algorithms for solving fuzzy relational equations in a probabilistic setting abstract we discuss methods suitable for finding approximate solutions to fuzzy relational equations defined in finite universe of discourse and taking the form of either a system of equations x k r y k k 1 2 k with r being unknown fuzzy relation or a single equation x r y with x being looked for the key idea of the proposed algorithms is to incorporate some statistics concerning the solution of a single equation and utilize it in the final process of approximation of the unknown fuzzy relation set numerical considerations support the established algorithms

numerical and applicational aspects of fuzzy relational equations we discuss numerical methods leading to a resolution of fuzzy relational equations which create a formal description of ill defined systems an applicability of the numerical approach is shown in solving some problems in fuzzy systems such as identification and control an idea of fuzzy discretization fuzzy chopping as a meaningful tool for handling various problems is also investigated

fuzzy linear optimization in the presence of the fuzzy relation inequality constraints with max min composition in this paper we study the new linear objective function optimization with respect to the fuzzy relational inequalities defined by max min composition in which fuzzy inequality replaces ordinary inequality in the constraints fuzzy inequalities enable us to attain the optimal points that are better solutions than those resulting from the resolution of the similar problems with ordinary inequality constraints denoted by x o i in this paper we have presented an algorithm to generate such optimal solutions also two examples are given to illustrate the algorithm presented and some real application of this work

two dimensional multidomain pseudospectral time domain algorithm based on alternating direction implicit method in order to deal with the stability problem restricted by the finite difference time domain fdtd and conventional pseudospectral time domain pstd the multidomain pstd algorithm based on alternating direction implicit adi technique is proposed in this paper this algorithm improves the stability and efficiency of conventional pstd while it maintains the accuracy and flexibility of conventional pstd for an accurate treatment of arbitrarily curved objects a compact matrix form is derived to effectively describe two dimensional adi multidomain pseudospectral time domain adi mpstd algorithm numerical results show an excellent agreement with analytical solutions as well as results obtained by the fdtd algorithm and fully demonstrate a remarkable improvement in stability and efficiency

solution of fuzzy equations with extended operations abstract assuming that is any operation defined on a product set x y and taking values on a set z it can be extended to fuzzy sets by means of zadeh s extension principle given a fuzzy subset c of z it is here shown how to solve the equation a b c or a b c when a fuzzy subset a of x or a fuzzy subset b of y is given the methodology we provide includes as a special case the resolution of fuzzy arithmetical operations i e when stands for or extended to fuzzy numbers fuzzy subsets of the real line the paper is illustrated with several examples in fuzzy arithmetic

fuzzy sets in pattern recognition methodology and methods abstract an objective of the paper is to discuss a state of the art of methodology and algorithms of fuzzy sets in the field of pattern recognition in real world recognition and classification problems we are faced with fuzziness that is connected with diverse facets of cognitive activity of the human being an origin of sources of fuzziness is related to labels expressed in feature space as well as to labels of classes taken into account in classification procedures an evident difference between a way of information processing by means of probability and fuzzy set theory and a way of interpretation of results is explained in detail in sequel methods of pattern recognition are studied in two main streams namely supervised and unsupervised learning different approaches to designing of classification schemes as e g relation calculus decision making approach etc are put into account a method of feature selection with the aid of fuzzy measure and fuzzy integral is introduced in clustering techniques methods relying on minimization of objective function are analyzed in detail also a problem of cluster validity expressed in terms of clustering indices is addressed numerical examples are also provided

applications of fuzzy relational equations for methods of reasoning in presence of fuzzy data the paper deals with implementation of a general scheme for fuzzy logic inference with the aid of fuzzy relational equations it is shown that antecedents and consequents are related in terms of a fuzzy relation various implementation methods of generalized modus ponens and modus tollens are put into extensive study it is pointed out that fuzzy relational equations form an effective and plausible tool for implementations of reasoning methods with fuzzy premises two examples of fuzzy controllers being a direct application of schemes of approximate reasoning are also discussed

direct and inverse problem in comparison of fuzzy data abstract discussed is the direct and inverse problem of comparison of two linguistic quantities specified as fuzzy sets relations in the same universe of discourse the proposed way of comparison makes use of some logical properties of connectives and implication operators stemming from the foundations of fuzzy sets a list of plausible equality indices is provided the inverse problem is formulated a set of its solutions is provided direct impacts of the discussed approach on development of fuzzy modelling are clearly indicated also it is pointed out how fuzzy sets of higher type or interval valued sets can be generated as a result of analysis of performance of any fuzzy model

fuzzy relation equations theory as a basis of fuzzy modelling an overview abstract we give a wide overview on the applications of fuzzy relation equations theory to decision making processes to the construction of preference relations and to knowledge engineering mainly fuzzy control and fuzzy pattern recognition general theoretical results and methodological aspects are pointed out

pictorial representations of fuzzy connectives part i cases of t norms t conorms and averaging operators part i of this paper shows a number of examples of existing t norms and t conorms which are known as good models of fuzzy set theoretic intersection and union and then the pictorial representations of t norms and t conorms are made with the aid of a computer several new examples of t norms and t conorms are also given averaging operators between min and max are summarized and their pictorial representations are also made

lower solutions of systems of fuzzy equations lower solutions of a system of fuzzy equations are determined by lower solutions of each equation the result is applied to fuzzy relation equations so that they are decomposed to simple fuzzy relation equations

relevancy of fuzzy model abstract a problem of relevancy of fuzzy models viz models with ability to cope with fuzziness is stated its importance in any identification scheme is underlined and discussed in light of some shortcomings existing in procedures of fuzzy model building two novel concepts viz equality index and confidence interval are introduced and utilized for fuzzy model evaluation a flexible way of interpreting results with the aid of fuzzy models is formulated by means of interval valued fuzzy sets fuzzy sets the clear importance of studies on the relevancy of fuzzy models for problem formulation and solution is pointed out

on some finite fuzzy relation equations abstract we give a necessary and sufficient condition for the resolution of a new type of finite fuzzy relation equation whose set of solutions is entirely characterized an example of application to approximate reasoning is also given

on the existence of solutions of systems of fuzzy equations abstract a necessary and sufficient condition for the existence of solutions of a fuzzy equation given by e sanchez is extended to systems of fuzzy equations

generalized solvability criteria for fuzzy equations abstract recently e sanchez has given a necessary and sufficient condition for the existence of a solution of a fuzzy a x c with the extension of a given binary operation in the universe of discourse using many valued logic we generalize this result and get a characterization also of the cases if a x c has only nearly a solution additionally we consider fuzzy relation equations and get similar results

fuzzy relational structures the state of art fuzzy relational structures built with the aid of fuzzy sets and fuzzy relations constitute a fundamental class of constructs arising in fuzzy set theory they give rise to fuzzy relation equations in this paper we will develop the concepts of single and multilevel relational structures different logic operations will be discussed in detail aspects of the knowledge representation will be studied and illustrated with several schemes of applications

approximate solutions of fuzzy relational equations in the previous chs we have discussed various problems concerning solutions of fuzzy relation equations obviously an underlying assumption is that there exists a nonempty set of solutions a situation may occurr and indeed it is quite common in which no solution exists nevertheless even in this case one might be interested to obtain an approximate solution and know to which extent it can be viewed as a solution this stream of investigations is particularly interesting for applicational purposes contrary to the topics already discussed in the previous chs this is a field of research which has not been developed enough so far it concerns studies on solvability properties of fuzzy relation equations in this ch we shall try to answer by using several techniques the following basic question how difficult is it to attain a situation in which the system of equations has solutions and then how to measure this property

analysis and synthesis of fuzzy systems by the use of probabilistic sets the paper deals with an application of probabilistic sets in system theory especially in identification problems in systems described by means of max min fuzzy relational equations the identification procedures discussed are based on some ideas of iterative clustering techniques isodata and fuzzy c means which lead to a concrete method of determination of probabilistic sets a vagueness function associated with the fuzzy relation of the system forms a validity indicator of the identification algorithm numerical examples containing fuzzy and nonfuzzy data form an illustration of the methods provided

neuron inspired learning rules for fuzzy relational structures abstract fuzzy relational equations are a suitable framework for fuzzy modelling however their constructive resolution methods suffer from known drawbacks in order to determine approximate solutions both conventional and adaptive gradient based learning methods are proposed namely for extended versions of fuzzy relational structures simulation results show that for both learning methods good approximate solutions are found however the adaptive version shows considerably quicker convergence rates the problem of process fuzzy identification using the proposed framework is then outlined as an illustrative application

fuzzy system identification via probabilistic sets abstract the paper deals with the applications of probabilistic sets in system theory especially in system identification and the design of fuzzy logic controllers fuzzy systems described by means of fuzzy relational equations and fuzzy systems are discussed the identification procedure is based on some ideas of clustering techniques and probabilistic sets numerical examples using fuzzy and nonfuzzy data are used to illustrate the theory

solvability of fuzzy relational equations and manipulation of fuzzy data abstract the problem of characterization of fuzzy relational equations with respect to their solvability property is studied some aspects of manipulation of fuzzy data with the aid of fuzzy relational equations are considered two stages of manipulation process are indicated combining pieces of evidence and inferring their mutual correspondence both of them are formulated and solved by the use of fuzzy relational equations the role of a solvability index is extensively studied numerical considerations give an illustration of the approach we propose and deal with some real data used for fuzzy controllers

some results in the setting of fuzzy relation equations theory abstract in this paper we point out certain theoretical aspects of fuzzy relation equations connecting some known results between composite and o fuzzy equations several theorems are obtained and some examples are provided too

operators preserving sesquilinear form abstract we investigate the class of operators preserving sesquilinear form we show some properties similar to those characterizing mappings which preserve the inner product our considerations are carried out in spaces with inner product structure as well as in normed spaces we also consider the class of additive mappings preserving orthogonality

solving nonlinear optimization problems with fuzzy relation equation constraints abstract an optimization model with a nonlinear objective function subject to a system of fuzzy relation equations is presented since the solution set of the fuzzy relation equations is in general a non convex set when it is not empty conventional nonlinear programming methods are not ideal for solving such a problem in this paper a genetic algorithm ga is proposed this ga is designed to be domain specific by taking advantage of the structure of the solution set of fuzzy relation equations the individuals from the initial population are chosen from the feasible solution set and are kept within the feasible region during the mutation and crossover operations the construction of test problems is also developed to evaluate the performance of the proposed algorithm

further relation equations with equality and difference composition operators in this ch we will introduce other composition operators which have a plausible logical interpretation they will be called equality and difference operator respectively we first define a notion of equality and difference of any two grades of membership of a fuzzy set and of a fuzzy relation in the sequel this concept will be utilized to form the respective composition operators and the related fuzzy equations afterwards we provide a method of resolution of these equations characterizing completely the set of the solutions all the fuzzy sets involved are defined on finite sets and with membership values in 0 1