A Fuzzy Choquet Integral with an Interval Type-2 Fuzzy Number-Valued Integrand

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1 A Fuzzy Choquet Integral with an Interval Type-2 Fuzzy Number-Valued Integrand Timothy C. Havens, Student Member, IEEE,, Derek T. Anderson, Student Member, IEEE, and James M. Keller, Fellow, IEEE Abstract Fuzzy integrals have been used to fuse the evidence or opinions from a variety of sources. These integrals are nonlinear combinations of the support functions and the possibly subective worth of subsets of the sources of information, realized by a fuzzy measure. There have been many applications and extensions of fuzzy integrals and this paper proposes a fuzzy Choquet integral, where the integrand takes an interval type-2 fuzzy number and the fuzzy measure is real numbervalued. Interval type-2 fuzzy numbers encode the second-order uncertainty in a fuzzy number. Type-2 fuzzy numbers have been been shown to be useful in many applications, including computing with words and control systems. We illustrate our method on several numerical examples as well as on a bioinformatics application. I. INTRODUCTION Let X = {x,x 2,...,x N } be a non-empty finite set typically of information sources or evidence [] and g : 2 X [, ] be a fuzzy measure with the following properties [2]: gø =, gx =; 2 If A B X then ga gb g is nondecreasing. The measure g is the possibly subective confidence or worth of each subset of information sources; hence property tells us that the worth of no sources, the empty set Ø, is and the worth of all sources, the universal set X, is. Property 2 follows intuition, in that two sources are worth at least as much as one, three sources are worth at least as much as two, and so on. A well-known fuzzy measure is the Sugeno λ-measure [2], which for sets A X and B X, such that A B =Ø, g λ A B =g λ A+g λ B+λg λ Ag λ B. It is proven that for a given set of densities g i, where g i = g{x i }, λ can be determined by solving λ += n +λg i,λ>. 2 i= The λ-measure is especially attractive because one only has to provide the measures of the singletons; the densities of the non-singleton sets are calculated using. Another T.C. Havens, D.T. Anderson, and J.M. Keller are with the Department of Electrical and Computer Engineering, University of Missouri, Columbia, MO 652, USA phone: , havenst@gmail.com, keller@missouri.edu, dtaxtd@missouri.edu. fuzzy measure that is built using the densities g i is the decomposable measure [3] ga =min, g i,a X. 3 i,x i A This measure has the added benefit that it is easy and fast to compute. There are many forms of the fuzzy integral []. In practice, fuzzy integrals are used for evidence fusion [4 6]. They combine sources of information by accounting for both the support of the question the evidence and the expected worth of each subset of sources as supplied by a fuzzy measure g. In this paper, we focus on the discrete fuzzy Choquet integral [7] proposed by Murofushi and Sugeno [8]. Let h : X [, be a real-valued function which represents the evidence or support of a particular hypothesis generally when dealing with sensor fusion problems, h : X [, ]. The discrete fuzzy Choquet integral is defined as C g h = N hx πi ga πi ga πi+, 4 i= where π is a permutation of X, such that hx π hx π2... hx πn, A πi = {x πi,...,x πn }, and ga πn+ =. In some cases, the evidence h cannot or should not be represented simply by numbers; h would be better represented as an interval-valued or fuzzy number-valued function. An example is the survey question, How many bottles of wine should I purchase for the reception?. Many people would answer this question with an interval, e.g. between 2 and 3. Thus an interval-valued h is more appropriate. In other situations fuzzy number-valued integrands would better suit the problem, e.g. around 25. Extensions of both the fuzzy Sugeno and fuzzy Choquet integral for both interval-valued and fuzzy number-valued integrands have been proposed in [9 ]. Let H =[h,h + ] and H be the interval-valued and fuzzy number-valued functions, respectively. The Choquet integrals of H and H are defined as C g H [ C g h,c g h + ] 5 C g Ha sup {α [, ]; a C g α H}, 6 where α H =[ α h, α h + ] is the closed interval of the levelcut of H at α [, ]. Notice that 6 is related to 5 by

2 TABLE I NOTATION X set of sources sensors, inputs, etc. x i ith source hx i evidence offered by x i h real-valued function H =[h,h + ] interval-valued function H type- fuzzy number-valued function H interval type-2 fuzzy number-valued function α H =[ α h, α h + ] level-cut of H at α g i numerical worth of x i g λ Sugeno fuzzy λ-measure [2] the representation theorem and extension principle [, 2], C g H = αc g α H. 7 α The output of a fuzzy Choquet integral with an intervalvalued integrand H is an interval and the output with a fuzzy number-valued integrand H is a fuzzy number. In this paper we extend 5 and 6 to interval type-2 fuzzy number-valued integrands. Type-2 fuzzy sets encode higherorder uncertainty by essentially blurring the membership function of a type- fuzzy set, thus allowing the membership function to be uncertain itself. Type-2 fuzzy sets have been found to be especially useful in the Computing with Words CW paradigm, proposed by Zadeh in [3], and in the most notable CW implementation, Mendel s perceptual computer [4 6]. Section II provides a detailed description and the necessary theoretical background of interval type-2 fuzzy sets. In Section III we develop the fuzzy Choquet integral for interval type-2 fuzzy integrands and in Section IV we offer several illustrative examples, including a bioinformatics application. We summarize in Section V. Please notice Table I which provides a list of the notation used in this paper. II. INTERVAL TYPE-2 FUZZY SETS Type-2 fuzzy sets T2 FSs and fuzzy numbers FNs [7] provide an additional level of uncertainty over Type- fuzzy sets T FSs, in that the T membership function MF has an underlying uncertainty in its values. Figure illustrates the relationship between a T FS and a T2 FS. The left view shows a T FS H for this section assume that H is equivalent to the fuzzy set Hx. For the T FS, the membership at a is exactly μa ; there is no second-order uncertainty to this value. The right view in Fig. shows how uncertainty can be applied to H to produce a T2 FS, as shown by the gray region around H although, in practice, T2 FSs are not ust blurred T FSs. The gray region is called the footprint of uncertainty FOU and this region defines the uncertainty in the shape of H. Thus for the T2 FS, the value of the membership μa is uncertain. In a generalized T2 FS, the membership μa would have a membership function that describes its T2 uncertainty along the vertical line that intersects with the gray region between μ and μ 2. However in this paper, we focus on interval type-2 fuzzy sets IT2 FSs [8, 9], where the uncertainty of the membership at a given μa μ T FS a H a μ μ 2 μ T2 FS Fig.. Left view shows a type- fuzzy set H. Uncertainty is applied to h in the right view shown by the gray region around H which results in a type-2 fuzzy set. The membership μa in the type-2 fuzzy set has an uncertain value between μ and μ 2. μ a H Embedded T FS UMF H UMF H LMF H Fig. 2. Interval type-2 fuzzy set, h; solid gray region Footprint of Uncertainty, solid line Upper Function, dotted line Lower Function, wavy line embedded type-fuzzy set [9] value a is uniform across a given interval. For example, the uncertainty of μa in the right view of Fig. would have equal membership across the interval [μ,μ 2 ]. We denote an IT2 FS with a tilde, e.g. H. Figure 2 is the quintessential illustration of an IT2 FS [9]. This plot shows that the FOU is bounded by the Upper Function UMF, which lies on the outside of the FOU, and the Lower Function LMF, which lies inside of the FOU. The UMF is the least certain embedded T FS within the IT2 FS; the LMF is the most certain embedded T FS within the IT2 FS. For the sake of completeness and to develop our notation, we now follow much of the theoretical development of IT2 FSs from references [9, 2]. For all a A the membership μa is an interval, thus we denote this membership function as μ a =[μ a,μ + a ]. Wecan write H as the union of μ a, a A H = /a, μ, 8 μ μ a a A where /a, μ indicates a secondary membership of at a, μ. Note that standard notation for union in the literature regarding IT2 FSs is ; however, to eliminate confusion with the Choquet integral notation we choose to use to denote union. Notice that this completely defines the IT2 FS H as the union of the vertical interval-valued sets μ a. The FOU a a

3 of H can be constructed as FOU H = μ a. 9 a A Reference [9] aptly describes this equation as the verticalslice representation of the FOU, as the FOU is constructed from all the vertical intervals μ a. The bounds of the FOU, the UMF and the LMF, are defined as UMF H FOU H, a A, LMF H FOU H, a A. As stated above, the UMF and LMF are fuzzy sets that bound the FOU. These sets are also examples of embedded FSs. The definition of an embedded IT2 FS is H e = [/μ a ]/a, μ a μ a. 2 a A This set has the shape of a T FS with a secondary membership of only at the values μ a μ a for each a. The companion of an embedded IT2 FS is an embedded T FS an example is the wavy line depicted in Fig. 2. An embedded T FS is defined as H e = μ a /a, μ a μ a, 3 a A where μ a is the membership value of the T FS H e at a. Embedded FSs are related by H e =/H e, 4 where this equation implies a T2 membership of along the embedded T FS H e. The embedded FSs in 2 and 3 are important in our definition of the fuzzy Choquet integral for an IT2 fuzzy integrand. The representation theorem from [2] states that H is the union of all its embedded IT2 FSs note that we have extended this to the continuous domain, H = H e, 5 where H e is an embedded IT2 FS in H. In this case, there is an infinite number of embedded IT2 FSs in H, hence the integral notation of the union. Furthermore, the FOU of H can be expressed as the union of the embedded T FSs, FOU H = H e. 6 Note that the union in 6 conforms to the definition of the union in [9]. Finally, H can be expressed as having a T2 membership of in its FOU, H =/FOU H. 7 With the above definitions, we now move on to our definition of the fuzzy Choquet integral. III. EXTENDED FUZZY CHOQUET INTEGRAL We begin our definition of the fuzzy Choquet integral for an IT2 FN-valued integrand by examining the problem. An interval-valued integrand H, as in 5, produces an intervalvalued result, while a FN-valued integrand H, as in 6, produces a FN. It follows that the Choquet integral with an IT2 FN-valued integrand H produces an IT2 FN. We define an IT2 FN as an IT2 FS where the UMF and LMF are type- FNs. Assume an IT2 FN-valued function H, e.g. a mapping of words in a CW system. Define the embedded T FN-valued functions He as the functions Hex i FOU Hxi, i,. Equation 7 shows that an IT2 FS can be constructed from its FOU and 6 shows that the FOU is the union of all its embedded T FSs; these results can also be shown for FNs. Thus, we define the FOU of C g H as the union of all C g He, FOU C g H C g He, 8 and 7 shows that C g H = where C g H is an IT2 FN. Proposition. If Z = Cg H then / FOU C g H, 9 UMF Z = C g UMF H, 2 LMF Z = C g LMF H, 2 where UMF H is the function where Hx i =UMF Hx i, i and LMF H is the function where Hx i =LMF Hx i, i. Proof. The UMF Hx i FSs are the upper-bound of the embedded T FSs in the FOUs of the evidence Hx i ; the LMF Hx i FSs are the lower-bound of the embedded T FSs in the FOUs of Hx i. The left-most points in the level-cuts α UMF Hx i,α [, ] represent the smallest values that the evidence can take for a given α and the right-most points in these level-cuts represent the largest values that the evidence can take. By definition of the FOU, the following property is true for all evidence H, α UMF Hx i α He x i 22 α He + x i α UMF + Hx i, i, where He is any [ embedded T FN-valued function in H and α UMF Hx i = α UMF Hx i, α UMF + Hx ] i. Additionally, the Choquet integral is monotonic; thus, C α g UMF H g He 23 g He + g UMF + H. Note that in the above property H e represents any combination of the embedded T FNs in the FOUs of the IT2 FNs produced by H.

4 Using the same logical steps as above, we can show that for a given set of evidence H, α He x i α LMF Hx i 24 α LMF + Hx i α He + x i, i, and, subsequently, C α g He g LMF H 25 g LMF + H g He +. Equation 6 shows that we can build C g UMF H and C g LMF H from the level-cuts, α UMF H and α LMF H. Thus, 23 and 25 together show that UMF Cg H = C g H = C g UMF H, 26 LMF Cg H = C g H = C g LMF H. 27 Remark. Proposition shows that we can build the UMF and LMF of FOU C g H by performing the Choquet integral on the respective functions on the UMFs and LMFs of the set of evidence H. Now we show that the FOU C g H is completely full. That is, the level-cut α C ghe is composed of the two closed intervals between the UMF and LMF of C g H. Proposition 2. Consider the level-cuts of UMF Cg H and LMF Cg H, where [ ] α UMF Cg H = α UMF C g H,α UMF + C g H, 28 [ ] α LMF Cg H = α LMF C g H,α LMF + C g H 29 and α [, ]. Then, α FOU C g H {[ ] = α UMF C g H,α LMF C g H 3, [ ]} α LMF + C g H,α UMF + C g H, α, α [, ]. The notation {[, ], [, ]} indicates the FOU is composed of two intervals. Proof. We begin by breaking down the extended Choquet integral as in Eqs. 5,7, FOU C g H = C g He = = α α αc g α H e α [ C g α H e,c g α H + e ]. 3 Taking the level-cut of 3 at a value β [, ] produces β α [ C g α He,C g α He + ] = β α α α [ C g α He,C g α He + ] = [ Cg β H e,c g β H + e ]. 32 This equation essentially is the collection of all possible intervals produced by the Choquet integral acting on the level-cuts of the embedded T evidence He. Thus, we prove the proposition by showing that the interval end-points C g β He and C g β He + are interval-valued, themselves. By the definition of the FOU, [ ] α He x i [ α UMF Hx i, α LMF Hx i ], 33 α He + x i α UMF + Hx i, α LMF + Hx i, i. 34 Moreover, because Hx i is an IT2 FN, [ α He x i = α UMF H x i, α LMF i] H x, 35 α H + e x i = [ α UMF + H x i, α LMF + H x i], i. 36 The Choquet integral is monotonic, thus [ ] C g α He = α UMF C g H,α LMF C g H, 37 [ ] C g α He + = α LMF + C g H,α UMF + C g H. 38 These equation show that collection of interval end-points in 32 are interval-valued at the bounds shown in 3, thus proving the proposition. Remark 2. Propositions and 2 show that, in practice, the extended Choquet integral can be calculated by performing the Choquet integral on the UMFs and LMFs of the IT2 FN-valued evidence. A constraint on the methods described here is that the UMFs and LMFs are FNs. In most real applications, the UMFs will be normal FNs have a maximum membership equal to. However, there are many IT2 systems where the LMFs are not normal. Figure 3 shows an example where the LMFs of the evidence are not normal. The methods we describe here will work for these cases as long as the maximum memberships of the LMFs are equal, that is sup LMF Hx =suplmf Hx 2 =...=suplmf Hx N, as is shown in Fig. 3. This is because the FOU of the IT2 FNs shown in Fig. 3 can still be expressed as the union of T FNs.

5 μ Fig. 3. The extended Choquet integral can be used with IT2 FNs that have non-normal convex LMFs if the maximum memberships of the LMFs are equal Hx C g H Hx 3.2 Hx Hx a gλ i = {.5,.5,.5} C g H Hx 3.2 Hx b gλ i = {.3,.5,.8} Fig. 4. Example of fuzzy Choquet integral with an IT2 fuzzy integrand; fuzzy λ-measure. In references [2, 22], the authors propose a method, called the linguistic weighted average, by which the fuzzy weighted average can be applied to IT2 FSs. They devise a scheme where the operation can be applied in the case where the LMFs are of different heights. We are examining their paradigm with the hope of generalizing the extended Choquet integral to this case. IV. NUMERICAL EXAMPLES Example. Figure 4 illustrates an example of our formulation of the fuzzy Choquet integral using the fuzzy λ- measure. The red IT2 fuzzy-numbers, Hx, Hx2, and Hx 3 are the evidence. Figure 4a shows the result C g H for the densities gλ i = {.5,.5,.5}; thus, this example is an average-like operator on H. Figure 4b illustrates C g H for the densities gλ i = {.3,.5,.8}. View b shows that the result reflects the fact that Hx3 has a higher density worth than the other evidence; thus, C g H is more similar to Hx 3. a TABLE II NUMERIC WEIGHTS OF THE RELIABILITY OF GO ANNOTATIONS [23] Evidence code Weight Traceable author statement TAS Inferred from sequence similarity ISS.8 Inferred from electronic annotation IEA.6 Non-traceable author statement NAS.4 Not documented ND. Not recorded NR. A. Gene Ontology similarity measure Reference [23] proposes a Choquet fuzzy integral-based similarity measure for genes and gene products described by Gene Ontology GO [24] annotations. Consider two genes, G = {T,...,T n } and G 2 = {T 2,...,T 2m }, where {T,...,T n } are the n GO annotations of gene G and {T 2,...,T 2m } are m GO annotations of gene G 2. Using these GO-based representations of G and G 2, the similarity of G and G 2 can be computed by considering the set of GO annotation pairs X = G G 2 = {T, T 2,...,T nm } as a finite set of information sources that support the similarity of G and G 2, where T k =T i,t 2 is a pair of terms. Assume that there exists a pairwise term similarity measure st k [, that represents the similarity of the pair of terms T k = T i,t 2 see [25 29] for more discussion on the similarity measure st k. It is easy to see that the evidences h of the similarity of G and G 2 are the nm pairwise similarity values, st k =h k. Each GO annotation has an associated evidence code that describes how that annotation was produced see for more information on these codes. Some annotation methods are more reliable; thus, these evidence codes allow biologists to weight the reliability of each annotation. However, these evidence codes are not numeric and, thus, cannot be directly applied to fuzzy integrals. Reference [3] proposes a method to encode words as IT2 FSs. This approach aggregates intervals into IT2 FSs, where each FS represents a word The intervals often are the result of a survey of experts. However, we do not yet have an IT2 fuzzy-valued fuzzy measure; we have a paper in preparation that addresses this topic. In [23], the authors associated a numeric weight with each evidence code; Table II contains these values. For each pair of terms T k = {T i,t 2 }, there are two associated evidence codes. We denote the numeric weights from Table II of these evidence codes as ct i and ct 2. Thus the confidence or worth of h k is g k = f ct i,ct 2, 39 where f is some aggregation operation maximum, average, or minimum on ct i and ct 2. We can now use or 3 to compute the entire fuzzy measure g over X. Example 2. GO-based similarity of two gene products This example computes the similarity of two gene products in the GP D94 data set [3] and, for comparison, mimics Example 4 in [23]. Consider the two gene products, G =

6 TABLE III GENE ONTOLOGY ANNOTATIONS OF GENE PRODUCTS IN EXAMPLE 2 GO Term Definition Evidence Gene product: AAH3569 GO:472 protein phosphatase activity TAS GO:647 protein amino acid dephosphorylation IEA GO:827 zinc ion binding NR Gene product: AAH2399 GO:472 protein phosphatase activity ISS GO:647 protein amino acid dephosphorylation NAS GO:6787 hydrolase activity NR a - Hx π, Hxπ2, Hxπ3, Hx π4 AAH3569 MTMR4 gene and G 2 = AAH2399 MTMR8 gene. The GO annotations and associated evidence codes of these gene products were extracted from the GO on December, 23. Table III contains the annotations of these gene products. The associated weights of G and G 2 are ct = {,.6,.} and ct 2 ={.8,.4,.}. The densities were computed by g k =min{ct i,ct 2 }, min {ct i,ct 2 } = The number-valued pair-wise GO term similarity matrix used in [23] is st i,t 2 = Thus, the sorted evidence are {h πi } = {st k } = {,,,,.,.,.33,.52,.58}, and the respective densities are {g i } = {.,.,.,.,.6,.4,.4,.8,.}. For our example, we picked IT2 FSs that represent the sorted evidence. These sets are shown in Fig. 5. In practice, one could use the methods described in [3] to aggregate intervals, created from multiple GO-based similarity measures, into IT2 FSs. We do not specifically address this in this paper. Figure 6 shows the resulting GO-based similarity of the gene products computed with the extended fuzzy Choquet integral proposed in this paper. View a shows the result using the decomposable fuzzy measure in Eq. 3. Note that for this measure the α =level-cut α= C g UMF H =.5, which is the same result that the number-valued Choquet integral similarity measure produced in [23]. View b shows the result using the fuzzy λ-measure. We prefer the λ- measure for real applications of the fuzzy Choquet integral as this measure adapts well to densities that have a sum greater than. V. CONCLUSION This paper proposed a fuzzy Choquet integral for an interval type-2 fuzzy number-valued evidence. We showed that the bounds of the footprint-of-uncertainty of the result could be calculated by computing the integral on the upper and lower membership functions of the interval type-2 evidence. This is an important aspect as this implementation b. - Hx π5, Hx π c.33 - Hx π d.52 - Hx π e.58 - Hx π9 Fig. 5. Interval type-2 FNs that represent the evidence of GO-based similarity of two gene products.

7 a Decomposable fuzzy measure, Eq b Fuzzy λ-measure Fig. 6. GO-based similarity of gene products, AAH3569 and AAH2399, computed with extended fuzzy Choquet integral and two different fuzzy measures. Red shows the evidence and green is the result. could be used in real applications with little additional computational overhead compared to a type- fuzzy Choquet integral. In fact, the big-oh complexity of the algorithms are equivalent. In the second example we extended the Gene Ontologybased similarity measure proposed in [23]. This shows that our method has a practical application. We also would like to emphasize that this shows future merit of using the Choquet integral with type-2 fuzzy numbers. In the Gene Ontology example, the worth of the evidence was expressed as linguistic evidence codes, to which we applied a numeric weight. We are currently surveying experts to produce interval type- 2 fuzzy numbers that represent each evidence code. We are also extending fuzzy measures to type-2 representations. This discussion leads to the prognostication of a generalized fuzzy integral and fuzzy measure that can address type-2 fuzzy numbers. Table IV illustrates how our proposed Choquet integral for interval type-2 fuzzy integrands is an instantiation of a linguistic fuzzy integral LFI ; our implementation is a LFI for a number-valued fuzzy measure and an interval type-2 fuzzy number-valued integrand evidence. We are currently working on the development of fuzzy integrals and fuzzy measures that fall within all the LFI cells of Table IV. The challenges in this regard include developing interval-valued, fuzzy number-valued, and interval type-2 fuzzy number-valued fuzzy measures. For example, how are fuzzy measure properties, enumerated at the beginning of this paper, extended to intervals, fuzzy numbers, and interval We follow the naming convention in [2, 22], where the authors proposed a linguistic weighted average. TABLE IV BREAKDOWN OF DIFFERENT TYPES OF FUZZY INTEGRALS; FI-FUZZY INTEGRAL, IFI-INTERVAL FUZZY INTEGRAL, FFI-FUZZY FUZZY INTEGRAL, LFI-LINGUISTIC FUZZY INTEGRAL Fuzzy Measure Evidence Numbers Intervals T FSs IT2 FSs Numbers FI IFI FFI LFI Intervals IFI IFI FFI LFI T FNs FFI FFI FFI LFI IT2 FNs LFI LFI LFI LFI type-2 fuzzy numbers? Interval-valued and fuzzy numbervalued fuzzy measures have been discussed [32, 33], but to our knowledge interval type-2 fuzzy measures have not. In conclusion, we propose LFIs as an instantiation of computing with words. Fuzzy integrals have been shown to be useful for information fusion [4 6], and we believe that the linguistic forms will be useful for fusion of evidence that is linguistic in nature. ACKNOWLEDGEMENT Timothy Havens and James Keller were supported in part by grants from the Leonard Wood Institute LWI and Army Research Office EV in support of the U.S. Army RDECOM CERDEC NVESD. Derek Anderson is a pre-doctoral biomedical informatics research fellow funded by the National Library of Medicine T5 LM789. REFERENCES [] M. Grabisch, Ed., Fuzzy Measures and Integrals: Theory and Applications. New York: Physica-Verlag, 2. [2] M. Sugeno, Fuzzy Automata and Decision Processes. New York: North-Holland, 977, ch. Fuzzy measures and fuzzy integrals: a survey, pp [3] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications. New York: Academic Press, 96. [4] M. Grabisch, Fuzzy Measures and Integrals: Theory and Applications. New York: Physica-Verlag, 2, ch. Fuzzy integral for classification and feature extraction, pp [5] J. Keller, P. Gader, and A. Hocaoglu, Fuzzy Measures and Integrals: Theory and Applications. New York: Physica-Verlag, 2, ch. Fuzzy integral in image processing and recognition, pp [6] S. Auephanwiriyakul, J. Keller, and P. Gader, Generalized Choquet fuzzy integral fusion, Information Fusion, vol. 3, p. 69, 22. [7] G. Choquet, Theory of capacities, Analles de l Institit Fourier, vol. 5, pp , 953. [8] T. Murofushi and M. Sugeno, An interpretation of fuzzy measure and the choquet integral as an integral with respect to a fuzzy measure, Fuzzy Sets and Systems, vol. 29, pp , 989.

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