a probabilistic fuzzy logic system for modeling and control in this paper a probabilistic fuzzy logic system pfls is proposed for the modeling and control problems similar to the ordinary fuzzy logic system fls the pfls consists of the fuzzification inference engine and defuzzification operation to process the fuzzy information different to the fls it uses the probabilistic modeling method to improve the stochastic modeling capability by using a three dimensional membership function mf the pfls is able to handle the effect of random noise and stochastic uncertainties existing in the process a unique defuzzification method is proposed to simplify the complex operation finally the proposed pfls is applied to a function approximation problem and a robotic system it shows a better performance than an ordinary fls in stochastic circumstance

maximal parabolic regularity for divergence operators including mixed boundary conditions abstract we show that elliptic second order operators a of divergence type fulfill maximal parabolic regularity on distribution spaces even if the underlying domain is highly non smooth the coefficients of a are discontinuous and a is complemented with mixed boundary conditions applications to quasilinear parabolic equations with non smooth data are presented

conditional density estimation using probabilistic fuzzy systems we consider conditional density approximation by fuzzy systems fuzzy systems are typically used to approximate deterministic functions in which the stochastic uncertainty is ignored we propose probabilistic fuzzy systems pfss in which the probabilistic nature of uncertainty is taken into account these systems take also fuzzy uncertainty into account by their fuzzy partitioning of input and output spaces we discuss an additive reasoning scheme for pfss that leads to the estimation of conditional probability densities and prove how such fuzzy systems compute the expected value of this conditional density function we show that some of the most commonly used fuzzy systems can compute the same expected output value and we derive how their parameters should be selected in order to achieve this goal the additional information and process understanding provided by the different interpretations of the pfs models are illustrated using a real world example

finding a weights restricted efficient extreme point and using it for solving molp problems this paper deals with solving a multiple objective linear programming problem a preferred weights vector from a restricted set of weights incorporating the case that the weighted problem may have unbounded solution is used for generating a preferred efficient extreme point the final solution can be found by changing interactively weights restrictions or comparing weights restricted efficient extreme points also the results can be used in vector maximum and interactive methods

a note on the conditional probability of fuzzy subsets of a continuous domain abstract the notion of a mass assignment corresponding to a discrete fuzzy set is generalised to include all fuzzy sets the definition is motivated by considering a possible mechanism by which an intelligent agent could extend set theoretic operations to fuzzy sets this same mechanism also suggests an alternative definition for the conditional probability of fuzzy sets it is noted that this definition is probability possibility consistent and it is shown that in the case where the probability distribution on the domain has a density function then the conditional probability of f given g can be expressed as the probability of f calculated relative to a conditional density function dependent on g this result is used to develop an analytical method for calculating conditional probabilities of fuzzy subsets of the reals where the membership functions of f and g and the density function are continuous piecewise linear functions such a method has applications to the development of a general semantic unification in fril

semiconormed possibility integrals and multi feature pattern classification abstract the semiconormed possibility integrals are proposed as a multi feature pattern classification model a semiconormed possibility integral is a nonlinear integration of a function and its corresponding non normalized possibility measures over feature space the function of an object s feature vector represents the possibilities with uncertainty that the object belongs to a class the uncertainty is due to the similar characteristics of objects from different classes and the distortion of the original characteristic information caused by feature data acquisition systems the uncertainty is assessed by the non normalized possibility measures a possibility measure of a feature is considered as the credibility of the feature to provide reliable information for pattern classification integration of a function and the possibility measures effectively reduces the uncertainty and improves the pattern classification results a pattern classification algorithm based on the semiconormed possibility integrals was used to classify a set of ellipse data and the well known iris data the classification results were compared with those obtained by using bayes classifier

representation and application of fuzzy numbers abstract in this paper we present the theoretical background of fuzzy numbers connected with the possibility and dempster shafer theories we describe some types of representation of fuzzy numbers and we study the notions of the distance and orders between fuzzy numbers based on these representations the second part is devoted to the application of fuzzy numbers in data analysis artificial intelligence and decision making

redefining chance constrained programming in fuzzy environment this paper presents an alternative definition of probability of fuzzy event in analogy to probability of nonfuzzy event we view probability of fuzzy event in terms of cumulative distribution of a random variable a multi objective imprecise chance constrained programming problem has been defined here which is an extension of classical chance constrained programming problem in fuzzy environment an illustrative numerical example is given

membership functions and probability measures of fuzzy sets the notion of fuzzy sets has proven useful in the context of control theory pattern recognition and medical diagnosis however it has also spawned the view that classical probability theory is unable to deal with uncertainties in natural language and machine learning so that alternatives to probability are needed one such alternative is what is known as possibility theory such alternatives have come into being because past attempts at making fuzzy set theory and probability theory work in concert have been unsuccessful the purpose of this article is to develop a line of argument that demonstrates that probability theory has a sufficiently rich structure for incorporating fuzzy sets within its framework thus probabilities of fuzzy events can be logically induced the philosophical underpinnings that make this happen are a subjectivistic interpretation of probability an introduction of laplace s famous genie and the mathematics of encoding expert testimony the benefit of making probability theo

deriving expected values from probabilities of fuzzy subsets abstract the problem of determining the expected value of a variable v on the basis of a fuzzy evidence of the type pr v is a is q is considered it is shown that if the set x of possible values of v is finite x x 1 x n and the membership function of the fuzzy probability q is upper semi continuous then the problem may be reduced to the analysis of a certain linear programming problem since this problem is of a special form it is possible to develope solution algorithms which are more effective than classical methods known in linear programming there are presented two algorithms using the first algorithm one may obtain the r cut of the fuzzy expected value for any fixed r 0 1 with the help of the second algorithm one may obtain the r cuts of the fuzzy expected value for all r 0 1 the presented approach is an essential improvement of the method proposed before that by yager 5

an efficient configuration for probabilistic fuzzy logic system a novel inference configuration is proposed to improve the computational efficiency and information loss in the probabilistic fuzzy inference process the probabilistic inference and the fuzzy inference are unified in one operation based on the continuous form of the probabilistic fuzzy set besides the faster inference operation it is able to produce fuzzy outputs in a complete probabilistic distribution that in turn will provide information about the approximation bound the computational analyses of six different fuzzy systems demonstrate the inference efficiency of the proposed method its effectiveness can be further demonstrated on the application to modeling of an industrial curing process the robust modeling performance discloses its potential in process modeling under complex environment

probability of fuzzy events defined as denumerable additivity measure abstract in this paper the probability of fuzzy events is defined as a denumerable additivity measure this definition is based on a non conventional approach of separativity between fuzzy subsets the presented measure fulfils all properties analogous to the properties of classical probability in the crisp case further the notion of conditional probability of fuzzy events complete fuzzy repartition and independent fuzzy events are defined by means of the probability measure considered here connections between all the above notions are presented in this paper too finally the bayes formula is proved for the fuzzy case

expected values from probabilities of fuzzy subsets abstract in many applications in decision theory and expert systems we may have information regarding the probability of some variable being equal to a fuzzy subset in this paper we provide a methodology for using this information to obtain the expected value of the variable

fuzzy equalization in the construction of fuzzy sets in this note we introduce and study a concept of fuzzy equalization fuzzy equalization concerns a process of building information granules that are semantically and experimentally meaningful the experimental relevance of a given information granule fuzzy set is directly linked with an encapsulation of a certain experimental evidence conveyed by the respective probability density function of available data we establish a detailed equalization algorithm developed for triangular fuzzy sets the study elaborates on the role of the fuzzy equalization in system design

fast semi analytic computation of elastic edge singularities abstract the singularities that we consider are the characteristic non smooth solutions of the equations of linear elasticity in piecewise homogeneous media near two dimensional corners or three dimensional edges we describe here a method to compute their singularity exponents and the associated angular singular functions we present the implementation of this method in a program whose input data are geometrical data the elasticity coefficients of each material involved and the type of boundary conditions dirichlet neumann or mixed conditions our method is particularly useful with anisotropic materials and allows to follow the dependency of singularity exponents along a curved edge

fuzzy stochastic linear programming survey and future research directions in many real life problems one has to base decision on information which is both fuzzily imprecise and probabilistically uncertain although consistency indexes providing a union nexus between possibilistic and probabilistic representation of uncertainty exist there are no reliable transformations between them this calls for new paradigms for incorporating the two kinds of uncertainty into mathematical models fuzzy stochastic linear programming is an attempt to fulfill this need it deals with modelling and problem solving issues related to situations where randomness and fuzziness co occur in a linear programming framework in this paper we provide a survey of the essential elements methods and algorithms for this class of linear programming problems along with promising research directions being a survey the paper includes many references to both give due credit to results in the field and to help readers obtain more detailed information on issues of interest

an interactive multiple objective linear programming method for a class of underlying nonlinear utility functions this paper develops a method for interactive multiple objective linear programming assuming an unknown pseudo concave utility function satisfying certain general properties the method is an extension of our earlier method published in this journal zionts s wallenius j 1976 an interactive programming method for solving the multiple criteria problem management sci 22 6 652 663 various technical problems present in predecessor versions have been resolved in addition to presenting the supporting theory and algorithm we discuss certain options in implementation and summarize our practical experience with several versions of the method

fuzzy cardinality and the modeling of imprecise quantification abstract the concept of cardinality of a fuzzy set has received attention from several researchers and has been defined in several apparently independent manners a systematic investigation of this notion is performed which unifies and improves previous attempts the cardinality of a fuzzy set viewed as a fuzzy integer is related to scalar cardinality indices the closely related question of the probability of a fuzzy event is dealt with lastly the usefulness of fuzzy cardinality for meaning representation of statements or queries involving fuzzy linguistic quantifiers is emphasized

probability on fuzzy sets and ϵ fuzzy sets abstract this research investigates the development of probability calculi for fuzzy sets abbreviated fs s two fs algebras are employed each as a basis for a fs probability calculus one employs minima for fs intersections and consequently maxima for unions the other uses a product rule for intersections and a standard probability like addition rule for unions the two derived fs probability theories are not identical and neither is identical with standard probability theory finally e fuzzy sets or almost standard sets are defined and investigated the main result of the last section is that while fs s do not satisfy all the laws of boolean algebra fuzzy sets that are almost standard almost satisfy these laws moreover the extent to which such laws are satisfied decreases as their complexity increases

solutions for quasilinear nonsmooth evolution systems in l p we prove that nonsmooth quasilinear parabolic systems admit a local solution in l p strongly differentiable with respect to time over a bounded three dimensional polyhedral space domain the proof rests essentially on new elliptic regularity results for polyhedral laplace interface problems for anisotropic materials these results are based on sharp pointwise estimates for green s function which are also of independent interest to treat the nonlinear problem we then apply a classical theorem of sobolevskii for abstract parabolic equations and recently obtained resolvent estimates for elliptic operators and interpolation results as applications we have in mind primarily reaction diffusion systems the treatment of such equations in an l p context seems to be new and allows by gauss theorem the proper definition of the normal component of currents across the boundary

on possibility analysis of fuzzy data this paper addresses the analysis of fuzzy data from a possibilistic perspective the concept of a double fuzzy variable is introduced and related notions are defined and investigated the properties and fuzzy parameter estimation of the possibility distribution are also investigated a maximum possibility likelihood principle is provided to estimate unknown fuzzy parameters a normal possibility distribution is used as an example and estimates of its fuzzy parameters are given the proposed possibility theory of fuzzy data constitutes a new approach to analyzing fuzzy data from a possibilistic point of view

fuzzy semi markov migration process in credit risk we explore the rating system used by credit agencies with a focus on problems that justify the use of fuzzy set theory we prove that a fuzzy market is viable if and only if an equivalent martingale measure exists from which we construct the forward probability measure and under which the discounted price of a default free bond is a martingale we model the evolution of credit migration of a defaultable bond as an inhomogeneous semi markov process with fuzzy states we study the effects of changing the real probability measure to a forward probability measure in addition we investigate the asymptotic behaviour of the survival probability in each fuzzy state given in the absence of default finally we discuss parameter estimation and calibration of the inhomogeneous markov chain with fuzzy states

systems control with generalized probabilistic fuzzy reinforcement learning reinforcement learning rl is a valuable learning method when the systems require a selection of control actions whose consequences emerge over long periods for which input output data are not available in most combinations of fuzzy systems and rl the environment is considered to be deterministic in many problems however the consequence of an action may be uncertain or stochastic in nature in this paper we propose a novel rl approach to combine the universal function approximation capability of fuzzy systems with consideration of probability distributions over possible consequences of an action the proposed generalized probabilistic fuzzy rl gpfrl method is a modified version of the actor critic ac learning architecture the learning is enhanced by the introduction of a probability measure into the learning structure where an incremental gradient descent weight updating algorithm provides convergence our results show that the proposed approach is robust under probabilistic uncertainty while also having an enhanced learning speed and good overall performance

discussion probability theory and fuzzy logic are complementary rather than competitive the relationship between probability theory and fuzzy logic has long been an object of discussion and some controversy the position articulated in this article is that probability theory by itself is not sufficient for dealing with uncertainty and imprecision in real world settings to enhance its effectiveness probability theory needs an infusion of concepts and techniques drawn from fuzzy logic especially the concept of a linguistic variable and the calculus of fuzzy if then rules in the final analysis probability theory and fuzzy logic are complementary rather than competitive

bell type inequalities for quasi copulas we present all bell type inequalities regarding at most four random events of which not more than two are intersected at the same time reformulating these inequalities in the context of fuzzy probability calculus leads to related inequalities on commutative conjunctors we prove that some of these inequalities hold for quasi copulas or copulas additionally for the important parametric family of frank t norms which are also copulas we identify for each of the inequalities the parameter values for which it is fulfilled thereby completing the results of pykacz and d hooghe

fuzzy subsets of the space of probability measures and expected value of fuzzy variable abstract we study the fuzzy subsets of the space of all probability measures and their applications to problems connected with fuzzy evidences first we investigate the properties of the fuzzy set of all probability measures for which the probability of fuzzy event a is equal to fuzzy probability q next using these investigations we calculate the expected value of the variable generated by the fuzzy evidence pr v is a is q in this paper we assume that a is a fuzzy number on the real line r

structural and functional quantization of vagueness abstract this paper proposes two methods of quantizing vague categories in terms of quantities whose identities have been either widely recognized in the fuzzy set theory literature or established by bayesian decision theory the methods are called structural quantization and functional quantization respectively the idea behind structural quantization is that for a comparative vague category like good scores there exists a preference ordering among the possible scores and how good a particular score is depends on how large the portion of examinees whose scores are lower than that score is functional quantization adopts the conclusion of bayesian decision theory that a rational agent s belief can be represented by a probability to quantize the term good score the method first assesses the two probabilities that respectively represent a rational agent s beliefs about john s score before and after learning that john has got a good score it then derives a membership function by comparing those two probabilities the equivalence of the two quantization methods is hypothesized the logic connectives not and and or are discussed in the setting of structural quantization while aggregation of information is addressed in the setting of functional quantization

measures of fuzzy sets in this paper a fuzzy set is considered as a point of a segment in riesz space using the daniell representation theorem of positive linear functionals on riesz space it is demonstrated that n a the s finite measure of fuzzy set a equals ua u n du some fundamental properties of measures and signed measures of fuzzy sets are discussed the law of large numbers in fuzzy occasions and a few useful properties of the conditional probability of fuzzy events are obtained

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

characterization of continuous t norms compatible with zadeh s probability of fuzzy events in this work we study the concept of p z compatibility which holds for those t norms for which zadeh s probability of fuzzy events fulfills kolmogorov s axioms whether a probability satisfies kolmogorov s axioms or not depends almost exclusively on the t norm and dual t conorm used to define intersection and union respectively of fuzzy sets in order to better understand fuzzy probability theory we provide a complete characterization of the p z compatible continuous t norms

medical diagnosis fuzzy sets and degrees of belief abstract a model simulating the medical diagnostic process is presented diagnostic groups are considered as fuzzy sets partial knowledge is quantified by belief functions the process is considered as the evaluation by the clinician of his degree of belief concerning the belonging of his patient to a fuzzy set given fuzzy and partial informations

characterization of measures based on strict triangular norms as a natural generalization of a measure space butnariu and klement 4 introduced t tribes of fuzzy sets with t measures they gave a complete characterization of t measures for a frank triangular norm t here we characterize t measures with respect to non frank strict triangular norms we show the specific roles of frank triangular norms and a newly introduced family nearly frank triangular norms

probability of a fuzzy event an axiomatic approach abstract axioms are proposed that could justify the natural definition of the probability of a fuzzy event initially given by zadeh they are based 1 on the postulate that the sum of the conditional probability of a fuzzy event and of its complement given any fuzzy event adds to one or 2 on soft independence for orthogonal sets with independent constitutive elements a general postulate is also required concerning the complement of a fuzzy set the classical definition of the operator representing the complement can also be deduced

paper evidence measures based on fuzzy information an organized body of results pertaining to the analysis of evidence when the available knowledge is pervaded with imprecision is provided the evidence theory of dempster and shafer is extended to the case of fuzzy observations and fuzzy events upper and lower possibilities of such events are derived by iterating the generation process of upper and lower probabilities as done by dempster both probabilistic and possibilistic models are developed in parallel these evidence measures are used for decision evaluation when the available knowledge is poor the classical model of decision making under uncertainty is thus extended to the case when the consequences of a decision are only roughly described and their probabilities of occurrence modeled by intervals or fuzzy numbers

a probabilistic and statistical view of fuzzy methods fuzzy set theory has primarily been associated with control theory and with the representation of uncertainty in applications in artificial intelligence more recently fuzzy methods have been proposed as alternatives to traditional statistical methods in statistical quality control linear regression and forecasting among other areas we review some basic concepts of fuzzy methods point out some philosophical and practical problems and offer simpler alternatives based on traditional probability and statistical theory applications in control theory and statistical quality control serve as our primary examples

from a gaussian mixture model to nonadditive fuzzy systems this work explores how a kind of probabilistic system namely the gaussian mixture model gmm can be translated to an additive fuzzy system we will prove the mathematical equivalence between the conditional mean of a gmm and the defuzzified output of a generalized fuzzy model gfm the relationship between a gmm and a gfm and the conditions for gmm to gfm translation will be made explicit in the form of theorems the work will then extend to special cases of the gfm specifically the mamdani larsen and takagi sugeno fuzzy models the possibility of reverse translation that is from a gfm to a gmm will also be discussed finally we will consider the generality of a gmm specifically how it can approximate other distribution functions

a heuristic simulated annealing algorithm of learning possibility measures for multisource decision making abstract in multisource decision making non normalized possibility measures are used to gauge the credibilities of each subset of information sources in order to reduce the uncertainty introduced by the sources and minimize decision error the credibility of individual information source i e the possibility density is objectively learned from subjectively chosen training data a learning algorithm of the possibility measures is constructed based on the ideas of simulated annealing and feasible direction the algorithm is designed for large training data set two multifeature image segmentation experiments and a comparison of learning speed are presented

linear programming under randomness and fuzziness this paper is concerned with the use of fuzzy sets theory well suited for nuanced human behavior to extend the applicability of linear programming in a way to let both randomness and fuzziness uncertainties be taken into account a fuzzy stochastic linear programming problem is considered and ways to handle it suggested the basic idea behind our approaches is to translate the original problem in deterministic terms via tools provided by fuzzy sets theory to get a classical one some conditions for the resulting problems to be solved without extra computational effort are stressed and illustrative examples given

a mass assignment theory of the probability of fuzzy events abstract mass assignment theory techniques for processing uncertainty in fril are reviewed the notion of the probability of a fuzzy event is introduced together with the t norm definition of conditional probabilities the latter is then shown to be probability possibility inconsistent an alternative theory of conditional probabilities based on mass assignments is presented together with a number of results illustrating some intuitive properties in particular the mass assignment theory of conditional probabilities is shown to be probability possibility consistent

convex combinations of nilpotent triangular norms abstract in this paper we deal with the open problem of convex combinations of continuous triangular norms stated by alsina frank and schweizer c alsina m j frank b schweizer problems on associative functions aequationes math 66 2003 128 140 problems 5 and 6 they pose a question whether a non trivial convex combination of triangular norms can ever be a triangular norm the main result of this paper gives a negative answer to the question for any pair of continuous archimedean triangular norms with different supports with the help of this result we show that a non trivial convex combination of nilpotent t norms is never a t norm the main result also gives an alternative proof to the result presented by ouyang and fang y ouyang j fang some observations about the convex combination of continuous triangular norms nonlinear anal 68 11 2008 3382 3387 theorem 3 1 in proof of the main theorem we utilize the reidmeister condition known from the web geometry

the mean value of a fuzzy number abstract the notion of expectation is defined for fuzzy numbers of intervals viewing them as consonant random sets this expectation is interval valued and remains additive in the sense of fuzzy interval addition as a by product some light is shed on the debate between probability and possibility theory and the problem of how to carry measure theoretic notions into the field of possibility theory

an overview of techniques for solving multiobjective mathematical programs multiobjective mathematical programming has been one of the fastest growing areas of or ms during the last 15 years this paper presents some reasons for the rapidly growing increase in interest in multiobjective mathematical programming a discussion of the advantages and disadvantages of the three general approaches articulation of the decision maker s preference structure over the multiple objectives prior to during or after the optimization towards multiobjective mathematical programming a nontechnical overview of many of the specific solution techniques for multiobjective mathematical programming and a discussion of important areas for further research the overview concentrates on those techniques which require an articulation of the decision maker s preference structure either during or after the optimization since these are the areas where most of the recent research has been conducted it differs from previous overviews in that in addition to the timing of the elicited preference informa

seminormed possibility integrals for application oriented modelling abstract semiconormed possibility integrals are proposed as a family of models for decision making which have some special properties due to the non normalized possibility measures the semiconormed possibility integrals are developed from the semiconormed fuzzy integrals and the non normalized possibility measures the non normalized possibility measures induced from the fuzzy sets have the property that the measure of the universe canbe less than equal to one when the universe consists of a set of information sources the possibility measure of a subset of information sources is considered as the credivility of the subset sources the non normalization is due to the fact that an information source in application introduces certainty during data acquisition therefore an information source in reality usually is not fully credible to improve the precision of modelling the credibility values of each information source are learned form training data in term of learning computation a non normalized possibility measure is much simpler to construct than a fuzzy measure finally two applications of the semiconormed possibility integrals are presented

a logic for reasoning about the probability of fuzzy events in this paper we present the logic fp l n l which allows to reason about the probability of fuzzy events formalized by means of the notion of state in a mv algebra this logic is defined starting from a basic idea exposed by hajek metamathematics of fuzzy logic kluwer dordrecht 1998 two kinds of semantics have been introduced namely the class of weak and strong probabilistic models the main result of this paper is a completeness theorem for the logic fp l n l w r t both weak and strong models we also present two extensions of fp l n l the first one is the logic fp l n rpl obtained by expanding the fp l n l language with truth constants for the rationals in 0 1 while the second extension is the logic fcp l n email protected allowing to reason about conditional states