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Fuzzy rough set models over two universes using textures
Abstract
The purpose of this paper is twofold: firstly, to exhibit the existence of a reasonable connection between fuzzy relations and (textural) fuzzy direlations in terms of fuzzy logical connectives; and secondly, to give a new perspective with respect to approximation operators due to fuzzy rough set models over two universes using textures. To this end, we investigate the serialities of fuzzy relations in terms of fuzzy logic connectives using fuzzy direlations. For different fuzzy logic systems, we show that the textural serialities provide alternative descriptions of the serialities of a fuzzy relation in terms of parameters. Without considering the motivation of the semantic link with the classical Pawlak rough sets, we obtain and discuss the natural fuzzifications of rough set models over two universes. Further, we execute that a textural fuzzy rough set model is an efficient template for the basic properties of the various fuzzy rough set models over two universes. For instance, we prove that the revised fuzzy rough set approximations with two universes are the natural extensions of the loose and tight approximations of De Cock et al.
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Dynamic relational systems have different forms in literature of rough set theory. They are divided into two parts by Pagliani as synchronic and diachronic dynamics. Synchronic dynamic case is related to the presence of a multi-sources. In the case of diachronic dynamics, it is supposed that changes occur in time. These changes are related to new objects, attributes or attribute values entered into the system. Here, we consider the dynamic systems which are related to the both of these types of dynamics. In a recent paper, the author studied on the results about reduct, definability and quasi-uniformity in dynamic relational systems. Here, the natural generalizations of these results are given. In particular, it is proved that weak and strong definabilities are preserved under pre-images with respect to direlation preserving difunctions between textural dynamic relational systems. Further, the connections between definable sets and the ditopology are determined by ditopologies of reflexive direlations. Then it is given some basic results on ditopologies induced by direlations and direlational quasi-uniformities. Morover, it is discussed on the connections between textural approximation spaces and direlational-quasi uniformities.
The notion of a rough set introduced by Pawlak has often been compared to that of a fuzzy set, sometimes with a view to prove that one is more general, or, more useful than the other. In this paper we argue that both notions aim to different purposes. Seen this way, it is more natural to try to combine the two models of uncertainty (vagueness and coarseness) rather than to have them compete on the same problems. First, one may think of deriving the upper and lower approximations of a fuzzy set, when a reference scale is coarsened by means of an equivalence relation. We then come close to Caianiello's C-calculus. Shafer's concept of coarsened belief functions also belongs to the same line of thought. Another idea is to turn the equivalence relation into a fuzzy similarity relation, for the modeling of coarseness, as already proposed by Farinas del Cerro and Prade. Instead of using a similarity relation, we can start with fuzzy granules which make a fuzzy partition of the reference scale. The main contribution of the paper is to clarify the difference between fuzzy sets and rough sets, and unify several independent works which deal with similar ideas in different settings or notations.
A novel fuzzy rough granular neural network (NFRGNN) based on the multilayer perceptron using back-propagation algorithm is described for fuzzy classification of patterns. We provide a development strategy of knowledge extraction from data using fuzzy rough set theoretic techniques. Extracted knowledge is then encoded into the network in the form of initial weights. The granular input vector is presented to the network while the target vector is provided in terms of membership values and zeros. The superiority of NFRGNN is demonstrated on several real life data sets.
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 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 of (X,R). Our definition satisfies the two important identities and , as well as a number of other interesting properties. We provide axiomatics to fully characterize those upper and lower approximations.
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.
This is the first of three papers which develop various fundamental aspects of the theory of ditopological texture spaces in a categorical setting and present important links with the theory of L-topological spaces. The authors begin by defining the notion of q-sets, which together with the p-sets considered earlier, enable the formulation of a powerful concept of duality. This plays an important role in the theory of direlations and difunctions, which is described here in detail. Difunctions are then taken as the morphisms of a category dfTex, whose objects are texture spaces. Several important subcategories are defined and the closely related construct fTex defined. Some properties of the functors between these categories are obtained.
In this paper, the dependence spaces are discussed for textural formal concepts considering the method given by Ma et al. A complete congruence on a complete lattice is an equivalence relation if it satisfies the infinite substitution property. More generally, a join-dependence and a meet-dependence space with respect to infinite domain of discourse are presented. Using the duality in textures, the closure and interior operators are defined to obtain the intensions and co-intensions of concept lattices, respectively. The main theorem for dual formal concept lattices given by Chen and Yao is stated. Further, it is shown that the co-intensions of a dual formal concept lattice can be obtained using an interior operator. Finally, the independency notion of Novotný for t-formal concepts is discussed.
Some fuzzy rough sets only consider relative error limit and others are still sensitive to mislabeled samples. Considering the absolute error limit, Fang and Hu [11] proposed granular variable precision fuzzy rough sets based on fuzzy (co)implications to remedy these defects, which are suitable to the databases with errors in conditional attribute(s) and decision making. However, there are some faults in the characterizations of granular variable precision fuzzy rough sets based on fuzzy (co)implications presented by Fang and Hu, such as false conclusions, insufficient and redundant conditions. In this paper, we further discuss the equivalent expressions of granular variable precision σ-lower and θ-upper approximation operators with the proper semicontinuity of fuzzy (co)implications. Based on those new equivalent expressions of granular variable precision σ-lower and θ-upper approximation operators, the composition of granular variable precision fuzzy rough sets based on fuzzy (co)implications is studied with respect to a general fuzzy relation. Moreover, we further study granular variable precision fuzzy rough sets based on fuzzy (co)implications to rectify those faults mentioned above.
The aim of this paper is to reveal the useful aspects of textures for generalized approximation spaces in rough set theory. To this end, we present a counterpart of rough sets over two universes for textures. We compare the textural results with the well-known basic properties of rough sets given in the literature. First, we define a t-seriality and a t-inverse seriality in textures. We show that almost all basic results with respect to rough set model over two universes can be formulated using textures. On the way, we give new results and observations due to lower and upper approximation operators which are not taken into consideration by the researchers. Moreover, we also discuss the revised approximation operators and attribute oriented formal concept lattices in a textural framework.
In this paper, we discuss formal context in textures. A texturing is a family of subsets of a domain of discourse satisfying certain conditions. Considering a t-formal context, we formulate the notions of extent and intent in terms of p-sets and q-sets. Then we define t-formal concept and t-formal co-concept. For complemented direlations, we prove that they give two different concept lattices which are dually isomorphic to each other. These lattices correspond to a concept lattice in the sense of R. Wille and a dual concept lattice given by Düntsh and Gediga, respectively. In particular, we observe that a texturing, as an imperfect collection of sets of a domain of discourse, provides a remarkable setting which still ensures information with respect to given system. Finally, we prove the main theorem of formal concept analysis for textures.
By means of a fuzzy coimplication operator J and a triangular conorm S, we set forth two pairs of (J,S)-fuzzy rough set models, which are generalizations of fuzzy rough sets. Then, according to the classifications of the coimplication operators (R-coimplicators and T-coimplicators), we investigate relationships among our proposed models and some existing rough set models. In this paper, by the idea of the PROMETHEE method, we apply (J,S)-fuzzy rough set models to make decisions with evaluation of fuzzy information. An example illustrates the feasibility and effectiveness of our proposed method to solve practical problems. By comparing the ranking results of the existing methods and our proposed method, we observe that the optimal selected alternative is the same, which means that there is a consistency among our proposed decision-making method and the existing methods. In addition, we find that the traditional methods may fail in some practical situations while our proposed method is still valid.
This paper introduces granular fuzzy rough sets in the view of fuzzy implicators and fuzzy coimplicators, and discusses the constructive and axiomatic approach to fuzzy granules based on fuzzy implicators and coimplicators. Moreover, we study the connection between fuzzy granules and fuzzy relations and discuss the relationship between existing granular fuzzy rough set models and that proposed in this paper. Considering the absolute error limit, we introduce the concept of the granular variable precision fuzzy rough sets based on fuzzy implicators and coimplicators. Then we present four propositions to ensure that the approximation operators can be efficiently calculated.
In this study, we introduce a textural counterpart of the unit operation of Wybraniec-Skardowska. A unit di-operation has two parts which are called a unit co-operation and a unit operation, respectively. In this respect, we have two types of symmetry. We show that the symmetricity of direlations is equivalent to the Galois connectivity of unit di-operations. For discrete textures, we determine the image co-operation of a unit co-operation. We prove that the symmetricity and duality are equivalent concepts for unit di-operations if one of the compounds is symmetric. We consider definability in terms of unit operations and unit co-operations. Further, we present a categorical discussion defining a category UN whose structures of objects are unit operations. We show that the category Rel whose objects are approximation spaces and morphisms are relation preserving functions can be embedded into UN.
In this paper, an approach for the fuzzy rough set models is presented using textures and a fuzzy version of the unit operations of Wybraniec–Skardowska. First, a fuzzy unit operation and fuzzy unit co-operation on fuzzy lattices are defined. It is proved that the well-known fuzzy rough set upper approximations as the approximation of Dubois and Prade are fuzzy unit operation. Using fuzzy direlations, an axiomatic system for fuzzy unit operations is studied and it is shown that fuzzy rough set systems obtained by different fuzzy logical connectives as Kleene–Dienes and Gödel implicators can be generated by the same textural fuzzy direlation. Finally, it is observed that the approximations of two different fuzzy rough set models together constitute two different Galois connections.
This book provides a uniform framework describing how fuzzy rough granular neural network technologies can be formulated and used in building efficient pattern recognition and mining models. It also discusses the formation of granules in the notion of both fuzzy and rough sets. Judicious integration in forming fuzzy-rough information granules based on lower approximate regions enables the network to determine the exactness in class shape as well as to handle the uncertainties arising from overlapping regions, resulting in efficient and speedy learning with enhanced performance. Layered network and self-organizing analysis maps, which have a strong potential in big data, are considered as basic modules,.
The book is structured according to the major phases of a pattern recognition system (e.g., classification, clustering, and feature selection) with a balanced mixture of theory, algorithm, and application. It covers the latest findings as well as directions for future research, particularly highlighting bioinformatics applications. The book is recommended for both students and practitioners working in computer science, electrical engineering, data science, system design, pattern recognition, image analysis, neural computing, social network analysis, big data analytics, computational biology and soft computing.
In this work, we discuss the neighbourhoods and approximation operators using p-sets and q-sets of a texture. Here, we show that the presections of a direlation correspond to lower and upper approximations in terms of successor neighbourhood operators while the sections of a direlation correspond to lower and upper approximations in terms of predecessor neighbourhood operators. For discrete textures, we observe that the weak forms of definabilities are preserved under the relation preserving bijective functions where the inverses are also relation preserving.
Ever since the first hybrid fuzzy rough set model was proposed in the early 1990’s, many researchers have focused on the definition of the lower and upper approximation of a fuzzy set by means of a fuzzy relation. In this paper, we review those proposals which generalize the logical connectives and quantifiers present in the rough set approximations by means of corresponding fuzzy logic operations. We introduce a general model which encapsulates all of these proposals, evaluate it w.r.t. a number of desirable properties, and refine the existing axiomatic approach to characterize lower and upper approximation operators.
The variable precision -fuzzy rough sets were proposed to remedy the defects of preexisting fuzzy rough set models. However, the variable precision -fuzzy rough sets were only defined and investigated on fuzzy ⁎-similarity relations. In this paper, the granular variable precision fuzzy rough sets with general fuzzy relations are proposed on arbitrary fuzzy relations. The equivalent expressions of the approximation operators are given with fuzzy (co)implications on arbitrary fuzzy relations, which can calculate efficiently the approximation operators. The granular variable precision fuzzy rough sets are characterized from the constructive approach, which are investigated on different fuzzy relations. The conclusions on the variable precision -fuzzy rough sets are generalized into the granular variable precision fuzzy rough sets.
The category Rel whose objects are all pairs (U,r)(U,r), where r is a relation on a universe U, and whose morphisms are relation-preserving mappings is a canonical example in category theory. One of the convenient categories for rough set systems on a single universe is Rel since the objects of Rel are approximation spaces. The morphisms of a ground category dfTex whose objects are textures can be characterized by definability. Therefore, we particularly investigate a textural counterpart of the category Rel denoted by diRel of textural approximation spaces and direlation preserving difunctions. In this respect, we prove that diRel is a topological category over dfTex and Rel is a full subcategory of diRel. In view of the textural arguments, we show that the preimage of a definable subset of an approximation space with respect to a relation preserving function is also definable in the category Rere of reflexive relations. Furthermore, we denote the category of all information system homomorphisms and all information systems by IS and we show that the category ISO of all information system homomorphisms and all object-irreducible information systems where the attribute functions are surjective is embeddable into Rel.
Many different proposals exist for the definition of lower and upper approximation operators in covering-based rough sets. In this paper, we establish relationships between the most commonly used operators, using especially concepts of duality, conjugacy and adjointness (also referred to as Galois connection). We highlight the importance of the adjointness condition as a way to provide a meaningful link, aside from duality, between a pair of approximation operators. Moreover, we show that a pair of a lower and an upper approximation operator can be dual and adjoint at the same time if and only if the upper approximation is self-conjugate, and we relate this result to a similar characterization obtained for the generalized rough set model based on a binary relation.
Some expanded fuzzy rough set models have been investigated to handle fuzzy databases with uncertain, imprecise and incomplete real-valued information. However, some of them are still sensitive to mislabeled samples and others have only considered relative errors. To remedy these defects, we propose a novel expanded fuzzy rough set model called the variable precision (θ,σ)(θ,σ)-fuzzy rough set model based on fuzzy granules. Considering the absolute error limit, we introduce the concept of the variable precision (θ,σ)(θ,σ)-fuzzy rough set firstly. Then we present a theorem to ensure that the approximation operators can be calculated efficiently. The basic properties of the model are also investigated, some of which are analogous to those of fuzzy rough sets. Furthermore, the degenerated variable precision (θ,σ)(θ,σ)-fuzzy rough set model is shown, and the difference and connection between the degenerated model and the variable precision rough set model introduced by Ziarko are studied. Finally, the definitions of β-fuzzy lower and upper approximation reducts are presented and the attribute reduction methods are proposed.
This paper aims to give a new perspective for definability in rough set theory. First, a counterpart of definability is introduced in textural approximation spaces. Then a complete field of sets for texture spaces is defined and using textural arguments, some new results are obtained for rough sets. It is shown that definability can be also discussed in terms of a complete field of fuzzy sets on a fuzzy lattice for the various fuzzy approximation spaces. It is also given a partial affirmative answer to an open problem posed by Wei-Zhi Wu in On some mathematical structures of T-fuzzy rough set algebras in infinite universes of discourse in Fundamenta Informaticae 108 (3–4) 2011 337–369.
We investigate in this paper approximate operations on sets, approximate equality of sets, and approximate inclusion of sets. The presented approach may be considered as an alternative to fuzzy sets theory and tolerance theory. Some applications are outlined.
This paper focuses on rough set models on two universes and reveals some new properties. A revised rough set model is proposed which provides a new selection of interval structures for uncertain reasoning using rough set theory and methods.
The authors continue the development of a theory of texture spaces, introducing complemented products and sums, and applying these in a series of representation theorems for fuzzy lattices and the lattices of double struck L sign-fuzzy sets, generalized fuzzy sets and intuitionistic sets. The second paper in this series will extend this enquiry by introducing subtextures and quotient textures and a second series of papers is under preparation which consider topological aspects of this correspondence.
Since introduction of the theory of rough set in early eighties, considerable work has been done on the development and application of this new theory. The paper provides a review of the Pawlak rough set model and its extensions, with emphasis on the formulation, characterization, and interpretation of various rough set models.
A theory of relations and corelations between the lattice of fuzzy subsets of a crisp set X and that of a crisp set Y is developed, based on the theory of relations and corelations between textures. In a series of examples it is shown that these notions generalize in a natural way the important concept of fuzzy relation from X to Y. Difunctions are also characterized and their relationship with known mappings between fuzzy sets is investigated.
The paper develops the machinery of a theory of sets whose boundaries are 'fuzzy'. It applies to situations involving uncertainty or ambiguity which is not statistical in nature, such as pattern classification, and some applications are discussed. Section titles include: L-fuzzy Sets; The Structure of L; Fuzzification; Categories and the Composition of Fuzzy Maps; Relations; Images; Decisions and Mappings between Lattices. (Author)
In this paper, a class of generalized fuzzy rough sets based on two universes are studied. Some new set-valued mappings and fuzzy set-valued mappings are introduced to discuss properties of the known model, and a new model for fuzzy rough sets is proposed which provides a new selection of interval structure for uncertainty reasoning using rough set theory. Some properties of the new model are revealed. The new model seems to be more natural in the sense that fuzzy sets are approximated by fuzzy sets on the same universe.
In rough set theory, the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties. Various fuzzy generalizations of rough approximations have been made in the literature. This paper proposes a general framework for the study of (I,T)-fuzzy rough approximation operators within which both constructive and axiomatic approaches are used. In the constructive approach, a pair of lower and upper generalized fuzzy rough approximation operators, determined by an implicator I and a triangular norm T, is first defined. Basic properties of (I,T)-fuzzy rough approximation operators are investigated. The connections between fuzzy relations and fuzzy rough approximation operators are further established. In the axiomatic approach, an operator-oriented characterization of rough sets is proposed, that is, (I,T)-fuzzy approximation operators are defined by axioms. Different axiom sets of T-upper and I-lower fuzzy set-theoretic operators guarantee the existence of different types of fuzzy relations which produce the same operators. Finally, an open problem proposed by Radzikowska and Kerre in (Fuzzy Sets and Systems 126 (2002) 137) is solved.
In rough set theory, the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties. Several authors have proposed various fuzzy generalizations of rough approximations. In this paper, we introduce the definitions for generalized fuzzy lower and upper approximation operators determined by a residual implication. Then we find the assumptions which permit a given fuzzy set-theoretic operator to represent a upper (or lower) approximation derived from a special fuzzy relation. Different classes of fuzzy rough set algebras are obtained from different types of fuzzy relations. And different sets of axioms of fuzzy set-theoretic operator guarantee the existence of different types of fuzzy relations which produce the same operator. Finally, we study the composition of two approximation spaces. It is proved that the approximation operators in the composition space are just the composition of the approximation operators in the two fuzzy approximation spaces.
In this paper it is shown that the lattice of intuitionistic subsets of a set X in the sense of D. Çoker may be represented as a special type of texture space, called an intuitionistic texture on X, and various characterizations are given. It is established that intuitionistic topologies are mapped to ditopologies on the corresponding texture, and some notions of compactness and stability are considered.
Kernel methods and rough sets are two general pursuits in the domain of machine learning and intelligent systems. Kernel methods map data into a higher dimensional feature space, where the resulting structure of the classification task is linearly separable; while rough sets granulate the universe with the use of relations and employ the induced knowledge granules to approximate arbitrary concepts existing in the problem at hand. Although it seems there is no connection between these two methodologies, both kernel methods and rough sets explicitly or implicitly dwell on relation matrices to represent the structure of sample information. Based on this observation, we combine these methodologies by incorporating Gaussian kernel with fuzzy rough sets and propose a Gaussian kernel approximation based fuzzy rough set model. Fuzzy T-equivalence relations constitute the fundamentals of most fuzzy rough set models. It is proven that fuzzy relations with Gaussian kernel are reflexive, symmetric and transitive. Gaussian kernels are introduced to acquire fuzzy relations between samples described by fuzzy or numeric attributes in order to carry out fuzzy rough data analysis. Moreover, we discuss information entropy to evaluate the kernel matrix and calculate the uncertainty of the approximation. Several functions are constructed for evaluating the significance of features based on kernel approximation and fuzzy entropy. Algorithms for feature ranking and reduction based on the proposed functions are designed. Results of experimental analysis are included to quantify the effectiveness of the proposed methods.
In this paper we point out that textures, simple textures and plain textures are, respectively, isomorphic to C-spaces, sober C-spaces and partially ordered sets. Using the isomorphism between textures and C-spaces, we show that subtextures, quotient textures, sums and products of textures can be identified as subspaces, quotient spaces, sums and products of C-spaces. Furthermore, it is shown that textures, except discrete textures, do not satisfy any separation axioms (T1,…,T4) other than T0, and so they are not pseudometrizable. Then we also give four useful characterizations of the complete distributivity condition of textures.
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.
This paper presents a general framework for the study of rough set approximation operators in fuzzy environment in which both constructive and axiomatic approaches are used. In constructive approach, a pair of lower and upper generalized fuzzy rough (and rough fuzzy, respectively) approximation operators is first defined. The representations of both fuzzy rough approximation operators and rough fuzzy approximation operators are then presented. The connections between fuzzy (and crisp, respectively) relations and fuzzy rough (and rough fuzzy, respectively) approximation operators are further established. In axiomatic approach, various classes of fuzzy approximation operators are characterized by different sets of axioms. The minimal axiom sets of fuzzy approximation operators guarantee the existence of certain types of fuzzy or crisp relations producing the same operators.
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.
In this paper we study conditions under which the implication operators satisfy the property I(x,c(x))=c(x) for all x∈[0,1],c being any strong negation. This study has led us to present different implication operator characterization theorems from automorphisms, obtaining a theorem similar to the one presented by P. Smets and P. Magrez [Int. J. Approx. Reasoning 1, 327-347 (1987; Zbl 0643.03018)], in which the strong negation c used is not generated by the same automorphism that generates the implication.
Fuzzy modal operators express interactions between binary fuzzy relations and fuzzy sets. Each of these operators is determined by a binary fuzzy relation (on a given universe) and transforms one fuzzy set (in this universe) to another one. In this paper, we define several fuzzy modal operators and provide characterisations of main classes of binary fuzzy relations by means of these operators. Interpretation of these characterisations is presented in the context of fuzzy modal logics. We show that these characterisations constitute the basis for determining characteristic axioms of particular classes of fuzzy modal logics.
In this paper, we introduce the notion of interval structures in an attempt to establish a unified framework for representing uncertain in- formation. Two views are suggested for the interpretation of an interval structure. A typical example using the compatibility view is the rough- set model in which the lower and upper approximations form an interval structure. Incidence calculus adopts the allocation view in which an inter- val structure is defined by the tightest lower and upper incidence bounds. The relationship between interval structures and interval-based numeric belief and plausibility functions is also examined. As an application of the proposed model, an algorithm is developed for computing the tightest incidence bounds.
A fuzzy implication, commonly defined as a two-place operation on the unit interval, is an extension of the classical binary implication. It plays important roles in both mathematical and applied sides of fuzzy set theory. Besides the basic axioms, there are many potential fuzzy implication axioms, among which eight are widely used in the literature. Different fuzzy implications satisfying different subgroups of these eight axioms can be found. However, certain interrelationships exist between these eight axioms. But the results remain incomplete. This paper aims to lay bare the interrelationships between these eight axioms. The result is instrumental to propose a classification of fuzzy implications.
Uninorms are an important generalization of t-norms and t-conorms, having a neutral element lying anywhere in the unit interval. A uninorm shows a typical block structure and is built
from a t-norm, a t-conorm and a mean operator. Two important classes of uninorms are characterized, corresponding to the use of the minimum
operator (the class U
min) and maximum operator (the class U
max) as mean operator. The characterization of representable uninorms, i.e. uninorms with an additive generator, and of left-continuous
and right-continuous idempotent uninorms is recalled. Two residual operators are associated with a uninorm and it is characterized
when they yield an implicator and coimplicator. The block structure of the residual implicator of members of the class U
min and of the residual coimplicator of members of the class U
max is investigated. Explicit expressions for the residual implicator and residual coimplicator of representable uninorms and
of certain left-continuous or right-continuous idempotent uninorms are given. Additional properties such as contrapositivity
are discussed.
Rough sets and fuzzy rough sets serve as important approaches to granular computing, but the granular structure of fuzzy rough sets is not as clear as that of classical rough sets since lower and upper approximations in fuzzy rough sets are defined in terms of membership functions, while lower and upper approximations in classical rough sets are defined in terms of union of some basic granules. This limits further investigation of the existing fuzzy rough sets. To bring to light the innate granular structure of fuzzy rough sets, we develop a theory of granular computing based on fuzzy relations in this paper. We propose the concept of granular fuzzy sets based on fuzzy similarity relations, investigate the properties of the proposed granular fuzzy sets using constructive and axiomatic approaches, and study the relationship between granular fuzzy sets and fuzzy relations. We then use the granular fuzzy sets to describe the granular structures of lower and upper approximations of a fuzzy set within the framework of granular computing. Finally, we characterize the structure of attribute reduction in terms of granular fuzzy sets, and two examples are also employed to illustrate our idea in this paper.
A new scheme of knowledge encoding in a fuzzy multilayer perceptron (MLP) using rough set-theoretic concepts is described. Crude domain knowledge is extracted from the data set in the form of rules. The syntax of these rules automatically determines the appropriate number of hidden nodes while the dependency factors are used in the initial weight encoding. The network is then refined during training. Results on classification of speech and synthetic data demonstrate the superiority of the system over the fuzzy and conventional versions of the MLP (involving no initial knowledge).