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LINZ 2002

23rd Linz Seminar on Fuzzy Set Theory

Analytical Methods and Fuzzy Sets

Bildungszentrum St. Magdalena, Linz, Austria February 5 – 9, 2002

Abstracts

Phil Diamond, Erich Peter Klement Editors

LINZ 2002—

ANALYTICAL METHODS AND FUZZY SETS

ABSTRACTS

Phil Diamond, Erich Peter KlementEditors

Printed by: Universitätsdirektion, Johannes Kepler Universität, A-4040 Linz

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Since their inception in 1979, the Linz Seminars on Fuzzy Sets have emphasized the developmentof mathematical aspects of fuzzy sets by bringing together researchers in fuzzy sets and establishedmathematicians whose work outside the fuzzy setting can provide direction for further research. Theseminar is deliberately kept small and intimate so that informal critical discussion remains central.There are no parallel sessions and during the week there are several round tables to discuss openproblems and promising directions for further work. LINZ2002 will be the 23rd seminar carrying onthis tradition.

LINZ2002 will deal with Fuzzy Analysis and its Applications. This very broad area, roughlyspeaking, involves looking at classical analysis when elements of vagueness and imprecision arepresent. It thus describes both fuzzification of established methods when important applications occur,and placing fuzzy structures as well-defined objects in classical functional analysis. The organizershope that the talks will provide a comprehensive mathematical framework both for pure techniquesand practical application of analytical methods utilizing fuzzy sets.

Phil DiamondErich Peter Klement

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Program Committee

Phil Diamond (Chairman), Brisbane, AustraliaDan Butnariu,Haifa, IsraelDidier Dubois,Toulouse, FranceLluis Godo,Barcelona, SpainSiegfried Gottwald,Leipzig, GermanyUlrich Höhle,Wuppertal, GermanyErich Peter Klement,Linz, AustriaWesley Kotz’e,Grahamstown, South AfricaRadko Mesiar,Bratislava, SlovakiaDaniele Mundici,Milano, ItalyEndre Pap,Novi Sad, YugoslaviaStephen E. Rodabaugh,Youngstown, OH, USAMarc Roubens,Liège, BelgiumLawrence N. Stout,Bloomington, IL, USAAldo Ventre,Napoli, ItalySiegfried Weber,Mainz, Germany

Executive Committee

Erich Peter KlementUlrich HöhleStephen E. RodabaughSiegfried Weber

Local Organizing Committee

Erich Peter Klement (Chairman), Fuzzy Logic Laboratorium Linz-HagenbergUlrich Bodenhofer,Software Competence Center HagenbergSabine Lumpi,Fuzzy Logic Laboratorium Linz-HagenbergSusanne Saminger,Fuzzy Logic Laboratorium Linz-Hagenberg

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Contents

Ulrich BodenhoferMonotonicity in vague environments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9

Dan Butnariu, Erich Peter KlementA proximal point method for optimization in the space of fuzzy vectors. . . . . . . . . . . . . . . . . . . . . . . . . 11

Phil DiamondA contemporary snapshot of fuzzy operational equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Martin KalinaNearness-based limits in multidimensional case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Eun Ju Jung, Jai Heui KimOn set valued and fuzzy stochastic differential equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Volker KrätschmerProbability theory in spaces of fuzzy sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41

Shoumei Li, Yukio OguraConvergence theorems for fuzzy valued random variables in the extended Hausdorff metric H∞ . . 44

Radko Mesiar, Anna Koles’arov’aOn some analytical properties of aggregation operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Endre Pap, Olga Hadži’cFixed point theory in probabilistic metric spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Susanne Saminger, Radko Mesiar, Ulrich BodenhoferT-transitivity and domination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55

Uziel SandlerDynamics of fuzzy systems: theory and applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Mamoru ShimodaA natural interpretation of fuzzy partitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

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Monotonicity in vague environments

ULRICH BODENHOFER

Software Competence Center HagenbergA-4232 Hagenberg, Austria

E-mail: [email protected]

Zadeh’s famous extension principle [8, 9, 10], as a general methodology for extending crisp conceptsto fuzzy sets, has served as the basis for the inception of new disciplines like fuzzy analysis, fuzzyalgebra, fuzzy topology, and several others. Most importantly, this fundamental principle allows toextend crisp mappings to fuzzy sets. Another well-known application—which is particularly impor-tant in fuzzy decision analysis and fuzzy control [4, 6]—is the possibility to define ordering relationsfor fuzzy sets.

This contribution is devoted to links between these two fields: we study in which way the mono-tonicity of a mapping is preserved by its extension to fuzzy sets. However, we do not restrict to thewell-known methodology of extending crisp orderings to fuzzy sets, but we start from the more gen-eral case that the domain under consideration is equipped with a fuzzy ordering [1, 2] induced bysome non-trivial fuzzy concept of indistinguishability. Even in such a case, it is possible to defineorderings of fuzzy sets in a way similar to the extension principle [1, 3].

First, we consider the classical case of orderings of fuzzy sets defined from crisp orderings bymeans of the extension principle. We show that the monotonicity of a mapping directly transfers toits extension. Furthermore, the same holds for the componentwise monotonicity ofn-ary operations.Next, we consider the general case that the universe is equipped with a fuzzy ordering induced bya fuzzy equivalence relationE. It is proved that the monotonicity of a mappingϕ is preserved byits extension if and only ifϕ is extensional with respect toE [5, 7]. However, it turns out that ananalogous correspondence does not necessarily hold forn-ary operations.

Acknowledgements

The author gratefully acknowledges support of the KplusCompetence Center Program which is fundedby the Austrian Government, the Province of Upper Austria, and the Chamber of Commerce of UpperAustria.

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References

[1] U. Bodenhofer. A Similarity-Based Generalization of Fuzzy Orderings, volume C 26 ofSchriftenreihe der Johannes-Kepler-Universität Linz. Universitätsverlag Rudolf Trauner, 1999.

[2] U. Bodenhofer. A similarity-based generalization of fuzzy orderings preserving the classicalaxioms.Internat. J. Uncertain. Fuzziness Knowledge-Based Systems, 8(5):593–610, 2000.

[3] U. Bodenhofer. A general framework for ordering fuzzy sets. In B. Bouchon-Meunier,J. Guiti’errez-R’ıoz, L. Magdalena, and R. R. Yager, editors,Technologies for Constructing Intel-ligent Systems 1: Tasks, pages 213–224. Springer, 2002. (to appear).

[4] E. E. Kerre, M. Mareš, and R. Mesiar. On the orderings of generated fuzzy quantities. InProc.7th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-BasedSystems (IPMU ’98), volume 1, pages 250–253, 1998.

[5] F. Klawonn. Fuzzy sets and vague environments.Fuzzy Sets and Systems, 66:207–221, 1994.

[6] L. T. K’oczy and K. Hirota. Ordering, distance and closeness of fuzzy sets.Fuzzy Sets andSystems, 59(3):281–293, 1993.

[7] R. Kruse, J. Gebhardt, and F. Klawonn.Foundations of Fuzzy Systems. John Wiley & Sons, NewYork, 1994.

[8] L. A. Zadeh. The concept of a linguistic variable and its application to approximate reasoning I.Inform. Sci., 8:199–250, 1975.

[9] L. A. Zadeh. The concept of a linguistic variable and its application to approximate reasoningII. Inform. Sci., 8:301–357, 1975.

[10] L. A. Zadeh. The concept of a linguistic variable and its application to approximate reasoningIII. Inform. Sci., 9:43–80, 1975.

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A proximal point method for optimization in the space of fuzzy vectors

DAN BUTNARIU1, ERICH PETER KLEMENT2

1Department of MathematicsUniversity of Haifa

IL-31905 Haifa, Israel

E-Mail: [email protected]

2Fuzzy Logic Laboratorium Linz-HagenbergDepartment of Algebra, Stochastics, and Knowledge-Based Mathematical Systems

Johannes Kepler UniversityA-4040 Linz, Austria


— abstract not available —

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A contemporary snapshot of fuzzy operational equations

PHIL DIAMOND

Mathematics DepartmentUniversity of Queensland

Brisbane, Queensland 4072Australia


1 Introduction

Early studies of fuzzy differential equations observed that there was a fundamentally different be-haviour from the crisp case: the solutions spread out as time evolved [14]. This seems to be an artifactof the Hukuhara derivative and Aumann integral of multivalued calculus, on which the fuzzy deriva-tive is based. Ultimately, the effect stems from the fact that Minkowski addition of sets is not, ingeneral, invertible.

Example 1.For simplicity, consider a fuzzy DE with interval values,

X′(t) =−AX(t), X(0) = X0 = [a,b], (1)

wherea > 0, A = [1− ε,1+ ε], 0< ε < 1 andX′ is the Hukuhara derivative. PutX(t) = [X (t),X (t)]and (1) becomes the system of ordinary DEs[

X ′

X ′

]=[

0 −1− ε−1+ ε 0

],

[X (0)X (0)

]=[

ab

].

Writing δ =√

1− ε2, the system has a solution

X(t) =[

X (t)X (t)

]= c1

[1+ ε

δ

]eδt +c2

[1+ ε−δ

]e−δt ,

where the constantsc1, c2 depend ona, b. Clearly, for almost alla, b, the solutionX(t) expandswithout bound ast increases, no matter how small isε (although, ifboth b→ a+ andε → 0+, thenc1 → 0). This behaviour contrasts with the crisp equation of exponential decayz′(t) =−z(t), z(0) =z0. Even a small incertitude in the parameters leads to vastly different and counterintuitive behaviour.Much the same happens with fuzzy integral equations and would appear highly likely in any form offuzzy equation which involves the Hukuhara fuzzy derivative and Aumann fuzzy integral.

In a seminal paper, Hüllermeier suggested how this unsatisfactory obstacle to modelling uncer-tainty could be overcome [11]. His insight was tointerpret the fuzzy DEX′(t) = F(t,X(t)), X(0) =X0 as afamily of differential inclusions

x′(t) ∈ Fβ(t,x(t)), x(0) = x0 ∈ X0β , 0≤ β≤ 1,

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whereFβ, X0β areβ-level sets. The set of pointsΣβ(t,X0

β ), of all solutions of theβ-th inclusion at timet, was the level set of a fuzzy set which could be regarded as the solution of the fuzzy DE.

If the interval valued example above is interpreted in this way, all solutionsx(t) are bounded,x(t)≤ x(t)≤ x(t), where

x′ =−(1+ ε)x, x(0) = a; x′ =−(1− ε)x, x(0) = b.

Consequently, the solution is regarded as the interval

Σ(t) =[ae−(1+ε)t , be−(1−ε)t

],

which, although retaining uncertainty, does suffer exponential decay to 0= 0 as t → ∞. This ismuch more satisfactory and intuitive behaviour.

This talk surveys some of the recent results using Hüllermeier’s interpretation applied to classesof fuzzy operator equations

X(t) = O(X)(t),

whereO(·) might be a differential or integral operator, define some sort of stochastic DE or delayDE, or even a control system. Such equations will be interpreted as families of inclusionsxβ(t) ∈O(xβ)(t), 0≤ β ≤ 1. The next section recalls some fundamental definitions relating to fuzzy setsand inclusions. The third section considers differential operators and the fourth studies Volterra typeintegral operators. The fifth section briefly surveys some work which is yet to appear and a conclusionsums up the talk.

2 Basic Concepts

2.1 Preliminaries

Let Dn denote the set of uppersemicontinuous (usc) normal fuzzy sets onIRn with compact connectedsupport. That is, ifX ∈ Dn, thenX : IRn → [0,1] is usc, supp(X) = ξ ∈ IRn : X(ξ) > 0 := [X]0 iscompact and there exists at least oneξ∈ supp(X) for whichX(ξ) = 1. Theβ−level set ofX, 0< β≤ 1,is

[X]β = ξ ∈ IRn : X(ξ)≥ β.

For brevity, level sets[X]β will usually be written asXβ. Clearly, forα ≤ β, Xα ⊇ Xβ. The level setsare nonempty from normality, and compact by usc and compact support. The metricd∞ is defined by

d∞(X,Y) = sup

dH(Xβ,Yβ

): 0≤ β≤ 1

,

whereX,Y ∈ Dn, anddH is the Hausdorff metric on compact subsets ofIRn. Then (Dn,d∞) is acomplete metric space [9]. If 0 is the origin inIRn, 0∈Dn denotes the fuzzy set such that 0(ξ) = 1 ifξ = 0 and is zero otherwise.

DefineEn ⊂Dn as those fuzzy sets with convex level sets. In particular, writeE = E1 = D = D1.If X ∈ En, write Xβ =

[X β,X β

].

To discuss solutions of inclusions, some function spaces are required.

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• Cn[0,T], the space of continuous functionsf : [0,T]→ IRn, with the norm of uniform conver-gence,‖ f −g‖∞ = max0≤t≤T | f (t)−g(t)|.

• L1n[0,T] is the space ofintegrable functionsfrom [0,T] to IRn, with metric‖ f −g‖1 =

∫ T0 | f (t)−

g(t)|dt.

• L∞n [0,T] is the space of measurable functions from[0,T] to IRn, bounded almost everywhere on

[0,T], with norm

‖ f‖L∞ = ess sup0≤t≤T

| f (t)|= infc≥ 0 : | f (t)| ≤ c a.e. 0≤ t ≤ T.

• ZT(IRn) = x(·) ∈C([0,T]; IRn) : x′(·) ∈ L∞([0,T]; IRn).

2.2 Differential and integral inclusions

Let Ω ⊂ IR× IRn be an open subset containing(0,x0) and letH : Ω → KnC be a compact, convex

setvalued mapping. The differential inclusion

x′(t) ∈ H(t,x(t)), x(0) = x0, (2)

is said to have a solutiony(t) on [0,T] if y(·) is absolutely continuous,y(0) = x0 andy(·) satisfies theinclusion almost everywhere (a.e.) in[0,T].

The integral inclusions are of Volterra type

x(t) ∈ h(t)+∫ t

0k(t,s)G(s,x(s))ds t∈ [0,T], (3)

where h : [0,T] → IRn is continuous, the matrix valued functionk : (s, t) : 0 ≤ s≤ t ≤ T →L1

n×n[0,T], G : [0,T]× IRn →KnC andG(·,x) has measurable selections. The integral is understood to

be in the sense of Aumann. A solution to (3) is a continuous functionx(·) ∈C[0,T] which satisfiesthe inclusion a.e. in[0,T].

Under mild conditions, (2) has solutions on[0,τ], 0 < τ ≤ T, [1, 3] and so does (3) [2]. Inboth, denote the set of solutions in[0,τ] by Σ(h;τ) and the attainability set for eachτ ∈ [0,T] byA(h;τ)= x(τ) : x(·)∈Σ(h;τ). It is known thatΣ(h;τ) andA(h;τ) are nonempty compact, connectedsets in their respective spaces [1, 2, 3]. For both,A(h;τ)⊂ IRn. With (2),Σ(h;τ) is a subset ofZτ(IRn),and for (3) it is a subset ofCn[0,τ].

3 Fuzzy Differential Equations

A convenient condition which guarantees existence of solutions to (2) is theboundedness assumption:there existb, T, M > 0 such that

• the setQ = [0,T]× (x0 +(b+MT)Bn)⊂Ω, whereBn is the unit ball ofIRn;

• H mapsQ into the ball of radiusM.

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Now, suppose thatF : IR× IRn → En and consider the fuzzy differential equation

x′ = F(t,x), x(0) = X0 ∈ En,

but identify it with the family of differential inclusions

x′β(t) ∈ Fβ(t,xβ(t)) xβ(0) = x0 ∈ X0β , 0≤ β≤ 1. (4)

Here, Ω is an open subset ofIRn+1 containing(0,X00 ), β ∈ I := [0,1] and F : Ω× I → Kn

C. Theboundedness assumption now holds if the setQ is as above andF mapsQ× I into the ball of radiusM. Denote the set of all solutions of (4) on[0,τ] by Σβ(x0,τ) and the attainable set byAβ(x0,τ) =x(τ) : x(·)∈ Σβ(x0,τ). ThenΣβ(x1,T) exists and is a compact subset ofZT(IRn), and each attainablesectionAβ(x1,τ), 0 < τ ≤ T, is a compact subset ofIRn [1]. It turns out that theAβ, Σβ build levelsets of fuzzy sets, which can then be regarded as solutions to the fuzzy DE.

Theorem 3.1 [10] Let X0 ∈ En and let Ω be an open set in IR× IRn containing0× supp(X0).Suppose that F: Ω→En is usc and write Fβ(t,x)∈Kn

C as theβ-level set for all(t,x,β)∈ IRn+1× [0,1].Let the boundedness assumption, with constants b, M, T, hold for all x0∈ supp(X0) and the inclusion

x′(t) ∈ F0(t,x), x(0) ∈ supp(X0). (5)

Then the attainable setsAβ(Xβ,T), β ∈ [0,1], of the family of inclusions

x′β(t) ∈ Fβ(t,xβ), xβ(0) ∈ Xβ := [X0]β, β ∈ [0,1], (6)

are the level sets of a fuzzy setA(X0,T) ∈Dn. The solution setsΣβ(Xβ,T) of (6) are the level sets ofa fuzzy setΣ(X0,T) defined onZT(IRn).

RemarksThe proof of the theorem relies on the Negoita-Ralescu characterization of fuzzy sets [13].The major part lies in showing

⋂∞i=1 Σβi

(Xβi,T) = Σβ(Xβ,T) for any nondecreasing sequenceβi → β in

[0,1], in the function spaceZT(IRn). This entails a sequential compactness argument. If the conditionthatF0 be bounded on[0,∞)×Γ, whereΓ⊆ IRn is open, is added to the conditions of the theorem, theinterval of existence and consequences extend to[0,∞).

This result opens the way for notions of periodicity and stability to be studied in the fuzzy context.See [5] for further details.

4 Fuzzy Volterra Integral Equations

If equalities, based on the Aumann integral, are used at each level set for such equations, similarunsatisfactory behaviour occurs as was the case for fuzzy DEs. This is unsurprising, since each fuzzyDE is equivalent to an Aumann integral equation [9]. However, if the equations are replaced by afamily of integral inclusions, a reasonable framework for modelling is obtained.

Supposev ∈ Cn[0,T] and F : [0,T]× IRn → En is strongly measurable (that is, each level setmapping is measurable) and majorized by anL1 function. Consider the fuzzy equation

x(t) = V(t)+∫ t

0k(t,s)F(s,x(s))ds, (7)

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interpreted as a family of integral inclusions

xβ(·) ∈Vβ +∫ ·

0k(t,s)Fβ(s,x(s))ds, 0≤ β≤ 1. (8)

Under mild but technical conditions on the kernelk(t,s), the solution setsΣβ(Vβ ; t) and attainabilitysetsAβ(Vβ ; t) are compact and connected [2], and are the level sets of fuzzy setsΣ andA , respectively[6]. Correctly interpreted, the result extends to the case whereV(t) is a fuzzy valued set overCn[0,T].

Moreover, using integral inequalities [12], and solving equations at extreme points of the bound-ary, exact or computational solutions can be found.

Example 2.Let (a;b)S denote a symmetric fuzzy with support[a,b] and consider the fuzzy equation

x(t) = (0;t2/2)S−∫ t

0(t−σ)(x(σ) ;3x(σ))Sdσ, quadt≥ 0,

or, equivalently for 0≤ β≤ 1,

xβ(t) ∈ [βt2/4,(1/2−β/4)t2]−∫ t

0(t−σ)

[(1+β)xβ(σ) ,(3−β)xβ(σ)

]dσ

That is,

xβ(t) ≥ βt2/4− (3−β)∫ t

0(t−σ)xβ(σ)dσ

xβ(t) ≤ (1/2−β/4)t2− (1+β)∫ t

0(t−σ)xβ(σ)dσ.

Taking Laplace transforms, noting the convolution integrals, using the result quoted on integralinequalities and simplifying,

β/2s(s2 +3−β)

≤ Xβ(s)≤1−β/2

s(s2 +1+β).

using partial fractions and taking the inverse transforms, the solution set consists of the fuzzy set withβ-levels the intervals, for 0≤ β≤ 1,[

β(1−cos(√

3−βt))2(3−β)

,(2−β)(1−cos(

√1+βt))

2(1+β)

].

5 Other Topics

5.1 Variation of constants formula

Analogues of this classical result have recently been developed [7]. Basically, what is required is afuzzy version of the state transition matrixΦ of a system. Recall that ifx′ = Ax, Φ(t) is the uniquematrix satisfyingΦ′(t) = AΦ(t), Φ(0) = I .

Now, givenx′(t) = Ax(t)+Bu(t), x(0) = x0,

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which can be rewritten as

x(t) = Φ(t)x0 +∫ t

0Φ(t−s)Bu(s)ds.

Suppose that the matricesA, B have fuzzy number entries and the initial conditionx0 ∈ En is fuzzy.Then this equation can be considered as a family of integral inclusions, provided that some meaningcan be ascribed toΦ(t). The family of inclusions

Φ′β(t) ∈ AβΦβ(t), Φβ(0) = I , 0≤ β≤ 1,

has solutions, for matricesV in the set of matricesAβ,

Φβ(t) =Y(t) : Y′ = VY, Y(0) = I

.

The Y(t) are found in the usual way: find a basis of vector solutionsv1(t), v2(t), . . . , vn(t) fromthe eigenvalue-eigenvector problem forV and form the matrixZ(t) = [v1 v2 . . . vn]. ThenY(t) =Z(t)Z(0)−1.

5.2 Controlling fuzzy dynamical systems

Variation of constants can be used for solving open loop control systems. A question arises as towhether some sort of feedback control is meaningful for control systems of the form

X′(t) = AX(t)+BU(t), X(0) = X0,

whereA, B are matrices andX0 an n-vector, all with fuzzy set entries overIR, in particular with afuzzy scalar, linear state feedback law of the formU(t) = K(t)X(t).

It turns out that an appropriate framework is given by set-theoretic approaches to robust control[15]. Again using the inclusion approach, the problem can be treated at each level set as a type ofcalculus of variations problem and can be solved by iterative numerical procedures. Matrix Riccatiequations are involved. For the one dimensional case, exact solutions have been found [8]. Thesecompare favourably with fuzzy versions of LQR control for such systems.

5.3 Miscellaneous results

A number of extensions to the above have been made, but are in a very preliminary stage of develop-ment. Integral equations of Hammerstein type have been studied, and the Volterra results generalized(Menshing Guo and Xiaoping Xue, personal communication). Very general operator equations in-volving measures and impulses are also being studied (Guo,loc. cit.).

It seems likely that existence theorems for stochastic differential inclusions can be used to studya fusion of the two uncertainties: fuzzy and random. Care has to be taken to specify which stochasticintegral is to be used, but some progress has been made using the Ito integral. The results are, as onemight expect, more complicated than others discussed here.

Some very recent progress has been made in filtering of fuzzy signals, governed by a fuzzy DEcontaining a white noise signal with a fuzzy covariance function. The inclusion approach is used,drawing upon results in robust filtering. These provide linear filters upper and lower bounds forthe covariance of an otherwise unspecified white noise disturbance. By considering the extremalequations of the system, bounds can be found for the estimator.

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6 Conclusion

Recent developments in the theory of fuzzy differential and fuzzy integral equations were surveyed.These treat equations as families of inclusions and interpret the solution sets as level sets of a fuzzysolution to the equation. Some applications were discussed and examples given.

References

[1] J.P. Aubin and A. Cellina,Differential Inclusions. New York: Springer Verlag, 1984.

[2] C. Corduneanu,Integral Equations and Applications. New York: Cambridge University Press,1991.

[3] K. Deimling,Multivalued Differential Equations. New York: Walter de Gruyter, 1992.

[4] P. Diamond, “Time-dependent differential inclusions, cocycle attractors, and fuzzy differentialequations,”IEEE Trans. Fuzzy Systems, vol. 7, pp. 734–740, 1999.

[5] P. Diamond, “Stability and periodicity in fuzzy differential equations,”IEEE Trans. FuzzySystems, vol. 8, pp. 583–590, 2000.

[6] P. Diamond, “Theory and applications of fuzzy Volterra integral equations”,IEEE Trans. FuzzySystems, to appear.

[7] P. Diamond, “Brief note on the variation of constants formula for fuzzy differential equations”,Fuzzy Sets and Systems, to appear.

[8] P. Diamond, “Design of optimally bounded linear state feedback laws for fuzzy dynamicalsystems”, FUZZ-IEEE 2001, December 2–5, Melbourne, Australia, 2001.

[9] P. Diamond and P. Kloeden,Metric Spaces of Fuzzy Sets. Singapore: World Scientific, 1994.

[10] P. Diamond and P. Watson, “Regularity of solution sets for differential inclusions quasi-concave in a parameter,”Applied Mathematics Letters, vol. 13, pp. 31–35, 2000.

[11] E. Hüllermeier, “An approach to modelling and simulation of uncertain dynamical systems,”Int. J. Uncertainty, Fuzziness & Knowledge-Based Systems, vol. 5, pp. 117–137, 1997.

[12] V. Lakshmikantham and S. Leela,Differential and Integral InequalitiesVol. 1. New York:Academic Press, 1969.

[13] C.V. Negoita and D.A. Ralescu,Applications of Fuzzy Sets to Systems Analysis. New York:Wiley, 1975.

[14] S. Seikkala, “On the fuzzy initial value problem,”Fuzzy Sets and Systems, vol. 24, pp. 319–330, 1987.

[15] S.D. Wang and T.S. Kuo, “Design of robust linear state feedback laws: ellipsoidal set-theoreticapproach,”Automatica, vol. 26, pp. 303–309, 1990.

18

Nearness-based limits in multidimensonal case

MARTIN KALINA

Dept. of MathematicsSlovak University of Technology

SK-813 68 BratislavaSlovakia


The concept of nearness-based limits has been introduced at the 7-th Fuzzy Days conference inDortmund (see [1]). Now, we are going to discuss the properties of such limits for functions havingmultidimensional domain.

We recall the definition of fuzzy nearness.

Definition F : R 2 → [0;1] is said to be a relation of fuzzy nearness iff

1. for anyx∈ R x F x = 1

2. for anyx,y x F y = y F x

3. for anyα∈]0;1[ and anyx there exist uniquexα > x andx−α < x such thatx F xα = α = x F x−α

4. for anyx there holds limy→∞ xF y = 0

5. for anyx≤ s≤ t ≤ z there holdsxF z≤ sF t.

If moreoverxF y = 1 iff x = y holds, then we callF the strict fuzzy nearness.

References

[1] Kalina M., Fuzzy limits and fuzzy nearness relation, In: Proc. of 7-th Fuzzy Days in Dortmund,October 2001, Springer, LNCS 2206, pp. 755 – 761.

19

On set valued and fuzzy stochastic differential equations

EUN JU JUNG, JAI HEUI K IM

Department of MathematicsPusan National University

Pusan 609-735Korea


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<=#$) #(A+ .( . #(3B<C $) + #(-,@$#D$& E$& .#(/$FG#HI) -( . B"$#%'&("$)*%J) ,$ .-(/'*.KL G0@ E"$FG M GHNO.O.FG"$#%'&("$)*%P) Q ./$ .,$)* R,$) #(.K

S4TEUVW.XYZ\[\]W ^_Y4Và& D<D/$) -, =3/$#(34 ./"FA#Ha$& 6) -( . b"$#%'&("$)*%=) ,$ .-(/'c)*dBe()*%=3/$) %) 3 =HN#(/

$& A .( . #(3<= .,=#HJ"$#%'&("$)*%G F)*.K!fg#(/$/$ 34#() -( F(hI) D)*=) <=34#(/',D) i<C $) "$#%'&("$)*%1 F)*j$#de4#$&6 PE g D"$#%'&("$)*%a) ,$ .-(/'D .<=#("$/'$ a$& ?<D/:$) -, B3/$#(34 ./"F(K2k#( #01) -$& A .<=) M0@#(/$lm#HEnA$& ./$#(o 8qpqrshI<=#(/$ B/$ % .,$ Fm$& A"$F#H?/'#(<t '.hj$&=)*.hg = /'#(<u/$)*e .hc&(Ce4 . .i .( . #(34 evF L e) 9k/'l(#0Mlo 8rsh4wM)*)cAxM<= .-,lv)so yqrshzc3-( .#(/$-() #(o 8.r+<DvFB#$& ./'.Kd|v J A<D/:$) -, E&q( =e4 . ."$) bevFb)g7~?-(/'o 8qpqrshzc3-( .#(/$-() #(o 8.rsK=J) <umJ) <+o r@J)*) . .01)*%O,o 8(8rc E "$#%'&("$)*%C) ,$ .-(/'hed)*#d%#()* ./P$& =<D/$) -, 3/$#(34 ./"FDg5:se4#( $.h,01&)*%'&6/$ 1 HN4) 633 )*%.$) #(.Kjà& 1$& .#(/$F#H#(/') /$FC) H:HN ./$ .,$)*v R,$) #(\&(je4 . . v$ .) ( . Fd .( . #(34 C) =%#("%5$) #(01) $&CHNO.O.Fd F)* . 6o rN5Kj;B3/$)*%*/ h>Eo pqrsh)+~?-(/'Go 8qpqrshzI/$)GM $%o 8qyqrB|,$#(%#v)%(o 8 r) ,$/$#% B$& %#(% .3P#HcHNO.O.FB/'#(</$)*e (hHNO.O.FA%#() $) #( 34 %5'$) #(GHNO.O.F<D/$) -, (KIà&)*1334 ./M .( . #(31=<=#(/$ J$e E ) $) #(+#H$& C M B"$#%'&("$)*%P) ,$ :-(/'4$&M)*a<=#(/$ E33 )*%.e (hG ./$) ( 1 1 B #(-,g#Hj#(<= J#H\$& E<=#,"1) <=34#(/',3/$#(34 ./$) M#Hg) -( . A"$#%'&("$)*%E) ,$ .-(/'*.K L +evFB) -6$& ) $) #(+3/$#(34 ./$) #H@$& 6 d b"$#%'&("$)*%) ,$ .-(/'h0@ 6"$FA$& =$& .#(/$F+HN#(/C d b"$#%'&("$)*%=) Q ./: .,$)*j R,$) #(.h"$#%'&("$)*%d) % ) #(3/$#(e .<DP+HNO.O.FA"$#%'&("$)*%C) Q ./$ .,$)*\ R,$) #(.Kà& C334 ./?)*P P#(J(1HN#( #0EM|v %5$) #(pD/$ %. *M#(<= C HN\/$ 'M#( P +/'#(</$)*e .h\ D$#wM)*)@bxM<= .-,lv)so yqr@mà& .#(/$ .<pK B <=) dm) <=34#(/',d <=3 #H?A m<D/$) -, (KA;i|v %5$) #(7hj0@ G G C "$#%'&("$)*%D) ,$ .-(/'g01) $&/$ 34 %5E$#m8:;) <= .) #(I>@/$#01)*<=#$) #(hjb)*$%$E#(<= D#Ha) 'E3/$#(34 ./$) .KG|v %5$) #(pB=/$ J/$ '1evFG(-DBJ) <+o rsKc;|v %5$) #(B=0@ E3/$#( ?$& E )*"$ .% dG)*R, . $#H\$& d#( $) #(GHN#(/MD M B"$#%'&("$)*%J) Q ./$ .,$)* R,$) #(Kc;A|v %5$) #(Ayh0@ J-() ( PHNO.O.F #(-,@#Hc|v %5$) #(+=K

TEJXv_^_^_V\X^_v 1e4 ?$& J a#H\ /$ 4v<Ce4 ./'a P$ D$¡ g%#(<=3 $ P3/$#(ee) ) "FD3(% (Kj¢? .#$

evF£ P¤ p¥cg$& JH_<=) F#Hj <= (/'e J M 6HN%5$) #(M G#( 01) $&B a)

20

$& JH_<=) F#H\#( .<=3"FG% #, Ge 'a#HcKk#(/B£ P¤ p¥c5h0@ E B P$ D¤ E@ 2K K , 8 p

wM ./$ P$ D¤ E6)*$& b3(% #HC P/$ P 2/'#(< /$)*e %'&!$& 6o r !h01& ./$ !6o ",r)*a$& E 34 %5'$) #(G#Hc/'#(< /$)*e "K#%$&!')(*&,+ T_S4T.-0/214365879;: =< b£ P¤ p ¥ )> ?A@G8CB BDFEHG0I KJ! ML*NOQP :0RS9 ?=< >ME TE VU e4 Pd g#H<= (/'e aHN%5$) #( XW Kj P%. AUY@ 9[Z[\ ]^_\ T>M`;a 9 01) $&6/$ 34 %5

$# ) H < bU dce ) <=3 F8gf <ih!8gfQj kU K@wM ./$ D8gf)*a$& d) )*%.$#(/1HN%5$) #(#H=cdK#%$&!')(*&,+ TT.-0/214365d79;:l ` 9 >? \ ? 9;]Â:6mbZ a \ T 9 @Ton_òT 9;:,\ IE P$ D¤ E P 8pB qFEr RS9 ? :0RS9;st9*9tuMv T : T8> ?kwi£ P¤ p¥cxTon Z[R.:0R > : U v I8> ?A@ \ ?_a mv I lyv T8@ 9[Z[\ ]^_\ T>M`;a 9z v:0Rdst9 T ^_9[Z;::2\ E

k#(/MvF6e |9 P$ D¤ E5h0@ E J$& *@ 9[Z[\ ]^_\ T>M`;a 9)Z a \ Ton st9 #H~bevF

x "8+ P$ D¤ E@HN#(/MvF)x $& ./$ E )*"MC) $ :s<= (/'e J3/$) $) #( c< c c #H

< 8b%'&G$&H "x < 8gf X S

9;] > st! EEEfg /$ F G)*cJ% #, e c#H P$ D¤ EjC)*c %#(<=34#,$e @01) $&=/$ 34 %5$# KgwM .% JevFà& .#(/$ .<pK phvHN#(/1vF*|9 P$ D¤ E5hv$& ./$ E )*"'1dB£ P¤ p¥ca%'&$& K L B) H=b)*M %#(<=34#,$e (hv$& .+) M&#(*a$& F; Ih01& ./$ P$& C% #,/$ )*a'l( .G01) $&B/$ 34 %5a$#$& E#(/$<K K

¢? .#$ devF* E@$& EH_<=) FG#HI #( .<=3"FG% #, Be 'M#Hc p! E| E@$& $#' ) "F6#Hc %'&B '101&)*%'&B/$ E*#=%#(v( $&v1) ,$ ./$*.K

k#(/MvF8c t p E5h0@ E

c t <DH23¡ f 0¢t 3£¤ M¥ 0¦ cP¤§

01& ./$ 0¢t ) H £¤ M¥ ¢ ¦ hG * cx c 3 ¡ f ¢ KsHxcu /$ e4#( hM$& . c t )*6$& mw?$#(/Q<= $/$)*%#HHc K k#(/c t6[¨ p E5h$& E R, ) "F

pK 8q ch ¨Pt h ¨ c t &#(* . 6o 8 rN5K

21

k#(/@E/'#(< V+£ P¤ p¥c5hK)*@%. 6 : ¤ ) H$& ./$ ?)*d+ P$ D¤ EI%'&$&x ¢ B gHN#(/MvF ¢ 01) $& ¢ 5K 9;] > st8 E EJk#(/PvFp £ P¤ p¥c5h4) P)*M0@ . lv#01B$& )*? :se4#( +) HgB#( FG) H$& JHN%5$) #( W v)*1) + P$ D¤ E5h)K (K 8 B P$ D¤ E . 6o yqrsh*o ro 8(8rN5K

P$ D¤ E /$ 3K P$ D¤ ! E=e4 +$& 3(% +#HE E /$ 3KF! E + :se4#( G/'#(< '1) A£ P¤ p¥c5K1k#(/ £ P¤ p¥j5h$& ¢M #Hi )* GevF

o Pr e 6o r < G%. BT 9;: > a n 9 @ Âst\gZ[9 T[TE01) $& d) 7 #(/ h\HN#(/&#(/ hj : K ¤ ?: E) H= o 7 XW p¥)*M= 1 HN%5$) #(+%'&$& 8q6HN#(/M mo 7 4 g)*M% #, G%#(v( 6) + K K h p(6HN#(/GvF7 o 7 =)*Gm/'#(< h1$&)*.h1HN#(/G P>@#(/$ .? '*c

E M 4 "!pc N O KL a 6?:s3/$#% $ J)*a%. s: ¤ ) H g)*g<= (/'e ?01) $&/$ 34 %5

$# DHN#(/G .( ./$F $# K L i B ?:s3/$#% $ b)*B%. &% ' ) H 4 bo 7 4 "!pc NFO o 7 gHN#(/c E5KwM)*)IbxM<= .-,lv)so yqrg&#0@ +$&EHN#(/dvFk i P$ D¤ EEbe:(4: .* | h

$& ./$ J )*"'M)*R, *)B P $¡=¤ Ea%'&$&

pK p( + ; 6o r 01& ./$ J$& C% #,/$ E)*a'l( .B01) $&+/$ 34 %5M$#D$& d#(/$< ) A P ¤ E5K@à&)*M/'#(< ,)

)*M%. 6$& M Z[\ ?A@ v:6v\ ?A> a 9tuo^_9[Z;: > :6v\ ?D#H~ -() ( . 0@ E .#$ E) MevF o 8 rsK- &/. U10+U # U ' 0 T32T L M G?:s3/$#% $ d)*M%. G s: ] > so:6v ?54> a 9 ) H

8q6HN#(/MvF # hS @)*1 < :se4#( h p( d)* s:;(3$ h ,6HN#(/ # # h o 76 r qaK .K

k#(/vFi/'#(< d < P$ D¤ ! E o 8 srJ G s:<D/$) -, (Km D ECe4 $& +3(% #HP s:;(3$ m/$ @ <= (/'e 3/$#% $ C%'&$&1HN#(/MvF h_6o 8 q r !K

à& J vM/$ 1)*M ./$) ( 6HN/$#(<:9c<=3 JK 8EK ) +)A~?-(/'o 8qpqrsK#%$&!')(*&,+ T<;T>= T[Ton ])9:0R > : % v T,>%? \ ?_? 9 4> :6v3 9,] > so:6v ?54> a 9,v ?=g EE 79;: U o E8 8rC> ?A@@ 9@? ? 9 U ` ml- : 5BA]p- : 5¤l >ME TEE r RS9 ? U v Tk>T 9;:C > a n 9 @ D] > so:6v ?54> a 9 EÂst\g\ I;EEfg /$ F E>F 6G )*1-() ( .evF

E>F 6G B P$ 76q¤ EE@ KB % q K .K22

# !XJH"* E>F G Kcà& .x" 6o 76 rHN#(/g#(<= E>F G Kj>@Fd$& M3/$#(34 ./$) #H\$& d%#() $) #( 34 %5'$) #(h0@ J&q(

"Q 6o 76 r4KBX6o 76 rBX6o % 76 r % q K .K

à&)*M&#0Mg$&"* E>F 6G h#$&

pK , E>F G | E>F 6G M#0 0@ =3/$#( C$& D%#(v( ./' (KDfg /$ F U D)*J-() ( .evF.U 4 o % 4 % 4 srsK "8 E>F 6G 6 F 6 G F G

F 6 G ) H % 4 N ) H % 4 G d=/'#(< /$)*e *GevF

8mp 6q % 4 o|v) % xpB 6 B 8K .K h4evFG$& %#(v( ) "FG/$-(<= ., h4) P&#(*k E>F G % /$ F(K L +0@ %.%.*% @$&

6o 76 r 6o 8mp 6 % 76 r 8mp 6 % q " K .K

à&)*1<= @$& E>F 6G | E>F G fg#(<Ce) ) -E$&)*I01) $& pK ,cD) -Cà& .#(/$ .< pK 8(h U P)*gd g :s<D/$) -, (Kà& E3/$#v#H\)*1%#(<=3 $ (K

T,W_[\vZ9.W Y]\.W ^_]B^_VW X_à& ?<D/$) -, M3/$#(34 ./"F#H$& P) -( . 6"$#%'&("$)*%1) ,$ .-(/')*g( ./$F) <=34#(/',gHN#(/I$&

"$Fm"$#%'&("$)*%6 F)*.KA|v) <=) */$ F$& <D/$) -, D3/$#(34 ./"F#H1$& G m"$#%'&("$)*%) ,$ .-(/'3*qFa) <=34#(/',1/$#( P) B 1 GDHNO.O.F"$#%'&("$)*%P F)*.Kj;$&)*1 %5$) #(0@ I) ,$/$#% g1 ) -(&,$ FJ) Q ./$ .,e\<=#(/$ I%#(v( .) ., ) $) #(J$&E$&#, c3/$ .v) #( FP . Go rsh*o 8(8rNM+3/$#( E$& <D/$) -, d3/$#(34 ./"F(KPk#(/J P ?:s3/$#% $ vh : 9#H +)*/$ g 3/$#% $ E$$)*"HNFv) -AHN#(/ .( ./$F# 4 4 K K KE = .#$ evF M$& = J#H@ c :; . %5$) #(?#H$& P g 6?:s3/$#% $ vKc¢?)*"$) -()*& cHN/$#(< $& P F G #H 4 . %5$) #("*B P$ ¤ Eg#Hc/'#(< gHN#(/MvFD # KL J ?:s3/$#% $ D)*J%. : ¤ ) Hg$& ./$ =)*JG3/$#% $ p Ea%'&$&x ¢ B 4 gHN#(/MvF # V ¢ 01) $& ¢ 4 o

23

9;] > st EMEP>@F=$& J$<= P/$-(<= .,a(@1 .<D/$lDpK hv0@ P%. . ?$&1C a 6?:s3/$#% $ d)*1 :se4#( G) HcG#( F6) H HB Eo

E /$ 3K ! E\e4 @$& 1 c#H K E /$ 3K! E\ :se4#( s:;(3$ <= (/'e J 1 B?:s3/$#% $ .K

k#(/MvF H@ ! E5h0@ d

q 6 ! ¤§

HN#(/M # h01& ./$ C)*M/$ B>@/$#01)*<=#$) #(G01) $& , =K .K#%$&!')(*&,+ TT\ s > ? m @ ! Eb> ?A@ # P :0RS9;st9k9tuMv T : T> ? dd£ P¤ p ¥ Ton Z[Rd:0R > : <F G [Âst\g\ I;E? /'"d&#0 $& | < P$ ¤ E5K= O 'Kà& .HN#(/dvFbxDh$& ./$ )*"d) $ :s<= (/'e 3/$) $) #( c< c c #H p d= 8 p %'&$&

O < 8gf q 6 à&va0@ E&q(

6o O r B O < 8gf q 6 h

< 8gf q 6 h < 8gf q 6

wM ./$ J0@ E&q(

< 8gf q 6 B

< ;8gf q 6 B < !o 8gfr¤ q 6 JL

< ¡D c 4 q JL

|v) % H@ ! E@ % 5h$& ./$ ? )*"'a%#("', %'&$&6o 8 q r JL & .%

< 8gf q 6 B

< ¡D c ¡D b

24

à&v0@ G&q( *6o O rH Hh w 01&)*%'&7) <=3 ) O < P$ ¤ E5Kà& ./$ HN#(/$ 8| < P$ ¤ Eo M#0 evFG$& = ) $) #(h 1)*P% #, A %#(<=34#,$e d01) $&/$ 34 %5?$# 'K|v#6evFBà& .#(/$ .< pK ph$& ./$ E )*"'? d£ P¤ p¥jM%'&B$& <F G 5K1à& d3/$#v#H)*M%#(<=3 $ (K

- &/. U10+U # U ' 0 T T à& P/'#(< da evF6à& .#(/$ .< K pC)*a%. G 2 " K #H x @ ! Ea01) $&A/$ 34 %5M$#6D/$ +>@/$#01)*G<=#$) #( 0@ d .#$ J) MevF d 8 q 6 K 9;] > st E EPJ) <GJ) <+o rGJ)*) . .01)*%O,o 8(8r BD 1 G) ,$ .-, daevF

d 4 6 % ¤§dà& J 1 %#() $) #( 34 %5'$) #(#H\C/'#(< 1)*a evFD) 'a . %5$) #(/$ .3/$ .:'$) #( pK p(5KMà&)*?<= 1$&?#(/P ) $) #(AevFB6 . %5$) #(+/$ .3/$ .,'$) #( <F G a)*<=#(/$ E%#(v( .) .,@$#D3/$#( J#(<= J3/$#(34 ./$) @#H dd ) l( P$& E<D/$) -, P3/$#(34 ./"F(K#%$&!')(*&,+ T32T n ^M^_\ T 9k:0R > : @ ! EEr RS9 ? :0RS9;st99tuMv T : Tk>T 9 n 9 ? Z[9 8 p \ I xTon Z[Rk:0R > : I \ s)9 > Z[R # P 4 e;2 4 J 8 p > ?A@

d ; 4 6 a ~ 8 p ¤§ >ME>ME K 8q

Âst\g\ I;EPk#(/\ (%'& # h) % dB£ P¤ p¥c5h$& ./$ @ )*"'\P R, .% O 8 p | <F G %'&J$& d ;2 O @ 8 p K K J . g .<=<DE8(K 8c) 6o yqrN5Kj|v) %

<F G q 6 % ¤§;

< 8gf q 6 c 8 p % )*1 :s<= (/'e P3/$) $) #(G#H

% ¤§ HN#(/JvF 8 p 5h$& ./$ C )*"' O P % 8 p %'&+$&8 O ) O W (% W

O < 8gf q 6

01& ./$ c 8 p % )*J :s<= (/'e C3/$) $) #(#H G 8 p % | 5KAà& ./$ 6 )*"'dAe R, .% % #H 8 p %'&$& O *25

O W DK .K Kcà&v@HN#(/MK K h d ;¤O 4 @ # 8 # 8 ; 4 6 @ # 8 8B % # 8 §| ; 4 6 @ # 8 8B % # 8S§| d o

|v) % H @ ! E5h$& ./$ E )*"'1D R, .% C 8 p #H %'&2$&GHN#(/G (%'& # 4 ;2 4 m d 8 p K zI k8 p D # 8 pB% à& .B$&)*? R, .% E)*1$& C ) /$ #( (KIà& J3/$#v#H\)*M%#(<=3 $ (K

9;] > st E¡EPà& .#(/$ .< K yJ&#0Mj$& d?)*af@("') -E/$ .3/$ .,'e (h)K (K h($& ./$ M )*"'= R, .% O d = 8 p #H s:I<= (/'e P3/$#% $ a%'&$& d ;SKO 4 @ = 8 p HN#(/M # =BK K K#%$&!')(*&,+ TT*79;: . @ ! EE r RS9 ?I \ sd91 9;som # P z 9)R > 9 d)g P$ ¤ ! EoÂst\g\ I;EJ|v) % m d . %5$) #(B#H dB/$ s:s<= (/'e (h dG)*+ :s<= (/'e (K`#3/$#( d G)*B9%#(v( 2 +) tK K hP) A)*B 6%) .,B$#!&#0 $& <F G )*%#(v( ievF2fg#(/$#( */$F28(K m) o yqrsK O <F G K à& .!$& ./$ /$ B"0@# :s<= (/'e 3/$) $) #( c< c c h < t t #H O ! 8 p , + 8 p % | 8 q 6 % %'&$& O

< 8gf O X

< 8 ¥ Xgk#(/MvF*xBB 8(h0@ E&q( O h 8j

< 8gf O 8j

< 8 ¥ B =h 8j4

26

< 8gf O 8j

< 8 ¥

F GF G F < < G 8 O

h 8j 01& ./$ F

G c ! (KI|v) % F G ~ 8 p 8 p % )*a :s<= (/'e 3/$) $) #(#H O h 8 j E8B B 8B B % |X O h 8 j <F G KcwM .% <F G )*a%#(v( K\k\) FD0@ ?3/$#( 1$&gHN#(/@ .( ./$F # h,$& ?/'#(< d)*g5:se4#( Kbk/$#(<tà& .#(/$ .<tK yB$& ./$ )*"' R, .% b x 8 p #H a%'&G$& 4 ;2 4 @ ~ 8 p d ; 4 6 @ = 8 p § K K >@Fà& .#(/$ .<pK pC) mo yqrsh0@ J&q(

6o d r d ¡D 3

4 6 ¡D

3F F G G F F G G ¢\ q 6 ¡D

3F F G G LF F G G 4 6 ¡D

3F F G G LF F G G q 6 3F F G G LF F G G q 3¡ F 6 G ¢ q

>@FJ$& @H_(%5j$& H@ ! E\C1 .<D/$ldK 8(hq$& @/$) -(&,j&C)* a)*) $ (KjwM .% evF1 .<D/$lGpK h d@)*1 :se4#( KIà& E3/$#v#H\)*1%#(<=3 $ (K

9;] > st EE?; à& .#(/$ .< K 70@ m3/$#( d P$ ¤ ! E5Kà&)*+) <=3 ) B$& F G <F G K9;7H_(%5 ha($<= G$& F G <F G K!à& .7$& ./$ B )*"'D O <F G27

%'&$& O F G h)K (K*6o O rq ex6o O r !KG|v) % dxig P$ ¤ p¥c5h\evF1 .<D/$lGpK h d K+ P$ ¤ E5KIà&va0@ J&q( 6o O rB 3

J 6o r 6o 3¡ F G F G ¢ r 6o d r

à&)*1)*1%#(,$/'()*%5$) #(Kcà& ./$ HN#(/$ P0@ E%.G01/$) $ J$& F G g)K (K h 8 F 6G q 6 % § #%$&!')(*&,+ TT \ s > ? m % ! E 1 d D Z[\ ? :6v ?_n \ nKT z v:0RVÂst\ `1D>M` v a v:6mp\ ? 9 EÂst\g\ I;EP H 8 p e4 G R, .% 6-() ( .(C) 7à& .#(/$ .<K yKx?) -+$& R, ) "F pK 8q5h0@ E&q(

d d 4 6 3 4 6 K p(

HN#(/P KJ|v) % 8 q 6 C)*P%#(,$) v#(M01) $&A3/$#(ee) ) "FG#( EHN#(/J )g E5h$& ./$ D/$ =3/$#(ee) ) "FAO. ./$#B '

, 8 p J%'&$&C8 4 6 P)*E%#(:

$) v#(?HN#(/J KEzI Kdà& . ¡D GB$& =/$) -(&,P&A)* #H K p(

%#(v( ./$-( I$#8=( W CHN#(/M KIà& J3/$#v#Hj)*1%#(<=3 $ (K

`#E3/$#( @$& M<D/$) -, a3/$#(34 ./"FC#Hd I D"$#%'&("$)*%a) ,$ .-(/'h0@ 1 1$& 1HN#( #01) -%#() $) #(K

f1~ ?¢?"à$~ fa5Kà& P3/$#% $ %@ ! Eg)*@ 6evF8 U K .K h01& ./$ HU o E8 8r H+g E1$$)*" @$&M$& ./$ E )*"'MD%#("', # D%'&$&

K , q 6 h /# K .K

HN#(/M # S#%$&!')(*&,+ T_ST = T[Ton ])9x:0R > : T> :6v T ?9 T \ ?A@ v:6v\ ? - 5 E,r RS9 ? d v TC>T 9;: > a n 9 @ D ] > so:6v ?54> a 9 EÂst\g\ I;EP %+g E@e4 E-() ( .GevF ,aB E=<D/$) -, % devF

% q 6 K .K

28

% % h K .K KIà& .hevFBfg#() $) #( fa5h % d)*1#( .-,$) ( P<D/$) :-, (Kc>@FD$& d ) $) #(B#H d5h0@ E%.B . P$&

F G E q 6 h 4 4 UK K 4 bo 7 P U § ;2 O @ O 4 V % 4 h U K K 4 mo 7 F F G G E F G E

>@FBà& .#(/$ .< pK 8(h dh U % U K .K Kax?) -à& .#(/$ .< pK h d h U d)*M6 :s<D/$) -, (K k/$#(< $&)*10@ J&q(

o d 76 rSh U o dh U 76 r 6 dh U#+$& o d 76 r 6 dE)K (K h dG)* :s<D/$) -, (KAà& 3/$#v#HM)*%#(<=3 $ (K 9;] > st EMEJk#(/1vF %+g E5h0@ J%.6'l( %+g E@$$)*"HNFv) - K ,I) fg#() $) #( fa1%'&$& q 6 KB q 6 à&)*P<= M$& 8 q 6 )*J . %5$) #(#H 8 q U 6 ,Kd|v#G0@ %.$qF$&fg#() $) #( fa@)*1#M#D"$/$#(-K

11#0 ) "$/'$ P$& E/$ 'aevF<= a#Hj=) <=3 E <=3 (K&*+ & T_ST HBg Ea$$)*"HNF

* 3 8p q HN#(/1 (%'& # =G 0¢\ CevF

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fg /$ F 0¢\ D)*P#( .-,$) ( (K L evFAà& .#(/$ .< 5K yK ) ;l( Ab'e4 ,o rsh4$&)*)*M :s<D/$) -, (Kcx?) -=s # @HN#(/$<C*0@ E%.B . J$&

¢\ 8h ¢\ q q 6 ¢? CevF ¢\ JK .K K?zIU o E8 8r\b e U K .K K?à& . C C$$)*"HNF K ,g) fg#() $) #( fa@01) $& 8(KIà&vaevFà& .#(/$ .< K 8gh dd)*1= M :s<D/$) -, (K

29

#%$&!')(*&,+ T_S T879;: ) 8 @ ! E%T> :6v T0I m \ ?A@ v:6v\ ? - 5 E!r RS9 ? z 9R > 93 q 6q ) q 6 B7

q 6

) q 6 K v

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#%$&!')(*&,+ T_ST879;: ) % @ ! EE)r RS9 ? v:iRS\ a @¤T q 6 ) q 6 BX q ) q Âst\g\ I;EP>@F9à& .#(/$ .< K yhg$& ./$ A )*"%#(,'e A R, .% ! p 8 p " 8 p ) P ) 5h/$ 34 %5$) ( . F+%'&A$&PHN#(/JvF # h 4 ;2 4 @ = 8 p , ) 4 ;2 " 4 @ 8 p d 4 6 ; 4 6 @ ~ 8 p ¤§

)d ) 4 6 ; " 4 6 a 8 p ¤§K K

c (3 ) H 4 k" 4 6 # 3 ) H 4 ~k" 4 6 §

x?) -à& .#(/$ .<pK pB) 2o yqra"0@#Ae()*%=3/$#(34 ./$) d<D < dh h < b =, \8gfih48gfQj¤,\8gfQj¤,48gfDHN#(//'#(< /$)*e " P!c , # h30

0@ J&q(

q 6 ) q 6 q 6q ) q 6 ¡ <D~ 3¡ F G F G ) H£¤ F + G F G ¢ ¦ 3£¤ F + G F G ) H¡ F G F G ¢ ¦ § ¡ 8gf 3¡ F G F G ) H£¤ F + G F G ¢ ¦ ¡h

8gf j 3£¤ F + G F G ) H¡ F G F G ¢ ¦ ¡

8gf 3 ) H q~k" q 6 ¡h 8gf j 3 ) H q~k" q 6 ¡ 8

p 3 ) H q~k" q 6 ¡h 8p 3 ) H q ." q 6 ¡h 8

p 8gf 3 ) H q~k" q 6 ¡ 8p 8gf j 3 ) H q~k" q 6 ¡ 8

p 8gf 3 ) H q~k" q 6 ¡h 8p 8gf j 3 ) H q~k" q 6 ¡ 8

p 3 ) H q~k" q 6 ¡h 8p 3 ) H q ." q 6 ¡h 8

p 3 ) H q~k" q 6 ¡3 ) H q~k" q 6 ¡ 8

p 3F F G G LF F G G ) HF F G G LF + F G G q~k" q 6 ¡

31

h 8p 3F F G G LF + F G G ) HF F G G LF F G G q~k" q 6 ¡h 8

p 3F F G G LF F G G ) HF F G G LF + F G G q~k" q 6 ¡ 3

F F G G LF + F G G ) HF F G G LF F G G q~k" q 6 ¡ 8p 3F F G G LF F G G ) HF F G G LF + F G G q~k" q ¡h 8p 3F F G G LF + F G G ) HF F G G LF F G G q~k" q ¡h 8

p 3F F G G LF F G G ) HF F G G LF + F G G q k" q ¡ 3

F F G G LF + F G G ) HF F G G LF F G q k" q ¡ 8p 3¡ F 6 G ) H£¤ + F 6 G ¢ ¦ ¡h 8p 3£¤ + F 6 G ) H¡ F 6 G ¢ ¦ ¡h 8

p 3¡ F 6 G ) H£¤ + F 6 G ¢ ¦ 3£¤ + F 6 G ) H¡ F 6 G ¢ ¦ § ¡ B 8p 3¡ F 6 G ) H£¤ + F 6 G ¢ ¦ ¡h 8p 3£¤ + F 6 G ) H¡ F 6 G ¢ ¦ ¡h 8

p 3¡ F 6 G ) H£¤ + F 6 G ¢ ¦ 3£¤ + F 6 G ) H¡ F 6 G ¢ ¦ ¡ <D~ 3¡ F 6 G ) H£¤ + F 6 G ¢ ¦ 3£¤ + F 6 G ) H¡ F 6 G ¢ ¦ § ¡ q ) q

à& E3/$#v#H\)*1%#(<=3 $ (K

9;] > st E 1 EE|v334#, $& ) @ ! EG$$)*"HNF fg#() $) #( fa5KM>@F

32

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)g$& ./$ J )*"'MG8:;) <= .) #(>@/$#01)*<=#$) #( d01) $& , =K .K h ) ) M)* s:;(3$ =D%#(,$) v#(c) K .Kj)K (K h,01) $&D3/$#(ee) ) "FC#( (h h4 W D( W h ) ) ) d d$$)*"HNF(K

Vh q h ) q 6 K .K

33

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z RS9;st9 U o E8 8r4Er RS9 ? :0RS9 T 9;:/ > a n 9 @k@ v 9;st9 ? :6v > a 9 n_> :6v\ ? - E ;5dR >¤Tp>n ? v n 9 T \ a n :6v\ ? E st\g\ I;E Xde4 JvF-() ( .Kc¢? PK .K(HN#(/a mo rsKjà& .6evF K v5hv0@ &q(

) ¢ g B 8h6 K B& .% ) qx @ ! E5K1>@F6$& ) $) #(+#Hj$& C ? B) ,$ .-(/'h0@ d%. ) q 66$&vM=%#(,$) v#(a3/$#% $

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35

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3 < o r o (r o

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36

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o ¢ <'r o ¢ r K .K

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40

Probability theory in spaces of fuzzy sets

VOLKER KRÄTSCHMER

Statistik und Ökonometrie, Rechts- und Wirtschaftswissenschaftliche FakultätUniversität des Saarlandes

D-66041 SaarbrückenGermany


1 Introduction

Statistical data are products of measuring predetermined items with respect to objects which are sam-ple points from fixed sample spaces. According to economic items each process of measurementis confronted with several types of vagueness: Stochastical vagueness by selecting objects from thesample space, epistemic (perceptive) vagueness since sample spaces may be specified unproperly,or incorrect results like false responses have been reported. Last but not least, theoretical economicconcepts have to be transformed into items to enable the process of measurement, which raises thequestion of adequacy due to the physical vagueness of many theoretical concepts.

As a consequence data inherit errors in measurement corresponding to the respective types ofvagueness: Random errors (stochastical vagueness), sampling errors (stochastical and epistemicalvagueness) and adequacy errors (physical vagueness). They can be defined within the terminology ofMathematical Statistics reformulating the measurement of items as parameter estimation. However,within the framework of Mathematical Statistics only the random errors are considered systematically.

In order to integrate different types of vagueness it has been proposed to represent outcomesof measurements by fuzzy subsets rather than vectors of real numbers. A desirable foundation ofMathematical Statistics with fuzzy observations needs then a suitable linkage of probability withfuzzy set theory.

Technically we shall investigate mathematical structures on fuzzy sample spacesX which are con-tained in the space of fuzzy subsets ofRk. We shall point out compatible pairs(d,F ) of a metricdto describe the random errors and aσ−algebraF to extend the methods of Mathematical Statistics.Theσ−algebraF should be generated by the metricd. So only Borel-σ−algebras on already knownmetrizable topological spaces(X,τ) are considered and compared. The discussion leads to some uni-fied concept of fuzzy-valued random variables called fuzzy random variables. A Strong Law of LargeNumbers, a Central Limit theorem and a Glivenko-Cantelli theorem for fuzzy random variables willbe formulated simultaneously with respect to some emphasized metrics. All of them are compati-ble with theσ−algebra which underlies the notion of fuzzy random variables. This gives reasonablelegitimation to use these metrics and the induced concept of fuzzy random variables.

41

2 Lp−spaces of fuzzy sets

In order to keep the line of reasoning comprehensive we shall restrict ourselves toFnococ(Rk), the

space of normal fuzzy subsets ofRk with convex, compact positiveα−cuts. Since every fuzzyset fromFno

coc(Rk) is uniquely determined by its support function, we can build in an obvious wayLp−subspaces ofFno

coc(Rk) with respect to the product measureλ1⊗ λSk−1of the Lebesgue-Borel

measureλ1 on [0,1] and the unit Lebesgue-Borel measureλSk−1on the euclidean unit sphereSk−1

in Rk. The spaceFnococp(Rk) (p ∈ [1,∞[) consists of allA ∈ Fno

cocp(Rk) with support functions being

λ1⊗ λSk−1−integrable of orderp, whereasFnococ∞(Rk) contains the fuzzy sets fromFno

coc(Rk) havingsupport functions which are essentially bounded with respect toλ1⊗λSk−1

. It can be shown that thefuzzy sets fromFno

coc∞(Rk) are exactly those fuzzy sets fromFnococ(Rk) with bounded support. Further-

more it turns out thatFnococ∞(Rk) is a dense subset of everyLp−spaceFno

cocp(Rk) with respect to therespectiveLp−metricρp.

3 Lp−Metrics

Identifying eachA ∈ Fnococp(Rk) with the λ1⊗ λSk−1−equivalence class of its support function, the

Lp−spaceFnococp(Rk) can be embedded into theLp−spaceLp([0,1]×Sk−1) generated byλ1⊗λSk−1

.

Then theLp−norm onLp([0,1]×Sk−1) induces the so calledLp−metric ρp on Fnococp(Rk). Some

fundamental properties of theLp−metrics will be collected. Especially, they are complete, and theyare also separable exceptρ∞.

4 Other metrics on theLp−subspaces ofFnococ(Rk)

Extending, dependent onp∈ [1,∞], the well knownLp,∞−metrics fromFnococ(Rk) to Fno

cocp(Rk) respec-tively, we shall introduce another class of metrics. They are all complete and induce, dependent onp,the same topology onFno

cocp(Rk) as the respectiveLp−metricρp.

Also the so calledSkhorod metric onFnococ(Rk) will be mentioned which induces a separable and

completely metrizable topology.

5 σ−algebras on spaces of fuzzy sets

We shall investigateσ−algebras on the sample spacesFnococ(Rk) and theLp−subspacesFno

cocp(Rk)(p ∈ [1,∞]). As a key concept aσ−algebra onFno

coc(Rk) will be suggested which covers the dif-ferent σ−algebras on theLp−spacesFno

cocp(Rk) generated by the introduced metrics. So we candefine aFno

coc(Rk)−valued random variable, calledfuzzy random variable, in a way which unifiesalready known alternative concepts. Particularly, theLp−metrics are compatible with the notion ofFno

cocp(Rk)−valued fuzzy random variable forp∈ [1,∞[.

42

6 Aumann expected value of fuzzy random variables

SinceFnococp(Rk)−valued fuzzy random variables can be identified asLp([0,1]×Sk−1)−valued random

variables forp∈ [1,∞[, limit theorems may be obtained by limit theorems for Banach-space-valuedrandom variables. For preparation this section provides conditions of integrability which ensures thatthe Aumann expected value of someFno

cocp(Rk)−valued fuzzy random variable can be identified withthe Pettis integral or even the Bochner integral.

7 Limit theorems for fuzzy random variables

Since there exists a general Glivenko/Cantelli-Theorem for separable metric random elements, onlyStrong Laws of Large Numbers (SLLN) and Central Limit Theorems (CLT) for fuzzy random vari-ables are of interest. Several versions of a SLLN with respect to some of the introduced metrics arealready known. Also different versions of CLT have been formulated. What is missing is a SLLN anda CLT with respect to the same separable metric, which ensures a Glivenko/Cantelli-Theorem too.

Applying limit theorems for random elements in separable Banach spaces of type 2, we shallformulate a SLLN and a CLT simultaneously with respect to eachLp−metricρp for p∈ [2,∞[.

8 Concluding Remarks

Summarizing the results, forp ∈ [2,∞[ Fnococp(Rk)−valued fuzzy random variables together with the

Lp−metricρp fulfill all the required properties such that probability theory in spaces of fuzzy subsetsof Rk may be regarded as well-founded. Since the limit theorems need the weakest assumptions in thecase ofp = 2 we would like to recommend the usage ofFno

coc2(Rk)−valued fuzzy random variablesand theL2−metricρ2.

For practicable statistical work on basis of fuzzy observations a further investigation of distribu-tions of fuzzy random variable is desirable. It might be useful to find standard classes and to developgeneral methods of Monte-Carlo simulation.

43

Convergence theorems for fuzzy valued random variables in theextended Hausdorff metricH∞

SHOUMEI L I1, YUKIO OGURA2

1Department of Applied MathematicsBeijing Polytechnic University

China

2Department of MathematicsSaga University

Japan

Embedding method is a typical method to discuss limit theorems of sequences of set valued and fuzzyvalued random variables. For the set valued case, for examples, Artstein and Vitale above mentionedused an embedding method to prove SLLN Hiai and Umegaki used the embedding method to obtainmartingale convergence theorems in 1975. For the fuzzy valued case, Klement, Puri and Ralescuobtained embedding theorems to prove SLLN in the sense of Extended Hausdorff MetricH1 and CLTwith respect to Extended Hausdorff MetricH∞ with Lipschitz condition in 1986. Puri and Ralescu alsoobtained a martingales convergence theorem inH∞ also with Lipschitz condition in 1991. Diamondand Kloden pointed out that all space of fuzzy sets with compact supports can be embedded intoanother Banach space in 1989 but they did not discuss isometric property with respect toH∞, whichis not easy to use for the proof of convergence theorems.

In this paper, we mainly give a new embedding method for fuzzy valued random variables withsame expectation and use it to prove a SLLN for fuzzy valued martingales in the sense ofH∞. Thenwe prove the CLT by using this new embedding theorem and the result on empirical process of Vander Vaart and Wellner. We would like to point that using the result on empirical process to prove CLTis Proske’s idea but we had some troubles on some inequalities of his proof. Here we rewritten theproof about CLT for fuzzy valued random variables. Throughout this paper we assume that fuzzy setsare not necessary to satisfy the Lipschitz condition.

44

On some analytical properties of aggregation operators

RADKO MESIAR1, ANNA KOLES’ AROV’ A2

1Department of Mathematics, Faculty of Civil EngineeringSlovak University of Technology



2Department of Mathematics, Faculty of Chemical and Food TechnologySlovak University of Technology



Binary non–decreasing operators on[a,b] ⊂ R with idempotent boundary pointsa, b will be simplycalled(binary) aggregation operators[2]. Without loss of generality we will suppose[a,b] = [0,1].During the study of sensitivity of aggregation operators [3] and some construction methods for ag-gregation operators [5], compare also [8], the following two analytical properties played a crucialrole:

(i) Lipschitz property

|A(x1,y1)−A(x2,y2)| ≤ |x1−x2|+ |y1−y2|, ∀x1,x2,y1,y2 ∈ [0,1],

(ii) Kernel property

|A(x1,y1)−A(x2,y2)| ≤max(|x1−x2|, |y1−y2|) , ∀x1,x2,y1,y2 ∈ [0,1].

Evidently, both these properties are a kind of continuity, and

kernel property⇒ 1–Lipschitz property⇒ continuity

but

kernel property: 1–Lipschitz property: continuity

Denote byC , L andK the classes of all continuous, 1–Lipschitz and kernel aggregation operators,respectively. All three classes are convex and closed under duality

Ad(x,y) = 1−A(1−x,1−y).

Moreover, the classesC andK are closed under compositionD = A(B,C). The classL does notpossess this property. However, for anyA ∈K andB, C ∈ L alsoD = A(B,C) ∈ L .

We list some other interesting properties of mentioned classes of aggregation operators:

45

(1) (i) TL ≤A ≤SL for all A ∈L (TL, SL - theLukasiewiczt–norm andt–conorm, respectively)

(ii) TM ≤ A ≤ SM for all A ∈K (TM = min, SM = max)

(2) (i) 0 (1) is the zero element ofA ∈ L if and only if 1 (0) is the neutral element ofA

(ii) 0 (1) is the neutral element ofA ∈K if and only if A = TM (A = SM)

(3) Any aggregation operatorA ∈K is idempotent, that is,A(x,x) = x for all x∈ [0,1].

The classK contains all Choquet and Sugeno integral based aggregation operators, including allweighted means and OWA operators. The classL ⊃K contains also all 2–copulas (dual copulas) [9].

A simple characterization of 1–Lipschitz aggregation operators is the following one:

An aggregation operatorA ∈ L if and only if the functionA : [0,1]2 → [0,1] satisfying for all(x,y) ∈[0,1]2 the equation

A(x,y)+ A(x,y) = x+y

is an aggregation operator.

Note that the previous equation is an analogue of the Frank functional equation [4] in the framework ofaggregation operators. By this characterization several other problems can be solved, for instance, dueto the non–continuity of uninorms, it is immediate that no uninorm can be solution to this equation,compare [1].

Evidently, ifA ∈L then alsoA ∈L . For allA ∈K we haveA ∈L , A is idempotent. However,A ∈Kif and only if A (and hence alsoA) is a shift invariant aggregation operator.

A crucial role in the structure of kernel aggregation operators plays the boundary of the unit square,B = [0,1]2\]0,1[2. Some of them can be uniquely determined >from their values on the boundaryB,compare [7]. In general, ifA ∈K then for allx∈ [0,1]

A(0,x)≤ x≤ A(x,1), A(0,x)+1−x≥ A(1−x,1),

A(x,0)≤ x≤ A(1,x), A(x,0)+1−x≥ A(1,1−x).

Vice versa, if the values of a kernel aggregation operatorA are known on the boundaryB, then forall x∈ [0,1] andε ∈ [0,1−x] we have

A∗(x,x+ ε)≤ A(x,x+ ε)≤ A∗(x,x+ ε),

where

A∗(x,x+ ε) = max(A(0,x+ ε),A(1− ε,1)− (1−x− ε)) ,

A∗(x,x+ ε) = min(A(x,1),A(0,ε)+x)

are kernel aggregation operators [6]. Analogous inequalities can be written forA∗(x+ ε,x) andA∗(x+ ε,x). The operatorsA∗ andA∗ are the weakest and the strongest kernel aggregation opera-tors respectively, coinciding withA on the boundaryB.

Acknowledgement

The investigation was supported by the grants VEGA 1/7076/20, 1/7146/20 and Action COST 274“TARSKI."

46

References

[1] T. Calvo, B. DeBaets, J. Fodor: The functional equations of Frank and Alsina for uninorms andnullnorms.Fuzzy Sets and Systems120(2001) 15–24.

[2] T. Calvo, A. Koles’arov’a, M. Komorn’ıkov’a, R. Mesiar: A review of aggregation operators.Int.Summer School on Aggregation Operators and Their Applications, AGOP’2001, July 10-13,2001, Oviedo, Spain.

[3] T. Calvo, R. Mesiar: Stability of aggegation operators.Proceedings of EUSFLAT’2001, Leices-ter, 2001, pp. 475–478.

[4] M. Frank: On the simultaneous associativity ofF(x,y) andx+y−F(x,y). Aeq. Math.19 (1979)194–226.

[5] A. Koles’arov’a, J. Mordelov’a: 1–Lipschitz and kernel aggregation operators.ProceedingsAGOP’ 2001, Oviedo, Spain, 2001, pp. 71–76.

[6] A. Koles’arov’a, J. Mordelov’a, E. Muel: Binary kernel aggregation operators and their marginals,submitted.

[7] J. L’azaro, T. Rückschlossov’a: Shift invariant binary aggregation operatorsProceedings AGOP’2001, Oviedo, Spain, 2001, pp. 77–80.

[8] J. Mordelov’a, E. Muel: Kernel aggregation operators.Proceedings AGOP’ 2001, Oviedo, Spain,2001, pp. 95–98.

[9] R.B. Nelsen:An Introduction to Copulas. Lecture Notes in Statistic 139, Springer, 1999.

47

Fixed point theory in probabilistic metric spaces

ENDRE PAP, OLGA HADŽI’ C

Institute of MathematicsUniversity of Novi SadYU-21 000 Novi Sad

Yugoslavia

E-mail: [email protected], [email protected]

1 Introduction

We present some recent results on t-norms which are closely related to some generalizations of thefixed point theorems in probabilistic metric spaces, obtained in [3]. We investigate closer the countableextension of t-norms and we introduce a new notion: the geometrically convergent (g-convergent) t-norm, which is closely related to the fixed point property. We prove that t-norms ofH-type and somesubclasses of Dombi, Acz’el-Alsina, Sugeno-Weber families of t-norms are geometrically convergent.We prove also some practical criterion for the geometrically convergent t-norms.

A new approach to the fixed point theory in probabilistic metric spaces is given in Tardiff’s paper[13], where some additional growth conditions for the mappingF : S×S→ D+ are assumed, andT ≥ TL , whereTL (x,y) = max(x+y−1,0). V. Radu [10] introduced a stronger growth condition forF than in Tardiff’s paper (under the conditionT ≥ TL ), which enables him to define a metric. Themetric approach also allows an estimation of the convergence with respect to the solution.

We present a fixed point theorem for a probabilisticq-contractionf : S→ S, where(S,F ,T) is acomplete Menger space,F satisfies Radu’s condition, andT is ag-convergent t-norm (not necessarilyT ≥ TL ). We prove in [3] a fixed point theorem for mappingsf : S→ S, where(S,F ,T) is a completeMenger space,F satisfy a weaker condition than in [10], andT belongs to some subclasses of Dombi,Acz’el-Alsina, Sugeno-Weber families of t-norms. We present a fixed point theorem for a generalizedprobabilisticC-contraction for multivalued mapping with an application in fuzzy metric space.

Notions and notations can be found in [3, 6, 9, 11].

2 Probabilistic B-contraction principles for single-valued mappings

A triangular norm (t-norm for short) is a binary operation on the unit interval[0,1], i.e., a functionT : [0,1]2 → [0,1] such that it is commutative, associative, monotone andT(x,1) = x. t-conormS isdefined byS(x,y) = 1−T(1−x,1−y).

Let ∆+ be the set of all distribution functionsF such thatF(0) = 0 (F is a nondecreasing, leftcontinuous mapping from[0,∞] into [0,1] with F(∞) = 1).

48

Definition 1. (a) A probabilistic metric space (in the sense of Šerstnev) is a triple(S,F ,τ) whereSis a nonempty set,F : S×S→ ∆+ is given by(p,q) 7→ Fp,q, τ is a triangle function, such that thefollowing conditions are satisfied for allp,q, r in S:

(i) Fp,p = H0; (ii) Fp,q 6= H0 for p 6= q; (iii) Fp,q = Fq,p; (iv) Fp,r ≥ τ(Fp,q,Fq,r).

(b) (S,F ,τ) is proper ifτ(Ha,Hb)≥ Ha+b (a,b∈ [0,∞)), where fora∈ [0,∞] we haveHa(x) = 0for x∈ [0,a] andHa(x) = 1 for x∈ (a,∞].

A Menger spaceis a triple(S,F ,T), where(S,F ) is a probabilistic metric space,T is a t-normand the following inequality holds

Fu,v(x+y)≥ T(Fu,w(x),Fw,v(y))

for everyu,v,w∈ Sand everyx > 0,y > 0.

We assume in the whole paper for the probabilistic metric spaces(S,F ,τ) that Range(F )⊂ D+,whereD+ = F | F ∈ ∆+, lim

x→∞F(x) = 1.

The first fixed point theorem in probabilistic metric spaces was proved by Sehgal and Bharucha-Reid (1972) for mappingsf : S→ S, where(S,F ,TM ) is a Menger space, andTM = min.

Definition 2. Let (S,F ,τ) be a probabilistic metric space. A mappingf : S→ S is a probabilisticq-contraction(q∈ (0,1)) if

Ff p1, f p2(x)≥ Fp1,p2

(xq

)for everyp1, p2 ∈ Sand everyx∈ R.

Theorem 3. Let (S,F ,TM ) be a complete Menger space and f: S→ S a probabilistic q-contraction.Then there exists a unique fixed point x of the mapping f and x= lim

n→∞f np for every p∈ S.

Further development of the fixed point theory in a more general Menger space(S,F ,T) wasconnected with investigations of the structure of the t-normT. Very soon the problem was in somesense completely solved. Namely, if we restrict ourselves to complete Menger spaces(S,F ,T), whereT is a continuous t-norm, then any probabilisticq-contractionf : S→ Shas a fixed point if and onlyif the t-norm is ofH-type.

If T is a t-norm,x∈ [0,1] then we writex(0)T = 1 and forn∈ N,x(n)

T = T(

x(n−1)T ,x

).

Definition 4. A t-normT is of H-type if the family(x(n)T )n∈N is equicontinuous at the pointx = 1.

O. Hadži’c and M. Budincevi’c (1978) proved the following fixed point theorem.

Theorem 5. Let (S,F ,T) be a complete Menger space, T a t-norm of H-type and f: S→ S a proba-bilistic q-contraction. Then there exists a unique fixed point x∈ S of the mapping f and x= lim

n→∞f np

for every p∈ S.

Since the t-normTM is of H-type, the fixed point theorem of Sehgal and Bharucha-Reid followsfrom Theorem 5.

V. Radu (1994) proved the following fixed point theorem.

49

Theorem 6. Any continuous t-norm T with the fixed point property is of H-type, where a t-norm Thas the fixed point property if and only if every probabilistic q-contraction f: S→ S, where(S,F ,T)is an arbitrary complete Menger space, has a fixed point.

A very interesting new approach to the fixed point theory in probabilistic metric spaces is givenin Tardiff’s paper (1992), where some additional growth conditions for the mappingF : S×S→D+

are assumed, andT ≥ TL (implies supx<1T(x,x) = 1 and then also continuity at(1,1)).

Theorem 7. Let (S,F ,T) be a complete Menger space and T a t-norm such that T≥ TL . If for everyu,v∈ S

∞∫1

ln(x)dFu,v(x) < ∞

holds, then any probabilistic q-contraction f: S→ S has a unique fixed point x and x= limn→∞

f np for

every p∈ S.

E. Parau and V. Radu (2001) proved the following theorem.

Theorem 8. If (S,F ,T) is a complete Menger space such that T≥ TL , then a probabilistic q-contraction f : S→ S has a fixed point if and only if there exist k> 0 and p∈ S such that

∞∫0

xk dFp, f p(x) < ∞.

3 Countable extension of t-norms

We investigate closer the countable extension of t-norms ([6], for other types of extensions see [8])and we introduce a new notion: the geometrically convergent t-norm, which is closely related to thefixed point property.

We prove in [3] that t-norms ofH-type and some subclasses of Dombi, Acz’el-Alsina, Sugeno-Weber families of t-norms are geometrically convergent. We obtained in [3] also some practicalcriterion for the geometrically convergent t-norms.

In the fixed point theory it is of interest to investigate the classes of t-normsT and sequences(xn)n∈N from the interval[0,1] such that lim

n→∞xn = 1, and

limn→∞

∞Ti=n

xi = limn→∞

∞Ti=1

xn+i = 1. (1)

It is of special interest for the fixed point theory in probabilistic metric spaces to investigate con-dition (1) for a special sequence(1−qn)n∈N for q∈ (0,1).

Proposition 9. If for a t-norm T there exists q0 ∈ (0,1) such that

limn→∞

∞Ti=n

(1−qi0) = 1, then lim

n→∞

∞Ti=n

(1−qi) = 1, for every q∈ (0,1).

50

Definition 10. We say that a t-normT is geometrically convergent (brieflyg-convergent, called alsoq-convergent for someq∈ (0,1)) if

limn→∞

∞Ti=n

(1−qi) = 1.

for every q∈ (0,1).

Since limn→∞

(1−qn) = 1 and∞∑

n=1(1− (1−qn))s < ∞ for everys> 0 it follows that all t-norms from

the familyT0 =

⋃λ∈(0,∞)

TDλ

⋃ ⋃λ∈(0,∞)

TAAλ

⋃T H

⋃λ∈(−1,∞]

TSWλ

areg-convergent, whereT H is the class of all t-norms ofH-type, and for other families see [6].

The following example shows that not every strict t-norm isg-convergent. LetT be the strict t-

norm with an additive generatort(x) =− 1log(1−x) . In this case the series

∞∑

i=1t(1−qi) for anyq∈ (0,1)

is not convergent.

4 Fixed point theorems related tog-convergent t-norms

We proved a fixed point theorem for a probabilisticq-contraction f : S→ S, where(S,F ,T) is acomplete Menger space,F satisfies Radu’s condition, andT is ag-convergent t-norm (not necessarilythe conditionT ≥ TL ). The metric approach allows an estimation of the convergence with respect tothe solution.

Theorem 11. Let (S,F ,T) be a complete Menger space and f: S→ S a probabilistic q-contractionsuch that for some p∈ S and k> 0

supx>0

xk(1−Fp, f p(x)) < ∞.

If t-norm T is g-convergent, then there exists a unique fixed point z of the mapping f and z= limn→∞

f np.

There are many corrollaries of the preceding theorem related the familyT0 of t-norms, see [3].Special non-additive measures, so calledS-decomposable measures (see [9]), generate a probabilisticmetric space in which Theorem 11 implies a specific fixed point theorem, see [3].

5 Multivalued contraction and an application in a fuzzy metric space

T. Hicks (1983) considered another notion of probabilistic contraction mapping than probabilisticq-contraction, which is incomparable with probabilisticq-contraction.

Definition 12. Let (S,F ,τ) be a probabilistic metric space andf : S→ S. The mappingf is a proba-bilistic C-contraction if there existsk∈ (0,1) such that for everyp,q∈ Sand for everyt > 0

Fp,q(t) > 1− t ⇒ Ff p, f q(kt) > 1−kt.

51

If f : S→ S is a probabilisticC-contraction and(S,F ,TM ) is a complete Menger space Hicksproved thatf has a unique fixed point.

As a multivalued generalization of the notion of a probabilisticC-contraction we introduced thenotion of a(Ψ,C)-contraction, whereΨ : R+ → R+, for R+ = [0,∞).

Let P (S) be the family of all nonempty subsets ofS.

Definition 13. Let (X,F ,τ) be a probabilistic metric space andf : S→ P (S). The mappingf iscalled a(Ψ,C)-contraction, whereΨ : R+ → R+, if for every p,q∈ S and everyx > 0 we have thefollowing implication:

Fp,q(x) > 1−x⇒ f or every u∈ f p there exists v∈ f q

such that Fu,v(Ψ(x)) > 1−Ψ(x).

If Ψ(x) = kx, x > 0, k∈ (0,1) then a(Ψ,C)-contractionf : S→ S is aC-contraction. LetP (S)cl

be the family of all nonempty closed subsets ofS.

Theorem 14. Let(S,F ,T) be a complete Menger space such thatsupx<1T(x,x) = 1, M ∈ P (S)cl, f :M → P (M)cl a (Ψ,C)-contraction, where the series∑∞

n=1 Ψn(s) is convergent for some s> 1. If f isweakly demicompact or

limn→∞

∞Ti=1

(1−Ψn+i−1(s)) = 1,

then there exists at least one element x∈M such that x∈ f x.

For some families of t-norms it can be obtained special fixed point theorems, see [3].

As a consequence of Theorem 14 we shall obtain a result on the existence of a fixed point for aclass of multivalued mappings in fuzzy metric spaces.

In [5] Kaleva and Seikkala introduced the notion of a fuzzy metric space. LetL,R : [0,1]×[0,1] → [0,1] be symmetric, nondecreasing in both arguments such thatL(0,0) = 0, R(1,1) = 1, Gthe set of all nonnegative upper semicontinuous, normal convex fuzzy numbers,X a nonempty set,d : X×X→G and0 be the fuzzy number defined by0(u) = 1 for u= 0 and0(u) = 0 for u 6= 0. Denote[d(x,y)]α = [λα(x,y),ρα(x,y)] (x,y∈ X, α ∈ [0,1]). The quadruple(X,d,L,R) is called afuzzy metricspaceif and only if (i)-(iii) hold, where

(i) d(x,y) = 0 if and only ifx = y.

(ii) d(x,y) = d(y,x), for all x,y∈ X.

(iii) d(x,y)(s+u)≥ L(d(x,z)(s),d(z,y)(u)), for all x,y,z∈ X,whenever

s≤ λ1(x,z), u≤ λ1(z,y), s+u≤ λ1(x,y)

andd(x,y)(s+u)≤ R(d(x,z)(s),d(z,y)(u)), for all x,y,z∈ X,whenever

s≥ λ1(x,z), u≥ λ1(z,y), s+u≥ λ1(x,y).

52

We shall suppose thatR is associative and satisfies the conditionR(a,0) = a, for everya∈ [0,1].This implies that the mappingT(a,b) = 1−R(1−a,1−b) (a,b∈ [0,1]) is at-norm.

Every Menger space(X,F ,T) is a fuzzy metric space(X,d,L,R) if

d(x,y)(u) =

0, u < sups; Fx,y(s) = 0= ux,y

1−Fx,y(u), u≥ ux,y,

and the functionsL andR are defined byL≡ 0 andR(a,b) = 1−T(1−a,1−b) (a,b∈ [0,1]), i.e.,R is the dual t-conorm forT.

The converse statement does not hold in general. But under an additional condition ond a fuzzymetric space(X,d,L,R) is a Menger space. This condition is the following: for allx,y∈ X

limu→∞

d(x,y)(u) = 0.

Then(X,F ,T) is a Menger space, whereT(a,b) = 1−R(1−a,1−b), for everya,b∈ [0,1] and themappingF is defined by(x,y∈ X, s∈ R)

Fx,y(s) = 0, for s≤ λ1(x,y), Fx,y(s) = 1−d(x,y)(s), for s≥ λ1(x,y).

If a fuzzy metric space(X,d,L,R) is given, then we call the above defined Menger space(X,F ,T)the associated Menger space.

Theorem 15. Let (X,d,L,R) be a complete fuzzy metric space such that

limu→∞

d(x,y)(u) = 0 for all x,y∈ X,

lima→0+

R(a,a) = 0 andΨ : R+ → R+ such that the series∑∞n=1 Ψn(u) is convergent for some u> 1. Let

f : X → P (X)cl be such a mapping that the following implication holds:

For every x,y∈ X and every u∈ f x there exists v∈ f y such that for every s> 0 d(x,y)(s) < s⇒1−Fu,v(Ψ(s)) < Ψ(s). If f is weakly demicompact orlim

n→∞R ∞i=n Ψi(s) = 0, then there exists x∈ X

such that x∈ f x.

For the proof see [3].

References

[1] O. Hadži’c, E. Pap (2000). On some classes of t-norms important in the fixed point theory,Bull. Acad. Serbe Sci. Art. Sci. Math.25, 15–28.

[2] O. Hadži’c, E. Pap (in print). A fixed point theorem for multivalued mappings in probabilisticmetric spaces and an application in fuzzy metric spaces,Fuzzy Sets and Systems.

[3] O. Hadži’c, E. Pap (2001)Fixed Point Theory in Probabilistic Metric Spaces. Kluwer Aca-demic Publishers, Dordrecht.

[4] T. L. Hicks (1983). Fixed point theory in probabilistic metric spaces,Univ. u Novom Sadu, Zb.Rad. Prirod.-Mat. Fak. Ser. Mat.13, 63–72.

53

[5] O. Kaleva, S. Seikalla(1984). On fuzzy metric spaces,Fuzzy Sets and Systems12, 215–229.

[6] E. P. Klement, R. Mesiar, E. Pap(2000).Triangular Norms, Kluwer Academic Publishers,Trends in Logic 8, Dordrecht.

[7] K. Menger (1942). Statistical metric,Proc. Nat. Acad. USA28, 535–537.

[8] R. Mesiar, H. Thiele (2000). OnT-quantifiers andS-quantifiers, in: V.Novak, I.Perfilieva,eds.,Discovering the World with Fuzzy Logic, Studies in Fuzziness and Soft Computing vol.57, Physica-Verlag, Heidelberg, 310–326.

[9] E. Pap (1995).Null-Additive Set Functions, Kluwer Academic Publishers, Dordrecht and IsterScience, Bratislava.

[10] V. Radu (1994).Lectures on probabilistic analysis, Surveys, Lectures Notes and MonographsSeries on Probability, Statistics & Applied Mathematics, No 2, Universitatea de Vest din Tim-isoara.

[11] B. Schweizer, A. Sklar(1983).Probabilistic Metric Spaces, Elsevier North - Holland, NewYork.

[12] V. M. Sehgal, A. T. Bharucha-Reid(1972). Fixed points of contraction mappings on proba-bilistic metric spaces,Math. Syst. Theory6, 97–102.

[13] R. M. Tardiff (1992). Contraction maps on probabilistic metric spaces,J. Math. Anal. Appl.165, 517–523.

54

T-transitivity and domination

SUSANNE SAMINGER1, RADKO MESIAR2, ULRICH BODENHOFER3

1Fuzzy Logic Laboratorium Linz-HagenbergDepartment of Algebra, Stochastics, and Knowledge-Based Mathematical Systems

Johannes Kepler UniversityA-4040 Linz, Austria


2Department of Mathematics, Faculty of Civil EngineeringSlovak University of Technology



3Software Competence Center HagenbergA-4232 Hagenberg

Austria


1 Introduction

Aggregation is a fundamental process in decision making and in any other discipline where the fusionof different pieces of information is of vital interest, e.g. in fuzzy querying.

Flexible (fuzzy) querying systems are usually designed not just to give results that match a queryexactly, but to give a list of possible answers ranked by their closeness to the query—which is partic-ularly beneficial if no record in the database matches the query in an exact way [14]. The closenessof a single value of a record to the respective value in the query is usually measured using a fuzzyequivalence relation, that is, a reflexive, symmetric andT-transitive fuzzy relation. Recently, a gen-eralization has been proposed [5] which also allows flexible interpretation of ordinal queries (such as“at least” and “at most”) by using fuzzy orderings [3]. In any case, if a query consists of at least twoexpressions that are to be interpreted vaguely, it is necessary to combine the degrees of matching withrespect to the different fields in order to obtain an overall degree of matching. Assume that we havea query(q1, . . . ,qn), where eachqi ∈ Xi is a value referring to thei-th field of the query. Given a datarecord(x1, . . . ,xn) such thatxi ∈ Xi for all i = 1, . . . ,n, the overall degree of matching is computed as

R((q1, . . . ,qn),(x1, . . . ,xn)

)= A

(R1(q1,x1), . . . ,Rn(qn,xn)

),

where eachRi is a T-transitive binary fuzzy relation onXi which measures the degree to which thevaluexi matches the query valueqi .

It appears natural to require that the functionA is an aggregation operator [7, 9, 13] and moreover,it would be desirable thatR is still T-transitive in order to have a clear interpretation of the aggregated

55

fuzzy relationR. Therefore, it is necessary to study which aggregation operators are able to guaranteethatRmaintainsT-transitivity.

It turns out that the preservation ofT-transitivity in aggregating fuzzy relations is closely relatedto the dominance of an aggregation operatorA with respect to the corresponding t-normT.

Let us recall some basic definitions.

Definition 1. [8] A (binary) operationA : [0,1]2 → [0,1] is called anaggregation operatorif A isnon-decreasing and the equalitiesA(0,0) = 0 andA(1,1) = 1 hold. Moreover, ifA is also associative,symmetric and has 1 as neutral element, then it is called a t-norm.

Definition 2. Consider a binary fuzzy relationR on some universeX and an arbitrary t-normT. R iscalledT-transitive if and only if, for allx,y,z∈ X,

T(R(x,y),R(y,z)

)≤ R(x,z). (1)

For more details on fuzzy relations, especially fuzzy equivalence relations and fuzzy orderingsand their properties, we recommend either original sources as [17, 1, 12], but also [10, 11, 4, 2].

Standard aggregation of fuzzy equivalence relations (fuzzy orderings) preservingT-transitivity isdone either be means ofT or TM (x,y) = min(x,y). Staying in the framework of t-norms, in fact anyt-normT∗ dominatingT can be applied to preserveT-transitivity, i.e. if R1,R2 are twoT-transitive,binary relations on a universeX , then alsoT∗(R1,R2) has this property (see [10]). Recall that trivially,for any t-normT, it holds thatT itself andTM dominateT.

As already mentioned above in several applications, other types of aggregation preservingT-transitivity are required [6]. Especially different weights (degrees of importance) of input fuzzyequivalences (orderings)R1 andR2 cannot be properly modeled by aggregation with t-norms, becauseof the commutativity . Therefore, we have to consider generalT-transitivity-preserving aggregationoperators.

Note that in the sequel we will deal with the aggregation of two givenT-transitive binary fuzzyrelationsR1,R2 acting on the same universeX. Our results can be easily modified for the case of theCartesian product ofT-transitive equivalence relations, as well as to the case of aggregating more thantwo T-transitive fuzzy relations such that the resulting output fuzzy relation will still beT-transitive.

2 T-Transitivity and Domination

Definition 3. [8] Let A,B be two aggregation operators. We say thatA dominatesB (A B), if andonly if, for all x,y,u,v∈ [0,1],

B(A(x,y),A(u,v)

)≤ A

(B(x,u),B(y,v)

). (2)

Observe thatA A if and only if A is bisymmetric. As already mentioned, for any t-normT,T T andTM T.

Further on we will denote the class of all aggregation operatorsA which dominate a given t-normT with DT = A | A T.

The following theorem generalizes the result from [10].

56

Theorem 4. Let |X|> 2. An aggregation operatorA preserves the T-transitivity of fuzzy relations onX if and only ifA ∈DT .

In the following, we will focus on the characterization of the systemDT . As already observed,T,TM ⊂DT . For any t-normT, some interesting properties ofDT can be found.

Proposition 5. Consider a t-norm T and the corresponding class of dominating aggregation operatorsDT . Then the following holds:

(i) For anyA,B,C ∈DT , alsoD = A(B,C) ∈DT .

(ii) If T is a continuous Archimedean t-norm with an additive generator f: [0,1] → [0,∞], then

for any p,q ∈ ]0,∞[, also the weighted t-norm Tp,q ∈ DT , with Tp,q(x,y) = T(

x(p)T ,y(q)

T

)and

x(p)T = f (−1) (p· f (x)) (see also [13, 9]).

Recall thatDT was discussed and characterized also in [16, 15] for the case thatT is a continuousArchimedean t-norm T with an additive generatorf : [0,1]→ [0,∞].

Proposition 6. [16] Under the circumstances given above,A ∈ DT if and only if there is a metric-preserving function H: [0,∞]2 → [0,∞] such that for all x,y∈ [0,1] :

f (A(x,y)) = H( f (x), f (y)).

Observe that Proposition 6 is in fact a corollary of Theorem 4. Indeed,H is metric preserving ifand only if it is a sub-additive function of two variables, i.e.

H(x+y,u+x)≤ H(x,u)+H(y,v)

for all x,y,u,v∈ [0,∞], what is, in fact, the domination of the sum operator overH.

3 Special Cases

We will now discuss three special cases of t-norms: TM = min(x,y),TP(x,y) = x · y (by isomorphism any strict t-norm can be covered [13]),TL (x,y) = max(x+ y−1,0)(by isomorphism, covering all nilpotent t-norms).

Proposition 7. The class of aggregation operators dominating the minimum t-norm TM is given by

Dmin = minf ,g | f ,g :[0,1]→ [0,1], non-decreasing,

f (1) = g(1) = 1, f (0) ·g(0) = 0,

whereminf ,g = min(

f (x),g(y)).

Evidently,A ∈ Dmin is symmetric if and only ifA(x,y) = f(

min(x,y))

for some non-decreasingfunction f : [0,1]→ [0,1] fulfilling f (0) = 0 and f (1) = 1. Note also, ifA ∈ Dmin, thenA = minf ,g,where f (x) = A(x,1) andg(y) = A(1,y) (for all x,y∈ [0,1]).

ConcerningTP andTL , though the classesDTP andDTL are completely characterized either byTheorem 4 or by Proposition 6, there is no counterpart of Proposition 7 in these cases. However, itis possible to give examples of these members of these classes, and of course, apply Proposition 5 toobtain new members.

57

Example 8. Observe thatx(p)TP

= xp and thus for allp,q∈ ]0,∞[, the operatorPp,q : [0,1]2→ [0,1],Pp,q(x,y)=xpyq is contained inDTP. Particularly, ifp+q = 1, thenPp,q is a weighted geometric mean (comparealso examples from [16, 15]).

However, observing that for all λ ≥ 1, the function Hλ : [0,∞]2 → [0,∞] ,Hλ(x,y) = (xλ + yλ)

1λ , is metric preserving, also any member of the Acz’el-Alsina family of t-norms

(TAA λ)λ∈[1,∞] (see [13]), is contained inDTP because of Proposition 6.

Example 9. Similarly, for all p,q ∈ ]0,∞[ ,Lp,q ∈ DTL , whereLp,q = TL p,q = max(0, px+qy+1− p−q). In particular, ifp+q = 1, Lp,q(x,y) = px+qy, i.e. anyweighted mean dominatesTL (compare also examples from [16, 15]).

Based onHλ any Yager t-normTYλ ∈DTL wheneverλ≥ 1.

4 Conclusions

An aggregation operatorA preservesT-transitivity of fuzzy relations if and only if it dominates thecorresponding t-normT (A ∈ DT). Although several methods for constructing aggregation opera-tors within a certain classDT have been mentioned, an explicit description ofDT could only beenpresented for the minimum t-normTM .

Acknowledgements

This work was partly supported by network CEEPUS SK-42 and COST Action 274 “TARSKI”. RadkoMesiar was also supported by the grant VEGA 1/8331/01. Ulrich Bodenhofer acknowledges supportof the KplusCompetence Center Program which is funded by the Austrian Government, the Provinceof Upper Austria, and the Chamber of Commerce of Upper Austria.

References

[1] J. C. Bezdek and J. D. Harris. Fuzzy partitions and relations: An axiomatic basis for clustering.Fuzzy Sets and Systems, 1:111–127, 1978.

[2] U. Bodenhofer. Representations and constructions of similarity-based fuzzy orderings. Submit-ted toFuzzy Sets and Systems.

[3] U. Bodenhofer. A similarity-based generalization of fuzzy orderings preserving the classicalaxioms.Internat. J. Uncertain. Fuzziness Knowledge-Based Systems, 8(5):593–610, 2000.

[4] U. Bodenhofer. Similarity-based fuzzy orderings: a comprehensive overview. InProc. EU-ROFUSE Workshop on Preference Modelling and Applications, pages 21–27, Granada, April2001.

[5] U. Bodenhofer and J. Küng. Enriching vague queries by fuzzy orderings. InProc. 2nd Int. Conf.in Fuzzy Logic and Technology (EUSFLAT 2001), pages 360–364, Leicester, UK, September2001.

58

[6] U. Bodenhofer and S. Saminger. Transitivity-preserving aggregation of fuzzy relations. Submit-ted toFSTA 2002.

[7] B. Bouchon-Meunier, editor.Aggregation and Fusion of Imperfect Information, volume 12 ofStudies in Fuzziness and Soft Computing. Physica-Verlag, Heidelberg, 1998.

[8] T. Calvo, A. Koles’arov’a, M. Komorn’ıkov’a, and R. Mesiar.A Review of Aggregation Operators.Servicio de Publicaciones de la University of Alcal’a, Madrid, 2001.

[9] T. Calvo and R. Mesiar. Weighted means based on triangular conorms.Internat. J. Uncertain.Fuzziness Knowledge-Based Systems, 9(2):183–196, 2001.

[10] B. De Baets and R. Mesiar. Pseudo-metrics andT-equivalences.J. Fuzzy Math., 5:471–481,1997.

[11] B. De Baets and R. Mesiar.T -partitions.Fuzzy Sets and Systems, 97:211–223, 1998.

[12] U. Höhle. Fuzzy equalities and indistinguishability. InProc. EUFIT’93, volume 1, pages 358–363, 1993.

[13] E. P. Klement, R. Mesiar, and E. Pap.Triangular Norms, volume 8 ofTrends in Logic. KluwerAcademic Publishers, Dordrecht, 2000.

[14] F. E. Petry and P. Bosc.Fuzzy Databases: Principles and Applications. International Series inIntelligent Technologies. Kluwer Academic Publishers, Boston, 1996.

[15] A. Pradera and E. Trillas. A note on pseudometric aggregation. Submitted toInt. J. GeneralSystems.

[16] A. Pradera, E. Trillas, and E. Castiñeira. On the aggregation of some classes of fuzzy relations. InB. Bouchon-Meunier, J. Guiti’errez-R’ıoz, L. Magdalena, and R. R. Yager, editors,Technologiesfor Constructing Intelligent Systems 1: Tasks. Springer, 2002 (to appear).

[17] L. A. Zadeh. Similarity relations and fuzzy orderings.Inform. Sci., 3:177–200, 1971.

59

Dynamics of fuzzy systems: theory and applications

UZIEL SANDLER

Jerusalem College of Technology91160 Jerusalem

Israel


Abstract

In this paper I present the dynamics theory which can operate with an imprecise knowledgeabout system states and with qualitative information about the forces influence. Such situation istypical for biology, economy, social and political science, where we don’t have, as rule, completeinformation about the system state and where our "fair" knowledge about the dynamics laws mustbe expressed in terms of "preferability", rather than in exact definition of the forces.

This approach was successfully applied to some problems of brain activity and the lymphocytecells maturation.

1 Introduction

There are a few equations in theoretical physics, which in a slightly different modification cover dy-namic problems of most physical systems. The same types of the equations appear, also, in the com-pletely “non-physical” areas like finance (Black-Scholes equation), political science (voting theory),ecology,etc. In this paper we show how the fuzzy logic mathematical apparatus applied to commoncausal principal together with some natural restriction on a system state space resolves this puzzle.

Another goal of this paper is concerned with the necessity to have dynamics theory which canoperate with an imprecise knowledge about system states and with qualitative information about theforces influence. This situation is typical for the above mentioned non-physical sciences. There, asa rule, we don’t have complete information about the system state and our "fair" knowledge shouldbe expressed in terms of "preferability", rather than in exact definition of the forces. For example,based on the observations and an intuitive experience, most experts will be consistent in formulationof the forces influence in this form: “if the system is in stateA then after a short time stateB will bepreferable than the others”. However the experts will be hindered and inconsistent if we ask them togive a more precise formulation of the dynamics laws, or even if we ask them to assign probabilitiesfor transitions from the given state to the others.

The main assumptions for this study are the following:

i) For most practical problems a system state space can be approximated by a Riemann manifold,but choice of the coordinates on the manifold is not unique, so the dynamics theory should becovariant under appropriate transformations of the coordinates.

60

ii) A system behavior is described in terms of "possibility" [1] that the system is in the neighborhoodUx of a given statex at a given timet. This possibility can be represented as a function ofthe domain and the time:m(U, t). This representation should be consistent with a commonlogic in a sense that at least for small domains the possibility that the system is in the domainU = U1

⋃U2; U1

⋂U2 = /0 must be equal to the possibility that the system is in the domainU1

or it is in the domainU2:m(U, t) =

S(m(U1, t);m(U2, t)),

whereS(m1;m2) denotes the logical connectiveor.

iii) The dynamics laws have "causal recursion" form such as: Possibility that a system is in someneighborhood of the state -x at the timet is determined by the possibility that: the system wasin a neighborhood of a statex1 at timet−δ and the transfer to thex-neighborhood during thetime intervalδ was possible,or it was in a neighborhood of a pointx2 at timet−δ and transferto thex - neighborhood during the timeδ was possible,or ... so on, for all possible states at thetime t−δ.

iiii) The dynamics is local, i.e. a system state cannot be changed significantly for a short time period.

In spite of the generality of the forms of the above mentioned assumptions, they dramatically reducethe possible types of the dynamics equations and lead to a deep relation between permissable form ofthe membership functionm(Ux, t) and representations of the logical connectivesor andand.

2 Fuzzy Dynamics

Denoting byPδ (Ux,Uxi , t) the possibility of the transition from the neighborhoodUxi of the pointxi

to the neighborhoodUx of the pointx during the timeδ, we can write the dynamics lawiii) in thesymbolic form:

m(Ux, t) = (1)

Sallxi (...;T (Pδ (Ux;Uxi , t) ;m(Uxi , t−δ)) ; ...),

whereT(P;m) denotes the logical connectiveand. Note, that the possibility of transitionPδ (Ux,Uxi , t)is determined by the forces acting on the system.

In order to find the dynamics equations we have to representS andT connectives and go to thelimit Ux → x (that is the membership function andS,T - connectivescan be determined for smallneighborhoodsUx only).

Admissible representations of theS andT connectives depend on the limiting properties of thefunctionm(Ux, t) in the limit Ux → x. Generally, there are two cases:

a)lim

Ux→xm(Ux, t) = µ(x, t) 6= 0.

b)lim

Ux→xm(Ux, t)≡ 0.

61

As these cases lead to substantially different descriptions of the system dynamics, they will be con-sidered separately.Case a) - dynamics of a fuzzy State GradeThe functionµ(x, t) can be interpreted

aspossibilitythat a system state isx at the timet. For definiteness we will call this function “StateGrade”.

Consider now the relationii) in the caseU1 → x; U2 → x and, therefor,U = U1⋃

U2 → x. In thislimit for continuousµ(x, t) we have:

S(µ,µ) = µ.

Together with such general properties like monotonicity, commutativity and with usual boundarycondition [2] this leads immediately to:

S(µ1,µ2) = max(µ1;µ2). (2)

For the connectiveand, however, we can take an arbitraryT-norm.

In the casea) the possibility of transition fromUxi toUx becomes a function of the statesxi , x andthe time:Pδ (xi → x, t). It is convenient to choose this function in the form:

Pδ(xi ,x, t) = G(v;x, t)),

wherev = x−xiδ andG(v;x, t) is the possibility that the system has the “velocity”v at the pointx at the

time t. Then, the symbolic expression (1) becomes an equation:

µ(x, t)≈

supv

T(G(v,x−vδ, t−δ);µ(x−vδ, t−δ)).

It can be shown (see [3]-[5] for details) that in the limitδ→ 0 this equation leads to the Liouville-typeequation:

∂µ∂t

=−∑i

∂H∂pi

∂µ∂xi

. (3)

The “Hamiltonian” -H is determined as:

H = (V(x,p, t) ·p),

with V(x,p, t) is obtained from the system:

∂G(V;x, t)∂V

= βp,

T(G(V;x, t);µ) = µ(x, t).

Equation (3) is equivalent to the Hamiltonian system for its characteristics:

dx(t)dt

=∂H∂p

, (4)

dp(t)dt

=−∂H∂x

. (5)

>From equations (3)-(5) it follows that the state grade remains constant on the characteristicsx(t):

µ(x(t), t) = µ(x(0),0),

for anyt.

Equations (4)-(5) describes a system evolution in an extended"Fuzzy state space’", which consistsof the two components:

62

α) the “physical” component -x

β) the “information” component -p

As the equation (5) is similar to the equation for the physical moment, we can callp - the moment ofpossibility. The direction of this momentp/|p| = ∇m(x, t)/|∇m(x, t)| is the direction >from a givenstate to the locally most preferable one.

It is important, that the HamiltonianH(x, p, t) can be obtained directly from the “linguistic” de-scription of the “forces” influence. The HamiltonianH can be more general than the common Hamil-tonians in physics, in particular, it may be a set-valued map, i.e. the system (4)-(5) describes bothdifferential equation and differential inclusion. The particular caseT(G;µ) = min(G;µ) has been con-sidered in [3],[4].Case b): dynamics of a fuzzy State DensityIn accordance with the requirement -

i) the functionm(U, t) should depend on invariant measure ofU only. The simplest such measure isρdU, wheredU is “volume” of U in a given coordinate system and multiplierρ makes the productρdU invariant under the admissible transformations of the coordinates.

In the caseb) this leads to the asymptotic:

limUx→x

m(Ux, t))g(ρ(x, t)dUx)

= 1, (6)

where the functiong(...) is continuous (but not necessary differentiable) near zero function andg(0) =0. We assume also thatρ is a smooth function ofx andt. As in physics the multiplier likeρ is called- density, we will callρ(x, t) as State Density. Generally speaking, the functiong could dependexplicitly on pointx, but, as we will see later, such dependence must be omitted in order to holdhomogeneity of the representations of the logical connectivesor andand for the whole state space.

Consider small domainsUx = Ux+ε⋃

Ux−ε with the volumes:

dUx = dUx−ε +dUx+ε, (7)

Possibility, that the system is inUx, is equal to:

g(ρ(x, t)dUx) = (8)

S(g(ρ(x− ε, t)dUx−ε);g(ρ(x+ ε)dUx+ε)).

Using (7), one has for the smoothρ(x, t) and smallε:

ρ(x)dUx =

ρ(x− ε)dUx−ε +ρ(x+ ε)dUx+ε

+o(εdU).

Thus, for the connectiveORwe have for the small and closedU1, dU2:

S(g(ρ(x1)dU1);g(ρ(x2)dU2))≈ (9)

g(ρ(x1)dU1 +ρ(x2)dU2).

63

In order to find out for an admissible form of the possibility of transition, let us assume now thatat the timet − δ the system was in the domainU0 with the possibility 1 and with the possibility 0elsewhere. It is obvious, that the possibility that at the timet the system will be in a domainUx isequal to the possibility that the system transferred toUx fromU0. That is in such situation the transitionpossibility -Pδ(Ux,Ux0, t) should be equal tom(Ux, t). Becausem(Ux, t) has of the asymptotic (6) wehave:

Pδ(Ux,Ux0)≈ g(ΓδdUx), (10)

whereΓδ depend on(x0;x; t), but doesn’t depend ondU.

In order to find admissible representation of the connectiveand, consider decomposition of thetransition fromUx0 toUx during the timeδ = δ1 +δ2 on the transitions fromUx0 to u1 during the timeδ1 and from u1 to Ux duringδ2, or >from Ux0 to u2 and from u2 to Ux, or ... so on, for all possibleintermediate domains (see Figure 1).

Figure 1: Decomposition of the transition possibility in the caseb)

In according with (10) we have:

Piδ1

= Piδ1

(Ux0,ui ) = g(Γi

δ1dui),

Piδ2

= Piδ2

(ui ,Ux) = g(Γi

δ2dUx

).

Then, for possibility of the “two-fold” transition fromUx0 to ui during the timeδ1 and>from ui toUx

during the timeδ2 one obtains:

T(Pi

δ1,Pi

δ2

)= g

(F(Γi

δ1dui ;Γi

δ2dUx

)),

with some unknown functionF . It follows from the figure and (9) that:

Pδ (Ux0,Ux) = S(...,T

(Pi

δ1,Pi

δ2

), ...)

=

= g

(∑

i

F(Γi

δ1dui ,Γi

δ2dUx

)),

As Pδ = g(ΓδdUx) this implies that:

ΓδdUx = ∑i

F(Γi

δ1dui ,Γi

δ2dUx

)=

64

=

(∑

i

f(Γi

δ1dui)

Γiδ2

)dUx.

In order that the last sum will converge for any partitiondui , we have:

f(Γi

δ1dui)

= const·Γiδ1

dui .

Without loss of generality we can putconst= 1. So

F(Γi

δ1dui ;Γi

δ2dUx

)= Γi

δ1Γi

δ2duidUx.

Thus, the logical connectiveand for the caseb) has to be defined as:

T(P1(Γ1dU1);P2(Γ2dU2)) = g(Γ1dU1Γ2dU2) . (11)

If the functiong is invertible we can write:

S(m1;m2) = g(g−1(m1)+g−1(m2)), (12)

T(P1;P2) = g(g−1(P1) ·g−1(P2)), (13)

so in the caseb) the logical connectivesor andand, at least infinitesimally, should be considered asthe pseudo-arithmetic sum and product [6],[7].

Therefore, in the given case the symbolic expression (1) leads to an equation:

g(ρ(x, t)dx) = (14)

g

∫Vx′

Γ(x,x′, t;δ

)ρ(x′, t−δ

)dx′

dx

.

Decomposing (14) with respect toδ leads to the equation (see [5] for details):

∂ρ∂t

+V (x, t) ·∇ρ =

D(x,t)∇2 +U (x,t)

ρ, (15)

with

U (x,t) = limδ→0

1δ

(∫Γδ (x−u,x,t)dnu−1

),

V (x,t) = limδ→0

1δ

∫Γδ (x−u,x,t)udnu,

D(x,t) = limδ→0

12δ

∫Γδ (x−u,x,t) |u|2dnu.

If the manifold consists ofM single-connected domains, this equation is transformed to:

∂ρσ

∂t+Vκ,i

σ (x, t)∇iρκ = (16)Dκ,i j

σ (x, t)∇i∇ j +Uκσ (x, t)

ρκ,

where indexesσ,κ = 1, ...,M describe the different single-connected domains.

The coefficients in these equations have the following meaning:D(x,t) describe the diffusion,V (x,t) - the flows andU (x,t) - the external fields. ForVσ ≡ 0 and complex valuedρ equation (16)corresponds to the Schreodinger equation and forDκ

σ ≡ 0 and quaternion valuedρ it corresponds tothe Dirac equation. For real valuedρ we have generalized Fokker-Plank equation.

65

3 Concluding Remarks

Equations (4)-(5) and (16) cover most dynamics equations successfully used in physics up to thepresent time. At the resent time the same types of the equations have been successfully applied to the“non-physical problems” in finance, political science, ecology, biology,etc.

It is interesting that there are “biological reasons” to use the above-mentioned representationsof the logical connectives. It is well known [8] that the elementary operations of a nerve cell areaccumulation and amplification of signals that correspond to the pseudo-arithmetic operations. Inorder to be able to use themax S-normthe neuron must be able to compare signals. Geneticallya neuron is able to compare several specific signals. In the process of elaboration of conditionalreflexes, living creatures develop the ability make a comparison among many other signals. It wasshown also that a single neuron could be taught to compare arbitrary signals [9]. Thus, comparisonmay become a basic neuron operation. For a certain kind of neuron a mechanism that could carry outmax-operations was proposed in [10].

Application of this approach to some problems of brain activity and the lymphocyte cells matura-tion were considered in [11] and [4].

References

[1] L.A.Zadeh, “Fuzzy sets as basis for a theory of possibility”,Fuzzy Set and Systems, v1, p.3-28,1978.

[2] D.Butnariu and E.P.Klement, “On Triangular Norm-Based Measures and Games with FuzzyCoalitions”, Kluwer Ac.Publ.,Dordrecht-Boston-London, 1993.

[3] Y.Friedman and U.Sandler, “Evolution of Systems under Fuzzy Dynamic Law”,Fuzzy Set andSystems, v84, p.61-74, 1996.

[4] Y.Friedman and U.Sandler, “Dynamics of Fuzzy Systems”,Inter.J. of Chaos Theory and Appli-cation, v2, No.3-4, p.5-21, 1997.

[5] Y.Friedman and U.Sandler, “Fuzzy Dynamics as Alternative to Statistical Mechanics”,Fuzzy Setand Systems, v106, p.61-74, 1999.

[6] R.Meisar and J.Rybarik , “Pan-operation structure”Fuzzy Set and Systems, v74, p.365-369,1995.

[7] E.Pap , “Null-Additive Set Function”, Kluwer Ac.Publ.,Dordrecht-Boston-London, 1995.

[8] Hebb, “Essay on Mind”, Lawrence-Erlbaum Assc., Hillsdale NJ, 1980.

[9] L.Tsitolovsky, “A model of motivation with chaotic neuronal dynamics”,BioSystems, v5(2),p.205-232, 1997.

[10] U.Sandler, “A Neural Networks with Multi-neuron”,Neurocomputing, v14, p.41-62, 1997.

[11] U.Sandler and L.Tsitolovsky, “Fuzzy Dynamics of Brain Activity”,Fuzzy Set and Systems,v121/2, p.237-245, 2001.

66

A natural interpretation of fuzzy partitions

MAMORU SHIMODA

Shimonoseki City UniversityShimonoseki 751

Japan


The aim of this talk is to present a natural interpretation of fuzzy partitions, which are naturally one-to-one correspondent to fuzzy equivalence relations.

In [1, 2, 3] we presented a new and natural interpretation of fuzzy sets, fuzzy relations, and fuzzymappings in a cumulative Heyting valued model for intuitionistic set thoery. By the interpretation wecan get most of the standard defining equations of basic notions and operations of fuzzy sets and fuzzyrelations, consider notions such as operations of fuzzy subsets of different universes, fuzzy relationsand mappings between fuzzy subsets, and make the meaning of Zadeh’s extension principle clear.

Let H be a complete Heyting algebra andVH be the cumulativeH-valued model. TheHeytingvalue‖ϕ‖ and thecheck setx are defined as usual. Foru,v∈VH , u andv aresimilar iff ‖u = v‖= 1.EveryA∈VH is called anH-fuzzy set, and for a crisp setX every subset ofX in VH is called anH-fuzzysubset of X. Themembership function of A on Xis the mappingµA : X−→H; x 7−→ ‖ x∈ A‖. There isa natural correspondence betweenH-fuzzy subsets ofX and mappings fromX to H, which preservesorder and basic set operations. IfR is anH-fuzzy subset ofX×Y, R is called anH-fuzzy relation fromX to Y, and it is called anH-fuzzy relation on Xin caseX = Y.

Theorem 1. An H-fuzzy relation R on X is an equivalence relation in the model iff for all x,y,z∈ X,µR〈xx〉= 1, µR〈xy〉= µR〈yx〉, and µR〈xy〉∧µR〈yz〉 ≤ µR〈xz〉 hold.

For anH-fuzzy equivalence relationR on X and for everyx∈ X, let [x] be the equivalence classof x (in the model) andPR = [x];x∈ X. Obviously[x] is anH-fuzzy subset ofX for eachx∈ X andPR is the set of all equivalence classes with respect toR.

Definition 2. A family P of H-fuzzy subsets ofX is called anH-fuzzy partition of Xif(a) for everyA∈ P there exists anx∈ X such thatµA(x) = 1,(b) for eachx∈ X there is a uniqueA∈ P such thatµA(x) = 1, and(c) for all A,B∈ P andx,y∈ X, µA(x)∧µA(y)∧µB(x)≤ µB(y).

For anH-fuzzy partitionP of X, the correspondingH-fuzzy equivalence relationRP onX can alsobe naturally defined, that is, for everyx,y ∈ X, ‖xRPy‖ = ‖∃A ∈ P(x ∈ A∧ y ∈ A)‖ holds. For twofamiliesP,Q of H-fuzzy sets,P andQ areequivalentif for every A∈ P there exists a uniqueB∈ Qsuch thatA andB are similar, and vice versa.

Theorem 3. Let X be a crisp set.(1) If R is an H-fuzzy equivalence relation on X and P= PR is the family of all equivalence classeswith respect to R, then P is an H-fuzzy partition of X, and R and RP are similar.(2) If P is an H-fuzzy partition of X and R= RP is the corresponding H-fuzzy equivalence relationon X, then P is equivalent to PR, the family of all equivalence classes with respect to R.

67

Hence there is a natural correspondence betweenH-fuzzy equivalence relations onX andH-fuzzypartition families ofX. The definition ofH-fuzzy partition seems to be different from any of precedingdefinitions of fuzzy partition in the literature.

References

[1] M. Shimoda, A natural interpretation of fuzzy sets and fuzzy relations, to appear inFuzzy Setsand Systems, submitted 1998, revised 2001.

[2] M. Shimoda, A natural interpretation of fuzzy mappings, submitted, 2000.

[3] M. Shimoda, A natural interpretation of fuzzy set theory, in:Proceedings of Joint 9th IFSA WorldCongress and 20th NAFIPS International Conference, 493-498, 2001.

68

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