generalized means as model of compensative connectives abstract empirical investigations have shown that instead of strong logical and and or connectives there is often some degree of compensation in between in this paper the generalized means are proposed as connective operators for fuzzy set theory which easily allow for modelling the degree of compensation in a natural manner including the arithmetic and geometric means as well as the maximum and minimum operators as special cases numerical examples with empirical data form an illustration of this concept some aspects of applications to decision making are discussed

triangular norms position paper ii general constructions and parameterized families this second part out of three of a series of position papers on triangular norms for part i see triangular norms position paper i basic analytical and algebraic properties fuzzy sets and systems in press deals with general construction methods based on additive and multiplicative generators and on ordinal sums also included are some constructions leading to non continuous t norms and a presentation of some distinguished families of t norms

generalized lagrangian dynamics of physical and non physical systems abstract in this paper we show how to study the evolution of a complex system given imprecise knowledge about the state of the system and the dynamics laws it will be shown that dynamics of these systems is equivalent to lagrangian or hamiltonian mechanics in a n 1 dimensional space where n is a system s dimensionality in some cases however the corresponding lagrangian is more general than the usual one and could depend on the action in this case lagrange s equations gain a non zero right side proportional to the derivative of the lagrangian with respect to the action examples of such systems are unstable systems systems with dissipation and systems which can remember their history moreover in certain situations the lagrangian could be a set valued function the corresponding equations of motion then become differential inclusions instead of differential equations we will also show that the principal of least action is a consequence of the causality principle and the local topology of the state space and not an independent axiom of classical mechanics we emphasize that our adaptation of lagrangian mechanics does not use or depend on specific properties of the physical system being modeled therefore this lagrangian approach may be equally applied to non physical systems an example of such an application is presented as well

pictorial representations of fuzzy connectives part ii cases of compensatory operators and self dual operators abstract as the continuation of part i in which pictorial representations of t norms t conorms and averaging operators were made with the aid of a computer this paper discusses compensatory operators and proposes generalized compensatory operators which can be obtained from averaging operators self dual operators are discussed which can be defined by using t norms t conorms and averaging operators finally symmetric sums are reviewed which are also self dual operators the pictorial representations of these operators are also made

a new parametric family of fuzzy connectives and their application to fuzzy control abstract it is of practical importance to have parametric families of fuzzy connectives which are easy to handle and which are rich enough in the situation under consideration this paper introduces a new axiomatic description to conjunctive and compensative connection of fuzzy sets the new parametric family of connectives is developed through the use of logical valuation for the use of logical valuation the dependence of the correcting variable described by linguistic rules is compared with the dependence of the correcting variable calculated by means of an analytical description of the fuzzy controller the use of the new parameterised fuzzy connective is demonstrated for a hydraulic servo drive system the control of the process shows that with the new parameterised fuzzy connective we reach the best result of control compared with the conventional connectives e g t norms t or averaging operators d

fuzzy stack filters their definitions fundamental properties and application in image processing a new fuzzy filter called fuzzy stack filter fsf is proposed to extend the filtering capability of conventional stack filter sf which is based on the positive boolean function pbf as its window operator we fuzzify the onset and off set of a given pbf to obtain two types of fuzzy pbfs then we adopt the architecture of threshold decomposition to develop this new fuzzy filter with a fuzzy pbf as its window operator each fuzzy pbf is associated with a set of control parameters therefore the original pbf can be estimated from above and below by two fuzzy pbfs with appropriate control parameters furthermore we can apply the fuzzy modifiers to modify the fuzzy pbfs such that the pbfs can be completely estimated by the fuzzy pbfs hence the stack filter is a special case of fuzzy stack filter since some control parameters are added in this new filter the neural learning algorithms can be easily developed under the flexibility of the given control parameters we first propose the fuzzy m n rank order filter to test our proposed learning algorithm in this simple learning algorithm we can remove the noise corrupted images very well in contrast to the filtering behavior of rank order filters we believe that the results presented will lead to more fruitful research on more advanced and powerful learning algorithms dedicated to the appropriate applications

searching for basic properties obtaining robust implication operators in fuzzy control abstract this paper deals with the problem of searching basic properties for robust implication operators in fuzzy control we use the word robust in the sense of good average behavior in different applications and in combination with different defuzzification methods we study the behavior of the two main families of implication operators in the fuzzy control inference process these two families are composed by those operators that extend the boolean implication implication functions and those ones that extend the boolean conjunction t norms and force implications in order to develop the comparative study we will build different fuzzy controllers by means of these implication operators and will apply them to the fuzzy modeling of the real function y x and two three dimensional surfaces we analyze whether one of these two properties extension of the boolean implication and extension of the boolean conjunction is sufficient for obtaining a good implication operator or whether some complementary properties are necessary next we analyze whether we can get basic properties for good implication operators presenting three basic properties for the so called robust implication operators

knowledge representation by conjunctive normal forms and disjunctive normal forms based on n variable m dimensional fundamental clauses and phrases abstract in this paper the n variable m dimensional fundamental phrases and clauses are used to completely build the universal basis for the general cnf and dnf furthermore the general n variable m dimensional cnf and dnf are also defined in order to construct the generally standard expressions of different axiom based cnfs and dnfs such as 11 axiom based boolean cnf and boolean dnf 9 axiom based fuzzy cnf and fuzzy dnf and 5 axiom based t norm t conorm cnf and t norm t conorm dnf especially the general algorithm to construct the general n variable m dimensional cnf dnf from any given dnf cnf is proposed in general the intrinsic relations among the axioms the space of the fundamental phrases clauses and the general cnf and dnf are briefly studied additionally some interesting applications of knowledge representation by using the axiom based cnf and dnf are also discussed finally some new conclusions are drawn based on the new algorithm and the applications

on equivalence classes of fuzzy connectives the case of fuzzy integrals an equivalence relation between connectives or operators in the framework of multiple criteria decision making is introduced two operators are said to be equivalent if they lead to the same ranking of alternatives some general results to find equivalence classes of operators are given through the concept of level surfaces in a second part these results are applied to the case of discrete fuzzy integrals which are considered here as n place operators first some general results on the situation of fuzzy integrals among fuzzy operators are given and then the equivalence classes of fuzzy integrals

fuzzy connectives based crossover operators to model genetic algorithms population diversity abstract genetic algorithms are adaptive methods which may be used to solve search and optimization problems genetic algorithms process a population of search space solutions with three operations selection crossover and mutation an important problem in the use of genetic algorithms is the premature convergence in a local optimum their main causes are the lack of diversity in the population and the disproportionate relationship between exploitation and exploration the crossover operator is considered one of the most determinant elements for solving this problem in this paper we present new crossover operators based on fuzzy connectives for real coded genetic algorithms these operators are designed to avoid the premature convergence problem to do so they should keep the right exploitation exploration balance to suitably model the diversity of the population

applicability of the fuzzy operators in the design of fuzzy logic controllers a study of the different roles played by the fuzzy operators in fuzzy control is developed in this paper the behavior of a very large amount of fuzzy operators is analyzed and a comparison of the accuracy of many fuzzy logic controllers designed by means of these operators is carried out in order to do that a comparison methodology is defined and two fuzzy control applications are selected the inverted pendulum problem and the fuzzy modeling of the real curve y x

new classes of threshold aggregation functions based upon the tsallis q exponential with applications to perceptual computing we introduce two new classes of single parameter aggregation functions based upon the tsallis q exponential qe function of nonextensive statistical mechanics these aggregation functions denoted qe aggregation facilitate simple modeling of the common human reasoning trait of threshold inference where either 1 at least one input must exceed a threshold in order to achieve a nonzero aggregation output or 2 if any one of the inputs exceeds a different threshold the aggregation output takes its maximum value we illustrate the thresholding behavior of these functions on interval type 2 fuzzy inputs using an example known in the literature as the investment judgment advisor we believe that the new qe class of aggregation operators will prove useful in extending the range of options available for the design of perceptual computing systems

fast solving method of fuzzy relational equation and its application to lossy image compression reconstruction a fast solving method of the solution for max continuous t norm composite fuzzy relational equation of the type g i j r sup t spl square a sub i sup t spl square b sub j i 1 2 i j 1 2 j where a sub i spl isin f x x x sub 1 x sub 2 x sub m bj spl isin f y y y sub 1 y sub 2 y sub n r spl isin f x spl times y and spl square max continuous t norm composition is proposed it decreases the computation time ijmn l t p to jm i n l p where l t and p denote the computation time of min t norm and relative pseudocomplement operations respectively by simplifying the conventional reconstruction equation based on the properties of t norm and relative pseudocomplement the method is applied to a lossy image compression and reconstruction problem where it is confirmed that the computation time of the reconstructed image is decreased to 1 335 6 the compression rate being 0 0351 and it achieves almost equivalent performance for the conventional lossy image compression methods based on discrete cosine transform and vector quantization

the law of large numbers for fuzzy variables under a general triangular norm extension principle this paper presents a law of large numbers for t noninteractive weakly noninteractive fuzzy variables when using a general t norm extension principle the result is that convergence to a crisp set is obtained for all archimediean t norm intersection operators this generalises a previous result of badard the result is proved by deriving it from a similar result for the law of large numbers for dependency bounds of sums of random variables dependency bounds are used when nothing is known of the joint distribution of two random variables apart from their marginals the bridge used to connect dependency bounds with fuzzy number operations is used to give a probabilistic interpretation of fuzzy number operations some numerical examples are presented and a new method of numerically calculating the sup t convolutions of fuzzy numbers is presented this new method generalises the well known result for the calculation of sup min convolutions in terms of the level sets of membership functions

on fuzzy associative memory with multiple rule storage capacity kosko s fuzzy associative memory fam is the very first neural network model for implementing fuzzy systems despite its success in various applications the model suffers from very low storage capacity i e one rule per fam matrix a lot of hardware and computations are usually required to implement the model and hence it is limited to applications with small fuzzy rule base in this paper the inherent property for storing multiple rules in a fam matrix is identified a theorem for perfect recalls of all the stored rules is established and based upon which the hardware and computation requirements of the fam model can be reduced significantly furthermore we have shown that when the fam model is generalized to the one with max bounded product composition single matrix implementation is possible if the rule base is a set of semi overlapped fuzzy rules rule modification schemes are also developed and the inference performance of the established high capacity models is reported through a numerical example

cooperative evolutionary learning of linguistic fuzzy rules and parametric aggregation connectors for mamdani fuzzy systems there are two tasks in the design of linguistic fuzzy models for a concrete application the derivation of the linguistic rule base and the setup of the inference system and the defuzzification method traditionally the derivation of the linguistic rule base has been considered the most important task but the use of the appropriate aggregation connectors in the inference system and the defuzzification interface can improve the fuzzy system behavior in this paper we take in consideration this idea we propose an evolutionary learning method to learn a linguistic rule base and the parametric aggregation connectors of the inference and defuzzification in a single step the aim of this methodology is to make possible a high level of positive synergy between the linguistic rule base and the aggregation connectors improving the accuracy of the linguistic mamdani fuzzy systems our proposal has shown good results solving three different applications we introduce a statistical analysis of results for validating the model behavior on the applications used in the experimental study we must remark that we present an experimental study with a double intention a to compare the behavior of the new approach in comparison with those ones that first learn the rule base and then adapt the connectors and b to analyze the rule bases obtained with fixed aggregation connectors and with the adaptive ones for showing the changes on the consequent rules changes on labels that produce a better behavior of the linguistic model than the classic ones

general fuzzy acceptors for syntactic pattern recognition abstract in this paper the behavior of general fuzzy acceptor over a commutative cl semigroup l is considered and the properties of equivalence relation are investigated from these results we can apply for pattern recognition of deformed images or for enhancement of distorted patterns

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

mode type operators on fuzzy sets abstract the purpose of this paper is to characterize the class of operators which are pointwise commutative associative compatible and averaging this class of operators coincides with the class of mode type operators

basic concepts for a theory of evaluation hierarchical aggregation via autodistributive connectives in fuzzy set theory abstract the aggregative operator introduced by combi 1982 in his analysis of basic concepts for a theory of evaluation in this journal is based on the axioms of associativity continuity and monotonicity in this paper an alternative approach is given where associativity is replaced by another axiom viz autodistributivity implying a kind of hierarchical aggregation instead of the former horizontal one the new set of properties yields a general class of operators for connecting fuzzy sets which are formally similar to the aggregative operators but are purely compensative in a strong sense moreover this modified approach generalizes a recent approach based on generalized means as compensative connectives it appears to be similar to expected utility approaches in decision making under uncertainty and seems to be a very promising tool to handle representation problems as they grow from other fields of human decision making e g of multicriteria analysis and of welfare theory

some algebraic properties of weakly non interactive fuzzy numbers abstract this study focuses on the concept of sup t norm convulution a generalized version of zadeh s extension principle the authors treat the algebraic structure of fuzzy arithmetic based on the sup t norm convolution and especially show some new properties comparing them with those of the conventional fuzzy arithmetic based on sup min convolution

vector l_0 sparse variable pca principal component analysis pca achieves dimension reduction by replacing the original measured variables with a smaller set of derived variables called the principal components sparse pca improves this with sparsity there are two kinds of sparse pca sparse loading pca slpca which keeps all the measured variables but zeroes out some of their loadings and sparse variable pca svpca which removes some measured variables completely by simultaneously zeroing out all their loadings because it zeroes out some measured variables completely svpca is capable of huge additional dimension reduction beyond pca while slpca keeps all measured variables and does not have this capability here we consider a vector l0 penalized likelihood approach to svpca and develop a penalized expectation maximization pem algorithm which remarkably in an l0 setting leads to a closed form m step and we provide a convergence analysis

a new look at solving a system of fuzzy relational equations abstract solving a system of fuzzy relational equations exactly has been studied for many years a usual understanding on the solvable conditions is that the input fuzzy sets must be normal and pairwise disjoint such understanding is reexamined in this paper we show that the usual pairwise disjoint condition is too conservative and a system of max t fuzzy relational equations can be solved exactly when the input fuzzy sets are semi overlapped a condition commonly found in most rule based system applications in addition the boundary condition of solvability with respect to the compactness of the input fuzzy sets is derived if it cannot be satisfied we show that the system of equations could still be solved almost exactly by specifying the t norm as drastic product the results have been applied to study the capacity of fuzzy relations

fuzzy integral in multicriteria decision making abstract this paper presents a synthesis on the use of fuzzy integral as an aggregation operator in multicriteria decision making definitions essential properties are given and compared to those of usual aggregation operators a quick survey on applications is given

fuzzy sets and their operations ii among the basic operations which can be performed on fuzzy sets are the operations of union intersection complement algebraic product and algebraic sum in addition to these operations new operations called bounded sum and bounded difference were defined by l a zadeh to investigate the fuzzy reasoning which provides a way of dealing with the reasoning problems which are too complex for precise solution this paper investigates the algebraic properties of fuzzy sets under these nex operations of bounded sum and bounded difference and the properties of fuzzy sets in the case where these new operations are combined with the well known operations of union intersection algebraic product and algebraic sum

data based fuzzy rule test for fuzzy modelling in the field of fuzzy modelling the exclusive consideration of the modelling error leads to problems concerning the handling of high dimensional applications and the interpretability of the resulting rule base to solve those problems a statistically motivated fuzzy rule test is proposed it decides if a fuzzy if then statement is a relevant rule or not in this way the problem of finding a good rule base can be reduced to the problem of finding good relevant rules

intelligent reduction methods based upon fuzzy set theory as a helpful tool for controlling industrial plants abstract in many real life situations we must separate relevant and irrelevant objects and situations a computer that advises operators in an industrial plant or controls subplants or the entire plant would be much more intelligent and efficient if it were able to reduce the number of parameters by determining which of them are relevant we describe a method to perform this difficult task based upon fuzzy set theory we also describe a computer implementation that has been tested successfully at dow benelux n v hence the proposed method and the implementation is usable we also demonstrate the method with a practical example

the mycin certainty factor handling function as uninorm operator and its use as a threshold function in artificial neurons abstract the uninorm operator class as defined by yager unifies the t norm and t conorm operator classes and allows special kind of aggregation that depends on the identity element in the paper the mycin certainty factor handling function f m is proved to belong to the uninorm operator class the shape of function f m is studied and its resemblance to artificial neuron s threshold functions is established the two variable function f m can be seen as an extension of typical neuron threshold functions to the three dimensional space the use of one of the two variables as a parameter gives the neuron tuning capabilities a specific class of artificial neural networks that cope with uncertainty and can use m as threshold function is also proposed

an approach to multi criteria decision making problems using probabilistic set theory abstract the method of multi criteria decision making under uncertainty and imprecision is presented in this paper the criteria represented by probabilistic sets may be aggregated using various operators such aggregation leads to the final decision set from which the essential information as mean value variance etc can be obtained applying different choice criteria the best alternative can be chosen the considerations are illustrated by means of numerical examples

basic concepts for a theory of evaluation the aggregative operator abstract starting with an explication of the aggregative concept and deducing a general structure which satisfies a number of minimal requirements properties of clustering the main features of a new mathematical theory called theory of evaluation are developed the theory sheds new light on such well known concepts as membership conjunction and disjunction and seems to be a very promising tool to handle representation problems as they grow from the fields of theory of fuzzy set and its many applications of human decision making and of multicriteria analysis

exotic spheres and curvature since their discovery by milnor in 1956 exotic spheres have provided a fascinating object of study for geometers in this article we survey what is known about the curvature of exotic spheres

deformed fuzzy automata for correcting imperfect strings of fuzzy symbols presents a fuzzy method for the recognition of strings of fuzzy symbols containing substitution deletion and insertion errors as a preliminary step we propose a fuzzy automaton to calculate a similarity value between strings the adequate selection of fuzzy operations for computing the transitions of the fuzzy automaton allows us to obtain different string similarity definitions including the levenshtein distance a deformed fuzzy automaton based on this fuzzy automaton is then introduced in order to handle strings of fuzzy symbols the deformed fuzzy automaton enables the classification of such strings having an undetermined number of insertion deletion and substitution errors the selection of the parameters determining the deformed fuzzy automaton behavior would allow to implement recognizers adapted to different problems the paper also presents algorithms that implement the deformed fuzzy automaton experimental results show good performance in correcting these kinds of errors

a randomized approach with geometric constraints to fingerprint verification abstract in this paper a fuzzy bipartite weighted graph model is proposed to solve fingerprint verification problem a fingerprint image is preprocessed first to form clusters of feature points which are called feature point clusters twenty four attributes are extracted for each feature point cluster the attributes are characterized by fuzzy values attributes of an input image to be verified are considered as the set of left nodes in a fuzzy bipartite weighted graph and the attributes of claimed template fingerprint image are considered as the set of right nodes in the graph the fingerprint verification problem is thus converted into a fuzzy bipartite weighted graph matching problem a matching algorithm is proposed for the fuzzy bipartite weighted graph model to find an optimal matching with a goodness score experimental results reveal the feasibility of the proposed approach in fingerprint verification

on the order of triangular norms comments on a triangular norm hierarchy by e cretu in cretu fuzzy sets and systems 120 2001 371 the members of the families of frank dubois prade yager and hamacher t norms respectively are compared in a pointwise way with the minimum the product and the lukasiewicz t norm all these results are well known and trivial moreover these families of t norms cannot only be compared with the three basic t norms above but all these families are monotone with respect to their index a fact which is also well known and straightforward to prove with the exception of the family of frank t norms whose monotonicity has been first proven in butnariu and klement triangular norm based measures and games with fuzzy coalitions kluwer dordrecht 1993

on the t transitivity of kernels abstract in this paper we present a view of kernels from a fuzzy set theoretical perspective indeed it turns out that kernels which are positive definite functions have to fulfill a consistency property given by the so called t transitivity of a fuzzy t equivalence relation with respect to the triangular norm t as a result we introduce a triangular norm t cos which is characterized as being the greatest one for which all kernels mapping to the unit interval and having constant 1 in its diagonal are t cos equivalences finally some applied relevant aspects are discussed as a starting point for further research

foundations of fuzzy sets abstract this paper gives an overview of the origins of fuzzy set theory and the problems for the foundations of fuzzy sets arising from those origins and current practice it then gives detailed accounts of categorical approaches using a closed structure to capture the fuzzy and connective within these categories weak forms of subobject representations provide an internal second order logic an approach to fuzzy real numbers and fuzzy topology is included to illustrate the use of this internal second order theory

on β precision aggregation in this paper we propose an extension of the current fuzzy connectives to precision aggregators which include a certain tolerance towards noise and measurement errors in the values of the aggregated criteria when their cardinality is large to do this we have set our extension proposal in the framework of yager s mam mom and mica operators and study the properties of the new extended operators a simple example illustrates the practical use of the extended aggregators