The existing group decision making techniques may not satisfy the order consistency for aggregation in some cases. The algorithm proposed in this paper overcomes the weaknesses of the existing techniques. The method determines the unknown preferences for group decision making in such a manner that the resulting matrix is T-consistent and order consistent simultaneously.

There has been a long history of using fuzzy-language equivalence to compare the behavior of fuzzy systems; however, the comparison at this level is too coarse. Recently, a finer behavioral measure, i.e., bisimulation, has been introduced to fuzzy-finite automata. However, the results obtained are applicable only to finite-state systems. In this paper, we consider bisimulation for general fuzzy systems, which may be infinite state or infinite event, by modeling them as fuzzy-transition systems (FTSs). To help understand and check bisimulation, we characterize it in three ways by enumerating whole transitions, comparing individual transitions, and using a monotonic function. In addition, we address composition operations, subsystems, quotients, and homomorphisms of FTSs and discuss their properties connected with bisimulation. The results presented here are useful to compare the behavior of general fuzzy systems. In particular, this makes it possible to relate an infinite fuzzy system to a finite one, which is easier to analyze, with the same behavior.

In this paper, a new fuzzy filter for the removal of random impulse noise in color video is presented. By working with different successive filtering steps, a very good tradeoff between detail preservation and noise removal is obtained. One strong filtering step that should remove all noise at once would inevitably also remove a considerable amount of detail. Therefore, the noise is filtered step by step. In each step, noisy pixels are detected by the help of fuzzy rules, which are very useful for the processing of human knowledge where linguistic variables are used. Pixels that are detected as noisy are filtered, the others remain unchanged. Filtering of detected pixels is done by blockmatching based on a noise adaptive mean absolute difference. The experiments show that the proposed method outperforms other state-of-the-art filters both visually and in terms of objective quality measures such as the mean absolute error (MAE), the peak-signal-to-noise ratio (PSNR) and the normalized color difference (NCD).

In this paper a new filtering framework for colour image sequences corrupted by random impulse noise is introduced. The proposed method consists of three successive filtering steps in order to find a good trade-off between detail preservation and noise removal. One hard filtering step, that should remove all the noise at once, would namely also remove a considerable amount of details. In the different noise detection steps, we make use of human knowledge represented in the form of fuzzy if-then rules. The detection of noisy pixel components is based on spatial and temporal information as well as on information from the other colour bands. Additionally, only detected pixel components are filtered by blockmatching based on an adaptation of the mean absolute difference (MAD) to noise. The other components remain unchanged. From the experimental results, it can be concluded that the proposed filter outperforms other state-of-the-art impulse noise filters.

Mathematical morphology is a theory to extract specific information such as edges and patterns from images. The original binary morphology, for binary black and white images, was extended to greyscale images, amongst others, by a fuzzy approach known as fuzzy mathematical morphology. This approach was based on the observation that greyscale images and fuzzy sets can be modelled in the same way. Recently, fuzzy mathematical morphology has been further extended based on extensions of classical fuzzy set theory. In this paper, we focus on the extension based on interval-valued fuzzy set theory, i.e., interval-valued fuzzy morphology, and we give an overview of the basic properties that hold in this model.

The concept of adjunction plays an important role in mathematical morphology. If the morphological operations, dilation and erosion form an adjunction in a complete lattice, then they, as well as the closing and opening constructed by them, will fulfill certain required properties in an algebraic context. In the context of fuzzy mathematical morphology, which is an extension of binary morphology to gray-scale morphology based on fuzzy set theory, we use conjunctions and implications to define fuzzy dilations and fuzzy erosions. In this paper, we investigate when these pairs of dilations and erosions form a fuzzy adjunction, which is also defined by an implication. We find that the so-called adjointness between a conjunction and an implication plays an important role here. Finally, we develop a theorem stating that a conjunction that is adjoint with an implication cannot only be generated by an R-implication but also by other implications. This allows the easy construction of fuzzy adjunctions. © 2009 Wiley Periodicals, Inc.

Integrating video coding and denoising is a novel processing paradigm, bringing mutual benefits to both video processing tools. In this paper, we propose a novel video denoising approach of which the main idea is reusing motion estimation resources from the video coding module for video denoising. In most cases, the motion fields produced by real-time video codecs cannot be directly employed in video denoising, since they, as opposed to noise filters, tolerate errors in the motion field. In order to solve this problem, we propose a novel motion-field filtering step that refines the accuracy of the motion estimates to a degree that is required for denoising. Additionally, a novel temporal filter is proposed that is robust against errors in the estimated motion field. Numerical results demonstrate that the proposed denoising scheme is of low-complexity and compares favorably to the state-of-the-art video denoising methods.

Klir introduced constrained fuzzy arithmetic (CFA) as a solution to the unnecessary precision loss when dealing with fuzzy quantities that represent linguistic variables. Since then, some attempts have been made to make CFA efficient, but none of these solutions is suitable for real-time applications. In this paper, we will propose a new CFA algorithm that can be used in such environments.

A significant part of real-world spatial information is affected by vagueness. For example, boundaries of non-administrative geographical regions tend to be ill-defined, while information about the nearness and relative orientation of two places is typically expressed through vague linguistic descriptions. In this paper, we propose a general framework to represent such information, using the concept of relatedness measures for fuzzy sets. Regions are represented as fuzzy sets in a two-dimensional Euclidean space, and nearness and relative orientation are expressed as fuzzy relations. To support fuzzy spatial reasoning, we derive transitivity rules and provide efficient techniques to deal with the complex interactions between nearness and cardinal directions.

Local search services allow a user to search for businesses that satisfy a given geographical constraint. In contrast to traditional web search engines, current local search services rely heavily on static, structured data. Although this yields very accurate systems, it also implies a limited coverage, and limited support for using landmarks and neighborhood names in queries. To overcome these limitations, we propose to augment the structured information available to a local search service, based on the vast amount of unstructured and semi-structured data available on the web. This requires a computational framework to represent vague natural language information about the nearness of places, as well as the spatial extent of vague neighborhoods. In this paper, we propose such a framework based on fuzzy set theory, and show how natural language information can be translated into this framework. We provide experimental results that show the effectiveness of the proposed techniques, and demonstrate that local search based on natural language hints about the location of places with an unknown address, is feasible.

When the time span of an event is imprecise, it can be represented by a fuzzy set, called a fuzzy time interval. In this paper, we propose a framework to represent, compute, and reason about temporal relationships between such events. Since our model is based on fuzzy orderings of time points, it is not only suitable to express precise relationships between imprecise events (ldquoRoosevelt died before the beginning of the Cold Warrdquo) but also imprecise relationships (ldquoRoosevelt died just before the beginning of the Cold Warrdquo). We show that, unlike previous models, our model is a generalization that preserves many of the properties of the 13 relations Allen introduced for crisp time intervals. Furthermore, we show how our model can be used for efficient fuzzy temporal reasoning by means of a transitivity table. Finally, we illustrate its use in the context of question answering systems.

In this paper, we introduce the idea of using motion estimation resources from a video codec for video denoising. This is not straightforward because the motion estimators aimed for video compression and coding, tolerate errors in the estimated motion field and hence are not directly applicable to video denoising. To solve this problem, we propose a novel motion field filtering step that refines the accuracy of the motion estimates to a degree that is required for denoising. We illustrate the use of the proposed motion estimation method within a wavelet-based video denoising scheme. The resulting video denoising method is of low-complexity and receives comparable results with respect to the latest video denoising methods.

Color image retrieval is becoming more and more important, and so is the quest for automated and reliable retrieval systems. In this paper we present and illustrate a new color image retrieval system, of which the novelty lies in the use of a fuzzy partition of the HSI color space and fuzzy similarity measures. The system has the advantage that the images don't have to be characterized in advance using several features, and it is quite flexible since the database images are not required to have the same dimensions.

Algorithms for clustering Web search results have to be efficient and robust. Furthermore they must be able to cluster a data set without using any kind of a priori information, such as the required number of clusters. Clustering algorithms inspired by the behavior of real ants generally meet these requirements. In this article we propose a novel approach to ant-based clustering, based on fuzzy logic. We show that it improves existing approaches and illustrates how our algorithm can be applied to the problem of Web search results clustering. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 455–474, 2007.

We present a family of algebraic structures, called rectangular bilattices, which serve as a natural accommodation and powerful generalization to both intuitionistic fuzzy sets (IFSs) and interval-valued fuzzy sets (IVFSs). These structures are useful on one hand to clarify the exact nature of the relationship between the above two common extensions of fuzzy sets, and on the other hand provide an intuitively attractive framework for the representation of uncertain and potentially conflicting information. We also provide these structures with adequately defined graded versions of the basic logical connectives, and study their properties and relationships. Application potential and intuitive appeal of the proposed framework are illustrated in the context of preference modeling.

Traditional rough set theory uses equivalence relations to compute lower and upper approximations of sets. The corresponding equivalence classes either coincide or are disjoint. This behaviour is lost when moving on to a fuzzy T-equivalence relation. However, none of the existing studies on fuzzy rough set theory tries to exploit the fact that an element can belong to some degree to several “soft similarity classes” at the same time. In this paper we show that taking this truly fuzzy characteristic into account may lead to new and interesting definitions of lower and upper approximations. We explore two of them in detail and we investigate under which conditions they differ from the commonly used definitions. Finally we show the possible practical relevance of the newly introduced approximations for query refinement.

In this paper, Ginsberg's/Fitting's theory of bilattices, and in particular the associated constructs of bilattice-based squares and triangles, is introduced as an attractive framework for the representation of uncertain and potentially conflicting information, paralleling Goguen's L-fuzzy set theory. We recall some of the advantages of bilattice-based frameworks for handling fuzzy sets and systems, provide the related structures with adequately defined graded versions of the basic logical connectives, and study their properties and relationships

Buckley and Qu proposed a method to solve systems of linear fuzzy equations. Basically, in their method the solutions of all systems of linear crisp equations formed by the alpha-levels are calculated. We propose in this paper a new method for solving systems of linear fuzzy equations based on a practical algorithm using parametric functions in which the variables are given by the fuzzy coefficients of the system. We show that our algorithm is much more efficient than the method of Buckley and Qu.

A new impulse noise reduction method for color images is presented. Color images that are corrupted with impulse noise are generally filtered by applying a grayscale algorithm on each color component separately or using a vector-based approach where each pixel is considered as a single vector. The first approach causes artefacts especially on edge and texture pixels. Vector-based methods were successfully introduced to overcome this problem. Nevertheless, they tend to cluster the noise and to receive a lower noise reduction performance. In this paper, we discuss an alternative technique which gives a good noise reduction performance while much less artefacts are introduced. The main difference between the proposed method and other classical noise reduction methods is that the color information is taken into account to develop (1) a better impulse noise detection method and (2) a noise reduction method that filters only the corrupted pixels while preserving the color and the edge sharpness. Experimental results show that the proposed method provides a significant improvement on other existing filters.

A new fuzzy filter is presented for the reduction of additive noise for digital color images. The filter consists of two subfilters. The first subfilter computes fuzzy distances between the color components of the central pixel and its neighborhood. These distances determine in what degree each component should be corrected. All performed corrections preserve the color component distances. The goal of the second subfilter is to correct the pixels where the color components differences are corrupted so much that they appear as outliers in comparison to their environment. Experimental results show the feasibility of the proposed approach. We compare with other noise reduction methods by numerical measures and visual observations. We also illustrate the performance of the proposed method as preprocessing step for edge detection.

Fuzzy temporal interval relations have been defined to support temporal knowledge representation and reasoning in the presence of vagueness. The most important impediment to use these fuzzy relations in real-world applications is the lack of a characterization that is both easy to implement and computationally efficient. In this paper, we provide such a characterization for the important class of piecewise linear fuzzy time intervals, which covers all types of fuzzy time intervals that we are likely to encounter in applications. Furthermore, we discuss a more elegant characterization for the special case of trapezoidally shaped fuzzy intervals.

In many current impulse noise models for images, corrupted pixels are replaced with values equal or near the maximum or minimum intensity values of the allowable dynamic range. In this paper, we present a new fuzzy filter for a more general noise model in which a noisy pixel has an arbitrary value in the dynamic range according to some underlying probability distribution. This filter consists of (1) a fuzzy detection method, where we investigate if a certain pixel position can be seen as noisy or not and (2) a fuzzy reduction method that reduces the noise while preserving the fine details (like edges and textures) of the image. Experimental results have shown that the proposed filter may be used for efficient removal of randomly valued impulse noise without distorting the useful information in the image.

Removing or reducing impulse noise is a very active research area in image processing. In this paper we describe a new algorithm that is especially developed for reducing all kinds of impulse noise: fuzzy impulse noise detection and reduction method (FIDRM). It can also be applied to images having a mixture of impulse noise and other types of noise. The result is an image quasi without (or with very little) impulse noise so that other filters can be used afterwards. This nonlinear filtering technique contains two separated steps: an impulse noise detection step and a reduction step that preserves edge sharpness. Based on the concept of fuzzy gradient values, our detection method constructs a fuzzy set impulse noise. This fuzzy set is represented by a membership function that will be used by the filtering method, which is a fuzzy averaging of neighboring pixels. Experimental results show that FIDRM provides a significant improvement on other existing filters. FIDRM is not only very fast, but also very effective for reducing little as well as very high impulse noise.

A new framework for reducing impulse noise from digital color images is presented, in which a fuzzy detection phase is followed by an iterative fuzzy filtering technique. We call this filter the fuzzy two-step color filter. The fuzzy detection method is mainly based on the calculation of fuzzy gradient values and on fuzzy reasoning. This phase determines three separate membership functions that are passed to the filtering step. These membership functions will be used as a representation of the fuzzy set impulse noise (one function for each color component). Our proposed new fuzzy method is especially developed for reducing impulse noise from color images while preserving details and texture. Experiments show that the proposed filter can be used for efficient removal of impulse noise from color images without distorting the useful information in the image.

Noise reduction is a well-known problem in image processing. The reduction of noise in an image sometimes is as a goal itself, and sometimes is considered as a pre-processing step. Besides the classical filters for noise reduction, quite a lot of fuzzy inspired filters have been proposed during the past years. However, it is very difficult to judge the quality of this wide variety of filters. For which noise types are they designed? How do they perform for those noise types? How do they perform compared to each other? Can we select filters that clearly outperform the others? Is there a difference between numerical and visual results? In this paper, we answer these questions for images that are corrupted with Gaussian noise

Interval-valued fuzzy sets constitute an extension of fuzzy sets which give an interval approximating the "real" (but unknown) membership degree. Interval-valued fuzzy sets are equivalent to intuitionistic fuzzy sets in the sense of Atanassov which give both a membership degree and a non-membership degree, whose sum must be smaller than or equal to 1. Both are equivalent to L-fuzzy sets w.r.t. a special lattice L*. In fuzzy set theory, an important class of triangular norms is the class of those that satisfy the residuation principle. In a previous paper5 we gave a construction for t-norms on L* satisfying the residuation principle which are not t-representable. In this paper we investigate the Smets-Magrez axioms and some other properties for the residual implicator generated by such t-norms.

In this paper we investigate whether the fuzzy arithmetic based on Zadeh's extension principle could be improved by redefining the fuzzy addition and multiplication so that the class of fuzzy numbers combined with these operations would constitute a field. We will prove that such a fuzzy arithmetic does not exist. This has important consequences for solving systems of linear fuzzy equations. Despite the lack of inverses, we propose a method to solve approximately such systems.

Rough set theory was introduced in 1982. Soon it was combined with fuzzy set theory, giving rise to a hybrid model, involving fuzzy sets and fuzzy relations, which appears to be a natural, elegant generalization. In this paper we reveal that in the fuzzification process an important step seems to be overlooked. The most fascinating part is that this forgotten step arises from the true essence of fuzzy set theory: namely, that an element can belong to a given degree to more than one fuzzy set at the same time.

We present an algebraic approach to fuzzy generalisation of rough sets. We take an arbitrary residuated lattice L as a basic algebraic structure. The L-fuzzy rough sets are defined using the product operator and its residuum provided by the residuated lattice L. Depending on classes of binary fuzzy relations, we define several classes of L-fuzzy rough sets and show properties of these structures.

In this paper, we have shown how the fuzzy set theory is used in establishing measures for image quality evaluation. Objective quality measures or measures of comparison are of great importance in the field of image processing. These measures serve as a tool to evaluate and to compare different algorithms designed to solve problems, such as noise reduction, deblurring, compression, etc. It is well-known that classical quality measures, such as the MSE (mean square error) or the PSNR (peak signal to noise ratio), do not always correspond to human visual observations. Therefore, several researchers are - and have been - looking for new quality measures, better adapted to human perception. In this paper, we show how the neighbourhood-based similarity measures can be combined with similarity measures for histogram comparison in order to improve the perceptive behaviour of these similarity measures.

The task of finding shortest paths in weighted graphs is one of the archetypical problems encountered in the domain of combinatorial optimization and has been studied intensively over the past five decades. More recently, fuzzy weighted graphs, along with generalizations of algorithms for finding optimal paths within them, have emerged as an adequate modeling tool for prohibitively complex and/or inherently imprecise systems. We review and formalize these algorithms, paying special attention to the ranking methods used for path comparison. We show which criteria must be met for algorithm correctness and present an efficient method, based on defuzzification of fuzzy weights, for finding optimal paths. © 2004 Wiley Periodicals, Inc. Int J Int Syst 19: 1051–1068, 2004.

Intuitionistic fuzzy sets form an extension of fuzzy sets: while fuzzy sets give a degree to which an element belongs to a set, intuitionistic fuzzy sets give both a membership degree and a nonmembership degree. The only constraint on those two degrees is that their sum must be smaller than or equal to 1. In fuzzy set theory, an important class of triangular norms and conorms is the class of continuous Archimedean nilpotent triangular norms and conorms. It has been shown that for such t-norms T there exists a permutation /spl phi/ of [0,1] such that T is the /spl phi/-transform of the Lukasiewicz t-norm. In this paper we introduce the notion of intuitionistic fuzzy t-norm and t-conorm, and investigate under which conditions a similar representation theorem can be obtained.

Abstract: Just like rough set theory, fuzzy set theory addresses the topic of dealing with imperfect knowledge. Recent investigations have shown how both theories can be combined into a more flexible, more expressive framework for modelling and processing incomplete information in information systems. At the same time, intuitionistic fuzzy sets have been proposed as an attractive extension of fuzzy sets, enriching the latter with extra features to represent uncertainty (on top of vagueness). Unfortunately, the various tentative definitions of the concept of an ‘intuitionistic fuzzy rough set’ that were raised in their wake are a far cry from the original objectives of rough set theory. We intend to fill an obvious gap by introducing a new definition of intuitionistic fuzzy rough sets, as the most natural generalization of Pawlak's original concept of rough sets.

Intuitionistic fuzzy sets constitute an extension of fuzzy sets: while fuzzy sets give a degree to which an element belongs to a set, intuitionistic fuzzy sets give both a membership degree and a non-membership degree. The only constraint on those two degrees is that the sum must be smaller than or equal to 1. In fuzzy set theory, an important class of triangular norms is the class of those that satisfy the residuation principle. In the fuzzy case a t-norm satisfies the residuation principle if and only if it is left-continuous. Deschrijver, Cornelis and Kerre proved that for intuitionistic fuzzy t-norms the equivalence between the residuation principle and intuitionistic fuzzy left-continuity only holds for t-representable t-norms.1 In this paper we construct particular subclasses of intuitionistic fuzzy t-norms that satisfy the residuation principle but that are not t-representable and we show that a continuous intuitionistic fuzzy t-norm [Formula: see text] satisfying the residuation principle is t-representable if and only if [Formula: see text].

A new fuzzy filter is presented for the noise reduction of images corrupted with additive noise. The filter consists of two stages. The first stage computes a fuzzy derivative for eight different directions. The second stage uses these fuzzy derivatives to perform fuzzy smoothing by weighting the contributions of neighboring pixel values. Both stages are based on fuzzy rules which make use of membership functions. The filter can be applied iteratively to effectively reduce heavy noise. In particular, the shape of the membership functions is adapted according to the remaining noise level after each iteration, making use of the distribution of the homogeneity in the image. A statistical model for the noise distribution can be incorporated to relate the homogeneity to the adaptation scheme of the membership functions. Experimental results are obtained to show the feasibility of the proposed approach. These results are also compared to other filters by numerical measures and visual inspection.

Fuzzy techniques can be applied in several domains of image processing. In this paper we will show how notions of fuzzy set theory are used in establishing measures for image quality evaluation. Objective quality measures or measures of comparison are of great importance in the field of image processing. Such measures are necessary for the evaluation and the comparison of different algorithms that are designed to solve a similar problem, and consequently they serve as a basis on which algorithm is preferred to the other. In this paper, we will show how similarity measures originally introduced to compare two fuzzy sets can be applied successfully in the domain of image processing.

In this paper we give an overview of classical and fuzzy-classical filters for image noise reduction. Together with our overview (2001) of fuzzy filters, this paper can be seen as a preparation to our comparative study of classical and fuzzy filters for image noise reduction.

In this paper we give an overview of existing fuzzy filters for image noise reduction. The paper is a sequel to our overview of classical and fuzzy-classical filters (2001), and can be seen as a preparation to our comparative study of classical and fuzzy filters for image noise reduction. We discuss the ideas behind and the construction of the following fuzzy filters: the "fuzzy inference ruled by else-action" filters, the "fuzzy control based" filters, and the GOA filter. Our goal is to give a consistent overview of these fuzzy filters, and to clearly show the mutual differences between them.

In this paper we present the results of a comparative study of classical and fuzzy filters for image noise reduction. The discussed fuzzy filters are classified, and their performance is compared with classical filters and evaluated by numerical and visual experiments.

According to the operation of decomposition (also known as representation theorem) (Negoita CV, Ralescu, DA. Kybernetes 1975;4:169–174) in fuzzy set theory, the whole fuzziness of an object can be characterized by a sequence of local crisp properties of that object. Hence, any fuzzy reasoning could also be implemented by using a similar idea, i.e., a sequence of precise reasoning. More precisely, we could translate a fuzzy relation “If A then B” of the Generalized Modus Ponens Rule (the most common and widely used interpretation of a fuzzy rule, A, B, are fuzzy sets in a universe of discourse X, and of discourse Y, respectively) into a corresponding precise relation between a subset of P(X) and a subset of P(Y), and then extend this corresponding precise relation to two kinds of transformations between all L-type fuzzy subsets of X and those of Y by using Zadeh's extension principle, where L denotes a complete lattice. In this way, we provide an alternative approach to the existing compositional rule of inference, which performs fuzzy reasoning based on the extension principle. The approach does not depend on the choice of fuzzy implication operator nor on the choice of a t-norm. The detailed reasoning methods, applied in particular to the Generalized Modus Ponens and the Generalized Modus Tollens, are established and their properties are further investigated in this paper. © 2001 John Wiley & Sons, Inc.

The author looks at the possibility to extend precise reasoning based on classical set theory to fuzzy reasoning based on fuzzy set theory by using Zadeh's extension principle. It is shown from the decomposition theorem in fuzzy set theory that the whole fuzziness of an object can be characterized by a sequence of local clear properties of that object. Hence, fuzzy reasoning can also be implemented by using a sequence of precise reasonings. Considering this ideal, we can translate the fuzzy relation "if A then B" in generalized modus ponens (GMP) rule into a corresponding precise relation between subsets of X and subsets of Y, then extend this corresponding precise relation to two kinds of transformations between fuzzy sets of X and fuzzy sets of Y by using of Zadeh's extension principle. Finally, two extension principle-based fuzzy reasoning methods, the GMP and generalized modus tollens fuzzy reasoning methods, are established, and some properties about these two fuzzy reasoning methods investigated.

We consider discrete possibilistic systems for which the available information is given by one-step transition possibilities and initial possibilities. These systems can be represented, or modelled, by a collection of variables satisfying a possibilistic counterpart of the Markov condition. This means that, given the values assumed by a selection of variables, the possibility that a subsequent variable assumes some value only depends on the value taken by the most recent variable of the selection. The one-step transition possibilities are recovered by computing the conditional possibility of any two consecutive variables. Under the behavioural interpretation as marginal betting rates against events these 'conditional' possibilities and the initial possibilities should satisfy the rationality criteria of 'avoiding sure loss' and 'coherence'. We show that this is indeed the case when the conditional possibilities are defined using Dempster's conditioning rule.

Fuzzy mathematical morphology is an extension of binary morphology to gray-scale morphology, using techniques from fuzzy set theory. In this paper, we will study the decomposition and construction of fuzzy morphological operations based on /spl alpha/-cuts. First, we will investigate the relationship between /spl alpha/-cuts of the fuzzy morphological operations and the corresponding binary operations. Next, we will review several ways to obtain fuzzy morphological operations starting from binary operations and /spl alpha/-cuts. The investigation is carried out in both the continuous and the discrete case. It is interesting to observe that several properties that do not hold in the continuous case do hold in the discrete case. This is quite important since in practice we only work with discrete objects.

Fuzzy mathematical morphology is an alternative extension of binary morphology to gray-scale morphology, using techniques from fuzzy set theory and fuzzy logic. The authors discuss a general logical framework for discrete morphology and investigate the decomposition of the fuzzy morphological operations in this framework.

This paper is devoted to the study of fuzzy weak orders, fuzzy quasi orders and fuzzy quasi-transitive relations in the context of group decision making. Transitivity is modelled by means of the minimum operator. In analogy to the numerical representation of classical weak orders, we consider a particular class of fuzzy weak orders that can be constructed from a numerical evaluation by means of Gödel implicator. We fully characterize the fuzzy relations that can be written as the intersection or union of a finite family of these so-called Gödel representable fuzzy weak orders.

Fuzzy information in expert systems and in fuzzy database systems is often represented by means of piecewise linear fuzzy quantities. In this article we describe an implementation of piecewise linear fuzzy quantities and how to perform fuzzy operations on them. This article can be seen as a natural continuation of Baekeland and Kerre [“Operations on piecewise linear fuzzy quantities: A theoretical approach,” Fuzzy Sets Syst., submitted], in which the mathematical properties of piecewise linear fuzzy quantities are investigated. © 1995 John Wiley & Sons, Inc.

In this paper we introduce a definition of consistency of the judgement matrix in the fuzzy Analytic Hierarchy Process (AHP) and give a general expression of all fuzzy weights under the condition of consistency. Finally, based on our discussion, the geometric average method is suggested for fuzzy weights calculation in the practical decision-making situation.

This article focuses on the axiomatization of fuzzy functional dependency (FFD) in a fuzzy relational data model, which is considered as a fundamental issue towards building a theory of fuzzy relational database design. As an inference basis, the FFD axiomatic system composed of four FFD inference rules is established, and proven to be correct in terms of FFD logical implication with respect to the relation scheme. Consequently, the concept of the FFD transitive closure is defined, and correspondingly a computational algorithm is developed. This helps to tell whether a specific FFD can be derived from a given FFD set using those inference rules. In addition, some details of the algorithm as well as the aspects of the computational complexity and correctness are investigated. Finally, the soundness and completeness of the axiomatic system are discussed. It is proven that the system is both sound and complete. Importantly, this result enables us to “equate” the concept of FFD logical implication to the concept of FFD derivation using the inference rules. © 1994 John Wiley & Sons, Inc.

This paper has to be considered as a guide to solving fuzzy relational equations on the unit interval. Although the number of publications on this topic is quite impressive, there doesn't seem to exist a handy structured overview of all types of equations and their solution procedures. Our overview starts with a thorough treatment of [Formula: see text] equations and systems of [Formula: see text] equations, with [Formula: see text] a continuous triangular norm. It is shown that these are the basic problems: all other equations, image and composition equations, can be reduced to these problems. We do not only structure well-known results, we also present some new insights in the solution procedures of fuzzy relational equations.

In this article we introduce an extension of Zadeh's compositional rule of inference in terms of the general rule of inference using a triangular norm extended to n arguments. Using this extension, all inferences schemes, crisp as well as fuzzy, based on the compositional rule of inference can be obtained in a uniform way. © 1993 John Wiley & Sons, Inc.

In fuzzy data modeling, there is a series of design issues that should be dealt with. This article concentrates on the definition of data redundancy in a fuzzy relational data model. Compared to the existing approaches, a general treatment is provided, enabling us to directly and rigourously handle the situation in which possibility distributions can appear as attribute values combined with closeness relations in domain elements. The treatment of fuzzy data redundancy, which we propose here, is mainly based on Zadeh's extension principle. In doing so, a natural incorporation of closeness relations into the operations defined by the extension principle is worked out. Certain desirable properties are achieved in our model, in terms of intuition and model design. © 1992 John Wiley & Sons, Inc.

In this article, we introduce a generalized extension principle by substituting a more general triangular norm T for the min intersection operator in Zadeh's extension principle. We also introduce a family of propositional logics, sup- T extension logics, obtained by the extension of classical-logical functions. A few general properties of these sup-T extension logics are derived. It is also shown that classical binary logic and the Kleene ternary logic are special cases of these logics for any choice of T, obtained by a convenient restriction of the truth domain. the very practical decomposability property of classical logic is furthermore shown to hold for the sup-min extension logic, albeit in a somewhat more limited form.

Due to a rapid expansion of most domains in recent times, there is a constant growing need for information. Together with this phenomenon, there has been an explosive growth of the amount of data needed and the corresponding means of data storage. In order to handle these large amounts of data and to realize a fast processing of the information asked for, more and more institutions and organizations have set up automized information processing and have built up their database. Appearing in all kinds of applications such as economical, social, political, medical, and governmental fields, databases have already proven their ability to reduce time and space with respect to the retrieval as well as to the storage of data and information. In many situations we have to deal with data which are given in imprecise form or which are only partially known or even totally unknown. We may expect that the construction of databases which can represent and manipulate fuzzy data will increase the application areas of database systems and improve the interface between men and machines. In this paper we have made a brief survey of the numerous applications of fuzzy set theory on data representation and information retrieval. The importance of fuzzy set theory with respect to information systems is illustrated by means of an already extensive bibliography containing more than 80 papers describing data systems that are somehow “fuzzy.” © 1986 John Wiley & Sons, Inc.