions in multi scale decision tables abstract granular computing and acquisition of if then rules are two basic issues in knowledge representation and data mining a formal approach to granular computing with multi scale data measured at different levels of granulations is proposed in this paper the concept of labelled blocks determined by a surjective function is first introduced lower and upper label block approximations of sets are then defined multi scale granular labelled partitions and multi scale decision granular labelled partitions as well as their derived rough set approximations are further formulated to analyze hierarchically structured data finally the concept of multi scale information tables in the context of rough set is proposed and the unravelling of decision rules at different scales in multi scale decision tables is discussed

dominance based rough set approach and knowledge reductions in incomplete ordered information system many methods based on the rough set to deal with incomplete information systems have been proposed in recent years however they are only suitable for the incomplete systems with regular attributes whose domains are not preference ordered this paper thus attempts to present research focusing on a complex incomplete information system the incomplete ordered information system in such incomplete information systems all attributes are considered as criterions a criterion indicates an attribute with preference ordered domain to conduct classification analysis in the incomplete ordered information system the concept of similarity dominance relation is first proposed two types of knowledge reductions are then formed for preserving two different notions of similarity dominance relations with introduction of the approximate distribution reduct into the incomplete ordered decision system the judgment theorems and discernibility matrixes associated with four novel approximate distribution reducts are obtained a numerical example is employed to substantiate the conceptual arguments

w2 1 regularity for solutions of the monge ampere equation in this paper we prove that a strictly convex alexandrov solution u of the monge ampere equation with right hand side bounded away from zero and infinity is w 2 1 mathrm loc this is obtained by showing higher integrability a priori estimates for d 2 u namely d 2 u llog k l for any k

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 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

efficient attribute reduction from the viewpoint of discernibility attribute reduction is an important preprocessing step in pattern recognition machine learning and data mining as an effective method for attribute reduction rough set theory offers a useful and formal methodology it retains the discernibility power of the original datasets thus attribute reduction has been extensively studied in rough set theory however the inefficiency of the existing attribute reduction algorithms limits the application of rough sets in this paper we first analyse the limitations of existing attribute reduction algorithms then a novel measure of attribute quality called the relative discernibility degree is proposed based on the discernibility theoretical analysis shows that this measure can find relative dispensable attributes and remain unchanged after removing the relative dispensable attributes and redundant objects in the process of selecting attributes this property can be used to reduce the search space and accelerate the heuristic process of attribute reduction consequently a new attribute reduction algorithm is proposed from the viewpoint of discernibility furthermore the relationships among the reduction definitions of the algebra view information view and discernibility view are derived some non equivalent relationships among these views of rough set theory in inconsistent decision tables are discovered a set of numerical experiments was conducted on uci datasets experimental results show that the proposed algorithm is effective and efficient and is applicable to the case of large scale datasets

semantic analysis of inductive reasoning abstract inductive learning is analysed from the semantic point of view processes of forming generalisations of concepts determined by examples of expert decisions are discussed it is claimed that since concepts are relative to background knowledge their inductive generalisations can be determined only approximately induction rules are defined providing a method of forming generalisations preserving positive and or negative instances of concepts

communication between information systems using fuzzy rough sets communication between information systems is a basic problem in granular computing and the concept of homomorphism is a useful mathematical tool to study this problem in this paper some properties of communication between information systems based on fuzzy rough sets are investigated the concepts of fuzzy relation mappings between universes are first proposed in order to construct a fuzzy relation of one universe according to the given fuzzy relation on the other universe the main properties of the mappings are studied the notions of homomorphism of information systems based on fuzzy rough sets are then proposed and it is proved that properties of relation operations in the original information system and structural features of the system such as approximations of arbitrary fuzzy sets and attribute reductions are guaranteed in its image system under the condition of homomorphism

attribute selection with fuzzy decision reducts rough set theory provides a methodology for data analysis based on the approximation of concepts in information systems it revolves around the notion of discernibility the ability to distinguish between objects based on their attribute values it allows to infer data dependencies that are useful in the fields of feature selection and decision model construction in many cases however it is more natural and more effective to consider a gradual notion of discernibility therefore within the context of fuzzy rough set theory we present a generalization of the classical rough set framework for data based attribute selection and reduction using fuzzy tolerance relations the paper unifies existing work in this direction and introduces the concept of fuzzy decision reducts dependent on an increasing attribute subset measure experimental results demonstrate the potential of fuzzy decision reducts to discover shorter attribute subsets leading to decision models with a better coverage and with comparable or even higher accuracy

cost sensitive rough set approach we consider test cost and decision cost simultaneously cost sensitive rough set was proposed and explored attribute reductions based on three different criteria were investigated cost sensitivity is an important problem which has been addressed by many researchers around the world as far as cost sensitivity in the rough set theory is concerned two types of important costs have been seriously considered on the one hand the decision cost has been introduced into the modeling of decision theoretic rough set on the other hand the test cost has been taken into account in attribute reduction however few researchers pay attention to the construction of rough set model with test cost and decision cost simultaneously to fill such a gap a new cost sensitive rough set approach is proposed in which the information granules are sensitive to test costs and approximations are sensitive to decision costs respectively furthermore with respect to different criteria of positive region preservation decision monotonicity and cost decrease three heuristic algorithms are designed to compute reducts respectively the comparisons among these three algorithms show us 1 positive region preservation based algorithm can keep the decision rules supported by lower approximation region unchanged 2 decision monotonicity based heuristic algorithm can obtain a reduct with more positive decision rules and higher classification accuracy 3 cost minimum based algorithm can generate a reduct with minor cost this study suggests potential application areas and new research trends concerning rough set theory