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

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

zeroes of the spectral density of the schroedinger operator with the slowly decaying wigner von neumann potential we consider the schrodinger operator mathcal l alpha on the half line with a periodic background potential and a perturbation which consists of two parts a summable potential and the slowly decaying wigner von neumann potential frac c sin 2 omega x delta x gamma where gamma in frac 1 2 1 the continuous spectrum of this operator has the same band gap structure as the continuous spectrum of the unperturbed periodic operator in every band there exist two points called critical where the eigenfunction equation has square summable solutions every critical point nu cr is an eigenvalue of the operator mathcal l alpha for some value of the boundary parameter alpha alpha cr specific to that particular point we prove that for alpha ne alpha cr the spectral density of the operator mathcal l alpha has a zero of the exponential type at nu cr

analysis of a hybridized interface stabilized finite element method for the stokes equations stability and error analysis of a hybridized discontinuous galerkin finite element method for stokes equations is presented the method is locally conservative and for particular choices of spaces the velocity field is point wise solenoidal it is shown that the method is inf sup stable for both equal order and locally taylor hood type spaces and a priori error estimates are developed the considered method can be constructed to have the same global algebraic structure as a conforming galerkin method unlike standard discontinuous galerkin methods that have a greater number of degrees of freedom than conforming galerkin methods on a given mesh we assert that this method is among the simplest and most flexible finite element approaches for stokes flow that provide local mass conservation with this contribution the mathematical basis is established and this supports the performance of the method that has been observed experimentally in other works

on characterization of generalized interval type 2 fuzzy rough sets this paper presents a systematic study of interval type 2 fuzzy rough sets integrating rough set theory with interval type 2 fuzzy set theory using constructive and axiomatic approaches from the perspective of a constructive approach a pair of lower and upper interval type 2 fuzzy rough approximation operators with respect to an interval type 2 fuzzy relation is defined the basic properties of the interval type 2 fuzzy rough approximation operators are studied using cut sets of interval type 2 fuzzy sets classical representations of interval type 2 fuzzy rough approximation operators are then presented and the connections between special interval type 2 fuzzy relations and interval type 2 fuzzy rough approximation operators are investigated adopting an axiomatic approach an operator oriented characterization of interval type 2 fuzzy rough sets is proposed in other words interval type 2 fuzzy rough approximation operators are characterized by axioms different axiom sets of interval type 2 fuzzy set theoretic operators guarantee the existence of different types of interval type 2 fuzzy relations that produce the same operators finally the relationship between interval type 2 fuzzy rough sets and interval type 2 fuzzy topological spaces is examined we obtain sufficient and necessary conditions for the conjecture that an interval type 2 fuzzy interior closure operator derived from an interval type 2 fuzzy topological space can associate with an interval type 2 fuzzy reflexive and transitive relation such that the corresponding lower upper interval type 2 fuzzy rough approximation operator is the interval type 2 fuzzy interior closure operator

on v filters and normal v filters of a residuated lattice with a weak vt operator in this paper we study further the filter theory of residuated lattices first we discuss the concepts of filters and normal filters of residuated lattices and propose some equivalent conditions for them then we introduce and investigate the notions of v filter and normal v filter of a residuated lattice with a weak vt operator and lay bare the formulas for calculating the v filters and the normal v filters generated by subsets finally we show that the lattices of v filters and normal v filters of a residuated lattice with a vt operator are both complete brouwerian lattices

probabilistic model criteria with decision theoretic rough sets in dealing with risk in real decision problems decision theoretic rough sets with loss functions aim to obtain optimization decisions by minimizing the overall risk with bayesian decision procedures two parameters generated by loss functions divide the universe into three regions as the decision of acceptance deferment and rejection in this paper we discuss the semantics of loss functions and utilize the differences of losses replace actual losses to construct a new four level approach of probabilistic rules choosing criteria ten types of probabilistic rough set models can be generated by the four level approach and form two groups of models two way probabilistic decision models and three way probabilistic decision models a reasonable decision with these criteria is demonstrated by an illustration of oil investment

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

rough sets and intelligent data analysis rough set based data analysis starts from a data table called an information system the information system contains data about objects of interest characterized in terms of some attributes often we distinguish in the information system condition and decision attributes such information system is called a decision table the decision table describes decisions in terms of conditions that must be satisfied in order to carry out the decision specified in the decision table with every decision table a set of decision rules called a decision algorithm can be associated it is shown that every decision algorithm reveals some well known probabilistic properties in particular it satisfies the total probability theorem and the bayes theorem these properties give a new method of drawing conclusions from data without referring to prior and posterior probabilities inherently associated with bayesian reasoning

a rough set approach to attribute generalization in data mining abstract this paper presents a method for updating approximations of a concept incrementally the results can be used to implement a quasi incremental algorithm for learning classification rules from very large data bases generalized by dynamic conceptual hierarchies provided by users in general the process of attribute generalization may introduce inconsistency into a generalized relation this issue is resolved by using the inductive learning algorithm lers based on rough set theory

an axiomatic approach of fuzzy rough sets based on residuated lattices rough set theory was developed by pawlak as a formal tool for approximate reasoning about data various fuzzy generalizations of rough approximations have been proposed in the literature as a further generalization of the notion of rough sets l fuzzy rough sets were proposed by radzikowska and kerre in this paper we present an operator oriented characterization of l fuzzy rough sets that is l fuzzy approximation operators are defined by axioms the methods of axiomatization of l fuzzy upper and l fuzzy lower set theoretic operators guarantee the existence of corresponding l fuzzy relations which produce the operators moreover the relationship between l fuzzy rough sets and l topological spaces is obtained the sufficient and necessary condition for the conjecture that an l fuzzy interior closure operator derived from an l fuzzy topological space can associate with an l fuzzy reflexive and transitive relation such that the corresponding l fuzzy lower upper approximation operator is the l fuzzy interior closure operator is examined

t type regular l relations on a complete brouwerian lattice abstract in this paper using a general infinitely distributive t norm t on a complete brouwerian lattice l we introduce the concept of t type regular l relations study some basic properties of them and give out some methods of calculating the maximal t type generalized inverse l relations of t type regular l relations

data classification based on tolerant rough set abstract this paper proposes a new data classification method based on the tolerant rough set that extends the existing equivalent rough set similarity measure between two data is described by a distance function of all constituent attributes and they are defined to be tolerant when their similarity measure exceeds a similarity threshold value the determination of optimal similarity threshold value is very important for the accurate classification so we determine it optimally by using the genetic algorithm ga where the goal of evolution is to balance two requirements such that 1 some tolerant objects are required to be included in the same class as many as possible and 2 some objects in the same class are required to be tolerable as much as possible after finding the optimal similarity threshold value a tolerant set of each object is obtained and the data set is grouped into the lower and upper approximation set depending on the coincidence of their classes we propose a two stage classification method that all data are classified by using the lower approximation at the first stage and then the non classified data at the first stage are classified again by using the rough membership functions obtained from the upper approximation set the validity of the proposed classification method is tested by applying it to the iris data classification and its classification performance and processing time are compared with those of other classification methods such as bpnn ofunn and fcm

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

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

positive approximation and converse approximation in interval valued fuzzy rough sets methods of fuzzy rule extraction based on rough set theory are rarely reported in incomplete interval valued fuzzy information systems thus this paper deals with such systems instead of obtaining rules by attribute reduction which may have a negative effect on inducting good rules the objective of this paper is to extract rules without computing attribute reducts the data completeness of missing attribute values is first presented positive and converse approximations in interval valued fuzzy rough sets are then defined and their important properties are discussed two algorithms based on positive and converse approximations namely mine rules based on the positive approximation mrbpa and mine rules based on the converse approximation mrbca are proposed for rule extraction the two algorithms are evaluated by several data sets from the uc irvine machine learning repository the experimental results show that mrbpa and mrbca achieve better classification performances than the method based on attribute reduction

rough set theory applied to lattice theory in this paper we intend to study a connection between rough sets and lattice theory we introduce the concepts of upper and lower rough ideals filters in a lattice then we offer some of their properties with regard to prime ideals filters the set of all fixed points compact elements and homomorphisms

fuzzy modifiers based on fuzzy relations in this paper we introduce a new type of fuzzy modifiers i e mappings that transform a fuzzy set into a modified fuzzy set based on fuzzy relations we show how they can be applied for the representation of weakening adverbs more or less roughly and intensifying adverbs very extremely in the inclusive and the non inclusive interpretation we illustrate their use in an approximate reasoning scheme

textures and covering based rough sets this paper introduces the relations between covering and dicovering in the framework of rough sets the natural counterpart of a covering in textures is given by a family of pairs of sets which is called a dicovering this leads us to two dual concepts which are called star and co star of a set respectively here we define dicovering approximation space and we discuss on covering and dicovering based rough sets in this context star corresponds to the upper approximation operator defined by zakowski and co star corresponds to the lower approximation operator studied by pomykala yao and zhu and wang

approximations and uncertainty measures in incomplete information systems there are mainly two methodologies dealing with uncertainty measurement issue in rough set theory pure rough set approach and information theory approach pure rough set approach is based on the concepts of accuracy roughness and approximation accuracy proposed by pawlak information theory approach is based on shannon s entropy or its variants several authors have extended the information theory approach into incomplete information systems however there are few studies on extending the pure rough set approach to incomplete information systems this paper focuses on constructing uncertainty measures in incomplete information systems by pure rough set approach three types of definitions of lower and upper approximations and corresponding uncertainty measurement concepts including accuracy roughness and approximation accuracy are investigated theoretical analysis indicates that two of the three types can be used to evaluate the uncertainty in incomplete information systems experiments on incomplete real life data sets have been conducted to test the two selected types the first type and the third type of uncertainty measures results show that the two types of uncertainty measures are effective

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

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

generalized fuzzy rough sets determined by a triangular norm the theory of rough sets has become well established as an approach for uncertainty management in a wide variety of applications various fuzzy generalizations of rough approximations have been made over the years this paper presents a general framework for the study of t fuzzy rough approximation operators in which both the constructive and axiomatic approaches are used by using a pair of dual triangular norms in the constructive approach some definitions of the upper and lower approximation operators of fuzzy sets are proposed and analyzed by means of arbitrary fuzzy relations the connections between special fuzzy relations and the t upper and t lower approximation operators of fuzzy sets are also examined in the axiomatic approach an operator oriented characterization of rough sets is proposed that is t fuzzy approximation operators are defined by axioms different axiom sets of t upper and t lower fuzzy set theoretic operators guarantee the existence of different types of fuzzy relations producing the same operators the independence of axioms characterizing the t fuzzy rough approximation operators is examined then the minimal sets of axioms for the characterization of the t fuzzy approximation operators are presented based on information theory the entropy of the generalized fuzzy approximation space which is similar to shannon s entropy is formulated to measure uncertainty in t generalized fuzzy rough sets a notion of fuzziness is introduced some basic properties of this measure are examined for a special triangular norm t min it is proved that the measure of fuzziness of the generalized fuzzy rough set is equal to zero if and only if the set is crisp and definable

boolean filters and positive implicative filters of residuated lattices the aim of this paper is to introduce the notions of boolean filters and positive implicative filters in residuated lattices and to investigate their properties several characterizations of boolean filters and positive implicative filters are derived the extension theorems of implicative filters and positive implicative filters are obtained the relations among boolean filters implicative filters and positive implicative filters are investigated and it is proved that boolean filters are equivalent to implicative filters and that every boolean filter is a positive implicative filter but the converse may not be true furthermore the conditions under which a positive implicative filter is a boolean filter are established

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