are grades of membership probabilities abstract a small sample is given of the difficulties with present day fuzzy set theory with respect to the requirements of a scientific theory and a summary of some probabilistic interpretations of grades of membership to be found in the literature the tee model for grades of membership is then suggested instead of starting out from mathematical postulates such as the assumed max and min operations for or and and the tee model uses a semantic physio logical psycho logical starting point by investigating the possibilities which a person has for assigning linguistic labels and partial grades of membership in a meaningful way the assignment of partial grades of membership is explained by the assumption that humans have the ability to take errors of observation and intersubject differences into consideration a membership value uex is identified with the subject s estimate of the probability that the label e g tall would be assigned to an object of exact attribute value uex in a natural language situation this situation being a labeling or a yes no one in which at least one of three possible sources of uncertainty is present almost all the accepted formulas of fuzzy set theory as well as the shapes of the grade of membership curves are then derived instead of having to be postulated however the derived formulas for or and and reduce to the max and min operations only in a number of special cases in another special case they reduce to the summation and the zero operation respectively the generally valid formulas are presented

uniformly ultimately bounded fuzzy adaptive tracking controllers for uncertain systems fuzzy adaptive tracking controllers for a class of uncertain nonlinear dynamical systems are proposed and analyzed the controllers consist of adaptive and robustifying components whose role is to nullify the effects of uncertainties and to achieve a desired tracking performance the interactions between the two components have been investigated the closed loop system driven by the proposed controllers is shown to be stable with all the adaptation parameters being bounded in particular the proposed controllers guarantee uniform ultimate boundedness of the tracking error and the time bound of the uniform ultimate boundedness is obtained an upper bound on the steady state tracking error is obtained as a function of the gain of the robustifying term and the parameters of the adaptive component the controllers are tested on an inverted pendulum and simulation results are included a comparison of the proposed controllers with the ones in the literature is conducted

a formal language approach for a 3d curve representation abstract a formal language approach for representing three dimensional 3d curves is presented based on the chain code for representing 3d curves defined in 1 we propose an approach for mapping 3d curves into strings this mapping allows us to have a unique curve descriptor which is invariant under translation and rotation also it is possible to use inverse and mirroring operators and the use of formal language techniques for 3d curve generation and analysis finally we present a result of this approach to represent and to generate polygonal sequences convergent to cube filling hilbert curves

the concept of grade of membership abstract the concept of grade of membership is bound up with the general problem of fuzzy reasoning in this paper an approach to the latter problem is outlined and the consequences for the notion of grade of membership are described particular care is taken from the beginning to base the theory on concepts which have an immediate practical meaning for this reason assertions rather than sentences are taken as fundamental an assertion being characterized in terms of utility and decision theory fuzzy sentences and degrees of truth and then fuzzy sets and grades of membership are defined in terms of assertions the definitions being illustrated with examples the problem of defining unary and binary operations connectives for assertions sentences and fuzzy sets is discussed finally some other aspects of the logic of assertions are outlined and the extent to which the concepts of fuzzy set and grade of membership can be incorporated into the theory is discussed

an interpretation of membership functions and the properties of general probabilistic operators as fuzzy set operators part i case of type 1 fuzzy sets abstract the grade of membership is assumed to be determined as the relative size of evidence grounds supporting a given element s membership compared to all the necessary reference grounds relevant to the judgement of membership grade and is interpreted as conditional probability conditioned by the given element by assigning the dependence relations of the evidence grounds to those of the fuzzy subsets the set operators are defined as probabilistic operators with dependence relations the properties of the operators are studied and it is found that with type 1 fuzzy subsets they satisfy all the basic properties satisfied by the crisp set operators illustrative examples are given describing the construction of membership functions and the selection of set operators in relation to the situations to be modelled the proposed operators a kind of parametrized operators with an index value of the dependence relations as a parameter cover the entire operators ranging from the so called max min operators to the bold union intersection operators an extension of the above operator range is also discussed

designing fuzzy sets with the use of the parametric principle of justifiable granularity this study is concerned with a design of membership functions of fuzzy sets the membership functions are formed in such a way that they are experimentally justifiable and exhibit a sound semantics these two requirements are articulated through the principle of justifiable granularity the parametric version of the principle is discussed in detail we show linkages with type 2 fuzzy sets which are constructed on a basis of type 1 fuzzy sets several experimental studies are reported which illustrate a behavior of the introduced method

epipolar parameterization for reconstructing 3d rigid curve abstract this paper describes a new method for reconstructing a 3d rigid curve from a sequence of uncalibrated images using 31 epipolar parameterization the approach can be divided into the following two steps first a nonlinear discrete method is presented for point by point reconstruction of the curve instead of whole curve reconstruction projective and euclidean geometric tools are used second the parametric representation of the curve is defined by 31 b spline curves a linear method is proposed to interpolate the reconstructed points to obtain a complete curve thus it is proved that the 31 curve interpolation is equivalent to determining a set of control points of 3d regularized b spline curves it is shown that the 3d epipolar parameterization is an efficient method for reconstructing a 31 curve from image sequences experimental results are presented for real data

part i modeling image curves using invariant 3 d object curve models spl minus a path to 3 d recognition and shape estimation from image contours this paper and its companion are concerned with the problems of 3 d object recognition and shape estimation from image curves using a 3 d object curve model that is invariant to affine transformation onto the image space and a binocular stereo imaging system the objects of interest here are the ones that have markings e g characters letters special drawings and symbols etc on their surfaces the 3 d curves on the object are modeled as b splines which are characterized by a set of parameters the control points from which the 3 d curve can be totally generated the b splines are invariant under affine transformations that means that the affine projected object curve onto the image space is a b spline whose control points are related to the object control points through the affine transformation part i deals with issues relating to the curve modeling process in particular the authors address the problems of estimating the control points from the data curve and of deciding on the best order b spline and the best number of control points to be used to model the image or object curve s a minimum mean square error mmse estimation technique which is invariant to affine transformations is presented as a noniterative simple and fast approach for control point estimation the best b spline is decided upon using a bayesian selection rule finally we present a matching algorithm that allocates a sample curve to one of p prototype curves when the sample curve is an a priori unknown affine transformation of one of the prototype curves stored in the data base the approach is tried on a variety of images of real objects

a framework for linguistic modelling a new framework for linguistic reasoning is proposed based on a random set model of the degree of appropriateness of a label labels are assumed to be chosen from a finite predefined set of labels and the set of appropriate labels for a value is defined as a random set valued function from a population of individuals into the set of subsets of labels appropriateness degrees are then evaluated relative to the distribution on this random set where the appropriateness degree of a label corresponds to the probability that it is contained in the set of appropriate labels this interpretation is referred to as label semantics a natural calculus for appropriateness degrees is described which is weakly functional while taking into account the logical structure of expressions given this framework it is shown that a bayesian approach can be adopted in order to infer probability distributions on the underlying variable given constraints both in the form of linguistic expressions and mass assignments in addition two conditional measures are introduced for evaluating the appropriateness of a linguistic expression given other linguistic information

additional criteria in fuzzy two level pattern recognition abstract pattern recognition with fuzzy sets is considered with regard to medical diagnostic problems an alternative interpretation of membership grade is suggested which expresses the degree of illness of a patient with respect to a certain disease a formal statement of a two level fuzzy classifier is presented the main emphasis is put on additional criteria designed to evaluate a fuzzy diagnostic decision for an unknown object two groups of criteria are formed decision making connected and domain dependent the decision making group of criteria can be viewed as related to the soft consensus measure in group decisions theory interconnections between criteria are discussed the notions proposed are illustrated with real medical data from the fields of aviation medicine and neonatology

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

a behavioral model for linguistic uncertainty abstract the paper discusses the problem of modelling linguistic uncertainty which is the uncertainty produced by statements in natural language for example the vague statement mary is young produces uncertainty about mary s age we concentrate on simple affirmative statements of the type subject is predicate where the predicate satisfies a special condition called monotonicity for this case we model linguistic uncertainty in terms of upper probabilities which are given a behavioural interpretation as betting rates possibility measures and probability measures are special types of upper probability measure we evaluate zadeh s suggestion that possibility measures should be used to model linguistic uncertainty and the bayesian claim that probability measures should be used our main conclusion is that when the predicate is monotonic possibility measures are appropriate models for linguistic uncertainty we also discuss several assessment strategies for constructing a numerical model

fuzzy sets and membership functions based on probabilities abstract since the introduction of fuzzy set theory in 1965 several attempts to establish the relationship between the grades of membership and the classical probability measures have been made it turns out that there are different sources of fuzziness that must be dealt with differently in the present paper we examine in detail two types of fuzziness namely the fuzziness due to classification in an under or overdimensioned universe and the fuzziness due to the intersubject differences in opinion for the former case the membership function is defined to be equal to the normalized distance from the point to the boundary of the set in a specific metric it is shown that this definition of the membership function is fully consistent with the max min operations for the union intersection however the membership function of the complement is defined differently from the usual one minus rule the fuzziness due to the intersubject differences turns out to be a simple averaging process and the explicit formulas for this case were derived several examples that illustrate the notions of fuzzy intervals and fuzzy numbers are given and the interpretations of the derived membership curves are presented the formulas for calculation of the membership function of a sum of fuzzy numbers and of a product of fuzzy numbers times a constant are derived an extended definition of the measure of fuzziness is presented and applied to the defined membership functions

conditional probability fuzzy sets and possibility a unifying view we propose an interpretation of fuzzy set theory both from a semantic and a syntactic point of view in terms of conditional events and coherent conditional probabilities during past years a large number of papers has been devoted to support either the thesis that probability theory is all that is required for reasoning about uncertainty or the negative view maintaining that probability is inadequate to capture what is usually treated by fuzzy theory in this paper we emphasize the role of conditioning in a proper framework i e de finetti s coherence to get rid of many controversial aspects moreover we introduce suitable operations between fuzzy subsets looked on as corresponding operations between conditional events endowed with the relevant conditional probability finally we show how the concept of possibility function naturally arises as a coherent conditional probability

a contour tracing and coding algorithm for generating 2d contour codes from 3d classified objects in 3d image data sets generated by voxel based classification each voxel is marked with a specific class label voxels of the same class label can form 3d objects of extremely complex shape interactively drawn regions are usually represented by their 2d region borders in order to combine automatically classified with interactively drawn regions a contour tracing and coding algorithm for generating optimized 2d contours from 3d classified objects is presented a special conversion algorithm allows a chain or a crack code representation an application to medical images shows the method s necessity and usefulness in dealing with highly complex regions

a behavioral foundation for fuzzy measures in savage 41 a behavioral foundation was given for subjective probabilities to be used in the maximization of expected utility this paper analogously gives a behavioral foundation for fuzzy measures to be used in the maximization of choquet expected utility this opens the way to empirical verification or falsification of fuzzy measures and frees them of their ad hoc character

measurement of membership functions and their acquisition abstract three basic views of the representation of membership functions are reviewed together with fundamental measurement of linguistic terms of linguistic variables the conclusion is that such measurements are either on an ordinal or an interval scale based on whether the appropriate axioms are validated by the empirical data with an allowance for stochastic variation the conjoint measurement is introduced for the case of multi dimensional linguistic variables whose linguistic terms are compositions of two or finitely many more component linguistic terms of distinct independent linguistic variables it is shown that any composition of distinct independent component linguistic terms of component linguistic variables by any t norm or s norm or any finite convext linear combination preserves the monotonic weak order property of the components in the composite once the scale properties of the measurement values of particular terms are validated composition procedures may be applied to the experimental data to obtain compound membership functions of fuzzy sets induced by meaningful representations of compositions of linguistic terms of linguistic variables finally four methods of membership acquisition and construction are reviewed from the perspective of fundamental measurement

fuzzy logic and the quality of assessment of portfolios abstract in this paper it is argued that competence like height is a fuzzy concept this paper attempts to show the advantages of using fuzzy logic in the domain of assessment of prior learning apl via portfolios of evidence this issue which presently is of great concern to vocational educators highlights the need for defining some procedure in order to rationalize decisions made by assessors dealing with such type of evidence the assessment of portfolios are very much perceived presently as the way forward in modern educational assessment as portfolios do not only address evidence for past achievement but also evidence for current achievement thus highlighting the relevance of fuzzy logic in this area may raise awareness among educators of the benefits of such methodology the paper argues that fuzzy logic could offer useful insight in the attempt to rationalize assessor s deliberations when facing portfolio types of evidence the paper also tried to refine the fuzzy logic methodology perceived as most adequate for portfolio assessment as well as raise some issues that need addressing the examples referred to are based on the british national council for vocational qualifications ncvq apl procedures

three dimensional digital line segments digital arcs in 3 d digital pictures are defined the digital image of an arc is also defined a digital arc is defined to be a digital line segment if it is the digital image of a line segment it is shown that a digital line segment may be characterized by the chord property holding for its projections onto the coordinate planes it is also shown that a digital line segment may not be characterized by its own chord property a linear time algorithm is presented that determines whether or not a digital arc is a digital line segment

interpretative versus prescriptive fuzzy set theory both traditional fuzzy set theory and the theory of subjective probabilities postulate their formulas the former because it does not accept a probabilistic interpretation of grades of membership the latter because it makes no connection between subjective probabilities and relative frequencies the interpretational theory of the tee model ascribes a well defined meaning to a membership value spl mu sub spl lambda in terms of probabilities which are limits of frequencies spl mu sub spl lambda is interpreted as the estimate by the subject of the probability that a given object would be assigned the label x in an everyday situation of uncertainty in contrast to the max min formulas used in many applications of fuzzy sets the postulated summation to 1 formula of all fuzzy clustering algorithms follows from the interpretative tee model this agrees with the author s view that fuzzy set theory should be defined as a theory which allows partial membership values of an object in a class or cluster not as a theory which uses specific mathematical operators the operators must be derived from the well defined interpretation of partial membership values

using extremal boundaries for 3 d object modeling the extremal boundaries of 3 d curved objects are the images of special curves drawn on the object and are called rims they are viewpoint dependent and characterized by the fact that the optical rays of their points are tangential to the surface of the object the mathematics of the relationship between the extremal boundaries and the surface of the object is studied this study makes it possible to design an algorithm for detecting those boundaries in the images that are likely to be extremal once this has been done one can reconstruct the rims and compute the differential properties of the surface of the object along them up to the second order if a qualitative description is sufficient the sign of the gaussian curvature of the surface along the rim can be computed in a much simpler way experimental results are presented on synthetic and real images the work provides a better understanding of the relationship between the apparent and real shape of a 3 d object as well as algorithms for reconstructing the local shape of such an object along the rims

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

the efficacy of fuzzy representations of uncertainty advocates of the theory of fuzzy sets as a system for representing uncertainty have based their case on five basic arguments these are 1 the reality hypothesis which holds that imprecision is an inherent property of the world external to an observer 2 the subjectivity hypothesis which holds that probability is an exclusively objective measure of uncertainty and that therefore subjective uncertainty can only be represented with fuzzy sets 3 the behaviorist hypothesis which claims that uncertainty systems should emulate rather than prescribe human behavior in the face of uncertainty 4 the probability as fiction hypothesis which claims that probability does not comprise a field of study in its own right and 5 the superset hypothesis which holds that fuzzy set theory includes probability as a special case and thus provides a richer uncertainty modeling environment we discuss and criticize all five we then criticize the argument that fuzziness represents a type of uncertainty distinct from probability and also the inordinate complexity of fuzzy methods we present a method for assessing the efficacy of fuzzy representations of uncertainty and apply this method in three examples

comparison of empirical and computed values of fuzzy conjunction abstract the necessity for further experiments such as in 4 for the foundation of fuzzy sets theory is shown generality of the experimental results on fuzzy conjunction in the paper by thole zimmermann and zisno 1979 4 for the other objects and language is verified for other conditions usually the minimum the product and the restricted sum operators are used as an intersection conjunction of fuzzy sets in fuzzy optimization f optimization on the basis of experiments it is shown that these operators are correct for f optimization their correctness is founded on the stability of the extremal point a decision in bellman zadeh s scheme under a positive linear mapping it is shown that these mappings exist between empirical conjunction and considered operators as a statistical result of linear regression analysis

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

unity and diversity of fuzziness spl minus from a probability viewpoint like the fields of probability and statistics fuzzy set theory is characterized by a variety of viewpoints adherents of fuzzy methods however consistently maintain that probability is not necessarily the optimal representation of uncertainty we rebut this view

a fuzzy c means variant for the generation of fuzzy term sets a fuzzy c means fcm variant is proposed for the generation of fuzzy term sets with overlap the proposed variant differs from the original mainly in two areas the first modification ensures that two end terms take the maximum and minimum domain values as their centers the second modification prevents the generation of non convex fuzzy terms that often occurs with the original algorithm the optimal number of terms and the optimal shape of the membership function associated with each term are determined based on the mean squared error criterion the exponential weight m used in the algorithm is found to greatly affect the shape of the membership function the effect of data size used for the generation of fuzzy terms is also discussed a generalized shaped function with a tunable parameter along with its complement is developed to fit all term sets generated by the fcm variant using various m values

a semantics for possibility theory based on likelihoods abstract in this paper a semantic basis for possibility theory based on likelihood functions is presented in some cases possibilities have been considered as approximations of shafer plausibility measures this approximation exchanges exactness of plausibility values for the simplicity of use of possibility values in this paper a different direction is followed possibility measures are considered as the supremum of a family of likelihood functions this is an exact interpretation not an approximation the minimum rule to combine possibility distributions is justified in this framework under general conditions conditions under which other rules can be applied are also studied

the generalized modus ponens considered as a fuzzy relation abstract the generalized modus ponens gmp is an inference rule in fuzzy logic with the following scheme if s1 is p1 then s2 is p2 but s1 is q1 hence s2 is q2 the objective is to determine q2 given p1 p2 and q1 thepredicate p1 in the antecedent of the if then rule is fuzzy hence the condition s1 is p1 to derive s2 is p2 is not strict therefore it should be possible to render a reasonable approximation of q2 when q1 diverges slightly from p1 a number of criteria for this inference rule will be presented in literature there are three basic assumptions for dealing with the gmp namely 1 the if then rule is implemented by an implication rule from some multivalued logic i e the if then rule is represented by a fuzzy relation 2 the premiss p1 can be strengthened or weakened to obtain q1 by taking a power of the membership function of p1 and 3 the conclusion q2 is derived by means of the max min composition rule it will be shown by a number of theorems that these three items are incompatible with the criteria furthermore it will be shown that it is arduous to construct an implication rule to meet the conditions 2 and 3 a way to restore this is to alter condition 2 instead of taking a power of the membership function it is also possible to change the parameters of this function but this also causes complications hence requirement 3 should be modified into some functional relation it will be shown how this can be accomplished in combination with the functions with parameters changed

2d tree object representation via the slope chain code abstract a method for representing 2d two dimensional tree objects is described this representation is based on a chain code which is called the slope chain code scc thus 2d tree objects are described by means of a chain of element strings suitably combined by means of parentheses these 2d tree objects correspond to naturally existing 2d tree structures this tree notation preserves the shape of trees and the shape of their branches allows us to know their topological and geometrical properties the proposed notation of 2d tree objects is invariant under translation rotation and optionally under scaling also it is possible to define a unique start vertex for each tree via the unique path in the tree using this notation it is possible to obtain the mirror image of any tree with ease furthermore two interesting properties of trees are presented the accumulated slope and the tortuosity tortuosity is a very important property of trees and has many applications in different fields in order to prove our method for representing 2d tree objects we obtain some tree descriptors of tree objects and compute their measures of accumulated slope and tortuosity finally we present some examples of 2d trees from the real world about echinoderm species identification

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

the three semantics of fuzzy sets abstract three main semantics for membership functions seem to exist in the literature similarity preference and uncertainty each semantics underlies a particular class of applications similarity notions are exploited in clustering analysis and fuzzy controllers uncertainty is captured by fuzzy sets in the framework of possibility theory the membership function of a fuzzy set is also sometimes a kind of utility function that represents flexible constraints in decision problems this paper advocates the claim that progress in operational semantics of membership functions presupposes that these distinct semantics be acknowledged and related to more basic measurement issues in terms of distance cost and frequency on which scientific traditions exist

digital representation schemes for 3d curves digital representation of three dimensional 3 d curves is considered two dimensional representation schemes are first reviewed then methods applicable to curves in three and higher dimensional spaces are described desirable properties of curve representation schemes are identified and used as a basis for qualitative and quantitative comparison of the various methods that have been suggested conclusions concerning the selection of a digital 3 d curve representation scheme for shape analysis and coding are drawn it is shown that grid intersect quantization is a poor choice for curve representation in 3 d space and that cube quantization that leads to 6 connected chain codes meets all the identified requirements and should be preferred

detection and characterization of cluster substructure i linear structure fuzzy c lines in part i a generalization of the fuzzy c means or fuzzy isodata clustering algorithms is developed necessary conditions for minimization of a generalized total weighted squared orthogonal error objective function lead to a picard iteration scheme which generates simultaneously i c fuzzy clusters in the data ii a set of c prototypical straight lines in feature space which best fit the data in a well defined sense iii a set of c prototpyical centers of mass on the c lines which characterize the core of each linear fuzzy cluster theoretical optimization is achieved using principal components of generalized within cluster fuzzy scatter matrices a convergence theorem for each algorithm in the infinite family is given the algorithms are exemplified by five numerical examples using both real and artificial data sets having essentially linear substructure in part ii the fuzzy c means and fuzzy c lines algorithms are shown to be special cases of a more general class of fuzzy algorithms the

symmetry detection in 3d chain coded discrete curves and trees symmetry detection in 2d and 3d shapes has been a classical problem in computer vision computational geometry and pattern recognition one of the reasons of this interest is because symmetry is related to the balance and harmony and in most cases it is considered like a beautiful proportion in this paper we present different methods to detect mirror and rotational symmetries in 2d and 3d open and closed curves and in trees represented by means of the orthogonal direction change chain code these methods detect both global and local symmetries they can also detect the number and position of the axes of symmetry they are simple easy to implement and fast to execute highlightssymmetry detection for 3d curves and tree objects via chain coding is presented symmetry is related to the balance and harmony the symmetry of plane and space filling curves is shown the symmetry detection on plane and space filling trees is computed

supplement to an interpretation of membership functions and the properties of general probabilistic operators as fuzzy set operators part i abstract basic idea of part i of an interpretation of membership functions and the properties of general probabilistic operators as fuzzy set operators 1992 mainly on the conceptual construction of membership functions is straightened out and illustrated by detailed examples the purpose of this article is to clarify unclear points in notations and concepts of mabuchi 1992 as a step toward the extension of the theory from type 1 to three valued fuzzy sets

when upper probabilities are possibility measures abstract a characteristic property is given for a pair of upper and lower probabilities induced by lower probability bounds on a finite set of events to coincide with possibility and necessity measures approximations of upper probabilities by possibility measures are discussed the problem of combining possibility distributions viewed as upper probabilities is investigated and the basic fuzzy set intersections are justified in this framework