calibrated propensity score method for survey nonresponse in cluster sampling weighting adjustment is commonly used in survey sampling to correct for unit nonresponse in cluster sampling the missingness indicators are often correlated within clusters and the response mechanism is subject to cluster specific nonignorable missingness based on a parametric working model for the response mechanism that incorporates cluster specific nonignorable missingness we propose a method of weighting adjustment we provide a consistent estimator of the mean or totals in cases where the study variable follows a generalized linear mixed effects model the proposed method is robust in the sense that the consistency of the estimator does not require correct specification of the functional forms of the response and outcome models a consistent variance estimator based on taylor linearization is also proposed numerical results including a simulation and a real data application are presented

on viewing fuzzy measures as fuzzy subsets we introduce the concept of a fuzzy measure discuss some of their basic properties and look at some notable examples we introduce a correspondence between a fuzzy measure on x and a fuzzy subset over the power set of x these fuzzy subsets are required to have special properties on the space 2x and we refer to these as measure type fuzzy sets we take advantage of this correspondence between a fuzzy measure and a fuzzy set to suggest the formulation of various set operations on fuzzy measures

model checking computation tree logic over finite lattices multi valued model checking is an extension of classical model checking used to verify the properties of systems with uncertain information content it is used in systems where the semantic domain of both the models of the systems and the specification is a de morgan algebra or more abstract structure such as semirings a common feature of de morgan algebras and semirings is that they satisfy the distributive law which is crucial for all of the multi valued model checking algorithms developed so far in this paper we study computation tree logic with membership values in a finite lattice which are called multi valued computation tree logic mctl we introduce three semantics for mctl over the multi valued kripke structure path semantics fixpoint semantics and algebraic semantics and prove that they coincide if and only if the underlying lattice is distributive furthermore we provide model checking algorithms with respect to the three semantics our algorithms show that mctl model checking problems with respect to the fixpoint and algebraic semantics can be solved in polynomial time in the size of the state space of the multi valued kripke structure the size of the lattice and the length of the formula while mctl model checking problem with respect to the path semantics is in pspace we also provide a lower bound for the mctl model checking problem with respect to the path semantics

algorithmic and logical characterizations of bisimulations for non deterministic fuzzy transition systems abstract bisimulation is a well known behavioral equivalence for discrete event systems and has been developed in fuzzy systems quickly in this paper we adopt an approach of the relational lifting that is one of the most used methods in the field of bisimulation research to define it for a non deterministic fuzzy transition system an o s 4 2 algorithm is given for testing bisimulation where s is the number of states and the number of transitions in the underlying transition systems two different modal logics are presented one is two valued and indicates whether a state satisfies a formula which is an extension of hennessy milner logic the other is real valued and shows to what extent a state satisfies a formula they both characterize bisimilarity soundly and completely interestingly the second characterization holds under a class of fuzzy logics in addition this real valued logic allows us to conveniently define a logical metric that captures the similarity between states or systems that is the smaller distance the more states alike although the work is inspired by the corresponding work in probabilistic systems it is obviously different in particular the real valued logic in this paper remains unexplored in fuzzy systems even in probabilistic systems

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

computation tree logic model checking based on possibility measures in order to deal with the systematic verification with uncertain information in possibility theory li and li 2013 introduced model checking of linear time properties in which the uncertainty is modeled by possibility measures xue lei and li 2011 defined computation tree logic ctl based on possibility measures which is called possibilistic ctl poctl this paper is a continuation of the above work first we study the expressiveness of poctl unlike probabilistic ctl it is shown that poctl in particular qualitative poctl is more powerful than ctl with respect to their expressiveness the equivalent expressions of basic ctl formulae using qualitative poctl formulae are presented in detail some poctl formulae that cannot be expressed by any ctl formulae are presented in particular some qualitative properties of repeated reachability and persistence are expressed using poctl formulae next adapting ctl model checking algorithm a method to solve the poctl model checking problem and its time complexity are discussed in detail finally an example is given to illustrate the poctl model checking method

a hierarchical neural model of matching abstract the issue of matching two fuzzy sets emerges as an essential design aspect in many algorithms of processing fuzzy information including fuzzy controllers pattern classifiers knowledge based systems etc this paper introduces a new model of matching the primordial features of the proposed model involve 1 i matching being carried out with respect to the grades of membership of the discussed fuzzy sets as well as some of their functionals like energy entropy transom 2 ii the concepts of hierarchies of matching in the model leading directly to a straightforward distinction between local and global levels of matching and 3 iii a distributed character of the model that is realized as a logic based neural network with enhanced learning capabilities

probabilistic propositional temporal logics we present two closely related propositional probabilistic temporal logics based on temporal logics of branching time as introduced by ben ari pnueli and manna acta inform 20 1983 207 226 emerson and halpern proceedings 14th acm sympos theory of comput 1982 pp 169 179 and emerson and clarke sci comput program 2 1982 241 266 the first logic ptlf is interpreted over finite models while the second logic ptlb which is an extension of the first one is interpreted over infinite models with transition probabilities bounded away from 0 the logic ptlf allows us to reason about finite state sequential probabilistic programs and the logic ptlb allows us to reason about finite state concurrent probabilistic programs without any explicit reference to the actual values of their state transition probabilities a generalization of the tableau method yields deterministic single exponential time decision procedures for our logics and complete axiomatizations of them are given several meta results including the absence of a finite model property for ptlb and the connection between satisfiable formulae of ptlb and finite state concurrent probabilistic programs are also discussed

linguistic quantifiers modeled by sugeno integrals since quantifiers have the ability of summarizing the properties of a class of objects without enumerating them linguistic quantification is a very important topic in the field of high level knowledge representation and reasoning this paper introduces a new framework for modeling quantifiers in natural languages in which each linguistic quantifier is represented by a family of fuzzy measures and the truth value of a quantified proposition is evaluated by using sugeno s integral this framework allows us to have some elegant logical properties of linguistic quantifiers we compare carefully our new model of quantification and other approaches to linguistic quantifiers a set of criteria for linguistic quantification was proposed in the previous literature the relationship between these criteria and the results obtained in the present paper is clarified some simple applications of the sugeno s integral semantics of quantifiers are presented

a linear regression model using triangular fuzzy number coefficients fuzzy regression analysis using fuzzy linear models with symmetric triangular fuzzy number coefficients has been formulated earlier the goal of this regression is to find the coefficients of a proposed model for all given input output data sets in this paper we extend the results of a fuzzy linear regression model that uses symmetric triangular coefficient to one with non symmetric fuzzy triangular coefficients this work eradicates the inflexibility of existing fuzzy linear regression models

on the uniqueness of possibilistic measure of uncertainty and information abstract it is demonstrated through a series of theorems that the u uncertainty introduced by higashi and klir in 1982 is the only possibilistic measure of uncertainty and information that satisfies possibilistic counterparts of axioms of the well established shannon and hartley measures of uncertainty and information two complementary forms of the possibilistic counterparts of the probabilistic branching or grouping axiom which is usually used in proofs of the uniqueness of the shannon measure are introduced in this paper for the first time a one to one correspondence between possibility distributions and basic probabilistic assignments introduced by shafer in his mathematical theory of evidence is instrumental in most proofs in this paper the uniqueness proof is based on possibilistic formulations of axioms of symmetry expansibility additivity branching monotonicity and normalization

diagnosability of fuzzy discrete event systems a fuzzy approach in order to more effectively cope with the real world problems of vagueness fuzzy discrete event systems fdess were proposed by lin and ying recently then we and cao and ying investigated the supervisory control of fdess independently in this paper we are concerned with another important issue of fdess the failure diagnosis more specifically 1 we propose a ldquofuzzy diagnosabilityrdquo approach by introducing a fuzzy diagnosability function to characterize the diagnosability degree which takes values in the interval 0 1 rather than 0 1 2 based on the observability of events we formalize the construction of the diagnosers that are used to perform fuzzy diagnosis 3 a number of basic properties of the diagnosers are investigated in particular we present a necessary and sufficient condition for failure diagnosis of fdess our results generalize the important consequences of the diagnosability for crisp discrete event systems dess introduced by sampath et al the newly proposed approach allows us to deal with the problem of diagnosability for both crisp dess and fdess 4 in addition a method for checking the fuzzy diagnosability for fdess is proposed also some examples are provided to illustrate the application of the diagnosability of fdess

stable and finite morse index solutions of toda system abstract we prove a liouville type theorem and a partial regularity result for stable and finite morse index solutions of toda system the main tools are integral estimates a monotonicity formula and blowing down analysis for such systems they may be split into two sub systems during the blowing down process some e regularity theorems are developed to deal with this problem

distribution based behavioral distance for nondeterministic fuzzy transition systems modal logics and behavioral equivalences play an important role in the specification and verification of concurrent systems in this paper we first present a new notion of bisimulation for nondeterministic fuzzy transition systems which is distribution based and coarser than state based bisimulation appeared in the literature then we define a distribution based bisimilarity metric as the least fixed point of a suitable monotonic function on a complete lattice which is a behavioral distance and is a more robust way of formalizing behavioral similarity between states than bisimulations we also propose an on the fly algorithm for computing the bisimilarity metric moreover we present a fuzzy modal logic and provide a sound and complete characterization of the bisimilarity metric interestingly this characterization holds for a class of fuzzy modal logics in addition we show the nonexpansiveness of a typical parallel composition operator with respect to the bisimilarity metric which makes compositional verification possible

on the concept of possibility probability consistency both probability and possibility may be seen as information about an experiment it is conceivable to have at some time these two forms of information about a same experiment and then the question of the relation between them arises at once in this paper some aspects of the concept of possibility probability consistency are studied the consistency is considered as a fuzzy property relative to the coherence between possibilistic and probabilistic information we analyse several measures of the degree of consistency and introduce an axiomatic to characterize them

on differential equations with interactive fuzzy parameter via t norms abstract in this paper we study fuzzy differential equations with uncertain parameters modeled by interactive fuzzy numbers the interactivity is formalized with the help of upper semicontinuous t norms to obtain solutions to fuzzy differential equations we use two different approaches the first uses a family of differential inclusions while the second is the fuzzification of the deterministic problem using the zadeh extension principle we show that the solutions obtained by the two methods are equal interactivity among the parameters and variable of the problem are studies with respect two methods we show that a hierarchy in uncertainties of the solutions arises when we chose one of the basic t norm the minimum the t norm of the product the t norm of lukasiewicz and t norm of the drastic product to model the interaction between parameters finally to illustrate the concepts introduced in the paper we study the malthusian model with interactive parameters

parameter estimations of normal fuzzy variables abstract although fairly much attention has been paid to theoretical applications of nahmias fuzzy variables little effort has been focused on the related parameter estimation issues this keeps down the practical use of fuzzy variables in this note we make a preliminary discussion on parameter estimation of normal fuzzy variables with parameters a b and distinguish three situations estimation of a with b known estimation of b with a known and estimation of unknown a and b formulae for calculating parameter estimates by use of practical collected data are presented

partial regularity of stable solutions to the supercritical equations and its applications abstract we prove the partial regularity of stable solutions of supercritical elliptic equations as an application we prove that any smooth stable entire solution to supercritical equations with p in a suitable range is radially symmetric

propensity score adjustment with several follow ups propensity score weighting adjustment is commonly used to handle unit nonresponse when the response mechanism is nonignorable in the sense that the response probability depends directly on the study variable a followup sample is commonly used to obtain an unbiased estimator using the framework of two phase sampling where the follow up sample is assumed to respond completely in practice the followup sample is also subject to missingness we consider propensity score weighting adjustment for nonignorable nonresponse when there are several follow ups and the final follow up sample is also subject to missingness we propose two methods one using calibration weighting and the other using a conditional likelihood using a so called reverse conditional probability both methods provides consistent estimates under correct specification of the response model a limited simulation study is used to compare the estimators the proposed methods are applied to the real data example in a korean household survey of employment

approximate bisimulations and state reduction of fuzzy automata under fuzzy similarity measures abstract for the purpose of reducing the number of states of fuzzy automata we study approximate bisimulations of fuzzy automata under fuzzy similarity measures which are induced by residuums of left continuous t norms for a real number 0 1 and a fuzzy similarity measure s we define approximate bisimulation under s between two fuzzy automata and prove that the degree of similarity between them is more than or equal to if there exists an approximate bisimulation between them furthermore we put forward the notion of surjective functional approximate bisimulations between two fuzzy automata under s using the notion of surjective functional approximate bisimulations under s the approximate bisimulations for a fuzzy automaton under s is defined as an equivalence relation with two additional properties according to an approximate bisimulation for a fuzzy automaton an aggregated fuzzy automaton has been constructed and the degree of similarity between them is also more than or equal to it is significant for us to find that there might not possess the greatest approximate bisimulation for a fuzzy automaton and we provide a polynomial time algorithm for computing all maximal approximate bisimulations finally we provide the conditions of the existence of the greatest approximate bisimulation

on the determination of strength of belief for decision support under uncertainty part ii fusing strengths of belief the problem of decision making under uncertainty is described and the difficulty of comparing actions with uncertain outcomes is noted we discuss the role of valuation functions which convert an uncertain outcome into a single representative value and hence aid in the comparison of uncertain alternatives the need for information about the strength of belief that a given outcome will occur is pointed out we then look at various methods for obtaining this information we consider in particular detail the situation in which the strength of belief is dependent on the outcome payoff

some relationships between possibility truth and certainty we discuss some relationships between the fuzzy subset concepts of possibility truth and certainty we show for a and b fuzzy subsets of x that the possibility of a given b is related to the truth of a given b through a sugeno like integral furthermore we show that if b is a normal fuzzy subset than poss a b is always at least as large as the certainty a b on the other hand if the maximum membership in b is less than 0 5 the certainty is always as least as large as the possibility

fuzzy set connectives as combinations of belief structures abstract consonant belief structures provide a representation for fuzzy sets owing to the fact that their plausibility measures are essentially possibility measures we note that two belief structures are equivalent if their plausibility and belief functions are equal this observation leads us to provide a multiple number of equivalent representations for any belief structure commensurate representations can be induced for two different belief structures by forcing the same number of focal elements with the same weights we show that if we represent two consonant belief structures in a commensurate manner then their aggregations are closed with respect to consonance provided that the additional requirement that the underlying probability distributions satisfy a condition of correlation is imposed the results of this work allow us to use belief structure representations for the manipulation of fuzzy subsets under various logical combinations all basic fuzzy set connectives can thus be interpreted in the framework of the theory of evidence

aggregating constraint satisfaction degrees expressed by possibilistic truth values in information systems one often has to deal with constraints in order to compel the semantics and integrity of the stored information or to express some querying criteria hereby different constraints can be of different importance a method to aggregate the information about the satisfaction of a finite number of constraints for a given data instance is presented central to the proposed method is the use of extended possibilistic truth values to express the degree of satisfaction of a constraint and the use of residual implicators and residual coimplicators to model the impact and relevance of a constraint the proposed method can be applied to any constraint based system a database application is discussed and illustrated

representation similarity measures and aggregation methods using fuzzy sets for content based recommender systems representation of features of items and user feedback and reasoning about their relationships are major problems in recommender systems this is because item features and user feedback are subjective imprecise and vague the paper presents a fuzzy set theoretic method ftm for recommender systems that handles the non stochastic uncertainty induced from subjectivity vagueness and imprecision in the data and the domain knowledge and the task under consideration the research further advances the application of fuzzy modeling for content based recommender systems initially presented by ronald yager the paper defines a representation method similarity measures and aggregation methods as well as empirically evaluates the methods performance through simulation using a benchmark movie data ftm consist of a representation method for items features and user feedback using fuzzy sets and a content based algorithm based on various fuzzy set theoretic similarity measures the fuzzy set extensions of the jaccard index cosine proximity or correlation similarity measures and aggregation methods for computing recommendation confidence scores the maximum minimum or weighted sum fuzzy set theoretic aggregation methods compared to the baseline crisp set based method csm presented the empirical evaluation of the ftm using the movie data and simulation shows an improvement in precision without loss of recall moreover the paper provides a guideline for recommender systems designers that will help in choosing from a combination of one of the fuzzy set theoretic aggregation methods and similarity measures

reliability estimation and design with insufficient data based on possibility theory early in the engineering design cycle it is difficult to quantify product reliability or compliance to performance targets because of insufficient data or information for modeling the uncertainties design decisions are therefore based on fuzzy information that is vague imprecise qualitative linguistic or incomplete the uncertain information is usually available as intervals with lower and upper limits in this work the possibility theory is used to assess design reliability with incomplete information the possibility theory can be viewed as a variant of fuzzy set theory the formal theories to handle uncertainty are first introduced using the theoretical fundamentals of fuzzy measures a computationally efficient and accurate hybrid global local optimization approach is subsequently described for calculating the confidence level of fuzzy response the method combines the advantages of the commonly used vertex and discretization methods subsequently the possibility theory is used in design a possibility based design optimization method is proposed where all design constraints are expressed in a possibilistic way it is shown that the method gives a conservative solution compared with all conventional reliability based designs obtained with different probability distributions finally a general possibility based design optimization method which handles a combination of random and possibilistic design variables is presented several numerical examples demonstrate the application of possibility theory in design

liouville type theorems for stable solutions of the weighted elliptic system abstract we examine the weighted elliptic system u 1 x 2 2 v v 1 x 2 2 u p in r n and prove liouville type theorems for the classical positive and nonnegative stable solutions in higher dimension in particular there are no positive stable solutions for any n 12 5 p 1 and 0 our proof is based on a combination of the bootstrap argument souplet s inequality and intermediate stability criterion which is used to obtain sharp results

liouville theorems for stable solutions of biharmonic problem we prove some liouville type results for stable solutions to the biharmonic problem delta 2 u u q u 0 in mathbb r n where 1 q infty for example for n ge 5 we show that there are no stable classical solution in mathbb r n when frac n 4 n 4 q le left frac n 8 n right 1

a note on similarity based definitions of possibility and probability probability theory is a rigorous theorywhich rests on a firm foundation the brilliant successes of probability theory are visible to all what is problematical relates to the basic question what is probability there are many answers but no unanimity what is suggested in this note are definitions of possibility and probability which are based on fuzzy logic more concretely the definitions are based on the concept of similarity similarity based definition of probability reduces to the traditional frequency based definition of probability when similarity is exact

multiplicity of solutions for a fourth order equation with power type nonlinearity let b be the unit ball in mathbb r n n 3 and n be the exterior unit normal vector on the boundary we consider radial solutions to delta 2 u lambda 1 rm sign p u p quad rm in b quad u 0 quad frac partial u partial n 0 quad rm on partial b where 0 for positive p we assume 5 n 12 and p frac n 4 n 4 or n 13 and frac n 4 n 4 0 such that if s there exists a radial weakly singular solution for s there exist infinitely many regular radial solutions and the number of radial regular solutions goes to infinity as s

using calibration weighting to adjust for nonresponse under a plausible model when we estimate the population total for a survey variable or variables calibration forces the weighted estimates of certain covariates to match known or alternatively estimated population totals called benchmarks calibration can be used to correct for sample survey nonresponse or for coverage error resulting from frame undercoverage or unit duplication the quasi randomization theory supporting its use in nonresponse adjustment treats response as an additional phase of random sampling the functional form of a quasi random response model is assumed to be known its parameter values estimated implicitly through the creation of calibration weights unfortunately calibration depends upon known benchmark totals while the covariates in a plausible model for survey response may not be the benchmark covariates moreover it may be prudent to keep the number of covariates in a response model small we use calibration to adjust for nonresponse when the benchmark model and covariates may differ provided the number of the former is at least as great as that of the latter we discuss the estimation of a total for a vector of survey variables that do not include the benchmark covariates but that may include some of the model covariates we show how to measure both the additional asymptotic variance due to the nonresponse in a calibration weighted estimator and the full asymptotic variance of the estimator itself all variances are determined with respect to the randomization mechanism used to select the sample the response model generating the subset of sample respondents or both data from the u s national agricultural statistical service s 2002 census of agriculture and simulations are used to illustrate alternative adjustments for nonresponse the paper concludes with some remarks about adjustment for coverage error

a monotonicity formula and a liouville type theorem for a fourth order supercritical problem we consider liouville type and partial regularity results for the nonlinear fourth order problem

replication variance estimation for two phase stratified sampling in two phase sampling the second phase sample is often a stratified sample based on the information observed in the first phase sample for the total of a population characteristic either the double expansion estimator or the reweighted expansion estimator can be used given a consistent first phase replication variance estimator we propose a consistent variance estimator that is applicable to both the double expansion estimator and the reweighted expansion estimator the proposed method can be extended to multiphase sampling

epsilon bisimulation relations for fuzzy automata this paper investigates the approximate bisimulation relations for fuzzy automata in order to study the approximate minimization problem of fuzzy automata for a small positive real number epsilon we introduce the notion of epsilon bisimulation relations between two fuzzy automata and prove that the behavior of a fuzzy automaton a differs by epsilon from the behavior of a fuzzy automaton b under a epsilon bisimulation relation between them also the notion of surjective functional epsilon bisimulation relations between two fuzzy automata is defined according to surjective functional epsilon bisimulation relations we discuss epsilon bisimulation relations for a fuzzy automaton a construction of aggregated fuzzy automaton by the given epsilon bisimulation for a fuzzy automaton is given furthermore we find that there might not exist the greatest epsilon bisimulation relation for a fuzzy automaton and we novelly give an effective algorithm to construct all maximal epsilon bisimulation relations for the given fuzzy automaton finally we point out that bisimulation relations for a fuzzy automaton are also epsilon bisimulation relations the conditions for the real number epsilon to ensure the existence of the greatest epsilon bisimulation are also discussed

a regularity theory of biharmonic maps in this article we prove the regularity of weakly biharmonic maps of domains in euclidean four space into spheres as well as the corresponding partial regu larity result of stationary biharmonic maps of higher dimensional domains into spheres c 1999 john wiley sons inc in this article we consider for simplicity the class of bihar monic maps from eu clidean domains to spheres we realize the standard spheres s k as unit vectors in r k 1 and consider maps u s k as vector valued functions that are contained in s k the energy functional for biharmonic maps is then j uj 2 dx a locally de fined biharmonic map is a map that is critical with respect to c ompactly supported variations we note that in the case where the domain has dimension four this en ergy functional is conformally invariant and hence conformal maps of euclidean four space are biharmonic in this sense we remark that this definition of bihar monic map depends on the embedding of the target space in euclidean space we

model checking of linear time properties in multi valued systems abstract in this paper we study the model checking problem of linear time properties in multi valued systems safety properties invariant properties liveness properties persistence and dual persistence properties in multi valued logic systems are introduced some algorithms related to the above multi valued linear time properties are discussed the verification of multi valued regular safety properties and multi valued regular properties using lattice valued automata are thoroughly studied since the law of non contradiction i e a a 0 and the law of excluded middle i e a a 1 do not hold in multi valued logic the linear time properties introduced in this paper have new forms compared to those in classical logic compared to those classical model checking methods our methods to multi valued model checking are accordingly more direct we give an algorithm for showing ts p for a model ts and a linear time property p which proceeds by directly checking the inclusion traces ts p instead of t r a c e s t s p a new form of multi valued model checking with membership degree is also introduced in particular we show that multi valued model checking can be reduced to classical model checking the related verification algorithms are also presented some illustrative examples and a case study are also provided

maximum entropy membership functions for discrete fuzzy variables due to the deficiency of information the membership function of a fuzzy variable cannot be obtained explicitly it is a challenging work to find an appropriate membership function when certain partial information about a fuzzy variable is given such as expected value or moments this paper solves such problems for discrete fuzzy variables via maximum entropy principle and proves some maximum entropy theorems with certain constraints a genetic algorithm is designed to solve the general maximum entropy model for discrete fuzzy variables which is illustrated by some numerical experiments

combining uncertainty and imprecision in models of medical diagnosis the paper presents a unified fuzzy probabilistic framework for modeling processes of medical diagnosis the two basic concepts of the dempster shafer theory i e focal elements and a basic probability assignment correspond to disease symptoms and the significance of an individual symptom in the diagnosis respectively the belief computation is related to diagnostic inference the final conclusion of the inference is the diagnosis with the greatest belief value fuzzy sets are used to describe focal elements it is shown how their membership functions and basic probability assignments are estimated on the basis of experimental data the interpretation of focal elements as fuzzy sets along with individual consideration of evidence imprecision and uncertainty of diagnosis are the essential new aspects of the presented method experimental studies have demonstrated the superiority of the proposed approach over some other modeling alternatives

unfair coins and necessity measures towards a possibilistic interpretation of histograms this short note defines a bijective mapping which turns a probability measure into a possibility measure this transformation has intuitive grounds especially the converse mapping corresponds to the usual assignment of identical probability values to elementary events of a given set when no information is available

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

sampling fuzzy discretization and signal reconstruction abstract the sampling theorem provides the condition under which a continuous time signal can be precisely represented and completely recovered from its instantaneous values equally spaced in time it gives a condition for the signal reconstruction problem the importance of this theorem pertains to the necessity of knowing when a continuous time signal is uniquely represented by its discrete version so that continuous time signals can be properly converted and handled by sampled data systems like digital computers the problem of the equivalence between either continuous time or discrete time signals and their fuzzy descriptions is analyzed here this encompasses the analysis of signal reconstruction when fuzzy systems are used as sampled data systems such as fuzzy filters fuzzy models or fuzzy controllers a necessary and sufficient condition is given for ensuring that the fuzzy system processes the legitimate representations of the signal presented at its interfaces furthermore a sufficient condition suitable for automatic tuning of membership functions is proposed and discussed the discussion encompasses the outline of some requirements that membership functions at fuzzy system interfaces should meet to allow humans to assess their semantic meaning

quantitative computation tree logic model checking based on generalized possibility measures we study generalized possibilistic computation tree logic gpoctl model checking in this paper which is an extension of possibilistic computation logic model checking introduced by li et al 2014 the system is modeled by generalized possibilistic kripke structures and the verifying property is specified by a gpoctl formula based on generalized possibility measures and generalized necessity measures the method of gpoctl model checking is discussed and the corresponding algorithm and its complexity are shown in detail furthermore the comparison between possibilistic computation tree logic and gpoctl is given finally a thermostat example is given to illustrate the gpoctl model checking method

uncertainty modelling for vague concepts a prototype theory approach an epistemic model of the uncertainty associated with vague concepts is introduced label semantics theory is proposed as a framework for quantifying an agent s uncertainty concerning what labels are appropriate to describe a given example an interpretation of label semantics is then proposed which incorporates prototype theory by introducing uncertain thresholds on the distance between elements and prototypes for description labels this interpretation naturally generates a functional calculus for appropriateness measures a more general model with distinct threshold variables for different labels is discussed and we show how different kinds of semantic dependence can be captured in this model

fuzzy logic and meteorological variables a case study of solar irradiance solar irradiance is an extreme case of an uncertain variable when measured on an hourly or shorter time interval in this paper solar irradiance has been considered as a case study for physical fuzzy modelling of a climate variable the uncertainty of the solar irradiance is treated as a fuzzy uncertainty whilst other variables are considered crisp the approach is robust as it does not rely on statistical assumptions and it is a possible alternative to modelling complex systems to our knowledge this is the first time that a physical model of a meteorological variable based on fuzzy numbers has been proposed previous rule based fuzzy meteorological models are only descriptive and cannot be extrapolated to non measured cases when compared with previous non fuzzy models of solar irradiance the fuzzy model shows an improved performance and when compared with experimental data the performance can be evaluated by fuzzy indices that take into account the uncertainty of the data and the model output

quantitative μ calculus and ctl defined over constraint semirings model checking and temporal logics are boolean the answer to the model checking question does a system satisfy a property is either true or false and properties expressed in temporal logics are defined over boolean propositions while this classic approach is enough to specify and verify boolean temporal properties it does not allow to reason about quantitative aspects of systems some quantitative extensions of temporal logics has been already proposed especially in the context of probabilistic systems they allow to answer questions like with which probability does a system satisfy a property we present a generalization of two well known temporal logics ctl and the calculus both extensions are defined over c semirings an algebraic structure that captures quantitative aspects like quality of service or soft constraints basically a c semiring consists of a domain an additive operation and a multiplicative operation which satisfy some properties we present the semantics of the extended logics over transition systems where a formula is interpreted as a mapping from the set of states to the domain of the c semiring and show that the usual connection between ctl and calculus does not hold in general in addition we reason about the complexity of computing the logics and illustrate some applications of our framework including boolean model checking

model checking fuzzy computation tree logic traditional temporal logics such as linear temporal logic and computation tree logic are widely used to specify properties of reactive systems model checking is a well established technique for verifying if a desired property described as a temporal logic formula holds over a reactive system model this paper presents fuzzy computation tree logic fctl a fuzzy extension of temporal logics by combining general fuzzy logic with computation tree logic and discusses its model checking problem first the notion of fuzzy kripke structures fkss and the syntax and semantics of their specification language fctl are introduced then we give a direct model checking algorithm for fctl over fks on the other hand we study when fctl model checking problem can be reduced to classical model checking ones and give a reduction method for an important fragment of fctl

uncertain robot environment modelling using fuzzy numbers abstract in this paper we present a fuzzy oriented methodology to model an uncertain geometric robot environment and to manipulate geometric uncertainties between robot coordinate frames we describe any geometric primitive of robot environment as a parameter vector in parameter space not only ill known values of the parameterized geometric primitives but the uncertain quantities of coordinate transformations are represented by means of fuzzy numbers restricted to appropriate membership functions for consistent interpretation about geometric primitives between different coordinate frames we manipulate these uncertain quantities using fuzzy arithmetic as an application we deal with a 2 dimensional straight line correspondence problem of a mobile robot vision system this example shows the usefulness of the proposed methodology

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

probabilities possibilities and fuzzy sets a formal analysis of probabilities possibilities and fuzzy sets is presented in this paper a number of theorems proved show the above measures have equal representational power when their domains are infinite however for finite domains it is proved that probabilities have a higher representational power than both possibilities and fuzzy sets the cost of this increased power is high computational complexity and reduced computational efficiency the resulting trade off of high complexity and representational power versus computational efficiency is discussed under the spectrum of experimental systems and applications

efficiencies of two stage systems with fuzzy data two stage dea data envelopment analysis models show the performance of individual processes and thus are more informative than the conventional one stage models for making decisions this paper extends this approach from deterministic to uncertain situations where the observations are represented by fuzzy numbers the extension principle is utilized to develop a pair of two level mathematical programs to calculate the lower and upper bounds of the a cut of the fuzzy efficiency by enumerating various values of a the membership functions of fuzzy efficiencies are constructed numerically it is found that the property of the system efficiency being equal to the product of the two process efficiencies which holds for the deterministic case also holds for the fuzzy case this property can be generalized to series systems with more than two processes an example of non life insurance companies in taiwan is used to explain how to calculate the system and process efficiencies and how to derive their relationship when the data is fuzzy

model checking of linear time properties based on possibility measure using possibility measure we study model checking of linear time properties in possibilistic kripke structures first the notion of possibilistic kripke structures and the related possibility measure are introduced and then model checking of reachability and repeated reachability linear time properties in finite possibilistic kripke structures are studied standard safety properties and regular properties in possibilistic kripke structures are introduced the verification of regular safety properties and regular properties using finite automata are thoroughly studied it has been shown that the verification of regular safety properties and regular properties in a finite possibilistic kripke structure can be transformed into the verification of reachability properties and repeated reachability properties in the product possibilistic kripke structure that is introduced in this paper several examples are given to illustrate the methods that are presented in this paper

possibilistic kalman filtering for radar 2d tracking abstract standard kalman filter skf introduced by kalman in the 60s has gained a non estimated importance in control as well as in robotics community its importance arises from the obtained optimal result in the sense of variance minimization under stochastic gaussian and unbiased perturbations and when the state model as well as the measurement model are precisely known however when the last requirement is relaxed such that one or more parameters governing the models are ill defined and rather given in terms of interval evaluations chen et al ieee trans aerospace electr syst 33 1 1997 251 259 have proposed interval kalman filter ikf by extending the arithmetic operations to interval calculus in this paper we rather assume that the uncertainty pervading some parameters of the models are given in terms of possibility distributions 21 this leads to a formulation of possibilistic kalman filtering pkf which agrees with ikf the same example of 2d radar tracking is tackled comparisons with ikf are investigated as well the influence of the modelling process on the performance of the filter besides the proposal permits to capture certainty qualified information which cannot be obtained from ikf

on some optimization problems under uncertainty abstract the last ten years show a diversification of the approaches for handling uncertainties in optimization problems by certain relaxations or on the contrary seeking a more conservative solution in this paper we present some new results and points of view in nonlinear inexact and fuzzy mathematical programming possible applications to capital budgeting of interrelated projects and to production planning are also discussed

characterization and comparison of sugeno and choquet integrals abstract starting from the concept of equiordered functions we characterize two types of fuzzy integrals that can be defined on any kind of fuzzy measure sugeno and choquet integrals the characterization theorems that we obtain allow us to carry out a comparative study of these two fuzzy integrals pointing out their formal similarities and their conceptual differences

some nonexistence theorems for semilinear fourth order equations in this paper we analyse the semilinear fourth order problem 2 u g u in exterior domains of n assuming the function g is nondecreasing and continuous in 0 and positive in 0 we show that positive classical supersolutions u of the problem which additionally verify u 0 exist if and only if n 5 and int 0 delta displaystyle g s over s 2 n 2 n 4 rm d s lt infty for some 0 when only radially symmetric solutions are taken into account we also show that the monotonicity of g is not needed in this result finally we consider the same problem posed in n and show that if g is additionally convex and lies above a power greater than one at infinity then all positive supersolutions u automatically verify u 0 in n and they do not exist when the previous condition fails

possibility and necessity in optimization abstract we employ possibility and necessity in fuzzy programming and in other optimization problems that traditionally use random variables the possibility approach to multiobjective programming produces the same results as fuzzy programming a detailed example is presented to illustrate the use of possibility and necessity in optimization problems that usually employ random variables to describe uncertainty

image segmentation using fuzzy homogeneity criterion abstract the approach proposed here is using fuzzy homogeneity vectors and the fuzzy cooccurrence matrix to segment images homogeneity vectors could be used to represent the homogeneous feature between a pixel and its neighbors and the degree of homogeneity could be determined by using a fuzzy membership function combining the homogeneity vectors and the fuzzy membership function we could extract the feature of an image which determines the fuzzy region width with the fuzzy region width we could use the fuzzy cooccurrence matrix to measure the fuzzy geometry properties of an image which is the fuzzy entropy values that can be employed for determinine the thresholds to segment images the advantages of this approach are as follows first homogeneity vectors take account of the spatial gray tone dependence thus using homogeneity vectors has a better noise tolerance second from the extracted feature the fuzzy region width could be determined automatically third because the fuzzy region width is decided by the feature of the image it could be adjusted according to the nature of the image fourth the dimension of the fuzzy cooccurrence matrix representing the image is significantly decreased but the properties of images are faithfully preserved a large number of experiments have been carried out on different kinds of images and good results have been achieved by the proposed method this method will have wide application in image processing

the gel fand problem for the biharmonic operator we study stable and finite morse index solutions of the equation delta 2 u e u if the equation is posed in mathbb r n we classify radial stable solutions we then construct nonradial stable solutions and we prove that unlike the corresponding second order problem no liouville type theorem holds unless additional information is available on the asymptotics of solutions at infinity thanks to this analysis we prove that stable solutions of the equation on a smoothly bounded domain supplemented with navier boundary conditions are smooth if and only if n leqq 12 we find an upper bound for the hausdorff dimension of their singular set in higher dimensions and conclude with an a priori estimate for solutions of bounded morse index provided they are controlled in a suitable morrey norm

fuzzy finite automata and fuzzy regular expressions with membership values in lattice ordered monoids we study fuzzy finite automata in which all fuzzy sets are defined by membership functions whose codomain forms a lattice ordered monoid l for these l fuzzy finite automata l ffa for short we provide necessary and sufficient conditions for the extendability of the state transition function it is shown that nondeterministic l ffa nl ffa for short are more powerful than deterministic l ffa dl ffa for short then we give necessary and sufficient conditions for the simulation of an nl ffa by an equivalent dl ffa next we turn to the closure properties of languages defined by l ffas we establish closure under the regular operations and provide conditions for closure under intersection and reversal in particular we show that the family of fuzzy languages accepted by dl ffas is not closed under kleene closure operation and the family of fuzzy languages accepted by nl ffas is not closed under complement operation furthermore we introduce the notions of l fuzzy regular expressions and give the kleene theorem for nl ffas the description of dl ffas by l fuzzy regular expressions is also given finally we investigate the level structures of l ffas our results provide some insight as to what extend properties of l ffas and their languages depend on the algebraic properties of l

semiring based constraint satisfaction and optimization we introduce a general framework for constraint satisfaction and optimization where classical csps fuzzy csps weighted csps partial constraint satisfaction and others can be easily cast the framework is based on a semiring structure where the set of the semiring specifies the values to be associated with each tuple of values of the variable domain and the two semiring operations and x model constraint projection and combination respectively local consistency algorithms as usually used for classical csps can be exploited in this general framework as well provided that certain conditions on the semiring operations are satisfied we then show how this framework can be used to model both old and new constraint solving and optimization schemes thus allowing one to both formally justify many informally taken choices in existing schemes and to prove that local consistency techniques can be used also in newly defined schemes

partial regularity of finite morse index solutions to the lane emden equation we prove regularity and partial regularity results for finite morse index solutions u h1 lp to the lane emden equation u u p 1u in

fuzzy ϵ approximate regular languages and minimal deterministic fuzzy automata ϵ accepting them abstract in this paper for a real number 0 1 we put forward the notion of fuzzy approximate regular languages and study their properties especially the pumping lemma in the context of this kind of languages by comparing sets of fuzzy approximate regular languages under the order of set inclusion we get an infinite hierarchy of fuzzy languages the smallest is the set of fuzzy regular languages and the biggest is the set of fuzzy languages and the sets of approximate regular languages are different for different 0 1 2 we also investigate whether operations closed in the set of fuzzy regular languages are still closed in the set of fuzzy approximate regular languages furthermore for a fuzzy approximate regular language f approximate equivalence relations for f are characterized in order to construct deterministic fuzzy automata accepting f if a fuzzy approximate regular language f is also a fuzzy regular language and accepted by a given accessible deterministic fuzzy automaton a then we give a polynomial time algorithm to construct at least one minimal deterministic fuzzy automaton accepting f by means of a finally we point out that the number of states of minimal deterministic fuzzy automata accepting a fuzzy regular language is smaller than or equal to that of minimal deterministic fuzzy automata accepting this language

fuzzy integrals on l fuzzy sets abstract the fuzzy measures and fuzzy integrals for zadeh s fuzzy sets were studied by qiao and wang 5 6 7 12 in this paper we consider the real valued fuzzy measures and fuzzy integrals for l fuzzy sets where l is a pseudocomplemented infinitely distributive complete lattice on a fuzzy algebra of l fuzzy sets the concepts of fuzzy measure and fuzzy integral are introduced some equivalent forms of fuzzy integrals on l fuzzy sets are given moreover we discuss several properties of fuzzy integrals on l fuzzy sets and prove some convergence theorems for a sequence of fuzzy integrals on l fuzzy sets

network of probabilities associated with a capacity of order 2 abstract this paper studies capacities of order 2 from a geometric point of view that is from convex polytope of probabilities defined by them this convex polytope is characterized by means of a network of probabilities representing relationships among vertices defined by the edges interesting properties may be obtained about the capacities from this network

quantitative model checking of linear time properties based on generalized possibility measures abstract model checking of linear time properties based on possibility measures was studied in previous work li and li 2013 however the linear time properties considered in the previous work was qualitative and was based on boolean logic the linear time properties containing possibility information we call it fuzzy linear time properties here which is based on fuzzy logic of system could not be represented and tackled using the previous methods we shall study quantitative model checking of fuzzy linear time properties based on generalized possibility measures in the paper both the model of the system as well as the properties the system needs to adhere to are described using possibility information to identify the uncertainty in the model properties the systems are modeled by generalized possibilistic kripke structures gpks in short and the properties are described by fuzzy linear time properties concretely fuzzy linear time properties about reachability always reachability constrain reachability repeated reachability and persistence in gpkss are introduced and studied fuzzy regular safety properties and fuzzy regular properties in gpkss are introduced the verification of fuzzy regular safety properties and fuzzy regular properties using fuzzy finite automata are thoroughly studied it has been shown that the verification of fuzzy regular safety properties and fuzzy regular properties in a finite gpks can be transformed into the verification of always reachability properties and repeated reachability persistence properties in the product gpks introduced in this paper several examples are given to illustrate the methods presented in the paper

advanced fuzzy inference engines in situation aware computing we focus on the very important family of context aware applications context aware computing relies on tasks like capturing sensing environmental parameters e g lightness location classifying context and inferring further knowledge about that context determine the situation the user is currently in however the relevant applications have to deal with the inherent imperfection of context sensing for decision making we propose an extension of context representation and inference for situation aware applications our model relies on fuzzy set theory to accommodate the imperfection of sensed context based on this model we have developed three fuzzy inference engines which rely on advanced semantics specialization mereological and compatibility relations we have evaluated the proposed engines through a series of experiments involving real users our findings indicate the strong points of the proposed context classification and inference processes

a possibilistic argument for irreversibility abstract our aim is to explicate the idea of irreversibility by a mathematical representation of irreversible processes occurring in possibilistic dynamical systems with this in mind solutions of the cauchy problem in the space of fuzzy points are examined to meet the correspondence argument solutions of the cauchy problem in the space of concentrated points are examined too

liouville type results for semi stable solutions of the weighted lane emden system abstract we examine the weighted lane emden system u 1 x 2 2 v p v 1 x 2 2 u q in r n where 1 p q and 0 and the weighted lane emden equation u 1 x 2 2 u p in r n where p 4 3 and 0 we prove that the system equation does not have the classical positive semi stable solution in dimension n 2 4 2 q 1 p q 1 0 n 2 2 2 p 1 p p 2 p in particular there is no positive semi stable solution of the weighted lane emden system if n 10 4 and 2 p q

user preferences discovery using fuzzy models user preferences discovery aims to learn the patterns of user preferences for various services or items such as movies preferences discovery is essential to the development of intelligent personalization applications based on decision and utility theories traditional approaches to preferences discovery explicitly query users about the behavior of value function or utility of every outcome with respect to each decision criterion consequently these approaches are generally error prone and labor intensive although implicit elicitation approaches have been proposed to address the above limitations extent approaches largely ignore multi valued nature of item features and uncertainty associated with item features and user preferences to address uncertainty due to vagueness and imprecision this research proposed a general framework for preferences discovery based on fuzzy set theories in addition new fuzzy models were created for preferences discovery and representation further an algorithm was developed to predict user preferences with uncertainty and visualization of item features user feedback and the discovered preferences helped improve the interpretation of the discovered knowledge the results of the simulation evaluation using a benchmark movie dataset revealed that the proposed preference discovery method 1 doubled the accuracy of preference discovery as compared to random prediction and 2 outperformed conventional techniques in making movie recommendation these findings suggest that fuzzy models are effective for preferences patterns discovery and personalized recommendation application

a set of practical experiments to validate the fuzzy kalman filter abstract the fuzzy kalman filter fkf was recently reformulated with a number of basic examples included in order to demonstrate its applicability we now look ahead to its practical implementation introducing deep analysis which includes very detailed simulations and complete real applications this will allow us to not only validate the fkf but also to draw important conclusions which will help users in the adjustment of the fuzzy filter parameters and the choosing of possibilistic distributions

monotonicity formulas for extrinsic triharmonic maps and the triharmonic lane emden equation abstract we derive monotonicity formulas and e regularity results for extrinsic triharmonic maps and triharmonic lane emden equations as an application we prove partial regularity results for both types of equations in the supercritical regime

observability and decentralized control of fuzzy discrete event systems fuzzy discrete event systems as a generalization of crisp discrete event systems have been introduced in order that it is possible to effectively represent uncertainty imprecision and vagueness arising from the dynamic of systems a fuzzy discrete event system has been modeled by a fuzzy automaton its behavior is described in terms of the fuzzy language generated by the automaton in this paper we are concerned with the supervisory control problem for fuzzy discrete event systems with partial observation observability normality and co observability of crisp languages are extended to fuzzy languages it is shown that the observability together with controllability of the desired fuzzy language is a necessary and sufficient condition for the existence of a partially observable fuzzy supervisor when a decentralized solution is desired it is proved that there exist local fuzzy supervisors if and only if the fuzzy language to be synthesized is controllable and co observable moreover the infimal controllable and observable fuzzy superlanguage and the supremal controllable and normal fuzzy sublanguage are also discussed simple examples are provided to illustrate the theoretical development

the evidential reasoning approach for mada under both probabilistic and fuzzy uncertainties abstract many multiple attribute decision analysis mada problems are characterised by both quantitative and qualitative attributes with various types of uncertainties incompleteness or ignorance and vagueness or fuzziness are among the most common uncertainties in decision analysis the evidential reasoning er approach has been developed in the 1990s and in the recent years to support the solution of mada problems with ignorance a kind of probabilistic uncertainty in this paper the er approach is further developed to deal with mada problems with both probabilistic and fuzzy uncertainties in this newly developed er approach precise data ignorance and fuzziness are all modelled under the unified framework of a distributed fuzzy belief structure leading to a fuzzy belief decision matrix a utility based grade match method is proposed to transform both numerical data and qualitative fuzzy assessment information of various formats into the fuzzy belief structure a new fuzzy er algorithm is developed to aggregate multiple attributes using the information contained in the fuzzy belief matrix resulting in an aggregated fuzzy distributed assessment for each alternative different from the existing er algorithm that is of a recursive nature the new fuzzy er algorithm provides an analytical means for combining all attributes without iteration thus providing scope and flexibility for sensitivity analysis and optimisation a numerical example is provided to illustrate the detailed implementation process of the new er approach and its validity and wide applicability

the possibilistic moments of fuzzy numbers and their applications this paper introduces the nearest weighted interval and point approximations of a fuzzy number and then suggests weighted possibilistic moments about these points of fuzzy numbers the possibilistic moments play an important role in fuzzy sets and systems specifically in physics mathematics and statistics we provide the definition of the moments of fuzzy numbers as well as the definition of moments in probability theory some of their applications are mentioned

fuzzy inference relation based on the theory of falling shadows abstract in this paper we establish a theoretical approach to define a fuzzy inference relation based on the theory of falling shadows the main characteristic of our definition of fuzzy inference relation is that it is semantically dependent in the sense that the formula of our definition will vary according to the correlation of the antecedent and the consequence of the given implication we shall show that the formulae of fuzzy inference relation given by ukasiewicz zadeh and the probability formula are consequences of our definition under three different correlations of the propositions

application of fuzzy naive bayes and a real valued genetic algorithm in identification of fuzzy model we present a method to identify a fuzzy model from data by using the fuzzy naive bayes and a real valued genetic algorithm the identification of a fuzzy model is comprised of the extraction of if then rules that is followed by the estimation of their parameters the involved parameters include those which determine the membership function of fuzzy sets and the certainty factors of fuzzy if then rules in our method as long as the fuzzy partition in the input output space is given the certainty factor of each rule is computed with the fuzzy conditional probability of the consequent conditioned on the antecedent by using the fuzzy naive bayes which is a generalization of naive bayes the fuzzy model involves the rules characterized by the highest values of certainty factors the certainty factor of each rule is the fuzzy conditional probability and it reflects the inner relationship between the antecedent and the consequent in order to improve the accuracy of the fuzzy model the real valued genetic algorithm is incorporated into our identification process this process concerns the optimization of the membership functions occurring in the rules we just involve the parameters of membership function of the fuzzy sets into the real valued genetic algorithm since the certainty factor of each rule can be computed automatically the performance of the model is shown for the backing truck problem and the prediction of mackey glass time series